### Cold Equations

The title, of course, evokes one of the all time notorious science fiction stories - from geek perspective perhaps the notorious SF story: 'The Cold Equations.' I have been in the bashing school regarding that story, on the grounds that the most basic safe operation procedures should have prevented it, and more broadly because it is anvilicious. (No, I won't link the Evil Website. If you want the link, google it.)

A fair literary response is that the anviliciousness is the point - people may argue about the story, but if you've read it you remember it.

This post is not about the story itself, but about those cold equations, specifically as they relate to reaching Earth orbit. And for that purpose, the grump about the story is, if anything, understated. Realistically, the spacecraft in the story should not have had anywhere a stowaway could hide in the first place. It would be like stowing away in a Formula I racing car.

The cold equations we are specifically interested in are handily available at the Atomic Rockets site. Orbital velocity in low orbit is about 7.8 km/s. Add the potential energy from being about 300 km up, and the kinetic energy needed to reach low orbit corresponds to about 8.2 km/s.

There are also some unavoidable losses from air friction and gravity. In a vertical launch, 1 g of your initial thrust just goes to hovering, adding nothing to your speed. A horizontal launch allows aerodynamic lift to do that work, but means more aerodynamic drag.

If your launch site is at low latitude you also get up a few hundred meters/second of rotational velocity as a freebie.

These variables are, well, variable, depending on vehicle configuration and launch site. But taken together, expect to burn some 9-10 km/s in delta v to reach orbit.

Now we can play with some (very crude!) virtual orbiters. Captain Obvious reminds you that these numbers are not remotely authoritative: for one thing, I routinely round off numbers to 2-3 significant figures.

The highest performance propellant mix that we can really count on is H2-O2, which is good for an Isp in the range of 420-455 seconds, corresponding to an effective exhaust velocity around 4.2-4.5 km/s. Performance in atmosphere is lower. Delicately ignore that for the moment.

Cutting to the chase - and in the best case - getting to orbit calls for a mission delta v equal to at least twice the drive's exhaust velocity. For an SSTO that corresponds to a mass ratio of e^2, or 7.39, or an 86 percent propellant fraction.

To simplistically model more conservative assumptions, again set effective exhaust velocity at 4.5 km/s (still ignoring atmosphere!), while equivalent mission delta v is 10 km/s. In that case the mass ratio rises to 9.23, for an 89 percent propellant fraction.

This is the truly cold equation, because it puts convenient space flight pretty much out of the running. In the ideal case, if your launch mass is 1000 tons, 860 tons of that will be propellants, with the remaining 140 tons for the tankage, thrust structure, engines, minor items such as the guidance package, and (oh yes) a payload. With more conservative assumptions you have 890 tons of propellants and 110 tons for everything else.

These proportions are not, in themselves, impossible. The first and second stages of the Saturn V had dry weights of less than 8 percent and 6 percent of loaded weight respectively. But the first stage used denser kerosene and LOX with much lower performance, while the second stage used H2-O2 but had initial acceleration of only 1.04 g in near-vacuum, and at sea level would have been unable to lift itself off the pad.

For the current state of the art, the dry weight of the SpaceX Falcon 9 first stage is about 7 percent of load weight, but it also uses kerosene and LOX. A tank for H2-O2 would have to be much bulkier - about 3 times the volume capacity - and thus much heavier.

The bottom line is that an expendable SSTO might be viable, but offers no advantage over two-stage expendables. Any saving in operational simplicity (no staging separation or second stage startup) would be have to be balanced against the extremely narrow margins of the design.

Note that both Americans and Russians used 'one and a half stage' designs for their experimental ICBM models, Atlas and R-7 Semyorka - both of which went on to very successful careers as space boosters. Their designs allowed all main engines to be started on the launch pad. But later developments added a true second stage, and modern generation boosters have at least two stages, often with boosters strapped to the first stage.

The Shuttle was (we must now say was!) essentially a 'one and a half stage' orbiter, with recovery of the solid boosters, engines, and payload bay, but expendable main propellant tank.

Getting to a recoverable SSTO rocket would require a tech revolution - either dramatically stronger materials, or dramatically more powerful propellants. Neither is impossible. But likewise, neither is foreseeably in the cards.

An airbreather ascent, as proposed for Skylon, does not call for quite so big a tech revolution, but still requires a couple of very big pieces of undemonstrated technology - jet engines operating up to Mach 5+, then efficiently shifting into rocket mode, and a huge, lightweight airframe capable of handling the heat loads.

Skylon is awesomely cool, but just as awesomely demanding. I can't quite rule it out, but I wouldn't want to rely on it.

Two stages makes it all a lot easier, which is why two-stage boosters are now typical. A fully reusable TSTO vehicle is almost certainly possible. Whether it would be viable - that is to say, competitive with modern generation expendables - is a much iffier question.

And because I've made you wait so long even for this much, I will take up recoverable TSTO, and its alternatives, in an upcoming post.

Related Post: 'The Cold Equations' came up here previously, in the comment thread of a post about what constitutes hard SF.

The image of the X-37 unmanned spaceplane comes from the Christian Science Monitor (which in spite of its name and affiliation has had a good reputation over the years).