Payload Fraction Derivation (DV1 w/payload, DV2 w/o)

Consider a vehicle carrying a payload that undertakes a first $\Delta v$, then drops off that payload and undertakes a second $\Delta v$ in the same overall vehicle configuration (tanks, engines, payload handling, etc.). It carries the propellant for both maneuvers, but only on the first maneuver does it have the added mass of the payload. This situation might be representative of:

1. a reusable lunar lander, based in lunar orbit, fully fueled by propellant delivered from the Earth and loaded with a payload from Earth, which then lands and unloads its payload, then returns to lunar orbit with nearly all its propellant expended.

2. a space tug that departs for geosynchronous orbit carrying a satellite, then returning to a low-Earth orbit for refueling and reloading.

Consider the mass of propellant ($m_\text{prop1}$) for the first $\Delta v$ and the mass of the propellant ($m_\text{prop2}$) for the second $\Delta v$ to be distinct amounts, carried in common tankage.

First, define the mass conditions at the beginning and end of $\Delta v_1$:

(1)    \begin{equation*} \eta_1 \equiv \exp(\Delta v_1/v_e) = \frac{m_\text{vehicle} + m_\text{prop1} + m_\text{prop2} + m_\text{payload}}{m_\text{vehicle} + m_\text{prop2} + m_\text{payload}} \end{equation*}

Alternatively, and just as importantly, the conditions bracketing $\Delta v_1$ can be described in terms of an initial mass:

     \begin{displaymath} \eta_1 = \frac{m_\text{initial}}{m_\text{initial} - m_\text{prop1}} \end{displaymath}

This expression can be conveniently rearranged to yield the propellant mass consumed by the vehicle in $\Delta v_1$:

     \begin{displaymath} m_\text{prop1} = m_\text{initial} \left(1 - \dfrac{1}{\eta_1}\right) \end{displaymath}

In a similar manner, we define the mass conditions at the beginning and end of $\Delta v_2$:

(2)    \begin{equation*} \eta_2 \equiv \exp(\Delta v_2/v_e) = \frac{m_\text{vehicle} + m_\text{prop2}}{m_\text{vehicle}} \end{equation*}

We can also express the conditions bracketing $\Delta v_2$ in another way, in terms of initial mass:

     \begin{displaymath} \eta_2 = \frac{m_\text{initial}/\eta_1 - m_\text{payload}}{m_\text{vehicle}} \end{displaymath}

     \begin{displaymath} \eta_2 m_\text{vehicle} = m_\text{initial}/\eta_1 - m_\text{payload} \end{displaymath}

     \begin{displaymath} \eta_2 m_\text{vehicle} + m_\text{payload} = m_\text{initial}/\eta_1 \end{displaymath}

     \begin{displaymath} m_\text{initial} = \eta_1\eta_2 m_\text{vehicle} + \eta_1 m_\text{payload} \end{displaymath}

The propellant mass consumed by the vehicle in $\Delta v_2$ can also be expressed in a manner analogous to prop1:

     \begin{displaymath} m_\text{prop2} = \left(1 - \dfrac{1}{\eta_2}\right)\left(\dfrac{m_\text{initial}}{\eta_1} - m_\text{payload}\right) \end{displaymath}

     \begin{displaymath} m_\text{prop2} = m_\text{initial}\left(\dfrac{1}{\eta_1} - \dfrac{1}{\eta_1\eta_2}\right) - m_\text{payload}\left(1 - \dfrac{1}{\eta_2}\right) \end{displaymath}

