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Wikipedia / Tsiolkovsky rocket equation

The classical rocket equation, Tsiolkovsky rocket equation, or ideal rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity and can thereby move due to the conservation of momentum. The equation is named afterand usually credited toKonstantin Tsiolkovsky, who derived and published the formula in 1903, though William Moore had outlined it as early as 1810 and elaborated further in a book published in 1813. Robert Goddard and Hermann Oberth also obtained the same result in 1912 and 1920, respectively. All four of them reasoned and derived the same model independently.

The maximum change of velocity (with no external forces acting) is:

where:

is the effective exhaust velocity)

Given the effective exhaust velocity determined by the rocket motor's design, the desired delta-v (e.g., orbital speed or escape velocity

The necessary wet mass grows exponentially with the desired delta-v.

History

The equation is named after Russian scientist Konstantin Tsiolkovsky who independently derived it and published it in his 1903 work.

The equation had been derived earlier by the British mathematician William Moore in 1810, and later published in a separate book in 1813.

American Robert Goddard independently developed the equation in 1912 when he began his research to improve rocket engines for possible space flight. German engineer Hermann Oberth independently derived the equation about 1920 as he studied the feasibility of space travel.

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 unexpended propellant".

Newton's second law of motion and where, with respect to the observer:

(conservation of linear momentum

is constant (known as Tsiolkovsky's hypothesis

the velocity of the rocket exhaust with respect to the rocket (the specific impulse, or, if measured in time, that multiplied by gravity hypothesis.

is the total working mass of propellant expended.

(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 delta-v may not always be the actual change in speed or velocity of the vehicle.

Other derivations

Impulse-based

is the initial mass minus the final (dry) mass,

Acceleration-based

Imagine a rocket at rest in space with no forces exerted on it (Newton's first law of motion). From the moment its engine is started (clock set to 0) the rocket expels gas mass at a constant mass flow rate R (kg/s) and at exhaust velocity relative to the rocket ve (m/s). This creates a constant force F propelling the rocket that is equal to R × ve. The rocket is subject to a constant force, but its total mass is decreasing steadily because it is expelling gas. According to Newton's second law of motion, its acceleration at any time t is its propelling force F divided by its current mass m

Now, the mass of fuel the rocket initially has on board is equal to m0 – mf. For the constant mass flow rate R it will therefore take a time T = (m0 – mf)/R to burn all this fuel. Integrating both sides of the equation with respect to time from 0 to T (and noting that R = dm/dt

Limit of finite mass "pellet" expulsion

.

pellets is the sum

.

this Riemann sum.

Special relativity

If special relativity is taken into account, the following equation can be derived for a relativistic rocket,) in the inertial frame of reference standing for the speed of light

Then, using the identity (here "exp" denotes the exponential function; see also Natural logarithm as well as the "power" identity at logarithmic identities (see Hyperbolic function

Terms of the equation

Delta-v

Delta-v (literally "change in velocity"), symbolised as Δv and pronounced delta-vee, as used in spacecraft flight dynamics, is a measure of the impulse that is needed to perform a maneuver such as launching from, or landing on a planet or moon, or an in-space 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.

Delta-v produced by reaction engines, such as rocket engines, 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, delta-v sums linearly.

For interplanetary missions, delta-v is often plotted on a porkchop plot, which displays the required mission delta-v 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 and is instead burned as propellant, 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 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/s2),

  • is the standard gravity, 9.80665m/s2 (in Imperial units 32.174ft/s2).

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). In what has been called "the tyranny of the rocket equation", there is a limit to the amount of payload that the rocket can carry, as higher amounts of propellant increment the overall weight, and thus also increase the fuel consumption. The equation does not apply to non-rocket systems such as aerobraking, gun launcheses, space elevators, launch loops, tether propulsion or light sails.

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.

Examples

Assume an exhaust velocity of of (Earth to LEO to overcome gravity and aerodynamic drag).

  • Single-stage-to-orbit = 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.

  • Two-stage-to-orbit of of = 0.648, therefore 64.8% of the remaining mass has to be propellant, which is 16.2% of the original total mass, 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% of the original launch mass is available for all engines, the tanks, and payload.

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

for each stage, gives:

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 liquid-fuel stage), the situation is more complicated.

See also

References