Sunday, April 5, 2015

Potholes on the Interplanetary Superhighway.

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 to bodies in the inner solar system 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 of the same textbook:

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. Here's a post entirely devoted to EML2.

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 wildly 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 boundary transfers leading from SEL2 to SML1? I don't think this particular highway exists.

To get a 1.52 A.U. 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 (though he likes to call them ballistic captures). Belbruno 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 heliocentric 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!

What About Ion Engines?

"What about ion engines?" a Belbruno defender might object. "They don't have the thrust to enjoy an Oberth benefit. So Belbruno's .7 km/s benefit is legit if your space craft is low thrust."

Belbruno & friends are looking at a trip from a nearly zero earth C3 to a nearly zero Mars C3. In other words from the edge of one Hill Sphere to another.

So to compare apples to apples I'll look at a Hohmann from SEL2 to SML1. I want to point out I'm not using Lagrange necks as key holes down some mysterious tube. They're simply the closest parts of neighboring Hill Spheres.

"Wait a minute..." says Belbruno's defender, "We're talking Hall thrusters. So no Hohmann ellipse, but a spiral."

Low earth orbit moves about 4º per minute. So a low-thrust burn lasting days does indeed result in a spiral. But Earth's heliocentric orbit moves about a degree per day while Mars' heliocentric orbit moves about half a degree per day. At this more leisurely pace, a 4 or 5 day burn looks more an impulsive burn. The transfer between Hill Spheres is more Hohmann-like than the spiral out of earth's gravity well.

Instead of a 1 x 1.524 AU orbit, the new Hohmann is  a 1.01 x 1.517 AU ellipse. The new Hohmann's perihelion is a little slower, the new aphelion a little faster.

Moreover, SEL2 moves at the same angular velocity as earth. So it's speed is about 101% earth's speed. Likewise SML1 moves at about 99.6% Mars' speed.

With this revised scenario, aphelion rendezvous delta V is now more like 2.4 km/s. Still, Belbruno's 2 km/s capture burn saves .4 km/s.

.4 km/s is better than chopped liver, right? Well, recall ion engines with very good ISP. I'll look at an exhaust velocity of of 30 km/s.

e2.4/30 - 1 = .083
e2/30 - 1 = .069

So given a 100 tonne payload, rendezvous xenon is 8.3 tonnes for Hohmann vs 6.9 tonnes for Belbruno's ballistic capture.

108.3/106.9 = 1.0134

We're adding a year to our trip time for a one percent mass improvement? Sorry, I don't see this a great trade-off.


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.

I'd like to thank Mike Loucks and John P. Corrico Jr. I've held these opinions for awhile but didn't have the confidence to voice them. Who am I but an amateur with no formal training? But talking with these guys I was pleasantly surprised to find some of my heretical views were shared by pros. Without their input I would not have had the guts to publish this post.


Magico said...

Found this, which I believe is trying to explain the idea.

Have to confess, that I'll need to read through it again before I get my head around it properly.

Hollister David said...

Magico, as I said above, there are low energy paths between the earth-moon and sun-earth Lagrange necks. But first from LEO you must invest 3.1 km/s (as mentioned in the paper you cited) to get the orbit near parabolic.

What I question is the existence of manifold tubes from the sun-earth Lagrange necks to the sun-Mars Lagrange necks.

The highways described wind about in the upper regions of earth's Hill Sphere. Where are the highways between planets?

Michael Driscoll said...

When reading your article the first thing that came to mind was the Aldrin Cycler. It has a period of n times 2.135 years. I don't understand orbital mechanics well enough to discuss it, but the wikipedia article has links to promising-looking analysis.

Hollister David said...

Michael, I might take a look at Aldrin Cyclers at some point. But they're not relevant to this post. They don't employ WSBs to or from L1 or L2 necks. And they take quite of bit of energy to establish.

People seem to conflate the so called superhighways with a variety of trajectories.

Anonymous said...

Here's your "beef":

Hollister David said...

Sorry, Anonymous. No beef there.

