I am trying to find the initial velocity to slingshot a planet around the sun and through a gap.
The green ball is the planet, and the yellow ball is the sun. In this trial I need to get the planet to go around the sun and through the gap at 278Gm. I have tried different approaches, but nothing seems to be even remotely correct. Anything under 20k m/s will land you in the sun and anything over 50k will slingshot you out of the system.
I want to know what formula to use so that I can solve this type of problem.
Answer
If you are in a circular orbit what you need is a Hohmann transfer, from Wikipedia:
In orbital mechanics, the Hohmann transfer orbit /ˈhoʊ.mʌn/ is an elliptical orbit used to transfer between two circular orbits of different radii in the same plane.
It works like this assuming the planet is in a circular orbit.
Then the amount of delta v needed to go from the green orbit to the yellow orbit is.
where units are
- v is the speed of an orbiting body
- μ=GM is the standard gravitational parameter of the primary body, assuming M+m is not significantly bigger than M (which makes vM≪v)
- r is the distance of the orbiting body from the primary focus
- a is the semi-major axis of the body's orbit.
Using an online calculator I deduce that the delta v you need is 25.07 km/s
This is independent of the mass of the planet.
Ok, let's start over with a different approach, what is the velocity exactly. Lets just use our trusted elliptical orbits.
Then using equations from this link you can calculate the speed at any point of a eclipse with,
v2=μ(2r−1a)
which leads to 44.31 km/s at perihelion.
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