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How Wolfe's tether spreadsheet works

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I plan to do a series of posts examining elevators and tethers. I will link to them as posts are completed:

Upper Phobos Tether
Lower Deimos Tether
Lunar Elevator
Pluto Charon elevator

They will be based on Chris Wolfe's spreadsheet for modeling tethers.

I'll try to explain how Wolfe's spreadsheet works.

Tensile strength

Density and tensile strength are important quantities for tether material. Tensile strength is measured in pascals.

A pascal is a newton per square meter, newton/(meter2). A newton is a unit of force, mass times acceleration.

Zylon has a tensile strength of 580 megapascals or 580 meganewtons per square meter. On earth's surface with it's 9.8 meter/sec2 acceleration, it would take a 591,836,735 kilogram mass to exert that much force. It would take a zylon cord with a cross section of one square meter to support this force. But that's more than half a million tonnes!

10 tonnes is more plausible payload for space cargo. A much thinner cord could support this. Cross section of a Zylon cord need only be 1.72e-9 square meters. If a circular cross section, cord would be about 47 micrometers thick. Strands of hair can be anywhere from 17 to 181 micrometers thick.



So number of newtons determines tether cross sectional area.

How many newtons?

How to figure number of newtons at the tether foot? First we set maximum payload mass as well as foot station mass. The default in Wolfe's spreadsheet is a ten tonne payload mass and a foot station massing 100 kilograms. But how many newtons does this 1,100 kilogram mass exert?

The net acceleration on this foot mass is acceleration from planet's gravity minus centrifugal acceleration minus moon's gravity.

(Click on illustration to embiggen)

This spreadsheet sets the origin at the planet center.
Tether foot radius is the foot's distance from planet center.
Barycenter radius is Orbital Radius * mass planet / (mass moon/(mass planet + mass moon)
Tether anchor radius is Orbital Radius - Moon Radius. The tether anchor is assumed to be at the near point of a tide locked moon.
Distance from Barycenter to Tether Foot is Tether Food Radius - Barycenter Radius.

The three force equations:
Gravity Planet = G * Mplanet / Tether Foot Radius2
Centrifugal Accelerationω2 * Distance from Barycenter to Tether Foot. ω is constant, it is the angular velocity of the orbit.
Gravity Moon = G * Mmoon / (Orbital Radius - Tether Foot Radius)2

Net acceleration is the sum of these three.


An illustration of the accelerations with net acceleration in red. Moon gravity is negative because it is pulling away from the planet. Centrifugal acceleration is also pulling away from the planet except left of the barycenter it is towards the planet. 

When a curve crosses the axis the value is zero. Centrifugal crosses the axis at the barycenter. In most cases barycenter will be beneath planet surface. The illustration above has an exceptionally large moon. 

Net acceleration crosses the axis at L1, at this point the three accelerations sum to zero. to the right of L1, net acceleration is towards the moon.

To approximate the tether we chop it into many small lengths:


To find tether volume in step 1, we multiply the cross section by length of step 1. (Recall cross sectional area is set by number of newtons coming from tether foot.) Multiplying this volume by tether density gives step 1 tether mass. Multiplying this mass by net acceleration gives us the newtons this length exerts.

Adding the newtons from step 1 to payload newtons means the next step has a thicker cross section. We multiply this new cross section by tether length * tether density * net acceleration to get newtons from the tether length along step two.

And so on.

Summing all the masses from each step gives us total tether mass.

This is an approximation. The finer we chop the tether, the closer the approximation. The spread sheets we'll be using cut the tether length into 1,000 parts.

Our sheet can be found here. It is a 1.7 megabyte file.

For an upper moon tether, anchor will be on the far side. Moon's gravity will be added instead of subtracted from planet's gravity. I'll label tether end "Tether Top" instead of "Tether Foot".  Otherwise, the spread sheet will be the same as the lower moon tether spreadsheet.









Lower Phobos Tether

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A Phobos tether can be built in increments, it is useful in the early stages. So there's no pressing need to build a huge structure overnight. I will look at various stages of a Phobos tether, examining mass requirements and benefits each length confers. To model the tether I am using Wolfe's spreadsheet. I will use Zylon with a tensile strength at 5,800 megapascals and density of 1560 kilograms per cubic meter. Here is the version of the spreadsheet with Phobos data entered.

7 kilometer lower Phobos tether - tether doesn't collapse but remains extended

At a minimum, the lower Phobos tether must extend far enough past Mars-Phobos L1 that the Mars-ward newtons exceed the Phobos-ward newtons. This will maintain tension and keep the elevator from falling back to Phobos.

I used Wolfe's spreadsheet to find location of tether foot where tether length Mars side of L1 balances tether length from Phobos to L1. That occurs when tether foot is about 6.6 kilometers from tether anchor:


So going past that a ways will give a net Marsward force.


At this stage tether to payload mass ratio is about .01. The tether length exerts negligible newtons compared to payload force. Therefore a payload descending the tether to Phobos' surface would exert enough force to collapse the tether, especially as it nears Phobos' surface. So a counterbalancing mass would be needed at the tether foot.

Benefits

Escape velocity of Phobos is about 11 meters/sec or about 25 miles per hour. A small rocket burn would be needed for a soft landing. This burn could kick up dust and grains of sand, some of which could achieve orbit. This would create an annoying debris cloud.

However a spacecraft could dock with a station at Mars Phobos L1 much the same way we dock with the I.S.S.  Payloads could then descend the tether and arrive at Phobos without kicking up debris.

It would also allow low thrust ion engines to rendezvous with Phobos.

It would also serve as a foundation which can be added to.

It would take a Mars Ascent Vehicle about 5 km/s to leave mars and rendezvous with this tether. Trip time would be about two hours, so the MAV could be small.

From this Phobos tether, a .55 km/s burn can send drop a lander to an atmosphere grazing periapsis. Aerobraking can circularize to a low Mars orbit moving about 3.4 km/s. If Phobos is capable of providing propellent, much of that 3.4 km/s could be shed with reaction mass.

In contrast, a lander coming from earth will enter Mars atmosphere at about 6 km/s. Since it takes about 14 km/s to reach this point, the lander will not have reaction mass to shed the 6 km/s. For more massive payloads like habs or power plants, shedding 6 km/s in Mars atmosphere is a difficult Entry Descent Landing (EDL) problem.

87 kilometer lower Phobos tether - copper pulls it's own weight

It would be nice to have power to the elevator cars. However copper only has a tensile strength of 7e7 pascals and density of 8920 kilograms per cubic meter. Have copper wire along the length of the Zylon tether would boost taper ratio. Using the spreadsheet, I set tensile strength and density to that of copper and lowered the tether foot until I got a taper ratio of 1.1. That gives a length of about 87 kilometers.


Benefits

Along this length of the tether, copper pulls it's own weight, as well as supports the payload. A massive power source can be placed at L1 -- at L1 there are no newtons either Phobos-ward or Mars-ward. A copper only tether of this length would be about .2 times that of payload mass.

Elevator cars can ascend this length without having to carry their own solar panels and battery.

If descending from L1 Mars-ward, Mars' gravity can provide the acceleration and no power source is needed.

Of course copper wires can be extended further but this would boost taper ratio as well as tether mass to payload mass ratio.

From this tether foot, it takes .54 km/s to drop to an atmosphere grazing orbit. Trip time is about two hours.

1,400 kilometer lower Phobos tether - release to an atmosphere grazing orbit


With Zylon, tether to payload mass ratio is .11. The tether mass is still a small fraction of payload mass.

Benefits

Releasing from the foot of this tether will send a payload to within a 100 kilometers of Mars' surface. Skimming through Mars upper atmosphere each periapsis will shed velocity and lower apoapsis.

Low Mars orbit velocity is about 3.5 km/s. The payload arrives at 4.1 km/s.

4,300 kilometer lower Phobos tether - payload enters atmosphere at 3 km/s.

With Zylon, tether to payload mass ratio is 2.55. Tether mass is almost triple payload mass.

Benefits

At 4,300 kilometers from Phobos, dropping a payload will have an atmospheric entry of 3 km/s, about .5 km/s less than low Mars orbit.

5800 kilometer lower Phobos tether - maximum length

Phobos orbit has an eccentricity of .0151. It bobs up and down a little. Mars' tallest mountain is about 25 kilometers tall. Given these considerations, tether can't be more than 5800 kilometers. Else the foot might crash into the top of Olympus mons.

With Zylon, tether to payload mass ratio is about 16.10.

Benefits

The tether foot will be moving about .57 km/s with regard to Mars. Mars Entry, Descent and Landing (EDL) is far simpler with .57 km/s. If Phobos is a source of propellent, much of that .57 km/s can be taken care of with reaction mass.

For an ascent vehicle, only a small suborbital hop is needed to rendezvous with the tether foot.








Upper Phobos Tether

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This is third in a series of posts that rely on Wolfe's model of tethers from tide locked moons. As with the Lower Phobos Tether post, I will look at possible stages of this tether examining tether to payload mass as well as benefits each stage confers.

7 kilometer upper Phobos tether - tether doesn't collapse but remains extended

I used Wolfe's spreadsheet to find location of tether top where tether length Phobos side of L2 balances the length extending beyond L2. This occurs 6.6 kilometers from the tether anchor. Having the tether extend 7 kilometers is sufficient to maintain tension.


Docking with a facility at the L1 or L2 regions is easier than landing on Phobos. In the words of Paul451: "Instead of a tricky rocket landing at miniscule gravity on a loosely consolidated dusty surface, you just dock with the L1-hub of the ribbon (same as docking with ISS), transfer the payload to the elevator card and gently lower it to the surface. Reverse trip to bring fuel from Phobos to your ship (Assuming ISRU fuel is available on Phobos.)"

937 kilometer upper Phobos tether - transfer to Deimos tether

Given tethers from two coplanar moons tidelocked to the same central body, it is possible to travel between the two moons using nearly zero reaction mass.

Above I attempt to show how peri-aerion and apo-aerion of elliptical transfer orbit matches velocity of the tether points this ellipse connects. Tether Vs are red, transfer ellipse'sVs are blue.


Above I try to explain the math for finding the tether lengths from Deimos and Phobos.

Trip time between the two tethers is about 8 hours.

Zylon taper ratio for a 937 kilometer tether length is 1.02. Tether to payload mass ratio is .0448. Or the tether is about 1/20 the mass of the payload.

I'll look at the Deimos tether in a later post.

Benefits

Easy travel between Deimos and Phobos is a benefit in itself. 

But this would be a huge help to ion driven Mars Transfer Vehicles.

I like the notion of reusable ion driven MTVs. Ion engines have have great ISP thus allowing a more substantial payload mass ratio. However they have pathetic thrust. Andy Weir's fictional Hermes spacecraft can accelerate at 2 millimeters/sec^2. Which actually is very robust ion thrust. However ithis is only medium implausible. Low thrust means little or no planetary Oberth benefit. Plus a lo-o-o-ng time to climb in and out of planetary gravity wells.



300 km above Mars surface in low Mars orbit, gravitational acceleration is about 3 meters/sec^2. For a 300 km altitude low earth orbit, gravitational acceleration is about 9 meters/sec^2. 2 mm/s^2 acceleration is less than 10^-3 of the gravitational acceleration at initial orbit velocity in both these case. However I will be kind and go with Adler's .856 * initial orbit velocity.

At 2 millimeters/s^2 it would take Hermes 38 days to spiral out of earth's gravity well from low earth orbit and 17 days to spiral out of Mars gravity well. Most of the slow spiral out of earth's gravity would be through the intense radiation of the Van Allen belts.

I was very disappointed when Neil deGrasse Tyson's trailer had Hermes departing from low earth orbit and arriving in Mars' orbit 124 days later.

Besides adding 10 km/s to the delta V budget, climbing in and out of gravity wells would add about two months to Hermes' trip time. Tyson's video describes an impossible trajectory.  I wish he'd fact check himself with the same enthusiasm he applies to others.

It would be much better for Hermes to travel between the edges of each gravity well. At least as close as practical to the edge. In earth's neighborhood, Hermes could park at EML2 between trips. In Mars' neighborhood, parking at Deimos would save a lot of time and delta V. From Deimos, astronauts and payloads can transfer to Phobos and then to Mars surface. In this scenario, Hermes' 124 day trip from earth to Mars is plausible.

2345 kilometer upper Phobos tether - Mars escape

If anchor in a circular orbit, escape velocity can be achieved if tether top is at a distance 2^(1/3) anchor's orbital radius. I try to demonstrate that here. Phobos is in a nearly circular orbit. To achieve escape, the tether would need to be 2435 kilometers long.

Zylon taper ratio: 1.11. Tether to payload mass ratio: .23. A little more than 1/5 of the payload mass.

Benefits:

Achieve mars escape.

6155 km kilometer upper Phobos tether - To a 1 A.U. heliocentric orbit

A tether this long can fling payloads to a 1 A.U. heliocentric orbit, in other words an earth transfer orbit.

Taper ratio: 1.8. Tether to payload mass ratio 1.6. The tether mass is nearly double payload mass.

Benefits

Catch/throw payload to/from earth.

7980 kilometer upper Phobos tether - to a 2.77 A.U. heliocentric orbit.

Zylon aper ratio: 2.53. Tether to payload mass ratio 3.21. A little more than triple payload mass.

Benefits:

2.77 A.U. is the semi major axis of Ceres. A tether this long could catch/throw payload to/from Ceres. But this doesn't take into account plane change because of Ceres inclination.

Even with plane change expense, this tether could be very helpful for traveling to and from The Main Belt.








Deimos Tether

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This is a fourth in a series of blog posts looking at various tethers using Chris Wolfe's model.

50 kilometer Deimos tether - minimum length to remain aloft.

Mars-Deimos L1 and L2 are about 14 kilometers from Deimos' surface. Another 26.5 kilometer length extended past these points would balance. Extending the tether 50 kilometers either way along with a counterweight would provide enough tension for the elevators to stay aloft.

Zylon taper ratio is 1. Tether to payload mass ratio: about .01. A ten kilogram tether could accommodate a thousand kilogram payload.

Benefits

There is no net acceleration at L1 and L2, so docking at ports at these locations would be like docking with the I.S.S.

This first step could serve as a scaffolding additional tether infrastructure could be added onto.

2942 kilometer lower Deimos tether - transfer to Phobos tether

Given an ~1000 upper Phobos tether and a ~3000 lower Deimos tether, it is possible to move payloads between the two moons with almost no reaction mass. The tether points connected by the ellipse match the transfer ellipse's velocities. See my Upper Phobos Tether post.



Zylon taper ratio: 1.01. Tether to payload mass ratio: .04. A one tonne tether could accommodate a twenty-five tonne payload.

Benefits

The idea of ion driven interplanetary vehicles excite me. The Dawn probe has demonstrated ion rockets are long lived and amenable to re-use. An ion rocket's fantastic ISP means a lot more mass fraction can be devoted to payload.

However ion rockets have pathetic thrust. They suck at climbing in and out of planetary gravity wells.

Here Mark Adler talks about ion rocket trajectories:

The fictitious Hermes from Andy Weir's The Martian can do 2 mm/sec^2 acceleration. Due to the need for a high alpha, I regard the Hermes as medium implausible but I will go with that number.

At Deimos' distance from Mars, gravitational acceleration is about  80 mm/s^2. The Hermes' acceleration over Mars gravitational acceleration at that orbit is about 1/40. A small fraction but a lot larger than the 10^-3 fraction Adler mentions.

Deimos moves about 1.35 km/s about Mars. With an impulsive chemical burn, it would take about .56 km/s to achieve escape. But with a 2 mm/s^2 acceleration, it would take about 5 days and and .8 km/s to achieve escape.

To spiral down to low Mars Orbit, it'd take Hermes more than 17 days and 3 km/s. So the Deimos rendezvous says about two weeks and more than 2 km/s delta V.

Once in heliocentric orbit, it is the sun's gravitational acceleration that we put in the denominator. Here is a chart of gravitational acceleration at various distances from the sun:


If the rocket's acceleration is a significant fraction of central body's acceleration, we can model burns as impulsive. The trajectory would be more like an ellipse than a spiral. At earth's distance from the sun., Hermes 2 mm/s^2 acceleration would be about a third the sun's gravity. At Mars, it's about four fifths. In the asteroid belt, Hermes acceleration exceeds acceleration from sun's gravity.

Ion rockets may not be great for climbing in and out of planetary gravity wells. But they're fine for changing heliocentric orbits, especially in the asteroid belt and beyond.

Fact checking Neil deGrasse Tyson

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Tyson is well known for fact checking movies, comics and other pop culture stuff. Here's giving Tyson a taste of his own medicine.



Tyson's trailer for The Martian

Hermes' impossible trajectory


Above is a link to Neil deGrasse Tyson's trailer for The Martian. At 1:15 of the vid, Tyson has the space ship Hermes departing from Low Earth Orbit (LEO). 124 days later he has Hermes arriving at Mars orbit (2:17 of the video).

Hermes is propelled with low thrust ion engines. In the book when Hermes is about to rendezvous with Watney's Mars Ascent Vehicle (MAV), Lewis says Hermes can do up to 2 mm/s2. This acceleration is also given online:



Two millimeters per second squared would require an extremely good alpha. But it's possible future power sources will deliver more watts per kilogram. So 2 mm/s2 is only medium implausible. I'll let this slide.

Problem is, low thrust ion engines really suck at climbing in and out of planetary gravity wells. From low earth orbit, it would take Hermes about 40 days to spiral out of earth's gravity well and about 20 days to spiral from the edge of Mars' gravity well to low Mars orbit. Two months spent climbing in and out of gravity wells destroys Andy Weirs' 124 day trajectory.

Given 2 mm/s2, the trajectory Tyson describes is flat out impossible.

A slow ride through the Van Allen belts.

At 1:50 of Tyson's video he talks about the danger of solar flares and how astronauts are vulnerable to radiation. Well, departing from LEO means a month long spiral through the Van Allen Belts. Not only does the long spiral wreck Weir's 124 day trajectory, it also cooks the astronauts.

Tyson enjoys some notoriety for fact checking fantasies like Star Wars or The Good Dinosaur. This leaves me scratching my head. Many of the shows he fact checks make no pretense at being scientifically accurate. However The Martian was an effort at scientifically plausible hard science fiction and thus is fair game. Same goes for Tyson's trailer.

A physically impossible trajectory along with cooking the astronauts earns Neil an F for fail. Tyson's effort at hard science fiction isn't any better than Gravity or Interstellar.


Neil's Five Points of Lagrange Essay

The Five Points of Lagrange was a Neil deGrasse Tyson article published in the April, 2002 issue of Natural History Magazine.

A few excerpts:


Popular usage has made "exponential" a general term for dramatic change. But a physicist should know the more specific mathematical meaning of the this word. Gravity falls with inverse square of distance, not exponentially.