Under the section Link Hypothesis, Topputo and friends write " Unfortunately, the R3BPs involved in this study, due to their small mass parameters, do not allow the manifolds to develop far enough to approach each other; hence in this case the manifolds do not intersect even in the configuration space."

That agrees with my observations.

Their favorable comparisons to Hohmann paths are misleading in that they ignore the possibility of exploiting the Oberth effect.

For example, let's look at going from an 165200 km circular earth orbit to a 123760 km Mars orbit (as shown in their table 4).

From a 165200 km orbit, a 1.12 km/s braking burn drops the ship to a 6638 km perigee. From there .618 km/s suffices for Trans Mars Insertion

At Mars .7741 km/s captures the ship into an 3697 km x 123760 km Mars orbit. A .447 apoaerion burn circularizes the orbit at 123760 km.

Add those 4 burns together and you get 2.96 km/s. Their path takes 3.97 km/s. That's about 1 km/s flushed down the toilet.

glenn robinson said...

At first glance, to the casual reader (most of us), the ITN sounds like a panacea. However,, you have to climb the Sun's potential hill if you are going outward. Traveling inward, coasting downhill, there are probably many ways to get most places, so long as you don't get too near a gravity well of an unwanted destination.

I never could understand why one would spend a fortune to escape one gravity well just to get trapped by another. Best situation of all is to stay away from the planets and just live and work among the minor bodies. IMHO, the only project that makes long term sense is to invest in a solar thermal economy in interplanetary space. The energy is free, the real estate is vast, and you can shoot surplus valuables downhill to earth to pay off the mortgage.

Until they invent anti-gravity, the planetary missions are pretty much one way and have no financial prospects.

Hollister David said...

Glenn, not sure what you mean by going down hill. Going down from a 1.5 A.U. circular orbit to a 1 A.U. circular orbit costs the same as goiong up for a 1 to a 1.5 A.U. orbit.

Down hill planetary orbits can take less delta v if you have the option of aerobraking at periapsis.

I agree with you on the minor bodies. If we make it to the Main Belt there's all kinds of potential for growth.

Chris said...

I haven't done my homework in orbital mechanics, but, would it be possible to turn a circular orbit into a highly elliptical obit with the same energy, but lower perihelion and higher aphelion? Thrusting sideways? Then aero/lithobrake at the destination? I think I would need to write my own Python or Matlab solver for this.

Hollister David said...

Chris, a circular orbit has zero flight path angle. That, velocity vector points in direction of local horizontal. A burn up (away from central gravitating body) or a burn down would change flight path angle. Such a burn combined with a retrograde component could result in a new velocity vector with the same magnitude. You would have a new elliptical orbit with the same semi-major axis and orbital energy. But the burn I describe takes energy. Starting from a 1 AU radius orbit doesn't give you a free ticket to a .5 x 1.5 AU orbit.

LocalFluff said...

What about airless bodies, like Mercury and Ceres, or for a spacecraft which is not suitable for aerobraking, does the "superhighway" save delta-v then? Or is using the Oberth effect alone enough to trump it?

Hollister David said...

Mercury or Ceres are also beyond the limits of near parabolic paths from SEL1 or SEL2. This would be true whether or not these bodies had an atmosphere.

Being smaller bodies, they have an even smaller mu. Mu in 3 body scenarios is (mass orbiting body/(mass orbiting body + mass central body). So the Sun Mercury and Sun Ceres L1 and L2 are even closer. Ceres' L1 and L2 are only 224,000 im for Ceres. A body nudged from the Lagrange necks would follow orbits nearly indistinguishable from Ceres.

Magico said...

Are there many Near-Earth asteroids, whose orbit path is close enough, that it could be accessed via this method?

That, at least, would allow potential ISRU refueling to continue on to a second target.

Other than that, I believe you're right regarding the Superhighway limitations.

Shane Ross said...

Hollister, yes, the superhighway definitely has its limitations. It was inspired by comets, which have plenty of time, but of course we don't for planning space missions (unless we take Deep Time approach).

