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I have been working on calculating all the the delta V needed to fly a rocket from one planet to another. I have worked out the delta V needed to raise the orbit to intercept the second planet, but once there, I cannot figure out how much delta V I would need to then circularize the orbit around the new planet.

crude MS paint of what I mean. X marks the burn spot

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  • $\begingroup$ What have you tried? Do you know the velocity at periapsis? Do you know the circular orbital velocity? $\endgroup$ – Russell Borogove May 17 '17 at 16:06
  • $\begingroup$ I can get all of those. I am looking more for a general fourmula. $\endgroup$ – user173724 May 17 '17 at 16:29
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    $\begingroup$ It should be clear that subtracting the circular velocity from the periapsis velocity is the delta-v required to change from flyby to circular orbit. $\endgroup$ – Russell Borogove May 17 '17 at 17:02
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    $\begingroup$ @RussellBorogove Is that it? Sweet, thats all I need. $\endgroup$ – user173724 May 17 '17 at 17:35
  • $\begingroup$ Sometimes the vis-viva equation is very helpful for this. $v^2 \ = \ GM(2/r-1/a)$ see article for more info: en.wikipedia.org/wiki/Vis-viva_equation#Equation $\endgroup$ – uhoh May 18 '17 at 1:42
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I'll use Earth to Mars as an example.

Here's a quick and dirty pic of an Earth to mars Hohmann:

enter image description here

For sake of simplicity I'm rounding quite a bit here.

At Mars rendezvous you can see the speed difference is about 2.5 km/s. We'll call this 2.5 km/s Vinfinity.

When the ship enters Mars sphere of influence, the path is better modeled by a hyperbola about Mars rather than an ellipse about the sun.

A hyperbola's velocity:

$V_{hyperbola}=\sqrt{V_{infinity}^2 + V_{escape}^2}$

A good memory device for remembering hyperbola's speed is thinking of Vinf and Vesc as the legs of a right triangle and the hyperbola's speed as the hypotenuse.

enter image description here

Vescape, a.k.a. escape velocity varies depending on distance from planet. The 11 km/s earth escape velocity near earth perigee. The 5 km/s Mars escape velocity also assumes a periapsis near Mars' surface.

Low Mars orbit is about 3.5 km/s. So to enter low Mars orbit, you subtract 3.5 from 5.6 km/s. You'd need about a 2 km/s burn.

Now the periapsis speed of a 300 x 574,000 km Mars capture orbit is about 4.8 km/s. So if a capture orbit is all you need, .8 km/s would suffice.

To calculate speeds at different points of various orbits, use the vis viva equation

$V=\sqrt{\mu (2/r - 1/a)}$

Where a is semi major axis of an orbit, mu is GM and r is distance from center of gravitating body.

If you don't want to look up different quantities and work out equations, you can use my Hohmann spreadsheet

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