Fun Fact - Specific Impulse

Specific impulse is a measure of an engine's efficiency, when that engine has nothing to push against. Thus we do not talk about the specific impulse of a car (with its tires pressed against the road), or a train on its track, or a submarine in the water, or a prop plane pulling itself through the air. The concept is most relevant to rockets, which must accelerate in the vacuum of space. Since there is nothing to push against, the engine must literally throw something (i.e. the propellant) behind, which pushes the ship forward via Newton's third law of motion: "For every action there is an equal and opposite reaction."

Let's begin with a simple example. You are on a space ship floating between the stars, and your only means of acceleration is a gun. Point it out the back and fire. The bullet races away from the ship, and the recoil pushes you forward. That's a perfectly good rocket engine, although it isn't very efficient. The fuel (gun powder) is not as energy rich as one would like, and the engine itself, the gun, is not optimised for thrust. It is designed to be a weapon. These are quite different goals - the astronaut wants lots of recoil, the hunter does not.

There are four parts to the assemblage: the engine (in this case the gun), the fuel (sulfur and carbon in the gun powder), the oxidizer (potassium nitrate in the gun powder), and the propellant (the bullet). The terms fuel and oxidizer have specific meaning to a chemist, regarding the flow of electrons, but that isn't important here. They are just two chemicals, A and B, that release a lot of energy when combined. In the case of gun powder they are already intermixed, so that the reaction runs as quickly as possible, creating an explosion rather than a slow burn. This is what you want from a gun, though not ideal for a rocket. Still, the explosion pushes the bullet out of the gun barrel at high speed and sends the rocket on its way. All other things being equal, a higher muzzle velocity translates into more thrust. See the earlier article about shooting a gun on the moon, and (perhaps) sending the bullet into orbit.

Impulse is the integral of force with respect to time. Remember that force is mass times acceleration. Assuming a constant mass, pull mass out of the equation and integrate acceleration with respect to time to get the change in velocity. Impulse is then mass times the change in velocity, or the change in mv, or the change in linear momentum. Specific impulse, denoted Isp, is the impulse derived per propellant mass. How much recoil do you get for each bullet? How much change in momentum does each kilogram deliver? Divide the change in mv by mass, and the unit remaining is velocity. In other words, specific impulse is measured in velocity. A kilogram of propellant increases the speed of a kilogram space ship by so many meters per second.

specific impulse can also be measured in seconds, but I will stay with velocity for now. If you happen to run into I expressed in seconds, multiply by G = 9.8 m/s2 to obtain I in meters per second. It's a simple conversion. Now where was I?

We only have so much propellant on board, so many bullets, so we want each kilogram to do as much work as possible. When you're out of propellant, you can't speed up or slow down, or even change direction. You're course is set. Thus a higher specific impulse is better.

When calculating Isp, the fuel is often considered part of the propellant. The bullet shoots out of the gun, but the expanding gases from the burnt gun powder, carbon dioxide, sulfur dioxide, and nitrogen, also jet out of the gun barrel. The fuel turns into more propellant, thus the gun powder and the bullet combine to form the denominator of the fraction that defines specific impulse. In other engines, such as the space shuttle, the fuel + oxidizer becomes the propellant in its entirety. There is no "extra" propellant. The combustion product is water vapor, which shoots out of the engine at high speed and pushes the rocket up, up, and away.

A jet engine combines fuel, carried on board the airplane, with oxygen in the air, to produce water and carbon dioxide, which forms the propellant. However, only the mass of the fuel is considered propellant, because the oxygen comes from outside. The jet fuel is carried aloft, and thus the jet fuel is the expensive commodity. A rocket, which leaves the atmosphere, must carry both fuel and oxidizer, and both chemicals are considered propellant. It would be interesting to design an airplane that flies through the Atmosphere of Titan, Saturn's largest moon. The "air" on Titan consists of methane and ethane, a mixture that we call "natural gas" hear on earth. If you are a typical home owner, methane enters your furnace through an underground pipe, burns with oxygen in the air, and heats your home in the winter. However, the entire atmosphere of Titan is natural gas. The plane must carry oxygen on board, its "gas tanks" full of liquid oxygen. O2 feeds into the engines and burns the hydrocarbons in the air, producing thrust. In this case the on-board oxygen is the propellant. Similarly, a motor boat, cruising about an ethane lake on Titan, carries liquid oxygen on board, which burns the hydrocarbons in the atmosphere, or in the lake itself, to turn a screw propeller and drive the boat about. As mentioned earlier, this engine has no specific impulse, since it pushes against the lake.

Return to the space ship powered by a gun. The momentum imparted to the ship is equal and opposite to the momentum of the bullet. Let the bullet have a mass of b and a muzzle velocity of v. The change in momentum produced by firing the gun is therefore b×v. Divide by b and the specific impulse is v. Under ideal conditions the specific impulse is the velocity of the bullet, or whatever you are throwing behind you to move forward. This is called the "effective exhaust velocity". It's really quite intuitive. If you can throw something behind you at high speed, then you need less of that stuff to move forward.

Let's put some numbers on the above. The .220 Swift has a very high muzzle velocity, 1,200 meters per second, and is available to the general public. It is used to hunt small game from a great distance, where accuracy is vital. No room for a bullet drop here. An experienced hunter can hit a groundhog at 375 yards. Since we are aiming at a squirrel, not an elephant, the bullet can be small. This reduces the recoil, which would otherwise be substantial given the high speeds involved. A space ship equipped with a .220 Swift and lots of ammunition has a specific impulse of 1,200. That's actually not too bad; after decades of research and development, and billions of dollars invested, the Space Shuttle main engine is only about 4 times as efficient. Gun powder, in the form of black powder, is still used in model rocket motors today.