I suspect that gravitational corridors which naturally connect Earth-bound orbits with Mars-bound orbits using no-fuel do exist, but might take a long time to achieve (I'm thinking thousands of years of flight time between the two planets). But using some propulsion, we may be able to reduce that time to several decades. This is probably still too much time for a mission, especially a manned mission, so for now, conventional approaches will probably continue to be used.

It would be an interesting study to try to map out the Earth-Mars natural connections. It would have to involve the concept of multiple gravity assists, or consecutive gravity assists, as discussed in

Shane Ross, 'evangelist' :)

Hollister David said...

Shane Ross, thank you for your comment.

Can comets ride the ITN between gas giants? Sure.
Can ships ride between Galilean moons? Yes.
But from Earth to neighboring planets? I am skeptical.

Here are some mass parameters:
Earth/Moon 1.22E-2
Sun/Jupiter 9.55E-4
Jupiter/Ganymede 7.79E-5
Sun/Neptune 5.15E-5
Jupiter/Europa 2.53E-5

Now some much smaller mass parameters:
Sun/Venus 2.45E-6
Sun/Earth 3.04E-6
Sun/Mars 3.23E-7

Let's look at using earth gravity assists to boost an orbit that's drifted from sun-earth L2. SEL2 has earth's angular velocity and is 1.01 A.U. from the sun. An object nudged from earth's influence will rise into an orbit having a semi major axis of 1.04 A.U. and have a period of 1.06 years.

Synodic period is 17.1 years. So we don't get an earth fly by until after 17.1 years at which time the object is 36º from perihelion. To get a fly by at the object's perihelion we'd need to wait 171 years.

Even with flybys at perihelion, the assist is fairly anemic when the object's .1 AU from the earth. If multiple assists could raise the aphelion to 1.52 A.U., it could take millennia. But I'm not even sure millennia is sufficient.

And at Mars' end we have an even tinier mass parameter.

Hollister David said...

Magico, I agree. If an NEO has a near parabolic trajectory, it can ride WSBs to take a barely hyperbolic trajectory to a barely elliptical orbit about the earth. I am hoping to see Planetary Resources or Deep Space Industries use the work of Ross and friends to park some resources in the Lunar Hill Sphere.

Chris said...

Long story short, outside the Earth-Moon system, the ITN doesn't offer anything over classical Oberth maneuver and gravity assist.

It seems to me, that the deltaV needed to move around the inner Solar system aren't so large anyway and the biggest problem is still in getting things from Earth to LEO.

I was thinking, what if we put a satellite in LEO, with a scoop to collect the tenuous atmospheric gases to use in an electric thruster, powered by solar arrays. If the satellite would be sufficiently heavy, maybe it could be used to lift payload out of Earth gravity well my momentum transfer, e.g. by a tether.

Hollister David said...

Chris -- Yes. Gradual accumulation and sudden release -- a capacitor would be an apt metaphor for a tether equipped with a high ISP/low thrust engine. But tethers are a little off topic for this post. Please feel free to comment on A Spiral of Tethers or Beanstalks Elevators, Clarke Towers

Hollister David said...

Besides the delta V to get to LEO, there's another large chunk -- delta V needed to get a high apogee, i.e. close to escape. This takes about 3.1 km/s from LEO.

The Genesis probes are often cited as a success story for ITN. But the ITNs were exploited after ~3.1 km/s had already been invested to give these probes a high apogee.

Shane Ross said...

There was a simulation study a couple decades ago where they looked at the possibility of ejecta from Mars (released due to asteroid impact) could reach Earth (at which point we'd call them 'meteorites').

Ejecta are often released from the planet with just barely enough energy to escape the planet's gravity, so they would be 'under the influence' of Lagrange point-style dynamics.

The study found that about 7% of the Martian ejecta made it to Earth within 10 million years. They also found that the fastest transfer was 16,000 years.

So this gives me confidence that a 'free' transfer is possible from a barely bound Mars orbit to a barely bound Earth orbit, but the travel time will be 10,000+ years.

