The mass of the neutron star in SMC X-1
A.K.F Val Baker, A.J. Norton & H. Quaintrell
Department of Physics and Astronomy, The Open University, Walton Hall, Milton Keynes, MK7 6AA
Introduction
• X-ray pulsars are accreting X-ray
binary systems where compact object
is a highly magnetized rotating
neutron star.
• Direct measurement of neutron star
mass if system is eclipsing.
• If their masses can be measured to
high accuracy, the equation of state for
nuclear matter may be constrained.
• Only 7 eclipsing X-ray binary pulsars
currently known.
The mass ratio q is defined as:
A value for i can be found from the following
geometrical approximation:
where Mx is the mass of the neutron star, Mo is the mass of the optical
companion, Kx is the semi amplitude of the neutron star’s radial velocity
curve, which can be found from pulse timing delays and Ko is the semiamplitude of the optical stars radial velocity curve, which can be found from
optical spectroscopy.
where a is the separation of the centres of mass of the two stars, qe is the eclipse half
angle and b is the ratio of radius of the optical companion to that of its Roche-lobe, RL.
RL has been found to be reasonably well fitted by the expression:
For a circular orbit it can be shown that:
where A, B and C are constants that
depend on W, the ratio of rotational
period of the giant star to the orbital
period (Rappaport & Joss1983).
and similarly,
where i is the inclination of the orbital plane to the line of sight and P is the
period of the orbit.
History of SMC X-1
Observations
• SMC X-1 is an eclipsing X-ray pulsar located in the Small Magellanic Cloud (SMC).
• Optical counterpart is the B0I supergiant Sk 160.
• Long quasi stable period of 50-60 days - believed to be result of quasi periodic obscuration of the neutron star
by a precessing accretion disk.
• Mode of mass transfer believed to have significant contributions from Roche-lobe overflow, as the stellar winds observed in Sk 160
are not strong enough to power accretion from the secondary onto the primary.
• Mass of the neutron star was first found using image-tube photographic spectroscopy.
• Reynolds et al. (1993) were the first to account for heating of the donor star by the X-ray flux from the neutron star.
• van Kerkwijk et al. (1995) pointed out the uncertainties introduced in this approach by not allowing for an accretion disk and
suggested that corrections for heating effects may be an over estimation.
• van der Meer et al. (2005) found a low value for the mass of the neutron star - did not account for heating corrections.
• August/September 2000
• 1.9m Radcliff telescope.
• Sutherland Observatory in South
Africa (SAAO).
• The grating spectrograph –
resolution of 0.5/Å pixel.
• 56 usable spectra spanning the
wavelength range 4300 – 5100Å
Median spectrum
of Sk 160
Analysis
• Each spectrum of Sk160 was cross-correlated against the median spectrum
of HR1174.
• Resulting radial velocities were heliocentric corrected.
• A simple sinusoid was fitted to a combination of our data and Reynolds et
al.’s data.
• Using the semi-amplitude found from this fit, the masses of the two stellar
components were found.
• Monte Carlo methods were used to determine the uncertainties on the
inferred values.
• An upper limit for the mass was obtained by assuming Sk 160 fills its
Roche-lobe.
• A lower limit for the mass was obtained by assuming i = 90o.
Raw radial velocity curve and best fit
X-ray heating corrections
• Radial velocity measurements of the optical companion in a binary
Heating corrected radial velocity curve and best fit
system reflect its motion about the centre of light.
• In Keplerian orbits the centre of light should be approximately
coincident with the centre of mass.
• Heating of the optical companion by the X-ray source can lead to a
variation between the centre of light and the centre of mass.
• Due to the heating and temperature dependence of spectral lines, the
shift in line centre may vary significantly.
• In this study we only used the helium lines for cross-correlation, so
these effects can be ignored.
• To correct for the heating effects we ran models using the
sophisticated light-curve synthesis program LIGHT2 (Hill 1988).
• The program generates non-Keplerian velocity corrections by
averaging a velocity based on contributions from weighted elements
of the giant stars projected stellar disk.
• The code was required to run through 3 iterations before convergence.
Discussion
References
• Our final value for the systemic velocity of SMC X-1, g =179.2±1.5 km s-1 is in excellent agreement with previously
obtained values (e.g. Hutchings et al. 1977).
• Our raw value for Ko and the corresponding upper limit on the neutron star mass, 1.020.10 M‫סּ‬, are both
comparable with those found by van der Meer et al. (2005).
• Previous studies assume the giant star is Roche-lobe filling, thus giving only upper limits to the stellar masses.
• Effects of X-ray heating on the inner hemisphere of Sk 160 are present but not dramatic (Reynolds et al. 1993).
• Non-Keplerian corrections, for Her X-1, found from both a diskless model and a disk model are in good agreement
(Reynolds et al. 1997), so we assume the lack of a disk in LIGHT2 does not affect our results significantly.
• We conclude that the mass of the neutron star in SMC X-1 lies in the range 0.92±0.09 M ‫ ≥ סּ‬Mx ≥ 1.22±0.10 M‫סּ‬.
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