The International Geophysical Year spurred the launch of the first artificial Earth satellites. The year was devoted to a study of the Earth and led to major gains in the field of geodesy. There is a good review on the subject at NOAA entitled Geodesy for the Layman. In the chapter on Satellite Geodesy there is a figure with a close-approach diagram similar to the one I used. I could not find a formula for the frequency vs time curve but that could mean at the time it was available elsewhere. Some of Guier's papers are cited in Kaula, Theory of Satellite Geodesy, 1966.
Doppler shifts were used to determine Sputnik's orbit in 1957. Subsequently, the US Navy's Transit satellites which were part of its NAVSAT system used the Doppler effect to determine positions on the Earth.
ESA, the European Space Agency, has made space more accessible to some new users as can be seen from the Vega VV02 launch Tuesday morning. Riding up with the Proba-V was Vietnam's VNREDSat 1A and Estonia's ESTCube-1 student satellite. Tracking satellites visually can be difficult because they have to be in sunlight to be seen and then daytime is too bright and at night this just leaves times just before sunrise and after sunset when they are not in the Earth's shadow. ESTCube-1 sends out coded messages on two amateur radio frequencies. This suggests the possibility that one can use the Doppler Effect to acquire some information about the satellite's location. Consider what happens when the satellite makes a close approach to a known position, x0, of the Earth's surface.
Over a short period of time the satellite's motion can be approximated by a uniform movement on a straight line. The range of the satellite is r and at some point in time it reached the close approach distance rm. As the satellite passes by the frequency received is affected by its relative motion and a plot of the frequency vs time looks something like the plot below which was done for an object emitting a steady tone as it passes by an observer on the ground. As the object approaches the observed frequency is higher than the emitted frequency and as it moves away it is lower.
To use this information we need to relate the shape of the frequency curve to range and velocity of the object. One can derive the following set of formulas where omega is the angular frequency and c the velocity of the transmitted wave.
One gets the formula used to plot the frequency curve which contains two unknown parameters, the minimum distance, xm here, and the velocity. The velocity can be found from the minimum and maximum ratios of the transmitted frequency and the observed frequency. The close approach distance can then be estimated by fitting it to the frequency observations. So it's not too difficult to estimate the range and velocity at the time of close approach. One may want to keep tabs on ESTCube-1 since it will be testing an e-sail tether in an attempt to make controlled changes to its orbital parameters.
As was mentioned in previous blogs the maximum precision available in Mathcad 11 is about 15 decimal places. The limit is somewhat fuzzy and the roundoff errors can accumulate with the sum of a large number of terms. If the roundoff error is δ=0.5x1015 for a number then the error for the sum of n numbers can be as much as n·δ so one needs to take the necessary care to avoid errors of this kind. I wrote some utilities to work with "big numbers" in Mathcad and did a check on the Mathcad's performance.
Note that Mathcad made a roundoff error on the sum of u and v above. The correlation matrix for the curve fit procedure involves the sum of squares and other even powers of xk which can bias the result. The expected value for the error of the sum is zero but the standard deviation of this sum is proportional to the square root of the number of data points.