4.1 Models for gravitational forces acting on a spacecraft

The primary force acting on a spacecraft in deep space is the force of gravity, specifically, the gravitational attraction of the Sun and, to a lesser extent, planetary bodies. When the spacecraft is in deep space, far from the Sun and not in the vicinity of any planet, these sources of gravity can be treated as Newtonian point sources. However, when the spacecraft is in the vicinity of a massive body, corrections due to general relativity as well as the finite extent and mass distribution of the body in question must be considered.

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 −5 𝒪 (c ) using a common parameterized metric. As this formalism forms the basis of modern spacecraft navigational codes, in the following, we provide a brief summary.

4.1.1 The parametrized post-Newtonian formalism

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 [230264265266267373412413414415416417Jump To The Next Citation Point419]. 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 [417Jump To The Next Citation Point] for details).

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

∑ [ ∑ ]2 ∑ 2 g = 1 − -2 μj-+ 2β- -μj − 1-+-2γ- μjr˙j- 00 c2 rij c4 rij c4 rij j⁄=i j⁄=i j⁄=i 2(2β-−--1)∑ μj-∑ μk- 1- ∑ ∂2rij- −5 + c4 rij rjk − c4 μj ∂t2 + 𝒪 (c ), j⁄=i k⁄=j j⁄=i
∑ α g0α = 2(γ-+-1)- μj˙rj-+ 𝒪 (c−5), (α = 1...3 ) c3 rij ( j⁄=i ) 2γ-∑ μj- − 5 gαβ = − δαβ 1 + c2 rij + 𝒪 (c ), (α,β = 1...3 ) (4.1 ) j⁄=i
where the indices 1 ≤ i,j ≤ N refer to the N bodies, rij is the distance between bodies i and j (calculated as |r − r| j i where r i is the spatial position vector of body i), μ i is the gravitational constant for body i given as μi = Gmi, where G is the Newtonian gravitational constant and mi is the body’s rest mass. The Eddington-parameters β and γ have, in this special case, clear physical meaning: β represents a measure of the nonlinearity of the law of superposition of the gravitational fields, while γ represents the measure of the curvature of the spacetime created by a unit rest mass.

The Newtonian scalar gravitational potential in Equation (4.1View Equation) is given by the 2 1∕c term in g00. Corrections of order 1∕c4, parameterized by β and γ, are post-Newtonian terms. In the case of general relativity, β = γ = 1. One of the simplest generalizations of general relativity is the theory of Brans and Dicke [60Jump To The Next Citation Point] that contains, in addition to the metric tensor, a scalar field and an undetermined dimensionless coupling constant ω. Brans–Dicke theory yields the values β = 1, γ = (1 + ω )∕(2 + ω) for the Eddington parameters. The value of β may be different for other scalar-tensor theories [8887389Jump To The Next Citation Point].

The PPN formalism is widely used in studies of relativistic gravitation [6364238Jump To The Next Citation Point239359Jump To The Next Citation Point387Jump To The Next Citation Point417Jump To The Next Citation Point]. The relativistic equations of motion for an N-body system are derived from the PPN metric using a Lagrangian formalism [387Jump To The Next Citation Point417Jump To The Next Citation Point], discussed below.

4.1.2 Relativistic equations of motion

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 [389Jump To The Next Citation Point]. 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 (c, G, 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 (4.1View Equation) 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 i due to the gravitational field of point sources, including Newtonian and relativistic perturbative accelerations [117387Jump To The Next Citation Point417Jump To The Next Citation Point], can be derived in the solar system barycentric frame in the form [27Jump To The Next Citation Point120237238240Jump To The Next Citation Point252]:

∑ μj(rj − ri){ 2(β + γ )∑ μl 2 β − 1∑ μk ( ˙ri)2 ( ˙rj)2 ¨ri = -----3----- 1 − ----2---- ---− ----2-- --- + γ -- + (1 + γ) -- j⁄=i rij c l⁄=i ril c k⁄=jrjk c c [ ]2 } ∑ − 2(1-+-γ)˙ri˙rj − -3- (ri −-rj)˙rj + -1-(rj − ri)¨rj + 3-+-4γ- μj¨rj- c2 2c2 rij 2c2 2c2 rij ∑ { } j⁄=i + 1- μj- [r − r ] ⋅ [(2 + 2 γ)˙r − (1 + 2γ)˙r ] (r˙ − ˙r ) + 𝒪 (c−4), (4.2 ) c2 r3ij i j i j i j j⁄=i
where the indices 1 ≤ j,k,l ≤ N refer to the N bodies and where k includes body i, 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 [240Jump To The Next Citation Point359387], and for the analysis of gravitational experiments in the solar system [388398417418Jump To The Next Citation Point420].

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 i can be expressed using zonal harmonics in the form [240Jump To The Next Citation Point]:

---μi-- ∑∞ J(ki)akiPk-(sin-𝜃) Uiobl = − |r − r| |r − r|k , (4.3 ) i k=1 i
where Pk(x) is the k-th Legendre polynomial in x, ai is the equatorial radius of planet i, 𝜃 is the latitude of the spacecraft relative to the planet’s equator, and (i) Jk is the k-th spherical harmonic coefficient of planet i.

In order to put Equation (4.3View Equation) 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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