Binary pulsar evolution

This is a Newtonian Kepler orbit with orbit-averaged, leading-quadrupole gravitational-radiation losses from the Peters model. Semimajor axis a and eccentricity e evolve together. Spin, tides, mass transfer, relativistic periastron precession, observed pulse timing and calibrated pulsar beams are not modeled. The named examples are approximate initial conditions, not precision timing fits.

Inputs and units. Each mass is 0.5–3 nominal solar masses, initial period 0.01–100 days, and initial eccentricity 0–0.9. These are the selected numerical and weak-field scope; parabolic orbits and strong-field merger are excluded. The exact nominal conversion GM☉ = 1.3271244×10²⁰ m³/s² follows IAU 2015 B3, with c = 299792458 m/s. A day is 86400 s and a year is 365.25 days. Myr means one million years; Gm means 10⁹ m, or one million km. No separately rounded G and solar mass are multiplied.

Secular evolution. Initially a₀³ = GM☉(m₁+m₂)P₀²/(4π²). Define β = (64/5)(GM☉³/c⁵)m₁m₂(m₁+m₂). Then ȧ = −βa⁻³(1+73e²/24+37e⁴/96)/(1−e²)⁷ᐟ² and ė = −(19/12)βa⁻⁴e(1+121e²/304)/(1−e²)⁵ᐟ². Kepler's law gives P = 2π√(a³/[GM☉(m₁+m₂)]) and Ṗ = 3Pȧ/(2a), in s/s. The chirp mass is (m₁m₂)³ᐟ⁵/(m₁+m₂)¹ᐟ⁵ in nominal solar masses.

For e₀ = 0, the formal lifetime is t_c = a₀⁴/(4β). At elapsed fraction f, a/a₀ = (1−f)¹ᐟ⁴ and P/P₀ = (1−f)³ᐟ⁸; e remains exactly zero. Eccentric systems use the full Peters integral, not an instantaneous enhancement multiplied into the circular lifetime. With k = 121/304 and z = (e/e₀)⁸ᐟ¹⁹, J(z) integrates u⁵(1+ke₀²u¹⁹ᐟ⁴)¹¹⁸¹ᐟ²²⁹⁹/(1−e₀²u¹⁹ᐟ⁴)³ᐟ² from 0 to z. The lifetime ratio is 6(1−e₀²)⁴J(1)/(1+ke₀²)³⁴⁸⁰ᐟ²²⁹⁹. The state follows from J(z)/J(1)=1−f and Peters' a(e) relation. A cached 4096-panel Simpson quadrature with local inversion provides a numerical approximation.

Two clocks. Calculate builds the evolution curve. The age field and slider select 0–95% of the formal lifetime; all current numbers and the curve marker refer to that same age. Remaining time is t_c(1−f). Changing age pauses the orbital preview. The formal lifetime extrapolates the point-mass model; no merger is drawn or predicted reliably by this approximation.

The separate preview holds that age fixed. Its phase is mean anomaly in turns, with 1 identical to 0. Kepler's equation E−e sin E = 2π phase determines the positions. The relative vector is a(cos E−e, √(1−e²) sin E); M1 and M2 lie at −m₂/(m₁+m₂) and m₁/(m₁+m₂) times this vector. The origin is the barycenter. The fixed initial scale makes shrinkage visible, with equal coordinate scales on both axes and illustrative star sizes. Play advances only phase at visual cycles/s; it does not advance evolutionary time or change the reported physical period. Switching away from the preview, hiding the tool or browser tab, and reset pause it. The planar diagram requires no WebGL.

Do not compare these rounded presets directly to a precise measured period derivative without the relevant timing fit and Galactic and kinematic corrections. This tool does not include spin-down or observed-timing corrections.

Sources: Peters (1964), section V, equations 5.6–5.14; Weisberg & Huang (2016), table 2 and section 4.3; IAU 2015 B3 nominal constants; Kramer et al. (2021), double-pulsar timing.