PHYS-1500 PHYSICAL MODELING
Class 14a: Orbital Motion
FALL 2006
NAME _________________________________
For this exercise, you are to construct a spreadsheet model of an artificial satellite orbiting Earth.
You can use a system similar to that used to calculate and graph trajectories of projectiles, but in
this case the force law will be a little different. For a gravitational attraction between Earth, of
mass M, and a satellite of mass m, the formula is,
Mm
Mm
F  G 2  G 2
,
r
x  y2
where G is the gravitational constant, and r is the distance between the centers of Earth and the
satellite. As the diagram below shows, the x component of the force is Fx = F cos , and the y
component is Fy = F sin .
However, as the next figure shows, cos  
x

r
x
x y
2
1
2
and
sin  
y

r
y
x  y2
2
.
Then, Fx   G
Mm
x  y2
2
x
and Fy  G
x2  y2
second law, a = F/m, we have,
F
M
a x  x  G 2
m
x  y2
x
Mm
x  y2
2
and a y 
y
x2  y2
Fy
. Since, according to Newton’s
 G
M
x  y2
y
.
m
x y
x  y2
These are the formulae that must be entered into the spreadsheet. Below is a copy of a
spreadsheet with those formulae, using the modified Euler method. Below that are the contents
of the first two rows of the spreadsheet, for columns A through G. Set up your spreadsheet to
match the one below, and then copy row 10 to rows 11 through 210.
2
2
2
2
.
A9 0
A10 =A9+$E$3
B9 =B5
B10 =B9+C10*$E$3
D9 =-($B$3*$B$4/(B9^2+E9^2))*B9/SQRT(B9^2+E9^2)
D10 =-($B$3*$B$4/(B10^2+E10^2))*B10/SQRT(B10^2+E10^2)
F9 =E5
F10 =F9+G9*$E$3
C9 0
C10 =C9+D9*$E$3
E9 0
E10 =E9+F10*$E$3
G9 =-($B$3*$B$4/(B9^2+E9^2))*E9/SQRT(B9^2+E9^2)
G10 =-($B$3*$B$4/(B10^2+E10^2))*E10/SQRT(B10^2+E10^2)
When the spreadsheet is complete, create a graph of y vs. x, i.e. column E vs. column B. You can
make your orbits look a bit more accurate, but not perfect, by adding scaling terms. They are
shown in column H. Cell H9 contains the formula, =(MAX(B9:B210)-MIN(B9:B210))/2, and
cell H10 contains, =-H9. Then graph both columns E and H vs. column B.
For best performance, choose t so that no more than one orbit is plotted. The program may give
strange results if the satellite is allowed to go around its path too many times. The program starts
the satellite in an orbit a distance R0 from the center of Earth, on the positive x axis, with a
velocity of v0 in the positive y direction.
2
1. For R0 = 6.50 ×106 m = 6500 km, calculate the value of v0 that should produce a circular
orbit. Hint: v0 
GM E
R0
v0 = _______________
units
2. Enter the value you just calculated in the spread sheet. Does it produce a circular orbit? List
the maximum and minimum distances of the satellite from the center of Earth. This can be
done easily by adding a new column that calculates r  x 2  y 2 , and then using the MAX
and MIN functions.
rmax = __________________
units
rmin = __________________
units
3. Now set R0 = 7.50 ×106 m = 7500 km, and repeat 1. and 2.
v0 = _______________
units
rmax = __________________
units
rmin = __________________
units
4. Now set R0 = 8.50 ×106 m = 8500 km, and repeat 1. and 2.
v0 = _______________
units
rmax = __________________
units
rmin = __________________
units
3
Class 14b: More Orbital Motion
For a satellite in orbit, both mechanical energy and angular momentum should be conserved.
Check that they are conserved in your model.
GMm
. Since we have not given the mass
r
E 1 2 GM
 2v 
of the satellite, m, it is just as easy to calculate the mechanical energy per mass,
.
m
r
Add a new column to the spreadsheet that calculates E/m.
Mechanical energy is given by, E  K  U  12 mv 2 
Angular momentum per mass, is given by,
L
 xv y  yv x . Add a column that calculates this
m
quantity.
1. For R0 = 7.50 ×106 m = 7500 km, and v0 = 7293 m/s, the value that produces a circular orbit,
find the maximum and minimum values of r, E/m, and L/m.
rmax = __________________
units
rmin = __________________
units
E/mmax = __________________
units
E/mmin = __________________
units
L/mmax = __________________
units
L/mmin = __________________
units
2. For R0 = 7.50 ×106 m = 7500 km, and v0 = 8000 m/s, find the maximum and minimum values
of r, E/m, and L/m.
rmax = __________________
units
rmin = __________________
units
E/mmax = __________________
units
E/mmin = __________________
units
L/mmax = __________________
units
L/mmin = __________________
units
4
3. For R0 = 7.50 ×106 m = 7500 km, and v0 = 6000 m/s, find the maximum and minimum values
of r, E/m, and L/m.
rmax = __________________
units
rmin = __________________
units
E/mmax = __________________
units
E/mmin = __________________
units
L/mmax = __________________
units
L/mmin = __________________
units
Does your model conserve mechanical energy and/or angular momentum?
5