Listening to the sound of satellites

Last week, components of the TUDelft satellite tracking station were delivered. This includes a radio, amplifiers, Software Defined Radio (SDR), a GPS-based clock and other cool electronics. We are now planning the integration of all the components in the existing ground station, which is used for the data downlink and command uplink of the Delfi-C3 and Delfi-n3Xt (both student designed and built satellites currently orbiting Earth). But how does this tracking (you speak of) works?

It starts with capturing and recording the communications signal of the satellite, which is transmitted continuously (if the Sun shines on the solar cells to produce energy). This is where all the fancy and complicated electronics play a role. (However, I am not an expert in this, so I will only explain the physics behind it). The high frequency (145.870 MHz) electromagnetic signal is down-converted to a signal that can be made digital, such that a computer can process it. First we put the signal through a Fourier Analyses to see its frequency spectrum (The pictures I showed many times on this blog):

So, here we see a part (zoom in) of the frequency spectrum of the recorded signal. The colors represent the amount of energy (sort of) of the signal at a particular frequency (x-axis) at a certain time (seconds after start of recording). What you can see is the characteristic Doppler-curve or S-curve. Due to the Doppler effect (just like sound of a passing ambulance), the recorded frequency of the signal is received higher than transmitted by the satellite when the satellite is flying towards you. The opposite happens when the satellite is flying away from you. The received frequency of the satellite is smaller than is transmitted by the satellite. So if the satellite was not moving with respect to you, you would see vertical straight lines in the figure above. 

Now comes the beauty of the physics! When we measure this frequency change, we can get information about the velocity of the satellite and its trajectory. Maybe you know it from your high-school physics book:

The difference in observed frequency (f) with respect to the transmitted frequency (f0) is the transmitted frequency multiplied with the ratio of the relative velocity (v) and the speed of the signal (c, which is the speed of light). The relative velocity is the change in distance in time between the transmitter and the receiver. In our case is the satellite transmitting and we are receiving. So in theory by measuring the received frequency and knowing the transmitted frequency (which we not exactly know) we can deduce the relative velocity of the satellite with respect to the ground station. We like to call this observable the range-rate.

Some of my students did this exercise using the signal of the Delfi-C3 that we recorded. With some assumptions on the transmitted frequency and neglecting atmospheric refraction, the following range-rate profile of the pass was obtained:

They used two different assumptions for the transmitted frequency (blue and red), but that is not what I want to show you. Look at the y-axis, the relative velocity is plus/minus 6 km/s. So at the beginning of the pass the satellite is flying with 6 km/s towards the ground station (lets assume the ground station is not moving). Then at around 460 seconds after the start of the recording the relative velocity between the satellite and the station is 0 m/s. This is exactly the Time when the satellite is at Closest Approach (TCA) or just above you. From then on the satellite will fly away from you. This change in relative velocity is not because the satellite slows down, stops, and speeds up. No, the velocity of the satellite is constant (well almost, but lets not go into the details). 

The change in relative velocity is because the satellite's location also changes. The following cartoon illustrates this.

What we measure is a component of the velocity of the satellite, the component towards us. At the horizon (when we start hearing the satellite), this component is largest due to the geometry of the setup. During the first part of the flight of the satellite this component becomes less (not well drawn in my cartoon, I am sorry for this), because the angle between the velocity direction  and the relative velocity component decreases. Just overhead, this angle is 0 degrees, which means that the velocity component in the direction of the satellite is zero. Also at this moment you are receiving the transmitted frequency of the satellite. The signal experiences no Doppler shift (This was assumption one of the students (blue)). However after this point, the relative velocity increases again, resulting in a change of received frequency. 

So by listening to the sound of satellites, you can predict its orbit and velocity. But how you transform range-rate observations into orbit determinations is a science in it self. I will need more than a few blogs to teach you this.