Hop's Blog

"What's the cost of propellant from earth vs getting it from the moon?" always comes up in discussions of lunar water. Or the cost of near earth asteroid propellant vs earth propellant.

Propellant is cheap, typically a small percentage of spacecraft expense. Spaceflight is expensive because vehicles are disposable. How much would a plane ticket cost if a 747 were thrown away each trip?

Well, how come we don't re-use our spaceships? It's due to constraints imposed by the rocket equation.

As delta-V budget climbs, dry mass fraction shrinks. We can't eliminate engine or payload mass. We cut dry mass by making walls thinner and structure more tenuous.

Thinner walls mean fragility. Designing upper stages is like designing egg shells.

Upper stages are like cascarónes, confetti eggs. While cascarónes are fragile by design, upper stages are fragile due to the constraints imposed by the rocket equation and high delta-V budgets. An upper stage plunging into the atmosphere is like a cascarón plunging onto a friend or relative's head. But the conditions of re-entering earth's atmosphere at 8 kilometers/second are much more extreme than the back of her mom's head.

Given propellant depots at LEO, GEO, and EML1 or 2, ferries between orbits would have delta V budgets of 4 km/s or less. Moving between orbits, they don't have to endure re-entry. Much less difficult mass fractions and eliminating the extreme conditions of re-entry make re-usable ferries doable.

But how would these ferries by fueled? Tankers from earth would have a delta V budget of at least 9.5 km/s. If the tankers are throw-away, it is simpler and cheaper to just use the tanker to deliver the payload rather than fueling a ferry to deliver a payload.

However if the fuel source is the moon's surface or an asteroid at EML1 or 2, the tankers have lower delta V budgets and thus much less difficult mass fractions. Given reusable tankers to supply fuel, reusable ferries make sense.

Moreover, given propellant in LEO, an upper stage returning to the earth's surface doesn't have to re-enter at 8 km/s. Given propellant in LEO it can refuel and shed some of it's orbital velocity via reaction mass instead of aerobraking. Eliminating the 8 km/s re-entry makes re-use of upper stages much less difficult.

So it's completely missing the point to compare the price of earthly propellant delivered to the moon's surface vs propellant mined on the moon. The object isn't to get water on the moon's surface. The object is to get propellant at various locations in cislunar space so the delta V budgets can be busted into manageable chunks.

By breaking the tyranny of the rocket equation, reusable ships become possible. Given easily reusable space ships, the economies of spaceflight are completely changed. This is the potential of lunar (or NEO) water.

This post was prompted by Robert Walker's comment: "I wonder what anyone here thinks about my idea for rotating carousels to provide gravity in the lunar colony? Not rotating entire hab, but just a thin shell of living quarters inside it, in a bigger hab if say a couple of hundred meters across, greenhouse domed, have like the living habs around the outside rotating continuously - perhaps on a track or something like that - at just the right speed for 1 g for the inhabitants. Smaller habs just rotate the entire room - and easier to construct than e.g. fairground rides on Earth because of the low gravity."

We know 0 g results in bone loss and other problems. We need gravity, but how much?

Is 2/5 g (Mars gravity) sufficient to keep us healthy? Or 1/6 g (moon gravity)? This is still not known. Our only data points are 0 g and 1 g. If a full g is needed, people on Luna or Mars bases would indeed living quarters on rotating carousels.

On the other hand, if lunar gravity is sufficient, no carousel is needed on the moon or Mars. That would also drasticly cut the minimum sized hab needed to keep workers healthy in a microgravity environment like on an asteroid.

The amount of artificial gravity felt in a spin hab is ω²r where ω is angular velocity in radians and r is spin hab radius. Obviously if 1/6 g does the job, the hab radius can be cut by a factor of 6.

Another quantity to look at is ω, angular velocity. Earlier it was believed 1 revolution per minute was the top angular velocity humans could comfortably endure. This combined with assuming a full gravity resulted in proposals like the behemoth Stanford Torus. (2 * pi / 60 seconds) * 894 meters = ~9.8 meters/second^2 or about 1 gravity.

If spinning doughnuts nearly 2 kilometers across are a prerequisite for asteroid miners, I wouldn't expect asteroid mining habs in this century or the next.

But is 1 revolution per minute really the top ω workers can endure? Research by James Lackner and Paul DiZio suggests workers could become acclimated to higher angular velocities. If workers can get used to 2 rpm that would cut needed radius four fold. 3 rpms would cut radius 9 fold. Here is a table showing hab radius (in meters) that be needed for various angular velocities and gravities:

	1 g	5/6 g	2/3 g	1/2 g	1/3 g	1/6 g
1 rpm	894	745	596	447	298	149
2 rpm	223	186	149	112	75	37
3 rpm	99	83	66	49	33	16
4 rpm	56	46	37	28	19	9

If lunar gravity is sufficient and workers can get used to 4 rpms, a 9 meter radius hab does the job!

Obviously a 9 meter radius spin hab is more doable than a 900 meter radius Stanford Torus.

If you hope for humans on surface of other planets or in asteroidal habs, it would be good to know what gravity humans need and what angular velocity they could get used to.

This is in response to Quantum G's question "Why do humans need to return to the Moon to get resources to make "consumables and propellant", if robots can be sent to do that instead?"

Just let autonomous and/or teleoperated robots do all the work. Who needs humans?

Quantum G should try working in an actual mine. As an ASU student, I spent four summers working in the Phelps Dodge copper mine in Ajo, Arizona. At the top of every bulletin board was Murphy's Law: "What Can Go Wrong, Will." And that was followed by many variations and corollaries of Murphy's Law.

Unlike a factory floor, mines are an uncontrolled, unpredictable environment. The unexpected can and does happen. When it does, human ingenuity is called for. You cannot write algorithms that anticipate every unforeseen problem.

Not that I'm against robots. See
Puppets, Telerobots & James Cameron,
Surgical Robots, and
Give NASA's SLS money to DARPA.
I believe improved robotics will be a major game changer when it comes to exploitation of space resources.

The moon is more amenable to tele robots than most locations in our solar system. At 384,400 kilometers from earth's surface, light lag latency is only 3 seconds. Since signal strength falls with inverse square of distance, lunar tele robots would enjoy much better bandwidth than machines on remote asteroids or Mars. Good bandwidth is important for immersive tele-presence as well as control of agile, dexterous robots.

And there are technologies that can mitigate a 3 second reaction time. For example Big Dog's balance or Google Car's collision avoidance.

Even so, a multitude of tasks are much easier with constant sensory feedback in real time. Things like finding a dropped hex nut. A 3 second light lag can make normally quick and easy chores time consuming and difficult. Robots controlled by humans in neighboring habs would be much more able than bots controlled from earth's surface.

And then there's the question of maintenance. Who maintains the robots?

Here is an article on mining giant Rio Tinto's "autonomous" robots. These driverless trucks move back and forth along well maintained and predictable routes. And they are closely monitored by nearby humans. Machines in less predictable environments such as the shovels are still human operated. And all the machines, whether "autonomous" or human operated, are maintained by humans.

Mines sans humans are still well beyond the state of art for earthly mines, much less mines in environments where we have zero operating experience.

Robots may reduce the need for human presence. But they won't completely eliminate the need for humans, not for a long while.

There is also important information to be gained from humans on the moon. What gravity do humans need to stay healthy? As I mention in What's the minimum spin hab?, this is still not known. If the moon's 1/6 gravity keeps humans healthy, that makes minimum spin habs for asteroid workers more than six times less massive. It would also indicate humans are okay living with Martian gravity.

Most of my models use circular coplanar orbits. But Jon Goff points out many asteroids have a healthy inclination. Departing for an asteroid from the moon's orbital plane often involves a big plane change. And plane changes can be expensive (see comments in What about Mr. Oberth?)

But easy plane changes is a big reason I love EML2. This takes some explaining.

Let's look at a 60 degree plane change but no change in speed. I pick 60º because it's easy-- the original and new velocity vector as well as the delta V vector are all sides of an equilateral triangle.

If your speed is about 8 km/s (as in low earth orbit), a 60 degree plane change costs about 8 km/s.

With higher orbits, plane changes are cheaper. At GEO (about 36,000 km above earth's surface), orbit speed is about 3 km/s and a 60 degree plane change costs 3 km/s.

EML2 is about 63,000 kilometers above the moon's surface. It's moving about .17 km/s with regard to the moon. So a 60 degree plane change at EML2 costs .17 km/s:

But wait, it gets even better!

My favorite route from LEO to EML2 is the one found by Robert Farquhar:

The orbit is time reversible. A .15 km/s braking burn at EML2 cuts speed with regard to the moon to .02 km/s. The allows the ship to fall deep into the moon's gravity well. With the a little Oberth help at perilune, another .18 km/s suffices to send the ship to an 182 km perigee.

Here's a single burn at EML2 that cuts speed to .02 km/s as well as doing a 60 degree plane change:

The .16 km/s braking/plane change burn is only .01 km/s more than the .15 km/s coplanar burn Farquhar calls for.

Where else can you get a 60º plane change for only .01 km/s?

EML2 has a tiny C3 with regard to both the moon and earth. This confers a big Oberth advantage. And the easy plane change is icing on the cake.

Those who advocate Mars settlement like to say Mars can be terraformed. First I will take a look at what it would take to terraform Mars.

How much air do we need to add to Mars?

From NASA's Mars Fact Sheet, surface density of the Martian atmosphere is about .02 kg/m³. Thats about 1.5% of Earth's surface air pressure of 1.27 kg/m³. Mars' atmosphere is virtually a vacuum.

Mars surface gravity is about 38% earth gravity. That means given an atmosphere of comparable temperature and composition, Mars atmosphere scale height is 264% earth atmosphere scale height. But Mars surface area is about about 28% that of earth's. 2.64 * .28 is about .75. To get comparable air density, we would need Mars' atmospheric mass to be about three quarters that of earth's atmosphere.

