Masses, Radii, and the Equation of State of Neutron Stars, 2016. A Puke(TM) Audiopaper

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Annual Reviews of Astronomy and Astrophysics.
Masses, Radii, and the Equation of State of Neutron Stars, 2016.

Feryal Ozel and Paulo Freire.

Department of Astronomy, University of Arizona, Tucson, Arizona, and:
The Max-Planck-Institute for Radio Astronomy, Bonn, Germany.

Annu. Rev. Astron. Astrophys. 2016. 54:401–40

This article’s doi: 10.1146/annurev-astro-081915-02332

Keywords. Neutron stars, dense matter, pulsars, pulsar timing, X-ray sources.

Abstract.
We summarize our current knowledge of neutron-star masses and radii.
Recent instrumentation and computational advances have resulted in a rapid increase in the discovery rate and precise timing of radio pulsars in binaries in the past few years, leading to a large number of mass measurements. These discoveries show that the neutron-star mass distribution is much wider than previously thought, with three known pulsars now firmly in the 1.9 to 2.0 Solar Mass range. For radii, large, high-quality data sets from X-ray satellites as well as significant progress in theoretical modeling led to considerable progress in the measurements, placing them in the 10 to 11.5 kilometers range and shrinking their uncertainties, owing to a better understanding of the sources of systematic errors. The combination of the massive-neutron-star discoveries, the tighter radius measurements, and improved laboratory constraints of the properties of dense matter has already made a substantial impact on our understanding of the composition and bulk properties of cold nuclear matter at densities higher than that of the atomic nucleus, a major unsolved problem in modern physics.

Section One. INTRODUCTION.
Our understanding of neutron stars has changed drastically since the first Annual Review article by Wheeler in 19 66 on this subject, when not a single neutron star was known and the discussion consisted of an entirely theoretical treatment of collapsed dense stars. The change has been most rapid in the past decade, when the discovery and precisely timed observations of pulsars have proceeded at an ever accelerating rate; the new-generation X-ray and gamma-ray telescopes have provided high-quality, large data sets, and a large body of theoretical work on neutron star emission properties and spacetimes have enabled significant recent developments in our ability to model these extreme objects and interpret the observations in solid frameworks.
We now know precise masses for approximately 35 neutron stars spanning the range from 1.17 to 2.0 Solar Masses and can pin down the radii of more than a dozen to the 10 to 11.5 kilometers range. The combination of the heaviest known neutron star mass with the existing radius measurements already places significant constraints on the cold dense matter equation of state, E o S, up to densities that are eight times the nuclear density.
Certain characteristics of neutron stars have been known for a while. Shortly after the discovery of pulsars, it became apparent that they are the observational manifestation of extremely compact stars made primarily of neutrons. It was also evident that, though the neutron degeneracy pressure played a role in supporting these compact objects against gravitational collapse, it was not sufficient to hold up a star beyond 0.7 Solar Masses and that repulsive nuclear forces were necessary in shaping their structure. Earlier calculations of the interiors treated nucleons with approximate potentials, predicting masses in the 0.5 to 3 solar mass range and radii between 7 and 20 kilometers.

Stars of such extreme compactness have central densities that are 5 to 10 times the nuclear saturation density of rho sat, equal to 2.8 times ten to the 14 grams per cubic centimeter.
Although the nuclear models improved over the years, the major advances have come from astrophysical observations and models because terrestrial experiments do not approach densities similar to that of neutron star cores, see Section 4 point 2. Theoretical work is guided by the well-developed effective field theory approaches and the rapidly developing quantum chromodynamics calculations, but there remain large uncertainties in our understanding of the actual compositions of the cores. Open questions include the composition of matter at high densities, such as the density at which matter needs to be described by quarks and no longer by nucleons, as well as the presence of strangeness or boson condensates in such matter; the isospin symmetry energy that defines the energy difference between normal and neutron-rich matter and, ultimately, the ability to calculate matter in Beta equilibrium from first principles; and the role and reliability of calculations of N-body interactions. Despite the large number of possibilities, each theoretical model, or its resulting equation of state, can be mapped into a mass-radius (M-R) relation by solving the general relativistic structure equations. This, in turn, allows measurements of the neutron star masses and radii to place strong constraints on the properties and interactions of cold, ultradense matter, as we discuss in this review.
These same observations also provide excellent tests of general relativity (GR) in the strong-field regime, using pulsars as well as neutron star surfaces as test beds.
The recent series of investigations of neutron star masses and radii are motivated not only by the aim of probing physics in new regimes but also because of the forefront role neutron stars play in many astrophysical phenomena. neutron stars are associated with numerous explosive, transient, and non-electromagnetic events, and neutron star properties play a role in shaping each one of them. The primary sources of gravitational waves that are expected to be detected with the gravitational wave detector Advanced LIGO, Laser Interferometer Gravitation-Wave Observatory, are the neutron star- neutron star and neutron star black hole mergers. These systems are also thought to be the central engines of short gamma-ray bursts. The dynamics of the mergers, the gravitational waveforms in late stages of coalescence, the lifetime of merger disks, and the resulting black hole formation timescales, as well as any accompanying bursts of neutrino, gamma-ray, and or optical emission, depend sensitively on the neutron star mass and radius and the E o S.
From a stellar life cycle and source population point of view, studies of neutron stars prove to be equally important. Different equations of state allow different maximum masses for neutron stars and, thus, determine the dividing line between neutron stars and black holes. This has a direct impact on the outcomes of supernova explosions and the nature of compact remnants, as well as on the numbers of neutron stars and black holes in the Galaxy. This, in turn, affects the number of observable compact object binaries, their properties, and possible merger gravitational-wave event rates. In fact, the supernova mechanism itself is affected by the equation of state of ultradense matter. The hot dense matter equation of state, an extension of the cold one probed by neutron stars, determines explosion conditions and is crucial for understanding core-collapse supernova explosions and the associated phenomena, including mass loss, r-process nucleosynthesis, and gravitational wave and neutrino emission.
The tremendous advances in the measurements of neutron star masses, neutron star radii, and the dense matter equation of state have come from a diverse array of techniques, applied to many different populations of neutron stars, and enabled by observations in all wavelengths from radio to gamma rays. For example, whereas the precise mass measurements have largely resulted from radio observations of pulsars, the radius measurements have almost exclusively been a result of X-ray observations of neutron stars in low-mass X-ray binaries, LMXB’s. In this review, we focus on the latest mass and radius measurements of neutron stars and their implications for the dense matter equation of state, as well as a number of questions that remain open

Section Two. NEUTRON-STAR MASS MEASUREMENTS.