"The Exchange of Impact Ejecta Between Terrestrial Planets", Gladman, Burns, Duncan, Lee, Levison : Science 271 (1387) [1996]

Shane Ross said...

Thanks for all the comments here. It's very stimulating. And by the way, the two-body approach to trajectories still reigns supreme. As someone who was in the trenches defending the three-body approach, I can tell you that the three-body approach is the one considered heresy. I often hears comments from NASA engineers like, "Really? We're going to use chaotic dynamics for spacecraft?"

In any case, I completely agree that the nonlinear dynamics of the three-body problem (which is all the 'ITN' is meant to invoke) is no panacea and much of the effort to get anywhere in the solar system is spent just getting into LEO, or to the edge of the Earth's gravity well.

The interesting *mathematical* question at that point is: Once at the edge of Earth's gravity well, where can I go? If I slightly change my initial conditions, or make slight course corrections, and if I have plenty of time, where can I direct a small mass?

Also, to clear up some confusion on the fuel consumption vs. time-of-flight trade-offs, here's a graphic I've used to illustrate the concept of low-energy paths that take a long time

The trajectories are shown in the Earth-Moon rotating frame. At the high fuel, low time limit is the Hohmann transfer. But if you get multiple gravity assists, the fuel consumption goes down, but made up for with more time. Note that these gravity assists are OUTSIDE the moon's sphere of influence, so are not the traditional two-body, patched-conics gravity assists. They involve the nonlinear dynamics of the three-body problem. Notice that there seems to be a theoretical lower limit to the fuel usage.

An analogous plot can be made for a transfer between moons of Jupiter, like here

Again, the gravity assists here are not the typical 'swingbys' as they are outside the sphere of influence of either moon.

A video version of a specific transfer is here:

Notice how the eccentricity of the spacecraft increases then decreases, while the entire time the semimajor axis is changing.

Now, I suspect an analogous plot of fuel vs. time could be made for a transfer between Earth and Mars (I haven't done this, but it's a good idea). Even though the mass parameters are small, the phase spaces may be connected -- it just may take thousands of years--like the ejecta from the ejecta exchange paper.

Anonymous said...

This might annoy you, not from a theorist but an "educator":

From the transcript:
"The idea of network of gravitational currents and flows, running throughout the solar system, shouldn’t really come as a surprise."

Hollister David said...

I wish folks would identify themselves as replying to "anonymous" can be ambiguous if there's more than one person posting anonymously. A nickname would suffice.

I had received an earlier message about this podcast. I didn't want to publicly criticize until I contacted Steve Nerlich.

Steve replied:

Hi Hop David,

Thanks. I read your blog, which I'm thinking might have come out around the time there were some spurious notions going around about using the Belbruno ballistic trajectory idea for a manned Mars mission - possibly mis-inspired by this Sci-Am article:, which did not suggest it was a solution for a manned mission. I don't have a strong opinion either way about the Mars ballistic trajectory idea. As I said to a recent critic, when we send people to Mars it will be via a Hohmann transfer orbit.

I think both Ross and Belbruno have been taken way out of context by alleged 'supporters', who have sometimes extrapolated their ideas to ridiculous extents. In writing my script I sought to make it clear you can't go uphill from a dead-stop (in a gravity-well sense) without burning fuel, but at a Solar System level you can gain velocity from gravity-assist manoeuvres (which do impart more energy towards the bottom of the well). The space-time 'flatspots" of LG points offer some (relatively-minor) opportunities to conserve fuel along the way and you can at least coast in a heliocentric orbit with the objective of finding a more advantageous point from which to depart one orbit to go to another (which will require fuel).

I agree that getting from Sun Earth L2 to Sun Mars L1 needs fuel. I was talking about how getting first from Earth L1 to Earth L2 in a long outward Solar System scale trajectory might represent a fuel-efficient interplanetary passage (assuming you started from Venus, say)

The whole thing is based on fairly knife-edge efficiency gains. The Hiten and ICE missions, with their crazy circuitous trajectories, suggest there is something in the Ross/Belbruno way of thinking, but of course both were re-purposed missions, using spacecraft that had expended most of their fuel in completing their primary missions. As I also said to a recent critic, there's no way you are going to purposefully launch a mission that will take decades or centuries to complete, for the sake of saving a bit of fuel. Fuel isn't that hard to come by.