The total mass of the Martian atmosphere is about 2.5 x 10¹⁶ kg. Earth's atmosphere is about 5 x 10¹⁸ kg. So to make Mars surface air density earth like, we'd need 3.6 x 10¹⁸ kg of air added to Mars.

But do we need sea level air density? No there are many people who live comfortably at higher elevations. This list of the world's highest cities show several places at around 5000 meter elevation. Granted the dwellers of the highest city La Rinconada don't live comfortably. But they demonstrate humans can live in air density half that of sea level. If half is sufficient, Mars only needs 1.8 x 10¹⁸ additional kilograms of air.

Would be Mars terraformers like to point at the frozen CO2 at the Martian poles. If Mars temperature is raised just a little, they hope the vaporized carbon dioxide would create a greenhouse effect that would cause more carbon dioxide to be vaporized. Their hope is that a runaway greenhouse effect could substantially boost Mars' atmosphere from frozen volatiles already in place.

According to Wikipedia, there is thought to be a 1 meter thick layer of CO2 at Mars north pole, a cap about 1,000,000 meters in diameter. At the south pole there is an 8 meter thick layer of CO2 over a cap having a 350,000 meter diameter. That's about 1.6 x 10¹² cubic meters of CO2. Dry ice has a density of 1.6 thousand kg/m³. If all of that CO2 is vaporized (an optimistic assumption) that totals about 2.5 x 10¹⁵ kg of atmosphere. Short by almost 3 orders of magnitude, a miniscule contribution toward the needed 1.8 x 10¹⁸ needed kilograms.

Zubrin and McKay believe runaway greenhouse could boost Mars atmosphere to 300 to 600 millibars. Besides the polar dry ice, they also mention CO2 in Martian regolith. I believe most of Zubrin's optmistic estimates are influenced more by wishful thinking than hard data. But for the sake of argument I'll grant 300 millibars of CO2. 300 millibars of CO2 is not breathable. But let's say green plants combine Martian water and CO2 to make sugars and starches plus oxygen. Taking the carbon out of 300 millibars of CO2 leaves about 220 millibars of oxygen. Earth's 1000 millibar atmosphere is 1/5 oxygen, so perhaps a 220 millibar oxygen atmosphere would be breathable. But it would also be an extreme fire hazard. Apollo 1 taught us a pure oxygen atmosphere isn't a good idea.

Even with Zubrin's very optimistic scenario, it seems we'd still need to import 1.5 1x 10¹⁸ kilograms of nitrogen.

Can we add to Mars' air with comets?

Zubrin and McKay suggest it'd take .3 km/s to nudge an ammonia asteroid in the outer solar system towards Saturn and then Saturn's gravity could throw the ammonia snowball Marsward.

"Consider an asteroid made of frozen ammonia with a mass of 10 billion tonnes orbiting the sun at a distance of 12 AU. Such an object, if spherical, would have a diameter of about 2.6 km, and changing its orbit to intersect Saturn's (where it could get a trans-Mars gravity assist) would require a DV of 0.3 km/s. If a quartet of 5000 MW nuclear thermal rocket engines powered by either fission or fusion were used to heat some of its ammonia up to 2200 K (5000 MW fission NTRs operating at 2500 K were tested in the 1960s), they would produce an exhaust velocity of 4 km/s, which would allow them to move the asteroid onto its required course using only 8% of its material as propellant. Ten years of steady thrusting would be required, followed by a about a 20 year coast to impact. When the object hit Mars, the energy released would be about 10 TW-years, enough to melt 1 trillion tonnes of water (a lake 140 km on a side and 50 meters deep). In addition, the ammonia released by a single such object would raise the planet's temperature by about 3 degrees centigrade and form a shield that would effectively mask the planet's surface from ultraviolet radiation. As further missions proceeded, the planet's temperature could be increased globally in accord with the data shown in Fig. 12. Forty such missions would double the nitrogen content of Mars' atmosphere by direct importation, and could produce much more if some of the asteroids were targeted to hit beds of nitrates, which they would volatilize into nitrogen and oxygen upon impact. If one such mission were launched per year, within half a century or so most of Mars would have a temperate climate, and enough water would have been melted to cover a quarter of the planet with a layer of water 1 m deep."

This scheme presupposes we could land a 20 gigawatt power source on a rock in the outer solar solar system. For comparison the Palo Verde Nuclear Power Plant, the largest nuclear power plant in the United States, produces about 3.3 gigawatts. So we're sending 6 Palo Verde Nuclear Power Plants out past Saturn. McCay's scheme stipulates using the comet's mass as reaction mass. So now we have a mining and transportation infra structure on the comet that digs up the ice and places this reaction mass in the nuclear rocket engine.

If we have the wherewithal to establish such infrastructure, we certainly have the ability to build habs on these rocks.

Asteroidal Real Estate

How much asteroidal real estate could 1.5 1x 10¹⁸ kilograms of air give us? An O'Neill cylinder 8 kilometers in diameter and 32 kilometers long would give us 804 square kilometers of real estate. Such a cylinder would have a volume of 1.6e12 cubic meters. On earth's surface, our air has a density of about 1.27 kg per cubic meter. So that volume at 1 bar density would be 2e12 kilograms of air.

1.5e18/2e12 = 750,000. Three quarters of a million O'Neill habitats. Recall each cylinder has 804 square kilometers of real estate. 750,000 * 804 km² = 603 million km². Mars' surface area is 145 million km². So if we put the asteroidal resources to use where they're at, we get 4 times as much real estate.

Some would point out that O'Neill cylinders are very extravagant pieces of mega-engineering. I completely agree! It's my belief that humans don't need a full g to be healthy, I believe .4 g (a little more than Mars' gravity) would suffice. In which case the hab radius could be 1.6 km. Such a hab would have only 321 km² of real estate but a volume only 2.6e11 cubic meters. 2.6e11 m³ * 1.27 kg/m³ = 3.3e11 kilograms. 1.5e18/3.3e11 = ~4.5 million. 4.5 million of the smaller O'Neill habitats. 4.5 million * 321 = 1460 million square kilometers. Or about as much real estate as 10 Mars planets.

If the goal is to provide more real estate and resources for humanity, terraforming Mars is an extravagant waste. We should ditch planetary chauvinism and go for the small bodies.

Patching Conics
Time and time again I've watched people patch conics by using straight addition and ignoring the Oberth benefit. It is a very easy mistake to make. I'll give an example of this error for an earth to Mars delta V budget.

At perihelion an earth to Mars Hohmann orbit is moving about 33 km/s. 3 km/s is needed to leave earth's 30 km/s heliocentric orbit and enter this transfer orbit. In a similar fashion it takes 2.5 km/s to leave Hohmann transfer and match velocities with Mars.

But before we go from one heliocentric orbit to another, we need to escape the planet's gravity well. Earth's surface escape velocity is about 11 km/s. Mars' surface escape velocity is about 5 km/s.

The novice will look at these 4 quantities and simply add them. 11 + 3 + 2.5 + 5 is 20.5. They'll tell you it takes about 20.5 km/s to get from earth's surface to Mars surface.

But to accurately patch conics you need to use the hyperbolic orbit that takes you out of the planet's sphere of influence to a heliocentric orbit. Hyperbolic orbit speed is sqrt(V_escape² + V_infinity²). But what's V_infinity? In this example it's the 3 km/s needed to go from earth's 30 km/s orbit to a Hohmann's 33 km/s. In Mars' neighborhood V_infinity is the 2.5 km/s needed to exit Hohmann and match velocities with Mars.

If you remember high school math, sqrt( a² + b²) should look familiar. It's the hypotenuse in the good old Pythagorean theorem! And that's what I use to visualize hyperbola speed:

The novice will tell once you've achieved the 11 km/s to escape earth's gravity well, you need another 3 km/s. The informed will tell you only another .4 km/s is needed.

For Mars to Earth the naive will tell you after you've reached 5 km/s to escape Mars then you need another 2.5 km/s to send the ship earthward. The savvy will tell you an additional .6 km/s is needed.

.4 vs 3 and .6 vs 2.5. In this case the novice method results in a 4.5 km/s overestimation of the delta V budget.

Erik Max Francis
I used to call this the Erik Max Francis Error. Delta V budgets from one planet to another would often come up in space usenet groups. Erik would use his Python BOTEC and give an answer to ten decimal places. People would ooooh and ahhhhh. Wow! Accurate to 10 significant figures! In reality Erik's answers were accurate to zero significant figures. I finally prodded him to correct his error. Now his BOTEC is accurate to 1 or 2 significant figures. So far as I know, he still gives answers to 10 decimal places.

Rune
Later I called it the Rune error. The total Vinf can be roughly estimated by subtracting Mars 24 km/s from earth's 30 km/s. And this is what Rune uses on the New Mars forum when Louis asks how much delta V is needed after you get out of earth's gravity well.

Brute force" trajectories would take about as much delta-v as is the difference between the orbital speeds of mars and earth, so about 29.8km/s (for earth) - 24km/s (for mars) = 5.8km/s

I have explained to the New Mars Forums many times that the speed of a hyperbola is sqrt(V_escape² + V_infinity²). Will Rune ever learn it? I doubt it.

But no matter, Rune's a member of Zubrin's cult. People expect Zubrinistas to be innumerate, nobody takes them seriously. But sadly they are loud and high profile. John Q Public can be misled into thinking they speak for all space advocates.

It's more damaging when someone in authority commits this error. Now I'm talking about Dr. Tom Murphy.

