-Rupsa Dasgupta
The theory of General Relativity, developed by Albert Einstein, is essential for grasping the fundamentals of the universe. It describes the gravitational interactions responsible for planetary movements and the motions of galaxies, stars, and our attraction toward the earth.
In recent years, scientists have discovered that distant galaxies accelerate away from us, proving that the universe is indeed expanding. Dark energy, a mysterious exotic force driving this expansion, has been the subject of extensive study ever since. This dark energy, along with an invisible but normally attracting matter called dark matter, makes up 96% of the universe (Ciufolini et al., 2012).
The Einstein field equations in general relativity describe gravitational interactions as the bending of spacetime brought about by various masses and energy (Ciufolini et al., 2012). For instance, the Earth’s mass generates a curvature in the fabric of spacetime, which is responsible for attracting the moon and other smaller satellites towards it. Any smaller test object which lies in the gravitational field of a second, larger object, travels along a geodesic in spacetime, where the larger body creates the curvature (in general relativity, a geodesic is essentially the trajectory of an object moving under the influence of gravity). Ideally, the moon and the earth’s artificial satellites would have followed a geodesic motion where gravity would be the only influencing factor. But their motion gets slightly deviated due to the effect of other non-gravitational forces such as atmospheric drag and thermal pressure. A test particle following geodesic motion must satisfy the following criteria: it should be electrically neutral, its angular momentum must be negligible, and the gravitational binding energy relative to the rest mass must be low. The size of the body itself must be small so that its motion is not unduly affected by the gravitational field fluctuations due to its volume (Ciufolini et al., 2012).
Einstein’s equations show us how the gravitational interaction between masses in spacetime results in their geodesic trajectory. General relativity is thus fundamentally based on geodesic motion. Now, let us discuss the approximation of the spacetime geodesic of a test particle, which is ultimately the aim of scientists who study the geometry of spacetime in the region surrounding a body.
In general relativity, an extended body brings forth a few problems. The equations of motion for an extended body are non-linear, and its internal structure makes approximations difficult. This internal structure is made of various media, where thermodynamic, tidal and kinetic factors also come into play. Moreover, several issues arise when an extended body is described as a test particle following geodesic motion since several Newtonian concepts like centre of mass, size etc. of an extended object have not crossed over to the realm of General Relativity.
Now we come to the Ehlers-Geroch Theorem, which states that small bodies with mass move in an almost geodesic motion, where gravity is not the only influencing factor (as it ideally should be). It assigns a geodesic γ to the trajectory of a broad object which has a sufficiently low gravitational field and thus creates a bridge between General Relativity and space experiments (Ehlers & Geroch, 2003). This helps us come to the conclusion that a small satellite’s movement is near-geodesic and serves as a suitable “laboratory” to test the theory behind relativity.
Because gravitational interactions are weak, the fabric of spacetime is perfect for testing gravitational and fundamental physics. But for that, a satellite should display the properties of a test particle as much as possible and should be sparsely affected by non-gravitational influences like atmospheric drag and radiation pressure. Moreover, it should be possible to calculate the position of such a satellite with great accuracy.
The best example of such a satellite is LARES (Laser Relativity Satellite) from the Italian Space Agency (ASI), which was launched by VEGA on the 13th of February 2012, circling the Earth’s orbit. The Satellite Laser Ranging (SLR) technique is used to measure its position to a high degree of accuracy. Short bursts of radiation are emitted from lasers on earth at small intervals and then reflected from reflectors in the satellite. The total time taken to cover the whole path by the laser helps us to measure the distance to the satellite at that particular instant, the error being of the order of millimetres.
But to test gravitational physics, the accuracy of the position of the satellite is not the only criterion. It also has to show the characteristics of a test particle. A test particle can be realised in space in either of two methods; a compact satellite with minimal drag, or a small-scale but highly dense body with a very low ratio between its area and mass.