Now we are positioned to calculate the total propellant load:

     \begin{displaymath} m_\text{prop} = m_\text{prop1} + m_\text{prop2} \end{displaymath}

substituting the definitions for prop1 and prop2

     \begin{displaymath} m_\text{prop} = m_\text{initial} \left(1 - \dfrac{1}{\eta_1}\right) + m_\text{initial}\left(\dfrac{1}{\eta_1} - \dfrac{1}{\eta_1\eta_2}\right) - m_\text{payload}\left(1 - \dfrac{1}{\eta_2}\right) \end{displaymath}

collecting terms and simplifying

     \begin{displaymath} m_\text{prop} = m_\text{initial} \left(1 - \dfrac{1}{\eta_1} + \dfrac{1}{\eta_1} - \dfrac{1}{\eta_1\eta_2}\right) - m_\text{payload}\left(1 - \dfrac{1}{\eta_2}\right) \end{displaymath}

     \begin{displaymath} m_\text{prop} = m_\text{initial} \left(1 - \dfrac{1}{\eta_1\eta_2}\right) - m_\text{payload}\left(1 - \dfrac{1}{\eta_2}\right) \end{displaymath}

Now let us define the vehicle’s “dry” mass entirely in terms of initial-mass-sensitive ($\phi$), propellant-mass-sensitive ($\lambda$), and payload-mass-sensitive ($\epsilon$) mass terms. This is a substantial simplification, but it should do for now.

     \begin{displaymath} m_\text{vehicle} = \phi m_\text{initial} + \lambda m_\text{prop} + \epsilon m_\text{payload} \end{displaymath}

     \begin{displaymath} m_\text{initial} = \eta_1\eta_2 m_\text{vehicle} + \eta_1 m_\text{payload} \end{displaymath}

substituting the definition of the vehicle’s mass in

     \begin{displaymath} m_\text{initial} = \eta_1 m_\text{payload} + \eta_1\eta_2(\phi m_\text{initial} + \lambda m_\text{prop} + \epsilon m_\text{payload}) \end{displaymath}

we collect terms related to the initial mass on the left hand side

     \begin{displaymath} m_\text{initial}(1 - \phi\eta_1\eta_2) = m_\text{payload}(\eta_1 + \epsilon\eta_1\eta_2) + \lambda\eta_1\eta_2 m_\text{prop} \end{displaymath}

     \begin{displaymath} \lambda\eta_1\eta_2 m_\text{prop} = \lambda\eta_1\eta_2 m_\text{initial} \left(1 - \dfrac{1}{\eta_1\eta_2}\right) - \lambda\eta_1\eta_2 m_\text{payload}\left(1 - \dfrac{1}{\eta_2}\right) \end{displaymath}

     \begin{displaymath} \lambda\eta_1\eta_2 m_\text{prop} = m_\text{initial} \left(\lambda\eta_1\eta_2 - \dfrac{\lambda\eta_1\eta_2}{\eta_1\eta_2}\right) - \lambda\eta_1\eta_2m_\text{payload}\left(\dfrac{\eta_2 - 1}{\eta_2}\right) \end{displaymath}

     \begin{displaymath} \lambda\eta_1\eta_2 m_\text{prop} = m_\text{initial} \left(\lambda\eta_1\eta_2 - \lambda\right) - m_\text{payload}\lambda\eta_1(\eta_2 - 1) \end{displaymath}

now substituting and collecting terms

     \begin{displaymath} m_\text{initial}(1 - \phi\eta_1\eta_2 - \lambda\eta_1\eta_2 + \lambda) = m_\text{payload}(\eta_1 + \epsilon\eta_1\eta_2 - \lambda\eta_1(\eta_2 - 1)) \end{displaymath}

further simplifying

     \begin{displaymath} m_\text{initial}(1 - (\phi + \lambda)\eta_1\eta_2 + \lambda) = m_\text{payload}(\eta_1(1 + \epsilon\eta_2 - \lambda(\eta_2 - 1))) \end{displaymath}

With all terms relating only to initial mass and payload mass, a general expression for payload fraction can at last be defined:

     \begin{displaymath} \dfrac{m_\text{payload}}{m_\text{initial}} = \dfrac{1 - (\phi + \lambda)\eta_1\eta_2 + \lambda}{\eta_1(1 + \epsilon\eta_2 - \lambda(\eta_2 - 1))} \end{displaymath}