Tuesday, May 19, 2015


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.

Farquhar Route

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.

Hop's Route from LEO to EML2

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 1.1153/2 years 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.


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.


Robert Clark said...

Your image showing a lunar gravity assist and Oberth effect burn at the Moon to lower the delta-v to get to EML2 reminded me of a question I've been thinking about.

If you use the Oberth effect at Earth on a Mars mission you can get a large benefit because the Earth has a high escape velocity of 11.1 km/s. This can significantly shorten your flight time.

However, I wanted also to shorten my flight time on the return trip. But using a similar idea of using the Oberth effect at Mars is not as effective because it only has an escape velocity of 5 km/s.

So I was thinking of using both a Phobos gravity assist and an Oberth effect burn at Mars. Here you would leave from some point in the Mars system analogous to EML2 or perhaps leave from Deimos and head towards Phobos.

Phobos has an orbital velocity around Mars of 2.1 km/s. And it is known you can get about twice the body's orbital velocity added onto your original velocity by using a gravitational slingshot effect, so this could add up to 4.2 km/s to your departure speed.

It wouldn't pay to use the Oberth effect also at Phobos because its escape velocity is so tiny. But I was thinking of combining the Mars Oberth effect at a 5 km/s escape velocity with the Phobos gravity assist boost of 4.2 km/s to get something comparable to the 11.1 km/s Oberth effect boost at Earth.

Here's how I'm thinking it might work. You leave say from Deimos and head towards Phobos getting a 4.2 km/s gravity assist. You schedule the departure so this slingshot around Phobos points you towards Mars.

But now how to calculate the Oberth effect at Mars? I'm thinking you would add on the 4.2 km/s Phobos gravity boost onto your delta-v burn and plug this into the Oberth formula. So if your burn was say 7 km/s, what you would plug into the formula would be 11.2 km/s.

Then with the 5 km/s Mars escape velocity, the Oberth formula Vinf = sqrt(2*(delta-v)*Vesc + (delta-v)^2) gives a Vinf of 15.4 km/s, more than double the 7 km/s burn speed.

Bob Clark

Hollister David said...

Bob, Phobos has a very shallow gravity help. I don't think it could be much help.

I'm not familiar with the equation in your last paragraph. Vhyperbola=sqrt(Vinf^2+Vesc^2). With a little re-arranging that's Vinf = sort(Vhyperbola^2-Vesc^2).

Delta V can vary quite a lot depending on what Mars orbit you want to enter.

Mars is a deep enough gravity well it can give a nice Oberth benefit.

Wicked_Inygma said...


Could you save further delta-v by utilizing ballistic capture?

What would the round-trip delta-v budget and and time table look like?

Hollister David said...

Wicked Inygma, the 3.11 km/s 74 day route used a form of ballistic capture as it slides into EML2. It does save some delta V.

Wicked_Inygma said...

Would ballistic capture be possible for the interplanetary space craft once it reaches Mars?

Hollister David said...

Inygma, note the orbit that slides into EML2 is near parabolic. The C3 is nearly zero.

Departing earth takes a decidely hypberbolic orbit with a Vinfinity of around 3 km/s.

Arriving at Mars from an Earth to Mars Hohmann the orbit is also decidedly hyperbolic with regard to Mars. The Vinfinity is 2.6 km/s.

The WSBs can do some interesting stuff in the earth moon neighborhood. It'd help for travel between Galilean Moons of Jupiter. Or the gas giants in the outer system can swap comets. But for travel between the small rocky bodies of the inner solar system? Not so much.

I talk about this on Potholes on the Interplanetary Super Highway

Wicked_Inygma said...

Even if the interplanetary ship does not use lunar volatiles as a fuel source it's still a good idea for the ship and the tanker to rendezvous at EML2. The tanker would be launched from Earth and most likely carrying methane and liquid O2. The rendezvous at EML2 does four things for you:

It first provides an efficient means to get most of the tanker's fuel nearly to escape velocity. Most of that fuel needs to reach escape velocity anyhow. Just a very small push from EML2 will move you to escape velocity. If the interplanetary ship were to load that fuel in LEO and head straight to Mars then it would be moving that fuel to escape velocity less efficiently. Since the tanker has several months to get to EML2 while the interplanetary ship is on its way back, the tanker can take advantage of the sun's gravity to reach EML2 very efficiently in 74 days.

Second is that because the tanker has taken on the role of moving fuel to almost escape velocity this reduces the requirements of the interplanetary ship. The tanker will require greater delta-v to do its job and the interplanetary ship will require less delta-v. Balancing the roles of the two crafts should provide more margin for both craft.

Third is that the interplanetary ship gets a bigger Oberth benefit when departing from EML2. It will drop down into Earth's gravity well on a highly eccentric orbit. The escape burn will occur when the ship is moving at 11 km/s so the Oberth benefit is substantially larger than that for an escape burn from LEO.

Finally, this trajectory allows you to use a reuseable Earth Departure Stage and have it return to EML2 after the escape burn at very little delta-v cost. This staging sheds mass off the interplanetary ship which will make it easier to do the later burns into Mars capture.

Steven Rappolee said...

I have a question that I do not know how to answer, How much Xenon would be needed to transfer 425 tons(The ISS) from LEO to EML2?
I am assuming that some of the lunar transfer orbits you discuss in your blog post would work here to
I asked this question the other night but not sure if I posted it correctly?


And a related post


Hollister David said...

Steven YellowDragon, please have patience. I screen all comments. Since I don't check my blog as often as I should, some comments languish awhile before being published.

How much of the 425 tons is power source? (Please see my post the need for better alpha). The ISS existing solar array wings wouldn't provide enough juice for Xenon engines unless you have a lot of time.

But I will assume a good power source and a good batch of Hall Thrusters. The slow ion spirals take more energy than Hohmann ellipses. I believe it'd take about 4 km/s delta V for something in LEO to reach an apogee where ballistic lunar capture is possible. I often call exhaust velocity of xenon Hall Thrusters 30 km/s although I understand this can vary depending on power source.

Exp(4/30) - 1 = .1426. And .1426 * 425 = 60.6. So you'd need 61 tonnes of xenon. But once again you also need to figure mass of power source and mass of the Hall Thrusters.

In space forums occasionally the notion of moving the ISS to a higher orbit comes up. And typically experienced aerospace engineers shoot the notion down. But I haven't paid too much attention to the arguments. I seem to recall the ISS is too flimsy to endure sustained thrust from chemical rockets. I don't know if that argument would hold for ion rockets.

Neil Jones said...

Thank you for this post. It was invaluable in making an analysis of my own. This may very well be the only place on the Internet where so much information about EML2 can be found.

By the way, this is my post, if you are curious