Pulsar mass measurements and tests of general relativity

Fig. 1: Precise neutron star mass measurements are useful for a variety of purposes. One of them is to constrain the macroscopic behaviour (in particular the relation between density and pressure, known as the equation of state, or EOS) of the cold, super-dense matter at the center of a neutron star. For each EOS (named in the figure) the relation between mass and radius for all neutron stars is indicated by its related curve. If a particular EOS predicts a maximum mass smaller than the largest measured NS mass (horizontal bars are for the three most massive pulsars tabulated below) then it is excluded.
Figure created by Norbert Wex. EOSs tabulated in Lattimer & Prakash (2001) and provided by the authors.

The table below lists all 40 precise (i.e, with a relative uncertainty smaller than 15%) NS mass measurements (highlighted in red) in a total of 31 stellar systems (one per row).

These are derived from radio timing and, in some cases, the combination of timing with optical measurements of their white dwarf companions. The masses and eccentricities are indicated within their 68.3% confidence limits, unless stated otherwise. I only list systems for which there is either no mass transfer nor significant mass loss, i.e., ``clean'' systems, where tests of general relativity are possible. This page is therefore not meant to be comprehensive, but merely my personal reference on precise NS masses and tests of gravitational theories.

I indicate in boldface the two constraints (types of constraints listed at the bottom) that yield the most precise mass values, or that were used to derive them in the literature. In cases where there are more than two constraints, one might be in the presence of a test of general relativity.

I list the total mass of the binary only if it is determined to better precision than the individual masses of the components. This normally only happens for eccentric systems, where the rate of advance of periastron can be measured precisely.

There are several systems with one single constraint on the masses. Many of these are millisecond pulsars in eccentric orbits in globular clusters, for which one can measure the total mass of the binary from the rate of advance of periastron. These are listed here.

If you find this table useful, then please cite the relevant original references. If not practical, then please cite this recent review on the topic. For a summary on millisecond pulsar masses, please cite this recent work on the topic. Thank you!

Latest changes:

2018 May 15: Added pulsar - neutron star system J0509+3801.
2018 March 6: Replaced the ``double neutron star system'' (DNS) term by the more precise ``Pulsar - Neutron star system'' (PSR-NS), in order to avoid confusion with the LIGO/Virgo systems.
2018 March 1: Updated parameters for PSR J1713+0737, in particular the new limits on violation of gravitational symmetries.
2018 Jan. 08: Updated masses from the NANOGrav Eleven-year Data Set (PSRs J1614−2230, J1713+0737, B1855+09, J1909−3744, J1918−0642 and J2043+1711).
2017 Nov. 22: Added pulsar - neutron star system J1757−1854.
2017 June 27: Updated mass of PSR J2222−0137.
2017 June 20: Added PSR J2053+4650.
2016 Nov. 15: Added PSR J1946+3417.

Acronyms used below:
NS - neutron star. A pulsar (PSR) is a NS for which we can detect periodic radio pulsations,
MSP - millisecond pulsar (P < 20 ms),
WD - white dwarf; He WD - Helium WD
MS - main sequence star,
GC - globular cluster,
GR - general relativity.