The vast majority of the precise mass measurements of neutron stars have been performed using radio observations of rotation-powered pulsars. Currently, more than 2,500 pulsars are known in the Galaxy, most of which are detectable as radio pulsars, but also some are observed in X-rays and an increasingly large number detected in gamma rays.
About 90 percent of radio pulsars are isolated. Their masses cannot be measured, because all of the current methods rely on precise tracking of the orbital motions through the arrival times of the observed pulsations. The remaining 250 pulsars are located in binary systems, three of these are in multiple component systems. We now focus exclusively on these.
Most of the pulsars in binaries are “recycled”: At some point during the evolution of their companions there was mass transfer from the companion to the neutron star. The mass transfer can, in principle, increase the pulsar mass significantly, but its clearest effect is the spin-up of the pulsar, to spin frequencies as high as 716 Hertz, and, by mechanisms that are poorly understood, a sharp reduction in the pulsar’s magnetic field, to values smaller than 10 to the power of 11 Gauss, 10 Mega Teslas. This process produces a wide variety of binary pulsars.
Those of greatest interest to us are the systems where both components are compact: the double neutron-star, DNS, systems and the millisecond pulsar white dwarf, MSP-WD, systems.
2 point 1. Timing Binary Pulsars.

The extraordinary long-term rotational stability of recycled pulsars and their fast rotation make them uniquely useful for timing. If a recycled pulsar is in a binary system, as often happens, then we can use this precise timing to measure its orbital motion with astounding precision.
In Newtonian gravity, the part of the orbital motion we can observe (generally along the line of sight only) can be described by five Keplerian parameters: the binary period Pb, the orbital eccentricity e, the projection of the pulsar’s semi major axis “A” sub psr onto the observer’s line of sight x sub psr equals “A” sub psr sine I over c, where i is the angle between the orbital angular momentum vector and the line of sight, and c is the speed of light, the time of periastron T sub zero, and the longitude of periastron omega.
For each new pulsar, these parameters are determined by fitting a radial velocity model to the observed spin periods, which are Doppler shifted by the orbital motion of the pulsar. This is similar to what can be done in spectroscopic systems. The unique feature of pulsars is that, by determining the correct rotational phase count, one can directly range the pulsar relative to the center of mass of the system. Measuring a pulse time of arrival with a precision of 1 microsecond translates into a ranging accuracy of 300 meters per measurement, which is much smaller than the astronomical sizes of the orbits. This provides Keplerian parameters many orders of magnitude more precise than those derived from Doppler shift measurements with the same data and is the reason why pulsars are so useful for such a wide variety of purposes, including uniquely precise mass measurements.
The binary mass function is then determined from the Keplerian orbital parameters:
The binary mass function f is equal to the cube of the companion M sub c sine inclination, divided by the Square of the total mass.
This equals four pi squared over the mass of the sun in time units times the cube of the projection onto the observers’ line of sight of the semi-major axis, divided by the square of the binary period.

Where the mass of the sun in time units, T Sol is equal to G time the solar mass divided by the cube of the speed of light, and is equal to 4.925490947 microseconds.
The product G times Mass Sol, where G is Newton’s gravitational constant and M Sol is the solar mass, is very precisely known despite the fact that neither G nor M sol is individually known to better than one part in a thousand, and:
M sub psr, M sub c, and MT, equal to M sub psr plus M sub c are the pulsar, companion, and total binary masses, respectively all expressed in solar masses.

This single equation has three unknowns, inclination, M sub psr, and M sub c. Therefore, in the absence of further information, the individual masses of the pulsar, M sub psr and the companion, M sub c cannot be determined despite the high precision in the determination of the Keplerian parameters.
If the projected semimajor axis of the companion’s orbit x sub c can be measured, either via timing, if the companion is a pulsar, or via phase-resolved optical spectroscopy, if the companion is a white dwarf or main-sequence star, one extra equation is obtained, equation two:
Q is defined as the ratio of the pulsar to the companion mass, M sub psr divided by M sub c, equal to the projections x sub c, over x sub psr.
Which is valid, to first post-Newtonian (PN) order, in all conservative theories of gravity. The number of binary systems in which we can measure q is small, but even for those systems, measuring f and q is not enough to determine the masses. This can be addressed in a few systems because the spectrum of a white dwarf companion allows, in some cases, a determination of its mass, M white dwarf, see Section 2 point 3 point 3.
Because of the small sizes of neutron stars and their neutron star or white dwarf companions compared with their orbits, both components can be treated as point masses. In this case, the “classical” tidal and rotational effects that complicate the motion of other astrophysical objects and hide the generally smaller, relativistic effects are simply absent.
This means that such systems are clean gravitational laboratories.
In such a system, a pulsar with good timing precision allows the detection and precise measurement of these small relativistic effects. These effects can be parameterized by post-Keplerian (PK) parameters, which, assuming that General Relativity is the correct theory of gravity and to leading PN order, are related to three of the Keplerian parameters (Pb, x, and e) and the system masses. For the five PK parameters that have been measured in the context of pulsar timing, these expressions are as follows (as in Stairs 2003, and references therein):
One. The rate of advance of periastron, denoted as omega dot, typically measurable in eccentric orbits. This is analogous to the anomalous perihelion advance in Mercury’s orbit, equation three:
Omega dot equals the minus five thirds root of P sub b over two pi times the two thirds power of the solar mass in time units times the total mass, divided by one minus eccentricity squared.

Two. The “Einstein delay,” gamma. In General Relativity, this is due in equal parts to the variations of the gravitational redshift and of special relativistic time dilation in an eccentric orbit, see equation four in the text:
Three. The orbital period decay P dot sub b due to the loss of orbital energy of the system caused by the emission of gravitational waves, this typically requires timing observations that span many years, equation five in the text.

Four. The range r and the shape s of the Shapiro delay, which is a delay in the reception of the radio pulses at the Earth that is due to the propagation of the radio signal in the curved spacetime near the companion star r equals The solar mass in time units times the mass of the companion, Equation six.
And s, equals sine of the inclination, equation seven in the text.
This effect is more easily detected in edge-on orbits.

Figure 1.
Mass-mass diagram for the J07 37 dash 30 39 system. The yellow areas are excluded by the mass functions of both pulsars, PSR’s J07 37 dash 30 39A and B. The lines represent the regions allowed by Equations 2 to 7 for the measured post-Keplerian (PK) parameters. All lines meet at a single point in the diagram, meaning, general relativity passes the tests posed by these four distinct constraints. This plot is important because it indicates that we get consistent mass measurements using any pair of PK parameters.

