April 23, 2020

The European Space Agency (ESA) will allocate funds for research within the project headed by UPWr: "Relativistic effects in the orbits of Galileo satellites". The contract with ESA was signed in April 2020.

The project is supposed to verify the validity of Einstein's general theory of relativity in terms of curvature of spacetime by massive objects – says Prof. Krzysztof Sośnica, the Project Manager. For the first time, the curvature of spacetime will be measured in a direct way that will allow us to enter the curvature value in metric units. To measure the curvature of spacetime we will be using the technique of satellite geodesy, and more precisely, the European Galileo satellite system. Until now, no one has metrically measured the direct curvature of spacetime, since these effects are relatively small in comparison to other forces impacting artificial satellites. On the other hand, measurements of extra-terrestrial objects are usually made using relative measurements, not direct measurements of distances, whose range is limited to circum-terrestrial objects.

*Illustrative drawing of the orbit in flat spacetime and spacetime curved by the Earth – by Krzysztof Sośnica*

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**What did Einstein predict in his theory?**

- Copernicus stated that all planets move in circles – explains Prof. Krzysztof Sośnica. Kepler, while observing the orbit of Mars, came to the conclusion that it is elliptical and the speed of Mars changes depending on the distance from the Sun. Newton formulated the law of universal gravitation and explained the source of planetary elliptical motion with the forces of gravity between individual planets, moons and the Sun. Newton's theory explains very well the majority of phenomena occurring in the Solar System and even outside of it. The very small effects observed in the change in Mercury's orbit orientation could not be explained using Newton's theory, but they were so small that some claimed they were due to measurement errors in Mercury's optical observations.

- Einstein turned the whole theory of gravity upside down, formulating the general theory of relativity – says Prof. Krzysztof Sośnica.

Einstein's theory states that the reason for the motion of planets is not the force of gravity but the curvature of spacetime caused by matter – that is mass and energy. Thus, curved spacetime tells objects how to move, while objects with mass tell spacetime how to curve.

- The theory explains many phenomena, most of which were confirmed after Einstein's death, including the aforementioned anomalies in Mercury's movement, gravitational lens, black holes, system dragging, i.e. spacetime vortices, time dilation or slowing the passage of time at massive objects, gravitational waves and even the expansion of the Universe. However, the effect that shows the orbits of celestial bodies to be not perfect ellipses, as would result from the theory of Kepler and Newton, but "pear-shaped", as results from the general theory of relativity, has not yet been directly measured – says Prof. Krzysztof Sośnica.

*Illustrative drawing showing the difference in shape and inclination of the nominal circular Galileo orbit and the orbit of satellites launched into elliptical orbits resulting from an accident
in the orbit*

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**Galileo – an orbital accident**

Galileo is a European navigation system – the equivalent of American GPS. The difference from the GPS system is that Galileo allows for three times more accurate positioning, has ultra-accurate atomic clocks on board – hydrogen masers, and is a civil system, not a military one, like GPS is. The Galileo system is financed by the European Union and ESA – hence also by Poland, as it is a member of both the organizations.

The first pair of fully operational Galileo satellites were to be launched into orbit on August 22, 2014. However, there was a failure during the final stage of the launch of the satellites by the Russian Soyuz rocket – fuel lines for the final orbit correction froze. As a result of the failure, the satellites, instead of being on circular orbits at an altitude of 23.225 km and an inclination of the orbit in relation to the equator at an angle of 56°, were in elliptical orbits at an altitude of 17,000 to 26,000 km and an inclination of 50°.

The variable height of the satellites above the Earth makes it impossible to use them for navigation due to the impossibility of sending information about the position of the satellites, because the standard data format of the so-called navigational message does not allow that. The satellites, however, proved to be fully operational. Initially, ESA planned to shut them down because their primary purpose could not be met. However, the scientific community issued a number of letters asking to keep the satellites active, because, as it turned out, they can be used for geodetic measurements, for determining the parameters of the Earth's rotational motion, and for verifying the effects connected with relativity theory.

*The number of visible GPS and Galileo satellites depending on geographical location and horizon obstruction – by T. Hadaś*

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**Relativistic orbit - a pear, not an ellipse**

General relativity theory is not directly used in determining the orbits of artificial Earth satellites – it is too complicated. It is the Newton's theory that is used with some corrections – relativistic accelerations. The main force impacting Earth's orbits is the Schwarzschild effect, which describes how general relativity theory changes the orbits of Earth satellites due to spacetime curves caused by spherical and massive Earth. Taking into account the Schwarzschild effect causes shifting the orbit by approximately 18 mm. However, the circular orbit remains a circular orbit, and the observed shift can be explained by a slight correction of constant gravity. So Newton's theory is quite sufficient for describing circular orbits.

