Consider a vehicle that undertakes a first , then picks up a payload and undertakes a second . It carries the propellant for both maneuvers, but only on the second maneuver does it have the added mass of the payload. This situation might be representative of:
1. a one-way lunar vehicle, arriving in low Earth orbit fully fueled but with no payload. It executes a trans-lunar injection burn and then a lunar orbit insertion burn, all with no payload. It then picks up a payload in lunar orbit and descends to the surface with the payload and lands with essentially all its propellant expended.
2. a reusable lunar lander, based on the surface of the Moon, fully fueled by lunar propellant but lacking any payload, which ascends to a lunar orbit and recovers a payload in that orbit, then descends to the surface, landing with nearly all its propellant expended.
3. a space tug that departs for geosynchronous orbit and recovers a satellite, then returns with it to a low-Earth orbit for repair.
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:
Yet another approach will later yield a useful relationship:
The propellant mass consumed by the vehicle in can also be expressed in a manner analogous to prop1:
Now we are positioned to calculate the total propellant load:
substituting the definitions for prop1 and prop2
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
Take equation(3) and multiply it through by :
then substitute the RHS for equation(5) in equation(4) and collect terms
With all terms relating only to initial mass and payload mass, a general expression for payload fraction can at last be defined:
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