Consider a vehicle carrying a payload that undertakes a first , then drops off that payload and undertakes a second 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 () for the first and the mass of the propellant () for the second to be distinct amounts, carried in common tankage.
First, define the mass conditions at the beginning and end of :
Alternatively, and just as importantly, the conditions bracketing can be described in terms of an initial mass:
This expression can be conveniently rearranged to yield the propellant mass consumed by the vehicle in :
In a similar manner, we define the mass conditions at the beginning and end of :
We can also express the conditions bracketing in another way, in terms of initial mass:
The propellant mass consumed by the vehicle in can also be expressed in a manner analogous to :
Now we are positioned to calculate the total propellant load:
substituting the definitions for and
collecting terms and simplifying
Now let us define the vehicle’s “dry” mass entirely in terms of initial-mass-sensitive (), propellant-mass-sensitive (), and payload-mass-sensitive () mass terms. This is a substantial simplification, but it should do for now.
substituting the definition of the vehicle’s mass in
we collect terms related to the initial mass on the left hand side
now substituting and collecting terms
With all terms relating only to initial mass and payload mass, a general expression for payload fraction can at last be defined:
We can compare this to our previous expression for payload fraction by assuming that and simplifying the result.
and see that for the same assumptions they are identical. A bit more insight can be obtained by remembering that the final mass fraction (FMF) is simply the inverse of the mass ratio, and that the propellant mass fraction (PMF) is one minus the final mass fraction:
Remember that this is just for the case where and thus .
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The case to which I’d really like a closed-form solution is the one where a lander with a crew module deploys some payload on the target surface before ascending back to rendezvous.
You can sorta-kinda model this with your φ, λ, and ε, coefficients, but then you’re doing trial-and-error on unrealistic versions of the coefficients themselves. I’d much rather be able to plug in the mass of the crew module I need, the mass of the payload I need to deploy, and then have coefficients control the dry mass as they were intended to.
This is going to be a hot topic for a while, given the efforts underway to define the Appendix P second-source HLS systems for Artemis. I suspect that most space nerds have lots of trial-and-error tools to fiddle with the trade space, but closed-form versions would be very handy.