Hohmann Transfer

[Fez]

I have an equation that shows the target speed you need to reach to have a successful hohmann transfer:

V = 1878968 * sqrt(2/RInitial-2/(RLowest + RHighest))

It's always worked in the past, but I recently tried to use it to find the target velocity for a transfer around the sun. When I plug in the numbers, I get a ridiculous answer of 16 m/s, and I don't know why. The weird thing is, I tried looking this formula up on Google to see if I missed something, and I cannot find it anywhere. Has anyone else used this formula?

[Fez]

The transfer orbit is from a lower altitude of 13,381,911,000 meters above the sun to 20,000,000,000 above the sun. When I plug those numbers in, I get 17 m/s as an answer.

 

[zeta function]

For the sun you must multiply it by 1.0827431828*10^9 instead of 1878968. That is the square root of the product of the gravitation constant and the sun's mass. The gravitational constant is about 6.67408*10^-11. If you were to this for Jool, it would be G (gravitational constant) multiplied by Jool's mass. It works the same for every planet/moon/body.

The constant is there because if you are orbiting a heavier body, it will have more gravity, and will require more delta-v to achieve the same result.

Edited June 14, 2018 by zeta function

[Fez]

So the constant 1878968 is only for Kerbin? If I want to do a Hohmann with another gravitational body, I have to use a different constant in my equation?

[Steel]

21 minutes ago, Fez said:

So the constant 1878968 is only for Kerbin? If I want to do a Hohmann with another gravitational body, I have to use a different constant in my equation?

Yes, that constant is actually dependent on the mass of the central body

Edited June 14, 2018 by Steel

[sevenperforce]

23 minutes ago, Fez said:

So the constant 1878968 is only for Kerbin? If I want to do a Hohmann with another gravitational body, I have to use a different constant in my equation?

If you were in a 13,381,911 km orbit around Kerbin and you were trying to transfer to a 20,000,000 km orbit, then yes, you'd need a 16 m/s burn to start the Hohmann. Assuming that the rest of your math is correct.

But you're talking about the sun. The sun's gravity is much stronger and so the numbers are totally different.

If you're crossing from one SOI into another SOI, then all bets are off.

 

[sevenperforce]

Specifically, if you're crossing from one SOI into another (for example, let's say you were trying to do a Hohman transfer from Kerbin into a higher orbit) then you could do it in two ways.

The inefficient way is to burn just outside of Kerbin's SOI, then circularize at Kerbin's orbital distance (but outside of Kerbin's SOI). Then you'd want to do your Hohman and you'd calculate as above, and you'd get a dV of 9,218 m/s. Yikes!

The better way to do it is to use the Oberth effect to your advantage. By burning to escape (in the right direction) out of low Kerbin orbit, you can add a lot of hyperbolic excess velocity (basically, the speed-at-which-you-leave-Kerbin) for a small additional burn.

Not going to lay out all the maths right here, but by way of example, suppose you want to go from Earth to Mars. It will cost you 3.22 km/s to escape Earth, at which point you would have to do another 3 km/s burn to get from Earth's orbit out to brush the orbit of Mars. However, if you combine the escape burn with a little extra hyperbolic velocity in LEO, then you only need a combined 3.82 km/s to get on your Mars-brushing orbit. That extra 0.6 km/s in LEO becomes an extra 3 km/s in interplanetary space.