Question

Imagine you are the orbital engineer for the first NASA space shot to Ceres, the largest known asteroid. Ceres’s nearly circular orbit around the sun has a radius of R = 2.77 AU. After being launched from the earth, the probe will initially be in a circular orbit around the sun with the same radius (r_{c} = 1.0 AU) and the same orbital speed (|\vec{v}_{c}| = 6.28 AU/y) as the earth’s orbit. The probe’s rocket engines will then fire briefly to increase the probe’s speed to that speed |\vec{v}_{c}| needed to put the probe into an elliptical orbit whose initial (and minimum) distance from the sun is r_{c} = 1.0 AU and whose final (and largest) distance from the sun is R = 2.77 AU (this is a Hohmann transfer orbit).(a) How long will it take the probe to get from the earth to Ceres in such an orbit? (b) What is the speed |\vec{v}_{c}| that the probe has to have just after firing its engines to be inserted into this orbit? (Assume the duration of the boost is short enough that its distance from the sun is still nearly r_{c} just after the engines have been fired.)

Answer 1

1. We are using Hohmann transfer orbit to transfer the probe from a circular orbit (of Earth) around the sun with the following characteristics:

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to the circular orbit of Ceres with an orbital radius of

and an orbital velocity of by means of a transfer elliptical orbit with the distances to the perigee and the apogee given as

The time taken for the orbit transfer is given by

where

with , the gravitational constant and , the mass of Sun.

You can derive equation (1) from Kepler's third law applied to the transfer orbit. The transfer time is half the period for the elliptical orbit. Now, plugging in the values, we get

2. The Hohman transfer is a two-impulse transfer. The speed of the probe in the final circular orbit (of Ceres) is given by

In addition, I am giving the details of the Hohmann transfer impulses:

The first impulse needed to put the probe into an elliptic orbit is

and the second impulse needed to put the probe from the elliptical orbit to the Ceres orbit is