Hands-On Math

Last Revision March, 2011

Last Revision March, 2011

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

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.

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.

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.

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