Wrong. Clarke's contribution was suggesting communication satellites be placed in geosynchronous orbit (GSO). A fantastic idea with tremendous impact. But Clarke wasn't the first to calculate the altitude of GSOs.

Herman Potočnik calculated the altitude of GSO in 1928.  It's possible this altitude was calculated even earlier. Newton might have done it.



Here Tyson seems to be talking about the so called Interplanetary Transport network. I went over this in Potholes on the Interplanetary Super Highway. The Weak Stability Boundaries (WSBs) emanating from Sun Earth Lagrange 1 (SEL1) or Sun Earth Lagrange 2 (SEL2) won't get you very far from earth's orbit. Same goes for the Sun Mars L1 and L2. Mass parameters between the sun and the tiny rocky planets are too small to be of much use.

The lack of an Oberth benefit largely compromises the advantage of SEL2's high location. A chemical Trans Mars Insertion (TMI) from Low Earth Orbit (LEO) is about 3.6 km/s. From SEL2, a direct burn to TMI takes about 3 km/s. An advantage of .6 km/s? No. When you consider that it takes a few months plus a 3.1 km/s LEO burn to reach SEL2, there is no advantage.

From SEL2 it's possible to do a multi burn TMI that enjoys the Oberth benefit. As described in What About Mr. Oberth?, a small burn can drop a craft deep in earth's gravity well where a perigee burn would confer a large Oberth benefit. But it takes a few months to fall earthward from SEL2. About the same Oberth benefit could be enjoyed by dropping from EML2 and this only takes 9 days.

Unlike SEL2, EML2 is close to the lunar cold traps or, potentially, asteroids parked in lunar orbit. These are possible sources of life support consumables and propellent. Close to EML2 in terms of time and distance as well as delta V.

I wish Tyson had recalled this article when he made his trailer for The Martian. The L1 and L2 regions sit on the edge of a planetary gravity well. Going from SEL2 to SML1 cuts climbing in and out of two deep wells. Weir's 124 day journey would have been plausible. But the Hermes would still have had a substantial delta V budget. It's not possible to ride the currents of space to "drift" from SEL2 to SML1.

Tyson on "idiot doctors"


Tyson says:

Somebody's diagnosed with terminal cancer. The doctor says you got 6 months to live. … Go to a 2nd doctor, you got 5 months to live. Go to a 3rd doctor, 7 months to live.  … What happens? You're alive a year later. … 3 years later the cancer's in remission. 5 years later, it's gone from your body.

You happen to have been a religious person … Here's what's astonishing - if you are that person you are more likely to believe that God cured you … than that you had 3 idiot doctors diagnose you.
Does someone surviving stage 4 cancer demonstrate the idiocy of doctors? No.

A competent doctor gives his patient the odds. He tells his patient the survival rates of other people in his condition.

Avi Bitterman notes:

When a doctor gives you a diagnosis that you have cancer and you are going to die in X amount of time, what the doctor is doing is giving you an average time of death with an amount of certainty given by the 95th or 99th percent confidence intervals of the time of death since diagnosis for people who were in your same situation. As with any bell-curve distribution there's bound to be outliers. That's just how a bell-curve works. If you manage to outlive that, it doesn't mean anything magical happened, but contrary to what Tyson suggests, it doesn't mean the doctor who made this prediction is stupid either, it just means a bell-curve works the way a bell-curve works. One would think this is something Tyson would know a thing or two about.
Indeed, statistics and bell curves crop up in most fields of science.


Are there more Tyson science bloopers?

I've only seen a fraction of Tyson's prolific output so I suspect I'm just scratching the surface. If readers know of more Tyson science bloopers, drop me a line.


Limits to growth, logistic vs exponential

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Malthusian growth model

The Malthusian growth model sees population growth as exponential.

P(t) = Poert
where
P=  P(0) is the initial population size,
r = population growth rate
t = time

Growth of microbe populations are often used to illustrate this. Let's say an amoeba will grow and divide into two amoeba after an day of absorbing nutrients.

Day 1: 1 amoeba
Day 2: 2 amoeba
Day 3: 4 amoeba
Day 4: 8 amoeba

And so on. Population doubles each day. Exponential growth is famous for starting out slow and then zooming through the roof.


On the left is exponential growth in cartesian coordinates. On the right in polar coordinates, radius doubles every circuit.

Malthus imagined a rapidly growing population consuming all their available food supply and then starving to death.

Logistic growth

Sometimes populations have suffered Malthusian disaster. More often rate of growth slows as the population approaches the limit that resources can support. This is logistic growth.

P(t) = Le-rt / (L +( e-rt - 1))

Where L is the maximum population local resources can support.


At the start, logistic growth resembles exponential growth. But as the population nears the logistic ceiling, growth tapers off. Above the blue boundary represents the limit to growth. In red is the logistic growth curve, the thinner black curve is exponential growth.

What slows growth?

In Heinlein's science fiction, war limits growth. This was also the foundation idea of Niven and Pournelle's The Mote In God's Eye -- War is the inevitable result of burgeoning populations.

The Four Horsemen of Apocalypse -- plague, war, famine and death are seen as natural outcomes of uncontrolled population growth.

A declining fertility rate is a less ominous way to step on the brakes. It is my hope most people will choose to have small families. And indeed, current trends indicate people are voluntarily having fewer kids. Still, there are skirmishes as various entities compete for limited resources.

Bad vs worse

A growing population, a growing consumer appetite, a limited body of resources. It doesn't take a rocket scientist to see growth must eventually level off.

Whether it levels off via the 4 horsemen or moderation and voluntary birth control, either option sucks.  It's disaster vs stagnation.

Alternatives?


Above is a Johnny Robinson cartoon from the National Space Society's publication.

I believe our solar system is possibly the next frontier. That has been the thrust of this blog since the start. If we do manage to break our chains to earth, it will be a huge turning point in human history, more dramatic than the settling of the Americas. The potential resources and real estate dwarf the north and south American land masses.

While settling the solar system allows expansion, it won't relieve population pressure on earth. Settlement of the Americas did not relieve population pressure in Europe, Asia and Africa. Mass emigration is impractical.

Rather, pioneers jumping boundaries starts growth within the new frontiers. I like to view the logistic growth spiral in polar form as a petri dish. When a population within a petri dish has matured to fill its boundaries, it sends spores out to neighboring petri dishes. Then populations within neighboring petri dishes grow to their limits.



The first petri dish still has a population filling the limit. They have not escaped the need to live within their means. I take issues with critics who say space enthusiasts want to escape to a new planet after earth has been trashed. Space enthusiasts know earth is fragile, more so than the average person. It is noteworthy that Elon Musk is pioneering planet preserving technologies such as electric cars and solar energy.

But even if mass emigration from Europe or Asia was not possible, the expansion into the Americas energized the economy and zeitgeist of the entire planet. It provided investment opportunities. Also an incentive to explore. This is the greatest benefit of a frontier. Curiosity is one of the noblest human qualities and I hope we will always want to see what lies over yonder hill. And that we will keep devising ways to reach the far side of the next hill. Satisfaction and contentment are for cattle. If we lose our hunger and wander lust we will no longer be human.




Liftport Lunar Tether

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This is the fifth in a series of posts using Chris Wolfe's spreadsheet to look at various elevators.

274,000 km Lunar Tether

This is based on the Ladder PDF written by Liftport  founder Michael Laine and Marshall Eubanks.


Eubanks and Laine suggest the use of Zylon or M5. This is why I've been using Zylon through out these tether posts. These gentlemen have invested a lot of time and effort researching elevators and tethers. If they like Zylon, I'll follow suit.

They propose launching the tether to EML1. From EML1, the tether anchor would descend moonward towards Sinus Medii on the lunar surface, 0º, 0º. The spent upper stage would drop with the tether foot earthward.

If the mass were tethers alone, the 264,000 length would be inadequate to keep the tether from collapsing to the moon. But spent upper stage acts as a counterweight to maintain tension.

Ratios earthside of EML1

A spent Centaur upper stage is about 2250 kilograms. This is the quantity I used for foot station mass. These newtons subtract from newtons available for payload. The Ladder PDF calls for 11 tonnes of Zylon. By trial and error I entered payload quantities until tether mass in my spreadsheet came to 11 tonnes.

In addition to foot station mass of 2250 kg, I got a maximum foot payload mass of 1640 kg.

Zylon taper ratio: 1.61. Tether mass to payload mass ratio: 8.05

Given the extreme the extreme length of this elevator, I expected a higher number than 8. But the net acceleration at the tether foot is only .0274 newtons per kilogram. With this acceleration, a 10 tonne mass would exert as much force as when my 62 pound dog sits on my lap.

Ratios moonside of EML1

But what sort of payload can this elevator support moonside of EML1?

At the anchor in Sinus Medii, my tether model's cross sectional area is 1.64e-8 square meters. Multipying this times Zylon's tensile strength gives ~95.4 newtons the tether can support. Net acceleration at this point is 1.4 meters/s^2 (mostly moon's gravity). 95.4 newtons/(1.4 m/s^2) = 68 kilograms. For a payload just above the moon's surface, the elevator can support 68 kilograms.

Tether to payload mass ratio: 161.

Let's say we wanted a 1 tonne elevator car capable of carrying 9 tonnes of cargo. We'd need a 1,610 tonne tether.

Benefits

Dropping a payload from 70,900 km earthward of EML1 would send a payload to to an atmosphere grazing orbit. Repeated perigee aerobraking passes could circularize the orbit. Shedding 3 km/s via repeated drag passes would require some thermal protection but not as much as the space shuttle which would shed 8 km/s over a very short time.

Thus lunar materials could be delivered to Low Earth Orbit (LEO) without using reaction mass.

Likewise, a 3 km/s LEO burn could deliver payloads to an apogee where orbit velocity matches tether velocity. Normal delta V from LEO to moon surface is about 6 km/s. So the elevator cuts about 3 km/s from the delta V budget for reaching the moon's surface. Cutting 3 km/s from delta V budget about doubles payload mass if using H/Lox bi-propellent.

Dropping a payload 160,000 km earth of EML1 would send a payload to an orbit with perigee as geosynchronous orbit altitude. At perigee the circularization burn is .95 km/s. Thus delta V between GSO and lunar surface is less than kilometer per second.

Some drawbacks

This is a very long tether. How fast can an elevator car move? Having copper wire along the length of the tether would boost taper ratio as well tether to payload mass ratio. For descent from EML1 to lunar surface, the tether to payload mass ratio is already 161.

So in addition to carrying gripping wheels and a motor, the elevator car must carry it's own power source. Photovoltaic arrays? There are solar powered golf carts. These aren't famous for their speed. There are Tesla cars whose lithium batteries can be charged by solar cells. These vehicles can move. It is also possible lithium batteries could be charged during an elevator cars down hill descent via regenerative braking. Downhill would be moonward or earthward from EML1. Movement towards EML1 would be uphill.

Batteries, solar arrays and/or regenerative brakes would boost elevator car mass and thus subtract from cargo mass.

Let's say the elevator car can move an average speed of 400 mph (644 kilometers/hour). A round trip along the length of this elevator and back would take about a month. If the elevator doubles payload mass delivered from LEO, it'd take about 160 months to recoup the investment of delivering tether mass from LEO.

And what justifies this investment? What are the benefits of a facility at Sinus Medii?

I'm a moon guy but it's the lunar poles I like. There are polar plateaus that enjoy near constant sunlight and very mild temperature swings. These plateaus neighbor permanently shadowed crater floors that might harbor rich volatile deposits. In situ CHON not only makes life support easier, but extra-terrestrial propellent could break the exponent in the rocket equation.

But Sinus Medii is at the equator. It's as far from the lunar poles as a lunar surface point can possibly be. We're stuck with two week nights, severe temperature swings and regolith drier than a bone.

Charles Radley has suggested mining He3.  I'm not holding my breath but what if we achieved fusion power? Here is John Schilling's take on fusion and lunar He3:
Helium-3 mining on the moon simply does not pass the arithmetic test. The highest 3He concentration ever recorded in lunar regolith is fifteen parts per billion, and the process by which it is deposited is inherently resistant to geologic concentration.
Assuming someone manages to invent a 3He fusion reactor that operates at 50% efficiency (giggle), that translates to net energy output of 4.5E6 joules per kilogram of high-grade regolith.
The energy output of a kilogram of the lowest grade of coal burned in a good 19th-century reciprocating steam engine, is about 4.5E6 joules per kilogram. And that doesn’t change if you substitute dried peat for the coal.
So, the proposal is to set up an enormous mining infrastructure on the Moon, and invent a fundamentally new kind of engine backed by fifty years of failed promises, for the sake of an energy source roughly as good as burning high-grade dirt in a type of engine obsolete for over a century.
And no, that analysis doesn’t change significantly if we include accessible reserves or environmental impact.
I understand that you want desperately to believe that there are immense riches to be had in space, as soon as the suits see the light and come up with the money. The good news is, this is probably true. But the list of great riches to be had in space, does not include lunar helium-3 (or helium-4, for that matter). The numbers do not add up, no matter what the glossy magazine articles say, and math trumps faith.

Other than fuel for fusion it is hard to imagine He3 markets that would justify the expense of a lunar tether and mine.

I admire Michael Laine. I believe tethers will play a part in making space transportation economical. I also like and admire Charles Radley as well as Marshall Eubanks. So it pains me to say this. At this point I am not enthusiastic about the Liftport Lunar elevator.

But there are other possible elevators in the moon's neighborhood.


Rotovator help with re-entry

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There are proposals for rotovators to catch payloads in low earth orbit and then throw them to higher orbits.

It occurs to me that rotovators used for throwing comm sats to GTO could also help the upper stage re-enter earth's atmosphere at a lower velocity.


1) Rotovator catches upper stage and payload in LEO.
2) Rotovator throws payload to higher orbit.
3) Rotovator drops upper stage into a suborbital orbit.

Step 3) accomplishes two things:

a) It restores some of the orbital momentum the tether lost in catching and tossing the payload.
b) It reduces re-entry velocity of the upper stage. Slower re-entry velocities make recovery and reuse of the upper stage less difficult.

Is this a good idea? Or another hare brained scheme? I'm tossing this out in several venues hoping knowledgeable folks will review it.




Hildas As Cyclers

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Hilda Asteroids - Red,   Sun Jupiter Trojans - Blue,   Main Belt - Green

The above image was made from screen captures of Scott Manley's beautiful animation Asteroids In Resonance With Jupiter.

Jupiter is the dot off to the left, the sun is the yellow dot in the middle. Within the Main Belt can be seen Mercury, Venus, Earth and Mars. I colored the different asteroids populations so we can tell them apart.

The Sun Jupiter Trojans have a 1 to 1 resonance with Jupiter. They co-rotate with Jupiter. The leading Trojans remain in a neighborhood 60 degrees ahead of Jupiter and the trailing Trojans stay in a neighborhood 60 degrees behind.

The Hildas have have 3 to 2 resonance with Jupiter meaning they circle the sun three times for every two Jupiter orbits. Jupiter's orbital period is about 12 years and the Hildas have 8 years periods.

The Hilda orbits only look triangular in Manley's animation because they're being viewed in a rotating frame. You can see Jupiter remains on the left side of the image. In an inertial frame, a Hilda orbit is an ordinary elliptical orbit with aphelion passing through the Trojans and perihelion passing through the main belt.


I envision the Hilda biomes playing a similar role as Marco Polo's caravans shuttling people and goods between east and west. But the Hildas travel between the Trojans and the Main Belt.

The would be a series of regular fly bys for a Hilda Cycler

1) Main Belt to trailing Trojans — 4 years.
2) Trailing Trojans to Main Belt — 4 years.
3) Main Belt to leading Trojans — 4 years
4) Leading Trojans to Main Belt — 4 years
5) Main Belt to Sun Jupiter L3 — 4 years. But there is no asteroid population at SJL3.
6) From SJL3 to Main Belt 4 years

Then back to step 1). The cycle repeats itself.

So not only can a Hilda be a go between between the Main Belt and Trojans, but it can also move stuff between the trailing and leading Trojan populations. Trailing to leading takes 8 years and leading to trailing takes 16 years.

As can be seen from Manley's animation, there is a steady stream of Hildas traveling the circuit. 

Delta V

The Hildas have a variety of eccentricities. I will look at a Hilda orbit having an eccentricity of .31. That would put the aphelion at 5.2 A.U. and the perihelion at 2.74 A.U. (The perihelion is in Ceres' neighborhood, Ceres' semi-major axis is 2.77 A.U.).

Assuming a circular, coplanar orbit at 2.74 A.U.,  it would take 2.6 km/s to leave a Main Belt Asteroid and board a Hilda.

Assuming a circular, coplanar orbit at 5.2 A.U., it would take 2.2 km/s to depart the Hilda and rendezvous with a Trojan.

However, coplanar orbits is a very optimistic assumption. Asteroids have a large variety of inclinations. Making a 10 degree plane change from a Hilda's orbit can cost 2 to 3 km/s.

Ways to mitigate delta V expense

Many asteroids spin about pretty fast. This plus their shallow gravity wells make them amenable to bean stalks, also known as space elevators. 

"Why would an asteroid need a space elevator?" I'm sometimes asked. The questioner will assert "It's very easy to get off an asteroid's surface, and getting off the body's surface is the only reason for an elevator." Which is wrong, of course.

Speed of a body on an elevator is ωr where ω is angular velocity in radians per time unit and r is distance from center of rotation. If r is large, the elevator can fling a payload at high velocity with regard to the asteroid. It is quite plausible for an asteroid's bean stalk to provide .5 to 1 km/s delta V.

Also an asteroid bean stalk allows rendezvous with an ion propelled space craft. Ion ships have great ISP but minute thrust. Soft landings with an ion craft are not possible on larger asteroids like Ceres, or Vesta.

And ion propelled ships are more viable in the outer system. When a ship's acceleration is a large fraction of the local gravity acceleration, an ion burn is more like a chemical impulsive burn. See General Guidelines for Modeling a Low Thrust Ion Spiral. In the outer Main Belt, the sun's gravity is about 1 millimeter/sec2. Sun's gravity at the Trojans is about .2 millimeters/sec2.

Jupiter's Trojans

Not much is known about Jupiter's Trojans. Japan hopes to launch a mission in the early 2020's. Another proposed mission is Lucy which would also launch in the early 2020's. 

These bodies are on average 5.2 A.U. from the sun and so receive only 1/27 the sunlight earth enjoys. For this reason I am hopeful they are rich in volatile ices. I'd give better than even odds they have lots of water and carbon dioxide ice. Nitrogen compounds like ammonia and cyano compounds are a possibility. Aside from earth, Nitrogen is in short supply throughout the inner solar system and these would be a great export to the Main Belt biomes.