e MT
Mass constraints Geodetic precession Notes
J0337+1715 365.953 001.62940
  1.4378(13) 0.19751(15)
i, q a First pulsar in a triple star system. Two companions are He WDs, hence the two companion masses. The two sets of Keplerian parameters are for the orbit of the inner system (containing MSP and WD) and the orbit of the outer WD with the inner system. q is measured independently for inner system. This is expected to yield an unprecedented test of the strong equivalence principle.
J0348+0432 025.561 000.10242 000.14098 0.0000020(10)   2.01(4) 0.172(3) q, mWD, GWa a PSR - He WD system. Largest precise NS mass. First massive NS in a relativistic orbit. GWa yields a radiative test of GR in a new extreme gravity regime.
J0437−4715 173.688 005.74104 003.36670 0.000019180(3)   1.44(7) 0.224(7) r, s a MSP - He WD system. Very high timing precision.
J0453+1559 021.843 004.07247 014.46679 0.11251847(8) 2.734(4) 1.559(5) 1.174(4) p, h3 u PSR - NS. Largest mass asymmetry for a DNS. Companion star is lightest NS known. Pulsar is most massive NS known in a DNS.
J0509+3801 013.065 000.37958 02.0506(3) 0.586400(6) 2.805(3) 1.36(8) 1.46(8) p, γ u PSR - NS. Highly eccentric, similar to the Hulse-Taylor pulsar.
J0621+1002 034.657 008.31868 012.03207 0.00245744(4) 2.32(8) 1.53+0.10−0.20 0.76+0.28−0.07 p, s a PSR - Massive WD system.
000.10225 001.41503
0.0877775(9) 2.58708(16) 1.3381(7)
R, p, γA, s, rA, GW, GB no
yes, yes
PSR - PSR. The only double pulsar known. Seven mass constraints, which determine the masses and yield 5 independent GR tests. No geodetic precession for older pulsar (A), indicating alignment of its spin axis with orbital angular momentum.
J0751+1807 287.458 000.26314 000.39661 0.0000005(11)   1.64(15) 0.16(1) s, GWa a MSP - He WD system, only case where mass is derived from s and assumption that GWa is as given by GR.
J1012+5307 190.267 000.60467 000.58182 0.0000012(3)   1.64(22) 0.16(2) q, mWD, GWa a MSP - He WD system. Useful constraints on dipolar GW emission.
J1141−6545 002.539 000.19765 001.85892 0.171884(2) 2.2892(3) 1.27(1) 1.01(1) p, γ, GWa yes Slow, young pulsar, formed after the companion, which is a massive WD.
B1534+12 026.382 000.42074 003.72946 0.27367740(4) 2.678463(4) 1.3330(2) 1.3455(2) p, γ, r, s, GW, G yes PSR - NS. Orbital decay (GW) has been measured precisely but does not provide a test of GR because of lack of precise knowledge of distance.
J1614−2230 317.379 008.68662 011.29120 0.000001333(8) 1.908(16) 0.493(3) r, s a MSP- Massive WD system, massive NS.
J1713+0747 218.811 067.82513, 032.34242 0.0000749402(6) 1.33+0.09−0.08,
h3, ς
r, s
a MSP - He WD system. This system surrently provides the best pulsar limit on the variation of the gravitational constant, on violations of the universality of free fall and on the α3 parameter.
J1738+0333 170.937 000.35479 000.34343 0.00000034(11)   1.47+0.07−0.06 0.181+0.007−0.005 q, mWD, GWa a MSP - He WD system. GWa yields the most stringent limits on scalar-tensor theories of gravity. This system also yields the best limits on Local Lorenz Invariance violation parameter α1
J1756−2251 035.135 000.31963 002.75646 0.1805694(2) 2.56999(6) 1.341(7) 1.230(7) p, γ, s, GW no PSR - NS. Low-mass NS companion. No geodetic precession for pulsar, indicating alignment of its spin axis with orbital angular momentum.
J1757−1854 046.518 000.18354 002.23781 0.6058142(10) 2.73295(9) 1.3384(9) 1.3946(9) p, γ, h3, ς GW u PSR - NS. Most accelerated binary, largest relative velocity, five measured PK parameters. Second-formed NS significantly more massive than first-formed NS.
J1802−2124 079.066 000.69889 003.71885 0.00000248(5) 1.24(11) 0.78(4) r, s a MSP - massive WD system.
J1807−2500B 238.881 009.95667 028.92039 0.747033198(40) 2.57190(73) 1.3655(21) 1.2064(20) p, ς, h3 u System formed through exchange encounter in GC NGC 6544. Most precise MSP mass. Companion is either a massive WD or NS, so this could be the first MSP - NS system.
B1855+09 186.494 012.32717 009.23078 0.00002170(3) 1.37+0.13−0.10,
r, s,
h3, ς
a MSP - He WD system. First measurement of Shapiro delay in a binary pulsar.
J1903+0327 465.135 095.17412 105.59346 0.436678409(3) 2.697(29)* 1.667(21)* 1.029(8)* p, h3, ς, q u First MSP outside GCs with eccentric orbit. MS companion. System likely originated as a triple. *Masses indicated to 99.7% C.L. as uncertainties are not Gaussian.
J1906+0746 006.941 000.16599 001.41995 0.0852996(6) 2.6134(3) 1.291(11) 1.322(11) p, γ, GW yes PSR - NS (likely), where we detect the younger pulsar (only 110 kyr old), not the older recycled NS as in most other DNSs.
J1909−3744 339.315 001.53345 001.89799 0.000000092(13) 1.48(3)
r, s
h3, ς
a MSP - He WD system. First MSP with a precise mass. Very high timing precision and high inclination. Useful constraints on dipolar GW emission.
J1910−5958A 306.167 000.83711 001.20605 <0.00001   1.3(2) 0.180(18) q, mWD, r, s a MSP - He WD system in GC NGC 6572.
B1913+16 016.940 000.32299 002.34178 0.6171340(4) 2.828378(7) 1.438(1) 1.390(1) p, γ, GW yes PSR - NS. The Hulse-Taylor binary. First binary pulsar discovered. First NS mass measurements. Orbital decay was the first GR test in a binary pulsar, and first radiative test anywhere. This showed that gravitational waves exist.
J1918−0642 130.790 010.91318 008.35047 0.000020340(18) 1.29+0.10−0.09 0.231(10) r, s a MSP - He WD system.
J1946+3417 315.444 027.01995 013.86907 0.134495389(17) 2.094(22) 1.828(22) 0.2556(19) p, h3, ς u Eccentric MSP - He WD system. MSP is massive.
J2043+1711 420.189 001.48229 001.62396 0.00000489(13) 1.38+0.12−0.13 0.173(10) r,s a MSP - He WD system.
J2053+4650 079.652 002.45250 008.80430 0.0000089(1) 1.40+0.21−0.18 0.86+0.07−0.07 h3, ς a MSP - massive WD system.
B2127+11C 032.755 000.33528 002.51845 0.681395(2) 2.71279(13) 1.358(10) 1.354(10) p, γ, GW yes PSR - NS. Formed through exchange encounter in GC NGC 7078 (M15). Acceleration of system in the cluster precludes further improvements in precision of radiative test of GR.
J2222−0137 030.471 002.44576 010.84802 0.000380940(3)   1.76(6) 1.293(25) r, s, p, GWa u Pulsar - massive WD system. Largest NS birth mass; companion is most massive WD in this list. Useful constraints on dipolar GW emission.
B2303+46 000.937 012.33954 032.6787 0.658369(9) 2.64(5) 1.24-1.44 1.2-1.4 p, mWD u Slow, young pulsar, formed after the companion, which is a massive WD.

