User:Sanchazo/sandbox

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Applicability

The rocket equation captures the essentials of rocket flight physics in a single short equation. It also holds true for rocket-like reaction vehicles whenever the effective exhaust velocity is constant, and can be summed or integrated when the effective exhaust velocity varies. The rocket equation only accounts for the reaction force from the rocket engine; it does not include other forces that may act on a rocket, such as aerodynamic or gravitational forces. As such, when using it to calculate the propellant requirement for launch from (or powered descent to) a planet with an atmosphere, the effects of these forces must be included in the delta-V requirement (see Examples below). The equation does not apply to non-rocket systems such as aerobraking, gun launches, space elevators, launch loops, or tether propulsion.

The rocket equation can be applied to orbital maneuvers in order to determine how much propellant is needed to change to a particular new orbit, or to find the new orbit as the result of a particular propellant burn. When applying to orbital maneuvers, one assumes an impulsive maneuver, in which the propellant is discharged and delta-v applied instantaneously. This assumption is relatively accurate for short-duration burns such as for mid-course corrections and orbital insertion maneuvers. As the burn duration increases, the result is less accurate due to the effect of gravity on the vehicle over the duration of the maneuver. For low-thrust, long duration propulsion, such as electric propulsion, more complicated analysis based on the propagation of the spacecraft's state vector and the integration of thrust are used to predict orbital motion.

Also, the equation strictly applies only to a theoretical impulsive maneuver, in which the propellant is discharged and delta-v applied instantaneously. Orbital maneuvers involving significantly large delta-v (such as translunar injection) still are under the influence of gravity for the duration of the propellant discharge, which influences the vehicle's velocity. The equation is most accurately applied to relatively small delta-v maneuvers such as those involved in fine-tuning space rendezvous, or mid-course corrections in translunar or interplanetary flights where the gravity field is relatively weak.

Nevertheless, the equation is useful for estimating the propellant requirement to perform a given orbital maneuver, assuming a required delta-v. To achieve a large delta-v, either $m_{0}$ must be huge (growing exponentially as delta-v rises), or $m_{1}$ must be tiny, or $v_{e}$ must be very high, or some combination of all of these. In practice, very high delta-v has been achieved by a combination of

very large rockets (increasing $m_{0}$ with more fuel)
staging (decreasing $m_{1}$ by throwing out the previous stage)
very high exhaust velocities (increasing $v_{e}$ )

Examples

Assume an exhaust velocity of 4,500 meters per second (15,000 ft/s) and a $\Delta v$ of 9,700 meters per second (32,000 ft/s) (Earth to LEO, including $\Delta v$ to overcome gravity and aerodynamic drag).

Single-stage-to-orbit rocket: $1-e^{-9.7/4.5}$ = 0.884, therefore 88.4% of the initial total mass has to be propellant. The remaining 11.6% is for the engines, the tank, and the payload. In the case of a space shuttle, it would also include the orbiter.

Two-stage-to-orbit: suppose that the first stage should provide a $\Delta v$ of 5,000 meters per second (16,000 ft/s); $1-e^{-5.0/4.5}$ = 0.671, therefore 67.1% of the initial total mass has to be propellant to the first stage. The remaining mass is 32.9%. After disposing of the first stage, a mass remains equal to this 32.9%, minus the mass of the tank and engines of the first stage. Assume that this is 8% of the initial total mass, then 24.9% remains. The second stage should provide a $\Delta v$ of 4,700 meters per second (15,000 ft/s); $1-e^{-4.7/4.5}$ = 0.648, therefore 64.8% of the remaining mass has to be propellant, which is 16.2%, and 8.7% remains for the tank and engines of the second stage, the payload, and in the case of a space shuttle, also the orbiter. Thus together 16.7% is available for all engines, the tanks, the payload, and the possible orbiter.