Their numbers are speculation. According to Wikipedia:

Estimates of the total number of Jupiter trojans are based on deep surveys of limited areas of the sky.[1] The L4 swarm is believed to hold between 160–240,000 asteroids with diameters larger than 2 km and about 600,000 with diameters larger than 1 km. If the L5 swarm contains a comparable number of objects, there are more than 1 million Jupiter trojans 1 km in size or larger. For the objects brighter than absolute magnitude 9.0 the population is probably complete. These numbers are similar to that of comparable asteroids in the asteroid belt. The total mass of the Jupiter trojans is estimated at 0.0001 of the mass of Earth or one-fifth of the mass of the asteroid belt.
Two more recent studies indicate, however, that the above numbers may overestimate the number of Jupiter trojans by several-fold. This overestimate is caused by (1) the assumption that all Jupiter trojans have a low albedo of about 0.04, whereas small bodies may actually have an average albedo as high as 0.12;[16] (2) an incorrect assumption about the distribution of Jupiter trojans in the sky. According to the new estimates, the total number of Jupiter trojans with a diameter larger than 2 km is 6.3 ± 1.0×104 and 3.4 ± 0.5×104 in the L4 and L5swarms, respectively. These numbers would be reduced by a factor of 2 if small Jupiter trojans are more reflective than large ones.[16]
The number of Jupiter trojans observed in the L4 swarm is slightly larger than that observed in L5. However, because the brightest Jupiter trojans show little variation in numbers between the two populations, this disparity is probably due to observational bias. However, some models indicate that the L4 swarm may be slightly more stable than the L5 swarm. 
The largest Jupiter trojan is 624 Hektor, which has an average diameter of 203 ± 3.6 km. There are few large Jupiter trojans in comparison to the overall population. With decreasing size, the number of Jupiter trojans grows very quickly down to 84 km, much more so than in the asteroid belt. A diameter of 84 km corresponds to an absolute magnitude of 9.5, assuming an albedo of 0.04. Within the 4.4–40 km range the Jupiter trojans' size distribution resembles that of the main-belt asteroids. An absence of data means that nothing is known about the masses of the smaller Jupiter trojans. The size distribution suggests that the smaller Trojans are the products of collisions by larger Jupiter trojans.

I'd love to see science fiction stores set on 624 Hektor.

Lunar Sky Hook

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Kim Holder has been urging me to do this blog post. Her comments in various forums have been helpful in thinking about this.

Vertical Lunar Tether In A Polar Orbit

This sky hook is a gravity gradient stabilized vertical tether. It's in a polar orbit so it will pass over the poles as well as the lower lunar latitudes.


Unlike an equatorial orbit, there are only two occasions during a lunar orbit where a tether's Vinf velocity vector is anti-parallel to the moon's velocity vector. So launch windows to earth would only occur each two weeks. That's still pretty often. These occasions are also good times to rendezvous with the tether.

Playing with earth moon three body simulations, polar orbits seem to remain stable up to a radius of around 20,000 kilometers. That is where I will set the anchor mass at the balance point of this sky hook. I believe this is far enough above the lunar surface that the mascons won't damage this tether's orbit.



Asteroid Anchor Mass Via a Keck vehicle

What to use for the anchor mass? With the asteroid retrieval vehicle proposed in the Keck Report, it is possible for a vehicle of moderate mass to retrieve a much larger mass to the earth moon neighborhood. The Keck authors believe a rock could be placed in high lunar orbit for around .17 km/s. A lunar orbit with a 20,000 km radius has a speed of around .5 km/s. I believe it would take around .7 km/s to park a rock in the orbit we want.

The Keck vehicle includes solar panel arrays and Hall ion thrusters. These would be great to have on a vertical tether. It takes awhile for ion engines to impart momentum, but given time they're about ten times as efficient as the best chemical rockets. A tether can build up momentum over time but release it suddenly. Thus they are a good way to enjoy an ion engine's great ISP and an Oberth benefit.

As well as adjusting the tether's orbit the Keck vehicle's solar arrays might also power elevator cars moving up and down the tether. If water is exported from from the lunar cold traps to the tether, the arrays might also crack water into oxygen and hydrogen bipropellent. There are a number of possible uses for this power source.

Upper Tether

The tether length above the anchor mass can be built in increments. I imagine the tether growing longer and more able with time. Here are three possible stages:

To EML2 or EML1



EML1 and 2 are about 65,000 km from the moon. To reach this apolune, we'd need an upper tether length of about 2700 kilometers. Using Wolfe's spread sheet, this tether length has a taper ratio of 1. With a safety factor to 3, tether mass to payload ratio is about .02.

This is pretty good. I believe this low stress tether length could accommodate copper wires to transmit power to the elevator cars.

Once at apolune, I believe it would take about .3 km/s to park the payload at EML2 or EML1.

EML2 is a good staging location should we want to travel to and from destinations beyond the earth-moon neighborhood.

To a Perigee at Geosynchronous Orbit 



Transfer orbit from GEO to the moon is about a an ~36,000 x 378,000 ellipse. Apogee speed is about .45 km/s. The moon's speed is about 1.02 km/s. So the tether needs to hurl a payload to a Vinf of (1.02-.45) km/s or about .57 km/s.

To achieve this Vinf our tether needs to be 12,200 km. Zylon taper ratio is 1.09. With a safety factor of three, Tether to payload mass ratio is about .167. So a ten tonne tether could accommodate a sixty tonne payload. This is still pretty good. A power cable along this length is also doable.

Perigee velocity of our transfer orbit is ~4.13 km/s. Geosynch orbit velocity is ~3.07 km/s. If the transfer orbit and destination geosynch orbit are coplanar, geosynch circularization would be about 1.06 km/s. But I expect that would be the exception rather than the rule. If the orbit inclinations differ by 20º, 1.6 km/s would be needed to park in geosynch.

To a Perigee at Low Earth Orbit.



A 300 x 378,000 km orbit has apogee velocity of ~.19 km/s. (1.02 - .19) km/s = .83 km/s.

To throw a payload to a trans earth orbit, our tether needs to impart a Vinf of .83 km/s. This takes a tether length of 19,200 kilometers. With a safety factor of three, Zylon taper ratio is 1.2. Tether to payload mass ratio is .38.

If perigee is through earth's upper atmosphere, aerobraking can provide a large part of the 3.1 km/s delta V for circularizing at LEO.

Lower Tether

Again, the tether length below the anchor mass can be built in increments. Incremental growth with time is more doable than trying to do the whole length in fell swoop. Here are some possible steps along the way.

To a Perilune at Low Lunar Orbit.


To drop a payload to a 90 km altitude perilune, length needs to be 7360 km. Given a safety factor of 3, Zylon taper ratio is 1.06. Tether to payload mass ratio is .15.

Velocity of transfer orbit's perilune is about 2.2 km/s. Low lunar orbit is about 1.6 km/s. It'd take about .6 km/s to circularize at low lunar orbit. 

To the Moon's Surface, Impact Velocity 1 km/s.



If the tether is extended to a length of 17890 km, tether foot altitude is about 370 km. Dropping a payload from this tether foot would result in a 1 km/s impact. 

Given a safety factor of two, Zylon taper ratio is 2.88. Tether to payload mass ratio is 26.87.

Note the safety factor is less than in the other scenarios. As we descend further into the moon's gravity well, stress climbs more rapidly. It would be more difficult to include copper wires for power along the lower parts of the tether.

To a Tether Foot Just Above the Moon's Surface.



Dropping the tether foot to an altitude of 10 kilometers gives us a length of 18,252 km. Safety factor of 2 and Zylon taper ratio is 3.72. Tether to payload mass ratio is  about 51.

Dropping from this tether foot, a payload would impact the lunar surface at .184 km/s. 

A .2 km/s payload delta V budget for soft landing seems quite doable. Likewise it would take about .2 km/s to launch a payload from the lunar to rendezvous with the tether foot.

However dropping the tether foot this far is considerably more ambitious than the other scenarios described above.

Travel About The Moon

Kim Holder noted such a tether might serve as transportation between locations on the moon.

Without a tether, going from pole to pole would take about 3.4 km/s: 1.7 km/s to launch and another 1.7 for soft landing. Going from equator to pole would take 1.53 km/s to launch and another 1.53 km/s for a soft landing, totaling 3.06 km/s. 

So a 18,000 km lower lunar tether length would make travel about the moon easier.

A Location to Process Asteroid Ore

It takes about .6 km/s to park ore from some of the more accessible asteroids in 20,000 km lunar orbit. If rendezvous with the tether top is doable, it could take considerably less.

I envision infrastructure accreting about the tether anchor mass 18,262 km above the lunar surface. Water, platinum, gold, rare earth metals, and other materials could be extracted at the anchor. Refined commodities could climb to the top of the tether and then tossed earthward.

A Synergy Between The Moon and Near Earth Asteroids

Moon and asteroid enthusiasts are often at odds with one another. They should be allies. In terms of delta V, it's a lot easier to park asteroids in lunar orbit than lower earth orbits. And given growing infrastructure in lunar orbit, the moon's surface becomes more accessible.

Tran Cislunar Railroad

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Three Orbital Tethers

This post revisits Orbital Momentum As A Commodity. But now I will examine these tethers using Wolfe's spreadsheet.

I envision 3 equatorial tethers to move stuff back and forth between LEO and the lunar neighborhood:




The location of these vertical tethers avoids zones of orbital debris:


The orange regions, LEO, MEO and GEO, have high satellite and/or debris density. Thus tethers in those regions would be more vulnerable to damage from impacts.

Dead Sats for tether anchors

Unless elevator mass is lot more than the payloads, the acts of catching or throwing could destroy the tether orbit. At first it looks like the need for a substantial anchor mass is a show stopper. But there are a large number of dead sats in equatorial orbits. By one estimate,  there's 670 tonnes in the graveyard orbit above geosynch.

The dead sats gathered might have functioning solar arrays. According to this stack exchange discussion, solar arrays degrade by 2 to 3% a year due to radiation, debris impacts and thermal degradation. Thus a 20 year old array could still be providing 50% to 66% of the power it delivered at the beginning of its life. The parabolic dishes for high gain antennas might also be salvageable.

Whether functioning or not, solar arrays as well as other paneling might be used as shades to keep propellent cold. If our tethers receive propellent from the moon or from asteroids parked in lunar orbit, shades would help with cryogenic storage.

Consolidating dead equatorial satellites would reduce their cross sectional area and help solve the problem of orbital debris.

Super GEO tether



The circular orbit pictured above is 10,000 km above Geosynchronous Earth Orbit (GEO). The lower part of the tether has a length of 7,000 km and the upper tether is 10,340 km in length.

A Space Stack Exchange answer estimates there are 670 tonnes of dead sats in the geosynch graveyard orbit.

Delta V to raise the dead sats to this higher orbit is about .28 km/s. This might be accomplished with ion engines. Also the elevator could be used to send some to the sats towards the lower MEO tether. This would help with the .28 km/s delta V budget.

Upper Super GEO Tether, 10,340 km long
Safety Factor3
Zylon taper ratio:1.38
Tether to payload mass ratio:.78
Tether top radius62,504 km
Tether top speed:3.3 km/s
Tether top net acceleration:.07 m/s2 (.007 g)
Payload apogee:384,400 km
Payload apogee speed:.53 km/s

The payload apogee is at lunar altitude and the payload's moving .53 km/s. The moon moves at about 1 km/s. So Vinf with regard to the moon is about .47.

Lower Super GEO tether, 7,100 km long
Safety Factor3
Zylon taper ratio:1.21
Tether to payload mass ratio:.47
Tether foot distance from earth45,000 km
Tether foot speed:2.4 km/s
Tether foot net acceleration:.07 m/s2 (.007 g)
Payload perigee:21,450 km
Payload perigee speed:5 km/s

The tether foot drops a payload to rendezvous with the MEO tether.

Sub MEO Tether


The circular orbit of the Sub MEO anchor mass is has a radius of 19,425 km. To get satellites from the super synchronous graveyard orbit to this orbit takes about 1.4 km/s. Some of that 1.4 km/s might be accomplished with the super GEO tether. Sending mass downward would help push the remaining GEO sats upward.

Upper Sub MEO Tether, 2,050 km long
Safety Factor3
Zylon taper ratio:1.30
Tether to payload mass ratio:.61
Tether top distance from earth21,450 km
Tether top speed:5 km/s
Tether top net acceleration:.3 m/s2 (.03 g)
Payload apogee:45,000 km
Payload apogee speed:2.4 km/s

The payload apogee radius and speed matches the foot of the super  GEO tether's radius and speed.
The top of this tether's radius and speed matches the payload perigee and speed sent from super GEO tether. The Sub MEO and Super GEO tethers can exchange payloads with minimal delta V at tether/payload rendezvous.

Lower Sub MEO tether.
Safety Factor3
Zylon taper ratio:1.35
Tether to payload mass ratio:.78
Tether foot radius17,375 km
Tether foot speed:4.1 km/s
Tether foot net acceleration:.38 m/s2 (.038 g)
Payload perigee:9,680 km
Payload perigee speed:7.3 km/s

The Low Sub MEO tether sends and receivse payloads to and from the upper Super LEO tether.

Super LEO Tether


The anchor mass is in a circular orbit of radius 9300 km.

Upper Super LEO Tether, 765 km long
Safety Factor3
Zylon taper ratio:1.4
Tether to payload mass ratio:.84
Tether top radius10,065 km
Tether top speed:7.1 km/s
Tether top net acceleration:.11 m/s2 (.011 g)
Payload apogee:17375 km
Payload apogee speed:4.1 km/s

The payload apogee is at lunar altitude and the payload's moving .53 km/s. The moon moves at about 1 km/s. So Vinf with regard to the moon is about .47.

Lower Super LEO tether, 450 km long
Safety Factor3
Zylon taper ratio:1.13
Tether to payload mass ratio:.29
Tether foot distance from earth8,844 km
Tether foot speed:6.2 km/s
Tether foot net acceleration:.7 m/s2 (.07 g)
Payload perigee:6,778 km
Payload perigee speed:8.3 km/s

Perigee altitude is about 300 km. Circular orbital speed at this atltitude is about 7.7 km/s. To send a LEO payload on it's way to the Super LEO tether would take about .6 km/s.

Sending a payload from the tether to LEO can take less than .6 km/s as the delta v needed for circularizing can be provided by aerobraking.

Total Tether Mass to Payload Ratio

We've looked at a total of 6 tether lengths, the upper and lower parts of 3 vertical tethers.

Tether Mass to Payload Mass Ratios & Lengths

  T/P
Length (km)
Upper Super GEO
  .78
10340
Lower Super GEO
  .47
  7100
Upper Sub MEO
  .61
  2050
Lower Sub MEO
  .78
  2050
Upper Super LEO
  .84
    765
Lower Super LEO
  .29
    450
Total:
3.77
22,755

Thus 38 tonnes of Zylon could accommodate 10 tonnes of payload. That's not too bad.

A much larger problem is the anchor mass needed for each tether. There are lots of dead sats just above GEO that could be gathered for the Super GEO tether anchor mass. But anchor masses for the sub MEO and super LEO tethers will be more expensive. This is a possible show stopper.

Facilitating Momentum Exchange

Using Hall Thrusters to restore momentum.

Sending mass from LEO to a lunar height apogee saps our tethers' orbital momentum. The momentum hit is somewhere around payload mass * 4 km/s. Orbital momentum can be restored gradually with ion thrusters. Hall Thrusters can expel xenon with a 30 km/s exhaust velocity.

Plugging these numbers into the rocket equation:

Propellent mass fraction = 1 - e -4/30 = ~.125.

About 1/8. So to make up for the momentum lost throwing 7 tonnes of payload, we'd need a tonne of xenon. Better than chemical but still expensive.

Lunar or NEA propellent as a source of up momentum.

Some Near Earth Asteroids (NEAs) can be parked in lunar orbit for as little as .2 km/s. Carbonaceous asteroids can be up to 40% water by mass (in the form of hydrated clays). There may be rich water ice deposits in the lunar cold traps. So far as I know, these are the most accessible potential sources of extra terrestrial propellent.

Catching propellent from higher orbits would boost a tether's momentum. Dropping this payload to a lower tether would also boost momentum.

Thus up momentum can be traded for down momentum. Xenon reaction mass to maintain tether orbits can be cut drastically with two way traffic.

Jon Goff's gear ratios

Jon Goff has pointed out it take some delta V to get propellent from the moon's surface to LEO. Thus only ~10% of propellent mined lunar cold traps would make it LEO. See his blog post The Slings And Arrows of Outrageous Lunar Transportation Schemes Part-1 Gear ratios.

Well, lunar propellent could be a source of down momentum for the Lunar Sky Hook I described recently. And a source of up momentum for the Trans Cislunar Railroad this blog post looks at. NEA propellent could also be a source of up momentum for the Trans Cislunar Railroad.

Using propellent as a source of tether up momentum I believe it's plausible for 40% of the lunar propellent to make it to LEO. In which case it becomes plausible to use reaction mass to mitigate the extreme conditions of re-entry.

Breaking the Genie's Bottle

The human race is a genie in a bottle. Given Tsiolkovsky's rocket equation, it's enormously difficult to cross the boundaries that confine us. But given infrastructure and resources at our disposal, we can build bridges to larger frontiers.



Pluto Charon Elevator

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Double Tidal Locking

Pluto and Charon are mutually tidally locked. That is, they both present the same face to the other planet all the time. They hover motionless in each other's sky. Pluto is in Charon synchronous orbit and Charon is in Pluto synchronous orbit.


A tether could be extended from Pluto's near point to Charon's near point. Since the orbit is so nearly circular, there would very very little flexing of this tether.

Minimum Tether to Remain Aloft

To remain aloft, a tether anchored to Charon would need to extend past the L1 point more than 10,000 kilometers to within nearly 2,500 kilometers of Pluto's surface.


This tether would be more than 15,000 kilometers long. Using Wolfe's Spreadsheet we find Zylon taper ratio is 1.13. Tether to Payload mass ratio is .88. This is with a safety factor of 3.

All The Way To Pluto

Extending the tether an additional 2,500 kilometers anchors it to Pluto's surface.


Taper ratio is about 1.7 and Tether to Payload mass ratio is 14.36.

Still acceptable but dramatically different from a tether only 2,500 shorter. This is because we dropped the tether foot into a much steeper part of Pluto's gravity well.


Net acceleration is .62 meters/second2 at the Pluto end of the elevator. Very close to Pluto's surface gravity. At the Charon anchor net acceleration is -.28 meters/second2. Very close to Charon's surface gravity. It is negative to indicate it's in the opposite direction from Pluto's gravity.

At L1 net acceleration is zero.

