## 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 previous article in this blog.

I posted the original version of this article to the CompuServe space forum in 1990. Since then, there have been a variety of advances, but nothing particularly revolutionary. Until now. We’ll look at several design equations for SSTO, and then examine the SpaceX numbers for Starship.

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’s feasible. This first look at the numbers is also known as a "back of the envelope" analysis, and this paper presents one for SSTO, in sort of a step-by-step style.

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 = m_{o}/m_{f}

In all these cases, m is mass. m_{o} is *initial* mass of the vehicle (fueled, with payload, on the pad), and m_{f} 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 = m_{f} / m_{o}.

The *propellant fraction* relates the mass of the propellant to the initial mass of the vehicle: z = m_{p} / m_{o}. (z is normally written as a lowercase zeta.)

The *payload fraction*, L, is simply: L = m_{pl} / m_{o}. m_{pl} is mass of the payload; L is normally written in lowercase script, but that makes l (el) look too much like 1 (one), so we’ll use L here.

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

So, we can now be a little more precise with regard to initial and final mass. Initial mass consists of propellant, payload, and structure. Final mass consists only of payload and structure, since the propellant needed to reach orbit has been consumed. (Note that some propellant always remains, but that effectively makes it deadweight. Or, if propellant is retained for further on-orbit maneuvering, then that effectively makes it useful payload.)

Writing this out:

m_{prop} + m_{pl} + m_{struc} = m_{o}

If we divide the whole thing by m_{o}, we can see how the mass fractions add up:

m_{p} m_{pl} m_{s} m_{o}

------ + ------- + ------- = ------- = 1

m_{o }m_{o }m_{o }m_{o }

Or:

z + L + s = 1.

This makes sense—the vehicle is made up of fuel, payload, and structure. And, we can derive another expression for Mass Ratio:

MR = 1 / (1 - z)

Without doing the algebra, but just remembering that the mass fractions all add up to one, we can see intuitively that it makes some sense: “everything” is the initial mass (one); “everything minus fuel” is the final mass (one minus z). We'll see the importance of these fractions later.

Finally, a note on reading the equations. As much 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, and the subscript follows. Enough preliminaries.

The advantage of SSTO is obvious—simplicity, which usually equates to reliability. It also usually implies 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 230 nm (426 km) circular orbit. (This is within the altitude band of the ISS.) 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 (V_{bo}) is the velocity needed to reach and remain in 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; for polar orbits, there’s no contribution or penalty). Vneeded is the actual velocity *change* that will occur.

For an eastbound launch at Earth’s equator, the Earth’s rotation would provide 464.5 m/sec, or about 1000 mph. Assuming an eastward launch from Kennedy Space Center, at latitude 28.5 degrees north:

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

In other words, 408 m/sec is provided by Earth's rotation if we launch *directly eastward* from Kennedy. (Note that launches to ISS are *not* directly eastbound—ISS is in a 51.6 degree inclination orbit, because that’s the latitude of the Baikonur Cosmodrome.)

Vburnout, in this example, is the speed of a satellite in a 230 nm circular orbit (V_{cs230}). The velocity of a satellite in a circular orbit is determined using:

V_{bo} is V_{cs230} = sqrt(mu/r) = sqrt((398,600 km3/sec2)/6804 km) = 7654 m/sec

I’m avoiding square-root signs and other symbols so we won’t have to worry about how a particular browser renders the equation. "mu" is Earth's gravitational constant; just trust me on this one. "r" is the distance of the satellite from Earth's center, so 426 km orbital altitude is added to 6378 km (Earth's radius) to get r = 6804 km. V_{bo} then equals 7654 m/sec.

Including Vlaunchsite, we find Vneeded is 7654 – 408 = 7246 m/sec. This is the *net velocity change needed*.

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

DeltaVdesign = 7246 m/sec + 1225 m/sec

DeltaVdesign = 8471 m/sec.

In other words, we must design our engine to generate a total change of velocity of 8471 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." (I’m not using significant figures here so the math is easier to follow.)

Now, another way to calculate DeltaVdesign is to use the mass ratio of the vehicle (m_{o}/m_{f}) and the exhaust velocity of the engine ("c"):

DeltaVdesign = c * ln(m_{o}/m_{f})

where m_{o}/m_{f }= MR, and Isp = c/g, as we learned in “Specific Impulse—What, besides thrust, gets us to orbit?” "ln" represents "natural logarithm."

At this point, we can see the two design parameters are Isp and MR = m_{o}/m_{f}. We just calculated the DeltaVdesign we need, so we want to use this new equation to find the necessary design parameters.