We must here emphasize that, like the Keplerian parameters, these are parameters that appear in the equations that describe the motion and timing of binary pulsars, such as those presented by Damour and Deruelle in 1985 and 1986. They are measured by timing a particular system and fitting the pulse arrival times. It is only when we adopt a specific theory of gravity that we can relate them to physical parameters of the components of the binary, in the case of General Relativity, and to leading PN order, we can relate them to their masses only, Equations 3 to 7. These constraints are displayed graphically in Figure 1. Note that there is nothing unique about the particular parameterization of Damour and Deruelle from 1985. As an example, instead of r and s, we can describe Shapiro delay with two less-correlated PK parameters, zeta and h3. Note, the Shapiro parameter zeta is the square root of one minus the absolute cosine of the inclination over one plus the absolute of the cosine of the inclination, and h3 is the third harmonic, found by a Fourier expansion of the binary mass function f.
2 point 2. Double Neutron-Star Mass Measurements.

The first system for which reliable mass measurements were made was PSR B19 13 plus 16, the first binary pulsar ever discovered. The compact orbit of this DNS, Pb equal to 0.32299 days, and its high orbital eccentricity e equal to 0.6171334(5) allowed a precise measurement of the PK parameter omega dot. Subsequently, Taylor et al in 1979 were able to measure two more PK parameters, gamma and Pb dot. The individual masses were determined from Equations 3 and 4, and the observed orbital decay was then used as a successful test of General Relativity. The General Relativity prediction for P b dot caused by gravitational wave emission (Equation 5) for the derived masses was consistent with the observed rate.

Table 1. Masses of double neutron-star systems and non-recycled pulsars.
The columns are: System, Total mass in solar masses, Pulsar Mass in solar masses, Companion mass in Solar masses and the mass constraint and references.

A number of other DNS’s have since been discovered. The eleven systems for which mass constraints are known are listed in the top two groups in Table 1, together with the latest references.
These systems are relatively rare, making up about 5 percent of the currently known population of binary pulsars. We list the total mass MT of each binary because, generally, that is much better known than the individual masses of the components. This happens because the orbits of all DNS’s are eccentric, so that omega dot (and MT) can be determined with high precision. In binaries where additional PK parameters are measured, we also list the individual neutron star masses and specify the parameters that were measured to derive the masses, for all these systems, Equation 1 can be used to determine the orbital inclination i. For one of these systems (J19 06 plus 07 46) it is possible for the companion to be a massive white dwarf instead. PSR B21 27 plus 11C is located in the globular cluster (GC) M15 and is very likely the result of an exchange interaction.
PSR B19 13 plus 16 is no longer unique in providing a stringent test of General Relativity. The orbital decay due to the emission of gravitational waves has been measured for five other DNS systems and, as we discuss below, a few millisecond pulsar white dwarf systems as well.
We also include in Table 1 two radio pulsars that have precise mass measurements even though they are not members of DNS systems. Their slow spins and higher magnetic fields indicate that they have not been recycled, making them resemble DNS’s rather than the millisecond pulsar white dwarf companions that we discuss in Section 2 point 3.

2 point 2 point 1. The double pulsar.
Among the DNS population, one system in particular stands out: J07 37 dash 30 39. This system is a unique laboratory for gravitational physics because it combines an unusual number of desirable features:
(a) With an orbital period of 2 hours 27 minutes, it is by far the most compact DNS known. Combined with the moderate orbital eccentricity of the system, this implies that the PK parameters omega dot, gamma and P B dot, are relatively large and easy to measure, promising one precise test of General Relativity.
(b) The system has a high orbital inclination, i around 89 degrees. This allows for a precise measurement of the two Shapiro delay parameters, r and s, thus providing two more independent tests of General Relativity.
(c) The second neutron star in the system is also an active radio pulsar, which is detectable at least some of the time. This makes it the only currently known double pulsar system and allows a precise measurement of the mass ratio, q, which “frees” one of the PK parameters from the role of mass determination and, therefore, allows one more test of General Relativity.
The four tests in this system provided by Equations 3 to 7 are remarkably stringent, but despite that, General Relativity has passed all of them. This can be seen in Figure 1, where all the regions compatible with the known mass constraints are consistent with the same pair of masses.
Such tests of gravity theories are not the topic of this review, we refer interested readers to Wex 20 14 for a comprehensive discussion. Nevertheless, systems like J07 37 dash 30 39 and Figure 1 are important for neutron star mass measurements because they demonstrate experimentally that any pair of mass constraints chosen from q and the PK parameters will generally yield consistent neutron star masses.
2 point 3. Millisecond Pulsars.

The term millisecond pulsars, MSP’s, refers to pulsars with spin periods in the range of P between 1.39 and 20 milli seconds and P dot less than 10 to the minus 19. These systems have been heavily recycled (and circularized) by a longlived accretion phase in an LMXB. The first millisecond pulsar by this definition, B19 37 plus 21, was discovered using the Arecibo telescope in 1982. At a 642-Hertz spin frequency, this was the fastest known millisecond pulsar for the subsequent 24 years. A great number of millisecond pulsars have been discovered since and, although about 20 percent are isolated, most of the remaining objects have white dwarf companions and very small orbital eccentricities.
The fast (and very stable) rotation of millisecond pulsars makes their timing significantly more precise than that for the pulsars in DNS’s. However, their low orbital eccentricities pose a problem for mass determination and tests of General Relativity, because, in these cases, the PK parameters omega dot and especially gamma cannot be measured accurately. Furthermore, most systems have orbital periods larger than 0.5 days, which makes a detection of the very small predicted values of P b dot nearly impossible. These difficulties made the measurement of millisecond pulsar masses a slow and difficult process. Four strategies have been followed over the past two decades, which have recently started producing a number of precise mass measurements.
2 point 3 point 1. Pulsars in globular clusters.

One way of getting around the issue of orbital circularity is to find millisecond pulsars in Globular Clusters. Because of the much higher stellar density, there are substantial gravitational interactions with nearby stars that, with time, can make an initially circular orbit acquire a substantial eccentricity. This allows the measurement of omega dot and, if there are no classical contributions to this effect, the total mass of the system MT for quite a few systems, see Table 2.
The first binary mass precise enough to be useful was that of PSR J00 24 dash 72 04H, a 3.21-millisecond pulsar in a 2.35 day, slightly eccentric e equal to 0.070560(3) orbit around a low-mass white dwarf in the GC 47 Tucanae, which yielded MT equal to 1.61 plus or minus 0.04 Solar masses. No further PK parameters were measurable; nevertheless, one can infer from combining Equations 1 and 3, and the mathematical fact sine i is less than or equal to 1) that M sub psr less than 1.52 Solar Masses.

Table 2. Masses of millisecond pulsars.