It's easy to see most of the stress newtons come from the close neighborhoods of Pluto or Charon. It might be worthwhile to build standard compressive towers at the elevator anchor points.

What's The Point?

Pluto's surface escape velocity is 1.2 km/s. Charon's surface escape velocity is .6 km/s. It's not that hard to get off the surface of Pluto or Charon. So what's the point of an elevator?

Space craft with very good ISP have meager thrust. With such space craft soft landings on Pluto or Charon would not be possible. Nor could they leave the surface of these planets.

But a low thrust craft could dock with the elevator at L1.

From L1 a small nudge could send passengers or cargo towards Pluto or Charon. And gravity would pull it the rest of the way down.

I believe Pluto Charon L1 would become  a major metropolis on the corridor between two major city states as well as a port to the rest of the solar system.

Will humans reach Pluto?

The Edge of Sunlight

Sunlight falls with inverse square of distance from sun. Asteroids 3 A.U. from from the sun will receive 1/9 of the insolation we enjoy on earth. Sun Jupiter Trojans at 5 A.U. will get 1/25 the sunlight. We could compensate by constructing large parabolic mirrors to harvest sunlight.

Giant parabolic mirrors could harvest sunlight for spin habs.


But Pluto  has a 30 A.U. by 49 A.U. orbit. And most of the time it dwells in the neighborhood of aphelion. 1/492 = about 1/2400. Mirrors for the KBO nation states would need to be vast. Mike Combs wrote a neat story featuring these sorts of mega mirrors. As much as I enjoy Mike's story, I don't think such monster mirrors are practical.

Fusion Power?

Will our technology achieve practical fusion power plants? Maybe. If so, that would vastly expand our possible frontiers.

The 4th Space Frontier

There's nothing like logistic growth ceilings to motivate opening a new frontier. As we settle and fill up one frontier, we start looking over the horizon. I'm going to make some wildly speculative predictions. This is a science fiction blog, after all.

1st space frontier: NEAs, Luna, Mars, Phobos and Deimos. This would give us one or two millennia of unrestrained growth.

2nd space frontier: The Main Belt. Three millennia of exponential growth. Ceres will be the capital of this United Federation of Main Belt Nation States. This frontier will open within a century or two after we establish a strong foothold on Mars/Phobos/Deimos.

3rd space frontier: The Sun Jupiter Trojans. The Hildas will be our ride from the Main Belt to the Trojans. It will take five hundred years to fill the Trojan petri dish.

4th space frontier: The Kuiper Belt as well as the icey moons of Saturn, Uranus and Neptune. As mentioned earlier, this would require practical power sources other than sunlight. Pluto will be the capital of the United Federation of Kuiper Belt Nation States. This frontier will take 10 millennia to expand into.

5th space frontier: The Oort. The nation states of the Oort will be separated by vast distances. They will be more isolated than even the nation states of the Kuiper. There is a strong incentive to become less reliant on trade and more self sufficient. 20 millennia of unrestrained growth. By the time we reach the outer Oort, nation states will be self sufficient biomes. There would be nothing preventing an outer Oort nation state from achieving solar escape velocity and leaving our sun's sphere of influence.

The Outer Oort Nation States will be natural generation star ships.


Charon Elevator through L2

Enough wild eyed fantasy. Back to mundane stuff like space elevators in the Kuiper Belt.

To maintain tension and remain , an elevator from Charon's far point through the Pluto Charon L2 would need to extend 41,000 kilometers. With a safety factor of 3, Zylon taper ratio would be 1.14. Tether to Payload mass ratio would be  about .3.

Small Problem: Styx

Pluto's moon Styx orbits at a distance of 42,600 kilometers from Pluto. Charon orbits at about 20,000 kilometers from Pluto. So a tether from Charon's far point can only extend about 22,000 kilometers before it runs the risk of an impact with Styx.

A counterweight would need to be placed on the elevator somewhere below the orbit of Styx. If placed just below the orbit of Styx, the tether top could impart a velocity of about .5 km/s. Which would be helpful for injection into heliocentric transfer orbits to other destinations in the solar system.

This elevator could also help with transportation between Charon and the other moons of Pluto: Styx, Nix, Kerboros and Hydra. 

An ion craft could also dock with Pluto Charon L2, so L2 could also serve as a port to the rest of the solar system. There are heteroclinic paths between L1 and L2 so transportation between the two elevators would be easy.

Pluto And I Share An Annivesary

Clyde Tombaugh discovered Pluto on February 18, 1930. February 18 is my birthday! So I guess it's only natural I'm interested in this body, we're practically twins.

General template for space elevators

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A Family of Conic Sections

Below is a general vertical space elevator. The conic sections are the paths payloads would follow if released from a point on the tether a distance r from body center. We choose our units so radius of the balance point is 1. The balance point is where centrifugal force matches gravity and that part of the elevator feels zero net acceleration.


This family of conic sections are coplanar, coaxial and confocal. Eccentricity is r3-1, setting r = 1 at the circular orbit of the balance point. (See this stack exchange answer for the math). In the yellow region are hyperbolic orbits. In the blue region are are elliptical orbits higher than the circular orbit at the balance point. In the orange region, the tether drops payloads into elliptical orbits lower than the circular orbit at the balance point.

A circle of eccentricity zero separates the orange and blue regions, radius of circle = 1.
A parabola of eccentricity 1 separates the blue and gold regions, radius of parabola's periapsis = 21/3

Here is the same graphic zoomed in:


Here is the graphic as a Scalable Vector Graphic. I am hoping science fiction writers and illustrators will download this resource and use it.

Scaling this graphic for a variety of scenarios:


The numbers are in kilometers. In the case of earth, the circular orbit is the geosynchronous orbit at an altitude of about 36,000 kilometers.

In general, radius of a synchronous orbit can be described as:

r = (Gm / ω2)1/3

Where ω is the body's angular velocity in radians, 2 pi radians/sidereal day.

Orbital Elevators

We usually think of an a space elevator anchored at the body's equator. An elevator can also be in a non synchronous orbit. Here the template is scaled to match the orbits of Phobos or Deimos:



Notice Phobos' tether foot  is above Mars surface. The foot is moving about .5 km/s with regard to Mars surface and therefore can't be anchored to Mars. Neither could a Deimos elevator be attached to Mars.

Orbital radius of Phobos is about 40% that of Deimos. So I cloned and shrunk Deimos' tether conics by 40%. I rotated the cloned family of conics by 180º.  The result is an interesting moiré pattern:


It was this pattern that led me to search for a common ellipse.

Eccentricity of the common ellipse:

e = (1 - (ωDeimosPhobos)1/2) / (1 + ωDeimosPhobos)1/2)

Periapsis and apoapsis of the common ellipse:

rperiapsis = (1 + e)1/3 rPhobos
rapoapsis = (1 - e)1/3 rDeimos

Here's a pic of the ellipse Phobos and Deimos share:


Thus it is possible to travel between Phobos and Deimos using nearly zero reaction mass.

Not just Phobos and Deimos

To be an anchor for a vertical elevator, a moon needs to be in a near circular orbit and tide locked to its planet. This describes most of the moons in our solar solar system. For two moons to share an ellipse, they need to be nearly coplanar. Again, most the moons in our solar system.

Here are the common ellipses between the moons of Saturn:


Judging by the two gas giants and two ice giants in our solar system, families of coplanar, tidelocked moons are common.

ZRVTOs

Zero Relative Velocity Transfer Orbits (ZRVTOs) - a term coined by Marshall Eubanks. In Marshall's words: "locations (and times, say for a Lunar and Terrestrial space elevator) where you drop things from one space elevator and they approach and hang motionless (for an instant) at a location on the other elevator.  ... what you would want for large scale movement of material."

Eubanks goes on to say "In practice, you might need a little bit of course correction delta-V to make up for radiation pressure, etc."

Also it would be rare for the elevators playing to catch to be perfectly coplanar. So a small plane change delta V expense will be the rule rather than the exception. Still the delta V budgets would be a small fraction of what it would take for normal lift off and insertion to Hohmann transfers.

Mini Solar Systems

Earlier I had looked at Mini Solar Systems, a notion I stole from Retrorockets. In our solar system Hohmann trip times between planets are on the order of months or years. Launch windows are typically years apart. But for a system of moons around a gas giant, trip times and launch windows are days or weeks. So a Flash Gordon paced story could take place without wildly improbable engineering.

GIELO and ELM

GIELO - Giant In Earth Like Orbit. ELM - Earth Like Moon. I have long been infatuated with this setting. Here is a painting I had done in 2001:


ELM the earth like moon is in the upper right. In the foreground a generation star ship is sending quad pod scout probes to investigate an artifact at the GIELO-ELM L4 region.

James Cameron's Avatar uses such a setting. Pandora is an ELM. I believe this setting could be developed a lot more. If ELM had sister moons and they were all tide locked, it would be a nice mini-solar system setting.

Icey moons with hospitable interiors.

Gas giants in Goldilocks zones aren't the only possibility. Temperature and pressure rise as we burrow deeper into a body. Earth might not be the only location in the solar system that has liquid water at a livable pressure. Thus the icey moons of our own solar system might eventually become "mini solar systems".

Planets of red dwarfs

And recently an approximately earth sized planet was found in the goldilocks zone of Proxima Centauri. Proxima Centauri is a small red dwarf star. The possibly earth like planet has an orbital radius of about 7.5 million kilometers and an orbital period of about 12 days. Planets about small red dwarfs are yet another possible "mini solar system" setting. Planets so close are likely tide locked to the star. Would atmospheric convection mitigate the temperature extremes between the night side and day side? I'm not sure. In any case, I believe there would be a comfortable region hugging the planet's frozen terminator. (By "frozen" I mean stationary).

Delta V and the rocket equation

The Retrorockets guy took a second look at mini solar systems. While trip times are short and launch windows frequent, it still takes a lot of delta V to insert to a Hohmann transfer. I was annoyed he used the incorrect Tom Murphy method of patching conics. But most his math is sound. He is correct that Tsiolkovsky's rocket equation would be a major pain in the mini solar system just as it is in ours. 

This is where a tether system comes into play. Given elevators on tide locked bodies and assuming most the bodies are nearly coplanar, travel between bodies could be done with very little reaction mass. It'd still take a lot of energy to move stuff up and down the elevators. But the difficult mass fractions imposed by the Tsiolkovsky's equation would no longer be a consideration.

Summary

Similar mathematical models and drawings can be used for a wide range of vertical tethers.

A popular misconception is that elevators are only good for getting off the ground. So it's a waste to build an elevator from a small body. But an elevator not only gets the payload off the ground, it can fling a payload towards a destination. The hyperbolic orbits portrayed in this post are especially interesting.

Space elevators would be especially useful in a system of tide locked moons. Or tide locked planets about a small star.

So far the only elevators I see portrayed in science fiction are from major planets. Like Kim Stanley Robinson's elevators in his Mars trilogy. Or Clarke's earth elevator in Fountains of Paradise. There are far more plausible elevators that could be very useful. These doable elevators could also provide many interesting settings.

Xenon

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Xenon The Noble Gas

Xenon is one of heavier Noble Gases

Screen capture from ChemicalElements.com

The noble gases are the orange column on the right of the periodic table. These are chemically inert. Which means they're not corrosive. This makes them easier to store or use.

Low Ionization Energy

Per this graph is from Wikipedia, Xenon has a lower ionization energy than the lighter noble gases.


Ionization energy for xenon (Xe) is 1170.4 kJ/mol. Ionization for krypton (Kr) is 1350.8 kJ/mol. Looks like about a 15% difference, right?

But a mole of the most common isotope of xenon is 131.3 grams, while a mole of krypton is 82.8 grams. So it takes 181% or nearly twice as much juice to ionize a gram of krypton.

Likewise it takes nearly 4.5 times as much juice to ionize a gram of argon than it does to ionize a gram of xenon.

The reaction mass must be ionized before it can be pushed by a magnetic field. Xenon takes less juice to ionize. So more of an ion engine's power source can be devoted to imparting exhaust velocity to reaction mass.

Big Atoms, Molar Weight

Low molar weight makes for good ISP but poor thrust. And pathetic thrust is the Achilles heel of Hall Thrusters and other ion engines. The atomic weight of xenon is 131.29 (see  periodic table at the top of the page).

Tiny hydrogen molecules are notorious for leaking past the tightest seals. Big atoms have a harder time squeezing through tight seals. Big whopper atoms like xenon can be stored more easily.

Around 160 K, xenon is a liquid with a density of about 3 grams per cubic centimeter. In contrast, oxygen is liquid below 90 K and a density of 1.1. So xenon is a much milder cryogen than oxygen and more than double (almost triple) the density.

Abundance

Ordinary atmosphere is 1.2 kg/m3 while xenon is about 5.9 kg/m3 at the same pressure. Xenon has about 4.8 times the density of regular air.

By volume earth's atmosphere is .0000087% xenon. 4.8 * .000000087 = 4.2e-7. Earth's atmosphere is estimated to mass 5e18 kg. By my arithmetic there is about 2e12 kg xenon in earth's atmosphere. In other words, about 2 billion tonnes.

Page 29 of the Keck asteroid retrieval proposal calls for 12.9 tonnes of xenon. Naysayers were aghast: "13 tonnes is almost a third of global xenon production for year! It would cause a shortage." Well, production is determined by demand. With 2 billion tonnes in our atmosphere, 13 tonnes is a drop in the bucket. We throw away a lot of xenon when we liquify oxygen and nitrogen from the atmosphere.

In fact ramping up production of xenon would lead to economies of scale and likely cause prices to drop. TildalWave makes such an argument in this Space Stack Exchange answer to the question "How much does it cost to fill an ion thruster with xenon for a spacecraft propulsion system?" TildalWave argues ramped up production could result in a $250,000 per tonne price. That's about a four fold cut in the going market price of $1.2 million per tonne.

Radon

If you examined the periodic table and ionization tables above you might have noticed there's a heavier noble gas that has an even lower ionization energy: Radon a.k.a. Rn.  Radon is radioactive. Radon 222, the most stable isotope, has a half life of less than 4 days. If I count the zeros on the Radon page correctly, our atmosphere is about 1e-19% radon -- what you'd expect for something with such a short half life. Besides being rare, it wouldn't last long in storage.

Where xenon excels

Great for moving between heliocentric orbits

Ion thrusters can get 10 to 80 km/s exhaust velocity, 30 km/s is a typical exhaust velocity. That's about 7 times as good as hydrogen/oxygen bipropellent which can do 4.4 km/s. But, as mentioned, ion thrust and acceleration are small. It takes a looong burn to get the delta V. To get good acceleration, an ion propelled vehicle needs good alpha. In my opinion, 1 millimeter/second2 is doable with near future power sources.

If the vehicle's acceleration is a healthy fraction of local gravity field, the accelerations resemble the impulsive burns to enter or exit an elliptical transfer orbit. But if the acceleration is a tiny fraction of the local gravity field, the path is a slow spiral.


Earth's distance from the sun, the sun's gravity is around 6 millimeters/second2. At Mars, sun's gravity is about 2.5 mm/s2 and in the asteroid belt 1 mm/s2 or less. Ion engines are okay for moving between heliocentric orbits, especially as you get out as far as Mars and The Main Belt.

Sucks for climbing in and out of planetary gravity wells

At 300 km altitude, Earth's local gravity field is about 9000 millimeters/second2. About 9 thousand times the 1 mm/s2 acceleration a plausible ion vehicle can do. At the altitude of low Mars orbit, gravity is about 3400 millimeters/sec2. So slow gradual spirals rather than elliptical transfer orbits. There's also no Oberth benefit.

At 1 mm/sec2 acceleration, it would take around 7 million seconds (80 days) to climb in or out of earth's gravity well and about 3 million seconds (35 days) for the Mars well.

Mark Adler's rendition of an ion spiral
where the thruster's acceleration is 1/000 that of local gravity at the start.

The general rule of thumb for calculating the delta V needed for low thrust spirals: subtract speed of destination orbit from speed of departure orbit.

Speed of Low Earth Orbit (LEO) is about 7.7 km/s. But you don't have to go to C3 = 0, getting past earth's Hill Sphere suffices. So about 7 km/s to climb from LEO to the edge of earth's gravity well.

It takes about 5.6 km/s to get from earth's 1 A.U. heliocentric orbit to Mars' 1.52 A.U. heliocentric orbit.

Speed of Low Mars Orbit (LMO) is about 3.4 km/s. About 3 km/s from the edge of Mars' Hill Sphere to LMO.

7 + 5.6 + 3 = 15.6. A total of 15.6 km/s to get from LEO to LMO.

With the Oberth benefit it takes about 5.6 km/s to get from LEO to LMO. The Oberth savings is almost 10 km/s.

10 km/s is nothing to sneeze at, even if exhaust velocity is 30 km/s. Climbing all the way up and down planetary gravity wells wth ion engines costs substantial delta V as well as a lot of time.

Elevators and chemical for planet wells, ion for heliocentric

So in my daydreams I imagine infrastructure at the edge of planetary gravity wells. Ports where ion driven driven vehicles arrive and leave as they move about the solar system. Then transportation from the well's edge down the well would be accomplished by chemical as well as orbital elevators.


Other possible sources of ion propellent.

Another possible propellent for ion engines is argon. Also a noble gas. Ionization energy isn't as good as xenon, but not bad. Mars atmosphere is about 2% argon. Mars is next door to The Main Belt. I like to imagine Mars will supply much of the propellent for moving about the Main Belt.

Lamentable Lagrange articles

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Gravity doesn't cancel at the Lagrange points

"There are places in the Solar System where the forces of gravity balance out perfectly. Places we can use to position satellites, space telescopes and even colonies to establish our exploration of the Solar System. These are the Lagrange Points."

From Fraser Cain's video on Lagrange points. A lot of pop sci Lagrange articles repeat and spread this bad meme. It just ain't so.

The 5 Lagrange points can be found in many two body systems. They can be Sun-Jupiter, Earth-Moon, Jupiter, Europa -- Any pair of dancers has this retinue of 5 Lagrange regions moving along with them. Above are the 5 Pluto-Charon Lagrange points. Also pictured are the gravity vectors these bodies exert. Pluto's gravity is indicated with purple vectors and these point towards Pluto's center. Charon's gravity is indicated with orange vectors and these point towards Charon's center.

For the gravity vectors to cancel each other, they need to be equal and pointing in opposite directions.

L1


The only L-Point where the gravity vectors pull in opposite directions is L1. And here the central body (Pluto) pulls harder than Charon. These two gravities don't balance out.


L3 and L2


Zooming in on the L3 and L2 points, we can see both bodies pull the same direction. These don't balance.