Professor Tom Murphy
This is from a graphic from Tom Murphy's Stranded Resources:

Murphy writes:

For instance, we travel around the Sun at a velocity of 30 km/s, while Mars sails at a more sedate 24 km/s. So to meet up with Mars, we have 6 km/s of extra velocity to burn, helping us up the hill. We speak of this as a Δv (delta-vee) adjustment to trajectory.

Same method as Rune for getting the total Vinf: Subtracting Mars' 24 km/s from earth's 30 km/s to get 6 km/s. An over estimation but not wildly inaccurate. And Murphy correctly shows earth's escape as 11 km/s and Mars escape as about 5 km/s.

But then Murphy straight up adds 11+6+5:

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

Sounds like he's being generous to the poor deluded space cadets by rounding 22 down to 20. But the distance from earth's C3=0 to Mars C3=0 is not 6 km/s. It's about 1 km/s. Here's Murphy graph corrected for the Oberth benefit:

For comparison, Murphy's erroneous graph is left in but a shade lighter. From surface of Earth to Surface of Mars is about 16 km/s.

"But wait!" a Murphy apologist might say. "Murphy didn't include the delta V needed to rise above earth's atmosphere. That's 1.5 to 2 km/s! That makes the budget more like 18 which can be rounded up to 20."

An atmosphere does indeed add to delta V for departing a planet. On the other hand, an atmosphere is a big help for planet arrival. Park in a capture orbit with periapsis velocity just a hair under escape. Position the periapsis in the planet's upper atmosphere. Each orbit at periapsis, atmospheric friction slows the ship. This is known as aerobraking. Almost all of Mars' 5 km/s descent can dealt with via aerobraking. Here is Murphy's graph corrected for atmospheric influence:

Now it takes 13 or 14 km's to reach escape. But with descent taken care of with aerobraking it only takes another 1 km/s to reach Mars' surface. A more realistic delta V budget from earth surface to Mars surface is about 14 or 15 km/s.

And in fact numerous Mars landers and orbiters have used this method. I am stunned that Murphy, a self proclaimed space insider, has never heard of aerobraking.

Is a 6 km/s error a big deal? Since the exponent of the rocket equation scales with delta V, it's a very big deal. Murphy himself would tell you exponential growth can be dramatic.

Above graph assumes hydrogen/oxygen bipropellent. Each 3 km/s added to the delta V budget about doubles propellent needed. Each 5 km/s added nearly triples the amount. Murphy's 20 km/s delta V budget would need a little more than triple the propellent of the actual 15 km/s delta V budget.

In my opinion the limits to growth is the most important issue facing us. Can space resources raise the ceiling on our logistic growth? If so, it is worthwhile to invest in building space infra-structure. If not, expensive space infrastructure is a waste of money. We should look at the question seriously. That is why I get bent out of shape when a so-called authority makes common mistakes that would embarrass a freshman aerospace student.

Tom Murphy's arguments against space are often cited in discussions of limits to growth. When I find such a discussion, I will chime in that a bright high school student could tear apart Murphy's arguments. The most recent visit was at Mike Stasse's Damn The Matrix. As usual, the Do The Math crowd response was insults, appeal to authority, but no math.

Judging by Murphy's fans, his blog should be renamed "Don't do the Math. Take my word for it because I'm a Ph. D."

In general, Murphy's followers do not use math. Rather they appeal to Murphy's authority. I am making Mike Stasse and his friend Maponos the poster children for this fallacious argument.

Is Murphy an authority? Does his PhD means he's qualified?

I responded with The Most Common Delta V error. High school seniors typically mispatch conics the same way Murphy does. His level of expertise is somewhere below Orbital Mechanics 101 for liberal arts majors.

Stasse passed this on to Tom Murphy himself. And got a reply! In the comments section Stasse quotes Murphy:

I don't dispute the more careful approach used on hopsblog. I put pieces together very simply, which may not represent more clever ways to manage interplanetary trajectories. That said, I stated clearly what I was doing, so that it's an easy job to pick it apart. I'm fine with that. I hope I never appealed to my authority as an orbital mechanics expert, because I am not.

Bold and underline added by me. Hear that, Stasse? By his own admission, Murphy's no expert. Perhaps Murphy hasn't appealed to his authority. But you and Maponos certainly have.

The quote goes on:

I just try to put scales on things and sort out roughly how hard things are. At the pace of a post a week (during that time)--on top of a busy job--I could not spend time polishing. 15 km/s (allowing a bit of rounding) is still frikin' hard, so my main point is barely scratched.

Yeah, Murphy's a busy guy and his hurried calculations are only rough approximations. Murphy and his apologists will claim his approximations aren't that far off and his points remain intact.

The point in this particular example doesn't stand. 15 km/s is about what it takes to put a geosynchronous communication satellite in place. This is doable as demonstrated by the large number of such sats. Murphy's 20 km/s is about what it takes to land on the moon's surface and come back. 15 km/s and 20 km/s are vastly different delta V budgets.

But 15 km/s vs 20 km/s is isn't the worst Murphy error. He's done much worse. From Grab That Asteroid! in Murphy's Stranded Resources post:

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. . . .

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, . . .

Does fetching an asteroid take twice the rock's mass in propellent?

Ratio of propellent to dry mass can be found with Tsiolkovsky's rocket equation:
(Mass propellent)/(dry mass) = e^{(delta V/exhaust velocity)} - 1

Let's see -- in Murphy's example delta V is 5 km/s. Exhaust velocity of oxygen and methane is about 3.4 km/s.

e^{(5 km/s / 3.4 km/s)} - 1 = 3.35. So for every ton of asteroid, we'd need more than 3 tons of propellent. At first glance it looks like Murphy is being kind and even under estimating propellent needed.

But methane and oxygen isn't the only propellent. Xenon from an ion engine has an exhaust velocity of around 30 km/s.

e^{(5 km/s / 30 km/s)} - 1 = .18

So about .2 tonnes (or 200 kilograms) of propellent to park a tonne of asteroid. 2/10 is not a "rough approximation" of 2.

But Murphy's error gets worse.

Murphy thinks we'd be lucky to find an asteroid outside of the Main Belt that takes 5 km/s to retrieve. Evidently he hasn't heard of Near Earth Asteroids. There are many asteroids that take much less.

The Keck study for retrieving an asteroid notes some asteroids take as little as .17 km/s. Let's plug in .17 km/s delta V:

e^{(.17 km/s / 30 km/s)} - 1 = .006

So 6 kilograms of propellent to park a tonne of Asteroid. Now Murphy's guesstimate of twice the asteroid's mass is off by a factor of about 350.

Six kilograms is about the mass of two chihuahuas. Two tons is about the mass of two large horses, big horses as in Budweiser clydesdales.

Yes, yes, Murphy's a busy guy and his numbers are rough approximations. Just as a chihuahua is roughly the same size as a large draft horse.

Murphy's figures are often off by several orders of magnitude his PhD notwithstanding.

Neither Stasse's appeal to authority nor Murphy's "rough approximation" defense salvage Murphy's arguments.

My notion of a reusable Earth Departure Stage (EDS) assumes a staging platform at Earth Moon Lagrange 2 (EML2). Propellent, water and air at EML2 might come from an carbonaceous asteroid parked in lunar orbit and/or volatiles in the moon's polar cold traps.

Pictured above is Robert Farquhar's route between EML2 and LEO. It's time reversible so it could be to or from EML2.

A .15 km/s burn at EML2 will drop a spacecraft to a perilune 111 km from the moon's surface. At this perilune the spacecraft is traveling nearly lunar escape velocity with regard to the moon and so enjoys an Oberth benefit. A .19 km/s perilune burn suffices to send the spacecraft earthward to a perigee deep in earth's gravity.

At perigee the spacecraft is traveling about 10.8 km/s, just a hair under earth escape. A burn at this perigee enjoys a huge Oberth benefit. A .6 km/s burn would suffice for Trans Mars Injection (TMI). So delta V from EML2 to TMI is (.15 + .19 + .6) km/s. I will round .94 km/s up to 1 km/s to give a little margin and also 1 is an easier number to type.

After TMI the EDS as well as it's payload is moving 11.5 km/s. To reuse the EDS we would need to return it to EML2.

Farquhar notes the trip from perilune to perigee takes about 140 hours. In that time the moon will advance 76º and the space craft 180º. So in my shotgun orbit simulator I set the perigee 104º ahead of the moon. My first try had pellets ranging from 10.7 to 10.9 km/s and then I'd narrow the blast to the pellets coming closest to the moon. After a few iterations I arrived at a perigee velocity of about 10.85 km/s. This gives an apogee of about 396,000 km and a period close to 2/5 that of the moon. After 50 days, the pellets return to a near moon fly by:

Thus braking about .6 km/s drop the EDS hyperbolic path to a trajectory where it will do a near moon fly-by after 50 days. At the near moon fly by it can do a .14 km/s burn for lunar capture. Then when it reaches an apolune near EML2, a .19 km/s burn to park at EML2.

Thus the EDS' delta V for returning to EML2 will be about 1 km/s.

This page still a work in progress, I'm getting good comments and info from a NASA spaceflight thread. Cryogenic boil off was an issue raised in that thread. A sixty day round trip goes well beyond what present hydrogen/oxygen upper stages do.

The United Launch Alliance has done work on hydrogen/oxygen upper stages that could do longer missions. See Advanced Cryogenic Evolved Stage (ACES). Cyrogenic boil off might be mitigated by Multi Layer Insulation (MLI). Another cooling device is a Thermodynamic Vent System (TVS). Those who live in the southwest are familiar with "swamp coolers" where water soaked pads cool by evaporation. In a similar fashion hydrogen boil-off can be used to cool the cryogens. The hydrogen boil off can be vented in a specific directions and used for station keeping or attitude control.

