Delta V to orbit

[Mjp1050]

How do I calculate how much delta-v it takes to get from the surface of a planet to orbit?

[Bill Phil]

Calculate orbital speed (assuming a circular orbit, it's v = sqrt(GM/r)).

Then you need to add the gravity loss. This is usually the acceleration due to gravity (for small changes in altitude, you can assume a constant value) multiplied by the time to launch and the average angle of ascent. This should approximate it.

[MinimumSky5]

There will also be atmospheric drag, but unless you fly far too fast on ascent or have a very non-aerodynamic ticket design, this should be fairly small. 

[K^2]

A very rough estimate for aerodynamic and gravity losses on ascent is 4gH/v_t, where H is scale height, g is surface gravity, and v_t is terminal velocity of your rocket near the surface. This is a semi-empirical result based on vertical ascent. It ignores many nuances of actual aerodynamics and gravity turn, but in practice works better than one ought to expect. At any rate, I'm not aware of any better estimate you can get from a simple formula. If you want more precise results, you'll need to do a lot of numerical work.

Consequently, a very rough estimate of total delta-V is sqrt(GM/r) + 4gH/v_t - v₀, where v₀ is your initial velocity due to rotation of the planet if you launch from equator. This will give you a good estimate for KSP, and a very rough estimate for real world. Still better than just the initial sqrt(GM/r), though.

[munlander1]

1 hour ago, K^2 said:

v_t is terminal velocity of your rocket near the surface

How do you find this exactly?

[Xd the great]

7 hours ago, munlander1 said:

How do you find this exactly?

Circular motion at an altitude

Aka a distance from the planets centre.

[Green Baron]

Terminal velocity is the velocity at which air resistance and gravity cancel each other out. Iow, the maximum velocity a free falling body reaches in the atmosphere. The formula involves calculating weight and drag, thus includes density and drag coefficient (~shape of the body) as variables and sounds like this:

https://www.grc.nasa.gov/WWW/K-12/airplane/termv.html

 

Edited September 4, 2018 by Green Baron
deleted "velocity as variable" when solving for velocity

[K^2]

21 hours ago, munlander1 said:

How do you find this exactly?

You drop the fully fueled rocket from great height, and see how fast it moves when it hits the ground. Or model the same on the computer, which tends to be safer.

For a simple, rough estimate, you can use a model where air resistance is F_D = (1/2) ρAv². Technically, there is going to be a drag coefficient in there, but taking it to be 1 puts you in the ball park. Here, ρ is density of air near the surface, A is area of relevant cross-section of the rocket, and v is velocity. At terminal velocity, this is equal to weight of the rocket. In other words, just solve ρAv_t² = 2mg, where m is mass of the rocket, and g is surface gravity again. Keep in mind that this formula doesn't work great for speeds approaching speed of sound, which is why the whole thing is a very rough estimate for real rockets.

If you are doing this in KSP, aerodynamics formula is the same, but area computations are a little wonky. I'm also not sure what it uses for air density and drag coefficients of different parts. My advice, go back to plan one and drop the rocket from great height nose first. See how fast it's going before it impacts the terrain, and use that. Better accuracy and more fun.

[Mjp1050]

Very helpful responses here. Thanks all!