Compact binary system dynamics at the second post-Newtonian order: Analytical formula of the coordinate time for eccentric and circular orbits

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Table 1

Input parameters related to six selected DNS systems, together with quantities estimated by us.

System	m₁ (M_⊙)	m₂ (M_⊙)	$\frac{\| m_{2} - {\bar{m}}_{2} \|}{m_{2}}$ $\frac{\left\|m_{2}-\bar{m}_{2}\right\|}{m_{2}}$	R₀ (R_⊙)	e₀	P (d)	$\dot{ω}$ $\dot{\omega}$ (deg yr⁻¹)	$\frac{\| \dot{ω} - \bar{\dot{ω}} \|}{\dot{ω}}$ $\frac{\|\dot{\omega}-\overline{\dot{\omega}}\|}{\dot{\omega}}$	Ref.
PSR B1913+16	1.44	1.39 ± 0.00(1)	1 × 10⁻⁵	4.53	0.617	0.32	4.227	7 × 10⁻⁵	1
PSR J0737-3039A/B	1.34	1.25 ± 0.00(001)	6 × 10⁻⁵	1.37	0.088	0.10	16.899	7 × 10⁻⁴	2
PSR J0453+1559	1.56	1.17 ± 0.00(4)	6 × 10⁻⁶	16.69	0.113	4.07	0.038	6 × 10⁻⁴	3
PSR J0509+3801	1.40	1.41 ± 0.00(1)	1 × 10⁻⁵	4.94	0.586	0.38	3.035	5 × 10⁻⁴	4, 5
PSR J0514-4002A	1.39	1.08 ± 0.00(1)	7 × 10⁻⁷	75.89	0.888	18.79	0.013	4 × 10⁻³	6
PSR J1518+4904	$1 . 47_{- 0.03}^{+ 0.03}$ $1.47_{-0.03}^{+0.03}$	$1 . 25_{- 0.03}^{+ 0.04}$ $1.25_{-0.03}^{+0.04}$	3 × 10⁻⁶	30.88	0.249	8.63	0.011	8 × 10⁻⁴	7

Notes. To appreciate the accuracy and power of our formula, we consider the relative difference between the theoretical and observed values. (1) Weisberg & Huang (2016); (2) Kramer et al. (2021); (3) Martinez et al. (2015); (4) McEwen et al. (2024); (5) Ding et al. (2024); (6) Freire et al. (2007); (7) Janssen et al. (2008).

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