Project 1: An Orbiting Satellite

A NASA satellite orbits the earth every 90 minutes. During an orbit, the satellite’s electric power comes either from solar array wings, when these are illuminated by the sun, or from batteries. The batteries discharge whenever the satellite uses more electricity than the solar array can provide or whenever the satellite is in the shadow of the earth (where the solar array cannot be used). If the batteries are overused, however, they can be damaged.*

You are to determine whether the batteries could be damaged in either of the following operations. You are told that the battery capacity is 50 ampere-hours. If the total battery discharge does not exceed 40% of battery capacity, the batteries will not be damaged.

(a) Operation 1 is performed by the satellite while orbiting the earth. At the beginning of a given 90-minute orbit, the satellite performs a 15-minute maneuver which requires more current than the solar array can deliver, causing the batteries to discharge. The maneuver causes a sinusoidally varying battery discharge of period 30 minutes with a maximum discharge of ten amperes at 7.5 minutes. For the next 45 minutes the solar array meets the total satellite current demand, and the batteries do not discharge. During the last 30 minutes, the satellite is in the shadow of the earth and the batteries supply the total current demand of 30 amperes.

i) The battery current in amperes is a function of time. Plot the function, showing the current in amperes as a function of time for the 90-minute orbit. Write a formula (or formulas) for the battery current function. Hint: The function is a piecewise defined function. The values for t between 0 and 15 are given by a sin function which should look like.

(ii) Calculate the total battery discharge (in units of ampere-hours) for the 90-minute orbit for Operation 1. Hint: The battery current function gives the rate at which the current is flowing. So how do you find the total discharge?

(iii) What is your recommendation regarding the advisability of Operation 1?

(b) Operation 2 is simulated at NASA’s laboratory in Houston. The following graph was produced by the laboratory simulation of the current demands on the battery during the 90-minute orbit required for Operation 2.

(i) Calculate the total battery discharge (in units of ampere-hours) for the 90-minute orbit for Operation 2. Hint: For some parts of the graph it is easy to find the total battery discharge. For the crazier looking part estimate it using the trapezoid approximation. To do that first estimate some values of the function using  Namely approximate the values and fill in the below table using the above graph.

Using the estimates in your table, now you can do a trapezoid approximation.

(ii) What is your recommendation regarding the advisability of Operation 2?

 

* Adapted from Amy C. R. Gerson, “Electrical Engineering: Space Systems,” in She Does Math! Real Life Problems from Women on the
Job, ed. Marla Parker, p. 61 (Washington, DC: Mathematical Association of America, 1995).

This project is taken from the book "Calculus: Single Variable" by Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum et al.