Ben Williams

Astro 510

12/09/98
Observational Determinations of
Neutron Star Masses and Radii

Placing observational constraints on the fundamental
properties of neutron stars was long thought to be improbable. Nobody was sure
of their existence outside the theoretical realm, and nobody knew how they could
be observed if they did exist. Now, with the knowledge of a significant number
of radio and x-ray pulsars and x-ray bursters, we have been able to directly
observe these high-density objects. Now that we know how to find them, we would
like to determine their fundamental properties, including their masses and
radii. Only with confident measurements of these quantities from empirical
evidence can we test current universal theories of physics. Through neutron star
studies, we can test theories of the equation of state of nuclear matter and the
accuracy of Einstein's general theory of relativity as the correct theory of
gravity. In this paper, I discuss the methods used by observers in order to
determine the masses and radii of neutron stars. The first section concerns mass
determinations of x ray pulsars in eclipsing binary systems. The second
discusses mass determinations of radio pulsars in binary systems, and the third
discusses the measurement of neutron star radii using x-ray bursters.

I. Mass measurements of eclipsing x-ray
binaries

Like measuring the masses of so many other astronomical
objects (including the sun and Earth), the best way to measure the mass of a
neutron star is by monitoring the effects that its gravity has on another
object, e.g. a binary companion. Therefore, when the first eclipsing binary
system containing an x-ray pulsar was found (Schreier et al. 1972), the day was
marked when we could make reliable mass determinations of neutron stars. By
careful monitoring of the cyclical Doppler shifts of the pulse period, we could
determine the period of the pulsar's orbit around its companion as well as its
radial velocity. An example of the data that show such velocities is shown in
figure 1a. These measurements put constraints on the mass of the x-ray pulsar's
companion, which is an optically bright star.

The mass function of a binary component, f(M) is derived
using Kepler's third law and the law of gravity and is given by:
f(MD) =
=

where P is the period of the orbit,  is the
radial velocity of the neutron star, and Mx and MD are the
masses of the neutron star and optical counterpart respectively. Clearly, this
function marks a lower limit on the mass of the optical counterpart, and a
similar relation may be derived for Mx. Therefore, once Vx
sini is determined using the Doppler shifts of the x-ray pulse period
and VD sini is determined using the Doppler shifts of
the spectral features of the optical companion, the ratio of the two masses will
be known. Then, if the system is an eclipsing binary, sini (the
inclination angle) can be modeled (it will be close to 1), and the masses of the
two counterparts can be determined. Figure 1b shows an example radial velocity
curve for an optical counterpart in an x-ray binary system. When compared to the
measurement of the radial velocity of the x-ray pulsar (figure 1a), the
measurement of the radial velocity of the optical companion is the more
uncertain measurement.
Figure 1: a) (above) Timing
measurements of SMC X-1 showing the periodic doppler shifts of the pulse period
(Joss and Rappaport 1984). b) (below) Shifts of the spectral features of the
optical companion of Vela X-1 (Strickland et al. 1997). Notice the difficulty in
determining radial velocities of the optical companion.

II. Mass measurements from radio pulsar
binaries

There is a second class of observable neutron star binaries,
known as radio pulsar binaries. These radio pulsar binaries are older systems,
where the companion of the pulsar is no longer visible because it has become
either a white dwarf or another neutron star whose pulsar beam does not sweep
past the Earth. Due to their ages, the pulsars in these systems have shorter
pulse periods and orbital periods than the pulsars in x-ray binaries. These more
frequently ticking clocks allow for more precise timing measurements. With
today's radio telescopes, we can measure the timing of these pulsars with
tremendous accuracy. Such precise measurements of the arrival times of the
pulses from these neutron stars have provided very accurate determinations of
the slight changes in the shapes and periods of their orbits. These
gravitational effects are due to the strong influence of general relativity in
these systems.

