Mass Ratio and Exhaust Velocity
The most basic factors in rocket science are the exhaust velocity (EV) of the propellants and the mass ratio (MR) of the vehicle. Various propellant combinations have differing exhaust velocities. The most common propellant combinations used today are LOX/RP-1 (Liquid Oxygen/Kerosene) and LOX/ LH2 (Liquid Oxygen/Liquid Hydrogen) as used in the Space Shuttle. These two have significantly different exhaust velocities. LOX/RP-1’s EV is about 7400 mph. LOX/LH2’s EV is 15000 mph.
Now of course a spacecraft must achieve a velocity of 17,500 mph to go into orbit. Not even LOX/LH2 could do that except for a unique physical phenomenon that allows a rocket to exceed the velocity of its exhaust if its empty weight at engine shutdown is sufficiently lower than its fully fueled weight at liftoff. That is its MR.
The basic MR equations were derived in 1903 by Konstantin Tsiolkovsky, who is revered in Russia as the Father of Russian rocketry. His equation is:
m0 is the initial total mass, including propellant, in kg (or lb)
m1 is the final total mass in kg (or lb)
ln is the natural logarithm e (2.7183)
vc is the effective exhaust velocity in m/s or (ft/s)
Δv is the change in velocity you desire in m/s (or ft/s) (for low Earth orbit [LEO], it would be about 25,000 ft/s).
Note that Tsiolkovsky approached this from the stand point of achieving a predetermined velocity. To determine what MR gives you the desired velocity, rewrite the equation as follows:
This equation indicates that a Δv of n times the exhaust velocity requires a mass ratio of en. For example, for a vehicle to achieve a Δv of 2.5 times its exhaust velocity, it must have a MR of e2.5 (approximately 12.2). The equation indicates that a "velocity ratio" of n requires a mass ratio of en.
Several years later, an American, Robert H. Goddard, wrote a treatise titled “A Method of Reaching Extreme Altitudes." In this he expanded on Tsiolkovsky’s work and proposed using liquid fueled rockets to reach the moon. Goddard came to be known as the Father of American Rocketry, and built and flew the world’s first liquid fueled rocket. Later, after WWII, he would design and develop a high altitude research rocket for the Army—the WAC Corporal—that would play a seminal part in the beginnings of the U.S. ICBM and space programs.
Theoretically, a rocket can achieve its exhaust velocity if its mass ratio is e—the natural logarithm 2.7183. However, basic physics makes that impossible. A rocket must fight air resistance on its first minute or two of flight, and gravity all the way to orbit. The reality of rocket science is that the MR must be near 3 to achieve exhaust velocity.
The original “big" rocket, the German V-2, which used LOX and the equivalent of RP-1, had a MR of just over 3. It was small by today’s standards, however, standing just 50 feet tall, so carried a limited propellant load. Its burn time was just about one minute. Its velocity at burnout was about 3000 mph.
It was a single stage rocket. Still, if a rocket’s MR is more than 3, it can exceed its EV. The Mercury/Atlas vehicle that orbited our first astronauts had a MR of 12.3. Even with the relatively low EV of its LOX/RP-1 propellant, that was sufficient for it to achieve orbital velocity. Unfortunately, it isn’t a matter of the MR multiplying the EV. The Atlas’ MR was four times that required to match its propellants EV, but its maximum possible velocity was not four times its EV. That’s because when EV is plotted against MR it generates an asymptotic curve, as shown in the illustration. This is a curve that comes ever closer to a line or point but never touches it—even if drawn to infinity. You can see the Atlas’ MR—12.3--barely gave it a delta v sufficient to reach LEO.