Column labels are System, Total mass in solar masses, Pulsar mass in Solar masses, Companion mass in solar masses, the mass constraint and references.

This was one of the earlier indications that the long recycling episodes that spin up millisecond pulsars do not necessarily make them very massive. Since the discovery of that system, most eccentric systems found in GC’s have shared the same characteristic: We can measure omega dot but no other mass constraints. The reason for this is that the most eccentric systems tend to be the ones with wider orbits, which are more easily perturbed by nearby stars. For these relatively wide orbits, it is very difficult to measure gamma and P b dot.
The two exceptions are two of the exchange encounter systems listed in Tables 1 and 2, B21 27 and J18 07. Both are located in clusters with collapsed cores, where exchange encounters are more likely. The orbital period of B21 27 is 8 hours and its orbital eccentricity is 0.681386, which allows measurements of omega dot, gamma, and even P b dot. The last system, J18 07, is particularly important for this discussion: It is a true millisecond pulsar with a spin period of 4.18 milli seconds, but despite the long accretion episode, its current mass is only 1.3655 Solar Masses. More than any other system, this millisecond pulsar demonstrates that full recycling can be achieved without a large mass increment.
Nevertheless, as the number of MT measurements accumulated, it became apparent that some systems, such as PSR’s J17 48 in Terzan 5, PSR J17 48 in NGC 64 40, and PSR B15 16 in M5, are quite massive but have very low-mass functions. This is only likely to happen if the pulsars and not the companions in these systems are massive. Indeed, by two thousand and eight, there was already strong evidence that a fair fraction of millisecond pulsars must have masses close to 2 Solar Masses. This statistical result was confirmed by the later millisecond pulsar mass measurements, described below.
2 point 3 point 2. Shapiro delay measurements.

For the vast majority of pulsar-white dwarf systems, the only measurable PK parameters are those related to the Shapiro delay. However, the fastest-spinning millisecond pulsars tend to have very low-mass helium white dwarf companions, which reduces the amplitude of the Shapiro delay. Therefore, in order to measure the masses in these systems, both a very good timing precision and a high orbital inclination are required.
The first millisecond pulsar-white dwarf system where the Shapiro delay was clearly detected was PSR B18 55. However, until recently, the timing precision in this system was not enough for a precise determination of the pulsar mass. This changed in two thousand and three with the discovery of PSR J19 09, a heavily recycled millisecond pulsar with P equal to 2.9 milli seconds and a light helium white dwarf companion. This system combines high timing accuracy with a high orbital inclination, which allowed, in 2005, the first precise measurement of the mass of a millisecond pulsar.
Since then, the number of millisecond pulsars discovered has been increasing significantly, see Section 5 point 1, and for that reason the number of precise Shapiro delay mass measurements has been accumulating, particularly for pulsars with more massive carbon-oxygen or oxygen-neon-magnesium white dwarf companions, such as PSR J18 02 or more recently PSR J22 22. The larger companion masses make the Shapiro delay easier to measure; this is why a disproportionate number of millisecond pulsars with well-measured masses have such massive companions. This does not apply as a rule to the millisecond pulsars themselves: The latter systems highlight the fact that many neutron stars are born with masses as low as about 1.2 Solar Mass.
Of these millisecond pulsars with massive white dwarf companions, the measurement that had by far the greatest impact was PSR J16 14. The originally reported mass of the pulsar was 1.97 Solar Masses, which was exceptional and unexpected among the millisecond pulsars with massive white dwarf companions. This mass measurement had implications for the neutron star mass distribution as well as for the neutron star equation of state, which are discussed in later sections, see Sections 2.5 and 4, respectively. Note that this measurement has since been slightly improved, which revised the mass to M equal to 1.928 Solar Masses, a value that is still consistent with the earlier value.

2 point 3 point 3. Spectroscopic mass measurements.

For some of the millisecond pulsars, particularly those with short orbital periods, the companion is optically bright. This has allowed reliable mass determinations of both components by studying the Balmer lines produced by hydrogen in the white dwarf atmosphere.
First, phase-resolved spectroscopy can measure how these lines shift with orbital phase, providing a measurement of the projected orbital velocity of the white dwarf. Comparing this with the orbital velocity of the pulsar measured in the timing, v sub psr equal to 2 pi times x sub psr over Pb, we can immediately determine the mass ratio, q defined as Pulsar mass divided by Mass white dwarf equal to v white dwarf over v pulsar.
Second, the width of the Balmer lines is related to the local gravitational acceleration, g equal to GM white dwarf over R squared white dwarf. Using model relations between Mass of white dwarf and Radius white dwarf for white dwarfs, both quantities can be determined from g. After this step Pulsar mass can be obtained trivially from q times the Mass of the white dwarf.
The first such measurement was made for PSR J10 12, later followed by the mass measurement for J19 10, a millisecond pulsar white dwarf system in NGC 67 52. To date, the two most important such measurements have been those of J17 38 and J03 48. For the former Pulsar mass equal to 1.47 plus 0.07 minus 0.06 Solar Masses and Mass white dwarf equal to 0.181 plus 0.007 minus 0.005 Solar Masses, this measurement was important because the relatively short orbital period, 8.5 hours allowed a measurement of P b dot, which agrees with the General Relativity prediction for these masses. Given the asymmetry in the binding energy of this system, this measurement introduced the most stringent constraints ever for several families of gravity theories, superseding the previous test from the pulsar-white dwarf system PSR J11 41.
For PSR J03 48, a pulsar with a white dwarf companion having an orbital period of only 2 hours 27 minutes, Antoniadis et al in 20 13 obtained a Pulsar mass equal to 2.01 Solar Mass and Mass white dwarf equal to 0.172 Solar Masses.
This confirmed, using a different method, that neutron stars could reach masses of the order of 2 Solar Masses. Furthermore, the short orbital period has allowed a measurement of P b dot, which again agrees with the General Relativity prediction and has already placed significant constraints on nonlinear phenomena predicted by some alternative theories of gravity for these massive neutron stars, such as spontaneous scalar-ization.
2 point 3 point 4. Triples, disrupted triples, and other monsters.