L4 and L5


Zooming in on the L4 and L5 points. Pluto pulls much harder. The angle between these vectors is 60º

The So-Called Centrifugal Force

There is a third player in these Lagrange tug of wars. What we used to call centrifugal force. This is not truly a force but rather inertia in a rotating frame. Here is an XKCD cartoon on this so called force:


Indeed, in a rotating frame, inertia sure feels like a force. The pseudo acceleration can be described as ω2r where ω is angular velocity in radians per time and r is distance from center of rotation. The vector points away from the center of rotation.

Putting Gravity and Centrifugal Force Together



Here's the same diagram but with centrifugal force thrown in (the blue vectors). Also the foot of the Charon gravity vectors are placed on the head of the Pluto gravity vectors -- this is a visual way to carry out vector addition.

For L1, Charon and Centrifugal Force are on the same team and they perfectly balance Pluto's gravity.

For both L2 and L3, Pluto and Charon are on the same team and they neutralize their opponent Centrifugal Force.

But what about L4 and L5? An observant reader may notice that the centrifugal force vector doesn't point away from Pluto's center. Adding Charon's tug to Pluto's tug moves the direction to the side a little bit.



Now the centrifugal force vector points from the barycenter. This is the common point of rotation around which both Pluto and Charon rotate. The same applies to L5.


L4, Charon's center and Pluto's center form an equilateral triangle.
The barycenter lies on the corner of a non-equilateral triangle.


And so it is with all the orbiting systems in our neighborhood. It is a 3 way way tug-of-war between centrifugal force, gravity of the orbiting body and gravity of the central body. Sometimes two players are on the same team, other places they switch. In L4 and L5 everyone pulls in a different direction. But in all 5 Lagrange points, the sum of the three accelerations is zero.



Matrices

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I enjoy linear algebra. I use it in making polyhedra and drawings. Here are a few visualizations of matrices. To keep it simple I use 2x2 matrices. 2x2 matrices act on vectors. The vertices of polygons in a plane can be described vectors, so these same transformations can be done to polygons that dwell in a plane.

Rotation Matrix


Rotating a polygon doesn't change it's area. The area remains the same. The determinant of this matrix is 1.


Proportional Scaling Matrix


Doubling size as well as height boosts a polygon's area by a factor of four. This determinant of this matrix is 4.


Non Proportional Scaling Matrix


This matrix stretches the width to twice the original and squeezes the height to half of what it was. Overall the area is unchanged. The determinant of this matrix is 1.


Shear or Skew Matrix

When I was using Macromedia Freehand, the graphics program called this transformation "skew". Then Adobe ate Macromedia and I was forced to use Adobe Illustrator. Illustrator calls it "shear".


This transformation transforms a horizontally aligned rectangle to a parallelogram with same base and height. Area remains unchanged. The determinant of this matrix is 1.

Flip Matrix


Making the first term in the main diagonal negative flips polygons about the y axis. Making the lower right term negative would flip polygons about the x axis.

Determinant is -1. Not sure what that means geometrically but absolute value of the area remains the same.

Illustrator Tool Box

The above matrices are various forms of tools from the Adobe Illustrator Tool Box. Here's a couple of screen caputres spliced together:


One tool compartment has the reflect and rotate transformations, the neighboring compartment has the scale and shear transformations. I'm not sure what the Reshape Tool is.

Hopefully I will have time and energy to add more to this matrix post soon.



Zylon Mars Elevator

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Mars Elevator With Conventional Materials

Mars spins nearly the same rate as earth (about a 24.62 hour day). Mars has about 1/9 earth's mass. At 17,000 kilometers, altitude of Mars synchronous orbit is less than half the altitude of geosynchronous orbit (about 36,000 kilometers).

These considerations have led some Mars enthusiasts to claim a Mars elevator made of conventional materials is possible. No bucky tubes or other science fiction material is needed, Kevlar will do. Is this true? I will take a look using Chris Wolfe's spreadsheet.

Safety Factor

In earlier blog posts using Wolfe's spreadsheet I used a safety factor of 1, a razor thin margin. The slightest scrape or nick will make the tether break. This is like drawing a pentagram to summon the demon Murphy's Law. No sensible entity would risk expensive payloads on such a narrow margin. Much less human lives. I hope to revise my earlier blog posts to include more sensible safety margins.

In later blog posts I looked at scenarios using a safety factor of 3. With this margin a portion of tether can lose up to 2/3 of it's mass without breaking.

In this post I'll use tables looking at a range of safety factors.  With a safety factor of 2, I cut tensile strength in half. A safety factor of 3 cuts tensile strength to a third. Which is a lot like cutting exhaust velocity in the rocket equation. Increasing an exponent can make tether thickness sky rocket.

Mars Equator to Mars Synchronous Orbit

This is the lower part of a Mars elevator. It exerts downward newtons that need to be balanced with upward newtons from elevator mass above Mars synchronous orbit.

Safety
 Factor 
Zylon
Taper
Ratio
Tether to
Payload
 Mass Ratio 
1
13
154
2
162
3191
3
2016
51824

Payload is mass of elevator car as well as elevator car's contents. The elevator car will need to include motors and power source.

Mars Synchronous to Sub Deimos Elevator Top

Elevator top is set 50 kilometers below Deimos' periapsis. This is to avoid collision. The counterweight and tether above Mars synchronous orbit must counterbalance the downward force of the lower elevator.

Safety
 Factor 
Zylon
Taper
Ratio
Tether to
Payload
 Mass Ratio 
Counterweight
to Payload
 Mass Ratio 
1
1.02
38
1200
2
1.03
955
14800
3
1.05
1761
180000


The Whole Shebang

Safety Factor 1

Assuming lifting a 10 tonne elevator car and contents from Mars' surface and given a safety factor of 1, we'd need 10 * (38 + 154) tonnes of tether material. That'd be 1,920 tonnes of Zylon. Perhaps worthwhile if the elevator had a vigorous through put. I think these are the numbers Mars enthusiasts are talking about when they talk about Mars beanstalks made of Kevlar.

Also needed would be a 12,000 tonne counterweight. That's about thirty times the mass of the I.S.S.. This to lift a 10 tonne elevator car from Mars' surface? The need for a stud hoss counterweight sinks the argument for a Mars elevator, in my opinion.

Safety Factor 2

10 * (162 + 955) = 11170. About 11 thousand tonnes of Zylon to lift a 10 tonne elevator car and contents.

We'd need a nearly 150,000 tonne counterweight.

I think it's pretty obvious a Zylon Mars elevator with a safety factor of two isn't worthwhile. I'm not going to bother looking at a safety factor of 3.

Benefits

The elevator top is moving at about 1.7 km/s. It needs another 1.6 km/s to achieve Trans Earth Insertion (TEI). From the surface of Mars it takes about 6 km/s for TEI. So the elevator cuts saves about 4.4 km/s off of trips to earth.

Obstacles

Given a sensible safety factor, a Zylon tether would need to be much more massive than the payload. The counterweight mass would dwarf the payload mass.

Mars neighbors the main asteroid belt. Some rocks from the belt make their way to Mars neighborhood. Collision with asteroidal debris could cut the tether. Given this elevator's 20,000 km length and healthy taper ratio, there is a large cross sectional area. This increases likelihood of an impact.

Also there is a chunk of Debris named Phobos which crosses the elevator's path every 10 hours or so.

Comparison to Phobos Elevator


A Phobos elevator dropping to Mars' upper atmosphere and extending to Trans Ceres insertion is about 13,700 km. This about 6,000 km shorter than the Mars elevator described above. It also has a smaller taper ratio. This makes for a smaller cross sectional area to intercept debris. Being anchored at Phobos, this elevator won't collide with Phobos. The top is well below Deimos. orbit.

This tether can provide Trans Ceres Insertion as well as Trans Earth Insertion.

It takes about a .6 km/s suborbital hop for a Mars ascent vehicle to rendezvous with this tether foot.

Using a safety factor of 1, the upper Phobos tether has a 3.21 payload to mass ratio. The lower Phobos tether has a tether to payload mass ratio of about 16.1. So from top to bottom, about twenty times the payload mass is needed in Zylon.

The Phobos takes about 1/10 of the Zylon mass for a mars elevator with a safety factor of one.

A sub Deimos Mars elevator can't throw payloads above Mars escape velocity.
But with higher taper ratio, it'd take ten times as much zylon mass than a Phobos elevator.
This is with a safety factor of 1.
A Zylon Mars elevator with better safety factors is impractical.

I hope to revisit the upper Phobos tether and lower Phobos tether pages and include safety factors of 2 and 3. I suspect with a higher safety factor that a Zylon tether from Phobos to Mars upper atmosphere may not be feasible.










Tahoe recreational math conference

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I've been invited to talk at a recreational math conference at Lake Tahoe April 28.

I will talk about the family of conics associated with a space elevator or a vertical orbital tether.

The presentation combines stuff from a number of my earlier blog posts. Here is the pdf I will use for this talk.

A screenshot from the cover of the booklet.

Pages 2 — 5 -- includes the visualizations I use to remember expressions for gravity and centrifugal acceleration. Which I use to show how canonical units are a whole lot easier to use than kilograms, kilometers and so on. For example the orbital periods or an orbit whose semi-major axis is k A.U. will have period k3/2 years. An asteroid whose semi-major axis is 4 A.U. will have a period of 8 years. A 9 A.U. semi-major axis gives a period of 27 years.

Pages 13 — 14 -- I believe my general method  for finding a ZRVTO orbit between tethers is new and original. If anyone knows of this method appearing in earlier publications, please give me a heads up.

Page 14 -- Most moons we know of are in nearly circular tide locked orbits. For such moons beanstalks through Planet-Moon L1 and L2 are plausible. I believe this would be a wonderful science device. But today's science fiction writers seems oblivious to any of the Lagrange points other than L4 and L5.

Pages 15 and 16 -- Shameless self promotion. I don't get any royalties on the Dover coloring books, I was paid a straight fee of $100 per page. Regardless, I'd like to see the books sell. A Dover editor placed his bets on me and I hope he is rewarded for this gamble. I do make money on T-shirt sales though. My T-shirts are available at Ajo-Copper-News.com  .



Space Meow Boys

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Sections of this long post:

1) Space Cadets
2) Space Meow Boys
3) Tom Murphy
4) James Nicoll
5) Charlie Stross
6) Opening A New Frontier Is Doable


Space Cadets


We are confined to a small, fragile planet. Being limited to a finite body of resources mean logistic growth. And we're rapidly approaching the ceiling to our logistic growth.

Opening a vast new frontier would allow growth for centuries or even thousands of years. Breaking free of Cradle Earth would be the most dramatic turning point in human history. If it’s possible then this goal is well worth pursuing.

But can we open the solar system to settlement and economic use? This is an open question in my opinion.

Some say space settlement is impractical. Be content with our limits, we’re told. Trying to push past our boundaries is a waste of time and we shouldn’t even try.

Civil, rational arguments are worth listening to. But some discussions are long on vitriol and short on math and physics.

Tarring With A Wide Brush

One dirty technique is tarring with a wide brush -- First find weak members in a group. Then hold up these members up as representative of the entire group. Give them a label.

Physics professor Tom Murphy does this. He holds up his clueless students as examples of space enthusiasts and tars us all with the label space cadets. Judging by the stories he tells, his students are some of the stupidest people on the planet. I suspect he teaches Astronomy 101 for Liberal Arts Majors.

Science fiction writer Charles Stross and book reviewer James Nicoll also like to use the label space cadet. They point to folks from Usenet who are long on wishful thinking and short on math skills. Their flavor of space cadet tends to be white and Libertarian.

Wrestling With A Pig

Friends tell me “Don’t wrestle with a pig. You both get dirty and the pig likes it.”  What they don’t realize is that I too am a pig. I love to wrestle in the mud!



I don't mind their dirty tricks. I'll do the same.

First I'll find nay sayers clueless in math and science. My label will be Space Meow Boys.


Space Meow Boys


Tom Murphy, James Nicoll and Charlie Stross are my examples of space meow boys.




Tom Murphy


Let's look at Murphy’s blog post Stranded Resources.

Murphy correctly puts a big emphasis on delta V and Tsiolkovsky’s rocket equation. But he sucks at calculating delta V. From his blog:

The next plot puts this in perspective, albeit only in simplified, approximate terms. The bottom of the plot represents the Earth’s ground. It takes 7.7 km/s of velocity to get to LEO (actually, it takes the equivalent of about 9.5 km/s because much effort is expended just climbing out, in addition to establishing the orbital speed). At 11.2 km/s, we’re free to take on the solar system.  The plot is based on minimum-energy Hohmann transfer orbits.



Each planet is represented by three dots: the top one being outside the planet’s grip in an identical solar orbit, the next one down at low-planet orbit (akin to LEO), and the lowest represents being at rest on the surface. For Saturn and Jupiter, these surface points are off the chart—so taxing is this requirement. And for these two, there’s no “there” there anyway to land on. Crudely speaking, we must have the means to accomplish all vertical traverses in order to make a trip. For instance, landing on Mars from Earth requires about 17 km/s of climb, followed by a controlled 5 km/s of deceleration for the descent. Thus it takes something like 20 km/s of capability to land on Mars, . . .

I bolded Murphy’s discussion of the Earth to Mars trip. Let’s look at his delta V.

He takes Earths 11.2 km/s escape velocity and adds in the ~6 km/s difference between Earth’s and Mars’  heliocentric orbits and then adds in Mars 5 km/s escape velocity. Which gives 22 km/s. Then Murphy leaves us with the impression he‘s being generous when he rounds down to 20 km/s

A first year aerospace student would cringe at Murphy’s bungled math. You don’t simply add Vescape and Vinfinity.

To get velocity of the hyperbolic orbit needed for TMI (Trans Mars Injection):

Vhyperbola = sqrt(Vescape2 + Vinfinity2)



A memory device is to think of Vescape and Vinfinity as the legs of a right triangle. Velocity of a hyperbolic orbit would be the hypotenuse.

Correctly patching conics get us 17 km/s from Earth surface to Mars surface

What About The Atmosphere?

Murphy points to a penalty imposed by Earth’s atmosphere:

It takes 7.7 km/s of velocity to get to LEO (actually, it takes the equivalent of about 9.5 km/s because much effort is expended just climbing out, in addition to establishing the orbital speed).

Yes, we suffer a loss of around 2 km/s to climb above the earth's atmosphere. There's some atmospheric friction as well as gravity loss during ascent. We'll give Murphy this 2 km/s. So our delta v budget goes up to 19 km/s.

But an atmosphere also offers the possibility of aerobraking. Is it possible Murphy hasn't heard of aerobraking? Or is he dishonestly focusing on the delta V penalties of an atmosphere while ignoring the benefits? The charitable judgement here is that Murphy is horribly clueless.

Aerobraking at the Mars end of an Earth to Mars trip can shave 6 km/s off the delta V budget. This takes our delta V budget down to 13 km/s. This is less than what it takes to park a satellite in geosynchronous orbit, something we routinely do.

Aerobraking at the Earth end of a Mars to Earth trip can shave 11 km/s off the delta V budget. This leaves a delta V budget of around 6 km/s for the Earth to Mars trip.

Grab That Asteroid!

Asteroid retrieval is a notion entertained by John S. Lewis, Planetary Resources, Deep Space Industries and others. If not retrieval of an entire asteroid, then retrieval of commodities from an asteroid.

Murphy argues against this using a ridiculous straw man scenario:

The asteroid belt is over 20 km/s away in terms of velocity impulse. If the goal is to use the raw materials for production on Earth or in Earth orbit, we have to supply about 10 km/s of impulse. We would probably try to get lucky and find a nickel-metal asteroid in an unusual orbit requiring substantially less energy to reel it in. So let’s say we can find something requiring only 5 km/s of delta-v. Our imagined prize will be a cube 1 km on a side, having a mass around 1013 kg. This is very small for an asteroid, but we need to moderate our ambitions. From a resource point of view, it’s still a lot. 
To get this asteroid moving at 5 km/s with conventional rocket fuel (or any “fuel” that involves spitting the mass elements/ions out at high speed) would require a mass of fuel approximately twice that of the asteroid. As an example, using methane and oxygen, (4 kg of O2 for every 1 kg of CH4), we would require two years’ of global natural gas production to be delivered to the asteroid (now multiply this by a large factor for the fuel to actually deliver it from Earth’s potential well). The point is that we would be crazy to elect to push the asteroid our way with conventional rockets.

Four things wrong this picture.

1) Murphy hasn't heard of NEAs? There are NEAs (Near Earth Asteroids) much closer to the Earth-Moon system. The Keck Report talks about NEAs that could be parked in a loose lunar orbit for as little as .17 km/s. 2006 RH120 was temporarily captured to the earth moon system with no delta V.

2) Murphy wants to use methane/oxygen bipropellant. This has an exhaust velocity of around 4 km/s in a vacuum. The Keck folks propose using xenon and Hall Thrusters. Exhaust velocity for this sort of ion engine can easily be 30 km/s.

3) A kilometer asteroid is far too large for practical rockets to retrieve. It would also be insanely dangerous. The Tunguska event likely came from an object between 60 and 200 meters in diameter. The Chixculub impact which wiped out the dinosaurs was thought to have been 10 to 15 kilometers. Perhaps a misdirected rock 1 kilometer in diameter wouldn't be an extinction level event. But it'd certainly cost trillions in property damage. The Keck folks talk about safety considerations at the bottom of page 15 of their report. They look at retrieving a 5 meter rock. Should a 5 meter rock fall earthward, it'd burn up harmlessly in the upper atmosphere.

4) Murphy assumes a metal rich asteroid. He could spend a few minutes Googling and find that water is the first commodity asteroid miners hope to exploit. Propellant not at the bottom of an 11.2 km/s gravity well would be a game changer that would reduce the cost of spaceflight. And cheaper spaceflight is a prerequisite to profitably exploiting asteroidal metals.

Plugging Murphy's 5 km/s delta V budget and 4 km/s exhaust velocity into the rocket equations tells use that we'd need more than two tonnes of propellant for every tonne of asteroid.

Plugging in .17 km/s delta V and 30 km/s exhaust velocity gives 6 kilograms of propellant needed to park a one tonne asteroid.



The fellow on the left is Tom Murphy. To the right is a self portrait.

Sometimes Murphy tries to excuse himself by pointing to his waffle words and furiously waving his hands. He seems to think words like "approximately" or "roughly" salvage his questionable claims.


Only 3 orders of magnitude off.

Refuel In Space?

The lunar cold traps are thought to have rich deposits of water ice as well as other volatile ices. These potential propellant sources are about 2.5 km/s from EML1 and EML2.

Here's some delta V maps focusing on EML1 and EML2:



There are also asteroid folks who hope to mine water from NEAs. See this Planetary Resources video or this Deep Space Industries video. Some NEAs are up to 40% water by mass and are only a small delta V nudge from being parked in lunar orbit. A water rich asteroid parked in lunar orbit would be even closer to EML1 and EML2.

