Studying Es’hail 2 Doppler

Es’hail 2, the first geostationary satellite carrying an Amateur radio payload, was launched on November 15. I wrote a post studying the launch and geostationary transfer orbit, and I expected to track Es’hail 2’s manoeuvres by following the NORAD TLEs. However, for reasons not completely known, no NORAD TLEs were published during the first two weeks after launch.

On November 23, people found Es’hail 2 around the 24ºE geostationary orbital slot by receiving its Ku-band beacons at 10706MHz and 11205MHz. On November 27, NORAD TLEs started being published, confirming the position of Es’hail 2 around 24ºE. Since then, it has remained in this slot. Apparently, this is the slot that will be used for in-orbit test before moving the satellite to its operational slot on 25.5ºE or 26ºE.

Since November 27, I have been monitoring the frequency of the 10706MHz beacon to measure the Doppler. A geostationary satellite is never in a fixed location as seen from the Earth. It moves slightly due to imperfections in its orbit and orbital perturbations. This movement is detectable as a small amount of Doppler. Here I study the measurements I’ve been doing.

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Es’hail 2 launch in GMAT

After a long wait by many Amateur radio operators, Es’hail 2, the first geostationary satellite carrying an Amateur radio transponder launched yesterday on a SpaceX Falcon 9 Full Thrust from the historical LC-39A pad in Kennedy Space Centre, which was used by many Apollo and Space Shuttle launches in the past.

Es’hail 2 is the second communications satellite operated by the Qatari company Es’hailSat. It was built by Mitsubishi Electric Corporation (MELCO). It carries several Ku and Ka band transponders intended for digital television, Internet access and other data services. It also carries an Amateur radio payload designed by AMSAT-DL, in collaboration with the Qatar Amateur Radio Society. The payload has two transponders, with S-band uplink and X-band downlink. One of the transponders is 250kHz wide and intended for narrowband modes, and the other one is 8MHz wide and intended for DVB-S and other wideband data modes.

SpaceX live-streamed the launch, and the recording can be seen in YouTube. Today, Space-Track has published the first TLEs for Es’hail 2 and the second stage of the Falcon 9 rocket. Here I look at these TLEs using GMAT.

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Geometry for DSLWP-B Moonbounce

I have already spoken about the Moonbounce signal from DSLWP-B in several posts. To sum up, DSLWP-B is a Chinese satellite that is orbiting the Moon since May 25. The satellite has an Amateur payload that transmits GMSK and JT4G telemetry in the 70cm Amateur satellite band. This signal can be received by well equipped groundstations on Earth, including the 25m radiotelescope at Dwingeloo, in the Netherlands (and also by much smaller stations).

The people at Dwingeloo publish the recordings that they make of the RF signal. In two of these recordings, the signal from DSLWP-B is received not only via the direct path, but also through a reflection off the Moon’s surface. The reflected signal is around 25dB weaker, usually has a different Doppler shift, and has a Doppler spread of around 50 to 200Hz.

What I find most interesting about this is that of all the days that Dwingeloo has observed DSLWP-B, in only two of them (on 2018-10-07 and 2018-10-19) the Moonbounce signal has been visible. Mathematically, using a specular reflection on a sphere model, whenever the satellite is visible directly, there is also a ray from the spacecraft that reflects off the lunar surface and arrives at the groundstation (see the proof here). Therefore, I think that there must be something about the particular geometry of the days 7th and 19th that made the Moon reflections relatively stronger and hence detectable. Here I use GMAT to study the orbital geometry when the reflections were received.

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DSLWP-B Moonbounce

If you have been following my latest posts, you will know that a series of observations with the DSLWP-B Inory eye camera have been scheduled over the last few days to try to take and download images of the Moon and Earth (see my last post). In a future post I will do a chronicle of these observations.

On October 6 an image of the Moon was taken to calibrate the exposure of the camera. This image was downlinked on the UTC morning of October 7. The download was commanded by Reinhard Kuehn DK5LA and received by the Dwingeloo radiotelescope.

