Scientists from UPWr and ESA have solved Einstein's puzzle

In an interview published on Głos Uczelni website, after the first presentation of the team's research results, following a year of analyses, Prof. Krzysztof Sośnica explained: – The general theory of relativity predicts many effects that are very non-obvious at first glance. It is very complicated and does not have a strict mathematical solution, but with certain assumptions many unexpected effects can be concluded. These include the aforementioned space-time vortexes, the rotation of the orbits of satellites and planets, changes in the passage of time, i.e. time dilation depending on the gravitational field and velocity (recently used in so-called chronometric geodesy), black holes and gravitational waves. It is fascinating to explore and confirm these unusual effects and new elements.

The general theory of relativity, presented by Albert Einstein over 100 years ago, still features many undescribed phenomena. Researchers from the Institute of Geodesy and Geoinformatics of UPWr, together with representatives of the European Space Agency (ESA), have published a paper that comprehensively describes what happens to artificial satellites orbiting the Earth and how the general theory of relativity affects the orbits and the movement of satellites. While describing the movement, they managed to discover some quite unexpected effects and predictions that had never been described in literature. The results of work of Prof. Krzysztof Sośnica were presented in October 2020 and March 2021 at a meeting of the ESA GNSS Science Advisory Committee (GSAC) of the European Space Agency as part of special guest papers, which generated a heated debate and have now been published in a scientific journal.

What affects artificial satellites?

The first effect concerning the change of Mercury's perigee position relative to the Sun was already described by Albert Einstein. This was one of the pieces of evidence that allowed the general theory of relativity to be widely accepted in the scientific community. Previously, some suspected that Mercury's strange motion was due to the presence of an extra planet between Mercury and the Sun, which was given the name Volcano. This hypothetical planet explained the change in Mercury's perigee position, but was never discovered. It was not until Einstein succeeded in explaining the disturbances in the motion of the planets through a new theory that described the relationship between time, space, gravity and matter in a comprehensive way.

–  But many other effects acting on the other parameters of the orbits have not yet been described in literature. Our publication fills this gap and presents a description of the perturbation of the parameters of the orbits and the change in the period in which the satellites orbit the Earth. We derived the effects in an analytical way and ran simulations to confirm the correctness of our predictions, – says Prof. Krzysztof Sośnica.

As the Head of the scientific discipline of civil engineering and transport and the Department of Satellite Geodesy at UPWr explains, the theory of relativity makes it possible to separate three main effects affecting the movement of satellites. The Schwarzschild effect is a consequence of the deflection of space-time by the mass of the Earth (treated as a regular sphere), the Lens-Thirring effect is a consequence of the rotation of the Earth around its own axis, which generates so-called space-time vortices that pull satellites with them, and the de Sitter effect, also called geodetic precession, is a consequence of the curvature of space-time by the Sun and the movement of satellites around the Earth moving around the Sun – it is therefore a consequence of the combination of the two motions.

– We described how these three effects affect the size and shape of the orbits and the orientation of the orbital plane relative to outer space. For the first time we showed how the size of the orbits of artificial satellites of the Earth changes due to the curvature of space-time caused by the Earth – says Prof. Sośnica and explains that scientists from UPWr and ESA discovered that the longer semi-axis of the orbit of all Earth's satellites decreases by 17.7 mm.

Surprising findings

– What surprised us is that this value is constant regardless of the altitude at which the satellite is orbiting. It does not matter whether it is 300 km as for low satellites or 36,000 km as for geostationary satellites. The change remains the same. We were also surprised by the sheer value of the change in the longer semi-axis of the orbit, as it is exactly twice the Schwarzschild radius, i.e. the radius of a black hole of the mass of the Earth – admits Prof. Krzysztof Sośnica.

As the UPWr researchers explain, if it were possible to squeeze the entire mass of the Earth into a sphere with a radius of 8.9 mm, then the Earth would become a black hole. Nothing could come out of it, not even light. The radius of a black hole is called the Schwarzschild radius or event horizon, from behind which no information can escape. The scientists studying the general theory of relativity discovered that the change in the semi-major axis of all Earth satellites is exactly twice the Schwarzschild radius.

– And even if the Earth collapsed and became a black hole, the effect on all the satellites of our planet would be -17.7 mm; no matter if the satellites were orbiting high or low above the black hole. We derived a formula for the change in orbit, which is described by a simple equation that is universal for all celestial bodies: -4GM/c^2, where G is the gravitation constant, M is the mass of the celestial body (e.g. Earth) and c is the speed of light in a vacuum – says Prof. Sośnica.

Logic? Not here

The second effect described in the researchers' paper for the first time is the effect of changing the shape of artificial satellites' orbits. Researchers from UPWr and ESA showed that the general theory of relativity changes the shape of orbits in the same way for elliptical and circular orbits. As Prof. Sośnica explains, all orbits undergo flattening, “elliptisation”, in a similar way.

– This surprised us, as logic dictates that the shape-shifting effect should be greater for elliptical orbits, and should be negligible for circular orbits. And yet that is not the case – says Prof. Sośnica and immediately tells us of a third effect, unexpected for the research team, that the value of so-called geodetic precession strongly depends on the angle of inclination of the Sun with respect to the plane of the satellite's orbit.

Millions? Not necessarily

The scientists showed that the effect of geodetic precession is greatest for geostationary satellites orbiting above the equator. Previously, no one paid attention to this because only the average effect was taken into account, not the actual effect resulting from the satellite-Earth-Sun geometry.

– NASA scientists designed the Gravity Probe B mission to confirm the effect of geodetic precession. The mission had an inclination angle relative to the equatorial plane of 90 degrees. And it was expensive – a total of 0 million was spent on it. We demonstrated that a much better solution would be to use satellites orbiting low above the equator and orbits over which the Sun is inclined at the maximum possible angle corresponding to the inclination of the ecliptic plane in relation to the equator. Then the mission would yield much better results in terms of the accuracy of the determined geodetic precession effect – stresses the Head of the Department of Satellite Geodesy at UPWr.

Finally, the researchers showed that the general theory of relativity in weak gravitational fields (neglecting the energy loss associated with gravitational waves), preserves the angular momentum of satellites and the energy of satellites orbiting the Earth over long intervals. At short intervals, the principles of conservation of energy and momentum, as well as Kepler's laws, are broken, which is particularly evident in the case of elliptical orbits.

For more information, see the paper written by a group of scientists from Poland, France, the Netherlands and Spain led by Prof. Krzysztof Sośnica from IGiG UPWr in the Celestial Mechanics and Dynamical Astronomy journal published by Springer Nature: K. Sośnica, G. Bury, R. Zajdel, K. Kazmierski, J. Ventura-Traveset, R. Prieto-Cerdeira, L. Mendes (2021) General relativistic effects acting on the orbits of Galileo satellites. Celestial Mechanics and Dynamical Astronomy 133, 14 (2021).