Recall the design fractions:

m_{prop} m_{struc } m_{pl}

z = ------------ s = ------------- L = -----------

m_{o} m_{o } m_{o}

where z + s + L = 1.

Using 1980’s technology, values for s ranged from 0.05 for solid fuel rockets to 0.2 for liquid fuel rockets, L was approximately 0.01, and z made 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, even though it’s now “old technology,” is still one of the most efficient engines available, with an Isp of 455 sec in vacuum. This Isp gives exhaust velocity 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)) = m_{o}/m_{f} = MR

2.71828(exp(8471/4464) = 6.64 = MR

With this engine, we *need* the ratio of the initial mass to the final mass to be 6.64. Unfortunately, if we use z = 0.8, s = 0.19, and L = 0.01, typical values, we calculate:

1 1

MR = ---------- = ------------ = 5

1 – z 1 – 0.8

Using these design fractions, the Isp is not high enough to get the required Mass Ratio of 6.64. We can reverse the equation to find what design fractions we do need:

z = 1 - (1/MR)

z = 1 - (1/6.64) = 0.849

A propellant fraction of 0.849 leaves 0.151 for the structure and payload, or, using the above choices, s = 0.150 and L = 0.01. That's a fairly low structural value. And, realistically, we probably want 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, 6.64, with our new payload fraction, L = 0.05, the same propellant fraction, z = 0.849 leaves only 0.101 for structure—50% 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 sec (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, now 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 fluorine 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 fluorine 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, T_{c}, and inversely proportional to molecular weight:

T_{c}

c = --------

M.W.

Since hydrogen is already the lightest element, increasing T_{c} is desirable. The limit here becomes nozzle temperature limitations.

T_{c} 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. So, a tradeoff is involved.

Other possibilities are similar to the National Aerospace Plane (NASP, follow-on to the X-30). 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 was to support SSTO. They never made it off the drawing board.

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’s 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 summary, 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.

In 1987, I concluded this paper with, “SSTO is a plan whose time is near.” At that time, is was more-or-less assumed that SSTO was simpler than staged designs and this simplicity would reduce launch costs. Thirty years later, every aspect of rocket technology has advanced. Reduction in launch cost, though, has come through a different route: *reusability*. And not just shuttle-style reusability, but a focused effort on reusability for the *sake* of cost reduction. That suggests that SSTO as conceived over the last few decades is probably *not* near.

If that’s true, then is there a place for SSTO at all? Yes, but probably not in launch from Earth. Return from Mars will need to be single stage, and likely not just to orbit, but to escape velocity and Earth landing.

According to SpaceX, an empty Starship tanker *can* actually achieve Low Earth Orbit without the booster, although it can’t get back. That’s not very practical, but just for fun, let’s take a look at those numbers.

Assume a launch from Cape Canaveral, directly eastbound. Recall from above that V_{needed} = 8485 m/sec, including losses.

Exhaust velocity c = Isp*g, so using an average Isp of 350 sec, c = 3433 m/sec. The LOX/Liquid methane Raptors are obviously not nearly as efficient as an SSME, so that makes the design challenge a bit harder. And, we’re really just guessing at the average Isp, since both sea level and vacuum engines will spend a large amount of time outside their optimal performance envelopes.

Calculating Mass Ratio:

e(exp(DeltaVdesign/c)) = m_{o}/m_{f} = MR

2.71828(exp(8471/3433)) = 11.73 = MR (needed)

From the 2019 Boca Chica presentation, Starship will tip the scales at around 110 tons empty and start off with 1200 tons of propellant. To get to orbit single stage, we won’t carry any payload, giving us a liftoff mass of 1310 tons.

That gives us a Starship MR of 1310/110 = 11.91

Elon Musk wasn’t kidding when he suggested an empty Starship could just make orbit. The design fractions work out to z = 0.916 and s = 0.084, with L = 0. Remember the earlier statement that s ranges from 0.05 to 0.2? That was for 1980’s technology. Small advances everywhere have helped, but as Musk points out an important factor is size--bigger ships result in smaller structural fractions. If we had engines with higher Isp, we might even be able to put useful payload into orbit—but those engines would need to be fueled by liquid hydrogen. LH_{2} is very low density, which would significantly increase the tank volume required, bumping up the structural fraction. This analysis also doesn't consider that three of Starship's Raptors have vacuum nozzles, which can't be used at sea level, so Starship as-designed wouldn't have enough thrust to get off the pad fully fueled.

These problems can be solved with redesign, of course. So, could it be done? Probably. But with reusability, it looks like we have a better path.

In the next article, we’ll take a look at staged designs.