Satellite Laser-Ranging as a Probe of Fundamental Physics

Satellite laser-ranging is successfully used in space geodesy, geodynamics and Earth sciences; and to test fundamental physics and specific features of General Relativity. We present a confirmation to approximately one part in a billion of the fundamental weak equivalence principle (“uniqueness of free fall”) in the Earth’s gravitational field, obtained with three laser-ranged satellites, at previously untested range and with previously untested materials. The weak equivalence principle is at the foundation of General Relativity and of most gravitational theories.

aluminum and gold in the gravitational field of the Sun (at a range of ~1.5 · 10 8 km) with a precision of ~10 −11 , and Braginsky and Panov 9 (1972) used aluminum and platinum in the gravitational field of Sun with a precision ~10 −12 . The 2012 test by the University of Washington 10 (the so-called "Eot-Wash" experiment) used a torsion balance to confirm the weak equivalence principle for beryllium-aluminum and beryllium-titanium test bodies in the field of the Earth to a precision of ~10 −13 . In April 2016 the space experiment MICROSCOPE of CNES (Centre National d'Études Spatial) was successfully inserted into orbit at an altitude of approximately 711 km. It was designed to test the equivalence principle to a precision of ~10 −15 comparing the motion of two proof masses, one of titanium-aluminum alloy, and one of a platinum-rhodium alloy. The science phase of the mission lasted for about two years. First results (agreement with the weak equivalence principle to parts in 10 14 ) were published in December 2017; the final measurements are expected this year (2019) 11 . Additional novel tests include one based on the time coincidence of gravitational radiation with electromagnetic observations from the LIGO event GW170817 12 . Also 13,14 , discuss the equivalence principle in the context of quantum systems.
Remarkably, GR even incorporates the strong equivalence principle 1 : gravitational energy (e.g. gravitational binding energy or the effective energy content of gravitational radiation) acts as a source (an active gravitational mass) for the gravitational field just like any other mass-energy, and responds to an external gravitational field (falls in that field) like any other passive gravitational mass. The strong equivalence principle has been validated by comparing acceleration of the Moon and the Earth toward the Sun using Lunar Laser Ranging, which measures the motion of the Moon relative to the Earth at the centimeter level. (The Earth's fractional gravitational binding energy is about twenty times that of the Moon). Differential acceleration of the two bodies would lead to polarization of the Moon's orbit; this has been excluded to parts in 10 13 by Williams, Turyshev and Boggs 15 . Lunar Laser Ranging improvements of an order of magnitude to ~1 mm at the Apache Peak station will contribute to even better lunar equivalence principle results 16 . Archibald et al. 17 studied the system PSR J0337 + 1715, and a gave a limit on the strong equivalence principle. The system consists of a triple: a tight (1.6 day) millisecond pulsar -white dwarf binary, in 327-day orbit about a distant white dwarf. Study shows the accelerations of the pulsar and its nearby white dwarf companion differ fractionally by no more than 2.6 × 10 −6 as they fall toward the distant white dwarf.
Weak equivalence may be violated if there is a weak (of roughly the same strength as gravity) fundamental field that couples to matter differently from the universality of gravity. For instance, some theoretical constructs suggest an almost massless scalar field which couples to the nucleon number, rather than to the total mass-energy of the object. This scalar gravity would therefore be composition-dependent (thus violating the weak equivalence principle) since the fractional nuclear binding energy is different among elements. Different gravitational theories can exhibit a breakdown of the weak equivalence principle depending on the range, for example compared to the range of Yukawa-type deviations from the inverse square law of gravitation 18 in a theory that couples to nucleon number. A composition dependent interaction between two bodies might be described by the following potential energy of a body 1 in the gravitational field of a body 2: The projection of the spacetime geodesic onto a spatial plane is, for example, an ellipse (with suitable coordinates). Here, the third spatial dimension is suppressed and the much smaller relativistic precession of the pericenter is not shown. (b) If there is a violation of the uniqueness of free fall, two bodies with the same initial conditions will not follow the same spacetime curves and their projections onto a spatial plane will, for example, be two different ellipses.