Besides these passive thermal control systems the ACES might also utilize a two stage turboBrayton cryocooler.
"This design was based on the Creare NICMOS cooler that has been flying on the Hubble Space Telescope for the last ~4 years. The turboBrayton cycle uses GHe as the working fluid and this cooled gas can be easily distributed to the loads (i.e. the 22K and 95K shields). The ACES cryocooler configuration, shown in Figure 3-1, has 3 compressors in series and 2 expansion turbines in parallel, one for the 22K load, and one for the 95K load".

In this ULA pdf an ACES 41 propellant tanker has 5 tonnes dry mass and 41 tonnes propellent. I will be much more conservative in my hypothetical reusable EDS. A Centaur has 2.25 tonnes dry mass, 21 tonnes propellent and 99.2 kilo newtons. I will use the same but boost the dry mass to 5 tonnes for MLI, cryocooling, solar arrays, etc.

Specs for Hop's EDS
5 tonnes dry mass
21 tonnes hydrogen/oxygen
99.2 kilo newtons thrust

After the EDS sends the payload on its way, it will need 1 km/s of propellent of delta V to return to EML2. Exhaust velocity of hydrogen and oxygen is about 4.4 km/s. Exp(1/4.4) - 1 is about .255. To get back the EDS' 5 tonnes of dry mass we'd need 1.3 tonnes of propellent.

So for the first leg of the trip we have (21 - 1.3) tonnes of propellent or 19.7 tonnes. The first leg is also a 1 km/s delta V budget. With a 1 km/s delta V budget, 19.7 tonnes of propellent can do 19.7tonnes/.255. That's about 77 tonnes. But recall 6.3 tonnes is EDS dry mass plus propellent for the return trip. That's (77 - 6.3) tonnes of propellent available for payload. Let's call that 70 tonnes.

This little EDS could impart Trans Mars Insertion (TMI) to 70 tonne payload. Two of these EDS stages could send a 140 tonne payload on its way to Mars. Wilson and Clarke imagine a Mars Transfer Vehicle (MTV) of 130 tonnes.

Of the MTVs 130 tonnes, about 60 tonnes is propellent and consumables. If propellent, water and air are available from an asteroid or lunar volatiles, it would only be necessary to send the MTV's 70 tonne dry mass to EML2.

Wilson and Clarke also call for two EDS stages (they call them TMS -- Trans Mars Stages). Their stages are 110 tonnes and not reusable.

130 + 2*110 = 350. 350 tonnes to LEO for each (non reusable) conventional MTV. Vs 70 tonnes to LEO for an MTV that relies on extra terrestrial propellent and consumables. And an MTV departing from and returning to EML2 would have a much lower delta V budget. Making the MTV reusable would be much more doable.

The EDS would zoom through the perigee neighborhood very quickly. Would it have enough time to do the burns and enjoy an Oberth benefit?

The EDS and payload would spend about 54 minutes in the shaded region above.

A 70 tonne payload plus a 26 tonne EDS total 96 tonnes. The thrust of the engine is 99.2 kilonewtons. Acceleration is newtons/kilograms. 96/99.2 is ~.96. .96 meters/second^2 is about a tenth of a g.

Delta V imparted is acceleration * time of burn. Recall the perigee burn is about .6 km/s or 600 meters/second. We solve for t.

a * t = v
.96 m/s^2 * t = 600 m/s
t = 600/.96 seconds = ~620 seconds, a little over 10 minutes. The ten minute neighborhood just preceding perigee is all close to 10.8 km/s.

After separating from payload, the EDS and it's return propellent mass 6.3 tonnes. 99.2 kilonewtons divided by 6.3 tonnes is 15.75 meters/second^2 or nearly two g's. The deceleration burn to brake the hyperbolic orbit to an elliptical capture orbit would take about 40 seconds.

Near Earth Asteroid Retrieval

The Near Earth Asteroid retrieval described in the Keck Report uses xenon as a propellent. The exhaust velocity would be 30 km/s. What possible use could an EDS with a measly 4.4 km/s exhaust velocity be for such a vehicle.

Along with xenon's high exhaust velocity comes very low thrust. It would the vehicle nearly two years to spiral from Low Earth Orbit (LEO) to escape velocity. A good part of that long spiral would be spent in the Van Allen Belts. Low Earth Orbit also has a relatively high debris density.

Low thrust rockets don't enjoy any Oberth benefit. So the spiral from LEO to C3=0 would take about 7 km/s. Recall the exhaust velocity of the xenon rockets is around 30 km/s. Exp(7/30) - 1 is .26. Using an EDS would leave the asteroid fetcher with about 33% more xenon.

Many NEAs are much closer than Mars in terms of delta V. So perigee burn would be much less than .6 km/s for TMI, probably more often in the neighborhood of .2 or .3 km/s.

When we get to other worlds how will we get from point A to B? There are no roads on Ceres. No rivers or oceans on the moon. No airports on Enceladus or even air for an airplane to glide on.

Until transportation infrastructure is built, suborbital hops seem the way to go. A suborbital hop is an ellipse with a focus at the body center. But with much of the ellipse below surface.

For a suborbital hop from A to B, how much velocity is needed? At what angle should we leave the body's surface? At what angle and velocity do we return?

Points A and B are two points in a Lambert Space Triangle. The triangle's third point C is the body's center.

Two of the sides are radii of a spherical body so the triangle is isosceles.

A point's distance from one focus plus distance to second focus is a constant, 2a. 2a is the ellipse's major axis. Since A and B are equidistant to the first focus, they're also equidistant to the second focus. Points A & B, the planet and ellipse are all symmetric about the ellipse's axis.

The payload returns to the body at the same angle and velocity it left.

There are a multitude of ellipses whose focus lies at the center and pass through points A and B. How do we find the ellipse that requires the least delta V?

Specific orbital energy is denoted ε.

ε = 1/2 v² - μ/r.

1/2 v² is the specific kinetic energy from the body's motion. - μ/r is the specific potential energy that comes from the object's distance from C, the body's center. For an elliptic orbit, | - μ/r | is greater than 1/2 v², that is the potential energy overwhelms the kinetic energy. If potential and kinetic energy exactly cancel, the orbit is parabolic and the payload's moving escape velocity.

So to minimize 1/2 v² we'd want to maximize | ε |.

It so happens that

ε = - μ/(2 a).

To grow |- μ/(2 a)| we shrink 2a. So we're going for the ellipse with the shortest possible major axis.

The foci of all these ellipses lie on the same line. Moving the focus up and down this line, it can be seen the ellipse having the shortest major axis has a focus lying at the center of the chord connecting A and B. That would be the red ellipse pictured above.

The angle separating A and B is labeled θ. Recall the black length and red length sum to 2a

2a = r + r sin(θ/2).

a = r(1 + sin(θ/2))/2.

Now that we know a, we can find velocity with the vis-viva equation:

V = sqrt(μ(2/a - 1/r)).

At what angle should we fire the payload?

It's known a light beam sent from one focus would be reflected to the second focus with the angle of incidence equal to the angle of reflection:

Angle between position vector and velocity vector is (3π - θ)/4. A horizontal velocity vector has angle π/2 from the position vector.
(3π - θ)/4 - π/2 = (3π - θ)/4 - 2π/4 =
(π - θ)/4

(π - θ)/4 is the flight path angle departing from point A. If A and B are separated by 180º (i.e. travel from north pole to south pole), flight path angle is 0º and the payload is fired horizontally. If A and B are separated by 90º (i. e. travel from the north pole to a location on the equator), flight path angle is (180º - 90º)/4 which is 22.5º

When A and B are very close, θ is close to 0. (180-0)/4 is 45º. When A and B are close the the payload would be fire at nearly 45º. With one focus near to the surface and the other at the body center, the ellipse would have an eccentricity of almost 1. The trajectory would look parabolic.

Here's a few possible suborbital hops on our moon:

Minimum energy ellipse from pole to pole is a circle. Launch velocity is about 1.7 km/s. It'd take the same amount of delta V for a soft landing at the destination. Trip time would be about 54 minutes

From the north pole to the equator would take a 1.53 km/s launch. Trip time would be about 27 minutes.

Period of a circular orbit (T) is 2 π * sqrt(r³/(Gm)) where m is mass of planet.

Mass can be expressed as ρ * volume where ρ is density.

2 π * sqrt(r³/Gm) = 2 π * sqrt(r³/(G * 4/3 *π * r^{3 *}ρ).

The r³ cancels out and we're left with T = sqrt(π/(G * 4/3 *ρ).

Thus trip times for a given separation rely solely on density. Period scales with inverse square root of density.

I did a spreadsheet where the user can enter angular separation on various airless bodies in our solar system. Delta Vs vary widely depending on size of the bodies. But trip times between comparable angular separations are roughly the same. This is because most the bodies have roughly the same density. Denser bodies like Mercury will have shorter trip times while icey, low density bodies will have longer trip times.

I got some help on this from a Nasa Spaceflight thread. My thanks to AlanSE and Proponent.

After time we would establish infrastructure on bodies and burrow into their volume. With tunnels we could reach various destinations with very little energy. I look at this in Travel On Airless World Part II.

This is a continuation of Travel on Airless Worlds where I looked at suborbital hops.

The surface of airless worlds will be exposed to radiation so it's likely the inhabitants would live underground.

Moreover, it is not as hard to burrow. The deepest gold mine on earth goes down about 4 kilometers. The heat and immense pressure make it hard to dig deeper. In contrast, the entire volume of a small body can be reached.

Courtney Seligman shows how to compute the pressure of a body with uniform density. The bodies we look at don't have uniform density but we'll use his method as a first order approximation.

Central pressure of a spherical body with uniform density is 3/(8 π G) * g * R²

Where G is universal gravitation constant, g is body's surface gravity and R is body's radius.

At distance r from center, pressure is (1 - (r/R)^2) * central pressure.

