Monday, May 24, 2010

Space Travel for Dummies, Part 6: Design Case Study


One of our satellites is missing. Sort of. Life might actually be easier if it had gone missing, but it's more like it's suddenly decided that it's done with its current job and wants a new career. Maybe it wants to be a sculptor. Or a dentist. Or just about anything except what it was built to do, which is sit in geosynchronous orbit and relay TV signals. I can't really blame it. If I had to hand off Survivor reruns for a living, I'd go nuts, too.

Galaxy 15 was, for five years, a perfectly unremarkable satellite. A standard product, almost identical to its siblings, it had functioned perfectly for nearly five years before wandering off to stretch its legs. No one really knows why, since it's no longer on speaking terms with ground control. But it's rapidly becoming a menace to navigation. At or around the end of May it's going to come close by the AMC-11 satellite, which is going to have to do some fancy dancing to avoid signal interference. Where Galaxy 15 goes from here is anyone's guess.

What we really need is a garbage truck. Unfortunately, no one seems to have built one yet. Which gives me an idea...

Over the next couple of weeks, we're going to go through a very basic, somewhat simplified vehicle design procedure. I honestly have no idea exactly what we'll end up with. But I expect to end up with something sufficiently detailed that I can "build" and fly it in Orbiter.

The first step is figuring out what kind of performance you'll need. This part isn't terribly difficult, as mathematics goes. Basically, you can figure out how much of a velocity change you need for each maneuver, and then add them all together to get a total performance budget.

To summarize: there are three equations we'll need, and one rule of thumb. Two of them involve the specific mechanical energy (SME) of the spacecraft. The SME is the sum of the spacecraft's kinetic and potential energy, divided by the mass.

(1) (SME) = (V)^2/2 - (MU)/(R)

V is the velocity, R is the distance from the spacecraft to the center of the Earth, and MU is the product of the Earth's mass and the universal gravitational constant. We can express the SME another way as well.

(2) (SME) = - (MU)/(2*A)

The variable A is the semi-major axis of the orbit. It is half of the sum of radius at perigee and the radius at apogee, and for a circular orbit it's equal to the constant radius of that circular orbit. Combining (1) and (2) above gives us three pieces of information: V1, the circular velocity of the parking orbit; V2, the required velocity of the low side of the transfer orbit; V3, the velocity at the high side of the transfer orbit; and V4, the velocity of the higher circular orbit.

It isn't necessarily obvious from (1) and (2) above, but one interesting thing about orbital mechanics is that even though higher-altitude orbits have more energy, the actual orbital velocities are smaller. Remember this, we'll use it later.

The third equation is the Law of Cosines:

(3) (Vc)^2 = (Va)^2 + (Vb)^2 - 2*Va*Vb*cos(G)

Here, Va and Vb are the velocities before and after a plane-change maneuver, G is the angle, and Vc is the required velocity change to go from Va from Vb. Note that if you're going from one circular orbit to another, Va = Vb. Plane-change maneuvers are incredibly expensive in terms of fuel. They are often unavoidable, though.

The rule of thumb I mentioned is that, to a good first approximation, launching into low Earth orbit requires as much fuel as accelerating 10,000 meters per second. Orbital speed is actually about 7,700 m/s, but you burn about 2,300 m/s worth of fuel overcoming gravity and atmospheric drag.

That said, the details of those computations make for tedious reading, so I'll simply present the results.

First, the ground rules. One: the vehicle will launch from, and recover to, Kennedy Space Center in Florida. Two: the vehicle will have a return payload of 8 tons, big enough to handle any current or projected communications satellite. Three: provision will be made for a crew of two, a commander and a pilot.

My first sequence of maneuvers went something like this:

(1) Launch into Low Earth Orbit (LEO): 10,000 m/s
(2) Plane Change into Equatorial Orbit: 3,804 m/s
[Loiter: up to 24 hours]
(3) Boost to Geostationary Transfer Orbit (GTO): 2,426 m/s
[Time of flight: 5h 16m 30s]
(4) Circularize at Geosynchronous Earth Orbit: 1,465 m/s
[Time on station: up to 13h 27m]
(5) Brake into GTO: 1,465 m/s
[Time of flight: 5h 16m 30s]
(6) Brake into LEO: 2,426 m/s
(7) Plane Change to KSC: 3,804 m/s
[Loiter: up to 24 hours]
(8) De-Orbit: 65 m/s

Total: 26,727 m/s (includes 5% reserve)

Now, the first thing we see here is that the plane change maneuvers eat up a huge part of our velocity budget. This is because the magnitude of a plane change maneuver scales directly with the velocity. They drink gas like nobody's business, which is why you avoid them whenever possible. However ... why am I doing it in a low orbit, when I could be doing it higher up, where the speed is lower? Partly because this first method was the obvious way ... and partly because I'm not sure I trust my ability to actually fly a fancier maneuver. Still, here's how the pros do it:

(1) Launch into LEO: 10,000 m/s
[Loiter: up to 24 hours]
(2) Boost to GTO: 2,426 m/s
[Time of flight: 5h 16m 30s]
(3) Plane Change and Circularize: 2,106 m/s
[Time on station: up to 13h 27m]
(4) Plane Change and Brake to GTO: 2,106 m/s
[Time of flight: 5h 16m 30s]
(5) Brake into LEO: 2,426 m/s
[Loiter: up to 24 hours]
(6) De-Orbit: 65 m/s

Total: 20,086 m/s (includes 5% reserve)

By combining the plane change and circularization at the top of your GTO ellipse, you save a whole 6 kilometers per second. That's huge. That's a tremendous amount of fuel you don't have to plan on hauling along with you. And when most of your weight is fuel, that adds up in a hurry.

Some of the times above are somewhat arbitrary. You're going to spend some time in low earth orbit waiting for the right launch window, and you're going to spend some time on your way back waiting to line up for re-entry. And you may or may not want a rest period in between your outward and inward GTO orbits. So, I'm planning on a nominal mission duration of three days. The absolute minimum would probably be 12 hours, but everything would have to line up perfectly. I'm going to budget a reserve here too, and allow for an extended duration of up to 7 days.

Here's the first fork in the decision tree: do we have a crew on board, or not?

There are good arguments both ways. On the one hand, hauling a crew along means you also have to haul along the stuff to keep them alive for the duration: food, water, oxygen. I was about to mention thermal control as well, but you'd have to do that for your electronics even in an unmanned craft. But an unmanned craft would be much, much lighter for the same mission specs.

But, since the point of this exercise is to design a simulator joyride anyway ... I've opted to include pilots. At least, for now. When we pick this up again next time, we'll see why orbit-capable rockets tend to be pretty big.


John said...

Can you go over this one more time, and this time, tell me in such a way that I can make my five year old daughter understand. She really does have an interest in what you are saying, but she is so simple sometimes.


Tim McGaha said...

I've added some navigation links to earlier posts of a more general nature. I'd meant to do this for some time, but hadn't gotten around to it yet.