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?" 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.

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.

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.

Friday, April 3, 2015

A spiral of tethers

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 21/3. 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.