This is the second cartoon delta-v map I've drawn. Clicking on the above can give a larger version.

My first cartoon map gave a lot more space to EML5 and little to L1 or L2. I knew of L4 and L5 through fiction like Gundam which was probably inspired by Gerard O'Neill's

*The High Frontier*. Since then I've become less interested in L4 and L5 and more interested in L1 and L2. This new map reflects that shift in focus.
I had heard of the Interplanetary Transport Network as well as Shane Ross, Martin Lo, and Edward Belbruno. But I knew almost nothing about the low delta V routes achieved with n-body mechanics. I had a vague notion that Lagrange points were involved but that was about it.

Then in 2009 I came across a thread in Nasa Space Flight entitled An Alternative Lunar Architecture. In that thread Kirk Sorensen wrote at length about EML2 and work done by Robert Farquhar. Farquhar's 3 body work was done in the late 1960's and early 70's, decades before Ross, Belbruno and other modern advocates of 3-body mechanics.

Here's one of the Farquhar graphics Sorensen posted to that thread:

This is a 9 day route from LEO to EML2 taking delta V of about 3.5 km/s. It's time reversible so .4 km/s can drop a payload from from EML2 to an atmosphere grazing perigee. There's a 4 day route to EML1 that takes 3.8 days. It was surprising to me that EML2 could be reached with less delta V even though it's on the far side of the moon.

Also on Nasa Spaceflight in 2009, Martijn Mmeijering started a thread looking at an online 3 body text book by Koon, Lo, Marsden and Ross.

It was in 2009 that I became more interested in L1 and L2.

There are routes between LEO and EML1&2 taking even less delta V, but these are time consuming. These are described by Andreas Stock's I

*nvestigation on Low Cost Transfer Options to the Earth-Moon Libration Point Region*. 3.1 km/s seems to be the minimum between LEO for EML1 as well as EML2. In the map above I've depicted these routes with darker brown branches.
EML1 moves slower than an ordinary earth orbit at that altitude. An EML1 object nudged a little earthward will fall into an approximately 100,000 by 300,000 km elliptical orbit about the earth. A .3 km/nudge suffices to send to it to a 36,000 km perigee where a 1 km/s burn can circularize the payload at geosynch orbit. A .7 burn can drop an EML1 payload to a LEO grazing orbit. If the LEO grazing orbit passes through the upper atmosphere, aerobraking can provide the 3.1 km/s needed to circularize at LEO.

EML2 moves faster than an ordinary earth orbit at its altitude. Nudged a bit away from the moon, a payload from EML2 will sail to a 1.8 million km apogee. The Sun Earth Lagrange 1 and 2 are 1.5 million kilometers from earth, so transfer from EML2 to SEL1 or 2 can be done with little delta V. Or an EML2 payload can sail through SEL1 or 2 completely out of earth's sphere of influence.

Nudge either EML1 or 2 a little moonward and they will fall into an approximately 5,000 x 60,000 km lunar orbit. Since the moon's rotating about the earth, a 60,000 km apolune can pass by both EML2 and EML1 over time. Thus it's possible to move between EML1 and 2 with very little delta V.

An object falling to a 300 km earth altitude from either EML1 or 2 will be traveling just a hair under escape when it reaches low altitude. Both EML1 and 2 make a complete circuit each 27.3 days so by timing your drop it's possible to choose longitude of perigee during a launch window. Plane changes are much less expensive at high altitudes so the velocity vector can be pointed in the right direction at perigee. Starting at EML2, injection into Mars or Venus Hohmanns can be done with around .9 km/s delta V (.4 km/sec to drop and a .5 km/s burn at perigee). On arriving at Mars, .7 km/s suffices to exit Hohmann for a 300 x 570,000 km Mars capture orbit.

A Near Earth Asteroid with a Vinfinity of 2 km/s or less can be dropped into an earth capture orbit using a lunar swing by. From there repeated lunar swing bys and little delta V can park the rock in high lunar orbit. Planetary Resources hopes to search for smaller rocks with their Arkyd orbital telescopes. If successful, they will likely find a multitude of rocks within .2 km/s of EML2.

Many NEAs are water rich and the cold traps at the lunar poles may have minable water deposits. So there are a number of potential propellent sources close to the earth-moon L1 and 2. Propellent sources high on the slopes of earth's gravity could break the exponent in Tsiolkovsky's rocket equation. This would give us mass fractions much easier to deal with. The highest delta V budget we'd have to endure is the 9.5 km/s from earth to LEO. Round trips between most other orbits would be in the neighborhood of 4 or 5 km/s.

Some notes on Venus: An earlier version of this map indicated delta V from Venus' surface to a capture orbit was 11.6 km/s. But then I came across an excellent delta V map by a fellow who calls himself Curious Metaphor. In a discussion of his map, I was convinced Venus' thick dense atmosphere would make for a slower, steeper ascent from the planet surface. I've added 20 km/s gravity loss between Venus' upper atmosphere and surface. In Venus' upper atmosphere I've added a location labeled

*Landis Land*. Named for Geoffrey Landis who noted there is a layer in Venus' atmosphere with earth like temperature and pressure. Moreover, a nitrogen/oxygen mix such as we breath would be buoyant in Venus' CO2 atmosphere. Landis is a scientist as well as a science fiction writer. Some of his fiction takes place on the cloud cities of Venus.
## 4 comments:

I really like this map. Do you have a copy of Andreas Stock's paper Investigation on Low Cost Transfer Options to the Earth-Moon Libration Point Region? The link appears to be broken now. If you do, please send me a copy at peter.m.mcarthur@gmail.com

In your "Inflated delta V's" post you gave the delta-v between highly elliptical orbits from Earth to Mars as 1.1 km/s. However, here to go from EM2 to a highly elliptical orbit around Mars is given as .9 + .7 = 1.6 km/s.

I would have thought to leave from EM2 would cost less delta-v since it is further out of Earth's gravity well.

In any case, it is interesting that to go from EM2 to a Mars transfer orbit, the required delta-v is so small, less than 1 km/s, whereas from LEO it's about 4 km/s.

Bob Clark

Bob, the capture orbits for departure are a best case scenario that would seldom (if ever) occur in the normal course of things.

For arrival, capture they can be used routinely, so long as peri-apsis passes through the upper atmosphere.

Following Prussing and Conway's method of Sphere of Influence (SOI), earth's SOI would be about 1,000,000 kilometers from earth's center. This would be apogee. Perigee would be 300 km, just above earth's atmosphere.

EML2 follows a roughly circular path 450,000 km from earth's center. If nudged loose from the moon's influence, an object at EML2 would fly to a 1,800,000 km apogee, beyond SOI. It is quite likely the sun would tear the object from earth's influence into a heliocentric orbit. Or if it does descend back to the earth, It would take around 4 months to return to a perigee where the Oberth benefit could be enjoyed.

So for routes from EML2 I use Farquhar's path to a perigee deep in earth's gravity. It takes a little more delta V but is still quite good.

Nydoc, I have sent you my copy of that pdf. I have also written the school asking permission to upload that pdf and make it available for download.

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