Question

1. Consider a Hohmann transfer orbit between Earth and Saturn (you may assume circular orbits). Describe the orbit in terms of its semi-major axis, period, and distance from the sun at aphelion and perihelion.

Answer 1

For a small body orbiting another much larger body, such as a satellite orbiting Earth, the total energy of the smaller body is the sum of its kinetic energy and potential energy, and this total energy also equals half the potential at the average distance (the semi-major axis):

Solving this equation for velocity results in the vis-viva equation,

where:

• is the speed of an orbiting body,
• is the standard gravitational parameter of the primary body, assuming is not significantly bigger than (which makes ),
• is the distance of the orbiting body from the primary focus,
• is the semi-major axis of the body's orbit.

Therefore, the delta-v (Δv) required for the Hohmann transfer can be computed as follows, under the assumption of instantaneous impulses:

to enter the elliptical orbit at from the circular orbit

to leave the elliptical orbit at to the circular orbit, where and r 2 are respectively the radii of the departure and arrival circular orbits; the smaller (greater) of and corresponds to the periapsis distance (apoapsis distance) of the Hohmann elliptical transfer orbit. Typically, is given in units of m3/s2, as such be sure to use meters, not kilometers, for and . The total is then:

Whether moving into a higher or lower orbit, by Kepler's third law, the time taken to transfer between the orbits is

(one half of the orbital period for the whole ellipse), where is length of semi-major axis of the Hohmann transfer orbit.

In application to traveling from one celestial body to another it is crucial to start maneuver at the time when the two bodies are properly aligned. Considering the target angular velocity being

angular alignment α (in radians) at the time of start between the source object and the target object shall be