# Orbit calculations in the solar system (journey to Mars)

• theanswerai

#### theanswerai

Homework Statement
The ship starts out in a circular orbit around the sun very near the Earth and has a goal of
moving to a circular orbit around the Sun that is very close to Mars. It will make this transfer
in an elliptical orbit as shown in bold in the diagram below. This is accomplished with an
initial velocity boost near the Earth ∆v1 and then a second velocity boost near Mars ∆v2.
Assume that both of these boosts are from instantaneous impulses, and ignore mass changes
in the rocket as well as gravitational attraction to either Earth or Mars. Don’t ignore the
Sun! Assume that the Earth and Mars are both in circular orbits around the Sun of radii RE
and RM = RE/α respectively. The orbital speeds are vE and vM respectively.
i. Derive an expression for the velocity boost ∆v1 to change the orbit from circular to
elliptical. Express your answer in terms of vE and α.
ii. Derive an expression for the velocity boost ∆v2 to change the orbit from elliptical to
circular. Express your answer in terms of vE and α.
iii. What is the angular separation between Earth and Mars, as measured from the Sun, at
the time of launch so that the rocket will start from Earth and arrive at Mars when it
reaches the orbit of Mars? Express your answer in terms of α.
Relevant Equations
centripetal forces, angular momentum, Kepler's 3rd Law
I cannot understand the solution at https://www.aapt.org/physicsteam/2015/upload/E3-2-5-solutions.pdf, because the solution is terse and skip steps (at least i think so). I figured out that the name of this transfer is "Hohmann-Transfer Orbit". A detailed walkthrough would be appreciated. If I am violating against any rules please comment, as this is my first post :-).

We cannot simply give you the answer. You need to make an attempt at solving the problem. Try writing down what you know about Keplerian orbits. For example, what would be the velocity of the ship at the beginning and at the end?