An elliptical orbit is necessary to verify the Schwarzschild effect, because then the distance between the satellite and Earth changes over time. In this case the accelerations resulting from Newton's theory have an inversely proportional dependence on the squared satellite's distance from the Earth, and the relativistic correction is inversely proportional to the square of the speed of light and the third power of the distance from Earth. Thanks to the variable distance between the Earth and the satellite, you can verify if Einstein was right. The Schwarzschild effect predicts that the large half axis of the Galileo satellites' orbit should be changed by 10 mm at the perigee and 26 mm at the apogee. In addition, the shape of the orbit changes – the orbit becomes more circular at the perigee and more elongated at its apogee, and consequently takes on a pear-like shape.

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**Why such research at the University of Life Sciences?**

A method of modelling Galileo orbits has been developed at the Institute of Geodesy and Geoinformatics, making it possible to determine orbits with previously unattainable accuracy. - The model was developed by Grzegorz Bury M Eng. as part of his doctoral thesis together with his supervisor, that is me, and the colleagues from the institute – says Prof. Krzysztof Sośnica.

The Galileo orbit model is hybrid – this means that most of the forces acting on the satellite are based on the known design properties of the satellites, while elements whose properties change over time - e.g. the coefficient of photon scattering by solar panels or solar wind gusts – are modelled with a set of additional variables calculated along with Kepler's parameters of the orbit describing the movement of the satellite in an elliptical or circular orbit.

The Galileo orbit model is based on the properties of individual body surfaces and solar panels of satellites, taking into account the coefficient of absorption, reflection and dispersion of incident photons. The development of the model was possible after the publication of metadata for the civil Galileo system containing details of the satellites’ structure. Photons falling on the satellite body are usually radiated in the same direction, while photons falling on thin solar panels can be radiated in all directions. The exact time the satellites were entering the Earth's shadow was taken into account, and how much of the solar disk is potentially obscured by the Moon from the satellite’s point of view. In addition, the model takes into account the pressure of photons reflected from the Earth's surface depending on seasons and the Earth surface cover (oceans, ice cover, continents). In addition, the micro-accelerations resulting from the thrust of the transmitting antenna sending the navigation signal through the satellites towards the Earth have been taken into account, because they cause the satellite to be shifted by about 10 mm for every 100 W of signal strength, and the signal strength of the latest Galileo is over 250 W. The lowest micro-accelerations affecting the satellite which are taken into account in the model are equal to 1/5000,000 of the weight of a mosquito on Earth, which is already significant when it comes to Galileo's precise orbits. Only such an accurate model of Galileo satellites allows us to retrieve millimetre effects from the orbits of Galileo satellites.

Galileo satellite orbits can be determined with two independent techniques – microwave techniques – the subject of Radosław Zajdel’s M Eng. doctoral dissertation – and laser ones. Laser measurements of distance to navigation satellites, including Galileo, are analysed by the Associated Centre for the Processing of Laser Measurements of Distance to Earth Artificial Satellites, based at UPWr., which since 2017, belongs to the international ILRS/NASA organization. – Since 2019, I have been a member of the ILRS steering committee, which attaches great importance to the highest possible accuracy of satellite measurements and maintaining the continuity of such observations – adds Prof. Sośnica.

*Illustrative drawing of the Galileo satellite movement around the Earth – from G. Bury’s doctoral dissertation*

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**ESA rewards and invites to cooperation**

- During Galileo Colloquium organized by ESA in September 2019 at the ETH Polytechnic in Zurich, Grzegorz Bury M Eng. presented the developed model of Galileo orbits in the orbit determination session, while I presented the preliminary results of research on measurements of spacetime curvature during the fundamental physics session – says Prof. Krzysztof Sośnica.

Both papers met with great interest primarily from the European Space Agency. ESA awarded them both first place among the best papers presented at Colloquium, and then offered to sign a contract for further research. Thus, the project was not obtained as a result of a competition, as is usually the case with scientific projects, but as a result of scientific work commissioned by ESA and dedicated for implementation exclusively at the UPWr.

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**Failure turned into spectacular success**

- Paradoxically, the orbital accident involving the launch of Galileo satellites into wrong orbits has opened up new scientific research opportunities that otherwise would not have been possible. This is a perfect example of how failure can be turned into spectacular success – says Prof. Krzysztof Sośnica. – The atomic clocks installed on board Galileo satellites have already confirmed the validity of Einstein's theory. Soon we will be able to see if the effects that Einstein predicted in his theory regarding the movement of artificial satellites and the shape of the orbit have confirmation in observations – adds Prof. Sośnica.

Prof. Krzysztof Sośnica from the UPWr Institute of Geodesy and Geoinformatics is the Project Manager of the project called: "General Relativistic Effects in the orbits of Galileo Satellites" ESA
Contract No. 4000130481/20/ES/CM. The contractors in the project are Grzegorz Bury M Eng. (determining Galileo orbits) and Radosław Zajdel M Eng. (Galileo data processing), while administrative
issues are supervised by Katarzyna Kopańczyk PhD Eng. from the UPWr Department of International Cooperation.

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