The measurements of these effects allow the observer to
discern many second order gravitational parameters which put tight constraints
on the masses of the components of the binary. The longitudinal advance of the
periastron of the neutron star's orbit, dw/dt, the changes in the gravitational
Doppler shift with transverse velocity, g, along with the system's orbital period spin-down rate, dP/dt, are
all observable for some of these systems. Some systems are only accurate enough
to measure one or two of them, yielding less tightly constrained mass
measurements. Nonetheless, all of these parameters put constraints upon the
masses of the objects in the system, and when all three are measured, very
accurate determinations of the masses are obtained. These observations mark the
first application of general relativity to precise astrophysical measurements,
and they have yielded the most accurate masses of neutron stars to date (see
e.g. Taylor and Weisberg 1982, Lyne and Bailes 1990, or Thorsett et al. 1993).
Figure 2 below shows such an accurate mass determination. Even though the
companion is completely invisible, by using all of the constraints from the
measured effects of general relativity, only a small range of neutron star and
companion masses is possible (Taylor and Weisberg 1982).

The masses determined using these observations show
excellent agreement with all other known neutron star masses. All of the neutron
stars whose masses have been measured with confidence are shown in figure 3. The
masses with the shortest error bars are those determined using these precise
timing observations of radio pulsars. Since a 1.2 solar mass core is needed to
produce a neutron star, one would expect the lowest masses to be at about this
mark, but the highest mass possible is unknown. These data suggest a fairly low
upper mass limit to the neutron star, inferring a soft equation of state for
nuclear matter and a narrow range of initial core masses that end their lives as
neutron stars.

Figure 2: Mass determination of PSR 1913+16. Notice
the consistency check with the rate of decrease of the orbital period (Taylor
and Weisberg 1982).
Figure 3: All known mass measurements
of x-ray and radio binaries (Thorsett et al. 1993). 4U 1700-37 is not a pulsar,
and could therefore be a black hole. The masses marked with an x have recently
been changed. The new mass is shown with a dot with approximate error bars, and
the reference is given.

III. Radius measurements using x-ray bursters

There is only one class of neutron star that allows
measurements of radius. These neutron stars are called x-ray bursters. X ray
bursters are very old neutron stars which have all but lost their magnetic
fields and are still orbiting around a companion in a close orbit. They accrete
material from the companion at a constant rate, and as shown in figure 4, they
build up layers of material onto their surfaces. Once this material reaches a
critical mass, it undergoes a thermonuclear runaway, causing a burst of
luminosity from the surface. Conveniently, many of these bursters are located in
globular clusters, which means that we know their distances quite well. This
knowledge of the distance, along with the other observable characteristics of
the burst, provides information about the radius of the neutron star.

Figure 4: Schematic for the mechanism which causes
x-ray bursts (Joss and Rappaport 1984).

The material that causes the
burst is dense, and therefore optically thick. The burst provides a rare
opportunity for the observer because it is the only time in the universe that
neutron stars radiate like a spherical blackbody, an extremely well understood
system. Using standard blackbody radiation formulae:

Lburst = 4 pi Rx2 s T4 = fx 4
pi
d2
Rx =

Since many bursters are found in globular clusters, their
distance (d) is known. We can directly observe the flux of the burst
(fx ), and we can infer the temperature (T) from the peak
wavelength of the spectrum of the burst. Therefore we can use a simple
calculation to estimate the radius of the neutron star surface. Plugging in
typical observed values, T=107, fx =
10-9 erg cm-2 s-1, d = 1022 cm,
yields a radius estimate of 10 km, as expected. In reality, more accurate
determinations, which take into account general relativity and the chemical
composition of the surface of the neutron star, can be made (see e.g. van
Paradijs and Lewin 1987, or Fujimoto and Gottwald 1989), but the basic concept
is the same. The answers obtained, e.g. 9.6 - 11.0 km (Fujimoto and Gottwald
1989), are close to the rough approximation.

Succinctly, observations of x-ray and radio binaries have
allowed astronomers to measure the masses and radii of neutron stars with a high
level of confidence. These measurements have led to precise tests of the
consistency of advanced theories in physics, including the equation of state of
nuclear matter and general relativity. The incredible consistency these
measurements have with predictions by these theories is an excellent indication
that our understanding of these exotic objects is accurate and moving in the
right direction.

References

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Joss, P. C. and Rappaport, S. A. 1984, ARAA,
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Lyne, A. G. and Bailes, M. 1990, MNRAS, 246,
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Reynolds, A. P. et al. 1997, MNRAS, 288,
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Schreier, E. et al. 1972, Ap. J. L., 172,
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Stickland, D., Lloyd, C., and Radziun-Woodham, A. 1997,
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