A satellite unaffected by the effect of drag is designed to minimize the effects of non- gravitational influences like radiation pressure. In this case, a proof mass is kept inside a cavity, near the satellite’s centre of mass. Control thrusters then drive the satellite, controlling the motion of the cavity and the proof mass such that the distance between its inner walls and the proof mass remains constant. Because of the closed cavity, the mass is not affected by any non-gravitational influences. When we look at Gravity Probe-B, an experiment in space conducted to test two predictions of General Relativity, the mean residual acceleration (the acceleration brought about by non-gravitational factors) realised was equal to 40 x 10-12 m/s2 (Everitt et al., 2011). This residual acceleration is an important metric to check the effect of drag. This is because in an ideal drag-free satellite, there is no residual acceleration, but in reality, some residual forces are always present in satellites, making them deviate from their ideal geodesic trajectory. Thus, the lower the residual acceleration, the lesser it is affected by drag (Lappas, Kostopoulos, 2020).
In the case of a satellite not immune to the effects of drag, the key to reducing non-gravitational perturbations is high density, coupled with a small area-mass ratio. Before the advent of LARES, the LAGEOS (Laser Geodynamics Satellites) had the lowest area-to-mass ratio and were the best available options to use as test particles. But the launch of LARES revolutionised the field. LARES is a satellite with a spherical laser range, like LAGEOS. But the difference lies in the materials used. LAGEOS is built with brass and aluminium, while LARES is made of an alloy of tungsten, that lowers its mass (it weighs 386.8 kg). It boasts of 92 retro reflectors, each of whose radius spans 18.2 cm. The ratio between its area and mass is smaller than LAGEOS by a scale of around 2.6. Its density is also extremely high. These features help reduce the effect of non-gravitational forces. The fact that it has high thermal conductivity also ensures that the non-gravitational perturbations are smaller than any other satellite previously observed.
The satellite’s residual along-track acceleration shows us how much it minimises the influences of non-gravitational forces like terrestrial and solar radiation pressure, atmospheric drag and thermal thrust. All these effects contribute to along-track acceleration, which is acceleration along the projection of the satellite’s path to the horizontal plane, signifying how much the radiation pressure, atmospheric drag and other such forces (besides gravity) influence its motion. So the lesser it is, the lesser these forces impact the satellite. In their calculations, scientists used the laser ranging data from LARES based on the first seven 15-day arcs, with the help of orbital analysis systems UTOPIA of UT/CSR (Centre for Space Research of the University of Texas, Austin), GEODYN II of NASA Goddard, and EPOS-OC of GFZ (Helmholtz Centre Potsdam GFZ German Research Centre for Geosciences) (Ciufolini et al, 2012).
It was found that the estimation of the errors due to the gravity model caused very little variation in the residual acceleration. For LARES, the scientists used the drag coefficient Cd (or the ratio between the drag acting on a body moving through the air and the product of its surface area and velocity). They used its best fit to model drag, and they saw that the best fit reflectivity coefficient C (or the ratio between the amplitudes of the reflected wave and the incident wave) was consistent with all their measurements. In the analysis spanning 105 days, GEODYN as well as UTOPIA saw that the residual along-track acceleration for LARES was as small as around 0.4 x 10-12 m/s2, while for the two LAGEOS satellites, it was between 1 to 2 x 10-12 m/s2. This effectively showed that the deviation of LARES from ideal geodesic motion was less than that of LAGEOS.
To further verify their findings, the scientists then fit the drag coefficient of LARES over the odd 15-day arcs and used its mean to find the average residual acceleration over the even arcs. Once more, their data matched, showing a residual acceleration of 0.4 x 10-12 m/s2 for LARES. This was even more astounding, given that the lower altitude of LARES in the earth’s atmosphere should have caused more gravitational perturbations (Ciufolini et al., 2012).
The reason why LARES has such a low residual acceleration compared to LAGEOS, despite its low altitude, is because it experiences lesser thrust due to thermal radiation. LARES is much more compact than LAGEOS since it has reflectors of 18 cm radius versus the 30 cm ones of LAGEOS. Its thermal conductivity is also greater since it is a solid sphere of one piece, compared to the three pieces of LAGEOS. LAREES’ cross-sectional area-mass ratio also adds to the low thermal thrust effects.