The theoretical expectation that the recycling pathway for fast millisecond pulsars result in highly circular orbits was corroborated by the fact that all fully recycled millisecond pulsars outside of GC’s observed until two thousand and eight had orbital eccentricities between ten to the minus seven and ten to the minus three. In two thousand and eight, the situation changed with the discovery of PSR J19 03. This fast-spinning millisecond pulsar, P equal to 2.15 milli seconds, is in a binary system with an about 1-Solar Mass main-sequence-star companion in a 95-day, eccentric orbit, e equal to 0.44. This system is thought to have originated from a hierarchical triple that became chaotically unstable, as suggested by observational data and numerical simulations. This finding suggested triple stellar systems containing millisecond pulsars might still be found in the Galaxy, which was later confirmed with the discovery of PSR J03 37, the first millisecond pulsar in a stellar triple system with two white dwarf companions. Note that the mass indicated in Table 2 for PSR J03 37 is based on these triple interactions.
Soon after the discovery of PSR J19 03, Bailes reported in 20 10 an “anomalous” orbital eccentricity of 0.027 for PSR J16 18, a 12-milliseconds pulsar first reported by Edwards and Bailes two thousand and one that is in a 22-day orbit with a low-mass white dwarf companion. More recently, in 20 13 Deneva et al, Barr et al, and in 20 15 Knispel et al, and Camilo et al reported the discovery of four more unusual binary millisecond pulsar systems with eccentric orbits:
PSR’s J22 34, J19 46, J19 50, and J09 55. These four systems are fully recycled with spin periods between 2 and 4.4 milli seconds, orbital periods from 22 to 32 days, and median companion masses M c approximately 0.25 Solar Masses, meaning, apart from their large orbital eccentricities in the range of 0.07 to 0.14, all parameters are compatible with the canonical recycling formation channel leading to an millisecond pulsar with a white dwarf companion.

The reason why these systems are mentioned at length here is that for nearly all of them, precise mass measurements have been, or will soon be, made. For PSR’s J19 03, J19 46, and J22 34, precise mass measurements have been obtained from the combination of omega dot and the Shapiro delay parameter h3. Note that for PSR J19 03, the uncertainty quoted in Table 2 corresponds to 99.7 percent confidence level, C L. A similar measurement will certainly be possible for PSR J19 50. For the triple system, even more precise masses for the millisecond pulsar and two white dwarf companions could be obtained from the 3-body effects in the system.
The masses in these systems are quite varied. Although the millisecond pulsars in the triple system J03 37 and in the eccentric binary system J22 34 have masses of 1.4378 Solar Masses and 1.393 Solar Masses, respectively, PSR J19 46 is in the massive neutron star group, with Pulsar mass equal to 1.832 Solar Masses. PSR J19 03 sits in the middle, with Pulsar mass equal to 1.667 Solar Masses, 99.7 percent C I. The error bars for this measurement are not Gaussian, and therefore the 68 percent error bar is not one third of this uncertainty.
2 point 4. Neutron Stars in X-ray Binaries.

Mass measurements have also been carried out for neutron stars with high and low stellar mass companions using the observations of these binaries in X-ray and optical wavelengths. We discuss the methods and results for these two classes below.
2 point 4 point 1. Neutron stars with high-mass companions.

For mass measurements of neutron stars with high mass companions, eclipsing X-ray pulsars, in which the companions block the X-rays from the pulsar for part of the orbit, form the primary targets. In these systems, X-ray observations of the pulsar yield the orbital period of the binary Pb, the eccentricity of the orbit e, the longitude of periastron omega zero, the semimajor axis of the neutron star’s orbit “A” X sine i, and the semi-duration of the eclipse theta sub e. In addition, optical observations of the companion star give its velocity amplitude K sub opt, its projected rotational velocity v rot sine i, and the amplitude of ellipsoidal variations “A”. From these observables, it is possible to solve for the fundamental parameters of the binary, which include the mass of the neutron star mass, Mass sub NS, the mass, radius, and rotational angular velocity of the companion M sub opt, R sub opt, and Capital Omega opt, and the inclination angle of the binary i.
These measurements are typically less precise than those obtained from radio pulsar timing, and several sources of systematic uncertainty in this technique are discussed by Ozel et al in 20 12 and Falanga et al 20 15. The ten neutron stars whose masses have been estimated in this way are listed in Table 3.
2 point 4 point 2. Neutron stars in low-mass X-ray binaries.

Spectral studies of thermonuclear X-ray bursts from neutron stars provide simultaneous measurements of their masses and radii. These mass constraints are less precise than those obtained from pulsar timing but still provide an estimate of neutron star masses in a different population. The details of the methods are discussed in Section 3 point 1, and the mass measurements for the six sources in this category are listed in Table 3.

Table 3. Masses of neutron stars in high-mass and low-mass X-ray binaries.

Column labels are System, Neutron Star mass in solar masses, error in solar masses and references.

This table also includes the LMXB Cygnus X 2, for which the mass of the neutron star was inferred using optical observations of its companion.

2 point 5. Neutron-Star Mass Distribution.

We show in Figure 2 the combined neutron star mass measurements in all of the above categories. It is already clear from this figure that current measurements show a broad range of neutron star masses, from approximately 1.1 to 2 Solar Masses. The differences between the neutron star masses in different categories are also evident. To study and characterize the mass distributions of these different classes in more detail, it is possible to use Bayesian statistical techniques on the currently available measurements. In particular, the three different categories of sources, namely, the DNS’s, the slow pulsars, meaning, the small spin period pulsars and neutron stars with high-mass companions, which are likely to be near their birth masses, and the recycled pulsars, which include all millisecond pulsars and the accreting neutron stars with low-mass companions), can each be modeled with Gaussian functions with a mean of M zero and a dispersion sigma, see equation 8 in the text.
Several studies have employed Bayesian techniques to measure the most likely values of the mean and dispersion for these systems. Figure 3 shows the inferred mass distributions for these different categories of neutron stars. The most likely values of the parameters for these distributions are the following: M zero equal to 1.33 Solar mass and sigma equal to 0.09 Solar Mass for the DNS’s, M zero equal to 1.54 Solar masses and sigma equal to 0.23 Solar Masses for the recycled neutron stars, and M zero equal to 1.49 Solar Masses and sigma equal to 0.19 Solar Masses for the slow pulsars.

Figure 2.
The most recent measurements of neutron-star masses. Double neutron stars, magenta, recycled pulsars, gold, bursters, purple, and slow pulsars, cyan, are included.

Figure 3.
The inferred mass distributions for the different populations of neutron stars.

A recent study also raised the possibility of two peaks within the recycled millisecond pulsar population, with the first peak at M equal to 1.39 Solar Masses and a dispersion sigma equal to 0.06 Solar Masses and a second peak appearing at Mass equal to 1.81 Solar Masses with a dispersion of sigma equal to 0.18 Solar Masses.
Among these inferred distributions, the narrowness of the DNS distribution stands out.
Although clearly not representative of neutron stars as a whole, as it was once thought, it probably points to a particular evolutionary mechanism that keeps the masses of neutron stars in these systems in a narrow range. Recent discoveries, such as the Double Neutron Star J04 53, indicate that the range of masses in DNS systems may also be wider than previously believed: the recycled pulsar has a mass of 1.559 Solar Masses, the heaviest known in any DNS, whereas the companion has a mass of 1.174 Solar Masses, the smallest precisely measured mass of any neutron star. We infer that the companion is a neutron star from the orbital eccentricity of the system, e equal to 0.112, which would not arise if it had slowly evolved to a massive white dwarf star.
2 point 6. Maximum Mass of Neutron Stars.