where −GM 1 M 2 /r is the standard Newtonian potential energy (representing the Newtonian gravitational theory as the lowest order approximation of GR), G is the gravitational constant, M 1 and M 2 are the masses of the two bodies, b 1 and b 2 are some composition dependent properties of bodies 1 and 2 defining the additional interaction, r is the distance between the two bodies and λ the Yukawa range of the interaction. The ratio b/M will in general be different for each body and thus bodies with different compositions will fall with different acceleration, violating the uniqueness of free fall. Furthermore, a measurable deviation from the universality of free fall may depend not only on the material of the proof masses, nucleon number, etc., but also, as in Eq. (1), on the range of the experiment (an effective change of  GM with distance) unless λ ≈ ∞. Therefore it is important to test the equivalence principle with different materials and at different ranges; an important aspect of the present determination is the distance scale involved. In our analysis we assume  GM is a universal constant and cast the problem entirely in terms of the universality (or not) of the ratio of satellite inertial mass to passive gravitational mass.

Test of Equivalence Principle: Laser-Ranged Satellites LARES, LAGEOS, and LAGEOS 2
We describe a test of the weak equivalence principle using for the first time freely falling high altitude laser-ranged satellites: LARES, made of sintered tungsten 19,20 ; and LAGEOS 21 and LAGEOS 2 22 , two almost identical satellites each composed of 57% aluminum shell/43% brass core by mass. These are materials never previously tested. Further details about the satellites are found in the Section Methods below. The number of well tracked dense laser ranged satellites is not large, so if new satellites meeting these criteria are launched their inclusion would improve our analysis to (at least partially) disentangle the weak equivalence result from a more controversial change of ⊕ GM with distance. [Such a gradient could be the result of a violation of the crucial theorem that a gravitating sphere acts as a point mass (shell theorem). The most general form for the force to fulfill the shell theorem, F(r) = Ar −2 + Λr contains 23 a cosmological constant Λ, which LARES and LAGEOS data constrain. Constraints on modified gravity laws, including Yukawa type, are essential for GR's Newtonian law as limit, as well as for understanding the dynamical features in the local group of galaxies and its vicinity (see, e.g. 24 )].
The self-gravities of all three satellites are negligible. By comparing the residual radial accelerations of these three satellites, we obtain a test validating the weak equivalence principle with an accuracy of ~10 −9 . The range of the test described here goes from ~7820 km from Earth's center (altitude 1450 km) for the LARES satellite to ~12200 km from Earth's center for LAGEOS and LAGEOS 2. Our test thus fills a distance gap not covered by the laboratory and Lunar Laser Ranging tests; any scale range in principle will constrain parameters entering the "fifth" force, phenomenology or coupled gravity models.

orbital Analysis and Results
We processed more than half a million normal points of the three satellites LARES, LAGEOS, and LAGEOS 2. The laser ranging normal points were processed using NASA's orbital analysis and data reduction software GEODYN II 25 , and validated by the orbital modelers UTOPIA 26 , and EPOSOC 27 . The data analysis was based on the Earth gravity model GGM05S 28  The GEODYN analysis includes Earth rotation from Global Navigation Satellite Systems (GNSS) and Very Long Baseline Interferometry (VLBI), Earth tides, solar radiation pressure, Earth albedo, thermal thrust, and lunar, solar and planetary perturbations. We analyzed the laser ranging data of the LARES, LAGEOS, and LAGEOS 2 satellites from February 2012 to December 2014. The laser ranging data for LARES, LAGEOS, and LAGEOS 2 were collected from more than 40 ILRS stations all over the world 31 .
