Exercises from "Fundamentals of Astrodynamics" Part 4

19 Jan 2020

Continuing with “Fundamentals of Astrodynamics” by Bate, Mueller, and White, but skipping around a little, it’s time for an easy one!

Exercise 1.6 Find an equation for the velocity of a satellite as a function of total specific mechanical energy and distance from the center of the earth.

Solution:

Skipping the derivation for now (I’ll save that for another post) the equation for specific mechanical energy, $\mathcal{E}$, is:

$$\mathcal{E} = \frac{v^2}{2} - \frac{\mu}{r},$$

where $v$ is the velocity, $\mu$ is the gravitational parameter, 1.40765x10¹⁶ ft³s^-2 for Earth, and $r$ is the radial distance from the center of the earth to the satellite. Simply rearranging for $v$ yields the desired equation:

$$\displaylines{ \frac{v^2}{2} = \mathcal{E} + \frac{\mu}{r} \\ \therefore v = \sqrt{2 \left ( \mathcal{E} + \frac{\mu}{r} \right )}. \blacksquare }$$ velocity specific mechanical energy