WHEN WILL FUSE BURN?: USING REAL ORBIT DATA
STUDENT ACTIVITY

 
Will a satellite remain in orbit forever? The answer is yes if the only force acting on it is the gravitational force of Earth so that it maintains its speed. Unfortunately, a satellite in low Earth orbit, like FUSE, must travel through the thin top layer of Earth's atmosphere. The small atmospheric drag forces slowly change FUSE’s orbit and cause it to fall to ever lower altitudes and into an increasingly thick atmosphere. Frictional heat ultimately leads to the satellite's fiery end. How long do you think FUSE can remain in orbit? A decade? A century? A millennium?

 
Here you will use real numbers collected by radar measurements to predict the lifetime of the FUSE satellite. If you have Internet access, you will also get the latest data for the FUSE orbit.

 
Adding the Latest Data Optional

 
To get the latest orbit frequency go to: http://www.heavens-above.com/orbitdisplay.asp?lat=39.3&lng=-76.6&loc=JHU&TZ=EST&satid=25791 where the revolutions/day are given for a specific date and time. For example, at 9:45:05 PM on  Monday, July 16, 2001, the number of revolutions per day was 14.40070553.  How can you figure out  the number of days since June 29,1999 corresponding to the time  9:45:05 PM on Monday, July 16, 2001?  One way is to look at the two-line orbital elements that are also given.

1 25791U 99035A   01197.90630513 +.00000543 +00000-0 +11321-3 0 03251
 2 25791 024.9814 048.9614 0011667 143.2067 216.9249 14.40070553108641

The number on the first line after the "A" is another way to write the date.  In this case, the date is approximately the 198th day in the year 2001. In the number 01197.90630513, the first two digits give the last two digits of the year and the rest of the number gives the day and fraction of day. There are 184 days between June 29, “day 0”, and the end of 1999. Then there are another 366 days to the end of 2000. Therefore to find the number to enter in the time column add 198 + 184 + 366 = 748. Notice that the number of revolutions per day is also given towards the end of the second line.

 
Another site for orbit data is  http://celestrak.com/NORAD/elements/index.html. In the category “Scientific Satellites”, click on “Space and Earth Science”. Scroll down the page until you find the entry for FUSE.
 

 To complete this part of the activity you will need to use two relationships.

• The time it takes FUSE to make one complete orbit, its period (T), is the reciprocal of its frequency, f.

T= 1 / f

Note that if f is measured in revolutions/day, 1/f will give the value of the period measured in days.

• Kepler’s Law of Periods states that for a satellite orbiting an object of mass M :

 T² = 4p² R³ /G M

T is the satellite's period, in seconds.
R is the distance between the centers of mass of the satellite and the object it is orbiting, in meters.
G is the gravitational constant. G = 6.67 X 10^-11 in SI units.
M is the mass of the orbited object, in kilograms.

Show that FUSE burns up when the orbital frequency has increased to 16.9 revolutions/day, assuming that the satellite is destroyed when its orbit decays to an altitude of 50 km (above the surface of Earth.)  Calculate the period corresponding to this orbit assuming Earth is a sphere of radius 6.37 x 10⁶ m and mass 5.98 x 10²⁴ kg.
 
 
 
 
 
 
 

Fitting the Data

 

1. Using values from the table above to the number of decimal places given, plot revolutions/day (dependent variable - vertical axis) versus time (independent variable - horizontal axis) and find the equation of the straight line that gives the best fit to the data. Use a graphing calculator or plot the points on graph paper, draw a straight line through the points and find the equation of a line that gives a reasonable fit. Write your equation in the space below.

 
 

2.  Use your best fit equation to predict the number of days since launch on June 24,1999 to reach  the frequency at which FUSE is destroyed: 16.9 revolutions/day. Convert your answer to years. Note that if you review your calculation, stating your answer in two significant figures is reasonable.

 

A lifetime of 215 years is predicted by a modeling program used by the FUSE mission control team. Your estimate for the lifetime of FUSE is based on data collected during its first few years in orbit.

3.  Do you expect your calculated lifetime to be greater or lower than a more realistic prediction? Why?