Tsiolkovsky rocket equation
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The Tsiolkovsky rocket equation, or ideal rocket equation, describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself (a thrust) by expelling part of its mass with high speed and thereby move due to the conservation of momentum. The equation relates the deltav (the maximum change of velocity of the rocket if no other external forces act) with the effective exhaust velocity and the initial and final mass of a rocket (or other reaction engine).
For any such maneuver (or journey involving a number of such maneuvers):
where:
 is the initial total mass, including propellant. The mass measurements can be made in any unit form (kg, lb, tonnes, etc). This is because the ratios will still be the same.
 is the final total mass without propellant, also known as dry mass.
 is the effective exhaust velocity,
 is deltav  the maximum change of velocity of the vehicle (with no external forces acting), and
 refers to the natural logarithm function.
(The equation can also be written using the specific impulse instead of the effective exhaust velocity by applying the formula where is the specific impulse expressed as a time period and is standard gravity ≈ 9.8 m/s^2.)
The equation is named after Russian scientist Konstantin Tsiolkovsky who independently derived it and published it in his 1903 work.^{[1]} The equation had been derived earlier by the British mathematician William Moore in 1813.^{[2]}
Contents
History
This equation was independently derived by Konstantin Tsiolkovsky towards the end of the 19th century and is sometimes known under his name, but more often simply referred to as 'the rocket equation' (or sometimes the 'ideal rocket equation').
While the derivation of the rocket equation is a straightforward calculus exercise, Tsiolkovsky is honored as being the first to apply it to the question of whether rockets could achieve speeds necessary for space travel.
Derivation
Consider the following system:
In the following derivation, "the rocket" is taken to mean "the rocket and all of its unburned propellant".
Newton's second law of motion relates external forces () to the change in linear momentum of the whole system (including rocket and exhaust) as follows:
where is the momentum of the rocket at time t=0:
and is the momentum of the rocket and exhausted mass at time :
and where, with respect to the observer:

is the velocity of the rocket at time t=0 is the velocity of the rocket at time is the velocity of the mass added to the exhaust (and lost by the rocket) during time is the mass of the rocket at time t=0 is the mass of the rocket at time
The velocity of the exhaust in the observer frame is related to the velocity of the exhaust in the rocket frame by (since exhaust velocity is in the negative direction)
Solving yields:
and, using , since ejecting a positive results in a decrease in mass,
If there are no external forces then (conservation of linear momentum) and
Assuming is constant, this may be integrated to yield:
or equivalently
 or or
where is the initial total mass including propellant, the final total mass, and the velocity of the rocket exhaust with respect to the rocket (the specific impulse, or, if measured in time, that multiplied by gravityonEarth acceleration).
The value is the total mass of propellant expended, and hence:
where is the propellant mass fraction (the part of the initial total mass that is spent as working mass).
(delta v) is the integration over time of the magnitude of the acceleration produced by using the rocket engine (what would be the actual acceleration if external forces were absent). In free space, for the case of acceleration in the direction of the velocity, this is the increase of the speed. In the case of an acceleration in opposite direction (deceleration) it is the decrease of the speed. Of course gravity and drag also accelerate the vehicle, and they can add or subtract to the change in velocity experienced by the vehicle. Hence deltav is not usually the actual change in speed or velocity of the vehicle.
If special relativity is taken into account, the following equation can be derived for a relativistic rocket,^{[3]} with again standing for the rocket's final velocity (after burning off all its fuel and being reduced to a rest mass of ) in the inertial frame of reference where the rocket started at rest (with the rest mass including fuel being initially), and standing for the speed of light in a vacuum:
Writing as , a little algebra allows this equation to be rearranged as
Then, using the identity (here "exp" denotes the exponential function; see also Natural logarithm as well as the "power" identity at Logarithmic identities) and the identity (see Hyperbolic function), this is equivalent to
Terms of the equation
Deltav
Deltav (literally "change in velocity"), symbolised as Δv and pronounced deltavee, as used in spacecraft flight dynamics, is a measure of the impulse that is needed to perform a maneuver such as launch from, or landing on a planet or moon, or inspace orbital maneuver. It is a scalar that has the units of speed. As used in this context, it is not the same as the physical change in velocity of the vehicle.
Deltav is produced by reaction engines, such as rocket engines and is proportional to the thrust per unit mass, and burn time, and is used to determine the mass of propellant required for the given manoeuvre through the rocket equation.
For multiple manoeuvres, deltav sums linearly.
For interplanetary missions deltav is often plotted on a porkchop plot which displays the required mission deltav as a function of launch date.
Mass fraction
In aerospace engineering, the propellant mass fraction is the portion of a vehicle's mass which does not reach the destination, usually used as a measure of the vehicle's performance. In other words, the propellant mass fraction is the ratio between the propellant mass and the initial mass of the vehicle. In a spacecraft, the destination is usually an orbit, while for aircraft it is their landing location. A higher mass fraction represents less weight in a design. Another related measure is the payload fraction, which is the fraction of initial weight that is payload.
