Satellite orbit

Ivan Cnop
icnop@vub.ac.be

Problem

Plot the orbit of a spy satellite watching over all countries in a zone of 45 degrees on either side of the equator.

Solution

Assume we have 20 revolutions per day.

This depends on the velocity (and  the height) of the satellite, but is fixed once and for all.

r = 1; a = 2.5; d =  .125;

a/d

Clear[x, y, z, A, B]

Run the satellite along the equator

x[t_] = r Cos[a t];
y[t_] = r Sin[a t];
z[t_] = 0

\(ParametricPlot3D[{x[t], y[t], z[t]}, {t, 0, 2\ π}
, 
      PlotPoints -> 100]\)

Rotate the orbit plane by 45 degrees

A=(({{1, 0, 0}, {0, Sqrt[2]/2, -Sqrt[2]/2}, {0, Sqrt[2]/2, Sqrt[2]/2}}));

\(ParametricPlot3D[A . {x[t], y[t], z[t]}, {t, 0, 2\ \ π}
, 
      PlotPoints -> 100, ViewPoint -> {3.888, \ 1.4, \ 0.950}]\)

Rotate of the globe

B[t_]=(({{Cos[d t], Sin[d t], 0}, {-Sin[d t], Cos[d t], 0}, {0, 0, 1}}))

Clear[curve]

curve[t_] = B[t] . A . {x[t], y[t], z[t]} // N

curve[t]

The action is given by a rather complicated 3x3 matrix, difficult to guess off-hand:

B[t] . A // MatrixForm

After   d 2 π   revolutions, i.e. after   3   hours:

ParametricPlot3D[curve[t], {t, 0, 2  π}, PlotPoints -> 400, 
      ViewPoint -> {3.888,  0.964,  0.950}]

After   d 8 π   revolutions, i.e. after   12   hours:

ParametricPlot3D[curve[t], {t, 0, 8  π}, PlotPoints -> 1200, 
      ViewPoint -> {3.888,  0.964,  0.950}]

Animation

\(Do[ParametricPlot3D[curve[t], {t, 0, j\ π/2}
, 
        PlotPoints -> 60\ \ j, ViewPoint -> {3.888, \ 0.964, \ 0.950}]
, {j, 
        7, 32}]\)

Question:

What if  a/d  is not a whole number, but rational (a fraction) ?
What if  a/d  is not a rational number, but irrational?

Clear[x, y, z, B, curve]

r = 1; a = 2.5; d = 1/π;

x[t_] = r Cos[a t];
y[t_] = r Sin[a t];
z[t_] = 0;

B[t_]=(({{Cos[d t], Sin[d t], 0}, {-Sin[d t], Cos[d t], 0}, {0, 0, 1}}))

curve[t_] = B[t] . A . {x[t], y[t], z[t]} // N

ParametricPlot3D[curve[t], {t, 0, 80  π}, 
      PlotPoints -> 2000, 
      ViewPoint -> {3.888,  0.964,  0.950}]

Exercise:

Replace 45 degrees by another angle and write the corresponding matrix A.
Repeat the above graphics.