If we include the acceleration due to the Earth's quadrupole moment (the Earth's oblateness measured by the J 2 coefficent 32 ), and the potential breakdown of the uniqueness of free fall, the radial acceleration a r of an Earth satellite can be written: [Note that a r is not the second time derivative r of the radial coordinate r. Consider for instance circular motion where the radius r is constant, but θ = −  a r r 2 ]. Here, for simplicity, we have included within m g /m i any breakdown of the uniqueness of free fall, for example of the type of the second term of Eq. (1). (m g is the passive gravitational mass of the satellite and m i its inertial mass; m g /m i is is a universal constant in GR and Newtonian Physics, equal to unity by choice of units). ⊕ M and ⊕ R are the Earth's mass and equatorial radius, r is the radial distance of the satellite from the Earth barycenter, and P 20 is the associated Legendre function, of degree 2 and order 0, of the satellite latitude (see Methods). The product ⊕ GM for the Earth is today measured 33 to be 398600.4415 km 3 /sec 2 (including the mass of the atmosphere) with an estimated relative (one-sigma) uncertainty of ~2 · 10 −9 . The Earth's dimensionless quadrupole moment 34 J 2 is equal to 0.0010826358 with a relative uncertainty of ~10 −6 to 10 −7 . According to the uniqueness of free fall, the ratio m g /m i is the same for every test body. Here we consider the possibility that such a ratio may be different for aluminum/brass of the LAGEOS satellites, at a distance of ~12220 km from the Earth center, and for the tungsten alloy of the LARES satellite, at a distance of ~7820 km from the Earth's center. On the basis of the LAGEOS, LAGEOS 2 and LARES laser-ranging observations, we then set an experimental limit on the deviation δ(m g /m i ): (2019) 9:15881 | https://doi.org/10.1038/s41598-019-52183-9 www.nature.com/scientificreports www.nature.com/scientificreports/  33 using the LAGEOS laser-ranging data. We normalize m g /m i = 1 for LAGEOS and the essentially identical (in both composition and altitude) LAGEOS 2. Thus δ(m g /m i ) can only appear in consideration of LARES. Eqs (9) and (10) below give the relations between the variations of the accelerations, and the parameter variations. The long term average residual accelerations for LARES are comparable to those for LAGEOS 2 even though LARES orbits at a much lower altitude (Fig. 2c) and LARES undergoes larger single-point excursions.
We observe the average residual radial accelerations: here the angle brackets "< >" are long term averages. The satellites' residual radial accelerations are mainly due to the errors δ ⊕ GM ( ), δJ 2 , δ(m g /m i ) and the measurement error in the radial distance, δr, of the three satellites. Recent studies of the best station performance in ranging to the LAGEOS and LARES satellites suggest one-sigma δr mm 2 for the LAGEOS satellites and δr mm 3 for LARES, which we adopt.
The method is to take the three equations for the residual radial acceleration of each of the three satellites (e.g., in Eq. (10) in Methods for LARES). These can be viewed as giving a vector of radial accelerations (Eq. (4)) equal to a square matrix M times a vector of measurement uncertainties: (δ(m g /m i ), δ ⊕ GM ( ), δJ 2 ), plus a term proportional to radial measurement uncertainty, + other errors. We invert this equation (multiply by M −1 ), which yields (Methods)): www.nature.com/scientificreports www.nature.com/scientificreports/ where ±1.1 · 10 −9 is the estimated systematic error principally due to the error in the measurement of the radial distance. Uncertainties in the modeling of the radial accelerations due to the errors in the Earth's spherical harmonics higher than the quadrupole moment, J 2 , and due to the errors in the modeling of atmospheric drag and of other non-gravitational perturbations, such as direct solar radiation pressure and Earth albedo, are included in the other errors above, and are much smaller. The combined residuals affecting δ(m g /m i ) are shown in Fig. 3. Equation (5) shows a confirmation of the equivalence principle for the three satellites with an accuracy of ~±10 −9 .

