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Astro 510
12/09/98
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:
= 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.
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).
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:
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
Fujimoto, M. Y. and Gottwald, M. 1989, MNRAS, 236, 545.
Joss, P. C. and Rappaport, S. A. 1984, ARAA, 22, 537.
Lyne, A. G. and Bailes, M. 1990, MNRAS, 246, 15.
Reynolds, A. P. et al. 1997, MNRAS, 288, 43.
Schreier, E. et al. 1972, Ap. J. L., 172, L79.
Stickland, D., Lloyd, C., and Radziun-Woodham, A. 1997, MNRAS, 286, L21.
Taylor, J. H. and Weisberg, J. M. 1982, Ap. J., 253, 908.
Thorsett, S.E. et al. 1993, Ap. J., 405, L29.
van Kerkwijk, M. H., 1995, A&A, 303, 497.
van Paradijs, J. and Lewin, W. H. G. 1987, A&A, 172, L20.