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Single Stage to Orbit Ships

by John McDonnell

Single Stage to Orbit: A First-Cut Analysis

Single Stage To Orbit (SSTO) is an idea that's been around for a long time, but so far has been technically unfeasible for a variety of reasons. This article discusses what SSTO is, what it requires technically, and ways we may achieve those technical goals in the future. The source is a presentation I gave in a rocket propulsion class in 1987. While my notes put together a (hopefully) coherent look at SSTO, it is by no means original technical research. Credit for that goes to all the practicing engineers who figured out this stuff so I could learn it in the classroom.

This article is geared towards individuals with an interest in space systems, who do not have technical training but would like to go "beyond the brochures." The only familiarity it assumes is with the parameter "specific impulse," written Isp. If that doesn't ring a bell, consult the file IMPULS.TXT.

(Keywords are PROPULSION THRUST SPECIFIC IMPULSE ROCKET ENGINE.) Generally speaking, one of the first steps after coming up with a concept is to do a first-cut analysis to see whether what is envisioned is technically doable. Later studies can then determine whether it is feasible. This first look at the numbers might also be known as a "back of the envelope" analysis, and this paper presents one for SSTO, in sort of a step-by-step style. First, some quick background.

In working with launch systems, several "design fractions" are used to relate different systems in the vehicle. First, let's define a term called Mass Ratio, MR = mo/mf. In all these cases, m is mass, and the o, f, etc. represents a subscript. mo is initial mass of the vehicle (fuelled, with payload, on the pad), and mf is the final mass of the vehicle (after the fuel is burned and previous stages dropped, if a staged vehicle). The mass fraction is simply the reciprocal: mass fraction = mf/mo.

The propellant fraction relates the mass of the propellant to the initial mass of the vehicle: z = mp/mo. (z is normally written as a lowercase zeta).

The payload mass fraction, l, is simply: l = mpl/mo (mpl is mass of the payload, l is normally written in script).

The structural fraction shows the relationship between the vehicle structure (tankage, engines, interstage, bus, etc.) and the overall vehicle mass: s = ms/mo. s is also called the deadweight fraction.

Now, applying some math, we can show that: z + l + s = 1. This makes sense--the vehicle is made up of payload, structure, and fuel. And, we can derive another expression for Mass Ratio: MR = 1/(1-z)

Without deriving the equation, we can see intuitively that it makes some sense: obviously, the final mass of the vehicle is related to payload.

We'll see the importance of these fractions later.

Finally, a note on reading the equations. Insofar as possible, I have translated the engineering notation of Greek and subscripts into phonetical versions, so you can simply read them as they appear. Generally, the variable appears first, or in capitals, etc., and the subscript follows. Enough preliminaries.

The advantage of SSTO is obvious--simplicity, which usually equates to reliability. It also usually means lower cost, easier manufacture, and simpler launch planning. The disadvantage is also clear--smaller payload than with a staged vehicle.

SSTO basically has two requirements--sufficient impulse to reach orbit, and, naturally, a thrust-to-weight ratio greater than one. The latter requirement means that many high specific impulse (Isp) engines are useless for SSTO. Currently, only chemical engines have the required thrust to be considered.

The first step is to determine the Isp required. We'll start with a goal of putting a payload in a 220 nm (410 km) circular orbit. To do this, the engine needs to give the payload a change in velocity, called delta-V. The delta-V necessary is delta-V- design. This is what we need from the engine to achieve orbit, and several factors influence what is ultimately needed for the design. Remember as we go along that this step's goal is to determine what we need from the engine itself.

Delta-V-design = Vneeded + Vlosses

where

Vneeded = Vburnout - Vlaunchsite

We'll examine each term and then assemble them. Vburnout is the velocity needed to reach the desired orbit. The rotation of the Earth provides some of that, so Vneeded is less as a result (assuming we are launching eastward--westward launches mean the Earth's rotation is taking away some initial speed). Vneeded is the actual velocity CHANGE which will occur.

Assuming an eastward launch from Kennedy Space Center, at latitude 28.5 degrees north:

Vlaunchsite = 464.5 * cos(latitude) = 408 m/sec

In other words, 408 m/sec is provided by Earth's rotation. Vburnout, in this example, is the speed of a satellite in a 220 nm circular orbit (Vcs220). The velocity of a satellite in a circular orbit is determined using:

Vbo is Vcs220 = sqrt(mu/r) = sqrt((398600 km3/sec2)/6786 km)

You can see my computer doesn't do square-root signs. "mu" is Earth's gravitational constant. "r" is the distance of the satellite from Earth's center. 408 km is added to 6378 km, Earth's radius, to get "r." Vburnout (Vbo) then equals 7664 m/sec. Including Vlaunchsite, we find Vneeded is 7256 m/sec. This is the net velocity change needed.

Now, "rule-of-thumb" losses due to drag and gravity range about 300 m/sec and 925 m/sec, respectively. Adding this in,

DeltaVdesign = 7260 m/sec + 1225 m/sec

DeltaVdesign = 8485 m/sec.

In other words, we must design our engine to generate a total change of velocity of 8485 m/sec. This number includes losses (1225 m/sec) due to drag and gravity, and gains due to the Earth's rotation giving us a free "push."

Now, another way to calculate DeltaVdesign is to use the mass ratio of the vehicle (mo/mf) and the exhaust velocity of the engine ("c"):

DeltaVdesign = c*ln(mo/mf)

where mo/mf = MR, and Isp = c/g, as we learned in IMPULS.TXT. "ln" is shorthand for "natural logarithm."