Finding the maximum mass of neutron stars is of particular interest in mass measurements because of its direct implications for the neutron star equation of states and neutron star evolution. The largest neutron star mass can rule out the equation of states that have maximum masses and fall below this value. The current record holder on this front is J03 48 with a mass of 2.01 plus or minus 0.04 Solar Masses.
There are also some studies of a particular class of millisecond pulsars called black widows, and their cousins redbacks, that have suggested higher neutron star masses. These millisecond pulsars irradiate and ablate their very low-mass companions. Although the pulsar timing provides the Keplerian parameters for the orbit, all other information about the masses in these systems is obtained from the modeling of the optical light curves, to determine orbital inclination, and the spectroscopy, to measure the mass ratio, of the companion star. Unfortunately, there are many difficulties in obtaining accurate measurements from these ablated companions. Even when using a model of an irradiated companion, the short timescale variability, the unevenly heated surface, and the observed asymmetry in the light curves significantly hinder the orbital inclination measurements.

Similarly, the difference between the center of light of the irradiated companion and its center of mass as well as the departure of the spectral features from assumptions of thermodynamic equilibrium introduce large uncertainties in the inferred mass ratios. These results are tantalizing and hint at the possibility of even more massive neutron stars than J03 48, but because of the systematics, we find that they are not yet as robust as the results from radio timing.
Section Three. RADIUS MEASUREMENTS.

Neutron Star radius measurements have progressed significantly in the past decade and a number of different techniques have been employed. Nearly all of the methods that are currently used rely on the detection of thermal emission from the surface of the star either to measure its apparent angular size or to detect the effects of the neutron star spacetime on this emission to extract the radius information.
The approaches can broadly be divided into spectroscopic and timing measurements. In addition, there are proposed methods for determining neutron star radii that show promise for the future but are not covered in this review, because there are currently limited data or because they provide only broad limits:
1. Quasi-periodic oscillations observed from accreting neutron stars,
2. Neutron star cooling,
3. Pulsar glitches, which help constrain the relative thicknesses of neutron star crust versus its core, and
4. Astero-seismology.

3 point 1. Spectroscopic Measurements.
Much like measuring the radii of normal stars, spectroscopic measurements of neutron star radii rely on determining their angular sizes by measuring a thermal flux F, modeling the spectrum to determine the effective temperature T eff, and combining this with a distance measurement D to obtain observed or apparent radii R sub obs, equation 9.
R sub obs, over D equals:
The square root of F bolometric, divided by sigma B times effective Temperature.
Where Sigma B is the Stefan-Boltzmann constant. Unlike stars, however, there are several complications that come into play in this process. First, owing to their extreme compactness, neutron stars gravitationally lens their own surface emission. This introduces mass-dependent corrections to the observed angular sizes, meaning, the physical radii are related to the observed radii via equation ten.
R obs over R equals the inverse square root of one minus 2 G M, over R c squared.

This equation holds as long as the radius is larger than the photon orbit 3 GM over c squared. If it is smaller, then the geometric radius observed at infinity would be equal to the square root of twenty seven times GM over c squared, independent of the stellar radius. The situation is even more complex in the case of neutron stars that spin moderately fast, because their space-time can no longer be described by a Schwarzschild metric, and there are further spin-dependent corrections to the observed angular sizes. Second, the surface emission may be contaminated by non-thermal emission from an isolated neutron star’s magnetosphere or by the disk surrounding an accreting one. Third, the magnetic field a neutron star possesses can be strong enough to cause temperature non-uniformity on the surface, which complicates the inference of the true stellar radius. Fourth, distances to neutron stars are, in general, more difficult to measure than those to normal stars.

Some of these challenges are overcome by detailed theoretical modeling of emission from neutron stars in General Relativity. Others are mitigated or eliminated by careful source selection, for example, sources with low magnetic fields, with low accretion luminosities, or located in GC’s with known distances, and by combining multiple observational phenomena to break the inherent degeneracies between neutron star mass and radius and to reduce the measurement uncertainties. With these criteria as a guide, thermal emission from LMXB’s in quiescence and during X-ray bursts have been the focus of recent work on neutron star radii.
3 point 1 point 1. Quiescent low-mass X-ray binaries.

The first group of sources on which radius measurement efforts have focused are the accreting neutron stars in LMXBs when they are in quiescence qLMXB’s. In quiescence, accretion ceases or continues at an extremely low level. This allows observations of the thermal emission from the stellar surface, which is powered by the re-radiation of the heat stored in the deep crust during the accretion phases. Because of the short, about minutes, settling time of heavier elements in the surface layers of neutron stars, the atmospheres of neutron stars in quiescence are expected to be composed of hydrogen. There are cases where the companion is hydrogen poor and a helium atmosphere may be appropriate, see below. The observed spectra of qLMXB’s are indeed quasi-thermal, with a weak power-law component at high energies that is likely due to residual accretion.
A number of qLMXB’s in GCs have been observed with the Chandra X-Ray Observatory and the X-Ray Multi-Mirror Mission-Newton, XMM-Newton. Their luminosities in quiescence are of the order of ten to the 32 to 33 ergs per second, equivalently ten to the 25 to 26 joules per second, making them very faint objects. In addition, the GC environments entail crowded fields. Because of this, the high angular resolution and low background of the modern X-ray instruments were crucial for obtaining spectroscopic constraints of their apparent angular sizes.
To make quantitative measurements from the observed spectra, radiative equilibrium models of neutron star atmospheres have been developed for a variety of surface compositions, surface gravities, and temperatures.
Owing to the low magnetic field strengths of neutron stars in LMXB’s, and the expectation of hydrogen atmospheres, un-magnetized hydrogen models have primarily been used in the spectral analyses to obtain observed angular sizes.
To date, reliable radius constraints have been obtained for eight qLMXB’s located in the GC’s M13, M28, M30, Omega Centaurus, NGC 63 04, and NGC 63 97, and two in 47 Tucanae. The observed (apparent) angular sizes obtained for these sources have been combined with Globular Cluster distances to measure their apparent radii. Under somewhat different assumptions, there have been several combined analyses of these measurements.
The most recent results are displayed as correlated contours on the neutron star M-R diagram, see Figure 4. The full M-R likelihoods and tabular data for these sources can be found online, hyper-text in original text. Several sources of systematic uncertainties that can affect the radius measurements have been studied. We discuss these in some detail below.

