Need help in calculating payload capacity to different orbits and inclination.

[Selective Genius]

The title is pretty self explanatory. I want to find the general equation to calculate the payload capacity of a particular rocket (with a certain delta-v) to  an orbit of x X y km with a Beta inclination around a planet of radius r and mass M. I hope to find a solution for an airless body before moving on to bodies with atmosphere.

Thanks in advance!

[Bill Phil]

A quick and dirty way of doing it is to take the required velocity for orbit and do 2d vector math to find how the initial rotation of the planet changes the target deltaV, add in gravity losses, and then use that as the target delta-v.

Then you will need the mass ratio of each individual stage and the specific impulse, then go into an iterative process. I’ve done some basic vehicle analysis this way.

You then split the delta v into stages and then run the numbers.

I recommend using the rocket equation but with one modification:

DeltaV = ISP * ln|(w+p)/(d+p)|

Where ISP is specific impulse (in m/s or Nm/kg), w is stage wet mass, d is stage dry mass, and p is payload mass. Solve for p:

First let’s divide both sides by ISP, then raise e to both sides. 

(w+p)/(d+p) = e^(DeltaV/ISP)

Let’s set e^(DeltaV/ISP) equal to R to simplify things.

I got:

p = (w-d*R)/(R-1)

Running with an S-IVB TLI gives about 51 tonnes, which is in the ballpark. 

Use p as the total stage mass plus the payload mass for the first stage.

[farmerben]

8 hours ago, Selective Genius said:

The title is pretty self explanatory. I want to find the general equation to calculate the payload capacity of a particular rocket (with a certain delta-v) to  an orbit of x X y km with a Beta inclination around a planet of radius r and mass M. I hope to find a solution for an airless body before moving on to bodies with atmosphere.

Thanks in advance!

https://en.wikipedia.org/wiki/Specific_orbital_energy

These are the droids you're looking for

[wumpus]

These equation are neatly solved without an atmosphere.  With an atmosphere it essentially requires simulation and numerical methods.  I'd simply use a stock "x m/s of delta-v needed to get through the atmosphere" and add it to the previously calculated values.  There really isn't an end to the rabbit hole of aerodynamics.