In this note, we derive the so called *Tsiolkovsky rocket equation* or simply *rocket equation*. It is given by

\begin{equation}

\label{eq:rocket}

\Delta v=v_e\ln\frac{m_0}{m_f}=I_{\mathrm{sp}}g_0\ln\frac{m_0}{m_f}

\end{equation}

where

- $\Delta v$ is the maximum change of velocity of the vehicle;
- $v_e=I_{\mathrm{sp}}g_0$ is the effective exhaust velocity;
- $g_0=9.8\ \mathrm{m}/\mathrm{s}^2$ is the gravitational acceleration of an object in a vacuum near the surface of the Earth;
- $m_0$, called
*wet mass*, is the initial mass, including propellant; - $m_f$, called
*dry mass*, is the final total mass without propellant.

The equation \eqref{eq:rocket} is named after the Russian scientist Konstantin Eduardovich Tsiolkovsky (September 5, 1857 – September 19, 1935). He is dubbed the father of Russian rocket science. It is also called *fuel equation*.

By the Newton’s second law of motion, the net external force $\vec{F}$ to the change in linear momentum $\vec{P}$ of the whole system (including rocket and exhaust) is

$$\vec{F}=\frac{d\vec{P}}{dt}=\lim_{\Delta t\to 0}\frac{\Delta\vec{P}}{\Delta t}$$

$\Delta\vec{P}=\vec{P}_2-\vec{P}_1$, where $\vec{P}_1=m\vec{V}$ is the momentum of the rocket at time $t=0$ and $\vec{P}_2=(m-\Delta m)(\vec{V}+\Delta\vec{V})+\Delta m\vec{V}_e$ is the momentum of the rocket and exhausted mass at $t=\Delta t$. Here, with respect to the observer, $\vec{V}$ is the velocity of the rocket at time $t=0$, $\vec{V}$ is the velocity of the rocket at time $t=\Delta t$, $\vec{V}_e$ is the velocity of the mass added to the exhaust and lost by the rocket during tim $\Delta t$, $m$ is the mass of the rocket at time $t=0$, and $m-\Delta m$ is the mass of the rocket at time $t=\Delta t$. The velocity of the exhaust $\vec{V}_e$ in the observer frame is related to the velocity of the exhaust in the rocket $\vec{v}_e$ by $$\vec{v}_e=\vec{V}_e-\vec{V}$$ or $$\vec{V}_e=\vec{V}+\vec{v}_e$$ Now, $\Delta\vec{P}$ can be written as $$\Delta\vec{P}=m\Delta\vec{V}+\vec{v}_e\Delta m-\Delta m\Delta\vec{V}$$ Since $\Delta m\to 0$ as $\Delta t\to 0$, we have \begin{equation}\label{eq:rocket2}\vec{F}=m\frac{d\vec{V}}{dt}+\vec{v}_e\frac{dm}{dt}\end{equation} If there are no external forces, then $\vec{F}=0$ i.e. $\frac{d\vec{P}}{dt}=0$ (conservation of linear momentum). \eqref{eq:rocket2} then becomes the separable differential equation \begin{equation}\label{eq:rocket3}-m\frac{d\vec{V}}{dt}=\vec{v}_e\frac{dm}{dt}\end{equation} Assuming that $\vec{v}_e$ is constant (Tsiolkovsky’s hypothesis) $v_e$, and integrating \eqref{eq:rocket3} we have $$\int_v^{v+\Delta v}dv=-v_e\int_{m_0}^{m_f}\frac{dm}{m}$$

where $v=|\vec{V}|$, $\Delta v=|\Delta\vec{V}|$, $m_0$ is the initial total mass and $m_f$ is the final mass. Finally, evaluating the integral yields the rocket equation \eqref{eq:rocket}.

From \eqref{eq:rocket}, we obtain

\begin{equation}

\label{eq:rocket4}

\frac{m_0-m_f}{m_0}=1-\frac{m_f}{m_0}=1-e^{-\frac{\Delta v}{v_e}}

\end{equation}

The equation \eqref{eq:rocket4} gives rise to the percentage of the initial total mass which has to be propellant. This tells us how efficient the rocket engine is as shown in the following example.

*Example*. Let us consider an SSTO (Single-Stage-To-Orbit) rocket. (Most rockets we are seeing are two-stage-to-orbit or three-stage-to-orbit ones.) The rocket uses liquid hydrogen/liquid oxygen for its propellant, so specific impulse is about $I_{\mathrm{sp}}=350$ s. The exhaust velocity is then given by $v_e=3.43$ km/s. $\Delta v$ needed to get the rocket to a 322 km high LEO (Low Earth Orbit) is 8 km/s. With these values \eqref{eq:rocket4} is evaluated to be

$$1-e^{-\frac{\Delta v}{v_e}}=0.9$$

This means that 90% of the initial total mass has to be propellant. The remaining 10% is for the engines, the fuel tank, and the payload. The payload would account for only about 1% of the initial total mass. This kind of rocket is obviously very inefficient and expensive.

In the Sci-Fi novella *The Wandering Earth* by Liu Cixin (there is also a movie of the same title on Netflix), the Sun will soon become a supernova and facing the ultimate cataclysmic extinction event, people on Earth turns their entire planet into a spaceship and attempt to relocate it to Proxima Centauri which is the closest star to the Sun (about 4.2 light-years). This is an extremely bold idea even in Chinese scale. (Well, they built the Great Wall!) Disappointingly though, in here, I showed using the rocket equation that it is not even possible for startship Earth to break away from its orbit around the Sun.