Mass ratio

The definition arises naturally from Tsiolkovsky's rocket equation:

$${\displaystyle \Delta v=v_{e}\ln {\frac {m_{0}}{m_{1}}}}$$

where

• Δv is the desired change in the rocket's velocity
• v_e is the effective exhaust velocity (see specific impulse)
• m₀ is the initial mass (rocket plus contents plus propellant)
• m₁ is the final mass (rocket plus contents)

This equation can be rewritten in the following equivalent form:

$${\displaystyle {\frac {m_{0}}{m_{1}}}=e^{\Delta v/v_{e}}}$$

The fraction on the left-hand side of this equation is the rocket's mass ratio by definition.

This equation indicates that a Δv of ${\displaystyle n}$ times the exhaust velocity requires a mass ratio of ${\displaystyle e^{n}}$. For instance, for a vehicle to achieve a ${\displaystyle \Delta v}$ of 2.5 times its exhaust velocity would require a mass ratio of ${\displaystyle e^{2.5}}$ (approximately 12.2). One could say that a "velocity ratio" of ${\displaystyle n}$ requires a mass ratio of ${\displaystyle e^{n}}$.

Sutton defines the mass ratio inversely as:[1]

$${\displaystyle M_{R}={\frac {m_{1}}{m_{0}}}}$$

In this case, the values for mass fraction are always less than 1.

• Rocket fuel
• Propellant mass fraction
• Payload fraction

Zubrin, Robert (1999). Entering Space: Creating a Spacefaring Civilization. Tarcher/Putnam. ISBN 0-87477-975-8.