Dynamics of Apsidal Motion in Non-Synchronous Binary Pulsars Coupled Orbit and Spin Evolution

TL;DR

This study investigates apsidal motion in three non-synchronous binary pulsars by numerically integrating Zahn's equilibrium-tide equations to quantify the roles of general relativity, stellar oblateness, and tidal interactions in orbital and spin evolution. It shows that gravitational-wave emission overwhelmingly drives orbital decay in the compact systems, while tides induce system‑specific synchronization and circularization timescales, with $t_{syn}\sim\mathcal{O}(10^3)$–$10^5$ yr and $t_{cir}\sim\mathcal{O}(10^5)$–$10^7$ yr for PSR 1913+16 and J0737‑3039A/B, and negligible tidal evolution for J0621+1002. The derived apsidal‑motion constant $k_2\approx0.1$ and tidal friction times (hours–days) align with neutron‑star structure models, and observed periastron advances broadly match general relativity, underscoring GR’s dominance in strong‑field binary pulsars and highlighting the need to include GW terms in long‑term orbital evolution analyses.

Abstract

The apsidal motion of a non-synchronous binary pulsar serves as a valuable probe of relativistic gravity, stellar stricture, and dynamical evolution of close binary systems, In this study, we investigate the combined influence of general relativity, stellar oblateness and tidal interaction on the apsidal motion of three binary pulsars: 1913+16, J0737-3039A/B, and J0621+1002. Zahn's tidal equations \cite{1977A&A....57..383Z, 1989A&A...220..112Z} were employed for numerical integrations to describe tidal effects and their role in orbital and spin evolution. We estimated the timescales for tidal synchronization and orbital circularization for each system. The results indicate that tidal effects play only a minor role in orbital decay compared with energy loss due to gravitational wave emission. This is evident in the compact system PSR 1913+16, where the orbital period decreases by approximately 76.5 $μ$s/yr as a result of gravitational radiation. The double pulsar J0737-3039A/B exhibits faster orbital evolution, with synchronization occurring in about 8.4$\times10{^3}$ years, whereas the wider system J0621+1002 shows negligible orbital change over timescales exceeding 10$^{10}$ years. The simulations demonstrate clear trends of decreasing semi-major axis and eccentricity, accompanied by an increase in spin rate among the binary pulsars studied. The derived apsidal motion constants [$k\simeq0.1$] are consistent with theoretical expected values, and the corresponding tidal friction times (between a few hours to several days) agree well with theoretical predication. These results emphasize the dominant role of relativistic effects in neutron star binaries and highlight the importance of including gravitational-wave terms long-term orbital evolution

Figures (8)

• Figure 1: Orbital visualization illustrating the effect of tidal torque as a function of the spin-orbit misalignment angle ($\theta$) for the three binary pulsar systems: J0621+1002 (Top), PSR 1913+16 (Middle), and J0737-3039A/B (Bottom). The plots show the orbital path (black ellipse/circle), the positions of the primary (blue/red circle) and companion (smaller circle), the spin axis (red arrow), the orbital angular momentum vector (perpendicular to the page), and the direction of tidal torque (green arrow) at different orbital phases. The visualization highlights how tidal torque varies with orbital phase and misalignment, with peak torque typically occurring near periastron (closest approach) where tidal forces are strongest. Note the different eccentricity scales and misalignment ranges depicted for each system.
• Figure 2: Side-by-side comparison of the normalized tidal torque as a function of the spin-orbit misalignment angle ($\theta$, in degrees) for the three systems: J0621+1002 (Top), PSR 1913+16 (Middle), and J0737-3039A/B (Bottom). The torque is zero when the spin and orbital axes are perfectly aligned ($\theta=0$) or perpendicular ($\theta=90^\circ$). As the misalignment angle increases, the torque magnitude changes, reaching its maximum at $\theta=45^\circ$, and changes sign (indicating a reversal in the direction of angular momentum transfer). Different lines may represent instantaneous torque, orbit-averaged torque, or specific misalignment ranges as indicated in the legend (ensure legend is clear in the actual figure). Note the different torque scales and potential highlighted regions (e.g., constrained range for J0737-3039A/B).
• Figure 3: Apsidal motion rate (degrees per year) as a function of orbital period (days) for the three binary pulsar systems: J0621+1002, PSR 1913+16, and J0737-3039A/B. The plot shows the total predicted apsidal motion rate (solid lines) and the separate contributions from General Relativistic effects (dashed lines) and tidal effects (dotted lines). Actual system parameters (observed orbital period and corresponding predicted apsidal motion rate) are marked with black dots. This figure demonstrates the dominance of GR effects over tidal contributions across different orbital period regimes.
• Figure 4: Evolutionary timescale curves for the three binary pulsar systems: J0621+1002 (Green), PSR 1913+16 (Blue), and J0737-3039A/B (Red). $Top panel$: Normalized semi-major axis ($a/a_{0}$) as a function with time (Million Years). $Middle panel$: Normalized eccentricity ( $e/e_{0}$) as a function with time (Million Years). $Bottom panel$: Normalized stellar spin angular velocity ($\Omega/\Omega_{0}$) as a function with time (Million Years). The curves illustrate how each parameter evolves over time in response to tidal interactions and system-specific dynamics, showing different rates of orbital decay, circularization, and spin synchronization.
• Figure 5: Histogram illustrating the distribution of predicted apsidal motion rates (degrees per year) for the three binary pulsar systems: J0621+1002, PSR 1913+16, and J0737-3039A/B. The histogram bins show the frequency or count of systems exhibiting apsidal motion rates within specific ranges, potentially highlighting the typical rates and spread for each system or category.