Tsiolkovsky rocket equation

The classical 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 after—and usually credited to—Konstantin 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 Herman 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 of the vehicle, ${\displaystyle \Delta v}$ (with no external forces acting) is:

$${\displaystyle \Delta v=v_{\text{e}}\ln {\frac {m_{0}}{m_{f}}}=I_{\text{sp}}g_{0}\ln {\frac {m_{0}}{m_{f}}},}$$ where:

• ${\displaystyle v_{\text{e}}}$ is the effective exhaust velocity (which is also equal to ${\displaystyle I_{\text{sp}}g_{0}}$)
  • ${\displaystyle I_{\text{sp}}}$ is the specific impulse in dimension of time;
  • ${\displaystyle g_{0}}$ is standard gravity;
• ${\displaystyle \ln }$ is the natural logarithm function;
• ${\displaystyle m_{0}}$ is the initial total mass, including propellant, a.k.a. wet mass;
• ${\displaystyle m_{f}}$ is the final total mass without propellant, a.k.a. dry mass.

Given the effective exhaust velocity determined by the rocket motor's design, the desired delta-v (e.g., orbital speed or escape velocity), and a given dry mass ${\displaystyle m_{f}}$, the equation can be solved for the required wet mass ${\displaystyle m_{0}}$: $${\displaystyle m_{0}=m_{f}e^{\Delta v/v_{\text{e}}}.}$$ The required propellant mass is then $${\displaystyle m_{0}-m_{f}=m_{f}(e^{\Delta v/v_{\text{e}}}-1)}$$

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

We can also express this as the ratio of fuel mass to payload mass: $${\displaystyle {\frac {m_{0}-m_{f}}{m_{f}}}=e^{\Delta v/v_{e}}-1}$$ and we see that it grows exponentially with ${\displaystyle \Delta v/v_{e}}$