Optimal engine:payload proportion for given gravity

Warning: These equations only generate upper limits, they don't account for atmospheric drag beneath Karman line (but it doesn't apply on Mercury, the moon, Phobos, Deimos, Io, Ganymede and Callisto) and assume gravity and payload mass are static.

These equations yield optimal quantities of engines per tonne of payload, solving the continuous problem of Spaceflight Simulator engine selection with fixed gravity. The calculator itself solves for the specific discrete combination that maximises delta-v in the.

Key: g is gravitational field strength (treated as constant), n is payload mass, f is fuel mass. Per engine, m is mass, y is thrust, c is fuel consumption.

With x engines at time t through operation, a stage's momentary upwards acceleration, per tonne of non-engine (payload and fuel) is x*y/((n+f-x*f*t)+x*m)*9.8-g.

The numerator is the momentary acceleration (the total thrust (x*y) per mass (n+f-x*f*t) times standard gravity (because thrust is measured in g's exerted per tonne but gravity is measured in m/s**2) minus gravity (g)), the denominator is the rate of fuel consumption. It simplifies to =(y/(n+m*x)-g/(9.8*x))/f.

As time t increases from 0 to f/(x*c), the delta-v is ∫_0^(f/(x*c)) x*y/((n+f-x*c*t)+x*m)*9.8-g dt = 9.8*y*ln((m*x+n+f)/(m*x+n))/c-g*f/(x*c) = (9.8*y*ln((m*x+n+f)/(m*x+n))-g*f/x)/c = (9.8*y*ln(1+f/(m*x+n))-g*f/x)/c.

This can't be solved for the x value corresponding with its maximum analytically, I don't think, but the Newton-Raphson method yields x'=(9.8*y*ln(1+f/(n+m*x))-(f*g)/x)/((9.8*f*m*y)/((n+m*x)^2+f*(n+m*x))-(f*g)/x**2) .

This is all very well for launch stages where mass can be unbounded, but most stages are going to be payloads for prior stages first, for which you want to maximise delta-v times the proportion of the prior stage's payload that this stage's payload occupies, (9.8*y*ln(1+f/(m*x+n))-g*f/x)/c*n/(n+f+m*x), which by the Newton-Raphson method can be computed x'=(x*(m*x+n)*(f+m*x+n)*(5*f*g-49*x*y*ln(1+f/(m*x+n))))/((m*x+n)*(5*f**2*g-49*m*x**2*y*ln(1+f/(m*x+n)))+f*(5*g*(2*m**2*x**2+3*m*n*x+n**2)-49*m*x**2*y)).

Here's a Desmos graph graphing each engine's effective specific impulse over altitude with a slider for planets, gravity decreases linearly beneath the surface and the Karman line and sphere of influence limits are displayed also.