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Scientists from UPWr and ESA have solved Einstein’s puzzle resulting from general relativity

28-03-2021

Although Einstein’s general theory of relativity is over 100 years old, it still contains a plethora of as yet undescribed phenomena. Scientists from the Institute of Geodesy and Geoinformatics (IGG UPWr) together with representatives of the European Space Agency (ESA) have published an article that comprehensively describes what happens with artificial satellites orbiting the Earth and how the theory of general relativity affects the orbits of satellites. The authors discovered some unexpected effects and predictions that have never been described in the literature before. Prof. Krzysztof Sośnica presented results from this study in October 2020 and in March 2021 at the meeting of the GNSS Scientific Advisory Committee of the European Space Agency (GSAC) as part of a special invited paper, which triggered a heated discussion. Now, these results have been published in Celestial Mechanics and Dynamical Astronomy by Springer Nature.

Theory of general relativity

General relativity results in a number of effects that could not be described using the principles of classical mechanics. The effects concern the concept of time, the curvature of the path along which light travels, and the motion of celestial bodies. Scientists from IGG UPWr and ESA looked at the effects of the movement of artificial Earth satellites.

The first effect of changing the position of Mercury’s perigee in relation to the Sun was already derived by Albert Einstein. This was one of the pieces of evidence that allowed for general relativity to be widely accepted in the scientific community. Previously, some suspected that Mercury’s strange movement was due to the presence of an additional planet between Mercury and the Sun, which was given the name Volcano. This hypothetical planet explained the shift in the position of Mercury’s perigee but had never been discovered. Only Einstein managed to explain the disturbances in the motion of the planets through a new theory describing the relations between time, space, gravity, and matter in a comprehensive manner.
However, many other effects affecting orbital parameters have not been described in the literature so far. The publication of scientists from UPWr and ESA fills this gap and presents a description of the perturbations of the orbital parameters and the change of the revolution period of satellites orbiting the Earth. Scientists derived the results in an analytical manner and conducted simulations confirming the correctness of their predictions.

The movement of artificial satellites in the theory of relativity

The theory of relativity allows distinguishing three main effects affecting the movement of satellites:
1. The Schwarzschild effect resulting from the deflection of space-time by the mass of the Earth (treated as a regular sphere),
2. The Lens-Thirring effect being a consequence of the Earth’s rotation around its axis, which generates the so-called space-time dragging which pulls the satellites,
3. The De Sitter Effect, also known as geodetic precession, which is a consequence of the Sun’s curvature of space-time and the movement of Earth’s satellites around the Sun - is, therefore, a consequence of combining two movements.
The article describes how all three effects affect the size and shape of the orbits as well as the orientation of the orbit plane with respect to outer space.

What relativistic effects have been described for the first time?

The first effect, which has not been described in the literature so far, is the change in the size of the orbits of artificial Earth satellites due to the curvature of space-time generated by the Earth.

It has been found that the semi-major axis of the orbit of all Earth satellites is reduced by 17.7 mm. The researchers were surprised to find that this value was constant no matter at what altitude the satellite is orbiting. Whether it is a low orbiting satellite at an altitude of 300 km or a geostationary satellite at an altitude of 36,000 km, the change is the same. An additional surprise was the value of the change of the orbit semi-major axis, because it is exactly twice the Schwarzschild radius, i.e. the radius of a black hole with the mass of the Earth.

Should the entire mass of the Earth be squeezed into a sphere with a radius of 8.9 mm, then the Earth would become a black hole. Nothing, not even light, could get out of it. The black hole radius is called the Schwarzschild radius or event horizon from which no information can escape. Scientists have found that the change in the semi-major axis of all Earth’s satellites is exactly twice the Schwarzschild radius. Even if the Earth were to collapse and become a black hole, the effect across all Earth’s satellites would be -17.7 mm; no matter if the satellites would orbit high or low above a black hole. The authors derived the formula for the orbit change, which is described by a simple equation universal for all celestial bodies: -4GM/c^2, where G is the gravity constant, M - the mass of a celestial body (e.g. the Earth), c - the speed of light in vacuum.

The second effect described for the first time is the effect of changing the shape of the orbits of artificial satellites. Scientists have found that General Relativity changes the shape of the orbits in the same way for elliptical and circular orbits. All orbits flatten, we may say elliptize, in a similar way. This is quite surprising, as logic would suggest that the shape-changing effect should be greater for elliptical orbits, while for circular orbits, it should be negligible.

The third unexpected effect was that the value of the geodetic precession strongly depends on the angle of the Sun inclination above the orbital plane. Scientists have shown that the geodetic precession effect is greatest for geostationary satellites. Previously, no one paid attention to this, because only the average de Sitter effect was taken into account, not the real one 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 of 90 degrees with respect to the equatorial plane. The Gravity Probe B cost in total $750 million. Scientists from IGG and ESA proved that a much better solution would be to use satellites orbiting at low inclination angles above the equator and over which the Sun is tilted at the maximum possible angle, corresponding to the inclination of the elliptical plane relative to the equator. Then, the mission would bring much better results in terms of the accuracy of the determined geodetic precession effect.

Ultimately, scientists showed that general relativity in weak gravity 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. However, at short intervals, the conservation of energy and momentum laws are broken, which is especially evident in the case of elliptical orbits.

More information can be found in the article by a group of scientists from Poland, France, the Netherlands, and Spain 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). https://doi.org/10.1007/s10569-021-10014-y .