Chapter 7
Part B - Does The Found Orbit Exhibit
the Keplerian Properties?
Review of Previous Topic
A reference ellipse was scaled up by a factor of 10^7. A focus was presumed to
be at Earth's centre and the scaled ellipse was presumed to represent the path of
an Earth satellite.
The altitude corresponding to the satellite altitude at perigee was provided to the
3D spreadsheet.
The period of the satellite was calculated using Newton's refinement of Kepler's
third law and the parameters of the scaled ellipse.
The tangential satellite velocity provided to the spreadsheet was selected such
that the period found by the spreadsheet agreed closely with the period that was
calculated from the parameters of the scaled ellipse.
The spreadsheet was then used to calculate the sequential positions versus time
of the satellite. The xy plot of these positions is reintroduced next.
The plot consists of 100 plotted output points beginning at coordinates 10, 0 and
ending at substantially the same position. This is accomplished by first choosing
the step size as an integer fraction of the period. If there are to be 100
output points per period, then some selected integer number of calculations must
be
specified for every output. As an example:
Step Size = Period/10,000
Steps per Output = 100
Number of Outputs = 100
The Three Conclusions of Kepler
Newton, with his Three Laws of Motion and his Theory of Universal Gravitation, was
able to show that Kepler's three conclusions, made about 50 years earlier, were
correct. The conclusions were:
1. The planets followed elliptical orbits.
Prior to Kepler, planet orbits were believed to be circular;
2. A line joining a planet and the sun sweeps out equal areas during equal intervals
of time;
3. A planet's period is proportional to the cube of the length of the semi-major
axis of its orbit.
If the implementation of the spreadsheet is correct then its calculated results
will allow the first two Kepler conclusions to be made.
Is the Satellite Orbit Elliptical?
How well this is substantiated should depend on the step size used in the calculations.
Fixing 100 outputs per period, several step sizes can be chosen for testing
by selecting Steps per Output.
The chosen test method is to compute and compare the sum of the distances from the
foci to each output position and to note the distance of the final point from the
starting point. The maximum and minimum values of the x coordinate are used to derive
the second focus
As the first example, with step size chosen as 1/1,000 of the period and with 10 steps
per output the results at the output points were:
The second focus was at ~ - 44.9 megametres.
The length of semi-major axis was ~ 32.45 megametres.
The average computed sum was ~ 64.89 megametres.
The computed sums had ~ 0.12% rms relative variation from
the average computed sum.
The end point was ~ 0.76 megametres from the starting point.
As the second example, step size is reduced by a factor of ten with the results:
The second focus was at ~ - 44.999 megametres.
The length of semi-major axis was ~ 32.499 megametres.
The average computed sum was ~ 64.999 megametres.
The computed sums had ~ 0.0013%
rms relative variation from
the average computed sum.
The end point was ~ 0.034 megametres from the starting point.
As the third example, step size is further reduced by a factor of ten, (calculations
every 0.583,090,505,733,156 seconds), with the results:
The second focus was at ~ - 44.99999 megametres.
The length of semi-major axis was ~ 32.49999 megametres.
The average computed sum was ~ 64.99999 megametres.
The computed sums had ~ 0.000013%
rms relative variation
from the average computed sum.
The end point was ~ 0.0047 megametres from the starting point.
It appears quite safe to conclude that the shape of the satellite path rapidly approaches that of an ellipse as step size is reduced.
Does a Line Joining the Satellite and Earth's
Centre Sweep Out Equal Areas in Equal Times?
As step size becomes small, the area of the triangular shaped wedge of space that
is bounded by the lines joining two adjacent calculation points to earth's centre
and the line joining the two points can be taken as representing the area swept
in each step.
The accumulation of these small areas between output calculations should vary little
from one output to another if Kepler's second observation is valid. The variation
can be expected to depend on step size.
Once more, 100 regularly spaced time intervals are employed to produce 100 locations
on the satellite path with the same 3 choices of step size as used when testing
for elliptical shape.
For the larger step size the average accumulated swath area at an output point was ~2.15*10^13 square megametres.
The relative rms variation in the accumulated areas
at the output points was ~ 0.034%.
With 1/10 that step size the average swath area was ~ 2.37 *10^13 square megametres.
The relative rms variation in the areas was ~ 0.00033%.
Reducing step size by a further factor of ten led to an average swath area of ~
2.39 *10^13 square megametres. The relative rms variation in the areas was ~ 0.0000033%.
It appears safe to conclude that equal areas are swept out in equal times.
Kepler's Third Conclusion, the 3D Spreadsheet and Newton
After employing Newton's version of Kepler's third conclusion to construct the orbit
of a satellite by scaling up an ellipse, we have shown with our 3D spreadsheet that
the satellite orbit is in accord with the first two of Kepler's conclusions when
step size is made small. In effect, we have shown that our 3D spreadsheet
behaves in accord with Newton's Three Laws of Motion and his Theory of Universal
Gravitation.