The function of any propulsion system is to make something move forward. Due to the conservation of momentum, this can only be accomplished by making something else move backwards. (This article ignores "reactionless" drives, as none are known to exist, and postulating them is difficult.) The heavier the stuff going backwards is, and the faster it's going, the faster you go. Generally, in order to design a spacecraft it's easiest to base your calculations off the force the drive is capable of, so that's what we'll try and calculate.
The first drive we will work on is a standard reaction drive. This is a drive where you take some material, and squirt it out the back of your ship (a chemical drive, a fusion drive, ion drive, etc). The basic relation we will use is the the one between change in momentum and force:
F = dp/dtIn other words, the the force exerted by or on a body is equal to the rate of change in it's momentum per unit time. This doesn't seem to help us much, but it's often easier to calculate the change in momentum than the force. Now all we need to know is how much stuff (mass) we're pushing out the back of the ship, and how fast it's moving away from us.
p = mv dp/dt = (dm/dt)*vwe'll take dm/dt as a paramater that describes the rate of reaction mass usage ("reaction mass" is a technical term for the stuff we make move backwards). We're assuming that we can use up stuff at any rate we want (not really fair from an engineering standpoint, you need pipes and stuff to get it into the drive, but that's irrelevant from a physics standpoint. So the only thing we need is the velocity the reaction mass is moving at relative to the vessel. Obviosly for our spacecraft we're going to want this to be as fast as possible, so that we get the maximum amount of thrust out of each kilogram of reaction mass spent. How do we know what velocity the reaction mass is travelling at? The easiest kind of a drive to make is a rocket. Basically a rocket is where you make your stuff hot, and use the heat to propel it out. In a gas (or any other material, but we're working with gasses here) , the molecules are in constant motion, and the speed at which they are travelling is determined by the temperature. In a thermally based drive, we put the reaction mass in a box that's open on only one side. Then any molecules that are travelling in the direction of the hole will leave the box, moving (relative to the box) at whatever velocity they're moving at due to temperature. In a gas, the average velocity of the molecules is detemined by:
v = sqrt(kT/m)where k is a constant (Boltzmanns constant = 1.38*10^-23) T is the temperature, and m is the mass of the molecule. So in order to increase the thrust we get out of each gram of reaction mass, we can do one of two things, we can increase the temperature, or we can decrease the mass of the molecule we're using. This makes Hydrogen (the lightest element) an ideal choice for reaction mass (and is why the space shuttles main engines carry more hydrogen than is needed to combust with the oxygen, the extra hydrogen makes the exhaust lighter, and makes the engines more efficient). Since it's pretty hard reduce the mass of the molecules below that of hydrogen, all we can do to improve the efficiency of our drive is to increase the temperature. The problem is that the specific impulse (specific impulse is a technical term for how much force a given amount of propellant can produce) of a drive is proportional to the square root of the temperature, so if we want to double the specific impulse, we need to quadruple the temperature. Quadrupling the temperature requires quadrupling the energy input.
The reason we want a high specific impulse is because we have to carry all of our reaction mass with us until we use it. This means that in addition to accelerating the ship, our drive has to accelerate the reaction mass we haven't used yet. In some drives this isn't a problem (for example the ion drives on satelites), usually because they aren't expected to do much accelerating, and therefore carry a very small amount of reaction mass in comparison to their total mass. For these drives getting the most amount of thrust out of a given amount of energy is most important. So, if you give a particle a given amount of energy (for example by accelerating it with an electric field) it's energy is given by:
E = 1/2 mV^2and it's momentum by:
p = mv p = sqrt(2*E*m)So, once again, doubling the specific impulse requires quadrupling the energy, but you want as massive a particle as possible.
The limiting factor in interstellar travel is usually reaction mass. In order to get anywhere in a reasonable amount of time, you need to accelerate more or less all the way, however the longer you plan on accelerating, the more reaction mass you need, and the more reaction mass that you carry, the more massive your ship is, and the more reaction mass you need to accelerate the reaction mass you haven't used yet. It's a vicious cycle. So most interstellar starship propulsion system designs use some method for getting around this. For example the bussard ramjet collects interstellar hydrogen (which it did not have to accelerate) along it's path, rather than carrying its reaction mass with it. Another idea is to use light pressure to accelerate a remote vehicle (eg the starwisp). With this design you do all the work at a stationary powerplant.