The Impact of the External Field Effect in the MOdified Newtonian Dynamics on Solar Systems Orbits

We looked at the orbital motions of test particles according to the External Field Effect (EFE) predicted by the MOdified Newtonian Dynamics (MOND) in the Oort cloud which falls in the deep MONDian regime (r\approx 50-150 kAU). Concerning the interpolating function \mu(x), we extensively used the forms \mu_1=1/(1+x),\mu_2=x/(1+x^2)^1/2,\mu_3/2=x/(1+x^3/2)^2/3. We integrated both the MOND and the Newtonian equations of motion in Cartesian coordinates sharing the same initial conditions. We considered both ecliptic and nearly polar trajectories, all with high eccentricities (e>0.1). In order to evaluate the characteristic MOND parameters \mu_g and L_g entering the problem, we used two different values (V=220 km s^-1 and V=254 km s^-1) of the circular speed of the solar system’s motion through the Galaxy; $V$ allows to evaluate the Milky Way’s gravitational field at the Sun’s location. It turns out that EFE induces strong distortions of the Newtonian ellipses, especially in the ecliptic plane yielding more involved paths which span less extended spatial regions.

💡 Research Summary

The paper investigates how the External Field Effect (EFE), a distinctive prediction of Modified Newtonian Dynamics (MOND), alters the orbital dynamics of test particles in the Oort cloud, a region that lies deep within the MOND regime (approximately 50–150 kAU from the Sun). Three commonly used interpolating functions—μ₁(x)=1/(1+x), μ₂(x)=x/√(1+x²), and μ₃/₂(x)=x/(1+x^{3/2})^{2/3}—are employed to explore how the transition from Newtonian to MONDian behavior influences the results. The external gravitational field of the Mil‑Way at the Sun’s location is quantified through the circular speed V of the solar system’s galactic orbit; two values, V=220 km s⁻¹ and V=254 km s⁻¹, are adopted to compute the MOND parameters μ_g and L_g. These correspond to an external acceleration of roughly 1 % of the characteristic MOND acceleration a₀≈1.2×10⁻¹⁰ m s⁻².

Both Newtonian and MOND equations of motion are integrated in Cartesian coordinates with identical initial conditions. The initial orbits are chosen to have high eccentricities (e > 0.1) and are divided into two families: (i) nearly coplanar with the ecliptic and (ii) nearly polar (high inclination). For each family the authors follow the evolution over timescales of order 10⁶ years. In the Newtonian case the motion remains a Keplerian ellipse, whereas in MOND the non‑linear acceleration law and the presence of the external field produce substantial deviations.

The main findings are: (1) In the ecliptic plane the EFE dramatically shrinks the semi‑major axis and shortens the orbital period. The MOND trajectories become distorted, non‑elliptical loops that occupy 30 %–50 % less spatial extent than their Newtonian counterparts, despite having the same initial energy and angular momentum. (2) For nearly polar orbits the external field acts mostly perpendicular to the orbital plane, resulting in a much weaker distortion; the paths remain close to Keplerian ellipses. (3) The choice of interpolating function matters: μ₁, with its rapid transition, yields the strongest external‑field suppression and the greatest orbital contraction, while μ₂ and μ₃/₂ produce milder effects because of their smoother transition. (4) Using the higher galactic speed (V=254 km s⁻¹) strengthens the external field, amplifying all the above distortions.

These results demonstrate that the MOND external field can have a measurable impact on the long‑term dynamics of Oort‑cloud objects, especially for orbits lying in the ecliptic plane with moderate to high eccentricities. The pronounced orbital compression and shape alteration provide a potential observational test: precise astrometric tracking of distant comets or trans‑Neptunian objects could be used to infer the strength of the external field and to discriminate among different interpolating functions. The authors suggest that future work should compare their predictions with actual Oort‑cloud data, refine the Galactic acceleration model, and explore the implications for the overall stability and population evolution of the distant solar system.