What is Murphy's argument against refueling in space?

He tells us it'd take a lot of delta V to get propellant from Jupiter or Titan.

Since the large delta-v’s required to get around the solar system require a lot of fuel, and we have to work hard to lift all that fuel from the Earth’s surface, could we just grab hydrocarbons from Jupiter or Titan and be on our way? 
Let’s say you arrived in Jupiter orbit running on fumes, relying on the gassy giant to restock your coffers. In order to get close enough to Jupiter, you’ll be skimming the cloud-tops at a minimum of 42 km/s. Getting 1 kg of fuel on board will require you to accelerate the fuel to the speed of your spacecraft, at a kinetic energy cost of 885 MJ. The energy content of methane is 13 kcal/g, or 54 MJ/kg. Oops. Not even enough to pay for itself, energetically. Get used to Jupiter. And I have completely ignored the fact that you need marry two O2 molecules to each molecule of methane, meaning you actually get only 11 MJ per kilogram of total fuel. Utterly hopeless.

No shit, Sherlock. Knock yourself out beating up this straw man.


Tom Murphy's argument is perhaps the stupidest straw man ever.

Momentum Exchange Tethers

In the comments section for Stranded Resources, Monte Davis writes:

At the level of fundamental elegance, you can’t beat tethers: instead of throwing away momentum in exhaust, you just keep re-using it as payloads are slung around — assuming tethers at all sources/destinations and an abundance of payloads. Before that, make-up energy could be supplied by spinning up tethers slowly with a low-thrust solar-electric or nuclear-electric drive.

Murphy replies to Davis:

I don’t follow the first point about not throwing away momentum in the form of exhaust in a tether system. Without throwing away momentum, you can gain none (and go nowhere). If stranded on a frictionless lake on a sled piled with bricks, the only way off is to hurl bricks away. If the bricks are tethered to you, you may be able to move about as mass is redistributed, but the center of mass will be in the same place always.

Momentum isn't thrown away. It's exchanged.

An orbital tether would not sit motionless like a brick on a frozen lake. It would drop after catching a payload from a lower orbit. It would also drop when throwing a payload to a higher orbit.

However an orbital tether would rise after dropping a payload to a lower orbit. It would also rise when catching a payload from a higher orbit.

With two way traffic an orbital tether could balance momentum draining maneuvers with momentum boosting maneuvers and thus maintain an orbit without huge amounts of propellant.

Also as Davis mentions, a tether can use ion engines. Ion engines can easily have 30 km/s exhaust velocities while the best chemical is around 4.4 km/s. This is a much more efficient way to restore momentum. With low thrust engines it would take a long time to build momentum but that would be okay if there were weeks between tether maneuvers.

Monte Davis is a science writer and editor who's worked for Omni, Discover, Psychology Today and other publications. He's got a chemistry degree from Princeton. In space forums Davis usually plays the devil's advocate against would be space colonizers.

Murphy could have invested 4 or 5 minutes Googling momentum exchange tethers. But he blows off Monte Davis as if he's one of his clueless students in Astronomy 101 for Liberal Arts Majors.


James Nicoll


James Nicoll reviews science fiction. An old Heinlein chestnut is "If you can get your ship into orbit, you're halfway to anywhere." Nicoll attempts to play with this notion at More Words Deeper Hole.

Apparently the subject line I was going to use is offensive so I will go with "halfway to anywhere" 
james_nicoll
april 1st, 2012 
Suppose it's the future and further suppose that space tourism actually takes off enough that there are excursions to the Moon akin to what we see in Antarctica. Although probably not the 37,000 people a year you see headed to Antarctica because going to the Moon is going to a crapton more expensive. 
Further, suppose
it occurs to someone whose life centers on ferrying rich bastards back and forth to the Moon that the delta vee to go from Low Earth Orbit (LEO) to Low Lunar Orbit (LLO) is about 8 km/s. It's the same the other way, assuming no aerobraking at the Earth end (No aerobraking at the Earth end means big mass ratios or some kind of fuel depot in LLO). That's considerably more delta vee than it takes to to Mars from the Moon and it further occurs to them it might be fun on the next trip home to leave the tourists on the Moon and take an unsheduled excursion to Mars.


How would you go about adapting a vehicle designed to do the LEO-LLO trip to a LLO-Mars trip? 
The first big issue is going to be air. Assuming a dozen passengers and three crew, and about a week to the Moon and back, the ship probably doesn't have more than 105 person-days of O2. Fast but still reasonably delta-vee conservative orbit to Mars is about 180 days. 
I suppose, this being fiction, you could do it the other way: the would-be Marsnaut needs 180 person-days, therefore the LEO-LLO transfer ship carries a couple of dozen passengers and some crew. That will at least get the Marsnaut to Mars alive.

Delta V

Let's start with James' delta V budget.

it occurs to someone whose life centers on ferrying rich bastards back and forth to the Moon that the delta vee to go from Low Earth Orbit (LEO) to Low Lunar Orbit (LLO) is about 8 km/s.

According to the Wikipedia delta V chart James snagged, it's 4.1 km/s from LEO to L4/5 and then .7 km/s to lunar orbit.

4.1 + .7 = 4.8, not 8.



A direct route from LEO to LLO would be more like 4 km/s.

For hard SF folks, 8 km/s from LEO to LLO is a glaring error. But it's no biggie for the English Lit types that participate in James' forum. They don't even notice.

Aerobraking

James stipulates

It's the same the other way, assuming no aerobraking at the Earth end (No aerobraking at the Earth end means big mass ratios or some kind of fuel depot in LLO). That's considerably more delta vee than it takes to to Mars from the Moon and it further occurs to them it might be fun on the next trip home to leave the tourists on the Moon and take an unsheduled excursion to Mars.

Why on earth would James stipulate no aerobraking? This is a very standard technique. Is this because his premise rests on LLO to Mars taking less delta V than LLO to LEO?

Maybe he's heard Mars folks say LEO to Mars is less delta V than LEO to the moon. Which is true enough if aerobraking is used. With no aerobraking we'd need to do any where from .7 km/s for Mars capture to a 6 km/s burn for a soft landing. Or else we'd sail right past Mars back into a heliocentric orbit.

Hohmann Launch windows

Here's the biggest howler:

and it further occurs to them it might be fun on the next trip home to leave the tourists on the Moon and take an unsheduled excursion to Mars.

An unscheduled excursion?! Unless James’ ferry guys have a huge delta V budget, the ship's doing a Hohmann transfer. Windows for Earth to Mars Hohmann open once each 2.14 years. Lots of pre-planning is needed to take advantage of these rare windows. A trip to Mars isn't something you do at the drop of a hat.

As usually happens, James post stimulates a lively conversation. Most of the participants don't notice the howlers. The biggest concern seems to be sufficient air and food for the long trip.

A problem they seem oblivious to is radiation. An 8 month trip would expose the passengers to a lot more GCRs and solar flares than the 4 day LLO to LEO trip. Much more radiation protection would be needed. A few meters of water are often suggested to protect the passengers from GCRs. A few meters of water around the ship exterior would be a lot more massive than the air, food and drinking water James and his friends were obsessing over.

At one time I regarded James was one of the more numerate participants in science fiction forums. But he’s been spending too much time with SJWs and English Lit folks. Not that I dislike social justice or English literature. But if James wants to talk hard SF, he needs to revisit some of his math and physics textbooks.


Charlie Stross


Charlie Stross was one of the participants in the Nicoll post I just fisked. In that forum he goes by the handle autopope.  Nicoll’s lack of math and science savvy was not noticed by Stross or most of those commenting.

Stross was also crowing that physics professor Tom Murphy shared his opinions, as if that validates his views.

But we shouldn’t condemn Stross because of the company he keeps. Instead, let’s look at his High Frontier Redux.

It starts out noting the outer solar system and Alpha Centauri are far away and settling these regions isn’t practical. This is like saying the Americas were out of reach for the early humans in Africa. But the Americas became accessible after humans spread across Asia and reached the Bering Strait.

To show the Kuiper Belt is forever beyond reach, Stross needs to demonstrate intermediate destinations aren’t within reach.

Later he does argue against colonizing neighboring bodies. But starting off with the most difficult, furthest destinations is wasting the reader’s time.

Let’s look at Stross’ argument against developing the moon:

What about our own solar system? 
After contemplating the vastness of interstellar space, our own solar system looks almost comfortingly accessible at first. Exploring our own solar system is a no-brainer: we can do it, we are doing it, and interplanetary exploration is probably going to be seen as one of the great scientific undertakings of the late 20th and early 21st century, when the history books get written. 
But when we start examining the prospects for interplanetary colonization things turn gloomy again. 
Bluntly, we're not going to get there by rocket ship. 
Optimistic projects suggest that it should be possible, with the low cost rockets currently under development, to maintain a Lunar presence for a transportation cost of roughly $15,000 per kilogram. Some extreme projections suggest that if the cost can be cut to roughly triple the cost of fuel and oxidizer (meaning, the spacecraft concerned will be both largely reusable and very cheap) then we might even get as low as $165/kilogram to the lunar surface. At that price, sending a 100Kg astronaut to Moon Base One looks as if it ought to cost not much more than a first-class return air fare from the UK to New Zealand ... except that such a price estimate is hogwash. We primates have certain failure modes, and one of them that must not be underestimated is our tendency to irreversibly malfunction when exposed to climactic extremes of temperature, pressure, and partial pressure of oxygen. While the amount of oxygen, water, and food a human consumes per day doesn't sound all that serious — it probably totals roughly ten kilograms, if you economize and recycle the washing-up water — the amount of parasitic weight you need to keep the monkey from blowing out is measured in tons. A Russian Orlan-M space suit (which, some would say, is better than anything NASA has come up with over the years — take heed of the pre-breathe time requirements!) weighs 112 kilograms, which pretty much puts a floor on our infrastructure requirements. An actual habitat would need to mass a whole lot more. Even at $165/kilogram, that's going to add up to a very hefty excess baggage charge on that notional first class air fare to New Zealand — and I think the $165/kg figure is in any case highly unrealistic; even the authors of the article I cited thought $2000/kg was a bit more reasonable. 
Whichever way you cut it, sending a single tourist to the moon is going to cost not less than $50,000 — and a more realistic figure, for a mature reusable, cheap, rocket-based lunar transport cycle is more like $1M. And that's before you factor in the price of bringing them back ... 
The moon is about 1.3 light seconds away. If we want to go panning the (metaphorical) rivers for gold, we'd do better to send teleoperator-controlled robots; it's close enough that we can control them directly, and far enough away that the cost of transporting food and creature comforts for human explorers is astronomical. There probably are niches for human workers on a moon base, but only until our robot technologies are somewhat more mature than they are today; Mission Control would be a lot happier with a pair of hands and a high-def camera that doesn't talk back and doesn't need to go to the toilet or take naps.

In Situ Resources

Stross is right that human habitats in space would be massive. But he imagines every kilogram of a lunar habitat would be brought up from earth’s surface. Evidently Stross has never heard of in situ resources. At the lunar poles there are thought to be volatile ices — water ice as well as carbon dioxide ice and nitrogen compounds. Water and air to breathe could be extracted from local resources. Habs could be covered with regolith for radiation protection.

Stross acknowledges that robots could establish infrastructure on the lunar surface. And in fact this is what Spudis and Lavoie advocate.

In Situ Resources and Delta V

Besides building habs and infrastructure to extract life support consumables, robots could also build propellant mines. Stross didn’t bat an eye when Nicoll stated LEO to LLO is 8 km/s. It is likely this science fiction writer has no notion what role delta V plays in the rocket equation.

Mass propellant / mass payload = e(delta V/Vexhaust) - 1.

Exhaust velocity of hydrogen/oxygen bipropellant is about 4.4 km/s. Now 3/4.4 is very close to ln(2).

That means when using oxygen/hydrogen, every 3 km/s added to the delta V budget doubles over all mass.

Starting with 1 tonne rocket dry mass plus payload,
For 3 km/s you’d need 1 tonne propellant.
For 6 km/s you’d need 3 tonnes propellant.
For 9 km/s you’d need 7 tonnes propellant.
And so on.

Overall mass grows exponentially with increasing delta V. The legend of Paal Paysam illustrates the dramatic quantities exponential growth can give. Krishna challenged a king to a game of chess wagering a chess board with 1 grain of rice on the the first square, 2 grains on the second, 4 on the third and doubling each subsequent square. The king calculated the numbers for the first few squares and accepted. Here’s an illustration of Krishna’s wager:



Breaking the rocket equation’s exponent into chunks has a dramatic effect on the amount of propellant used. With each propellent depot, the delta V budget starts over:


We can start back to 1 grain of rice at each propellant depot.
Mount Everest is visible in this version, no longer covered with rice.


Delta V from earth’s surface to LEO is about 9.5 km/s. LEO to lunar surface is about 6 km/s. The additional 6 km/s boosts four fold the mass that needs to be parked in LEO.

If the ship could refuel in LEO, that would cut GLOW (Gross Lift Off Weight) four fold.

Here’s a delta V map focusing on EML2 and LEO. Moon to LEO is about 3 km/s using aerobraking.




But savings on propellent isn’t the chief advantage here. With an extraterrestrial propellant source, inter orbital tankers and ferries could move between orbits without ever suffering the extreme conditions of an 8 km/s re-entry into earth’s atmosphere.

Also with delta V budgets on the order of 4 km/s, inter orbital vehicles can devote a higher mass fraction to structure. Present day upper stages have less mass fraction than an aluminum Coke can. Which makes durable structure and adequate thermal protection very difficult if not impossible.


A racing bike vs a mountain bike.
With a racing bike we want to minimize mass.
But a racing bike is fragile while a mountain bike is durable and rugged.
When an upper stage has a 4% dry mass fraction, durability is not an option.

Elon Musk and Jeff Bezos seem well on their way to developing economical, reusable booster stages. Bezos wants to help establish lunar propellant mines. If Bezos, Bridenstine et al successfuly export lunar propellant to LEO, upper stages could refuel before re-entry into the atmosphere. Reuse of upper stages is much more plausible if re-entry velocity is 4 km/s or less.

Space Elevators

Stross mentions the possibility of Space Elevators.

Arthur C. Clarke popularized the notion with his novel Fountains of Paradise. Clarke, Asimov and Heinlein were writers from the great generation. They had some physics and tech savvy as well as an optimistic can-do attitude.

Baby boomer SF writers are more about bleak dystopias and cautionary tales. Like main stream pop culture they rely on sex and glorifying substance abuse to sell their product. With a few exceptions, SF writers from my generation tend to suck at math and physics. Hopefully younger science fiction writers will pick up the mantles of Heinlein and Clarke.

A space elevator was a good idea in the time of Clarke. Since then we’ve massive amounts of junk into Low Earth Orbit (LEO). Here is a panel from the Hubble telescope that spent 14 years in LEO:



See this Space Stack Exchange discussion on orbital debris.

The extreme height of a space elevator gives it enormous cross sectional area. Much more cross section than the panel pictured above. So even if we could manufacture long strands of Bucky tubes with insanely high tensile strength, the elevator would be severed by impacts.

However full blown Clarke towers have smaller cousins: orbital tethers. Being a lazy baby boomer writer, Stross seems content to rehash tired 1970s SF ideas. It is possible Stross has never heard of orbital tethers.

Orbital tethers can be placed in orbits relatively free of debris. They would be much shorter than a full blown Clarke Tower and would suffer much less stress. They could be made from existing materials like Zylon. I talk about orbital tethers at Trans Cislunar Railroad. Given two way traffic, a tether could harvest up momentum from higher orbits and trade it with the down momentum of lower orbits. Thus with two way traffic a tether could impart delta V with little expenditure of energy and propellant.

Orbital tethers could also be anchored on Phobos and Deimos.

Given tethers of modest mass, payloads can be exchanged between Phobos and Deimos via a Zero Relative Velocity Transfer Orbit (ZRVTO).



Given a somewhat more substantial tether, a Phobos tether could throw payloads down to a 1 A.U. perihelion (in other words, a transfer orbit to earth) or to a 3 A.U. aphelion (in other words a transfer orbit to the Main Belt).

An upper Phobos tether capable of launching payloads to various regions of our solar system needn't be that massive.



A Phobos tether extending to Mars upper atmosphere would drop payloads into Mars atmosphere at .6 km/s. About mach two, the Concorde Jet would routinely do this through a much thicker atmosphere. This about 1/10 the velocity landers from earth normally enter Mars' atmosphere. A Phobos tether descending to Mars' upper atmosphere isn’t practical using Zylon but would certainly be doable if they manage to manufacture long lengths of Bucky tubes.

Summary of Stross' Errors

Stross gives us numbers assuming all propellant and hab mass comes from earth's surface.

Using in situ resources most of the hab mass can be made from materials at hand.

More importantly there's the possibility of in situ propellant. This can drastically cut delta V budgets. Which cuts propellant and energy needed. It also makes robust, reusable vehicles possible.

Momentum exchange tethers are doable. This would further reduce energy and propellant needed to travel between space destinations.


Opening A New Frontier Is Doable


It is possible to establish infrastructure that would greatly reduce the cost of traveling about in space.

Yes, it would be expensive but it is doable. Dennis Wingo's book Moonrush documents several examples of government/private enterprise partnerships establishing massive transportation and communication infrastructure. The trans continental railroad was such a collaboration.

NASA administrator Jim Bridenstine has expressed his desire to work with SpaceX and Blue Origin to establish lunar and cislunar infrastructure. It is possible this could come to pass.

But the effort would have better prospects for adequate funding if the public perceived it as possible. The space meow boys have used bad math and silly straw man arguments to strengthen the public perception that this is pie in the sky.

The first steps towards opening the space frontier would be establishing infrastructure on other bodies. Semi-autonomous tele-robots are dropping in price while becoming more capable. British Petroleum has used R.O.V.s to build oil wells on the sea floor. It is possible to build the initial space infrastructure without a human presence.

Once robots have established infrastructure to extract propellant and keep humans alive, the cost of human presence plummets.

Why does Murphy argue so vehemently against a new frontier? He's worried that we'd be okay with trashing the earth if we had the option to move.  Bill Maher makes the same argument.

Maher and Murphy are giving us a false dichotomy. We can do both. We need to work to preserve our home as well as open new frontiers. Space advocates are more aware than the average person that our precious planet is finite and fragile.

For example Musk is also working on solar energy and electric cars in addition to his rockets. Bezos is advocating moving destructive mining and manufacturing out of our ecosphere.

Musk and Bezos are doing more for a sustainable future than a million space meow boys.


Asteroid Day

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On June 30 in 1908 the Tunguska object took out a good chunk of forest in Siberia.