Cees Bassa observed that in the waterfalls of the recordings made in Dwingeloo a weak Doppler-shifted signal of the DSLWP-B GMSK signal could be seen. This signal was a reflection off the Moon.

As far as I know, this is the first reported case of satellite-Moon-Earth (or SME) propagation, at least in Amateur radio. Here I do a Doppler analysis confirming that the signal is indeed reflected on the Moon surface and do some general remarks about the possibility of receiving the SME signal from DSLWP-B. Further analysis will be done in future posts.

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Lunar orbit perturbations and DSLWP-B

Ever since simulating DSLWP-B’s long term orbit with GMAT, I wanted to understand the cause of the periodic perturbations that occur in some Keplerian elements such as the excentricity. As a reminder from that post, the excentricity of DSLWP-B’s orbit shows two periodic perturbations (see the figure below). One of them has a period of half a sidereal lunar month, so it should be possible to explain this effect from the rotation of the Moon around the Earth. The other has a period on the order of 8 or 9 months, so explaining this could be more difficult.

DSLWP-B eccentricity

In this post I look at how to model the perturbations of the orbit of a satellite in lunar orbit, explaining the behaviour of the long term orbit of DSLWP-B.

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DSLWP-B long-term orbit evolution

In my last post I commented that one of the motivations for the periapsis raise manoeuvre of DSLWP-B on July 20 was to prevent the satellite from crashing into the Moon in a few months. When Wei Mingchuan BG2BHC told me this, I found it a bit surprising, since I had the impression that the periapsis had been raising naturally since the satellite was injected in lunar orbit on May 25. Thus, I decided to propagate the orbit for a period of 2 years using GMAT and study the long-term effects in the Keplerian elements.

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Update on DSLWP-B eclipse manoeuvre

A few days ago I discussed the manoeuvre performed by DSLWP-B in preparation for the lunar eclipse. The manoueuvre raised the periapsis of DSLWP-B by around 385km. Wei Mingchuan BG2BHC has now informed me that the manoeuvre was performed on 20 Jul 2018 10:47:09.657. There were two motivations for this manoeuvre: first, to avoid eclipse, as I showed in the previous post; second, as Wei tells me, to prevent DSLWP-B from crashing into the Moon in a few months (more on this in a future post).

Wei doesn’t know the delta-v used for the manoeuvre, but estimating it is an easy exercise using GMAT, which is what I will do in this short post. In this simulation I am taking the orbital state for DSLWP-B from the first line of the 20180714 tracking file published in dslwp_dev. I will assume that the manoeuvre was a prograde burn performed at apoapsis that raised the periapsis by 385km. The GMAT script I have used is lunar_eclipse_manoeuvre.script.

First I propagte the orbit to the date mentioned by Wei. I note that the spacecraft is a little short of apoapsis, so I propagate to apoapsis, which happens at 20 Jul 2018 10:49:33.178 UTC. Then I propagate to periapsis and take note of the periapsis radius, which is 3030.91km. Finally, I use GMAT to estimate a burn that will achieve a periapsis radius of 3415.91km using a differential corrector.

The differential corrector finds a delta-v of 17.2m/s. The iterations of the differential corrector can be seen in the figure and text below. A more difficult exercise is to find a burn that stitches together the orbits described by the 20180714 and 20180727a tracking files. I will leave this as an exercise for the reader. Something very similar was done in DSLWP-B’s journey to the Moon: part II.

Differential corrector solving for the periapsis raise burn

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Comparison of DSLWP-B orbit determination with tracking files

In the last post I compared the results of my orbit determination for DSLWP-B using one lunar month of Doppler data with the observations in the VLBI experiment done on June 10. In this post I will compare my orbit determination with the tracking files published by Wei Mingchuan BG2BHC in gr-dslwp. These tracking files are produced from the orbit determination performed by the Chinese Deep Space Network using two-way S-band Doppler measurements.