Discussion
Our test of the weak equivalence principle (uniqueness of free fall) using the laser ranged satellites LARES, LAGEOS and LAGEOS 2 fills a gap in the tests of this principle fundamental to Einstein's gravitational theory of General Relativity. Some alternative theories of gravitation predict deviations from the uniqueness of free fall that are enhanced at certain ranges depending on a typical scale length and are enhanced for different materials. Previous tests of the weak equivalence principle were Earth laboratory tests in the gravitational field of Earth (at a distance of ~6370 km from the center of the Earth) and at the MICROSCOPE distance of ~7000 km, and Earth laboratory tests in the gravitational field of the Sun (at ~1.5 × 10 8 km). There were no tests at a range between 7820 km and 12270 km prior to the present test using the LAGEOS and LARES satellites. Furthermore the uniqueness of free fall was never previously confirmed comparing test bodies made of aluminum/brass and tungsten, such as the LAGEOS and LARES satellites. Our test has confirmed the validity of the weak equivalence principle for these metals over ranges 7820 km and 12270 km, to accuracy of one part per billion. Also, since LARES differs both in composition and in orbital radius from LAGEOS and LAGEOS 2 (which are very similar to one another in these properties) our observation can be viewed as constraining (δG)/G, again to ~ one part in 10 −9 , over the range 7820 to 12270 km.

Methods
The orbits 19 of the three satellites are almost circular, with very small orbital eccentricity. The LAGEOS satellite has semimajor axis 12270 km (altitude 5890 km), orbital eccentricity of 0.0045 and orbital inclination of 109.84°; LAGEOS 2 has semimajor axis 12160 km (altitude 5780 km), orbital eccentricity of 0.0135 and orbital inclination of 52.64°; LARES, semimajor axis 7820 km (altitude 1450 km), orbital eccentricity of 0.0008 and orbital inclination of 69.5°. In this analysis we assume all three satellites are in circular orbits. All three are passive, spherical, www.nature.com/scientificreports www.nature.com/scientificreports/ laser-ranged satellites. LAGEOS was launched in 1976 by NASA, and LAGEOS 2 in 1992 by ASI, the Italian Space Agency, and NASA. They are two almost identical spherical passive satellites covered with corner cube reflectors to reflect back laser pulses emitted by the stations of the satellite laser ranging (SLR) network of the International Laser Ranging Service (ILRS) 31 . SLR allows measurement of the radial position of the LAGEOS satellites with a median accuracy of the order two millimeters over an Earth-surface to satellite distance of ~6000 km. LARES is a satellite of ASI, launched in 2012 by ESA, the European Space Agency, with the new launch vehicle VEGA. LARES was designed to approach as closely as possible an ideal test particle 20 . This goal was achieved by minimizing the surface-to-mass ratio of the spherical satellite (the smallest of any artificial satellite), by reducing the number of its parts, by avoiding any protruding components, and by using a non-magnetic material. LARES carries laser retro-reflectors similar to those on LAGEOS and LAGEOS 2. Since LARES is at lower altitude, it is accessible to more ranging stations. Some of these stations have slightly reduced timing accuracy (compared to those that range to LAGEOS). As a result, the median accuracy of positioning of LARES is at the roughly the three millimeter level.
The classical gravitational potential of a spheroid, such as the Earth 32 , can be written: where ⊕ M is the Earth's mass, ⊕ R its equatorial radius, J 2 its quadrupole moment, G the gravitational constant, r is the radial distance from the origin and P 20 is the Legendre associated function of degree 2 and order 0: 20 2 where φ is the latitude and in the expression (6) we have neglected higher order P n0 terms. If there is a violation of the weak equivalence principle, the ratio m g /m i of the gravitational mass to the inertial mass may be different between the aluminum/brass of the LAGEOS satellites and the tungsten alloy of the LARES satellite. Furthermore a composition dependent interaction between two bodies may depend on the distance between the two bodies and on the range of the interaction as described by Eq. (1). We have then indicated with δ(m g /m i ) in Eq. (2) any breakdown of the uniqueness of free fall including, for example, one of the type of the second term of Eq. (1). The radial acceleration of a satellite, such as LAGEOS and LARES, can thus be written by Eq. (2) above. Therefore, the leading terms of the residual, instantaneous unmodeled radial accelerations of the LAGEOS and LARES satellites can be written: ) is itself universal). Similarly, we assume a universal value of J 2 , with a possible offset δJ 2 between its observed, fiducial value, and its true value; δJ 2 is also universal.
