Es’hail 2 differential Doppler measurements

Since I published my Es’hail 2 Doppler measurement experiments, Jean Marc Momple 3B8DU has become interested in performing the same kind of measurements. The good thing about having several stations measuring Doppler simultaneously is that you can perform differential measurements, by subtracting the measurements done at each station. This eliminates all errors due to transmitter drift, since the drift is the same at both stations.

Of course, differential measurements need to be done with distant stations, to ensure different geometry that produces different Doppler curves in each station. Otherwise, the two stations see very similar Doppler curves, and subtracting yields nothing.

The good thing is that Jean Marc is in Mauritius, which, if you look at the map, is on the other side of the satellite compared to my station. The satellite is at 0ºN, 24ºE, my station is at 41ºN, 4ºW, and Jean Marc’s is at 20ºS, 58ºE. This provides a very good geometry for differential measurements.

Some days ago, Jean Marc sent me the measurements he had done on December 22, 23 and 24. This post contains an analysis of these measurements and the measurements I took over the same period, as well as some geometric analysis of Doppler.

It would be interesting if other people in different geographic locations join us and also perform measurements. As I’ll explain below, a station in Eastern Europe or South Africa would complement the measurements done from Spain and Mauritius well. If you want to join the fun, note a couple of things first: The Doppler is very small, around 1ppb (or 10Hz). Therefore, you need to have everything locked to a GPS reference, not only your LNB. Also, the change in Doppler is very slow. The Doppler looks like a sinusoidal curve with a period of one day. To obtain meaningful results, continuous measurements need to be done over a long period. At least 12 hours, and preferably a couple days.

The raw measurements done by Jean Marc can be seen in the figure below. He wasn’t able to record continuously during the three days of measurement, but the characteristic sinusoidal Doppler curve is well visible, meaning that this data is appropriate for further analysis.

Note the frequency jump that happens around 2018-12-22 08:20 UTC. This is probably caused by Jean Marc’s equipment not being correctly locked in frequency at the start of the measurements. Thus, we will discard the measurements done before this point.

The figure below shows the processed measurements by Jean Marc and myself. I have used averages of 5 minutes and subtracted the mean to remove the transmitter frequency offset.

As you can see, I don’t have measurements for a large part of December 23 and 24. During those days I was monitoring the Amateur transponder tests, so I stopped my frequency measurements. Although there is not a lot of temporal overlap between Jean Marc’s measurements and mine, there are still enough measurements to have a good idea of what is happening.

In the figure above I have also included the Doppler predicted by NORAD TLEs (using only one TLE: the one which had the closest epoch). You can see that the Doppler seen by Jean Marc’s station and by my station is very similar. In what follows, I’ll do a geometric analysis to explain why and to show what can be gained by using differential measurements (besides eliminating the transmitter drift). I already did some geometric remarks in my previous post about Es’hail 2 Doppler, so perhaps it’s better to read those first.

The figure below shows the position of Es’hail 2 during the measurement period in latitude, longitude and altitude coordinates. To compare them, note that an movement of 0.1º in latitude or longitude at geostationary orbit corresponds to 74km approximately. This means that the movement in altitude has an amplitude of 7km, and the movements in latitude and amplitude have an amplitude of 1km.

Besides the amplitude of each of the movements, it is also important how sensitive is the line of sight vector to movements in each of these directions, as I remarked in my previous post. This depends on the projections of each of these directions onto the line of sight vector, which depends on the geometry given by the geographic location of the groundstation.

For EA4GPZ, the projections for altitude, latitude and longitude are respectively 0.992, -0.108 and 0.059. For 3B8DU, they are 0.994, 0.059 and -0.088. With this data, we can plot the amount of Doppler that is caused by the movements in each of the directions, so that the total Doppler is the sum of three components. This is shown in the two figures below.

For the geometry of EA4GPZ, we see that most of the Doppler is due to altitude changes, while the movements in latitude and longitude contribute in small equal magnitudes.

The situation for 3B8DU is very similar, except that the signs of the Doppler due to latitude and longitude changes are opposite, due to the two stations been located on opposite sides of the satellite.

As we have seen, both stations have a sensitivity to altitude movements close to one. This is indeed the case for all the stations on the surface of Earth, due to the fact that the geostationary radius is much larger than the Earth radius. This means that the Doppler due to altitude changes will be very similar in all stations. By subtracting, we can eliminate most of the contributions of this component and study the contributions from latitude and longitude changes.

The figure below shows the geometry of the differential Doppler between EA4GPZ and 3B8DU. As we expected, most of the contribution of altitude changes is eliminated. The contributions of latitude and longitude changes result approximately opposite, so the final full Doppler is something small.

The reason for this is that the differential Doppler between EA4GPZ and 3B8DU is sensitive to movements in the northwest-southeast direction, while the satellite was moving on a northeast-southwest direction, as the figure below shows (the dot indicates the position at the start of the measurement interval). Thus, a station located in Eastern Europe or South Africa would complement well the geometry of EA4GPZ and 3B8DU by giving sensitivity in the northeast-southwest direction.

The differential Doppler measurements between EA4GPZ and 3B8DU are shown in the figure below. As remarked above, simultaneous measurements with both stations are only available for small periods of time, unfortunately.

Still, we see that the available measurements don’t match the predictions by the TLE. Since the experimental setup is very precise (everything is GPDSO locked, we are performing differential measurements to eliminate transmitter frequency offset, and the rate of change of Doppler is very slow, so precise time-tagging of measurements is not critical), this clearly indicates an error in the TLEs. Thus, we are improving upon the TLEs with our Amateur orbit determination system.

Note that, as shown in the figure above, Es’hail 2 main movement was along a northeast-southwest line, and our measurement setup is sensitive to movements orthogonal to that line. This means that we’re measuring small deviations from the dominating trend in latitude-longitude movement (which is caused by the inclination, the RAAN and the argument of perigee). These small deviations are very sensitive to the Keplerian parameters and orbital perturbations, so it is not very surprising that the TLEs don’t match well our differential measurements.

It would be very interesting to continue these differential measurements over a window of a couple of days, and adding a third station that provides sensitivity to the northeast-southwest direction.

The calculations in this post have been done in this Jupyter notebook.

Update 2018-12-31: I have just noticed that I had another set of measurements from Jean Marc. The complete set of measurements can be seen below.

I have redone all the plots above including these measurements.

Also, to show how sensitive is the differential Doppler to small changes in the TLEs, I have redone all the calculations using the nearest TLE to 2018-12-24 (the above plots use the nearest TLE to 2018-12-23). Note that the behaviour of the satellite position in latitude and longitude is rather different.

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