Last time, we took up the plight of the wayward Galaxy 15 satellite, and started looking into how we'd go about building what amounts to a space-going garbage truck. Someone's going to have to build one eventually. We may as well see how hard it's going to be. We got a good start on that last time, by figuring out we would need a delta-V (DV) budget of 20,086 meters per second to accomplish the stated mission.

This is directly related to the amount of fuel you have to burn to get from here to there and back again. About a hundred years ago, Konstantin Tsiolkovskii became interested in the mathematics of space travel, and got to work figuring out how to compute the performance of a space-going rocket. The equation he derived was:

(DV) = (Ve) * ln ( Mi / Mf )

where (DV) is the velocity change of the rocket, (Ve) is the exhaust velocity, (Mi) is the initial vehicle mass, and (Mf) is the vehicle mass at burnout. In some texts, you'll see the ratio (Mi/Mf) written as (R). The exhaust velocity (Ve) is a function of the rocket engine you're using. Depending on who you talk to and how they're feeling that day, you'll hear this equation called either the Rocket Equation or Tsiolkovskii's Equation.

This gives us enough information to compute the ratio of the initial mass to the burnout mass of the rocket, which isn't quite enough information for us to estimate a takeoff gross weight for the rocket. The burnout mass of the rocket will consist of two different kinds of things: things that have a fixed mass, and things whose mass can "float". Things that have a fixed mass are your crew, your payload, the life support system, things like that. You know how much they weigh, and that's independent of how big or small the rocket ends up being. Things that can "float" are everything else: structure, engines, plumbing. Eventually, you have to nail hard numbers down for everything, of course; but you have to start somewhere.

Now, we can expand the mass ratio computation a little bit to reflect this:

(Mi/Mf) = ( Me + Mp + MP ) / ( Me + MP )

where (Me) is the empty or dead weight, (Mp) is the mass of propellant, and (MP) is the payload mass. Here, "payload" means anything whose mass is fixed. So, if it's a multistage rocket, the "payload" is the entire upper stage.

This leaves us with a small problem: if we know values for (MP) and (DV), we still don't know what either (Me) or (Mp) are. That is, we have one equation and two unknowns. In order to go any further, we have to make an assumption about (Me). There are a few different approaches, but the easiest is to assume that (Me) is some fraction of the total stage weight. For this example, we are going to use a fraction of 0.1. This is an exceedingly optimistic value, but will serve to illustrate the point. Substituting:

(Mi/Mf) = ( Mp + 0.9 * MP ) / ( 0.1 * Mp + 0.9 * MP )

Given (DV) and (MP), we can solve for (Mp), and then we can compute the overall vehicle mass.

For this example, the fixed-mass items are:

(1) Crew cabin: 3,000 kilograms

(2) Payload: 8,000 kilograms (one satellite up, one satellite down)

So, MP = 11,000 kilograms, at least on Earth return.

GUESS 1: We're going to try to use a single stage to do the job. I basically know this won't work, but we'll try it anyway. Using DV=20,086 and MP=11,000, we throw the numbers in and ... we get a negative number for gross weight. Well, that just tells us you can't get there from here. (It's a standard engineering sanity check -- when your equations return nonsense, you've either made an arithmetic error, or you've assumed something to be possible that really isn't.)

GUESS 2: Now, we'll try a two-stage vehicle, one to put the stack into orbit, the second to get us the rest of the way there and back. This works out to approximately DV=10,000 for each stage. For the second stage, we get an initial weight of ... another negative number. So that won't work, either.

GUESS 3: We're up to three stages now. Splitting our DV up three ways gives us about 7,000 to cover with each stage. For the third stage, we get ... hey, it's a positive integer! Third stage weighs in at 134 tons. Second stage, 1,637 tons. Which gives us a grand total of 19,971 tons gross weight at liftoff.

At this point, I would like to observe that the Saturn V rocket tipped the scales at 6.5 million pounds on the pad, or 3,250 tons. It is the largest object known to have flown under its own power. So, our garbage truck tips the scales at about six times the weight of the biggest thing that's ever flown in the entire history of forever.

Yeah, that sucker ain't never gonna get built. Even

*I'm*not

*that*crazy.

We're giving up on this concept at three stages. Theoretically, there's no reason we couldn't go to four or even five stages; and the gross weight would probably come down some more. But there are two problems. One, you reach a point of diminishing returns. You don't realize enough mass reduction to make the additional complexity worthwhile. Another is operationally-driven. With three stages, each stage is recoverable and reusable, at least in principle. With a fourth stage, you're just about committed to leaving at least some hardware "out there", which kind of defeats the purpose of the vehicle.

Well, that stinks. But we're not quite done, yet.

Let's think about what it is we really need to do. We need to get to and from low earth orbit. Then we need to get to and from geosynchronous orbit. There's no law that says the same vehicle has to do

*both*.

So: we have two different vehicles. One is robotic, and runs from LEO to GEO. The other has a crew, and flies from the ground into orbit. They meet at a depot, where the shuttle drops off fuel and upward-bound satellites, and the truck refuels and takes fresh satellites up, bringing expended satellites back for disposal. Further, since the truck never needs to come back to Earth, it doesn't need a high-thrust engine, so we can use lower-thrust but higher-efficiency engines.

Let's take a first pass at RoboTruck: it carries 10,000 kg worth of fixed-mass items, and uses electric engines with an exhaust velocity of approximately 20,000 m/s. To get to GEO and back, it needs a total DV capacity of 10,000 m/s. Putting those numbers into our above equation gives us a total RoboTruck weight of right at 18 tons. Quite reasonable.

Now, our ground-to-orbit ship needs to be able to carry either a full RoboTruck, or a satellite plus some amount of fuel. Let's say, 20 tons total. It needs a DV capacity of 10,000 m/s. Putting this into our equations yields a third-stage mass of 54 tons, second-stage total mass of 115 tons, and total pad weight of 389 tons.

*This,*we can do. It's still fairly hefty, but not so much so that it beggars the imagination. All the bits are recoverable and reusable, so we aren't contributing to the debris problem. And it's conceivable with current technology. On the down side, it's sufficiently expensive that its benefit doesn't justify the cost of actually bending metal to build it.

Which brings up one last question: what's the price point where something like this

*would*be worth building? My guess is that engines would have to be much, much more efficient

*and*powerful. Say, if you were able to build a high-thrust engine that was almost as efficient as an electric rocket ... able to generate exhaust velocities of about 10,000 m/s and still be powerful enough to lift itself. Such a vehicle with three stages would weigh in at only a hundred tons or so, and a single-stage model would "only" tip the scales at 300 tons or thereabouts.

Such engines become a real possibility, once we unlock the secret of controlled nuclear fusion. But that's a story for another day ... we hope.

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