EML2 plane change

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.

Edit: Isaac Kuo has pointed out that earth has a 30 km/s vector coplanar to the ecliptic plane. A hyperbolic orbit departing earth might have a v infinity of 3 or 4 km/s. When you add a 3 or 4 km/s vector tilted 60 degrees to the 30 km/s vector, you are only inclined 4 or 5 degrees from the ecliptic plane. Still, this is a substantial plane change! This post is incomplete as I've only looked at departure from EML2. Hopefully, I'll soon have time to look at burns at perilune and perigee burns and the possible trans asteroid orbits.

Who needs humans?

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, 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.

What's the minimum spin hab?

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 need 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.

While we're talking about effects of microgravity, let's take a look at cosmonaut Valeri Polyakov. From a SpaceDaily article: "Polyakov's space flight had lasted 438 days (bettering a year by more than two-and-a-half months). Yet upon return, his health was not much different than other cosmonauts' after a long flight. After those first steps, he completely readapted to gravity within two months. Moreover, his bone loss had been very low, only around 7 percent in some of his weight-bearing bones," 

 Granted, only a small fraction of us have the self discipline to adhere to Polyakov's exercise regimen. But he demonsrates that exercise can mitigate microgravity bone loss.


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.

Scott Manley did a nice video on spin habs.

Arrgh! It's not the cost of the fuel

"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.