Fig. 1 The red curve denotes the change in distance between a test particle moving along a spacetime geodesic and a similar particle affected by the 0.4 x 10-12 m/s2 along-track acceleration of LARES. The blue curve shows the change in distance between a test particle and the 1 x 10-12 m/s2along-track acceleration of LAGEOS. These show that LARES has more agreement with the geodesic motion predicted by General Relativity.
[Source: Ciufolini, I., Paolozzi, A., Pavlis, E., Ries, J., Gurzadyan, V., Koenig, R., Matzner, R., Penrose, R., & Sindoni, G. (2012). Testing general relativity and gravitational physics using the lares satellite. The European Physical Journal Plus, 127(11). https://doi.org/10.1140/epjp/i2012-12133-8%5D
Thus, after careful measurements and calculations, it has been found that the geodesic motion of a satellite, a prediction of General Relativity, has been accurately manifested in LARES. It is to date the most appropriate satellite available to carry out tests of gravitational physics in space. Therefore, due to the exceptional features of LARES, it, along with the LAGEOS satellites, will be employed to determine the frame-dragging effect which Einstein wrote about in his papers. This effect implies that the frame of reference around any mass will be “dragged” around it. Thus, this new and improved measurement method will be more accurate than when only the LAGEOS satellites were used.
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ABOUT THE AUTHOR
Rupsa Dasgupta
The author is in their second year at St. Xavier’s College, studying physics and mathematics.
We would like to thank Dr.Soma Sanyal, Associate Professor, University of Hyderabad for reviewing the article and for her valuable inputs.
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Main Reference
- Ciufolini, I., Paolozzi, A., Pavlis, E., Ries, J., Gurzadyan, V., Koenig, R., Matzner, R., Penrose, R., & Sindoni, G. (2012). Testing general relativity and gravitational physics using the lares satellite. The European Physical Journal Plus, 127(11). https://doi.org/10.1140/epjp/i2012-12133-8
References
- Ciufolini, I., Gurzadyan, V. G., Penrose, R., & Paolozzi, A. (2013, February 21). Geodesic motion in general relativity: Lares in Earth’s gravity. arXiv.org. Retrieved May 13, 2022, from https://doi.org/10.48550/arXiv.1302.5163
- Pardini, C., Anselmo, L., Lucchesi, D. M., & Peron, R. (2016, November). Estimation of the Perturbing Accelerations induced on the LARES satellite by Neutral Atmospheric Drag. ResearchGate. Retrieved May 12, 2022, from https://www.researchgate.net/publication/309766334_Estimation_of_the_Perturbing_Accelerations_Induced_on_the_LARES_Satellite_by_Neutral_Atmosphere_Drag
- Ehlers, J., & Geroch, R. (2003, September 16). Equation of motion of small bodies in relativity. arXiv.org. Retrieved May 13, 2022, from https://doi.org/10.48550/arXiv.gr-qc/0309074
- Everitt, C. W. F., DeBra, D. B., Parkinson, B. W., Turneaure, J. P., Conklin, J. W., Heifetz, M. I., Keiser, G. M., Silbergleit, A. S., Holmes, T., Kolodziejczak, J., Al-Meshari, M., Mester, J. C., Muhlfelder, B., Solomonik, V., Stahl, K., Worden, P., Bencze, W., Buchman, S., Clarke, B., … Wang, S. (2011, May 17). Gravity probe B: Final results of a space experiment to test general relativity. arXiv.org. Retrieved July 13, 2022, from https://arxiv.org/abs/1105.3456
- Lappas, V., & Kostopoulos, V. (2020). A Survey on Small Satellite Technologies and Space Missions for Geodetic Applications. In V. Demyanov, & J. Becedas (Eds.), Satellites Missions and Technologies for Geosciences. IntechOpen. https://doi.org/10.5772/intechopen.92625
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