What is the pressure 4 kilometers below earth's surface?
Earths's radius R is 6378000 meters. r is that number minus 4000 meters. g is about 9.8 meters/sec^2.
Plugging those numbers into
(1 - (r/R)^2) * 3/(8 π G) * g * R²
gives 2120 atmospheres.

Besides pressure, heat also discourages us from burrowing deeper. So it might be possible to dig deeper on cooler worlds but for now we'll use 2120 atmospheres as the limit beyond which we can't dig.

3/(8 π G) * g * R² gives different central pressures for various worlds:

Ceres center is 1430 atmospheres, well below our 2120 atmosphere limit. And Ceres is a cooler world than earth. We would be able to tunnel clear through the largest asteroid in the main belt. Since smaller asteroids would have smaller central pressure, we would be able to tunnel through the centers of every asteroid in the main belt.

Imagine a mohole going from a body's north pole to south pole:

The diagram above breaks the acceleration vector into vertical and horizontal components. The mohole payload has the same vertical acceleration components as an object in a circular orbit with orbital radius R, R being body radius.

Somone jumping into this mohole could travel to the opposite pole for zero energy. Trip time would be the orbital period: 2 π sqrt(R³/μ).

Other chords besides a diameter could be burrowed. I like to imagine 12 subway stations corresponding to the vertices of an icosahedron:

The red subway lines to nearest neighbors would correspond to the 30 edges of an icosahedron .

Green subway lines to the next nearest neighbors would correspond to the 30 edges of a small stellated dodecahedron.

And there could be 6 diameter subway lines linking a station to stations to their antipodes.

The energy free travel time of all these lines would be the same as the diameter trip time:
2 π sqrt(R³/μ).

It would be possible have a faster trip time than 2 π sqrt(R³/μ). A train could be accelerated during the first half of the trip and decelerated the second half. During the second half, energy could be recovered using regenerative braking.

Most of small bodies in our solar system have internal pressures that don't prohibit access. But in some cases central pressure exceeds 2120 atmospheres. We'd be able to burrow only so deep. Here's my guesstimate of the maximum depth for various bodies:

Luna 40 kilometers
Mars 15 kilometers
Ganymede 77 kilometers
Callisto 97 kilometers
Europa 55 kilometers
Titania 318 kilometers
Oberon 61 kilometers
Pluto 200 kilometers
Haumea 82 kilometers
Eris 108 kilometers

Here is the spreadsheet I used to look at internal pressures.

The top four kilometers of earth's surface is only a tiny fraction of the accessible mass in our solar system.

Lately this blog has been getting some hits from the Kerbal Space Program forum.

This looks like a good game. It seems based on the patched conics approach to orbital mechanics. It's good to see a popular game teaching users concepts like Hohmann or bi-elliptic transfers, sphere of influence, etc.

The art is appealing. Descriptions are entertaining. I purchased a copy for $27.00. It might be a way to become acquainted with some folks who share my orbital mechanics hobby. Hope it's a good investment!

Using the Kerbal Wiki I whomped up a HohmannKSP Spreadsheet. A few people like my spreadsheets for our solar system. Hopefully I'll be making some useful spreadsheets for this game.

Usual disclaimers apply:

My spreadsheets assume circular, coplanar orbits. Some of the game orbits are inclined and eccentric.

I occasionally make mistakes -- data entry as well as arithmetic errors. I'd be grateful if users check my efforts.

Doug Plata recently suggested some possible space exploration topics. All of them are very interesting.

Dr. Plata has good ideas and invests a great deal of time and effort looking at them. He's involved in two excellent websites: LunarCOTS.com and CisLunarOne.com .

July and August is a slow period for the Ajo Copper News, the weekly newspaper my sister and I publish. Most people with money and good sense leave Ajo, Arizona for the summer months. Hopefully I will have time to examine some of Plata's topics in my blog over the next few months.

Here they are:

Partial vs full reusability

Falcon 9 has nine engines on the first stage and one engine on the second stage. So, if only the first stage is reused, it would seem to me that 9/10 engines would be recovered. That's got to be a huge reduction in launch cost right there, yes? Just how much? Certainly achieving even partial reusability would make SpaceX even more competitive that it already is. If the Falcon Heavy were to be partially reusable, reusing the lateral boosters would mean only 18 out of 28 engines would be recovered unless the central core could be reused as well.

Propulsion service options

For a cis-lunar transportation system we most often think of fuel depots in LEO. One problem with this is the need for fuel depots in multiple LEO planes with those depots being used only occasionally. However, if propellant were coming from ice harvesting operations at the lunar poles, then conceivably an OTV could bring propellant into any LEO inclination just prior to a launch into that orbit from Earth. However, in this scenario, do we even need LEO depots? Why couldn't the OTV dock with the launched satellite and then use its own engines to boost the satellite to GTO or even GEO? Do we need fuel depots or could propulsion service be enough?

Power options for lunar mining

Say you are wanting to do ice harvesting operations in a lunar polar permanently shadowed crater with the rim of such having a peak of eternal light (PEL). Great, but there's still potentially kilometers of distance between the source of power and the ice harvesting operations site. How to deal with that gap? RTGs? Laser beaming of power? Drive a rover laying a cable down the side of the crater? Hop the lander from the rim to the floor while draping a (superconducting) wire? Or forget a solar panel farm at the PEL and crack the water at a fuel depot in orbit? An interesting trade analysis.

Aerobraking

OTVs tend to be painted as broad, turtle-shaped craft. But how do you launch and assemble such a thing? Can aerobraking be done about as easily with a cylindrical-shaped OTV? How about heat flaps popping out giving more surface area and control? Necessary? If one skims high enough in the atmosphere does one even need a heatshield? What about using a lifting body form?

Travel times further out into the solar system

So if we develop the ability to safely send humans to Deimos, how much longer would it take to send them to Vesta, Ceres, and a Moon of Jupiter?

---------

ROCKETRY

Heavy Lift vs Single Stage vs Reusable vs Gun

Air launch

Partial vs Full reusability

Chemical versus liquid rockets

CIS-LUNAR TRANSPORTATION INFRASTRUCTURE

Propulsion service options

Power options for lunar mining

Aerobraking

ORBITAL DYNAMICS

The orbital dynamics of a Phobos vs Deimos vs surface mission

HUMAN FACTORS

Mass calculations of open-loop, vs closed chemical, vs ECLSS

How could an RP5 be provided?

How do the space radiation numbers compare between locations (i.e. LEO, free-space, lunar surface, Phobos, Mars?)

Animal studies

Partial gravity options

COLONIZATION

O'Neillian vs lunar colony - Where first?

Travel times further out into the solar system

The largest space forum I know of is forum.nasaspaceflight.com. There are a lot of knowledgeable people who participate including a number of professional aerospace engineers.

I enjoy this forum but in my opinion there is a bias for NASA sponsored HLV.

For example, the forum has subsections devoted to manned missions to Mars, the Moon and Near Earth Asteroids. There are a number of ways such missions could be accomplished. But evidently Chris Bergin feels SLS is the only option:

Notice all the HSF (Human Space Flight) missions come under the HLV/SLS/Orion/Constellation heading.

I mentioned to Chris Bergin that there may be other routes. For example an architecture based on propellent depots might get us to the moon. Chris retorted that some of the ULA depot guys participate in NSF and they don't bad mouth SLS. Well, one of ULA's parent companies is Boeing. Umm, Chris, maybe there's a reason ULA employees would be hesitant to criticize SLS.

I posted a cartoon to NSF:

Chris found my use of the "pork" offensive. He also didn't like didn't like my portrayal of Senator Shelby.

There are a few things Apollo, Ares and SLS have in common:
1) They're very large rockets
2) They're completely expendable.

Since they're big, that means big expense. Since they're not reusable, that big expense will be incurred each and every trip.

If every mission is going to cost a few billion, we are not going to colonize the Moon, Near Earth Asteroids or Mars. Settlement would take a long, sustained effort and this sort of expense just isn't sustainable.

If Shelby et al are trying to sell SLS as a way to open a new frontier, they are committing fraud.

So what is the goal of NASA's human space flight program? The occasional flag & footprints publicity stunt doesn't justify the expense. If NASA's human space flight program is all about jobs in Florida, Texas and Alabama, it should be axed.

I make no apologies for my cartoon.

I was lamenting to a friend that science fiction is ignoring robotic advances and the near term possibilities they create. He replied "You must read Clive Cussler. I wouldn't call him a science fiction writer but he uses ROVs a lot. DARPA, the military and mining entities show up in most of his stories. I know you will like the characters from NUMA."

I took his advice and am now 3 books into the adventures of Dirk Pitt and friends. Not high literature by any means. Just entertaining, satisfying adventure yarns. And Cussler does his homework. The printed words in his books are the tip of a large iceberg. It is obvious many hours of research lie beneath the surface of each story. Cussler's interests are eclectic. He likes to study engineering and technology. Also biology, oceanography, chemistry, geology, archeology, art, history, culture, food, religion, etc., etc. Each book I've learned new stuff from many different fields.

Why am I so fired up about ROVs, AUVs, etc? It is my belief advancing robotics will be the game changer that opens the door to space, the final frontier. Cussler's stories are more relevant to space exploration than most current science fiction.

Already remotely operated robots are doing work in places too dangerous or hard to reach for human workers. This technology is being advanced by many players: DARPA, NOAA, British Petroleum, Rio Tinto, the military and others.

It is interesting that Google bought up the best performers in a recent DARPA robotics competition. Google has also invested in Planetary Resources and SpaceX as well as funded the Google Lunar X-Prize. Dot com billionaires opening the door to space could be a rich vein for story tellers.

On my wish list: Cussler taking a look at the void that lies between us and our neighbors in the solar system. Asteroids and planets are islands and continents in an ocean that extends past all horizons.

