Tsiolkovsky Rocket Equation

What the equation says

The Tsiolkovsky formula gives the speed an apparatus develops under the thrust of a rocket engine, constant in direction, when no other forces act; this speed is called the characteristic velocity. In its common form V = I·ln(m0/m1), I is the engine's specific impulse (thrust divided by the per-second mass flow of propellant), m0 the initial mass and m1 the final mass after the propellant is spent (P4, P17, P18, P21, P24). The ratio m0/m1, sometimes called the Tsiolkovsky number, expresses the relative propellant reserve. The formula can be obtained by integrating Meshchersky's differential equation for a material point of variable mass (P61).

Derivation and publication

Tsiolkovsky derived the formula in the manuscript "Rocket" on 10 (22) May 1897 and published it in the May 1903 issue of Nauchnoe Obozrenie (P32, P752). In the 1903 original it appears in a dimensionless mass-ratio form: V is the rocket's final velocity, V1 the velocity of the ejected elements relative to the rocket, M1 the mass of the rocket without explosive substances and M2 the mass of those substances (P36, P41, P50, P53, P56, P59). The familiar modern form writes it with specific impulse and the natural logarithm of the mass ratio.

Mass ratio and multistage rockets

Solving the formula for the ratio of initial to final mass gives a ratio that grows exponentially with the required velocity, so the structural mass, tanks and engine also grow. This is what shows mathematically that a single stage cannot practically reach orbital velocities, and why the equation is summed per stage for a multistage rocket. For a multistage rocket the final velocity is the sum of the velocities obtained separately for each stage, with the total initial mass of all later stages added to each stage's masses (P90). For sequentially thrusting stages the equation applies to each stage, with the initial mass being the total mass after discarding the previous stage and the final mass the total mass just before discarding the stage concerned, and the specific impulse may differ from stage to stage (P1875).

Losses in real flight

In real flight the speed a rocket develops is usually below the characteristic velocity because of losses caused by gravity, drag and other factors (P159). Gravitational losses are the largest part of the total. The largest share of aerodynamic losses falls on the first stage's flight segment, where the rocket passes through the lowest and densest layers of the atmosphere, while the largest share of control losses falls on the second stage's segment, where the rocket turns from vertical to horizontal flight and the thrust vector deviates most from the velocity vector (P166, P200, P427, P478).

Use in design

Derived at the end of the nineteenth century, the formula remains an important part of the mathematics used in rocket design, in particular for determining basic mass characteristics (P287). Rearranged, it gives the fuel mass a single-stage rocket needs for a given characteristic velocity at a given payload mass, specific impulse and structural coefficient (P385). The formula also yields a criterion: k must exceed eV/I − 1, otherwise the set velocity cannot be reached at any propellant expenditure (P411, P417). Such calculations are performed not only when choosing a configuration but also in verification calculations as the design is detailed (P675).

Generalizations

For a rocket flying at a velocity close to the speed of light a generalized Tsiolkovsky formula holds; for a photon rocket the specific impulse equals the speed of light (P677, P700, P703).

Priority and naming

The equation is named after Russian scientist Konstantin Tsiolkovsky, who independently derived and published it in 1903 (P903). The British mathematician William Moore derived it earlier in 1810 and published it in a book in 1813, while Robert Goddard obtained the same result independently in 1912 and Hermann Oberth about 1920; all four reasoned and derived the same model independently (P789, P904). Tsiolkovsky is honored as the first to apply the derivation to the question of whether rockets could reach the speeds needed for space travel (P906).

The equation holds when the effective exhaust velocity is constant, and many rocket dynamics studies rest on that hypothesis; it accounts only for the reaction force of the engine and excludes aerodynamic or gravitational forces, which must be added to the delta-v requirement when launching from a planet with an atmosphere (P1254, P1832).

Notation and later standing

Tsiolkovsky used Russian letters for his own system of writing mathematical formulas, to the point that scientific editors of editions in the 1930s to 1960s had to convert his formulas to conventional notation (P2202).

His work was formally acknowledged when he was elected to the Socialist Academy and granted a government pension (P2058); in 1920 he joined the Russian Society of Amateurs of World Studies, received a personal pension from the Soviet government and in 1932 the Order of the Red Banner of Labour (P2092).

The formula has been reproduced on a Polish postage stamp of 1963, a Nicaraguan stamp of 1971 from a series on ten mathematical formulas, and the margins of a 2002 Belarus postal block (P743).

Related topics

The equation became the theoretical starting point for Korolev and the designers of the Soviet space program. Because the required propellant mass grows exponentially with the desired velocity change, large launch vehicles still combine multistage configurations with lightweight structures.