On the Symmetries and Rank Deficiency Problems in the BepiColombo Relativity Experiment

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UNIVERSITY OF PISA

MASTER OF SCIENCE IN MATHEMATICS

Master Thesis

On the Symmetries and Rank Deficiency Problems

in the BepiColombo Relativity Experiment

12 June 2020

Candidate:

Sofia Murgia

Supervisors:

Prof. Giacomo Tommei Dr. Giulia Schettino

Dr. Daniele Serra

Examiner:

Prof. Giovanni F. Gronchi

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Introduction

Mercury is the least explored planet in the inner solar system. Compared to the other planets, it is difficult to reach: even more energy is needed than sending a mission to the outer solar system. This is due to the fact that, along its journey towards the Sun, a spacecraft needs to decelerate, because the solar gravitational force increases with the square of the distance. On the other hand, since Mercury is difficult to observe from the Earth, due to its close proximity to the Sun, for an in-depth study of the planet and its environment it is necessary to operate a spacecraft around the planet.

The investigation of Mercury has particular importance for various aspects: being the planet the closest to the Sun, it has a privileged position for the analysis of the gravitational effects of a massive body like the Sun. Furthermore, being Mercury a now-quiescent body, it preserves evidence of the early history of the terrestrial planets, thus, its exploration can allow to understand their origin.

After the NASA’s Mariner 10 mission, which, from November 1973 to March 1975, provided the first detailed data and close-up images of Mercury, the planet remained unexplored for the following decades, until the launch of the NASA Discovery class mission MESSENGER in August 2004. MESSENGER orbited Mercury from 2011 until 2015, studying the planet’s chemical composition, geology, and magnetic field.

The next scheduled mission for the exploration of Mercury is the joint ESA/JAXA mission BepiColombo, executed under the ESA leadership and part of the Horizon 2000+ program. It was launched in October 2018 and the arrival at Mercury is foreseen for December 2025, after 7 years of cruise. The mission aims to study the composition, geophysics, atmosphere, magnetosphere and origin of the planet. The Mercury Orbiter Radio science Experiment (MORE) is one of the experiments on-board BepiColombo, whose scientific goals concern both the geodesy and geophysics of Mercury and some tests of fundamental physics. In particular, taking advantage from the fact that Mercury is the best-placed planet to investigate the gravitational effects of the Sun, MORE will allow an accurate test of relativistic theories of gravitation. Such test consists in processing the radio observations (range, range rate) between the MORE on-board transponder and the on-ground antennas in order to obtain an accurate estimate of some Parameterized Post-Newtonian (PPN) parameters, whose knowledge can allow to discern the validity of the theory of General Relativity from alternative theories of gravity.

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iv INTRODUCTION The challenging scientific goals of MORE can be fulfilled only by means of the combination of very accurate tracking, which is possible thanks to state-of-the-art on-board and on-ground instrumentation, and precise orbit determination of the spacecraft and of both Mercury and the Earth. The Celestial Mechanics Group of the University of Pisa is in charge of the data analysis of the MORE experiment. The parameters estimation is performed together with the orbit determination, by means of an iterative procedure based on a classical non-linear least squares fit (LS), implemented in a dedicated software, ORBIT14, developed at the Department of Mathematics of the University of Pisa.

In this Thesis we aim to investigate a critical issue of the MORE relativity experiment: the presence of symmetries affecting the outcome of the experiment. Generally speaking, symmetries in an orbit determination problem lead to rank deficiencies in the normal matrix of the LS fit, which can result in a significant degradation of the solution. Thus, it is especially important to understand whether we are in the presence of symmetries and develop suitable methods to handle them. The investigation is undertaken by means of both theoretical considerations and numerical simulations, using the ORBIT14 software.

Until now, a detailed study of the symmetries of the BepiColombo-MORE experiment has never been performed: the existence of certain symmetries was assumed as a hypothesis and simulations were carried out adopting suitable strategies to handle these symmetries. In this Thesis we start an extensive analysis to investigate the actual presence of rank deficiencies.

When we consider the complex scenario of the MORE experiment, the spectral analysis of the normal matrix leads to results that are difficult to interpret. As a consequence, we proceed in abstract terms, by progressive approximation, studying test cases. We start analyzing a set-up in which the presence of an exact symmetry is guaranteed by theory. Then, we add to this basic scenario a list of perturbations and we study separately their influence on the symmetry. More in detail, the basic scenario consists in a 3-body problem between Mercury, the Sun and the Earth Moon Barycenter (EMB) in addition to a 2-body problem accounting for the motion of the spacecraft around Mercury. As for the perturbations, we consider the influence of the fact that some state vectors (that of the Moon and the other planets, as well as that of the on-ground antenna) are not directly computed by the ORBIT14 software, solving the related equation of motion, but, rather, they are provided by the Jet Propulsion Laboratory ephemerides or the IERS (International Earth Rotation and Reference Systems Service) orientation data.

This Thesis is structured as follows:

Chapter 1. After having recalled the fundamental principles of General Relativity, we

introduce the PPN formalism. Then, we briefly review the main experimental tests that have been performed in the past in order to constrain the value of the PPN parameters. We conclude with a discussion on possible future improvements that can be achieved by exploiting the data provided by the ongoing space missions.

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INTRODUCTION v

Chapter 2. We describe the MORE experiment, with a special focus on the relativity

tests. We analyze the Mercury-centric dynamics of the spacecraft and the heliocentric dynamics of Mercury and the EMB, describing all the effects that come into play at the required level of accuracy. In the final part, we explain how to compute the range and range rate observables, taking into account the relativistic effects.

Chapter 3. We review the theoretical aspects of orbit determination and how to apply

them to the MORE experiment, as well as how to interpret the results in terms of uncertainties in the determination of the desired parameters. We briefly explain the implementation of the orbit determination and parameter estimation in the ORBIT14 software. We perform a first simulation and we analyze the results, suggesting the presence of an approximate rotational symmetry.

Chapter 4. After recalling the theoretical concepts related to symmetries, we discuss the

effectiveness of two strategies for curing the rank deficiency: the descoping strategy and the use of constraints to inhibit the symmetry. In particular, the second part of the Chapter contains the original work and a discussion of the obtained results. This Thesis ends with some indications on possible issues to analyze in future work for a better understanding of the phenomena we studied.

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Chapter 1

Foundations of theories of gravity

When General Relativity (GR) was proposed, its experimental confirmation was almost a side issue. Einstein did calculate observable effects of GR, such as the perihelion advance of Mercury and the deflection of light, but he regarded such empirical questions as secondary compared to the inner consistency and elegance of the theory. On the contrary, experimental gravitation arouses considerable interest in today’s astrophysics, with increasing efforts to test the theory’s predictions, both at the solar system level and at cosmic scales (see, e.g., [1] for a discussion).