My podcast was largely an attempt to communicate how space-time geometry and gravity-assist manoeuvres might be used to some advantage in space-travel. I think that is the ITN-idea in a nutshell. It is based on knife-edge efficiency gains, requiring huge amounts of time to complete - and so is, by-and-large, an impractical solution to space-travel.



Listening to his podcast I believe he reinforces the notion that the ITN is a panacea. Perhaps that wasn't his intent. Evidently Steve knows the ITN's not practical for travel from earth to neighboring planets.

As I said, I'm more enthusiastic about using these devices in the earth moon neighborhood. I am working on a blog post titled "EML2"

Robert Walker said...

Hi Hop David,

I've been alerted to this page by Standing Space on my Science20 blog. You might like to read the discussion there.

It's my understanding that the ITN is a sequence of patched together 3 body solutions. Same idea as the Voyager gravity assists, but they only used patched together 2 body solutions so they could only find a small subset of the possible solutions. By using 3 body solutions instead they got a much wider range of gravity assist type solutions - with the patched together 2 body cases as special cases of the more general ITN.

Also - it doesn't use Earth's Moon for gravity assist to get to Mars or wherever. It only uses the Moon as a 3 body Earth Moon Spacecraft (ignore the sun) to get from the Earth Mon L1 to the Sun Earth L2. Because it uses patched together 3 body problems - it then ignores the Moon from then on. You are in the same situation you would be if you got to L2 directly from Earth with a big delta v boost, but it just took less fuel to get there because of the use of the Moon.

For the long timescale orbits from Earth to Mars it uses repeated flybys of Earth. Which you can do with a slow departure from EL2, so you spend a fair while in vicinity of Earth so that gives options to flyby Earth several times so boosting each time.

If a particle just reaches escape velocity from Earth - or from Mars - it can get from one to the other over timescales of centuries or more likely millions of years. That's how the Mars meteorites get from Mars to Earth. Once they reach escape velocity from Mars, then eventually they will all eventually, within about 20 million years, end up either back on Mars or hitting Earth, or they can hit the Sun, various other possibilities. Just through multiple flybys. After 20 million years approximately all the debris is cleared out, and some of it impacts Earth in that timescale. That's how we get meteorites from Mars on Earth, about 1 meteorite in 100 is from Mars.

With the ballistic transfer - that's just a reduction of delta v - and it arrives in Mars orbit a bit ahead of Mars. So then as Mars gradually catches up with it, it then gets captured into a distant orbit around Mars. So no need for delta v for capture by Mars, so that's the main saving there - at the Mars end. And you can also depart Earth for Mars at any time not just for the Holmann transfer launch window. At cost of taking several months longer to get to Mars. And then once you are captured in a distant orbit around Mars in that way - you still need the delta v to get down to a low orbit - but - the big plus there is that the delta v can be applied over any timescale you like. You don't have one single critical big delta v that you have to get right to get into orbit around Mars. And you can also use gentle thrust. With that method you could use an ion thruster to first get captured by Mars and then slowly spiral down to whatever orbit you are interested in around Mars.

Hollister David said...

Robert, the reason I looked at the earth moon system is to compare the distance of L1 and L2 from the central body. SEL1 and 2 are 99% and 101% of an A.U. In contrast EML1 and 2 is more like 5/6 and 7/6 of an L.D. The range of paths emanating from an L1 or L2 neck are much less dramatic when the μ is only .00000304.

An 1 A.U. x 1.52 A.U. ellipse has Vinfinities of 3 km/s (earth departure) and 2.6 km/s

Between SEL2 and SML1 is a 1.01 A.U. x 1.51 A.U. ellipse. Vinfinities are about the same. The big difference is you have zero Oberth benefit. From LEO i takes around 3.1 km/s to get to SEL2 and then it'd take another 3 km/s for Trans Mars Insertion for a total of around 6 km/s just to get you on your way to Mars. TMI directly from LEO is around 3.6 km/s.

