Cillian Thomas - 05/02/2022
Most recently, a video emerged from SpaceX showing the stage separation of the Falcon 9 during the Cosmo Skymed 2 mission, as seen from the ground. Ultimately the mission was a complete success, launching satellites into orbit for the Italian government.
An image from the ground showing the first stage separating from the second stage of the Falcon 9
Source: Twitter
Why is this done, however? Why would SpaceX throw part of their rocket away mid-flight?
There are two main reasons. These are that single stage vehicles must drag inert mass along with them as they ascend which reduces the payload capacity, and also due to the fact that the most optimum geometry of nozzle changes with altitude. Let’s start with the main reason, extra inert mass.
Assuming a rocket engine ignites in free space to give a ‘velocity increment’ from an initial velocity of 0 to a final velocity, the velocity increment can be expressed as the following $^{[1]}$ :
$$
\Delta u = c \hspace{0.1cm}ln(1/MR)
$$
Where $\Delta u$ = velocity increment m/s, $c$ = effective exhaust gas velocity of the nozzle, and $MR =$ mass ratio of the rocket, which is the final mass with respect to the initial mass, $m_f/m_0.$
This mass ratio can also be expressed in terms of the initially loaded propellent, where the propellant mass fraction is denoted $\zeta$ , and $\zeta = m_p / m_o$. It can be shown that $\zeta = 1 - MR$.
This yields an exponential relationship between the propellent mass fraction $\zeta$ , and the velocity increment ( $\Delta u$) that the rocket can obtain.
$$ \Delta u = c \hspace{0.1cm}ln(1/MR) = c \hspace{0.1cm}ln(1/(1-\zeta)) $$
| $\zeta$ | $\Delta u$ |
|---|---|
| 0 | 0 |
| 0.1 | 0.105360516 |
| 0.2 | 0.223143551 |
| 0.3 | 0.356674944 |
| 0.4 | 0.510825624 |
| 0.5 | 0.693147181 |
| 0.6 | 0.916290732 |
| 0.7 | 1.203972804 |
| 0.8 | 1.609437912 |
| 0.85 | 1.897119985 |
| 0.9 | 2.302585093 |
| 0.95 | 2.995732274 |
| 0.99 | 4.605170186 |
It can be seen from the above graph that as the propellent mass fraction increases for a given rocket design, the maximum velocity that can be acheived increases exponentially. When the propellant mass fraction is increased from 0.85 - 0.9, there is a 22% increase in velocity that can be obtained. This is significant since the kinetic energy is proportional to the square of velocity.