The modern history of experimental relativity can be divided roughly into four periods: Genesis, Hibernation, Golden Era and Quest for Strong Gravity [1]. The Genesis (1887-1919) comprises the period of the two well-known experiments of relativistic physics: the Michelson-Morley experiment [2] and the Eötvös experiment (see, e.g., [3]); followed by the two major confirmations of GR: the deflection of light (see, e.g., [4]) and the perihelion advance of Mercury (see, e.g., [5]). Afterwards, there was a period of Hibernation (1920-1960), during which technological and experimental possibilities did not manage to keep up with theoretical work. Starting from 1960 there was a change of direction with a great number of astronomical discoveries as quasars, pulsars, cosmic background radiation. New technologies were developed, as atomic clocks, radar and laser ranging, space probes: a brand-new type of experiments became possible. As a consequence, experimental gravitation experienced a Golden Era (1960-1980) characterized by a systematic effort to understand the observable predictions of GR, to compare them with the predictions of alternative theories of gravity and to perform new experiments to test them. All the results supported GR and most of the alternative theories of gravity (see Section 1.1.3) fell by the wayside. Since that time, the field has entered what might be termed as a Quest for Strong Gravity. The term "strong" in GR alludes to the critical distinction between strong and weak gravity, determined by the quantity ε ∼ GM/(R c2), where G is the Newtonian gravitational constant, M is the order of magnitude of the mass involved in the phenomenon, R is the characteristic distance scale and c is the speed of light. For instance, near the event horizon of a black hole or in the case of the expanding observable universe, it holds ε ∼ 1 while for neutron stars ε ∼ 0.2 :

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2 CHAPTER 1. FOUNDATIONS OF THEORIES OF GRAVITY these are the regimes of strong gravity. For the solar system, ε < 10−5: this is the regime of weak gravity.

1.1 The equivalence principles

The Equivalence Principle (EP) has historically played an important role in the develop-ment of gravitation theories. Newton regarded this principle as a cornerstone of mechanics and Einstein adopted it as a basic element in his development of GR. Nowadays, the principle of equivalence can be regarded as the foundation, not merely of Newtonian gravity or of GR, but of the broader idea that spacetime is curved.

Newton came up with the idea of the equivalence principle by stating that the quality of a body named mass is proportional to the weight. In a modern language it states the equality of the coefficient m_i in Newton’s second law, F = m_ia, called inertial mass, which

describes how the particle responds to the applied force and the gravitational mass m_g that is the coefficient in Newton’s law of universal gravity, F = mgg, which determines the

overall magnitude of the gravitational force exerted on the particle by the external bodies. This principle has some important consequences, first of all the fact that the free fall of any object in the same gravity field depends only on its initial state and not on the composition or structure (universality of free fall). Furthermore, it follows that it is impossible to detect the difference between a uniform static gravitational field and a uniform acceleration: free-fall and inertial motion are physically equivalent. As it will be shown in Section 1.1.3, this property allows the geometrical description of spacetime, which is at the basis of GR.

In the modern usage, the equivalence principle can take the following forms, depending on the context: weak, Einstein or strong equivalence principle.

1.1.1 Weak equivalence principle (WEP)

The weak form of the EP is limited to strong and electroweak interactions. More in detail, a rigorous formulation of WEP neglects the particle’s gravitational potential, excludes self-gravity effects and the tidal force. Furthermore, the experimental setup (including observer) must not influence the test particle’s motion. Taking into account all the above considerations, the formulation of the WEP is : “In a homogeneous gravitational field, the acceleration of a freely-falling, structureless test particle is independent of the particle’s properties, i.e. its mass, composition, or thermodynamical state” [1]. As a result, the worldline of a freely-falling test particle in a given gravitational field depends only on the particle’s initial position and velocity. Thus, all test particles in this gravitational field will undergo the same acceleration, independently of their properties. Provided that the external gravity field is homogeneous, a freely-falling observer should find the other freely-falling particles, in the observer’s immediate proximity, moving at uniform velocities relative to him. Thus, from the viewpoint of a freely-falling observer, the mechanics of particles in

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1.1. THE EQUIVALENCE PRINCIPLES 3 free fall is indistinguishable from their mechanics in the absence of gravity. This enables us to cast the WEP into another form: “In a homogeneous gravitational field, the laws of mechanics in a freely-falling reference frame are the same as in the inertial reference frame in the absence of gravity.”

For 36 years the experiment by Braginsky and Panov (1972, [6]) had remained the most accurate test of the WEP. According to their measurement the relative difference between the inertial and gravitational masses did not exceed 10−12. This result was replaced only in 2008 by Schlamminger et al. [32] who were able to decrease this difference to 3 × 10−13.

1.1.2 The Einstein equivalence principle (EEP)

WEP is formulated basically for mechanical motion of test particles provided with rest mass. Einstein proposed that it would be natural to extend this principle from mechanics to electrodynamics and to any other type of non-gravitational fundamental interactions. He suggested that in the close vicinity of a freely-falling observer, all non-gravitational laws of physics are indistinguishable from the same laws formulated in the inertial reference frames. In modern terms, these ideas were summarized by Robert Dicke and resulted in what has come to be called the Einstein equivalence principle (EEP):

1. the weak equivalence principle holds;

2. the outcome of any local non-gravitational experiment is independent of the velocity of the freely-falling reference frame in which it is performed;

3. the outcome of any local non-gravitational experiment is independent of where and when in the universe it is performed.

The word local in the formulation of the EEP means that with a necessary degree of precision, the external gravity field can be assumed as static and homogeneous. The second and the third part of the EEP are called local Lorentz invariance and local position invariance, respectively. In addition to the tests of the WEP, the EEP can be tested by searching for variation of dimensionless constants and mass ratios.

1.1.3 Metric theories and General Relativity

The only theories of gravity that can fully embody EEP are those that satisfy the postulates of metric theories of gravity, which are (see, e.g., [1]) :

1. spacetime is endowed with a symmetric metric;

2. the trajectories of freely falling test bodies are geodesics of that metric;

3. in local freely falling reference frames, the non-gravitational laws of physics are those written in the language of Special Relativity (SR).

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4 CHAPTER 1. FOUNDATIONS OF THEORIES OF GRAVITY In fact, if EEP is valid, then in local freely falling frames the laws governing experiments must be independent of the velocity of the frame. In particular, the values for the various atomic constants must be independent of location and time. The only laws we know of that fulfill this request are those that are compatible with SR. Furthermore, in local freely falling frames, test bodies appear to be unaccelerated, that is, they move on straight lines: such locally straight lines simply correspond to geodesics in a curved spacetime.

GR is a metric theory of gravity, as many other alternative theories of gravitation, including the Brans-Dicke theory and its generalizations (see [8]).

The empirical evidence supporting EEP implies the conclusion that the only theories of gravity that are feasible are metric theories, or possibly theories that are metric apart from very weak or short-range non-metric couplings (as string theory). From now on we shall turn our attention exclusively to metric theories of gravity. With this assumption, other gravitational fields beside the metric could, in principle, exist, but their role must be that of contributing to the spacetime curvature associated with the metric, because matter could not couple to these fields. Thus, matter may create these fields, which, in turn, may generate the metric together with the matter, but they cannot act back directly on the matter. Matter responds only to the metric.

The property that all non-gravitational fields should couple in the same manner to a single gravitational field is called universal coupling. As a consequence, one can consider the metric as a property of the spacetime itself rather than as a field over spacetime. In fact, its properties may be measured and studied using a variety of different experimental devices, composed of different non-gravitational fields and particles, and, due to universal coupling, the results will be independent of the device.

We observe that the condition (3) in the definition of metric theories implies that, where gravity can be neglected, physics is Lorentz invariant as in SR. That is, SR is a suitable model, for practical applications, whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall, we observe that there are no global inertial frames; instead, there are approximate inertial frames moving alongside freely falling particles. Moreover, the straight time-like lines are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity implies a change in spacetime geometry, which has to be no longer globally Minkowskian, but only locally.

The concepts introduced so far can be formally defined within the framework of Differ-ential Geometry. In the following we use some of the basic tools of DifferDiffer-ential Geometry, a more formal approach is beyond the scope of this Thesis and can be found, for example, in [29] or [30].

What distinguishes one metric theory from another is the number and kind of gravitational fields it contains in addition to the metric and the equations that determine the structure and evolution of these fields. From this viewpoint, a metric theory of gravity can be purely

dynamical or prior-geometric [1]. By purely dynamical metric theory we mean any metric

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1.1. THE EQUIVALENCE PRINCIPLES 5 least, one of the other fields in the theory. By prior geometric theory, we mean any metric theory that contains fields or equations whose structure and evolution are given a priori, independently of the structure and evolution of the other fields in the theory. These absolute elements typically include flat background metrics η or cosmic time coordinates. GR is an example of purely dynamical theory since it contains only one gravitational field, the metric

g, and its structure and evolution are governed by partial differential equations, the Einstein field equations: Rµν− 1 2gµνR = 8πG c4 Tµν,

where Rµν is the Ricci curvature tensor, R is the scalar curvature, Tµν is the stress-energy

tensor, gµν is the metric tensor, G is Newton’s gravitational constant and c is the speed

of light in vacuum. Both the Ricci tensor and the scalar curvature are obtained from the Riemann curvature tensor via appropriate contractions of indices. The Riemann curvature tensor R is a tensor field on M which measures locally how a pseudo-Riemannian manifold differs from being the Euclidean space. In particular, it contains all the information concerning the curvature of the space.

The stress-energy tensor is a quantity that describes the density and the flux in spacetime of both energy and momentum, generalizing the stress tensor of Newtonian physics. It is the source of the gravitational field in the Einstein field equations of GR, exactly as the mass density is the source of such a field in Newtonian gravity.

1.1.4 Strong equivalence principle (SEP)

By discussing metric theories of gravity from a general point of view, it is possible to draw some conclusions about the nature of gravity in different models. Let us consider a local freely falling frame, small enough that inhomogeneities in the external gravitational fields can be neglected, but large enough to include a system of gravitating matter and its associated gravitational fields. For example, the system could be a star, a black hole or the solar system. To determine the behavior of the system we must calculate the metric, which depends upon the boundary conditions: as a matter of fact, since the metric is coupled directly or indirectly to the other fields of the theory, its structure and evolution will be influenced by those fields and in particular by the boundary values taken on by those fields far from the local system. Thus, the gravitational environment in which the local gravitating system resides can influence the metric generated by the local system and, consequently, the results of local gravitational experiments may depend on the location and velocity of the frame relative to the external environment. Of course, local non-gravitational experiments are unaffected since the gravitational fields they generate are assumed to be negligible, thus the dependence on the boundary condition is negligible itself.

We can now make the following statements about different kinds of metric theories. • A theory which contains only the metric g yields local gravitational physics which

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6 CHAPTER 1. FOUNDATIONS OF THEORIES OF GRAVITY is independent of the location and velocity of the local system. This follows from the fact that the only field coupling the local system to the environment is g and it is always possible to find a coordinate system in which g takes the Minkowski form at the boundary between the local system and the external environment (neglecting inhomogeneities in the external gravitational field). Thus, the asymptotic value of

g is constant, i.e. it is independent of the location, and it is asymptotically Lorentz

invariant, thus independent of the velocity. GR is an example of such a theory. • A theory which contains the metric and dynamical scalar fields (ϕ, A) yields local

gravitational physics which may depend on the location of the frame but which is independent of the velocity of the frame. This follows from the asymptotic Lorentz invariance of the Minkowski metric and of the scalar fields, but in this case the asymptotic values of the scalar fields may depend on the location of the frame. An example is Brans-Dicke theory (see [8]), where the asymptotic scalar field determines the effective value of the gravitational constant, which can vary as ϕ varies. On the other hand, a form of velocity dependence in local physics can enter indirectly if the asymptotic values of the scalar field vary with time on cosmological scales. Then, the rate of variation of the gravitational constant could depend on the velocity of the frame.

• A theory which contains the metric and additional dynamical vector or tensor fields or prior-geometric fields yields local gravitational physics which may have both location and velocity-dependent effects. An example is the Einstein-aether theory (see, e.g., [9]), which contains a dynamical time-like vector field; the large-scale distribution of matter establishes a frame in which the vector has no spatial components and systems moving relative to that frame can experience motion-dependent effects.

These ideas can be summarized in the strong equivalence principle (SEP), which states that:

1. WEP is valid for self-gravitating bodies as well as for test bodies;

2. the outcome of any local test experiment is independent of the velocity of the (freely falling) apparatus;

3. the outcome of any local test experiment is independent of where and when in the universe it is performed.

The distinction between SEP and EEP lies in the inclusion of bodies with self-gravitational interactions (planets, stars) and of experiments involving gravitational forces (Cavendish experiments, gravimeter measurements). Note that SEP contains EEP as the special case in which local gravitational forces are ignored. The above discussion of the coupling of auxiliary fields to local gravitating systems indicates that if SEP is strictly valid, there must be one and only one gravitational field in the universe, the metric g. These arguments have

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1.2. THE PARAMETERIZED POST-NEWTONIAN FORMALISM 7 not been proven yet, but it has been found empirically that almost every metric theory other than GR introduces auxiliary gravitational fields, either dynamical or prior geometric, predicting violations of SEP at some level. GR seems to be the only viable metric theory that embodies SEP completely [1].

SEP requires, among other things, that the gravitational constant G be the same at all times and everywhere in the universe. Experimentally, its validity was confirmed with good precision through timing of binary pulsars (see [11]) and, to a lesser extent, by laser ranging to the Moon (see [10]). In conclusion, it is still unknown if SEP is exactly fulfilled in GR.

1.2 The parameterized post-Newtonian formalism

As discussed in Section 1.1.3, what distinguishes one metric theory from another is the particular way in which matter and possibly other gravitational fields generate the metric. The comparison between different metric theories of gravity and between theories and experiments becomes particularly simple within the assumption of slow-motion, weak-field

limit. This approximation, known as the post-Newtonian limit, is accurate enough to include

most solar system tests expected in the near future. It turns out that, in this limit, the spacetime metric g predicted by every metric theory of gravity has the same structure. Choosing a suitable coordinate system, called nearly globally Lorentz coordinate system, it can be written as an expansion about the Minkowski metric η_µ,ν = diag(−1, 1, 1, 1) in terms of dimensionless gravitational potentials of varying degrees of smallness, which are constructed from the matter variables.

The only way in which one metric can differ from another is in the numerical values of the coefficients that appear in front of the metric potentials. The parameterized post-Newtonian (PPN) formalism inserts parameters in place of these coefficients, parameters whose values depend on the theory under study. In the current version of the PPN formalism, ten param-eters are used, chosen in such a manner that they measure or refer to general properties of metric theories of gravity. Under reasonable assumptions about the kind of potentials that can be present at post-Newtonian order, one finds out that ten PPN parameters exhaust the possibilities (see, e.g., [12]). In the following, we summarize the crucial points of this formalism.

Coordinate system.

The framework uses a nearly globally Lorentz coordinate system (t, x1, x2, x3). We are

using three-dimensional, Euclidean vector notation, and all coordinate arbitrariness ("gauge freedom") has been removed by specialization of the coordinates to the standard PPN gauge (see, for example, [31] for all the details).

Matter variables.

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8 CHAPTER 1. FOUNDATIONS OF THEORIES OF GRAVITY comoving with the gravitating matter;

• vi₌ dxi

dt : coordinate velocity of the matter;

• w_i velocity components of the PPN coordinate system relative to the mean rest-frame of the universe;

• p : pressure as measured in a local freely falling frame momentarily comoving with the matter;

• Π : internal energy per unit rest mass, which includes all forms of non-rest-mass, non-gravitational energy, e.g., energy of compression and thermal energy;

• PPN parameters: γ, β, ξ, α1, α2, α3, ζ1, ζ1, ζ3, ζ4.

The parameter γ describes how much space-curvature is produced by the unit rest mass and it parameterizes the velocity-dependent modification of the two-body interaction, while the modification of the non-linear three-body general relativistic interaction depends upon the Eddington parameter β, which describes how much nonlinearity there is in the superposition law for gravity (see [53]). They are the only non-zero PPN parameters in GR. The preferred-location effects associated with the gravity field are controlled by ξ. The parameters α1, α2, α3 account for possible preferred frame effects associated with the

motion of the solar system with respect to a hypothetical privileged reference frame of the universe violating the Lorentz invariance. Lastly, ζ1, ζ2, ζ3, ζ4 (and also α3) give information

about the possible violation of conservation of the total momentum of the solar system – in particular all of these quantities must be zero in order to have conservation laws.

There would be other three other PPN parameters, which are the components of the velocity of motion of the solar system with respect to the privileged frame of the universe:

w = (w1, w2, w3). Concerning these parameters, the standard procedure is to assume

that their value refers to the specific reference frame in which the microwave background radiation is isotropic (see [13]).

We show below in Table 1.1 the values that these parameters take in the following cases: • in GR;

• in any theory of gravity that possesses conservation laws for total momentum, called

semi-conservative;

• in any theory that in addition possesses global conservation laws for angular momen-tum, called fully conservative .

Metric.

Following [68], the coefficients of the metric in the PPN formalism can be obtained as follows. First of all, the quantities which should be considered as ”small” need to be identified. In

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1.2. THE PARAMETERIZED POST-NEWTONIAN FORMALISM 9

Table 1.1: The PPN parameters and their significance (α3 has been shown twice to indicate that it is a

measure of two effects).

the solar system, the Newtonian gravitational potential U is such that G U/c2 is everywhere smaller than 10−5. Planetary velocities are related to U by the virial theorem, which yield

v2∼ U . The matter making up the Sun and planets undergoes the pressure p, but since the bodies are in hydrostatic equilibrium, this pressure is comparable to ρ U (more precisely, it holds that ∇p = ρ∇U , where ρ is the density taken as constant), which leads to p/ρ ∼ U . From thermodynamics, other forms of energy (compressional energy, radiation, thermal energy, magnetic energy, etc.) are related to pressure by p V ∼ E ∼ ρ Π V , where V is the characteristic volume and, thus, the ratio of energy density to rest-mass density Π is related to U by Π ∼ c2_{p/ρ ∼ U . Thus, we can consider those quantities as belonging to the same}

order of smallness, denoted by O(ε), that is:

G U/c2 ∼ v2/c2 ∼ p/(ρ c2) ∼ Π/c2∼ O(ε).

Now, writing the Lagrangian L for a single particle in a metric gravitational field, we can verify that Newtonian physics is given by an approximation for L which takes into account

O(ε) terms. From the previous Section, we can infer that Post-Newtonian physics must

involve in L terms of the next order O(ε2).

We note that the odd-half-order terms, for example O(ε1/2) or O(ε3/2), contain an odd occurrence of velocities v or of time derivatives ∂/∂t. Thus, these factors change sign under time reversal, consequently, odd-half-order terms must represent dissipative processes in the system. Assuming the conservation of rest-mass energy and conservation of energy in the

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10 CHAPTER 1. FOUNDATIONS OF THEORIES OF GRAVITY Newtonian limit, terms of O(ε1/2) and of O(ε3/2) cannot appear in the Lagrangian L.

In order to express L at the O(ε2) level, we need to include the various components of the metric up to the appropriate order: the knowledge of g₀₀ must be of the order O(ε2),

g0j of the order O(ε3/2) and gij of the order O(ε). From calculations, choosing units so that

G = c = 1, it follows that (see, e.g., [35]):

g00= − 1 + 2U − 2βU2− 2ξΦW + (2γ + 2 + α3+ ζ1− 2ξ)Φ1+ + 2(3γ − 2β + 1 + ζ₂+ ξ)Φ₂+ 2(1 + ζ₃)Φ₃+ 2(3γ + 3ζ₄− 2ξ)Φ₄+ − (ζ1− 2ξ)A − (α1− α2− α3)w2U − α2wiwjUij+ (2α3− α1)wiVi+ O(ε3); g0i= − 1 2(4γ + 3 + α1− α2+ ζ1− 2ξ)V1− 1 2(1 + α2− ζ1+ 2ξ)Wi− 1 2(α1− 2α2)wiU + − α2wjUij+ O(ε5/2); gij = (1 + 2γU )δij + O(ε2). (1.1) Moreover, the metric potentials take the form (see, e.g., [35]):

U = Z _ρ0 |x − x0_|d 3_x0_, _U ij = Z _ρ0_{(x − x}0₎ i(x − x0)j |x − x0_|3 , A = Z _ρ0_[v0_{· (x − x}0_)]2 |x − x0_|3 , Φ₁ = Z _ρ0_v02 |x − x0_{| d}3_x0 , Φ2= Z _ρ0_U0 |x − x0_{| d}3_x0, Φ₃ = Z _ρ0_Π0 |x − x0_{| d}3_x0 , Φ4= Z _p0 |x − x0_{| d}3_x0, Vi = Z _ρ0_v0 i |x − x0_{| d}3_x0 , Wi = Z _ρ0_[v0_{· (x − x}0_{)](x − x}0₎ i |x − x0_|3 d3_x0 . (1.2)

1.3 Test of post-Newtonian parameters

In the following, we briefly describe the main experimental tests that have been performed so far to constrain the value of PPN parameters.

1.3.1 Test of the parameter γ

In 1964 Shapiro showed that a radar pulse from the Earth to a target body passing near the Sun would experience a time delay because of the spacetime curvature [33]. Shapiro’s original formulation was derived from the Schwarzschild solution of the Einstein field equations, that is a particular solution obtained under the assumption that the mass that create the gravity field is spherical and its electric charge and angular momentum are all zero. The solution is a well-known approximation used for describing slowly rotating astronomical object such as many stars and planets. Using the first order approximation of the PN

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1.3. TEST OF POST-NEWTONIAN PARAMETERS 11

Figure 1.1: An example of SC configuration, lengths are not in scale.

formalism (1-PN) we express the time delay as follows:

∆t_S = (1 + γ)G M c3 ln _r e+ rp+ rep re+ rp− rep , (1.3)

where M is the solar mass, c is the speed of light, re is the Sun-Earth distance, rp is

the Sun-body distance, rep is the Earth-body distance. In a superior conjunction (SC)

configuration, i.e. when the Sun-Earth-Probe and Sun-Probe-Earth angles are both smaller than 5◦ (see Figure 1.1), the one-way delay can be approximated as follows:

∆tS= (1 + γ) G M c3 ln _4r erp b2 (1.4)

where b is the impact parameter, that is the distance between the Sun and the Earth-body line, supposed to be rp. The delay is larger during a SC, thus it is a convenient

configuration for a test of γ. Since one does not have access to a "Newtonian" signal against which to compare the round-trip travel time of the observed signal, it is necessary to do a differential measurement of the variations in round-trip travel times as the target passes through superior conjunction and to look for the logarithmic behavior of Equation (1.4). In order to do this accurately, one must take into account the variations in the round-trip travel time due to the orbital motion of the target relative to the Earth. This can be done by using radar-ranging and possibly other data available of the target taken when it is far from SC (i.e., when the time-delay term is negligible) to determine an accurate ephemeris for the target. The ephemeris can be used to predict the PPN coordinate trajectory r_p(t) near SC, then combining that trajectory with the one of the Earth re(t) to determine the

Newtonian round-trip time and the logarithmic term in Equation (1.4).

The resulting predicted round-trip travel times in terms of the unknown coefficient

1

2(1 + γ) are, then, fit to the measured travel times using the method of least squares (see

Section 3.2) and the quantity 1₂(1 + γ) is, thus, estimated (see, e.g., [36] for details). The most important source of error in SC experiments (SCE) is due to the plasma contained in the solar corona, which perturbs the radio signal causing an additional time delay, proportional to the inverse of the square of the signal frequency. Because of its

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12 CHAPTER 1. FOUNDATIONS OF THEORIES OF GRAVITY extreme variability, the plasma delay is very difficult to model. Nevertheless, because of its dispersive character, it can be directly estimated and removed from the observables if the spacecraft supports a multifrequency link (see [14]): by exploiting the frequency dependence of the refraction index, the differences between the delay and Doppler measurements 1 done in the different bands provide information on the solar plasma content along the path followed by the radio waves. In this way, most of the measurement errors introduced, in a single channel, by the plasma can be removed.

The best estimate of γ to-date is the one obtained with the Cassini-Huygens SCE of 2002. For the first time in that occasion the data processing took advantage of the Ka-band (∼ 32 GHz) for the radio tracking of the Cassini probe and the availability of the X/X, Ka/Ka and X/Ka multifrequency link allowed to completely remove the solar plasma noise. This led to the current estimate of γ = 1 + (2.1 ± 2.3) × 10−5 [34]. A similar experiment will be performed by BepiColombo during the cruise phase to Mercury: the simulations predict a significant improvement in the estimate (see [36] for details).

Closely related to the time delay is the phenomenon of the deflection of the light, which is the change of path of light as it passes near a massive body: instead of following a straight line as expected from classical physics, GR predicts that it follows a geodesic of the spacetime. Historically it was the first experimental confirmation of GR: in 1919 Eddington’s observation of the bending of optical starlight during a solar eclipse helped to make Einstein famous (see [15]).

A ray of light, which passes by the Sun at a distance b, is deflected by an angle:

δϑ = 1 2(1 + γ) 4 G M c2_b 1 + cos Φ 2 ; (1.5)

where M and b are as above and Φ is the angle between the Earth-Sun line and the

incoming direction of the photon. For an almost exact superior conjunction it holds that

b ∼ Rs, Φ ∼ 0 and δϑ ∼ 1.007505 × (1 + γ)/2. The prediction of the full bending of light by

the Sun was one of the greatest successes of GR, however the experiments of Eddington and his co-workers had only 30 percent accuracy. Similar experiments performed in the subsequent years did not achieve better results: the value of γ was estimated between one half and twice the GR value, depending on the case, and the accuracies were low [1] .With the advent of the very long base interferometry and of dedicated space missions, the same result was achieved with errors of 1% or smaller and more recently to the 10−4 level [36].

1.3.2 The perihelion shift of Mercury and limits on β

The fascinating story of Mercury’s contribution to the tests of the theory of gravitation starts in 1859 with the announcement by Le Verrier that, after the perturbing effects of the planets on Mercury’s orbit had been accounted for and after the effect of the precession of

1

the measurements of the Doppler effect, that is, the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It depends by the relative difference in velocity between the observer and the source (that is, the range rate).

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1.3. TEST OF POST-NEWTONIAN PARAMETERS 13

Figure 1.2: Keplerian elements. In particular, we note the argument of periapsis ω (named argument of

perihelion in the case of an heliocentric orbit), the inclination i and the longitude of the ascending node Ω.

the equinoxes on the astronomical coordinate system had been subtracted, it remained in the data an unexplained advance in the perihelion of Mercury. The modern value for this discrepancy is 43 arcseconds per century. This misfit between theory and orbital observations could be approached in two ways: putting into question the gravitation theory or assuming its validity and seeking for a "missing mass". To explain the excess in Mercury’s perihelion drift, Le Verrier adopted the latter standpoint: he postulated the existence of a planet, named Vulcan, between the Sun and Mercury. Soon, he realized that Vulcan should be a belt of many small bodies rather than a single object (see, e.g., [16]). At the end of the 19th century, after several decades of unsuccessful search for Vulcanoids, the problem had a brilliant solution through adopting a radical change of the gravitation theory: although Mercury’s perihelion drift was not the primary topic for Einstein to complete his GR theory, he discovered that his theory exactly explained the observed drift, computed in a more complete and precise way by Newcomb. Some months later, de Sitter computed precisely enough the drift within a fully relativistic formulation of planetary motion and concluded that there was an excellent agreement with the observations (see [17]).

The predicted advance per orbit is represented by the longitude of perihelion ˜ω = ω + Ω cos i (see, e.g., [35]), where ω is the argument of perihelion, i is the inclination of the

orbit and Ω is the longitude of the ascending node (see Figure 1.2).

Including both relativistic PPN contributions and the Newtonian contribution resulting from a possible solar quadrupole moment and choosing units such that G = c = 1, the predicted advance is given by:

∆˜ω = 6πm p ₁ 3(2 + 2γ − β) + 1 6(2α1− α2+ α3+ 2ζ2)µ + J2R2 2mp , (1.6)

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14 CHAPTER 1. FOUNDATIONS OF THEORIES OF GRAVITY mass of the two-body system, respectively; p = a(1 − e2) is the semilatus rectum of the orbit, with the semi-major axis a and the eccentricity e; R is the mean radius of the oblate body

(the Sun); and J2 is a dimensionless measure of its quadrupole moment (for details cfr.

[1]). The first term in 1.6 is the classical relativistic perihelion shift, which depends upon the PPN parameters γ and β. The second term depends upon the ratio of the masses of the two bodies; it is negligible for Mercury, since µ = m_merc/Ms∼ 2 × 10−7 . The third term

depends upon the solar quadrupole moment J₂. Assuming that the Sun rotates uniformly with its observed surface angular velocity, so that the quadrupole moment is produced by centrifugal flattening, one may estimate J₂ ∼ 10−7_{: this agrees reasonably well with values}

inferred from rotating solar models that are in agreement with observations of the normal modes of solar oscillations; the latest inversions of data give J₂= (2.246 ± 0.022) × 10−7 [21]. Substituting standard orbital elements and physical constants for Mercury and the Sun we obtain the rate of perihelion shift ˙˜ω, in seconds of arc per century:

˙˜ ω ∼ 42.0098 × ₁ 3(2 + 2γ − β) + 3 × 10 −4 J2 10−7 . (1.7)

Combining the data of Mercury’s orbit provided by Messenger spacecraft with the knowledge of the Cassini bound on γ, a bound on the parameter β was obtained (see [21]):

β − 1 = (−1.62 ± 1.8) × 10−5. (1.8)

Further analysis could push this bound even lower, although knowledge of J2 would have

to improve simultaneously (see, e.g., the discussion in [1]).

1.3.3 The test of the SEP

Will ([1]) pointed out that many metric theories of gravity (perhaps all except GR) can be expected to violate one or more aspects of SEP, as detailed in the following.

The Nordtvedt effect: limits on η

In 1968 Nordtvedt showed that many metric theories of gravity predict that massive bodies violate the weak equivalence principle - that is, fall with different accelerations depending on their gravitational self-energy (see [26]). For a spherically symmetric body, the acceleration from rest in an external gravitational potential U has the form:

a =mp m ∇U , mp= 1 − η Eg m , η = 4β − γ − 3 − 10ξ − α1+ 2 3α2− 2 3ζ1− 1 3ζ2, (1.9)

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1.3. TEST OF POST-NEWTONIAN PARAMETERS 15 where E_g is the negative of the gravitational self-energy of the body (E_g > 0) and η is

a coefficient named Nordtvedt parameter. This violation of the massive-body equivalence principle is known as the Nordtvedt effect. The effect is absent in GR (η = 0) but present in scalar-tensor theory. The existence of the Nordtvedt effect does not violate the results of laboratory experiments, since for laboratory-sized objects it holds that Eg/m ≤ 10−27, far

below the sensitivity of current or future experiments. However, for astronomical bodies,

Eg/m may be significant (3.52 × 10−6 for the Sun, 4.64 × 10−10 for the Earth, 1.88 × 10−11

for the Moon). If the Nordtvedt effect is present (η 6= 0) then the Earth should fall toward the Sun with a slightly different acceleration than the Moon. This perturbation in the Earth-Moon orbit leads to a polarization of the orbit that is directed toward the Sun as it moves around the Earth-Moon system, as seen from Earth. This polarization represents a perturbation in the Earth-Moon distance of the form

δr = 13.1 × η cos(ω0− ωs)t (1.10)

where ω0 and ωs are the angular frequencies of the orbits of the Moon and Sun around the

Earth.

Since August 1969, when the first successful acquisition was made of a laser signal reflected from the Apollo 11 retroreflector on the Moon, the Lunar Laser Ranging (LLR) experiment has made regular measurements of the round-trip travel times of laser pulses between a network of observatories and the lunar retroreflectors. The Nordtvedt parameter

η along with several other important parameters of the model can be, then, estimated within

a least squares fit. The results of LLR cannot be considered as a conclusive test of SEP until one eliminates the possibility of a compensating violation of WEP for the two bodies, because the chemical compositions of the Earth and Moon differ in the relative fractions of iron and silicates. To this end, the Eöt-Wash group carried out an improved test of WEP for laboratory bodies whose chemical compositions mimic that of the Earth and Moon. The results reduce the ambiguity in the LLR bound; it follows that the bound on the Nordtvedt parameter can be deduced as |η| = (4.4 ± 4.5) × 10−4 [37]

Constancy of the Newtonian gravitational constant

Most theories of gravity that violate SEP predict that the locally measured Newtonian gravitational constant may vary with time as the universe evolves. If G changes with cosmic evolution, its rate of variation should be of the order of the expansion rate of the universe, i.e. ˙G/G ∼ H0, where H0 = 7 × 10−11 yr−1 is the Hubble expansion parameter (see, e.g.,

the discussion in [1]). Several observational constraints can be placed on ˙G/G, bounding

the present rate of variation or bounding a difference between the present value and a past value. The first type of bound typically comes from LLR measurements, planetary radar-ranging measurements and pulsar timing data. The second type comes from studies of the evolution of the Sun, stars and the Earth, Big-Bang nucleosynthesis and analyses

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16 CHAPTER 1. FOUNDATIONS OF THEORIES OF GRAVITY of ancient eclipse data. The bounds coming from the last category are sharper, but they depend on the adopted model for the Big Bang and on the assumption made on the variation of G. The experiments within the solar system are typically capable of constraining only possible variations of the product GM, thus any improvement in the knowledge of ˙G/G

beyond 10−13needs an independent knowledge of the actual mass loss from the Sun, due to both the radiation of photons and neutrinos (∼ 0.7 × 10−13yr−1) and due to the solar wind (∼ 0.2 × 10−13 yr−1) [1]. Considering the parameter µ= GM, the current bound is

8 × 1015 cm s−2 [39] and its rate is limited by ζ = ˙µ/µ= 4.3 × 10−14 [41].

Preferred-frame and preferred-location effects

In the post-Newtonian limit, preferred-frame effects are governed by the values of the PPN parameters α1, α2, α3 and some preferred-location effects are governed by ξ. The

most important effects are variations and anisotropies in the locally-measured value of the gravitational constant which lead to a series of anomalous events: Earth tides and variations in the Earth’s rotation rate, contributions to the orbital dynamics of planets and the Moon, self-accelerations of pulsars, contribution to the torques on the Sun and on spinning pulsars that could be seen as variations in their pulse profiles.

A tight bound on α₃equal to α₃∼ 4×10−20_{was obtained by placing limits on anomalous}

eccentricities in the orbits of a number of binary millisecond pulsars [38].

Constraints on α₁, α2 are inferred from the latest determinations of the observationally

admitted ranges for any anomalous solar system planetary perihelion precessions. Removing the a priori bias of unmodeled or mismodeled standard effects (such as the general relativistic Lense – Thirring precessions and the classical rates due to J2) the studies allowed to infer

α1= (−1 ± 6) × 10−6, α2 = (−0.9 ± 3.5) × 10−5 (see [40]).

1.3.4 Tests of post-Newtonian conservation laws

Conservation laws involve PPN parameter ζ₁, ζ2, ζ3, ζ4 and α3. Only three of them have

been constrained directly with any precision: ζ2, ζ3 and, as we said above, α3. Concerning

the other parameters, the constraint on ζ1 can be derived indirectly from the Equation 1.9,

describing the Nordtvedt effect. Concerning ζ₄, which is related to the gravity generated by fluid pressure, there is strong theoretical evidence that it is not really an independent parameter – in any reasonable theory of gravity there should be a relation between the gravity produced by kinetic energy ρv2, internal energy ρΠ and pressure p (see, e.g., [27]). From such considerations, it follows the additional theoretical constraint:

6ζ₄= 3α₃+ 2ζ₁− 3ζ₃. (1.11)

Concerning ζ₃, a theory of gravity described by PPN formalism that violates conservation of momentum, but that obeys to Equation 1.11, is such that the electrostatic binding energy

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1.4. FUTURE IMPROVEMENTS OF THE CONSTRAINTS 17 the form

mA= mP +

1 2ζ3Ee.

The resulting limit on ζ3 from the lunar experiment is ζ3< 10−8 (see [35]).

Regarding ζ2, the most meaningful limit is obtained from the observations of the binary

pulsar PSR1913+16. In this case, the analysis is based on another consequence of a violation of conservation of momentum: a self-acceleration of the center of mass of a binary stellar system (see [1] for details). The limit |α₃+ζ₂| = 4×10−5has been deduced from observations. Since α₃ has already been constrained to a significantly lower accuracy (see Section 1.3.3), we obtain a strong bound on ζ2< 4 × 10−5 (see [28]).

1.4 Future improvements of the constraints on PPN

param-eters

At the moment, there are many space missions which are providing data or which will provide data in the near future, that can be employed to achieve an improvement of the PPN parameters, of J₂ and of ˙G/G. Some examples are detailed in the following.

• Possible improvements can be obtained by analyzing the data already available from ended missions. This is the case of MESSENGER, a NASA robotic spacecraft that orbited the planet Mercury between 2011 and 2015, studying Mercury’s chemical composition, geology, and magnetic field. The mission obtained a huge data set of range measurements for the determination of Mercury’s ephemeris, which has been exploited in the last years to estimate both the gravimetry and rotational state of Mercury (in particular the Mercury gravity field, spin-pole axis, rotation period, Love number and also ephemeris, see [22]) and some parameters related to GR and the evolution of the Sun (see [21]).

• Regarding different strategies used to analyze mission data, in [24] it has been pre-liminarily shown that a strategy based on the simultaneous data analysis of several missions might lead to a further reduction of the formal uncertainties on the parameters of interest, via an effective reduction of their correlations. A first step towards a possible future combined analysis works can be seen in [25]: here the simulations are based on a data analysis strategy that considers only one mission at a time, eventually employing as a-priori the results of other missions. The semi-analytical model used for the covariance analysis consists in the computation of the perturbations on planetary orbits by solving the Hill’s equations.

• Another ongoing experiment, already underlined in Section 1.3.3, is the Lunar Laser Ranging (LLR). It aims to provide precise observations of the lunar orbit that can

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18 CHAPTER 1. FOUNDATIONS OF THEORIES OF GRAVITY contribute to a wide range of scientific investigations. In particular, the highly ac-curate measurements of the distance between the Earth and Moon provide a crucial information used to determine whether, in accordance with the Equivalence Principle, both of these celestial bodies are falling towards the Sun at the same rate, despite their different masses, compositions, and gravitational self-energies. That is, the sensitivity of the lunar orbit to the equivalence principle comes from the acceleration of the Earth and Moon by the Sun: any difference in those accelerations due to a failure of the equivalence principle causes an anomalous term in the lunar range with the 29.53 day synodic period. The amplitude would be proportional to the difference in the gravitational to inertial mass ratios for Earth and Moon. It is expected that future improvements in the accuracy of LLR tests, due to an extension of the data set with existing stations and to a new APOLLO instrument, will have the capabil-ity to improve bounds on PPN and related parameters, in particular on η and ˙G/G [37].

• A different type of experiment that should soon produce interesting new tests of gravity concerns triple system containing a radio pulsar. The only previously known system of this kind, named PSR B1620-26, shows only weak interactions. Recently, it is under study (see [23]) the case of PSR J 0337 + 1715, a millisecond pulsar (a neutron star with a rotational period smaller than about 40 milliseconds) in a hierarchical triple system with two other stars. The system is characterized by strong gravitational interactions, which allow to infer the masses of the pulsar (1.4368 ± 1.3 × 10−3)M and the two

white dwarf companions, (0.19751 ± 1.5 × 10−4)M and (0.4101 ± 3 × 10−4)M, as

well as the inclinations of the orbits (both about 39.2◦). The gravitational field of the outer white dwarf strongly accelerates the inner binary containing the neutron star, and the system will thus provide an ideal laboratory in which testing the strong equivalence principle. There have been other strong-field SEP tests performed using millisecond pulsar and the Galactic field as the external perturbing field, but, in the case of J 0337 + 1715, the perturbing field (that of the outer white dwarf) is six to seven orders of magnitude larger, greatly magnifying any possible SEP violation effects.

• Gaia is a space observatory of the ESA, launched in 2013 and expected to operate until 2022. The spacecraft is designed for astrometry: it measures the positions, distances and motions of stars with unprecedented precision. The mission aims to construct by far the largest and most precise 3D space catalog ever made. So far it has collected astrometric data for about 1.7 billion objects in our galaxy [19], mainly stars, but also planets, comets, asteroids and quasars. The estimation of the PPN parameter γ is also one of the objectives of Gaia: by measuring the gravitational deflection due to the Sun and the parallax shift of such objects, Gaia is expected to measure γ with an accuracy of 2 × 10−6 (see [20]), one order of magnitude better than Cassini.

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1.4. FUTURE IMPROVEMENTS OF THE CONSTRAINTS 19

• BepiColombo is an ongoing space mission to the planet Mercury, whose goal is to perform a comprehensive study of Mercury, including characterization of its magnetic field, magnetosphere, and both interior and surface structure. Thanks to full on-board and on-ground instrumentation capable to perform very precise tracking from the Earth, the Mercury-centric orbit of the spacecraft and the heliocentric orbit of Mercury can be determined with unprecedented accuracy. Starting from the radio observations (range, range rate) it is possibile to perform an estimation of the parameters of interest,

which, in the case of the relativity experiment, are the PPN parameters and some related parameters.

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Chapter 2

Tests of gravity with Radio Science

Experiments

2.1 The BepiColombo mission

BepiColombo is a space mission for the exploration of the planet Mercury, jointly developed by the European Space Agency (ESA) and the Japan Aerospace Exploration Agency (JAXA). The mission includes two spacecraft (see Figure 2.1): the Mercury Planetary Orbiter (MPO), shown in Figure 2.2, mainly dedicated to the study of the surface and the internal composition of the planet (see, e.g., [43]) and the Mercury Magnetospheric Orbiter (MMO), shown in Figure 2.3, designed for the study of the planetary magnetosphere (see, e.g., [44] for details). The two orbiters have been launched together in October 2018 on an Ariane 5 launch vehicle from Kourou (French Guiana) and they are being carried to Mercury by a Mercury Transfer Module. The arrival at Mercury is scheduled for December 2025, after 7.2 years of cruise and one flyby of the Earth, two flybys of Venus and six flybys of Mercury

1_{. After the arrival, the orbiters will be inserted in two different polar orbits: the MPO on a}

480 × 1500 km orbit with a period of 2.3 h, while the MMO on a 590 × 11639 km orbit. The nominal duration of the mission in orbit is 1 year, with a possible 1 year extension. The Mercury Orbiter Radio science Experiment (MORE) is one of the experiments on-board the MPO spacecraft.

The scientific goals of MORE concern both fundamental physics tests and the geophysics of Mercury. In particular, the radio science experiment will provide the determination of the gravity field of Mercury ([45]-[48]) and its rotational state ([49]-[51]), in order to constrain the planet’s internal structure (gravimetry and rotation experiments). Moreover, from the fact that Mercury, being the innermost planet, is the best placed planet to investigate the gravitational effects of the Sun, MORE will allow an accurate test of relativistic theories of gravitation (relativity experiment) ([52]-[59]).

1

The spacecraft’s current position and its entire journey can be visualized at: https://www.cosmos.esa. int/web/bepicolombo/home. An animation showing BepiColombo’s launch and cruise to Mercury, including the separation of the two science orbiters, can be seen at: https://sci.esa.int/s/wQmQErw

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22 CHAPTER 2. TESTS OF GRAVITY WITH RADIO SCIENCE EXPERIMENTS

Figure 2.1: The configuration of the spacecrafts in BepiColombo mission.

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2.1. THE BEPICOLOMBO MISSION 23

Figure 2.3: BepiColombo MMO’s science instruments.

The global experiment consists in a very precise orbit determination of both the MPO orbit around Mercury and the orbits of Mercury and the Earth around the solar system Barycenter (SSB), performed by means of state-of-the-art on-board and on-ground instru-mentation (see [61]). In particular, the on-board transponder will collect the radio tracking observables (range, range rate2), up to a goal accuracy (in Ka-band) of about σr = 15 cm

at 300 s for one-way range and σ_r_˙ = 1.5 × 10−4 cm/s at 1000 s for one-way range rate. To appreciate how valuable is the opportunity provided by the BepiColombo mission, we need to consider the following features. First, the orbit of the MPO spacecraft around Mercury has an altitude between 400 and 1500 km, that is, its orbit is not very eccentric with respect to previous missions to Mercury. This will be possible since the investigation of the Mercury’s magnetosphere – which requires a highly elliptic orbit – is carried on by a separated orbiter (the MMO), so that the MPO may reside on a much less eccentric orbit. Second, the range and range rate measurements will be performed by using a full 5-way link to the MPO (see [61]). As we have already mentioned in Section 1.3.1, the differences between the delay (range) and Doppler (range rate) measurements done in the different channels (in the Ka band, in the X band and in mixed mode with both) provide information on the solar plasma content along the path followed by the radio waves between the Earth and the spacecraft. By exploiting this information, the perturbations due to the solar plasma can be modeled and, thus, most of the measurement errors they introduce in the problem can be removed. The expected performances correspond to an improvement of about two orders of magnitude with respect to what was possible with the previous technology [53].

Third, an accelerometer (Italian Spring Accelerometer - ISA [60]) is placed on-board the MPO. This is a key point, since the non-gravitational perturbations due to the Sun around Mercury are strong: both the flux of visible light and infrared are one order of magnitude larger compared to the case of the Earth. The resulting radiation pressure effects are very difficult to model, also due to the fact that the spacecraft reflects and radiates back

2 _{with the expression “range” we mean the distance between the spacecraft and the tracking station,}

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24 CHAPTER 2. TESTS OF GRAVITY WITH RADIO SCIENCE EXPERIMENTS in a very complex way. As a result, the propagation of the orbit of an artificial satellite around Mercury turns out to be affected by large errors, thus the presence of an on-board accelerometer becomes essential to perform a precise orbit determination. From a practical point of view, the readings of the accelerometer are simply added to the equations of motion containing the gravitational terms, as additional accelerations, making feasible the orbit determination process.

The Relativity experiment

The aim of the MORE relativity experiment is to investigate the dependence of the equation of motion from the PPN parameters and other related parameters. In this way, by isolating the effect that each parameter has on the motion, it is possible to constrain the parameter value within some accuracy threshold, testing the validity of GR predictions.

In the framework of metric theories of gravitation, each violation of the principles of GR is reflected in a corresponding modification of the spacetime metric, affecting the observational model in different ways: the connection between local and global coordinate systems, the equations of motion and the propagation of the electromagnetic signal. In practice, it turns out that the coordinate transformation effects are too small to investigate PPN effects [53]. For instance, the transformation between the barycentric and geocentric coordinate systems alters the position of an Earth station by about 6 cm in GR: though this effect will be taken into account, its dependence on the PPN parameters can be neglected. Thus, the experiment aims at investigating the dependence of the equations of motion from the PPN parameters and the propagation effects. Concerning the equations of motion, the satellite motion is best referred to the planet it is orbiting around. Indeed, the solar system state vector of the spacecraft, that enters in the definition of the observables (see Section 2.4), can be decomposed in the local motion of the satellite (see Section 2.2) and the motion of Mercury’s center of mass in the solar system (described in details in Section 2.3). Both these components are affected by relativistic perturbations in their own way. It can, however, be observed that the satellite local dynamics alone, i.e. the motion around Mercury’s center of mass, cannot be exploited to constrain these phenomena (see [53]). Indeed, the largest effect is the geodetic precession due to the solar gravity field, while the perihelion drift due to Mercury’s relativistic monopole gravity is significantly smaller. The geodetic precession causes a constant rotation of the spacecraft orbit with respect to distant stars, resulting in a displacement of about 30 cm over one month [53]. This value is far too small to be detected, therefore the local dynamics of the satellite around Mercury is modeled with sufficient accuracy by purely Newtonian equations of motion. It is only the heliocentric motion of the Mercury’s center of mass that is capable of constraining the PPN parameters. The observations related to the Earth-Mercury range and range rate will be compared with a theoretical model depending on the PPN parameters, appearing in the equations of the heliocentric motion. Moreover, the light propagation relativistic effects in both range and range rate need to be taken into account: this effect is most noticeable when Mercury gets