The meaning of δr is different. It is the mean value of the uncertainty in the radial distance of each satellite from the Earth center of mass, mainly due to errors in the determination of the Earth center of mass, biases in laser ranges, errors in the modeling of the dispersion of the laser pulses by the troposphere, and uncertainties arising from determining the precise position of the retroreflector with respect to the center of mass of the satellite. Since these are mean values of uncorrelated errors, there are different values of δr for each satellite. We take 3 mm for LARES and 2 mm for the LAGEOS satellites.
With Eq. (8), we can write for LAGEOS (here angle brackets "< >" are long term averages): The coefficient of δJ 2 in Eq. (9) is the average over an orbit of the P 20 Lagrange associated function of sinφ, where φ is the latitude of the satellite. This function can be written as a function of the orbital inclination I and the true anomaly f: Therefore the average value of P 20 over one orbital period is: www.nature.com/scientificreports www.nature.com/scientificreports/ which is used in Eq. (9).
A similar expression to Eq. (9) holds for LAGEOS 2. However for LARES we take into account a possible deviation of m g /m i from unity, so we have an additional term proportional to δ(m g /m i ):  (9) and (10) above in terms of the normalized residual accelerations: Here P 40 is the associated Legendre function of degree 4 and order 0. The estimated 28 uncertainty δJ 4 is δJ 4 ~ 2 × 10 −11 . This uncertainty is ~a factor of fifty below our ranging error. The periodicity of the various lunar and planetary perturbations averages their effect below the ranging error. The largest non-gravitational perturbation on the LAGEOS and LARES satellites, including the various radiation pressure perturbations and particle drag, is due to direct solar radiation pressure which produces mainly periodical effects. Further, its magnitude is × − 10 6 the average radial acceleration due to J 2 and the uncertainty in the modeling of the radiation pressure on LAGEOS is less than 1% 36 . Even if the direct solar radiation always acted to produce a constant radial acceleration, it and its uncertainty would be negligible with respect to the corresponding ones due to J 2 and δJ 2 . (The non-gravitational perturbations are almost three times smaller on LARES than on LAGEOS, since the cross-sectional-area-to-mass ratio of LARES is about 2.7 times smaller than that of LAGEOS). A term not written in the dimensionless acceleration equation (Eq. (11)) would be proportional to J 2 × δI; but the uncertainty in this term J 2 × δI is much smaller than the ranging error (~10 −9 ). [In solutions we find variances in inclination less than .
As noted, we set δ(m g /m i ) to zero for LAGEOS and LAGEOS 2. We also make the approximation that their radii are the same. Then the matrix in Eq. (12) is www.nature.com/scientificreports www.nature.com/scientificreports/ We now address the last term in Eq. (11), the column matrix of (2δr/r) i , which gives the magnitude of the uncorrelated errors for the satellites. These are the largest remaining uncontrolled errors.
Consider the 2δr/r-induced errors in the equivalence principle term: We use the quotes on the operators "+" because in fact the error δr is uncorrelated between satellites, so we add these errors by quadrature. Also, though this term has an explicit "−" sign, it is actually stochastic, so contributes "±" to the errors. Thus the full statement of our result is: The physical result from these calculations is our statement of the equivalence principle: δ(m g /m i ) = 2.0 × 10 −10 ± 1.1 × 10 −9 among the three satellites, consistent with the result zero to within the ~10 −9 fractional accuracy of the determination.

Data availability
The laser-ranging data of LARES, LAGEOS and LAGEOS 2 are available at the NASA's archive of space geodesy data CDDIS (Crustal Dynamics Data Information System) 37 .