A few days ago, Bert Hubert, the creator of galmon.eu, discovered a sinusoidal oscillation in the clock drift \(a_{f1}\) parameter of the broadcast ephemerides of Galileo satellites. This variation has a frequency that matches the orbital period of 14 hours and 7 minutes. At first, I suggested that it might be caused by relativistic effects, which are given by\[-\frac{\sqrt{\mu}}{c^2}e\sqrt{A}\sin E,\]where \(\mu\) is the Earth’s gravitational parameter, \(c\) is the speed of light, \(e\) is the eccentricity, \(A\) is the semi-major axis, and \(E\) is the eccentric anomaly. In fact, the order of magnitude of the oscillations that Bert was seeing seemed to agree with this formula.

However, then I realised that this relativistic effect is not included in the broadcast clock model. It needs to be included back by the receivers. Therefore, it shouldn’t appear at all in the broadcast clock. Something didn’t seem quite right. This post is an in-depth look at this problem.