A space shuttle launch is pretty expensive. Exactly how expensive depends on who you ask, but if you divide the yearly cost of the program by the number of launches you get something in the neighborhood of half a billion dollars. That's actually pretty trivial by federal budget standards, but it's still not chump change. It's a measure of just how difficult it is to get to space. Let's try to put that number in perspective by running a few numbers describing the cost of energy. Think of it as a Fermi problem.

An orbiting shuttle has potential energy by virtue of its height above the ground. It also has kinetic energy by virtue of its very considerable orbital speed. It's a one-line calculation to figure our how much once you look up some numbers.

Potential energy is m*g*h, which for typical shuttle takeoff weights and operating altitudes gives something like 2.13 x 10^{11} J. The acceleration g due to gravity does vary with altitude, but at shuttle altitudes the variation is small enough that we can overlook it for this rough of a calculation.

Kinetic energy is the usual mv^{2}/2. Plugging some more fairly typical figures I come up with a kinetic energy of 3.27 x 10^{12} J. The kinetic energy thus dominates the total energy, which is by adding the two is about 3.38 x 10^{12} J.

Here's the interesting comparison. Here in Texas, electricity costs about ten cents per kilowatt hour. How much would a space shuttle launch worth of energy cost at that rate? Well, there's 3,600,000 J in a kilowatt hour, so doing the division we see that much energy costs about $90,750.

Obviously this is a grossly unfair comparison. Evewn simple technology like an incandescent light bulb turns only a very small fraction of its electricity into visible light. Launching a shuttle is not every remotely a matter of just turning electricity into altitude and speed. But it does suggest that there is some room for improvement. I think the ideal method would be a space elevator system, where you actually would be getting pretty close to turning electricity into an orbit. Unfortunately that's a long way away if indeed it ever becomes practical.

Hmm. Probably would have made a good quiz problem, given just the orbital altitude and separately deriving the orbital velocity from that. Unfortunately that chapter was a month ago. Maybe next semester!

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I have often wondered whether we could get better energy efficiency for strictly non-living payloads using a particle accelerator system. Since it's magnetic, I'd expect a pretty efficient electricity to acceleration ratio. I presume that putting all the acceleration up front would not only kill any astronauts but would squish their soft fleshy bodies flat... but perhaps a robotic launch vehicle could survive? Is this system feasible?

The kinetic energy of a lump of coal in low Earth orbit is its enthalpy of combustion. The choice launch platform is the (already flattened) top of Mt. Kilimanjaro elevation 15,000 feet and 3°4'33"S. (Cape Canaveral is elevation 9 feet and 28°29′20″N). NASA is an ass.

Tabulated heats of combustion for coals range from 2.4 - 3.2x10^7 J/kg. The all-purpose heat of combustion rule: 5 eV for every molecule of O2 used. This gives 4x10^7 J/kg for pure carbon oxidized to CO2, and is consistent. For an object in low Earth orbit v^2 = Rg, and the kinetic energy/kilogram is Rg/2. With R = 6x10^6 m (from the center of mass) and g =10 m/s^2 that works out to 3x10^7 J/kg. A rocket uses 100X as much energy to insert a kilogram of payload into low Earth orbit.

The reusable (two exceptions noted) Space Scuttle is at least 3X as expensive/boosted mass as a disposable Saturn V three-stage booster in constant dollars. How can that be? The vast majority of the Space Scuttle's payload is the Space Scuttle. Recovery and refurbishment costs are "off budget".

Forget a space elevator. The minimum energy path upward is a spiral. If it conducts electricity it will Roman candle with the first geomagnetic storm (inductance of a half-turn coil). The fastest elevator on Earth is 37.7 mph (Taipei 101).

So I guess that rules out a sky hook as well?

Jesse:

No, it's not (very) feasible. It's not very hard to create a linear accelerator which can boost small payloads into LEO. The problem is acceleration, for linear accelerator with 10 km length acceleration will be:

S = a * t^2/2

a = v_orb / t

so:

S = v_orb * t /2

Substituting real values:

10km = 8 km/s * t /2

t~= 3 seconds

a ~= 3 km/s^2 ~= 300g.

That's FAR too much even for most cargo payloads. You'll need an accelerator with something like 1000km length to launch humans.

The method discussed in #1 and #4 was used in Jules Verne's "From the Earth to the Moon". They built a large, vertical cannon in the ground - at or near Cape Canaveral in Florida !! if I recall correctly - and fired the bullet-shaped projectile to the moon using gun cotton as an explosive.

The dimensions are in the book, and the answer is worse (a lot worse!) than what Alex worked out. They would have been very flat astronauts soon after launch.

Other random comments:

Virial theorem. In circular orbit, KE = GMm/(2r), PE = -GMm/r, and total E = -GMm/(2r). The KE in low earth orbit is abut half of what it takes to escape from the surface since r is not much bigger than the radius of the earth.

You ignored air friction, which is the main problem with launching a satellite as a projectile from the surface by giving it all of the required energy in advance. Well, that and getting the velocity pointed the right way.

If you decide to store that electricity (or whatever) on the spaceship, you have to include the mass of the storage system when figuring out the specific impulse of the proposed power system. Using coal in your Space Scuttle (see

http://en.wikipedia.org/wiki/Coal_scuttle

if you didn't get Uncle Al's joke) might be better than NiMH or LiIon batteries, or even capacitors. I'll leave that for your students to work out.

In fact, a slingshot or railgun alone can never work, because you can't put something in orbit with a single impulse from the ground. Once you stop accelerating it, it is in orbit (neglecting air friction), and will return 97 minutes later to the same point, going in the same direction. Uh-oh, everybody at the launch site duck!

Except of course unless it's moving horizontally at the release point, it would have to go through the Earth to get there. And including air friction only makes it worse. You need additional delta-v sometime after launch. Ideally half an orbit later, but it could be some other time. So anything you launch has to either have a working propulsion system, or rendezvous with something already up there which has a working propulsion system.

As Dennis Moore would say, "This redistribution of energy is trickier than I thought."

Uncle Al:

You're forgetting two important things, it's not height above sea level, it's radius from the center of the Earth. The equatorial bulge means that local "g" is lower near the equator. The second part is the "free" kinetic energy from the Earth's rotation closer to the equator. There's also the unspoken third factor of failed launches coming down over the ocean rather than inhabited areas.

Ah, Bill... We are in agreement that Mt. Kilimanjaro massively trumps Cape Canaveral as the planetary choice launch platform for every technical reason. Tanzania's annual GDP of $15 billion implies the US could purchase the entire country for less than the cost of the War on Drugs.

The fact that the Shuttle would be many times more expensive than disposable rockets was known as soon as the first cost overruns, several years before first takeoff, brought the message home: it is less expensive to throw away sophisticated machinery than to bring it back and refurbish it.

When the infamous tiles kept falling off, there was a great opportunity to say "for every reentry we are bringing back a building with wings, maybe this is not such a good idea", but politics prevailed.

When the calculations of financial viability for the shuttles required about a launch per week, lifting more cargo than anyone ever thought usable, it was a great moment to say "stop".

When the logistics of one launch per week proved impossible, NASA could have diversified its operation with small rockets for most lifting, but they decided to go ahead, keeping a once-per-month schedule at the expense of security, and look what happened.