Effective exhaust velocity
The effective exhaust velocity is often specified as a specific impulse and they are related to each other by:
where
 is the specific impulse in seconds,
 is the specific impulse measured in m/s, which is the same as the effective exhaust velocity measured in m/s (or ft/s if g is in ft/s^{2}),
 is the acceleration due to gravity at the Earth's surface, 9.81 m/s^{2} (in Imperial units 32.2 ft/s^{2}).
Applicability
The rocket equation captures the essentials of rocket flight physics in a single short equation. It also holds true for rocketlike 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 deltaV requirement (see Examples below). The equation does not apply to nonrocket 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 deltav applied instantaneously. This assumption is relatively accurate for shortduration burns such as for midcourse 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 lowthrust, 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.
Examples
Assume an exhaust velocity of 4,500 meters per second (15,000 ft/s) and a of 9,700 meters per second (32,000 ft/s) (Earth to LEO, including to overcome gravity and aerodynamic drag).
 Singlestagetoorbit rocket: = 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.
 Twostagetoorbit: suppose that the first stage should provide a of 5,000 meters per second (16,000 ft/s); = 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 of 4,700 meters per second (15,000 ft/s); = 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.
Stages
In the case of sequentially thrusting rocket stages, the equation applies for each stage, where for each stage the initial mass in the equation is the total mass of the rocket after discarding the previous stage, and the final mass in the equation is the total mass of the rocket just before discarding the stage concerned. For each stage the specific impulse may be different.
For example, if 80% of the mass of a rocket is the fuel of the first stage, and 10% is the dry mass of the first stage, and 10% is the remaining rocket, then
With three similar, subsequently smaller stages with the same for each stage, we have
and the payload is 10%*10%*10% = 0.1% of the initial mass.
A comparable SSTO rocket, also with a 0.1% payload, could have a mass of 11.1% for fuel tanks and engines, and 88.8% for fuel. This would give
If the motor of a new stage is ignited before the previous stage has been discarded and the simultaneously working motors have a different specific impulse (as is often the case with solid rocket boosters and a liquidfuel stage), the situation is more complicated.
Common misconceptions
When viewed as a variablemass system, a rocket cannot be directly analyzed with Newton's second law of motion because the law is valid for constantmass systems only.^{[4]}^{[5]}^{[6]} It can cause confusion that the Tsiolkovsky rocket equation looks similar to the relativistic force equation . Using this formula with as the varying mass of the rocket seems to derive Tsiolkovsky rocket equation, but this derivation is not correct. Notice that the effective exhaust velocity doesn't even appear in this formula.
See also
 Deltav budget
 Oberth effect applying deltav in a gravity well increases the final velocity
 Spacecraft propulsion
 Mass ratio
 Working mass
 Relativistic rocket
 Reversibility of orbits
 Variablemass systems
References
 ↑ К. Э. Циолковский, Исследование мировых пространств реактивными приборами, 1903. It is available online here in a RARed PDF
 ↑ Moore, William; of the Military Academy at Woolwich (1813). A Treatise on the Motion of Rockets. To which is added, An Essay on Naval Gunnery. London: G. and S. Robinson.
 ↑ Forward, Robert L. "A Transparent Derivation of the Relativistic Rocket Equation" (see the right side of equation 15 on the last page, with R as the ratio of initial to final mass and w as the exhaust velocity, corresponding to v_{e} in the notation of this article)
 ↑ Plastino, Angel R.; Muzzio, Juan C. (1992). "On the use and abuse of Newton's second law for variable mass problems". Celestial Mechanics and Dynamical Astronomy. Netherlands: Kluwer Academic Publishers. 53 (3): 227–232. Bibcode:1992CeMDA..53..227P. ISSN 09232958. doi:10.1007/BF00052611. "We may conclude emphasizing that Newton's second law is valid for constant mass only. When the mass varies due to accretion or ablation, [an alternate equation explicitly accounting for the changing mass] should be used."
 ↑ Halliday; Resnick. Physics. 1. p. 199. ISBN 0471037109.
It is important to note that we cannot derive a general expression for Newton's second law for variable mass systems by treating the mass in F = dP/dt = d(Mv) as a variable. [...] We can use F = dP/dt to analyze variable mass systems only if we apply it to an entire system of constant mass having parts among which there is an interchange of mass.
[Emphasis as in the original]  ↑ Kleppner, Daniel; Robert Kolenkow (1973). An Introduction to Mechanics. McGrawHill. pp. 133–134. ISBN 0070350485.
Recall that F = dP/dt was established for a system composed of a certain set of particles[. ... I]t is essential to deal with the same set of particles throughout the time interval[. ...] Consequently, the mass of the system can not change during the time of interest.