3 point 1 point 1 point 1. Atmospheric composition.

The majority of qLMXB’s for which optical spectra have been obtained show evidence for H alpha emission, indicating a hydrogen-rich companion. Although none of these spectra have been obtained for GC qLMXB’s, assuming that sources in GC’s have similar companions to those in the field led to the use of hydrogen atmospheres when modeling quiescent spectra.

Figure 4.
The combined constraints at the 68 percent Confidence Level over the neutron-star mass and radius obtained from (a) all neutron stars in low-mass X-ray binaries during quiescence and (b) all neutron stars with thermonuclear bursts. The light gray lines show mass relations corresponding to a few representative equations of state, see Section 4 point 1 and Figure 7 for detailed descriptions and the naming conventions for all equations of state.

There is one source among the six that have been analyzed in detail, for which there is evidence to the contrary. There is only an upper limit on the H alpha emission from the qLMXB in NGC 63 97 using Hubble Space Telescope observations. Because of this, this source has been modeled with a helium atmosphere and the corresponding results are displayed in Figure 4.
3 point 1 point 1 point 2. Non-thermal component.

Assuming different spectral indices in modeling the non-thermal spectral component also has a small effect on the inferred radii. The low counts in the spectra do not allow an accurate measurement of this parameter; however, a range of values have been explored in fitting the data.
3 point 1 point 1 point 3. Interstellar extinction.

Because of the low temperature of the surface emission from qLMXB’s, the uncertainty in the interstellar extinction has a non-negligible effect on the spectral analyses. Different amounts of interstellar extinction have been assumed in different studies. A recent study explored different models for the interstellar extinction in the analysis of the qLMXB’s in omega Centarus and NGC 63 97 and found statistically consistent results with small differences in the central values but larger differences in the uncertainties. The relevant uncertainties have been incorporated into the results shown in Figure 4.
3 point 1 point 2. Thermonuclear bursts.

Neutron Stars in LMXB’s show a phenomenon called thermonuclear Type one X-ray bursts, in which the accreted material undergoes a helium flash that consumes the fuel that is spread over the neutron star surface. The observed X-ray luminosity rises rapidly, in a timescale of approximately 1 seconds, which corresponds to the diffusion timescale from the burning layer to the neutron star surface.

The energy is then radiated away on a timescale of about 15 to 50 seconds for the majority of bursts during the so-called cooling tails. In a subset of the cases, the luminosity reaches the Eddington luminosity where the radiation forces match or exceed the gravitational forces and lift the photosphere off of the neutron star surface. These photospheric radius expansion, P R E, events have characteristic signatures in which the photosphere is observed to reach several times the neutron star radius and the temperature has a second rise.
Several approaches have been developed and used to determine the neutron star radius by making use of thermonuclear bursts. Van Paradijs in 19 79 introduced using the apparent angular size obtained during the cooling tails of the bursts as a way to obtain correlated M-R constraints. Subsequently, researchers discussed different ways in which the degeneracy between the radius and the mass can be broken through multiple spectroscopic measurements. A number of studies since have used a combination of the apparent angular sizes, the Eddington fluxes obtained from the P R E bursts, and the source distances to measure the neutron star radius and mass. In particular, the apparent angular sizes were measured from the flux and the temperature obtained from time-resolved spectroscopy, whereas the Eddington limit is measured at the moment when the photosphere touches back down onto the stellar surface. The Eddington flux is related to stellar parameters M and R and distance D via equations eleven and twelve.
K sub es is the electron scattering opacity, and the last term in the equation arises from temperature corrections to electron scattering.
X is the hydrogen mass fraction of the atmosphere.
The quantity “A” sub g is equal to 1.01 plus 0.067 times the effective gravity in units of 10 to the 14 centimeters per second squared.
And the effective surface gravity is given by equation fourteen, which is:
G sub eff is equal to GM over R squared multiplied by a General relativity correction.

Because the dependence of the Eddington flux on the stellar mass and radius, Equation 11, is different from that of the apparent angular size, Equation 10, combining these two spectroscopic measurements breaks the degeneracies between the stellar parameters introduced by General Relativity effects.
Atmosphere models during bursts are used for the interpretation of spectra and the correct measurement of effective temperatures. A number of such models have been developed, addressing the Comptonization of photons by the hot surface electrons with increasing levels of sophistication and have been calculated for a range of surface compositions and effective gravities.
The deviations from a blackbody at the effective temperature of the atmosphere are quantified by the so-called color correction factor fc is equal to Tc over Effective temperature, which is then folded into these measurements.
Finally, the measured angular sizes are subject to additional general relativistic corrections due to moderately high spin frequencies observed in neutron stars with thermonuclear bursts. In particular, the effects of the quadrupole moment and the ellipticity of the neutron star on the neutron star spacetime as well as the rotational broadening of the thermal spectra can be calculated in the Hartle-Thorne metric and lead to corrections in the inferred angular size of the order of about 10 percent.

This approach has been applied to the neutron stars in the LMXB’s, listed in the text.
Ozel et al in 20 16 performed an updated analysis with several improvements, using new statistical methods and applying the theoretical corrections discussed above, and uniformly incorporated systematic uncertainties into the measurements. The resulting 68 percent C L contours over the mass and radius for these six neutron stars are shown in Figure 4. When combined, these measurements indicate neutron star radii in the 9.8 to 11 kilometer range, which is consistent with the results obtained from the analyses of qLMXB’s.
Another approach that has been utilized for radius measurements is to determine the spectral evolution of the color correction factor in PRE bursts. Because the spectral distortions depend on the effective surface gravity g eff and the emitted flux, the change in the color correction factor with luminosity as the burst cools down from an Eddington flux can provide a measure of the stellar mass and radius. The application of this method to 4U 17 28, 4U 17 24, 4U 16 08, and recently three additional sources led to radii measurements that range from very small, R less than 6 kilometers in the case of 4U 17 28 to quite large, 13 kilometers less than R less than 16 kilometers for 4U 16 08. The data selection used in the latter studies is discussed below.
One final approach is to compare the observed properties of the bursts, such as their recurrence and decay times, and their peak fluxes to the expectations from theoretical light curve models to infer a surface redshift for the neutron star. These constraints can then be combined with the spectral evolution during the burst cooling tails to obtain constraints on the neutron star radius. Unfortunately, the theoretical models in this case are not very predictive and fail to explain the light curves of almost any of the nearly 50 sources extensively studied with the Rossi X-Ray Timing Explorer. Therefore, the application of this method has been limited to the one source, GS 18 26, where models can approximately match the light curve properties, and resulted in the limits R less than 6.8 to 11.3 kilometers for a Mass less than 1.2 to 1.7 Solar Masses, which is also consistent with the results obtained from qLMXB’s and with PRE bursts see Figure 4.
As with the qLMXB’s in GC’s, several sources of systematic uncertainty exist in the spectroscopic measurements of thermonuclear bursts. Their effects on the radius measurements have been studied in recent years, and these are discussed in some detail below.
3.1.2.1. Distances.

