## Spaceplanes by the Numbers

Continuing this line of thought, assume the spaceplane has reached an altitude of nearly 100,000 ft and a speed of well over 3,000 mph. As the air begins to thin, the aerospike rocket is gradually brought online. This uses the same combustion chamber as the ramjet, keeping weights to a minimum. By the time the spaceplane has reached 4,000 mph and 150,000 ft, it is operating on rocket engines alone. It is now up to the rockets to take the ship to 17,500 mph and 528,000 ft. This translates into about a 25,000 ft/s DV, including 4,000 ft/s for remaining drag and gravity losses, and 1,000 ft/s as a hedge factor in case earlier assumptions are off. Normal ground-launched rockets require about 30,000 ft/s DV to reach orbit; so we have gained a DV advantage of 5,000 ft/s, or 3,400 mph with these air-breathing techniques. This might not seem like a lot, but let us look at the numbers, beginning with the exponential form of the rocket equation.

With this formula, it is possible to calculate the required mass ratio R for a given DV and specific impulse. We will use a I of 450 s, yielding an effective exhaust velocity c of 14,480 ft/s. We will also try some other numbers, just for the fun of it. Take a peek at Table 8.1, and then we will discuss what it all really means.

The first three rows show the trend at constant specific impulse as DV is reduced from 30,000 to 20,000 ft/s. Exhaust velocity depends directly on specific impulse, and so it remains the same also. Notice the right column. By reducing DV by one third, mass ratio is reduced by almost one half. In other words, there is a huge advantage in gaining as much airspeed as possible before igniting the rocket engines. Building a spaceplane with a mass ratio of 4 is a lot easier than building one with a mass ratio of 8. But since we earlier decided on a DV requirement of 25,000 ft/s, we will settle on the R = 5.62 design. That is still a lot easier than the R = 7.94 case.

 Table 8.1 Required mass ratios for various DV at various I sp sp v ' Exhaust velocity Mass ratio 30,000 25,000 20,000 25,000 25,000 450 450 450 400 350 14,480 14,480 14,480 12,870 11,260 7.94 5.62 3.98 6.98 9.21

The last two rows show the importance of specific impulse and exhaust velocity. As these values are lowered, required mass ratio shoots up. Although it might just be possible to build a spaceplane with a mass ratio of 7, building one with R = 9.21 would be very difficult, because 89.1% of the light-off weight would have to be propellants. This leaves only 10.9% for ship and payload. Aiming for a mass ratio of 5.62, we find that the propellant mass fraction at light-off has declined to 462/5.62 = 82.2%, leaving 17.8% for the ship and its payload. Let us work the numbers and see if they hang together. Assume a gross light-off weight (GLOW) of 1,000,000 lb - a nice round number. Of this, 822,000 lb are propellants - liquid oxygen and liquid hydrogen. The specific impulse will be an assumed 450 s in efficient aerospike engines. The burn-out weight is 178,000 lb, which includes the ship, crew, and cargo. Now just plug and chug:

AV = (450 s) (32.174 ft/s2) loge (1,000,000 lb/178,000 lb) AV = (14,478 ft/s) loge (5.62) AV = 25,000 ft/s

The numbers work. By limiting the engines to 2 G's throughout the entire burn to orbit, the spaceplane will accelerate from 5,000 to 25,000 ft/s in 5 min 11 s. The Space Shuttle takes 8 min 30 s to reach orbit, incurring 3 min 19 s more in gravity losses than our spaceplane will. In actual operations, the initial acceleration will start out at less than 2 G's and wind up at more than 2 G's. But this analysis at least provides a basic comparison.

How do all these numbers compare to the Space Shuttle? The Shuttle system has a gross lift-off weight of 4.5 million pounds, including the 58,000-lb ET with its 1.6 million pounds of propellants, the two SRBs weighing 1.3 million pounds each, and the Shuttle orbiter with a lift-off weight of 240,000 lb, including a 55,000-lb payload. Fully 57.5% of the GLOW is in the SRBs. Another 37.2% is in the filled ET. The liquid oxygen alone accounts for 30.7% of the launch weight. The remaining 5.3% is in the fully loaded delta-winged orbiter. The 55,000-lb payload makes up only 1.2% of the total (Table 8.2).

