An Effective Rotational Correction to Newtonian Gravity for Extended Bodies

Newtonian gravity usually treats interacting bodies as point masses and ignores effects related to their internal structure and rotation. However, real physical bodies are extended in size and often rotate, which can slightly modify their gravitational interaction. In this work, an effective correction to the Newtonian gravitational force is studied for two extended rotating bodies. Starting from the exact Newtonian grav-itational potential for continuous mass distributions and using the large separation approximation, a higher-order correction term is derived. This correction depends on the moments of inertia of the two bodies and shows a force dependence proportional to R -6. The result is interpreted as an effective rotational or quadrupole-level correction rather than a new gravitational force. Order-of-magnitude estimates are calculated for planetary systems and compact astrophysical systems such as binary neutron stars. While the correction is extremely small for planetary scales, it becomes more relevant in compact systems where high precision is required. The analysis remains consistent with Newtonian gravity and the weak-field limit of general relativity.