Two of the three techniques discussed above rely on a measurement of source distances to obtain neutron star radii. For the sources that reside in Galactic Cluster’s, the cluster distances are utilized.
These are subject to the same uncertainties as those used for the qLMXB measurements. Other techniques have also been considered to measure distances to bursting neutron stars. In most cases, these measurement techniques yield either reliable lower limits or source distances with relatively large uncertainties, which dominate the resulting uncertainties in the radius measurements.
3.1.2.2. Detection of outliers.

As with all physical measurements, identification of outliers in data sets that may contaminate the results is an important issue. Guver et al in 20 12 developed a data driven Bayesian Gaussian-mixture approach to identify the outliers in the spectroscopic burst data without any theoretical biases. The resulting selected data were used in the radius measurements reported by Ozel et al in 20 16. In an alternative approach, Poutanen et al and Kajava et al in 20 14 performed data selection by requiring that the observations follow theoretical expectations.

This approach resulted in a selection of a different, and much smaller, subset of bursts and led to the measurement of a stellar radius R greater than 12 kilometers for 4U 16 08 and to a Radius in the range of 10.5 kilometers to 12.8 kilometers for the three sources used by Nattil et al in 20 16. The consistency of the burst evolution with theoretical expectations and the limitations of the latter method when applied to Rossi X-Ray Timing Explorer data are discussed in detail by Ozel et al in 20 15.
3 point 2. Radii via Pulse Profile Modeling.

NS radii can also be measured or constrained by analyzing the properties of periodic brightness oscillations originating from temperature anisotropies on the surface of a spinning neutron star. The amplitudes and the spectra of the oscillation waveforms depend on the neutron star spacetime, which determines the strength of the gravitational light bending the photons’ experience as they propagate to an observer at infinity, as well as on the temperature profile on the stellar surface and on the beaming of the emerging radiation. Using theoretical models for the emerging radiation, the properties of the brightness oscillation can, therefore, be used to probe the stellar spacetime and to measure its radius and mass.
The theoretical work on neutron star spacetimes has enabled increasingly more precise calculations of the effects of the gravitational lensing on the surface photons, which are used for the analyses of the waveforms. Earlier approximations appropriate for non-spinning neutron stars were supplemented by adding in the effects of Doppler shifts and aberration or those of frame dragging for slow spins up to 300 Hertz. At even faster spins, the oblateness of the neutron star and the quadrupole moment of its spacetime on the photon trajectories have been accurately accounted for. For stars near break-up, greater than 1 kilo Hertz, fully numerical solutions of the spacetimes become necessary.
The waveforms depend also on the location and the size of the hot spots, the beaming pattern of the radiation emitted from the stellar surface and the line of sight of the observer with respect to the rotation axis. Modeling the shapes and amplitudes of the waveforms, therefore, can yield constraints on a combination of all of these parameters and result in measurements of these parameters with correlated uncertainties. The challenge here is to reduce the number of unknowns and to eliminate the correlated uncertainties between the parameters in order to obtain a measurement of the neutron star radius. The size of the hot spot has a minor effect when it is sufficiently small, so this parameter can be eliminated in some cases. The beaming of radiation can also be calculated from atmosphere models.
Analysis of the oscillation waveforms has been performed to explore the properties of numerous types of neutron stars. Although this method yielded useful results on the surface emission properties of slow pulsars and magnetars, constraints on the neutron star radius and spacetime come, in particular, from the analysis of oscillations from accretion-powered millisecond pulsars, rotation-powered millisecond pulsars, and thermonuclear X-ray bursters. These three classes have distributions of spin frequencies that differ from one another, as shown in Figure 5. Therefore, spin effects need to be incorporated to different levels when modeling each population. We discuss the radius measurements from each of these classes individually below.
3 point 2 point 1. Rotation-powered pulsars.

Even though the emission from rotation-powered millisecond pulsars is largely nonthermal and is dominated by their magnetospheres, a number of sources show a clearly detected thermal component in the soft X-rays.

Figure 5.
The cumulative distribution of spin frequencies of rotation-powered millisecond pulsars, millisecond pulsars, accretion-powered millisecond pulsars, and accreting neutron stars that show thermonuclear burst oscillations.

X-ray data from a number of millisecond pulsars, obtained with Roentgen-satellit (ROSAT), Chandra X-Ray Observatory, and XMM-Newton, have been analyzed using hydrogen atmosphere models for the thermal emission from a polar cap. In these models, pulse profiles are calculated based on the beaming of radiation predicted by the theoretical models.
An analysis of PSR J0 437, assuming a mass of 1.4 Solar Masses, yielded bounds on the radius in the R equal to 6.8 to 13.8 kilometers range, 90 percent C L. A subsequent measurement of the pulsar mass at 1.76 Solar Masses increased the lower limit on the neutron star radius to R greater than 11.1 kilometers, 99.9 percent C L, but this result is likely to be revised again in view of the most recent measurement of the pulsar mass (see Table 2). Similar analyses for PSRs J00 30 and J21 24 lead to lower limits on their radii of 10.7 kilometers, 95 percent C L, and 7.8 kilometers, 68 percent C L, respectively, assuming a pulsar mass of 1.4 Solar Mass.
There are a number of complexities that affect the interpretation of the pulsar waveform data from millisecond pulsars. First, the non-thermal magneto-spheric emission is subtracted from the total observed emission in the X-rays to allow for modeling of the thermal surface component from the polar caps. The thermal component itself is typically modeled with two separate regions with different temperatures, each emitting with a hydrogen atmosphere spectrum. The pulse profiles have been modeled with two polar caps, but the waveforms have required a non-antipodal geometry and led to additional uncertainties from introducing an additional offset parameter. The footprints of polar caps in the studies performed to date have assumed circular caps, but the effect of the shape of the polar cap on the waveforms has not yet been explored.
Finally, the angles that specify the observer’s line of sight and the colatitude of the polar caps with respect to the stellar spin axis are assumed to be known independently. The uncertainties introduced by the errors in these angles have not been factored into the quoted radius constraints.

Figure 6
The radii constraints obtained from analysis of the waveforms from accretion-powered and ro