This is a topic suggested by Doug Plata. What impact will partial reusability have on efforts to settle and exploit space?

Reusable Booster stages.

SpaceX is working on a reusable booster stage. This has potentially enormous savings.

Why is reusing a booster such a big deal? Some might think getting above the atmosphere is a minor challenge compared to achieving orbital velocity. Lets take a look at hows and whys of vertical ascent.

The flight path angle is the angle between horizontal and the velocity vector.

If earth were an airless world, horizontal launches would be optimal. In other words the flight path angle would be zero.

But earth has an atmosphere. To avoid a long trip through the atmosphere a flight angle closer to vertical is called for.

Taking off from the earth at 8 km/s, a nearly vertical flight path angle vs horizontal take off.

Low earth orbit velocity is about 8 km/s. If a spacecraft achieved this velocity at earth's surface with a zero flight path angle, nearly a quarter of it's orbit (about 10,000 kilometers) would be through earth's atmosphere.

Most meteorites burn up in the mesosphere about 70 km up. Air density at this altitude is less than a thousandth of sea level. Orbital velocity at sea level would subject the rocket to extreme temperatures.

Dynamic pressure is another quantity to consider. Dynamic pressure is often denoted with the letter q. The maximum dynamic pressure a spacecraft endure is referred to as max-Q. The max-Q of the space shuttle was about 33 kilo-pascals. A severe hurricane has a dynamic pressure of 3 kilo-pascals.

8 km/s at sea level would give a dynamic pressure of about 40,000 kilo-pascals.

Before making the major horizontal burn to achieve orbital velocity, we must get above the dense lower atmosphere. The shortest path through the atmosphere is a vertical ascent.

But a vertical ascent incurs gravity loss.

Earth's surface gravity is 9.8 meters/sec^2. Each 102 seconds spent in vertical ascent costs 1 km/s delta V. Gravity loss is a major expense associated with ascent.

To minimize ascent time, a high thrust to weight ratio (T/W) ratio is desirable. The more oomph a booster stage has, the less time gravity loss is incurred.

A booster stage with more rocket engines will have a higher thrust to weight ratio. The Falcon 9 booster has 9 Merlin engines as compared to the second stage which has only 1.

Since a booster has 9 engines and the upper stage 1, would reuse mean 90% savings?

The upper stage also needs avionics, a power source, propellent tanks etc.. So I'd be surprised if the upper state is 10% of the expense. My guess would be more like 1/6. Still a 5/6 savings would be substantial.

But even a 5/6 savings wouldn't be realized by re-use. Still unknown are refurbishment costs. Also unknown is how many times a booster can be re-used.

I give better than even odds SpaceX's reusable booster will cut launch costs by 50%.

Reusable Upper Stage

After the booster stage has lifted the spacecraft above the atmosphere, the upper stage provides the horizontal burn to achieve orbital velocity. This take about 8 km/s.

Tsiolkovsky's rocket equation and an 8 km/s delta V budget mandate the upper stage is about 90% propellent and 10% dry mass. The smaller dry mass fraction means more tenuous structure and less thermal protection. It is hard to see how an upper stage could endure the extreme conditions of an 8 km/s re-entry into earth's atmosphere.

I would bet against SpaceX achieving a reusable upper stage.

Reusable Capsule

A capsule doesn't need a huge delta V budget. Just enough to lower it's perigee so it passes through the upper atmosphere. With a delta V budget less than 1 km/s, a capsule can have robust structure as well as a substantial heat shield.

I give SpaceX better than even odds at achieving a reusable Dragon capsule.

What does re-use do to economies of scale?

An item can be much cheaper if many units are mass produced on an assembly line. With mass production, design and development is amortized to a marginal expense.

If the average rocket engine is re-used 10 times, we would need at least a ten fold market increase to maintain economies of scale.

Could re-use lower prices enough to boost the market ten fold or more? I am not sure this would happen. What's the market for launch vehicles? Communication sats, surveillance and weather sats, occasionally ferrying passengers to the I.S.S. It's not clear cutting launch costs by half or even two-thirds would explode this market.

Economies of Scale with Re-use

The are possible new markets such as space tourism or mining. I don't expect those markets to take off so as a launch costs millions.

But what if the entire package was re-usable? The upper stage as well as booster and capsule? Reducing the cost by another order of magnitude opens many new markets: orbital hotels, lunar and asteroid mining, bases on the moon and Mars, etc..

But for upper stage re-use we would need propellent sources other than from the bottom of earth's gravity well. We would need orbital infra-structure: ferries between the various orbits and regions in our earth moon neighborhood: LEO, GEO, EML1, EML2 and DRO.

Establishing this mining and transportation infra-structure could provide the initial market. Once infra-structure is established, development of space would proceed like a snow ball rolling down a hill.

In my opinion partial re-use isn't sufficient to get the ball rolling. But it's an important step toward achieving full re-use. What happens after full re-use? If we can cut expenses down to the point where propellent is the dominant cost, I'd expect the market to explode at an exponential rate.

First off let's look at the great granddaddy of vertical tethers, the Clarke tower.

For a vertical tether in circular orbit, there's a point where the net acceleration is zero. Above that point, so called centrifugal force exceeds gravity. Below that point, gravity exceeds so-called centrifugal force. If a payload is released on this point of on the tether, it will follow a circular orbit alongside the tether. This point I call the Tether Center.

In this case, the tether center is at geosynch height, about 42,000 km from earth's center. I set 42,000 km to be 1. What path does a payload follow if released from the tether below the center?

It will be a conic section. Call the conic's eccentricity e. Call the distance from tether point r.

If dropped from below center, r = (1-e)^1/3.
If released from above center, r = (1+e)^1/3.

Here's my derivation. Mark Adler also gives a nice demonstration in the comments on that post.

This is true of any vertical tether in a circular orbit.

If there are two prograde, coplanar vertical tethers at different altitudes, there's an elliptical path between them where the perigee velocity matches a point on the lower tether and apogee velocity matches a point on the upper tether.

If a payload is released from the lower tether at the correct time, it will rise to the upper tether which will be moving the same velocity as the payload at apoapsis. Rendezvous can be accomplished with almost no delta V. Cargo can be exchanged between tethers with almost no reaction mass.

Let r for the release point above the tether be (1+e)^1/3 and release point below the tether be (1-e)^1/3. Then both the larger and smaller ellipse will be the same shape.

Center of the above tether is 8000 km. I tried to place it above the dense orbital debris regions of low earth orbit. The tether is 461.6 kilometers long. Dropping from the foot will send a payload to a 150 km attitude perigee. Throwing a payload from the tether top will send a payload to a 9780 km apogee.

From a 150 km altitude orbit, it takes about .33 km/s to send a payload to the tether foot.

Both ellipses have the same eccentricity, about .0864

I repeatedly clone, scale by 126% and rotate 180º:

By ascending and playing catch with a series of tethers, a payload might make it's way from LEO to the vicinity of the moon:

But there's a problem with this scheme. A tether loses orbital momentum each time it catches a payload from below. Ascending and throwing to a higher orbit also saps orbital momentum. How do we keep these tethers from sinking?

Imagine resources parked in lunar orbit. Maybe propellent mined from the lunar poles. Or perhaps platinum from an asteroid parked in a lunar DRO. To send cargo to earth's surface or low earth orbit would entail catching from a higher orbit, descending and dropping to a lower orbit:

If cargo is moved down as well as up, momentum boosting maneuvers can be balanced with momentum sapping maneuvers.

Thus mass in high orbits are sources of up momentum. This itself could be a commodity, a way to preserve orbits of momentum exchange tethers.

This tether spiral scheme cuts tether length, especially in regions of high debris density and the Van Allen Belts.

In this illustration successive ellipses vary by a factor of 2^1/3. Other rates of expansion are possible. Let k be the ratio of one ellipse apogee to the apogee below. k = (1+e)^4/3/(1-e)^4/3. Thus we can wind the spiral tighter or loosen it by choice of ellipse eccentricity.

Wikipedia describes the Interplanetary Transport Network as "… pathways through the Solar System that require very little energy for an object to follow." See this Wikipedia article. They also say "While they use very little energy, the transport can take a very long time."

Low energy paths that take a very long time? I often hear this parroted in space exploration forums and it always leaves me scratching my head.

The lowest energy path I know of is the Hohmann orbit. Or if the destination is noticeably elliptical, a transfer orbit that is tangent to both the departure and destination orbit. Although I think of bitangential transfer orbits as a more general version of the Hohmann orbit.

Bitangential Transfer Orbit

The transfer orbit is tangent to both departure and destination orbit.

The Hohmann transfer is the special case where departure and destination orbits are circular.

Illustration from my pdf on tangent orbits.

In the case of Mars, a bitangential orbit is 8.5 months give or take a month or two. Is there a path that takes a lot longer and uses almost no energy? I know of no such path.

L1 and L2

The interplanetary Superhighway supposedly relies on weak stability or weak instability boundaries between L1 and/or L2 regions. Here is an online text on 3 body Mechanics and their use in space mission design. The authors are Koon, Lo, Marsden and Ross. Shane Ross is one of the more prominent evangelists spreading the gospel of the Interplanetary Super Highway.

The focus of this online textbook is the L1 and L2 regions. From page 10:

L1 and L2 are necks between realms. In the above illustration the central body is the sun, and orbiting body Jupiter. L1 and L2 are necks or gateways between three realms: the Sun realm, the Jupiter Realm and the exterior realm.

Travel between these realms can be accomplished by weak stability or weak instability boundaries that emanate from L1 or L2. From page 11:

My terms for various Lagrange necks

First letter is the central body, the second letter is the orbiting body.

Earth Moon L1: EML1
Earth Moon L2: EML2

Sun Earth L1: SEL1
Sun Earth L2: SEL2

Sun Mars L1: SML1
Sun Mars L2: SML2

Since I'm a lazy typist that is what I'll use for the rest of this post.