In February of 2016 the United Nations approved a resolution stating

30 June International Asteroid Day to observe each year at the international level the anniversary of the Tunguska impact over Siberia, Russian Federation, on 30 June 1908 and to raise public awareness about the asteroid impact hazard

A good time to talk about a project on my wish list. An orbital asteroid devoted to finding  asteroids.

A wide field infrared scope much like WISE. But unlike WISE positioned at SEL1 or SEL2 so the earth isn't a major heat source.

A scope that can make simultaneous observations in visual wavelengths as well as infrared. This would tell us the asteroid's albedo from which we'd get a good estimate of size.

A scope that points towards the inner solar system. For various reasons asteroids within the earth's orbit are very hard to see from earth's surface. An orbital scope pointing towards the inner solar system would give us an inventory of a body of objects we presently know almost nothing about.

I was surprised and pleased to learn such a telescope had already been proposed. NEOCam. Principal Investigator Amy Mainzer.

We're presently getting a pretty good inventory of Chixculub size rocks. But Tunguska sized rocks are much harder to see. And there's a bunch more rocks this size. NEOCam would help us get a better handle on potential city killers.

Another potential NEOCam benefit: It could inventory potential asteroids for mining companies like Planetary Resources or Deep Space Industries.

NASA administrator Jim Bridenstine is enthusiastic about developing space as a source of resources and enabling economic growth.  He's show interest in lunar poles as low hanging fruit.

Near Earth Asteroids are also a low hanging fruit.  If Bridenstine's goal is to expand our economic activity into deep space, NEOCam is a great investment.

A few days ago Marshall Eubanks commented:

NEOCam is not in good shape. Over a year ago it was given one additional year of fairly minimal support. About all that was said about it in this month's SBAG meeting was that it "Continues in extended Phase A" - i.e., on life support.

I hope this changes. We need a telescope devoted to asteroid discovery.

Orbital Mechanics Coloring Book 2nd edition

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Costs

The folks making the first edition went belly up. So the first edition is out of print. I want to do a second edition. The makers of the first edition would print it in batches of around 50 (I believe) and cost of printing was $6.00/book. Suggested Retail Price was $10.00.

Checking my Amazon author's page over the years I believe around 2000 coloring books were sold. When you get into quantities in the thousands, set up costs of traditional printing are amortized over more units and traditional lithographic printing is cheaper.

I recently asked for quotes:


Given the small cost difference between 56 and 64 pages, I've decided to go for 64 pages.

The first edition was 40 pages. Given the new book has 24 more pages and the cost is a lot less, I believe it's a good gamble I can sell 4000 2nd edition coloring books.

I'm thinking of an Suggested Retail Price of $4.99 and to retailers I would charge $2.99.

Sadly the past decade or two hasn't been kind to the newspaper industry. Our weekly newspaper is still hanging on but my sister and business partner likes to say we're in the buggy whip business. I am thinking of attempting a Kick Starter to fund the printing.

New in the second edition

Given 24 more pages I can add a lot of extra stuff. I've kept most of the original 40 pages and added:

Page 18


In the section on Kepler's 2nd Law I've added a visualization that helps show r X v is twice the area swept out over a given time period. That specific angular momentum is twice the area of the ellipse per orbital period.

Page 22


Page 22 attempts to portray my visualization that helps me remember centrifugal acceleration is ω2r.

Pages 28 and 29



Attempts to explain radians and to show circular motion is ωr where ω is angular velocity in radians.

Pages 30 to 35

Are devoted to orbital vertical tethers. I am going to try to start calling these Sarmount tethers as I have recently learned Eagle Sarmount proposed these in the 1990s.

Perhaps science fiction device but I like them any way. The geometry and math associated with these is pleasing, in my opinion. Here are two pages from this section:


Pages 49 - 51

Are about the Oberth Benefit and EML2


Pages 51 - 52

Are about the rocket equation and mass fractions.


Page 64

Will be resources that have helped me. Books, websites, forums. Atomic Rockets, NasaSpaceflightForums, Space Stack Exchange, Tough SF and others. I am adding to this list as more occur to me.

Pages 53 to 63

Still to be made. Any suggestions?

Front and back cover


Front and back cover will be a couple of my more playful drawings. Text for delta V map to be added in. Also Suggested Retail Price, ISBN number and barcode. The section on Dandelin Spheres remains in the coloring book and the front cover will one of the Dandelin drawings colored in.

The earlier book was labeled a workbook. The printers of the first edition told me this was because ISBN numbers for workbooks are free. However I want this book to be playful as well as informative. In my opinion something categorized as a coloring book is more marketable than a workbook.

Inside front cover

Will be my favorite equations. The Vis Viva Equation will be at the top. I've been thinking of making a reference sheet to pin to the wall next to my computer. This would serve.

Here is the coloring book as of early March (4.8 MB pdf, not too big). Reviews would be appreciated. Steven Pietroban invested a fair amount of time looking over the first edition and found many small errors and a few substantial errors. Given my tendency to make misteaks, I'm sure there are errors hiding in my more recent effort. A heads up would be much appreciated if you see something wrong.

My email is hopd at cunews dot info.






Bridenstine's Why The Moon Matters

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Back in December 29, 2016 Bridenstine made a blog post "Why The Moon Matters". Bridenstine was representative of Oklahoma at the time.

Sadly the post was taken down when Bridenstine left his post as representative and became NASA administrator. But I recently found the post using the Wayback Machine.

Bridenstine's reasons were pretty the same arguments made by lunar scientist Paul Spudis (RIP).

I post it here for historical reference. Copying and pasting:

Jim's Blog

Why the Moon Matters

by Rep. Jim Bridenstine

f t # e
Washington, December 29, 2016 0 comments
On July 20, 1969, the free world won the space race when an American flag was planted on the Moon. Twelve Americans walked on the Moon during the Apollo program, resulting in a treasure trove of knowledge not only about the Moon, but about the universe.  Even better, by demonstrating the United States’ political, economic, and technological prowess, it played a part winning the Cold War. In 1983, Ronald Reagan introduced the Strategic Defense Initiative to defend the free world from nuclear ballistic missiles. While many called it destabilizing, and even suggested it was impossible to achieve, the Soviet Union took it very seriously, made every effort to eliminate it, and spent whatever it took to compete. They eventually went bankrupt.  SDI, while not fully implemented, was a geopolitical success built on the technical credibility provided by Apollo. As Ronald Reagan predicted, “We win. They lose.”

Through SDI, the Brilliant Pebbles program was born as a space based system to track and destroy ICBMs. Years later, in 1994, a Brilliant Pebbles satellite was repurposed to orbit and map the Moon. That mission, called Clementine, tested military sensors and made history when it provided evidence of lunar water ice. Later experiments by NASA and other space agencies indicated billions of tons of water ice at each lunar pole.

This single discovery should have immediately transformed America’s space program. Water ice not only represents a critical in situ resource for life support, but it can be cracked into its components, hydrogen and oxygen, to create the same chemical propellant that powers rockets.

All of this is available on a world that has no atmosphere and a gravity well that is 1/6th that of Earth. In other words, standard aerodynamic limitations do not apply, permitting the placement of the propellant into orbit either around the Moon or around the Earth.

From the discovery of water ice on the Moon until this day, the American objective should have been a permanent outpost of rovers and machines, with occasional manned missions for science and maintenance, in order to utilize the materials and energy of the Moon to drive down the costs and increase the capabilities of American operations in cis-lunar and interplanetary space.

Water ice on the Moon could be used to refuel satellites in orbit or perform on-orbit maintenance. Government and commercial satellite operators could save hundreds of millions of dollars by servicing their satellites with resources from the Moon rather than disposing of, and replacing, their expensive investments. Eventually, the customers of Direct TV, Dish Network, internet broadband from space, satellite radio, weather data, and others could see their bills reduced and their service capacities greatly increased.

While most satellites are not currently powered by liquid oxygen and liquid hydrogen, next generation satellite architectures could utilize lunar propellant if low-cost in-orbit servicing were available. Commercial operators will follow if the United States leads with its own constellations.  Such leadership would require a whole-of-government approach with the interagency support of the newly reconstituted National Space Council. The objective is a self-sustaining, cis-lunar economy, whereby government and commercial operators save money and maximize the utilization of space through the use of lunar resources.

This is also the first step for manned missions deeper into our solar system. A permanent human presence on other celestial bodies requires in situ resource utilization. The Moon, with its three-day emergency journey back to Earth, represents the best place to learn, train, and develop the necessary technologies and techniques for in situ resource utilization and an eventual long term human presence on Mars. Fortunately, the Space Launch System and Orion will start testing in 2018. This system, with a commercial lander, could quickly place machines and robots on the Moon to begin the cis-lunar economy. With the right presidential guidance, humans could return in short order as well; this time, to stay.

There are other economic benefits to a permanent presence on the Moon. Utilization of lunar oxides for in situ additive manufacturing (3-D printing) could sustain and develop lunar operations. If economical, we should pioneer the extraction of highly valuable platinum group metals and the ability to transport them back to Earth. The development of practical solar power satellites that beam energy directly to all areas of the Earth is made possible through the use of the resources of the Moon. Research on this concept is already being done in Japan, as well as at the Naval Research Lab here in the United States. The United States government should lead the way in retiring risk for these endeavors with the intent to empower commercial companies to sustain the cis-lunar economy. This could fundamentally alter the economic balance of power on Earth.

As the cis-lunar economy develops, competition for locations and resources on the Moon is inevitable. The Chinese currently have landers and rovers on the Moon. The United States does not. Very soon, the Chinese will be the first of humanity to explore the far side of the Moon and place robots at the poles. As my friend Congressman Bill Posey says, “They are not going there to collect rocks.” China has its own manned space station. The United States’ commitment to the International Space Station ends in 2024. China has a domestic capability to launch its Taikonauts into orbit. The United States relies on Russia. American adversaries are testing antisatellite weapons and proliferating satellite jamming, spoofing, and dazzling technologies. It is time for the United States to re-posture and assert true space leadership.

It must be stated that constitutionally, the U.S. government is required to provide for the common defense. This includes defending American military AND commercial assets in orbit, many of which have the dual role of providing commercial and military capabilities. The same applies for assets on and around the Moon. The U.S. government must establish a legal framework and be prepared to defend private and corporate rights and obligations, all keeping within the 1967 Outer Space Treaty. The United States must have cis-lunar situational awareness, a cis-lunar presence, and eventually must be able to defend freedom of action in space. Cis-lunar development will proceed with American values and the rule of law if the United States leads.

Space utilization has transformed the human condition, including how we communicate, navigate, produce food and energy, conduct banking, predict weather and perform disaster relief. While many of these gains are a result of private investment and commercial markets, they are only possible because the United States government took the lead and retired risk for these capabilities. Today, we are experiencing a space renaissance. The first launch of the Space Launch System is less than two years away. In 2021, we will use the Orion capsule to send astronauts beyond low Earth orbit for the first time since the 1970s. Commercial launch vehicles are maturing and commercial deep space habitats are currently in development. A renewed focus on utilizing the Moon can help further these advances and achievements. The choices we make now can forever make America the preeminent spacefaring nation.




Wish list for space video games.

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Kerbal Space Program

The Kerbal Space Program has demonstrated video games can be a very effective teaching tool.

It used to be very frustrating trying to talk about orbital mechanics. Use a word like "perigee" and eyes would glaze over as the audience tunes out.

But now there are many KSP players who are comfortable with terms like The Oberth Benefit, Bi Elliptic Transfers and other what use to be arcane, esoteric notions.

I'm hoping for more scientifically accurate games to make their way into pop culture.

Wish Number 1: Shotgun N-body simulations

Readers of this blog may know I'm a little obsessed with EML2 (Earth Moon Lagrange 2).

Many of my delta V numbers from EML2 assume dropping from EML2 using the Farquhar route and then insertion to a transfer orbit when moving ~11 km/s at perigee.

The Farquhar Route

However Farquhar's well done graphic is a simplification. A ship departing from from the EML2 region would not depart from a point. Rather it would drop from a halo or lissajous orbit about EML2. There are a multitude of possible orbits in this region.

Dropping from different parts of a halo orbit will result in different longitudes and latitudes for a perigee. And if you're doing a perigee burn for injection to a Mars transfer orbit, you want to be at a specific location and heading a specific direction when making your burn. 

Years ago Tom Powell helped me build a shotgun Java n-body sim from Bob Jenkins' code. blast many pellets in the general direction of your target. See which pellets pass closest and then narrow the shotgun blast.


Above is an attempt to show the shotgun concept. A fellow going by the handle "Impaler" was annoying me in a space forum. First I blast the varmint with broad scattering of buckshot. Then I more thoroughly pepper his backside by narrowing the blast between pellets 5 and 7.

By successively narrowing buck shot from earth to EML2 I found this route 3.1 km/s route from LEO to EML2:


The sim included earth, moon and sun. A lunar swing by boosts apogee on the way out. And then sun's tidal influence serves to raise perigee to EML2 altitude.

There are some serious limitations to my shot gun sim. There are only a few very limited scenarios that Tom Powell and I set up, very specific times and places. I would like to be able to have the user get location and velocity of a body at any time. I under stand there's software called SPICE that does this but I don't know how to use it.

Also JAVA seems obsolete. It's very difficult for me to use my own pages any more.

And my sims cans only specify the initial burns. Once you have a transfer path to a location it'd be nice to be able to do burns to park at that location.

I would use such a tool to learn how to move between different loosely bound lunar orbits.

Perhaps dropping from one EML2 halo orbit during a launch window wouldn't put the perigee in the right place for an injection burn. How much delta V would it take to move to a more favorable halo orbit?

Supposedly there are heteroclinic paths between halo orbits EML1 and EML2. A shotgun sim might help a player find these paths.

Halo orbits about EML1 and EML2 are part of a family of orbits that also include Near Rectilinear Halo Orbits (NHROs). If a lunar gateway is placed in an NHRO, it'd be fun and useful to explore different lunar orbits you could enter from an NHRO.

Wish Number 2: Tunneling on small bodies

For planets and large moons we are limited to exploiting only the thin outer shell of a body. Heat and pressure prohibit us from tunneling deeply.

But the entire volume is accessible for a small body.

If we could exploit the entire volume of Ceres, the dwarf planet could make Trantor look like Dogpatch.

I am hoping some planetary scientists and geologists could build tools to guesstimate how deeply we could tunnel given a body's surface temperature, radius and mass.

Wish Number 3: Tensile towers on small bodies

Also known as space elevators.

I'd be so happy to see a world building game use something like Wolfe's spreadsheet to examine effort and materials needed for various elevators. The user could input tensile strength, planet's angular velocity and body mass to examine various scenarios.

For example given Ceres' high angular velocity and shallow gravity well, Ceres synchronous orbit is only 706 kilometers above Ceres' surface. Materials needed for a Ceres or Vesta elevator are only a tiny fraction of what a Clarke Tower from earth or Mars would need.

Besides elevators from synchronous orbits it would also be good to enable users to build from planet-moon L1 and L2 points. 

The Mars Phobos and Mars Deimos are the moon elevators I like most. But elevators from L2 or L2 are usable in any family of tide locked moons.

Various tethers enabling ZRVTOs between Saturn's moons.

A gas giant's family of moons with tethers could be a rich setting for dramatic stories.

Hohmann trip times and launch windows between moons are on the order of days and weeks so it'd be possible to have a fast paced story without resorting to implausible engineering.

Dramatic situations might include missing a tether catch and being trapped in an orbit that won't rendezvous with a  tether catcher until the passengers have died from using up air, food or water. Or terrorists could sever a tether. There are many possibilities.

Wish Number 4: Compressive towers on small bodies.

Given lower gravity it's possible to build taller structures even given constraints imposed by a material's compressive strength. Likewise, sky scrapers are less plausible if a body has greater surface gravity.

I'd like the user to be able to specify a material's compressive strength and body gravity to get maximum plausible height for structures.

This would be especially useful for elevators between mutually tidally locked bodies like Pluto and Charon. Compressive towers built from the surfaces of Pluto and Charon could extend a fair distance towards Pluto-Charon-L1 and the also the Pluto Charon barycenter. This would considerably reduce the stress on the elevator and thus reducing the mass of the materials needed.

Any other ideas?

Well made multi user games could be a way to educate as well as stimulate interest in space exploration. If a reader has any other suggestions please comment. I screen comments for spam but I eventually post what I believe are worthwhile comments. It's unlikely an actual video game developer will ever read these but there's no harm in day dreaming.






The wrong question. Who invented calculus, Newton or Leibniz?

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Science historian Thony Christie (also known as @rmathematicus) had a number of essays on the Scientopia website. Scientopia bit the dust and I thought the essays were lost. But Bryan Kelly showed me they still exist on the Wayback Machine.

This essay by Christie is one I often refer to. I copied and pasted it from the Wayback machine with Tony Christie's permission.

Ich bin ein Gastblogger II: The wrong question.

I’m an alien

I’m a legal alien

I’m an Englishman in Nürnberg1

Being an English historian of mathematics resident in Germany I have been often asked, over the years, by people who know a little about the history of mathematics, “Who invented the calculus, Newton or Leibniz?” This is probably the most famous argument about priority of discovery and possible plagiarism in the history of science and still able to provoke nationalist sensibilities 300 years after the fact. Now as I mentioned in my first post this was the first theme in the history of mathematics that caught my attention and over the years I have devoted a considerable amount of time and effort to investigating the subject. There are two possible answers to the question. The short semi-correct answer is, both of them. The much longer and much more correct answer is nobody, calculus wasn’t invented by a single person but evolved piece by piece over more than two thousand years. What follows is not a history of calculus but a very bare and incomplete skeleton naming some of the important stations between the first appearance of concepts considered central to the calculus and the work of Newton and Leibniz.

The fundamental idea behind the infinitesimal integral calculus is first recorded in the so-called method of exhaustion of the Greek mathematician Eudoxus of Cnidus who flourished at the beginning of the fourth century BCE and is used for a handful of proofs by Euclid in his Elements. Refined by possibly the greatest of all Greek mathematicians, Archimedes, it became a powerful tool for the determination of areas and volumes as well as centres of gravity and most famously for his, for the time, highly accurate determination of the value of P, the relation between the circumference and diameter of a circle. The Greeks were also nominally aware of the problem of determining tangents to given curves, the fundamental concept of the differential calculus, but it did not play a significant role in their mathematical considerations. No further progress was made in antiquity before the general decline in learning beginning in the 2nd century CE and it was first in the High Middle Ages that integration returned to European mathematics.