At this point, we can see the two design parameters are Isp and MR = mo/mf --we just calculated the DeltaVdesign we need, so we want to use this second equation to find the necessary design parameters.

Recall the design fractions:

mprop mstruc mpl

z = ------- s = ----------- l = -------

mo mo mo

where z+s+l = 1.

Using current technology, s ranges from 0.05 for solid fuel rockets to 0.2 for liquid fuel rockets, l is approximately 0.01, and z makes up the remainder, from about 0.8 to 0.95. These fractions are the key to determining what a given vehicle design can do.

The SSME, the most advanced engine currently available, has an Isp of 455 sec in vaccuum. This Isp gives c = 4464 m/sec.

We rewrite the above equation and solve for MR (if you're not into math, "e" raised to an exponent is the inverse of the natural logarithm--you can trust the calculator on this one):

e(exp(DeltaVdesign/c)) = mo/mf = MR

2.71828(exp(8485/4464) = 6.69 = MR

We need a ratio of the initial mass to the final mass of 6.69. Unfortunately, if we use z = 0.8, s = 0.19, and l = 0.01, typical values, we calculate:

1

MR = ------- = 5

1 - z

Using these design fractions, the Isp is not high enough to get the required Mass Ratio of 6.69. We can reverse the equation to find what design fractions we do need:

z = 1 - (1/MR)

z = 1 - (1/6.69) = 0.851

A propellant fraction of 0.851 leaves 0.149 for the structure and payload, or, using the above choices, s = 0.139 and l = 0.01.

That's a pretty low structural value. And, realistically, we need a larger payload. Using these design fractions, a 1 million kg booster, 1.5 times the size of a Titan III, could only lift a 10000 kg payload, no better than a Titan. We'd be as well off with a proven, staged design. Let our design goal be l = 0.05--pretty high, but enough, say, to justify the cost of designing and fabricating the SSTO rocket.

But now to get the required Mass Ratio, 6.69, with our new payload fraction, l = 0.05, the same propellant fraction, z = 0.851 leaves only 0.099 for structure--much lower.

Worse, SSME Isp at sea level is only 363 sec, which gives

c = 3560 m/sec ==> MR = 10.84 to orbit

Taking an average Isp of 410 (remember, this is just a quick first look):

c = 4000 m/sec ==> MR = 8.34 to achieve orbit. This requires z = 0.880 and leaves 0.120 for payload and structure, or, recalling our desired 0.05 payload, only 0.070 for structure.

This is an unrealistically small structural fraction for a liquid fuel rocket, and a liquid fuel system is what gives the high Isp we used. However, we could possibly develop a lighter structure- -new materials may allow liquid fuel rockets with a deadweight fraction down around 0.1. This would allow us to achieve orbit. But the light structure may be expensive to build and difficult to manufacture, undercutting a reason for switching to SSTO launchers. That's one possibility.

Experiments with hydrogen and flourine have indicated an average Isp increase of 15 seconds over hydrogen and oxygen. This gives a MR requirement of 7.70, or s = 0.080. However, the added difficulties of storing and pumping flourine probably wouldn't help us get that low a structure weight.

An alternative is to increase Isp by increasing c. Exhaust velocity (and hence Isp) is proportional to the combustion chamber temperature, Tc, and inversely proportional to molecular weight:

Tc

c = ------

M.W.

Since hydrogen is already the lightest element, increasing Tc is desirable. The limit here becomes nozzle temperature limitations. Tc can be increased by increasing chamber pressure. The problem with increasing chamber pressure is, of course, fabricating engines which can operate at those pressures. The SSME chamber pressure is 3000 psi, and it pushes the edge of our technology.

And, while increasing chamber temperature does increase exhaust velocity, c, here the danger is that going too high causes dissociation of the exhaust gas molecules, robbing performance. A tradeoff is involved.

Other possibilities are similar to the National Aerospace Plane- -using jet engines, ramjets, or scramjets along part of the ascent effectively increases the propellant available without increasing the propellant weight or propellant fraction. The engines, however, add to the deadweight fraction, so their performance would have to add enough to the Isp to offset the increase in structure weight. In fact, the entire concept of the NASP and the British/ ESA HOTOL concept is SSTO.

Finally, of course, is the possible development of new fuels. In the 60's it was estimated that the theoretical Isp of LOX/LH2 was around 360 seconds--now it is up to 455 seconds. While chemical propulsion will never reach the specific impulse of some other systems, new fuels may well push the Isp up enough to make SSTO feasible.

In conclusion, the concept of SSTO requires advances in nearly every aspect of propulsion technology to be workable: better fuel performance, higher chamber pressures and temperatures, and lighter structures. In any given area the advances push the theoretical limit. But, combined, the advances needed in each area are small, and within our reach.

SSTO is a plan whose time is near.

Postscript:

1. This paper was done back in 1987. If you have new information which supercedes any of the data included here, I would be interested in updating the file.

2. If anyone has numbers on the proposed Space Ship Experimental (SSX), I'd like to see how they meet this "first cut." (And other vehicles as well.)

3. Comments on the paper's readability are welcome. I'm aiming this at everyone, so if something didn't get translated to English very well, I'd like to know.

4. Finally, if you have info on how to make a "second cut," I'd definitely like to hear from you.

John McDonnell

 
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