EML1 and 2

I am very excited about the earth-moon Lagrange necks. They've been prominent in many of my blog posts and I plan to devote another blog post to EML2 soon.

EML1 and 2 are about 5/6 and 7/6 of a lunar distance from earth:

Both necks move at the same angular velocity as the moon. So EML1 moves substantially slower than an ordinary earth orbit would at that altitude. EML2 moves substantially faster.

It takes only a tiny nudge and send objects in these regions rolling about the slopes of the effective potential hills. Outside of the moon's influence they tend to fall into ordinary two body ellipses (for a short time).

Here's the ellipse an object moving at EML1 velocity and altitude would follow if the moon weren't there:

An object nudged earthward from EML will fall into what I call an olive orbit.

It's approximately 100,000 x 300,000 km.

In practice an EML1 object nudged earthward will near the moon on the fifth apogee. If coming from behind, the moon's gravitational tug can slow the object which lowers perigee.

Here is an orbital sim where the moon's influence lowered perigee four times:

I've run sims where repeated lunar tugs have lowered perigees to atmosphere grazing perigees. Once perigee passes through the upper atmosphere, we can use aerobraking to circularize the orbit.

Orbits are time reversible. Could we use the lunar gravity assists to get from LEO to higher orbits? Unfortunately, aerobraking isn't time reversible. The atmosphere can't increase orbital speed to achieve a higher apogee. And low earth orbit has a substantially different Jacobi constant than those orbits dwelling closer to the borders of a Hill Sphere.

So to get to the lunar realm, we're stuck with the 3.1 km/s LEO burn needed to raise apogee. But once apogee is raised, many doors open.

There are low energy paths that lead from EML1 to EML2. EML2 is an exciting location.

Without the moon's influence, an object at EML2's velocity and altitude

would fly to an 1,800,000 km apogee. This is outside of earth's Hill Sphere!

In the above illustration I have an apogee beyond SEL2. But by timing the release from EML2, we could aim for other regions of the Hill Sphere, including SEL1.

Here is a sim where slightly different nudges send payloads from EML2:

See how the sun bends the path as apogee nears the Hill Sphere? From EML2 there are a multitude of wildy different paths we can choose. In this illustration I like pellet #3 (orange). It has a very low perigee that is moving about 10.8 km/s. And it got to this perigee with just a tiny nudge from EML2. Pellet # 4 is on it's way to a retrograde earth orbit. Most of the other pellets are saying good bye to earth's Hill Sphere.

I am enthusiastic about using EML1 and EML2 as hubs for travel about the earth-moon neighborhood. But a little less excited about travel about the solar system.

We've left Earth's Hill Sphere. Now what?

Recall that EML1 and 2 are ~5/6 and 7/6 of a lunar distance from the earth. SEL1 and 2 are much less dramatic: 99% and 101% of an A.U. from the sun. Objects released from these locations don't vary much from earth's orbit:

Running orbital sims gets pretty much the same result pictured above.

Mars is even worse:

Are there weak instability boundaries leading from SEL2 to SML1? I don't think this particular highway exists.

To get a 1.52 aphelion, we need a departure Vinfinity of 3 km/s. To be sure EML2 can help us out in achieving this Vinfinity. In other words we could use lunar assists to depart on a Hohmann orbit. But a Hohmann orbit is different from the tube of weak instability boundaries we're led to imagine.

And once we arrive at a 1.52 aphelion. we have an arrival 2.7 km/s Vinfinity we need to get rid of.

Pass through SML1 at 2.7 km/s and you'll be waving Mars goodbye. The Lagrange necks work their mojo on near parabolic orbits. And an earth to Mars Hohmann is decidedly hyperbolic with regard to Mars.

What about Phobos and Deimos? The Martian moons are too small to lend a helpful gravity assist. We need to get rid of the 2.7 km/s Vinf and neither SML1 nor the moons are going to do it for us.

Mars ballistic capture by Belbruno & Toppotu

Edward Belbruno is another well known evangelist for the Interplanetary Superhighway. He cowrote this pdf on ballistic Mars capture.

Here is a screen capture from the pdf:

The path from Earth@Departure to Xc is pretty much a Hohmann transfer. In fact they assume the usual departure for Mars burn. Arrival is a little different. They do a 2 km/s circularization burn at Xc (which is above Mars' perihelion). This particular path takes an extra year or so to reach Mars.

So they accomplish Mars capture with a 2 km/s arrival burn. At first glance this seems like a .7 km/s improvement over the 2.7 km/s arrival Vinf.

Or it seems like an advantage to those unaware of the Oberth benefit. If making the burn deep in Mars' gravity well, capture can be achieved for as little as .7 km/s.

Comparing capture burns it's 2 km/s vs .7 km/s. So what do we get for an extra year of travel time? 1.3 km/s flushed down the toilet!

Zero energy trips my a**.

Summary

The virtually zero energy looooong trips between planets are an urban legend.

I'll be pleasantly surprised if I'm wrong. To convince me otherwise, show me the beef. Show me the zero energy trajectory from an earth Lagrange neck to a Mars Lagrange neck.

Until then I'll think of this post as a dose of Snopes for space cadets.

Escher loved spirals, the Droste effect, and recursive stuff that suggests infinity.

I was looking at a golden rectangle with a series of squares marked off and thought it'd make for a neat variation on Escher's print of hands drawing hands.

What the heck is Alpha?

What Alpha am I talking about? A power source's ratio of mass to power. The first time I ran into this quantity was in a NASASpaceFlight discussion of VASIMR's 39 day trip to Mars.

Kirk Sorensen worked for NASA 10 years. He has Master's degrees in aerospace as well as nuclear engineering. He is co-founder of Flibe Energy, a company that hopes to build thorium nuclear reactors. I don't always agree with him but I believe he has some expertise in this field. For the time being I am talking his word for it.

So this Magic Alpha, .50 kg/kWe, what is that? Here's an attempt to portray it:

This Ford Focus has a 160 horsepower engine. One horsepower = ~750 watts, so the engine is about a 12,000 watt power source. Not only the engine but also gasoline and oxygen. Pictured with the Focus is Dominique who masses about 60 kilograms. If she were a power source replacing the engine, gasoline and oxygen, we'd have an alpha of .5 kg/kW.

An electric car like the Tesla uses a battery. But just as a gas engine must make periodic stops to gas up, a Tesla must be frequently recharged. Enroute to Mars there are no gas stations and no electric outlets.

Later in the same NASA Space Flight thread, Sorensen says:

I don't really care if Samim Anghaie is crook but his 0.5kg/kWe number is a fantasy. Why FCD builds his VASIMR sales case on that number when all other reputable electric propulsion researchers have rejected it (even though makes their thrusters look incredible too) is beyond my understanding.

Indeed. Such a power source would make Hall thrusters look great. So far as I know Franklin Chang Diaz and Samim Anghaie are the only folks whose schemes rely on such an alpha.

Thermal Watts vs Electric Watts

39 day VASIMR trips to Mars are mentioned on Page 42 of The Plundering of NASA by Rick Boozer. Boozer argues that SLS and Orion are pork barrel make work programs and that money could be better spent on SpaceX and other programs. In general I agree with Boozer but was disappointed to see his endorsement of VASIMR.

So I asked Boozer about the Magic Alpha. Boozer came back with Project Nerva, a nuclear thermal rocket. He wrote:

Project NERVA claimed up to 5 GW possible with total mass of 38,600 kg. That works out to .00008 kg per W or .008 per kW. According to that Sorenson is incorrect.

NERVA's output is thermal watts. Thermal and electric watts are two very different things.

A nuclear electric power plant must first convert thermal watts to electric watts. But that's not the only problem.

The plant must dump waste heat. Massive cooling towers have become an icon for nuclear energy. From Wikimedia:

Earthly nuclear power plants can use water to carry off waste heat. In space there are no neighboring streams a nuclear power plant can use. In fact vacuum is a great insulator. A nuclear electric power source would need massive radiators.

I mentioned to Boozer that thermal and electric power sources were very different things.

He replied:

David, I just don't know where you are coming from. There you went earlier lecturing me about the difference between thermal and electrical energy which is something that I teach physics students all the time. If I wasn't competent in physics my students wouldn't be making the high grades they are and I couldn't have got my Master's in astrophysics.

I was hoping Boozer would demonstrate Sorensen was wrong. Sadly, pointing to his students' good grades and his degree did absolutely nothing to demonstrate the plausibility of Diaz' Magic Alpha. I was convinced of one thing though: Rick Boozer isn't credible.

I did not bother reading past page 42 of The Plundering of NASA.

What's the best plausible Alpha?

I search space forums for discussions of low mass power sources. So far as I can tell, thin film photovoltaics show the best promise. Roll Out and Passively Deployed Array (RAPDAR) might deliver 250 kW/kg. RAPDAR's thin film solar cells use an Elastic Memory Composite (EMC) for support and structure. Rolled up and cooled, the EMC will fit in a small volume and thus can fit under a fairing. When the sun warms it, the EMC will expand to the shape it needs to be.

On a Nasa Space Flight thread space entrepreneur Jeff Greason opined:

While I won't speak to this specific design, more generally I am quite convinced that thin film solar approaches 1 kW/kg are definitely possible near term. However there is very little serious work going on, and packaging such systems for launch and deploying them without spoiling the mass is not at all trivial.

But do keep thinking -- it is not crazy, at least 1 kW/kg rather than two.

Thin film solar is extremely fragile, however, so the packaging is really challenging.

That's the rub, packaging. How useful are acres of Saran Wrap® with no structure? There needs to be a supporting frame to keep the film spread. It also needs to be kept pointing towards the sun so the supporting frame needs to be attached to gimbals and motors. What is the Alpha including supporting structure, gimbals and motors?