However earlier than that there were interesting developments in Kerala in West India. At its core calculus is about summing infinite converging series, diverging series can’t be summed, and in the 17th century several important series representing important geometrical constants such as P and trigonometrical functions such as sine and cosine were analysed and discussed by European mathematicians and named after their supposed discoverers such as Gregory, Leibniz and Newton. The series had however already been discovered and analysed by the so-called Madhava or Kerala school of mathematics founded by Madhava who flourished in the second half of the 14th century. The same mathematicians also made extensive use of the method of Archimedes to determine areas and volumes. Attempts have been made to prove the hypothesis that the further development of the calculus in the 17th century was stimulated by Jesuit missionaries bringing knowledge of the work of the Kerala School to Europe, however despite extensive research no evidence of transition has been found up to now. In the Early Middle Ages Islamic mathematicians were also aware of and used Archimedean methods.

In the 14th century the Oxford Calculatores proved the mean speed theorem, which is usually attributed to Galileo, and in the next century Oresme proved it graphically (drawing graphs two hundred years before Fermat and Descartes!) and integrating the area under the graph. In the 16th century the works of Archimedes experienced a renaissance in Europe and many of the leading mathematicians devoted themselves to determining centres of gravity using his methods. The 17th century sees an acceleration in the application of what would become the calculus. Kepler used integration to prove his second law of planetary motion, the areas law, basically summing segment of the ellipse and letting them become smaller and smaller until infinitesimal. However as he had no concept of limits even he was aware of the fact that he was claiming to be able to add areas after they had ceased to exist! This piece of highly dubious mathematics contributed to the fact that the second law was still rejected long after the first and third laws had been accepted. In fact the second law was only finally accepted in 1672 when Nicolas Mercator provided a new more reliable proof. Kepler also used a form of integral calculus in his small pamphlet on determining the volume of wine barrels, a work that is often mentioned in a mocking tone but is actually an important milestone in the history of the calculus. The developments now come thick and fast with Galileo, Cavalieri (a pupil of Galileo’s), Grégoire de Saint-Vincent (a Jesuit mathematician who first gave the method of exhaustion its name), the Frenchmen Roberval, Fermat, Pascal and Descartes, the Dutchman van Schooten and in Britain John Wallis, Isaac Barrow and James Gregory all making significant contributions. It was also in the 17th century with the development of the science of mechanics that the differential calculus came to the fore with the problem of finding tangents to curves in order to determine rates of change. Many people in the list above made major contributions to the solution to this problem. Fermat is sometimes referred to as the “father of calculus” because he was the first mathematician to use what we now call the h-method (a method that I have to explain regularly to my private maths pupils) to determine first derivatives of functions. However like Kepler he has no real concept of a limit and just lets his ‘h’ (in his case its actually an ‘e’) disappear at the appropriate moment without explanation!

I hope I have said enough to make it clear that there was an awful lot of calculus around before Newton and Leibniz even considered the subject, so what did they do? It is often claimed that their major contribution was the discovery of the fundamental theorem of the calculus, i.e. that integration and differentiation are inverse operations but even this is not true. The theorem first appears in an implied form in the work of James Gregory and more explicitly in that of Isaac Barrow both of which are explicitly cited by both Leibniz and Newton in their own work. Newton and Leibniz collected up the strands scattered throughout the work of the mathematicians listed above and collating, sorting and standardising create a coherent body of work that we now call infinitesimal calculus but even their effort where actually only a milestone along the route. Finding sums of numerous infinite series and determining integrals and derivatives of many functions proved a very difficult process and many 18th century mathematicians won their spurs by solving a particularly difficult problem in the now developing analysis, most notably Leonard Euler. However one central and absolutely fundamental problem still remained, neither Leibniz nor Newton had a limit concept and their rather cavalier attitude to elimination of infinitesimals led to Bishop George Berkeley’s famous and very justified retort about ghosts of departed quantities. This problem was not really solved until the German mathematician Karl Weierstraß came along in the 19th century.

I have entitled my post “The wrong question” because I personally thing that in any area of science the question as to who discovered/invented a particular discipline, method, theory etc is almost always displaced. We shouldn’t be asking who invented the calculus Leibniz or Newton but rather what did Leibniz and Newton contribute to the on going evolution of that branch of mathematics that we now call the calculus? All branches of science (and I consider mathematics to be a science, see my last guest post here next week), all theories all discoveries have long evolutionary histories and individuals only make contributions to those histories they don’t write the whole history alone.

Let’s take a very brief look at another example where people tend to express themselves as if one individual had produced a major scientific theory complete in one go, like Athena springing fully armed from the head of Zeus, the theory of relativity. If one were to take the popular accounts literally then Einstein dreamt up the whole affair whilst travelling to his work at the Patent Office in Bern on the tram. However the theory of relativity also has a long history. The principle of the relativity of motion to a frame of reference can be found in the works of Galileo, to whom it is oft falsely attributed, but it can also be found in Copernicus’ De revolutionibus and two thousand years earlier in the works of Euclid. The central discussion as to whether time and space are absolute or relative can be found in the Leibniz Clarke correspondence at the beginning of the 18th century with Samuel Clarke basically fronting for Newton. Einstein own work was largely prompted by the incompatibility of the theories of Newton and James Clerk Maxwell, a problem much discussed and analysed in the 19th century. Einstein famous discussion of synchronicity of clocks is foreshadowed by a similar discussion in the 19th century by the operators of railway networks.  Moving from special to general relativity we have the contributions of Minkowski, Hilbert and others.

To close I have made much use of the concept of evolution in this post and anybody who regularly reads John Wilkins at Evolving Thoughts will know that the biological theory of evolution has a long history before Darwin published that book 150 plus years ago and readers of Larry Moran or the fearsome P Z Myers will know that modern evolutionary theorists object to being called Darwinians because the theory of evolution has evolved since Charles’ day. To recap, it is wrong to ask who invented or discovered a scientific discipline or theory, one should instead ask what did a given individual contribute to the theory or discipline in question?

For those who wish to know more about such things as the method of exhaustion or the fundamental theory of calculus then the articles at Wikipedia are mostly OK. On the individual mathematicians and their contributions to the history of calculus a visit to MacTutor is recommended.

For those who prefer books, you can read about the details of the priority dispute between Leibniz and Newton in definitive form in Rupert Hall’s “Philosophers at War” or in more popular form in Jason Bardi’s “The Calculus Wars”. A very general popular account of the history of infinite in mathematics is Ian Stewart’s “Taming the Infinite” a much more challenging book on the history of the infinite in mathematics is David Foster Wallace’s “Everything or More”.

On the history of calculus the standard works are, in ascending order of technical difficulty, Carl B. Boyer’s “The History of the Calculus”, Margaret E. Baron’s “The Origins of the Infinitesimal Calculus” and C. H. Edwards Jr.’s “The Historical Development of the Calculus”.

There is a chapter on the Kerala School in George Gheverghese Joseph’s “The Crest of the Peacock”. Joseph has also written a complete book on the subject his “Passage to Infinity”. For a corrective to some of Joseph’s more exaggerated claims I recommend reading the relevant parts of Kim Plofker’s “Mathematics of India”.

“The Leibniz-Clarke Correspondence” has been edited and annotated by H.G. Alexander and anybody interested in the connections between 19th century train time tables and Einsteins Theory of Relativity should read Peter Galison’s excellent “Einstein’s Clocks and Poincare’s Maps”

If you actually read and digest all of the above then you can start writing your own blog posts on the history of calculus.

1) With apologies to Sting!

Orbital Tethers as Momentum Capacitors

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Some years ago I was knocked off my chair when I was soldering a flash for our camera. The flash was powered by two AA batteries. How could such a dinky power source pack such a wallop?

It was because of the flash's capacitors. A capacitor will build up a charge over time and then release the accumulated charge suddenly. In this case the flash would deliver a very bright and brief flash of light when the camera shutter was open.

The orbital tether as a capacitor for momentum.

An ion engine can have an exhaust velocity of ~30 kilometers per second. That is nearly eight times that of the best chemical exhaust, around 4 kilometers per second. That means a much smaller exponent in the rocket equation. When we're taking exponents, scaling by 1/8 can make a huge difference in delta v delivered per kilogram of propellent.

The problem is the ion engine's dinky thrust. A chemical rocket can slam you back in your seat with 4 or 5 g's. But an ion engine's delicate push is barely perceptible, like the push of a feather. It takes a long time to build up delta V which makes it difficult to enjoy an Oberth benefit. It can also mean a trip lasting months for a trip that would take hours via a chemical rock

But a tether with an ion engine can take months or weeks between catches or throws to build up momentum.  So while it takes a long to build up momentum, it can release or impart it suddenly with a Catch or a throw.

So a tether can impart a brief and powerful change in momentum even with the ion rocket's barely perceptible thrust. It is like a capacitor but for momentum instead of electricity.

More tether stuff

It's been a long time since I did a post on tethers. So I'm going to toss in some other random bits that have accumulated in my head over the years.

Musk and Carmack on orbital tethers.

Back in 2016, shortly after SpaceX had landed a booster on an earth platform, John Carmack tweeted:


Elon replied:


I was delighted to see this exchange. I've been hanging around spaceflight forums since the 90s and Carmack has long been a big name in new space. Carmack and Armadillo Aerospace were X-Prize winners in 2006. I don't think I need to review what Musk has been doing.

Carmack, Fear and Dread


Carmack and ID software made a very successful computer game set on the Martian moon Phobos. The names of the Martian moons (Phobos and Deimos) means Fear and Dread. Which is very appropriate for the computer game Doom.

I love the idea of using the Martian moons as settings for science fiction stories. I believe they will be great assets in humanity's effort to settle the solar system. I've done a number of blog posts on Phobos and Deimos:

Phobos, Panama Canal of the Inner Solar System.

Upper Phobos Tether

Lower Phobos Tether

Deimos Tether.

But if I'm trying to sell Phobos and Deimos maybe I shouldn't be mentioning Doom. Oh well.

ZRVTO between Phobos and Deimos

Thinking about elevators anchored on Phobos and Deimos it occurred to me there would be a Zero Relative Velocity Transfer Orbit (ZRVTO) between the moons' tethers.


The transfer orbit's velocity at periapsis matches the speed of Phobos' tether top. Velocity at apoapsis matches the foot of the Deimos tether. Thus passengers and cargo could be exchanged between the moons using very little propellent.

I hope this idea will eventually be used.

ZRVTOs in other settings

There can be ZRVTOs in other settings. The tether anchor masses need to be tide locked in circular, coplanar orbits. Which describes a lot of the moons of Jupiter, Saturn, Uranus and Neptune. I look at this in Mini Solar Systems.

In Trans Cislunar Railroad I look at ZRVTOs between possible tethers in earth orbit:



In some ways tether mass is like propellent mass in the rocket equation — tether mass goes up exponentially with increasing delta V.

But tossing payloads between tethers breaks the the delta V budget into chunks. And thus greatly reduces the needed tether mass.

How much mass is saved with with ZRVTOs?

The above system tosses payloads up to the moon. For kicks I decided to see what would happen if I made a single LEO tether long enough to throw payloads to the moon.

I placed an anchor mass 1000 kilometers above earth's surface and tweaked tether length until the apoapsis of a flung payload was a lunar distance (384,400 km).


Taper ratio about 35. The tether would need to be about 106 times as massive of the payload it throws given a safety factor of three. It would need to about 1,842 kilometers long.

Let's compare that to my system of tethers in the Tran Cislunar Railroad:


So my system of tethers is actually about 5,000 kilometers longer than LEO tether capable of slinging payloads to the moon. That's a little disappointing. But tether to payload mass is less than 4.

So there is more than a 25 fold savings in tether mass. That is gratifying.

"Would only matter if it was extremely big"

Recall Musk's reply to Carmack was "Would only matter if it was extremely big".

The tether anchor would need to be a lot more massive than the payloads it handles. Or else the act of catching or throwing a payload would destroy the tether's orbit.

That's not an issue for tethers anchored on planetary moons. But it seems like a show stopper for tethers in earth orbits. Or maybe not....

Big Balls of Dead Sats

There is a lot of dead sats that could be harvested for a momentum bank. As of 2015 it was estimated that there were 670 tonnes of dead sats in the graveyard orbit just above geosynchronous orbit. 

Gathering these into a single anchor mass would vastly reduce their surface area and those lessen the likelihood of impacts generating orbital debris.

Many of them still have solar panels that can provide electricity. There are high gain antenna dishes. Some of the harvested momentum mass might even be useful.

What about LEO?

In 2016 I was wondering how we could provide a massive anchor for a LEO tether. But since then Elon Musk and SpaceX have been launching StarLink, a huge constellation of communication satellites.

I expect at the present time the plan is send an aging StarLink satellite down to the upper atmosphere and let it become a shooting star. But couldn't satellites in similar orbits be gathered together to form a momentum bank? If so, Musk has already made huge deposits into an LEO momentum bank.

And the StarLink satellites have a lot of solar panels as well as ion engines that could be salvaged.

Some ZRVTO equations

Some equations to figure lengths of tethers that accomodate Zero Relative Velocity Transfer Orbits. 


Lower tether length would be L1((1+e)^(1/3) - 1).
Upper tether length would be L2(1 - (1-e)^(1/3))

Momentum Exchange

At the outset of this post I mentioned tethers could impart momentum gradually built up by ion engines whose exhaust velocity is a lot higher than chemical rockets.

If there is downward traffic as well as upward that could greatly reduce the argon or xenon propellant used by the ion engines.
Momentum boosting maneuvers
Catching payloads from above or dropping them into lower orbits
Momentum depleted maneuvers
Catching payload from below or tossing them into higher orbits

These could be balanced to achieve most of delta v needed, in my opinion.

Propellent Mass as cost driver?

Reducing propellant mass should be a top priority. In even the best circumstances Gross Lift Off Weight (GLOW) from earth's surface will be dominated by propellant.

Charlie Stross was indignant when I was ridiculing space naysayers. He said he wouldn't dignify my criticisms with a public response. But then proceeded to make several thoughtful public responses. Link. I quote:

"My cost estimate was for near-future transport to LEO.

"Contemporary civil airlines' operating costs are on the order of triple the cost of fuel. (Equal shares: fuel, airframe depreciation and maintenance, and crew/ground support costs.)

"If you want to do it for less than triple the fuel cost, you need to beat the standards of a viciously competitive industry that's been trying to pare costs for around a century.

"SpaceX currently cite the cost of fuel for a Falcon 9 as ~$200,000. So my BOTE would get us to $600,000 for a ~20 ton payload. I gather they're currently quoting about $60M, so there' someroom for improvement."
That's from a comment Charlie made in 2018. And I would agree that spaceflight dominated by propellent cost is optimistic. You would also need very durable, reusable rockets that don't require a great deal of maintenance.

What could we import from above?

For a momentum exchange tether to work you need two way traffic. So what mass from above could provide up momentum?

Lunar propellent might be the first import from above. In 2010 India's lunar orbiter Chandrayaan 1 found evidence of massive ice deposits on the lunar poles Link. The late lunar geologist Paul Spudis would argue an off earth source of propellent could confer a commercial and military advantage to the power that controls it Link. Former NASA administrator Jim Bridenstine also made the same argument Link

Jon Goff voiced some objections to lunar propellent. Given the delta V between the lunar surface and and earth orbits, only a small fraction of lunar propellent would arrive to supply propellent depots in earth orbits. If propellent were delivered by conventional rockets. 

However more would arrive if lunar propellent were delivered via momentum exchange tethers. And they would provide up momentum for the momentum exchange tethers and reduce the need for propellent mass. 


A photo of Starship thermal tiles from

The Starship upper stages will likely re-enter at a much higher velocity than the booster stages. 35 kilo pascals is typical max Q for ascent. But for for descent 90 kilo pascals is common. Will Starship be able to economically refurbish after re-entry? Unlike the Space Shuttle Starship has a stainless steel hull. It looks like the thermal tiles are mechanically attached rather than glued on.

SpaceX likely has improved on thermal protection since the Space Shuttle Days. But I still expect re-uses after an 8 km/re-entry to be difficult.

If the upper stage could refuel in Low Earth Orbit (LEO), the upper stage could re-enter with an even lower velocity than the booster stage. Re-use would be far less difficult.  Re-usability may even become so advanced that cost of transportation would be triple the fuel costs, as Charlie Stross imagines in his best case scenario.

The Heteroclinic Zone

There are a family of loosely bound lunar orbits where a little delta V and use of earth's tidal influence can make a big change to the orbit. It's possible to move from Earth Moon Lagrange 2 (EML2) to  Earth Moon Lagrange 1 (EML1) with only a tiny burn. These low delta V routes between lunar orbits are called heteroclinic paths. They talk about these paths in chapter 3 of Dynamical Systems, the Three-Body Problem and Space Mission Design by Koon, Lo, Marsden and Ross.

EML1 is around .3 km/s from the top tether I mentioned in ZRVTOs in Other Settings earlier in this post.

From EML it's easy to reach EML2

EML2 is only .9 km/s from Trans Mars Insertion using the Farquhar route.


I am a little obsessed with EML2. I have a post devoted to this Lagrange point.

NHROs

Included in The Heteroclinic Zone are Nearly Rectilinear Halo Orbits NHRO. I like these orbits for a number of reasons.


The perilune of these orbits are near the lunar poles. The lunar poles and cold traps are where I daydream of lunar propellent mines. So orbital insertion from the propellent mines to The Heteroclinic Zone is quite doable. 

I like to imagine advanced propellent mines with a rail guns that launch propellent into NHROs. 

A distant apolune is at the other end of an NRHO. It travels slowly in this region giving it lots of hang time over the lunar poles. It can give long periods line of sight periods to the lunar cold traps. In some ways this is like an earthly Molniya orbit. This would be useful in the early stages of a propellent mine when construction is being done by remotely controlled robots.

Phobos lending a hand with Mars EDL


A 1340 kilometer tether from Phobos could drop payloads into periaerion skimming Mars atmosphere. The payload would enter Mars atmosphere at about 3.6 km/s. Entry from an Earth to Mars Hohmann would be about 5.5 km/s. (3.6/5.5)^2 = ~.43. So less than half the kinetic energy to be shed.

And it may be doable to mine oxygen from Phobos minerals so propellent could also lend a hand in shedding velocity.

A 5680 km/s tether from Phobos would allow entry to Mars atmosphere about about .6 km/s However a Phobos tether going deep in Mars' gravity well is more difficult. Given Zylon and a safety factor of 3, taper ratio would be 84 and tether to payload mass ratio would be 640. See my lower Phobos Tether post.

Park Trans Planetary Vehicles at the edge of gravity wells
.

Vehicles that can keep humans alive for months must be massive. Vehicles for shorter trips can be much smaller.

What is the point of launching a trans Mars vehicle from earth's surface and landing it on Mars' surface? It would be like using a huge Mac truck to deliver a pizza from a restaurant to a customer's front porch.

Far better, in my opinion, to park trans planetary vehicles at EML2 or Phobos. Then there is no need for a thermal protection system to endure re-entry. And there is no need for the ship to be 90% propellent to climb up steepest slopes of planetary gravity wells.








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