2.1 Compact binaries with neutron stars

Double NSs have been discovered because one of the components of the binary is observed as a radio pulsar. The precise pulsar timing allows one to search for a periodic variation due to the binary motion. This technique is reviewed in detail by Lorimer [238]; applications of pulsar timing for general relativity tests are reviewed by Stairs [386].

Basically, pulsar timing provides the following Keplerian orbital parameters of the binary system: the binary orbital period Pb as measured from periodic Doppler variations of the pulsar spin, the projected semimajor axis x = asin i as measured from the semi-amplitude of the pulsar radial velocity curve (i is the binary inclination angle defined such that i = 0 for face-on systems), the orbital eccentricity e as measured from the shape of the pulsar radial velocity curve, and the longitude of periastron ω at a particular epoch T 0. The first two parameters allow one to construct the mass function of the secondary companion,

4π2x3 (M sini)3 f(Mp, Mc ) = --2---= ----c------2. (1 ) P bT⊙ (Mc + Mp )
In this expression, x is measured in light-seconds, T ≡ GM ∕c3 = 4.925490947 μs ⊙ ⊙, and M p and Mc denote masses of the pulsar and its companion, respectively. This function gives the strict lower limit on the mass of the unseen companion. However, assuming the pulsar mass to have the typical value of a NS mass (for example, confined between the lowest measured NS mass 1.25 M ⊙ for PSR J0737–3039B [241Jump To The Next Citation Point] and the maximum measured NS mass of 2.1 M ⊙ in the NS–WD binary PSR J0751+1807 [291]), one can estimate the mass of the secondary star even without knowing the binary inclination angle i.

Long-term pulsar timing allows measurements of several relativistic phenomena: the advance of periastron ˙ω, the redshift parameter γ, the Shapiro delay within the binary system qualified through post-Keplerian parameters r, s, and the binary orbit decay ˙ Pb. From the post-Keplerian parameters the individual masses Mp, Mc and the binary inclination angle i can be calculated [46].

Of the post-Keplerian parameters of binary pulsars, the periastron advance rate is usually measured most readily. Assuming it to be entirely due to general relativity, the total mass of the system can be evaluated:

( )5∕3 2∕3 2∕3 ω˙= 3 2π- T⊙--(Mc-+-Mp--)--. (2 ) Pb (1 − e2)
High values of the derived total mass of the system (≳ 2.5M ⊙) suggests the presence of another NS or even BH2.

If the individual masses, binary period, and eccentricity of a compact binary system are known, it is easy to calculate the time it takes for the binary companions to coalesce due to GW emission using the quadrupole formula for GW emission [309Jump To The Next Citation Point] (see Section 3.1.4 for more detail):

( P )8 ∕3 ( μ ) −1 ( M + M )− 2∕3 τGW ≈ 4.8 × 1010 yr -b- ---- --c----p- (1 − e2)7∕2. (3 ) d M ⊙ M ⊙
Hereμ = MpMc ∕(Mp + Mc ) the reduced mass of the binary. Some observed and derived parameters of known compact binaries with NSs are collected in Tables 2 and 3.


Table 2: Observed parameters of double neutron star binaries.
















PSR P P b a sin i 1 e ˙ω ˙P b Ref.
[ms] [d] [lt-s] [deg yr–1] [× 10–12]








J0737–3039A 22.70 0.102 1.42 0.088 16.88 –1.24 [49Jump To The Next Citation Point]
J0737–3039B 2773 [241]
J1518+4904 40.93 8.634 20.04 0.249 0.011 ? [290]
B1534+12 37.90 0.421 3.73 0.274 1.756 –0.138 [455388]
J1756–2251 28.46 0.320 2.75 0.181 2.585 ? [98]
J1811–1736 104.18 18.779 34.78 0.828 0.009 < 30 [242]
J1906+0746 144.07 0.116 1.42 0.085 7.57 ? [239]
B1913+16 59.03 0.323 2.34 0.617 4.227 –2.428 [155]
B2127+11C 30.53 0.335 2.52 0.681 4.457 –3.937 [7333]


















Table 3: Derived parameters of double neutron star binaries










PSR f (m ) M + M c p τ = P ∕(2P˙) c τ GW
[M ⊙] [M ⊙] [Myr] [Myr]





J0737–3039A 0.29 2.58 210 87
J0737–3039B 50
J1518+4904 0.12 2.62 9.6 × 106
B1534+12 0.31 2.75 248 2690
J1756–2251 0.22 2.57 444 1690
J1811–1736 0.13 2.6 1.7 × 106
J1906+0746 0.11 2.61 0.112 300
B1913+16 0.13 2.83 108 310
B2127+11C 0.15 2.71 969 220












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