Precision orbital dynamics and apsidal precession in eccentric exoplanetary systems

Orbital dynamics in compact and eccentric exoplanetary systems provide a promising laboratory for testing relativistic effects beyond the Solar System. The increasing precision of photometric, radial-velocity, and transit-timing observations has significantly improved the characterization of extrasolar planetary orbits, opening the possibility of probing subtle relativistic signatures in regimes inaccessible through Solar System observations alone. In particular, apsidal and periastron precession measurements have become increasingly relevant for studies of orbital evolution, dynamical stability  in compact planetary systems. Highly eccentric and short-period exoplanets are especially suitable for such investigations because relativistic corrections are expected  in strong orbital curvature regimes.In this work, we investigate relativistic periastron shifts in a sample of eccentric exoplanets using a computational framework based on effective weak-field relativistic orbital dynamics derived from axisymmetric gravitational geometries. Our study focuses on identifying orbital regimes where relativistic contributions may become observationally relevant while naturally recovering the standard general relativistic behavior for nearly circular systems. We develop a Python code in order to compute apsidal precession rates directly from observed orbital parameters, including eccentricity, semi-major axis, orbital period, and stellar mass, and compare the resulting predictions with standard post-Newtonian expectations. This allows us to provide  a systematic exploration of the dependence of relativistic precession signals on orbital properties and provides a basis for future incorporation of additional observational constraints, e.g., transit-timing variations and long-term orbital monitoring. Our results indicate that eccentricity, orbital period, and semi-major axis jointly play a central role in determining the magnitude of relativistic apsidal precession signals. We also compare the resulting precession rates with standard relativistic predictions and investigate the observational viability of detecting such effects in current and future exoplanet surveys.