The tracking files contain a listing of the position \(x\) and velocity \(v\) vectors for DSLWP-B in ECEF coordinates. The entries are given at intervals of one second. The tracking files can be used directly to compute Doppler as received in a groundstation. In fact, if the ECEF coordinates of the groundstation are \(x_0\), then \(R = \langle x – x_0, v\rangle/\|x-x_0\|\) is the range-rate, and so the Doppler can be computed as \(D=-fR/c\), where \(f\) is the downlink frequency and \(c\) is the speed of light in vacuum. Here I have used this method to compute the Doppler according to the tracking files.

All the tracking files published so far have been considered in this study, except for the first two, which contained an incorrect anomaly at epoch. The figure below shows the residuals between the Doppler measurements made by Scott Tilley VE7TIL and my orbit determination (called “new elements”) and each of the tracking files. It seems that the residuals are quite similar.

The figure below shows the difference between the Doppler according to each of the tracking files and the Doppler predicted by my orbit determination.

It seems that the match is quite good for the central days, but not so good towards the edges. My orbit determination is numerically propagated from a single set of elements at MJD 28264.5 for the whole period, while the tracking files probably use different sets of elements that are propagated numerically over a few days only. Therefore it might happen that my orbit determination is affected by some accumulative error due to numerical integration or an inaccuracy in the force model.

Validation of DSLWP-B orbit determination using VLBI observations

In my last post I presented my orbit determination of DLSWP-B using one lunar month of S-band Doppler measurements made by Scott Tilley VE7TIL. In this post, I will use the delta-velocity measurements from the VLBI experiment on 2018-06-10 to validate my orbital elements.

In the figures below, I compare between the sets of data: The old elements, obtained in this post:

DSLWP_B.SMA = 8762.40279943
DSLWP_B.ECC = 0.764697135746
DSLWP_B.INC = 18.6101083906
DSLWP_B.RAAN = 297.248156986
DSLWP_B.AOP = 130.40460851
DSLWP_B.TA = 178.09494681

The new elements obtained in my last post:

DSLWP_B.SMA = 8765.95638789
DSLWP_B.ECC = 0.764479041563
DSLWP_B.INC = 23.0301858287
DSLWP_B.RAAN = 313.64185464
DSLWP_B.AOP = 113.462338342
DSLWP_B.TA = 178.5519212

The elements obtained from the 20180610 tracking file published by Wei Mingchuan BG2BHC in dslwp_dev. This tracking file contains a list of ECEF position and velocity vectors for DSLWP-B. The first entry is taken as the orbital state and the orbit is propagated in GMAT, as done in this post. It would also be possible to calculate the delta-velocity directly from the ECEF data, but the results would be fairly similar and I already have a script to do it with GMAT orbit propagation.

A degree 2 polynomial fit to the VLBI observations. It turns out that the delta-velocity during the VLBI experiment can be approximated fairly well by a parabola, so it makes sense to use this as a reference. Note that this also implies that this set of delta-velocity measurements alone would be insufficient to perform orbit determination, as a degree 2 polynomial gives us 3 coefficients, while we would need to determine 6 parameters for the orbital state. Adding delta-range would only give us an extra variable, so orbit determination using VLBI would need several sets of measurements well space in time so that the orbit can be observed at different anomalies.

The figures below show a comparison between the four sets of data and the VLBI measurements.

The RMS errors are respectively 0.248m/s for the old elements, 0.167m/s for the new elements, 0.156m/s for the tracking file, and 0.137m/s for the polynomial fit. Thus, we see that the newer elements represent a good improvement over the older elements. The tracking file gives a slightly better result than the new elements. However, the new elements should give good results over a long time span of over 20 days, as we have seen in the previous post, while the orbital parameters derived from the tracking files tend to change often.

The Jupyter notebook used to make these calculations has been updated in here.