A Mars ballistic capture is doable if the path is near parabolic with regard to Mars. So you'd need a low eccentricity heliocentric ellipse with a semi major axis in the neighborhood of 1.52 A.U.

If the heliocentric transfer ellipse has a perihelion in the neighborhood of 1 A.U., the path will be decidedly hyperbolic with regard to Mars. Ballistic capture isn't doable.

Robert Walker said...

Hop - I've found out more about the ballistic transfer method. The original paper is here: Earth--Mars Transfers with Ballistic Capture.

They say in that paper that it's already been used for three missions from Earth to Moon, for capture into lunar orbit with much less delta v than you'd need for Holman transfer.

The way they explain it there is that the three body interactions cause the v∞ of your spacecraft relative to the Moon to change from positive to negative - while for Holman transfer of course it has to remain positive throughout until you apply the thrust for more delta v.

The spacecraft get captured into a temporary orbit around the Moon. A bit like the way that sometimes NEAs get captured into temporary orbit around the Earth for a few months.

For those also, the v∞ of the NEA relative to Earth has to change from positive to negative, which would be impossible with patched together 2 body interactions.

It's a temporary capture, would eventually leave that orbit, so you have to use some delta v to spiral down to a lower stable orbit.

But that extra delta v is a lot less than the amount needed for Holman transfer because your spacecraft already has a negative v∞

So, they are saying they've found a solution that works in the same way for Mars.

(I commented on the article as well, repeating same comment here as you may have missed it):

Hollister David said...

Robert, you link to Belbruno and Toppotu's paper.

You evidently didn't notice the part of this post subtitled Mars ballistic capture by Belbruno & Toppotu. I link to their pdf in the opening paragraph of that section.

And once again, some mass parameters:

Earth/Moon 1.22E-2
Sun/Jupiter 9.55E-4
Jupiter/Ganymede 7.79E-5
Sun/Neptune 5.15E-5
Jupiter/Europa 2.53E-5

Now some much smaller mass parameters:
Sun/Venus 2.45E-6
Sun/Earth 3.04E-6
Sun/Mars 3.23E-7

The earth-moon mass parameter is about 4,000 times greater than the sun-earth mass parameter and about 38,000 times greater than the sun-Mars mass parameter.

And the ballistic savings of lunar missions are somewhat exaggerated. I talk about the earth moon system in my post EML2

Robert Walker said...


Sorry. I did read that section but hadn't made the connection, forgot the names of the authors and didn't realize this was the same paper.

Looking more closely at that paper, I wonder if it's been misreported. (And if this is right then I need to update my article also to reflect it).

In section 6, they find that it is better than Holman transfer for orbits down to a periapsis radius of a little under 30,000 km. That's twice the distance of Deimos.

And the actual savings summarized in table 3, and they say saving is about 25% if you want to transfer to an orbit that has closest point to Mars of 200,000 km - of course rare that you want to do that.

The news stories have summarized this as saying that the ballistic transfer gives a 25% saving over Holman, but as you say, for a close orbit, the Holman gives a saving over ballistic.

I'm still interested in ballistic if this is correct, because of its planetary protection value - that it means there's no chance of hitting Mars as a result of applying too much delta v due to a miscalculation or misfiring rocket for the insertion maneuver. Plus ability to send materials to Mars at almost any time, not just every two years.

Hollister David said...

Robert, paths from Sun Earth L2 can easily reach aphelions of 1.07 A.U. And a lunar swing by can drop a hyperbola with 1 km/v Vinfinity to earth capture orbit. So there's a huge number of NEAs that can be captured to the earth moon system. This has implications not only for earth defense but also using asteroids as resources. With a few tiny, well timed burns it's practical to park an NEA with a C3 just above zero to an earth capture orbit with a C3 just below zero.

But while I'm enthusiastic about ballistic NEA capture, I still say using WSBs to Mars isn't practical.