Finding Delta V to circularize orbit after a transfer orbit.

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.

• What have you tried? Do you know the velocity at periapsis? Do you know the circular orbital velocity? May 17 '17 at 16:06
• I can get all of those. I am looking more for a general fourmula. May 17 '17 at 16:29
• 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. May 17 '17 at 17:02
• @RussellBorogove Is that it? Sweet, thats all I need. May 17 '17 at 17:35
• 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
– uhoh
May 18 '17 at 1:42

I'll use Earth to Mars as an example.

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

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.

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
The text passage contains a discussion about calculating the delta V required to circularize an orbit around a new planet after a transfer orbit. It includes a specific question about the formula for this calculation and provides an answer using the vis-viva equation and a hyperbola's velocity formula.
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Given a text passage, determine:
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Finding Delta V to circularize orbit after a transfer orbit.

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.

• What have you tried? Do you know the velocity at periapsis? Do you know the circular orbital velocity? May 17 '17 at 16:06
• I can get all of those. I am looking more for a general fourmula. May 17 '17 at 16:29
• 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. May 17 '17 at 17:02
• @RussellBorogove Is that it? Sweet, thats all I need. May 17 '17 at 17:35
• 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
– uhoh
May 18 '17 at 1:42

I'll use Earth to Mars as an example.

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

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.

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
user
The text passage contains a discussion about calculating the delta V required to circularize an orbit around a new planet after a transfer orbit. It includes a specific question about the formula for this calculation and provides an answer using the vis-viva equation and a hyperbola's velocity formula.

TYPE: SINGLE_QUESTION_WITH_ANSWER