The mass constraints and the requirements for their measurement are:
i: Mass estimate based on measurement of deviations from Keplerian orbits caused by many-body interactions via pulsar timing. This requires a number of massive components in the system larger than 2.
R: Mass ratio derived from pulsar timing. This requires a double pulsar system, only one case known (the J0737−3039 system).
q: Mass ratio derived from pulsar timing and optical spectroscopy. This requires a companion WD bright enough for spectroscopic radial velocity measurements.
mWD: White dwarf mass (optical). This requires a companion WD bright enough for detailed spectroscopic modeling.
p: Rate of advance of periastron (timing). This requires significant orbital eccentricity.
γ: Einstein delay (timing). This requires a combination of significant orbital eccentricity and compactness.
s: Shape parameter of Shapiro delay (timing). All Shapiro delay parameters require the system to have a high orbital inclination and good timing precision.
r: Range parameter of Shapiro delay (timing).
ς: Orthometric ratio of Shapiro delay (timing) - From the Freire & Wex (2010) reparameterization of the Shapiro delay.
h3: Orthometric amplitude of Shapiro delay (timing) - From the Freire & Wex (2010) reparameterization of the Shapiro delay.
GW: Orbital decay (timing), caused by energy loss due to gravitational waves. Detecting this requires a compact orbit.
GWa: Same as GW, but for an asymmetric system, like a pulsar-WD system. This introduces stronger constraints on alternative theories of gravity.
G: Rate of geodetic precession.
A subscript refers to the pulsar for which we measure the parameter (this is only an issue for the J0737−3039 system).
For the column on the detection of geodetic precession, the meaning of the entries is:
a: The spin of this pulsar is expected to be aligned with the orbital angular momentum of the system. This means that geodetic precession should not cause detectable changes in the pulse profile.
u: The spin of this pulsar is not expected to be aligned with the orbital angular momentum of the system, but geodetic precession is too slow to cause detectable changes in the pulse profile.
In the following cases, the pulsar spin is not expected to be necessarily aligned with the orbital angular momentum of the system and geodetic precession should be fast enough to produce detectable changes in the pulse profile:
yes: Such changes are indeed observable, confirming the spin-orbit misalignment.
no: No changes in the pulse profile are detected. This suggests the pulsar spin is aligned with the orbit after all.

Fig. 2: Measuring the masses of the components of a binary pulsar is a necessary first step for testing general relativity and other gravity theories with that system. Here we display graphically how suitable are different binary pulsars for such tests, in particular radiative tests of gravity theories. In the horizontal axis we plot the NS mass, for PSR - NS systems we plot the mass of the observable NS. In the vertical axis we plot the relative orbital velocity, which measures the strength of relativistic effects. The color scale indicates the figure of merit for dipolar gravitational wave emission (DGW), a phenomenon predicted by the best studied alternative theories of gravity. The circles indicate all ``clean'' (no mass loss or transfer) binary pulsars with well-determined masses, listed in the table below. Filled circles are pulsars with weakly self-gravitating companions (e.g. white dwarfs). PSR - NS systems are presented as hollow circles because they are less suitable for detecting DGW, as both binary components have similar binding energies. PSR J0348+0432 is in a unique position to constrain the occurrence of this phenomenon. Figure created by Norbert Wex.

Paulo C. C. Freire