The LH2 tank is 2.706 times the volume of the LOX tank. The LH2, at a density of 0.5922 lb/gal, makes up 14.45% by weight, and the LOX, at a density of 9.491

 Component Lift-off weight (lb) Empty weight (lb) Propellants (lb) SRBs (1,300,000 lb each) 2,600,000 400,000 (both) 2,200,000 (both) External tank 1,680,222 58,500 1,621,722 Orbiter 240,000a 151,205 33,545b Total 4,520,222 609,705 3,855,267 Payload 55,250

"Includes 55,250-lb payload b(240,000 - 55,250 - 151,205) lb = 33,545 lb

"Includes 55,250-lb payload b(240,000 - 55,250 - 151,205) lb = 33,545 lb

Table 8.3 Space Shuttle liquid propellants

Propellant Weight Volume Density Tank

LOX 1,387,457 lb 146,181 gal 9.491 lb/gal 54.6 x 27.6-ft LOX tank

LH2 234,265 lb 395,582gal 0.5922lb/gal 97.0x27.6-ftLH2 tank

°Total 1,621,722 lb 541,763 gal 153.8 x 27.6-ft External tank lb/gal, accounts for 85.55% by weight of the total propellants (Table 8.3) . This information will help us define the parameters of our own spaceplane, using the Space Shuttle as a baseline.

Using the Shuttle propellant data above, and beginning with a total propellant weight of 822,000 lb, we find that we will need 200,570 gal of LH, weighing 118,779 lb, and 74,093 gal of LOX weighing 703,221 lb. The task is to efficiently package these propellants into a spaceplane weighing no more than 178,000 lb, including engines, passengers, crew, and cargo. These propellants will take up almost exactly half the volume taken up by the Shuttle's ET propellants, but in exactly the same relative proportions. We do not even consider using solid propel-lants because of their inefficiency, both in terms of specific impulse and operations. In our spaceplane, the liquid oxygen makes up 70.3% of the 1 million pound lightoff weight, 85.5% of total propellant weight, but takes up only 27% of the total 274,663-gal propellant volume. This is because it is 16 times denser than the liquid hydrogen. For its part, the hydrogen makes up 11.9% of the GLOW, 14.5% of total propellant weight, but requires 73% of the propellant tankage due to its low density. At light-off of the rocket engines, 17.8% of the GLOW is in the spaceplane, passengers, crew, and cargo, the other 82.2% being in the propellants.

How does the propellant mass fraction of our spaceplane compare to that of the Shuttle? The lower this number, the better. In the following simple exercise, we show how propellant mass fraction f is derived. The starting point is mass ratio R, which is the combined spaceship and propellant masses (ms + mp) divided by the spaceship mass alone (ms).

There are two different ways to calculate propellant mass fraction: using mass ratio alone, as just derived, or dividing total propellant weight by gross lift-off (or light-off) weight:

Let us see if we get the same number for our spaceplane. If everything is correct, then we should:

fp = (5.62 - 1)/5.62 = 0.822 f = 822,000/1,000,000 = 0.822

So far, so good. But how does this compare to the Space Shuttle?

More good news: our spaceplane has a better (lower) propellant fraction than the Shuttle; so it should have a better (higher) payload fraction as well. We can also work the problem backwards, and figure out the effective mass ratio of the Space Shuttle so that we can compare that to our spaceplane.

The effective mass ratio of the Shuttle system is 6.797, significantly higher than our own 5.62. The Shuttle achieves this higher mass ratio, of course, by staging. But it also requires this higher mass ratio because it has to lift a much greater weight off the launchpad.

With the help of the numbers we have, a spaceplane can be designed from scratch. The first step is to just guess at the overall dimensions. We know that the total propellant volume is half of what is carried in the Shuttle's ET. And we know that the ET weighs 58,000 lb empty, while the Space Shuttle weighs about three times as much. This compares favorably to our own unfueled spaceplane, which weighs 178,000 lb with crew, cargo, and passengers. To make the ship economical, it should have a good payload capacity of something like 10-20 tons.

That is a starting point. Engineers must figure out how to package all components, integrate the tanks, structure, thermal protection system, and so on, together in a realistic design. It goes without saying that the fuel and oxidizer tanks should follow the same design as those in the Saturn S-II stage, in which they were efficiently sandwiched together with a narrow honeycomb structure in between. Propellant densities will dictate the overall dimensions, and when the first spaceplane begins service as a tanker, its prime cargo should be water. It is safe, has a high density (8.345 lb/gal), and can be turned into LOX (9.491 lb/gal) and LH2 (0.5922 lb/gal), the best chemical rocket propellants in the universe. Water's high density means that it does not require huge, heavy tanks. The dual use concept for the passenger-carrying spaceplane should be taken very seriously. The gross takeoff weight will be the GLOW plus the weight of fuel required to take the spaceplane from the runway to light-off altitude.