A Hohmann orbit is the trajectory of a spacecraft between two nearly circular
orbits requiring the least fuel. In this simulation we will study a Hohmann
orbit of a space probe between the orbits of Earth and Mars. It is an ellipse with
the perihelion (the point closest to the sun) at the orbit of Earth, and the
aphelion (the point farthest from the sun) at the orbit of Mars.
To achieve the correct ellipse the velocity of the spacecraft must be increased by the correct amount at the correct time. At the correct time the angle Earth-Sun-Mars must have a value given by a formula in the right column. See the figure below. This angle is zero when the simulation starts.
In the right column you will find formulas to calculate the Hohmann orbit.
The assumptions made to derive these formulas are:
1) The orbits of the planets are circular and in the same plane
2) The space probe starts so far from Earth that the gravitational influence of Earth on the space probe can be neglected
3) The velocity increase of the space probe is instantaneous
4) The gravitational attraction of Mars on the space probe when it approaches Mars is neglected.
In practice one must allow for all of these factors, and the time of launch, velocity increase and the exact orbit must be calculated numerically.
To simplify, Earth is not included in the simulation, but the space probe initially follows Earth's orbit. The orbits of Earth and Mars are assumed to be perfect circles.
Hohmann orbit formulas
Angle Earth-Sun-Mars at launch
where r1 and r2 are the radii of the two circular orbits.
Time of launch
Speed in inner circular orbit:
Speed increase to reach an orbit of radius r2:
Time from inner to outer orbit: