Of particular interest is the possibility that a celestial mechanics experiment might help distinguish between different theories of gravity. There exists a method, the parametrized post-Newtonian (PPN) formalism, that allows one to describe various metric theories of gravity up to using a common parameterized metric. As this formalism forms the basis of modern spacecraft navigational codes, in the following, we provide a brief summary.

In 1922, Eddington [113] developed the first parameterized generalization of Einstein’s theory, expressing components of the metric tensor in the form of a power series of the Newtonian potential. This phenomenological parameterization has since been developed into what is known as the parametrized post-Newtonian (PPN) formalism [230, 264, 265, 266, 267, 373, 412, 413, 414, 415, 416, 417, 419]. This formalism represents the metric tensor for slowly moving bodies and weak gravitational fields. The formalism is valid for a broad class of metric theories, including general relativity as a unique case. The parameters that appear in the PPN formalism are individually associated with various symmetries and invariance properties of the underlying theory (see [417] for details).

The full PPN formalism has 10 parameters [418]. However, if one assumes that Lorentz invariance, local position invariance and total momentum conservation hold, the metric tensor for a system of point-like gravitational sources in four dimensions at the spacetime position of body may be written as (see [389])

The Newtonian scalar gravitational potential in Equation (4.1) is given by the term in . Corrections of order , parameterized by and , are post-Newtonian terms. In the case of general relativity, . One of the simplest generalizations of general relativity is the theory of Brans and Dicke [60] that contains, in addition to the metric tensor, a scalar field and an undetermined dimensionless coupling constant . Brans–Dicke theory yields the values , for the Eddington parameters. The value of may be different for other scalar-tensor theories [88, 87, 389].

The PPN formalism is widely used in studies of relativistic gravitation [63, 64, 238, 239, 359, 387, 417]. The relativistic equations of motion for an -body system are derived from the PPN metric using a Lagrangian formalism [387, 417], discussed below.

Navigation of spacecraft in deep space requires computing a spacecraft’s trajectory and the compilation of spacecraft ephemeris: a file containing the position and velocity of the spacecraft as functions of time [389]. Spacecraft ephemerides are computed by orbit determination codes that utilize a numerical integrator in conjunction with various input parameters. These include an estimate of the spacecraft’s initial state vector (comprising its position and velocity vector), adopted constants (, , planetary mass ratios, etc.) and parameters that are estimated from fits to observational data (e.g., corrections to the ephemerides of solar system bodies).

The principal equations of motion used by orbit determination codes describe the relativistic gravitational acceleration in the presence of the gravitational field of multiple point sources that include the Sun, planets, major moons and larger asteroids [360]. These equations are derived from the metric Equation (4.1) using a Lagrangian formalism. The point source model is adequate in deep space when a spacecraft is traveling far from those sources. When the spacecraft is in the vicinity of a planet, ephemeris programs also compute corrections due to deviations from spherical symmetry in the planetary body, as well as the gravitational influences from the planet’s moons, if any.

The acceleration of body due to the gravitational field of point sources, including Newtonian and relativistic perturbative accelerations [117, 387, 417], can be derived in the solar system barycentric frame in the form [27, 120, 237, 238, 240, 252]:

where the indices refer to the bodies and where includes body , whose motion is being investigated.These equations can be integrated numerically to very high precision using standard techniques in numerical codes that are used to construct solar system ephemerides, for spacecraft orbit determination [240, 359, 387], and for the analysis of gravitational experiments in the solar system [388, 398, 417, 418, 420].

In the vicinity of a celestial body, one must also take into account that a celestial body is not spherically symmetric. Its gravitational potential can be modeled in terms of spherical harmonics. As Pioneers 10 and 11 both flew by Jupiter and Pioneer 11 visited Saturn, of specific interest to the navigation of theses spacecraft is the gravitational potential due to the oblateness of a planet, notably a gas giant. The gravitational potential due to the oblateness of planetary body can be expressed using zonal harmonics in the form [240]:

where is the -th Legendre polynomial in , is the equatorial radius of planet , is the latitude of the spacecraft relative to the planet’s equator, and is the -th spherical harmonic coefficient of planet .In order to put Equation (4.3) to use, first it must be translated into an expression for force by calculating its gradient. Second, it is also necessary to express the position of the spacecraft in a coordinate system that is fixed to the planet’s center and equator.

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