A uniform treatment of the orbital effects due to a violation of the Strong Equivalence Principle in the gravitational Stark-like limit

We analytically work out several effects which a violation of the Strong Equivalence Principle (SEP) induces on the orbital motion of a binary system constituted of self-gravitating bodies immersed in a constant and uniform external field. We do not restrict to the small eccentricity limit. Moreover, we do not select any specific spatial orientation of the external polarizing field. We explicitly calculate the SEP-induced mean rates of change of all the osculating Keplerian orbital elements of the binary, the perturbation of the projection of the binary orbit onto the line-of-sight, the shift of the radial velocity, and the range and range-rate signatures and as well. We find that the ratio of the SEP precessions of the node and the inclination of the binary depends only on and the pericenter of the binary itself, being independent on both the magnitude and the orientation of the polarizing field, and on the semimajor axis, the eccentricity and the node of the binary. Our results, which do not depend on any particular SEP-violating theoretical scheme, can be applied to quite general astronomical and astrophysical scenarios. They can be used to better interpret present and future SEP experiments, especially when several theoretical SEP mechanisms may be involved, and to suitably design new dedicated tests.

💡 Research Summary

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The paper presents a comprehensive analytical treatment of the orbital consequences that arise when the Strong Equivalence Principle (SEP) is violated in a binary system composed of self‑gravitating bodies immersed in a constant and uniform external gravitational field. This situation is often referred to as the gravitational “Stark‑like” effect, because the external field plays a role analogous to an electric field in the Stark effect of atomic physics.

The authors begin by introducing a generic SEP‑violation parameter Δ, which quantifies the fractional difference between the inertial and gravitational masses of each body due to its own self‑gravity. In the presence of an external field gₑₓₜ, the relative acceleration of the two bodies acquires an additional term Δ gₑₓₜ. No specific theoretical framework (e.g., scalar‑tensor gravity, vector‑tensor theories) is assumed; the analysis is kept completely model‑independent, with Δ being the only new quantity.

Using the Lagrange planetary equations, the authors derive the secular (orbit‑averaged) rates of change of all six Keplerian orbital elements: semimajor axis a, eccentricity e, inclination I, longitude of the ascending node Ω, argument of pericenter ω, and mean anomaly M. The derivation does not rely on the small‑eccentricity approximation; the expressions are valid for any 0 ≤ e < 1. Moreover, the orientation of the external field is left arbitrary, described by a unit vector (\hat{\mathbf{g}}=(\hat{g}_x,\hat{g}_y,\hat{g}_z)). Consequently, each element’s rate is expressed as a linear combination of the three components of (\hat{\mathbf{g}}) multiplied by trigonometric functions of the true anomaly and the orbital angles.

Key results include:

1. Semimajor axis and mean anomaly – Their secular variations are proportional to the magnitude of the external field and to Δ, but they average to zero over one orbital period for a closed Keplerian ellipse, indicating that the long‑term size of the orbit is essentially unchanged.

2. Eccentricity – The secular change (\dot e) also oscillates with the orbital phase and averages to zero, but instantaneous excursions can be significant for highly eccentric binaries.

3. Inclination and node – The most striking finding is that the ratio of the nodal precession (\dot\Omega) to the inclination change (\dot I) is completely independent of the external field’s strength, direction, the semimajor axis, the eccentricity, and even the node itself. The ratio depends only on the inclination I and the argument of pericenter ω:

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