What's so important about specific impulse? To answer that, consider going faster and faster in a rocket. You need more fuel of course, but you have to lift that fuel off the ground, and then you have to accelerate it before it is used. To go faster and farther, you need even more fuel, but then you need more fuel to lift and accelerate that fuel, and more fuel to carry that fuel along, and so on. The rocket grows exponentially in size, weight, and cost. Here is some math to support this intuition.

By conservation of momentum, a tiny amount of mass hurled out the back of the rocket at velocity I has to equal the rocket's mass, at that time, multiplied by its change in velocity. This implies the following difference equation, over a very small time interval t.

m × dv = -I × dm

The minus sign accounts for the fact that the change in mass is negative, fuel is being burned as propellant is thrown out the back. Turn this into a differential equation with respect to time, then integrate from time t0 to t1, velocity v0 to v1, and mass m0 to m1, the rocket with all its fuel and the capsule that finally is traveling out into space.

mv′ = -Im′

v′ = -Im′/m

v = -I log(m)

v1 - v0 = -I (log(m1) - log(m0))

delta v = -I log(m1/m0)

delta v = I log(m0/m1)

Assume the mass of the payload is m1. This could be a probe, or a capsule with three astronauts and three years of supplies headed to Mars, or a multigenerational ship the size of a city carrying a self-sustaining colony to another star. Whatever it is, it's mass is m1. The mass of the fueled rocket, with all its stages, ready to go, is m0. If you want to impart a certain velocity to your payload, the amount of fuel and propellant, m0/m1, grows exponentially with the desired velocity. This is the ideal rocket equation, also called the Tsiolkovsky rocket equation, named after Tsiolkovsky who published it in 1903. (Earlier derivations, dating back to 1813, have since been discovered.)

This is a rather pessimistic result. A better engine, with a higher specific impulse, allows your rocket to carry less fuel, at less cost, with less risk, but the exponential relationship persists. And this doesn't count the cost of getting the rocket, and all that fuel, off the ground and away from earth's gravity, or decelerating once you reach your destination. The space shuttle weighs 165,000 pounds, or 220,000 pounds with a payload, while the shuttle plus tanks plus fuel weighs 4.4 million pounds. That's a ratio of 20, and the shuttle doesn't even break orbit. Launching a small group of humans away from Earth and out to Mars is at the limit of our technology. Assume a mass ratio of 30, to obtain a delta v of 11 km/sec, earth's escape velocity. This is about 0.00004 times the speed of light. If we wanted to send the same space craft to the nearest star at half the speed of light, (and that would still require 8 years to get there), then delta v is multiplied by 12,000, and the mass ratio is raised to the 12,000 power. 30 to the 12,000 is an unthinkable number, much larger than all the mass and energy in the universe. Ain't gonna happen.

Take a step back and suppose your rocket has the size and mass of the planet Jupiter. this is still unthinkable, but just play along. Jupiter is 2E27 kg, thus its ratio, compared to the mass of the space shuttle, is 2E22. This is about 30 to the fifteenth power. A jupiter sized rocket can only increase the speed of the space shuttle by a pathetic factor of 15, 165 km/sec, or 0.0006 times the speed of light. It still takes 8,000 years to reach the nearest star, and don't forget, you, and your descendents, need oxygen and water and food for the journey, so your space ship is actually much larger than the space shuttle. Bottom line: we're not going to the stars any times soon, in fact we can't conceive of any way to get there at all. The only physical contact we can make with another star, using today's technology, or any technology in the foreseeable future, is a basketball sized sphere with some spores inside, sent to a watery planet orbiting a nearby star in hopes of seeding life. spores so protected might survive the hundred thousand year journey through the vacuum of space, and the fiery trip through the planet's atmosphere, and the impact on land or at sea. Just might. See the next article.

Ok - so we're not leaving the solar system any time soon, but can we improve on chemical rockets? Perhaps. In 2003 the European Space Agency launched Smart-1, a lunar orbiter, from French Guiana. A traditional chemical rocket carried Smart-1 up to earth orbit, but at that point a much more efficient engine took over. The probe contains no fuel; the energy comes from the sun. Solar panels generate an electric field, which ionizes xenon and then accelerates the charged ions out the back, somewhat like a linear particle accelerator. Liquid xenon is the on-board propellant, 50 liters at 150 atmospheres of pressure, having a mass of 82 kilograms. The Hall-effect thruster realizes a specific impulse of 16,000, more than 3 times the impulse of the space shuttle engines. The tradeoff for this high efficiency is low thrust, accelerating the probe at just 0.00002 G. Obviously this isn't going to get anything off the ground. We will always need chemical rockets for launch. However, this gentle thrust over many months took Smart-1 from earth orbit to lunar orbit at very low cost. building on this technology, NASA has an ion thruster under development that will have a specific impulse of 90,000. This could be used by the Jupiter Icy Moons Orbiter as it travels from moon to moon using a minimum of propellant. The Dawn spacecraft also uses ion thrusters, as it gently accelerates out past the orbit of Mars to visit the asteroids Ceres and Vesta.

More exotic rockets that take advantage of nuclear fission, hydrogen fusion, and even antimatter are on the drawing board. These could attain a specific impulse of one million or more - but at this point we can't even build a prototype. High impulse notwithstanding, such rockets, when carrying anything bigger than a breadbox, are still restricted to the solar system and the Kuiper belt. multiplying I by a million, or even a billion, will not overcome the exponential factor in the rocket equation and take us to the stars.

Further Reading

Muzzle velocity and the .220 Swift
Impulse and specific impulse
Tsiolkovsky rocket equation
Propeller
Space shuttle mass, fueled and empty
Jupiter mass
Smart 1 lunar orbiter
Dawn spacecraft
Antimatter rockets