A thermal rocket is a rocket engine that uses a propellant that is externally heated before being passed through a nozzle to produce thrust, as opposed to being internally heated by a redox (combustion) reaction as in a chemical rocket. Thermal rockets can theoretically give high performance, depending on the fuel used and design specifications, and a great deal of research has gone into a variety of types. However, aside from the simple cold gas thruster and steam rocket, none have proceeded past the testing stage. Theory For a rocket engine, the efficiency of propellant use (the amount of impulse produced per mass of propellant) is measured by the specific impulse (), which is proportional to the effective exhaust velocity. For thermal rocket systems, the specific impulse increases as the square root of the temperature, and inversely as the square root of the molecular mass of the exhaust. In the simple case where a thermal source heats an ideal Monatomic gas reaction mass, the maximum theoretical specific impulse is directly proportional to the thermal velocity of the heated gas: where is the standard gravity, is Boltzmann's constant, T the temperature (absolute), and m is the mass of the exhaust (per molecule). For reaction mass which is not monatomic, some of the thermal energy may be retained as internal energy of the exhaust, and this equation will be modified depending on the degree of dissociation in the exhaust, frozen-flow losses, and other internal losses, but the overall square-root proportionality will remain. A more detailed equation for the maximum performance of a thermal rocket can be found under de Laval nozzle or in Chung. Thus, the efficiency of a thermal engine is maximized by using the highest feasible temperature (usually limited by materials properties), and by choosing a low molecular mass for the reaction mass. Cold gas thruster The simplest case of a thermal rocket is the case in which a compressed gas is held in a tank, and is released through a nozzle. This is known as a cold gas thruster. The thermal source, in this case, is simply the energy contained in the heat capacity of the gas. Steam rocket A steam rocket (also known as a "hot water rocket") is a thermal rocket that uses water held in a pressure vessel at a high temperature, such that its saturated vapor pressure is significantly greater than ambient pressure. The water