How will we deploy acres of Saran Wrap® from a small volume that fits within a fairing?

However Greason's optimism is somewhat reassuring. Being a bonafide space-cadet, I cling to optimistic opinions as long as I can.

Why is Alpha such a big deal?

As mentioned at the beginning of this post, a great alpha would make ion thrusters a more formidable tool. A big cut to parasitic mass would give ion thrusters better acceleration as well as great ISP.

Good Alpha would also make ISRU more plausible. Readers of my blog know I'm hung ho on use of extra-terrestrial propellent, either near earth carbonaceous asteroids or frozen volatiles in the lunar cold traps.

Let's say we do mine water in the moon's neighborhood and we want to crack it to hydrogen/oxygen bi-propellent. Cracking a mole of water (18 grams) takes 287000 joules. A tonne of water is 55555 moles. 55555 moles*287000 joules/mole =13166666667 joules. If we wanted to crack 10 tonnes of water per day, we'd need a 1.5 mega-watt power source. And that doesn't include refrigerating the cryogens.

Cracking water isn't the only ISRU electricity hog. Just about all extra-terrestrial mining and industry will need lots of juice.

In the 50's and 60's NASA and the military provided big incentives to miniaturize electronics as low mass and small volume circuitry is a pre-requisite for rockets and missiles. I believe miniaturizing a power source should be a top goal for NASA. If we hope to settle space, a better Alpha should be given a higher priority than Apollo redux.

Earth Moon Lagrange 2 or EML2 is one of 5 locations where earth's gravity, moon's gravity and so called centrifugal force all cancel out. It lies beyond the far side of the moon at about 7/6 of a lunar distance from earth.

Infrastructure at any of these 5 locations could be kept in place with a small station keeping expense. Other high earth orbits would be destabilized by the influences of the earth, moon or sun.

Of these 5 locations, EML2 is the closest to escape. How close?

Specific orbital energy is given by

v²/2 - GM/r

v: velocity with regards to the earth
G: gravitational constant
M: mass earth
r: distance from earth center

Here's specific orbital energies for a few orbits:

For EML1 and EML2 I'm looking at resulting earth orbits for payloads nudged away from Luna's Hill Sphere.

Most the energy is getting from earth's surface to Low Earth Orbit (LEO). Then another huge chunk is getting from LEO to escape.

EML2 is right next door to escape (aka C3=0). If the goal line is Trans Mars Injection, EML2 is on the 9 yard line.

EML2's orbital energy is about -180,000 joules per kilogram. How much is that? Well, Kattie is standing next to the small generator which provides electricity for our business during power outages during the summer monsoons. It would take this 20 kilo-watt generator 9 seconds to crank out 180,000 joules.

An EML2 payload nudged away from Luna would rise to an 1.8 million km apogee. An ordinary earth orbit at 450,000 kilometers from earth's center would move about .94 km/s. But since EML2 is moving at the moon's angular velocity, it is traveling 1.19 km/s. Earth's Hill Sphere is about 1.5 million km in radius. So depending on timing, an EML2 nudge could send a payload out of earth's sphere of influence into a heliocentric orbit.

Another possibility is the sun's influence could send a payload back towards the earth with a lower perigee:

All of these pellets were nudged from EML2. The sun's influence has wrested most of these from earth's influence. But check out pellet number 3 (orange). The sun's influence has dropped this pellet to a perigee deep in earth's gravity well. For a .1 km/s nudge from EML2 we can get a deep perigee that can give a very healthy Oberth benefit. However, such a route takes about 100 days.

Using an lunar gravity assist along with an Oberth enhanced burn deep in the moon's gravity well, EML2 is 9 days and 3.5 km/s from Low Earth Orbit (LEO):

This route was found by Robert Farquhar.

Bi-Elliptic Transfers

It radii of two different orbits differ by a factor of 11.94 or more, a bi-elliptic transfer takes less delta V than Hohmann. EML2 radius / LEO radius is about 67, so LEO to EML2 could definitely be a beneficiary of bi-elliptic.

From LEO, a 3.1 km/s burn gets us to a hair under a escape. A multitude of elliptical orbits fall under this umbrella!

As you can see, it takes almost as much to get as high as EML1 as it does to reach a 1.8 million km apogee. I chose 1.8 million as an apogee since a 450,000 x 1,800,000 km ellipse at perigee has the same altitude and speed as EML2. At perigee a payload can slide right into EML2 with little or no parking burn.

What's needed is an apogee burn to raise perigee to 450,000 km. A 6738x1,800,000 km ellipses moves very slow at apogee, a mere .04 km/s. A 450,000 x 1,800,000 km ellipse doesn't move much faster at apogee, about .3 km/s So a .26 km/s apogee burn suffices to raise perigee.

So the total budget is .26 + 3.1555 km/s. This 3.42 km/s delta V budget is better than a Hohmann but about the same as Farquhar's 9 day route.

But recall apogee is beyond earth's Hill Sphere. With good timing, the sun can provide the apogee delta v.

Here's a route I found with my shotgun orbital sim:

LEO burn is about 3.11 km/s. Payload passes near the moon on the way out, boosting apogee and rotating line of apsides. The sun boosts apogee as well as perigee. Coming back the pellets slide right into EML2 (the circular path alongside the Moon's orbit).

This LEO to EML2 route took 74 days and 3.11 km/s.

EML2 and Reusable Earth Departure Stages.

Using the Farquhar route, it takes about .4 km/s to drop from EML2 to a perigee moving just under escape velocity. At this perigee .5 km/s will give Trans Mars Insertion (TMI). After the departure stage separates from the payload it's pushing, it can do a .5 km/s braking burn to drop to an ellipse with a near moon apogee. Once at the moon, another .4 km/s takes the EDS back to EML2.

For massive craft moving from between earth's neighborhood and other heliocentric orbits, it makes little sense to climb down to Low Earth Orbit (LEO) and back each trip. It saves time and and delta V to park at EML2 on arrival. If EML2 becomes a stop for interplanetary space craft, a reusable EDS is a good way to depart the earth/moon neighborhood.

I talk about this in more detail at Reusable Earth Departure Stages.

EML2 and Fast Transits

Here's a pic of a Non Hohmann Mars transfer:

Mars and earth orbits are approximated as circular orbits. A Hohmann orbit will have a 1 A.U. perihelion and a 1.52 A.U. aphelion. The transfer orbit above has perihelion .7 A.U. and aphelion 1.53 A.U. Semi-major axis of this orbit is (.7 + 1.53)/2 A.U. or 1.115 A.U. Orbital period is 1.115^3/2years which is about 1.18 years.

The trip to Mars isn't the entire orbital period though, just the turquoise area swept out from departure to destination. The turquoise area is 31.5% of the ellipse's area. 31.5% of 1.18 years is about 135 days or about 4.4 months.

Departure and arrival Vinf are indicated by the red arrows. These are the vector differences between the transfer orbit's velocity vector and the planet's velocity vector at flyby. A change in direction accounts for most of the Vinf. I'm assuming Mars and Earth are in circular orbits with a zero flight path angle. Therefore the direction difference between vectors can be described with the flight path angle of the transfer orbit's velocity vector.

In this case the earth departure Vinf is 11.3 km/s. That's a big Vinf! But if falling from EML2, only a 4.9 km/s perigee burn is needed. This is doable.

At Mars the Vinf is 5.14 km/s. But a periaerion burn of 2.57 km/s brakes the orbit into an (3697x2345 km ellipse. This orbit could be circularized via periaerion drag passes through the upper atmosphere. Since this ellipse has a period less than a day, orbit could be circularized in a few weeks.

An upper stage can have a 8 km/s delta V budget. Recall it takes about .4 km/s to fall from EML2. Therefore let's try to find a route that takes about 7.6 km/s from perigee burn to periaerion burn.

Trial and error with my Non Hohmann Transfers spreadsheet gives:

With chemical rockets departing from EML2, I believe 4 month trips to Mars are doable.

EML2 Proximity to Possible Propellent Sources.

In terms of delta V, time and distance EML2 is quite close to several possible propellent sources.

There are thought to be frozen volatiles in the lunar cold traps. Some craters at the lunar poles have floors in permanent shadow. Temperatures can go as low as 30 K. Volatiles that find their way to the cold traps would freeze out and remain. There may be rich deposits of H₂0, CO₂, CH₄, NH₃ and other compounds of hydrogen, carbon, oxygen and nitrogen. These would be valuable for life support as well as propellent.

The moon's surface is about 2.5 km/s from EML2.

Also there are proposals to retrieve asteroids and park them in lunar Deep Retrograde Orbits (DROs). DROs are stable lunar orbits that can remain for centuries without station keeping. Planetary Resources would like to retrieve water rich carbonaceous asteroids. Carbonaceous asteroids can contain up to 20% water by mass in the form of hydrated clays. They can also contain compounds of carbon and oxygen.

LDROs would be about .4 km/s from EML2.

Summary

EML2 would make a great transportation hub. Not only for travel to destinations throughout the solar system but also within our own earth moon neighborhood.

Arrgh! It's not the cost of the fuel

What's the minimum spin hab?

Who needs humans?

EML2 plane change

Terraforming Mars vs Orbital Habs

How much air do we need to add to Mars?

Can we add to Mars' air with comets?

Asteroidal Real Estate

The most common delta V error

Murphy's reply

Reusable Earth Departure Stage

Travel on airless worlds

Travel on Airless Worlds Part II

Kerbal Space Program

Space topics from Dr. Plata

Will I be banned from Nasa Space Flight Forum?

Clive Cussler - Two Thumbs Up

Partial vs full reusability

A spiral of tethers

Potholes on the Interplanetary Superhighway.

A Golden Escher tribute

The Need for a Better Alpha

EML2