Transcript A0620-00 poster

The Masses and Evolutionary State of the Stars
in the Dwarf Nova SS Cygni
Edward L. Robinson & Martin A. Bitner
Department of Astronomy, University of Texas at Austin, Austin, TX 78712, USA
(appeared 2007, ApJ, 662, 564)
ABSTRACT
1. INTRODUCTION
As the brightest dwarf nova and one of the brightest cataclysmic variables of any kind, SS Cygni
has been extensively observed. Its outbursts, for example, have been continuously monitored
since 1896 and their properties are the gold standard against which accretion disk instability
models for dwarf nova outbursts must be compared (Cannizzo & Mattei 1992; Lasota 2001).
The basic properties of SS Cyg are thought to be well established. The primary star is a white
dwarf, the secondary is a K4-5 V star, and the orbital period is P = 0.27624 d. According to the
Ritter and Kolb (1998) catalog, the masses of its two component stars MK = 0.704 M and MWD =
1.19 M. However, standard techniques for measuring the masses of binary stars do not work for
SS Cyg because the radial velocity curve of the white dwarf is not measured and the orbital light
curve does not show eclipses. Previous measurements of the masses of its stars have, therefore,
been forced to rely on dubious assumptions about the properties of the accretion disk or about the
interior structure of the K star.
We report here a measurement of the masses using a method that was first developed for black
hole X-ray binaries and does not require these assumptions. The technique has three steps (see,
eg, Casares 2005) :
1) The amplitude of the secondary's radial velocity curve, KK, and the orbital period give the mass
function
3
3
M WD sin i
KK P

(1  q)2
2πG
We have obtained new spectroscopic observations of SS Cyg. Fits of synthetic
spectra for Roche-lobe-filling stars to the absorption-line spectrum of the K star
yield the amplitude of the K star's radial velocity curve and the mass ratio:
KK = 162.5 ± 1.0 km s-1 and q = MK/MWD = 0.685 ± 0.015. The fits also show
that the accretion disk and white dwarf contribute a fraction f = 0.535 ± 0.075 of
the total flux at 5500 Å. Taking the weighted average of our results with
previously published results obtained using similar techniques, we find KK =
163.7 ± 0.7 km s-1 and q = 0.683 ± 0.012.
The orbital light curve of SS Cyg shows an ellipsoidal variation diluted by light
from the disk and white dwarf. From an analysis of the ellipsoidal variations we
limit the orbital inclination to the range 45° < i < 56°. The derived masses of the
K star and white dwarf are MK = 0.55 ± 0.13 M and MWD = 0.81 ± 0.19 M,
where the uncertainties are dominated by systematic errors in the orbital
inclination.
The K star in SS Cyg is 10% to 50% larger than an unevolved star with the same
mass and thus does not follow the mass-radius relation for Zero-Age MainSequence stars. Its mass and spectral type are, however, consistent with models
in which the core hydrogen has been significantly depleted.
4. ANALYSIS OF THE ORBITAL LIGHT CURVE
The mean V-band orbital light curve of SS Cyg is shown in Figure 4 (Voloshina & Khruzina
2000). The basic double-humped variation is produced by ellipsoidal variations of the K star. The
extra amplitude and asymmetry of the hump at phase 0.75 is produced by a bright spot on the
outer edge of the disk, but the hump at phase 0.25 is relatively unaffected by the spot and
measures the amplitude of the ellipsoidal variations.
2) The rotational period of the secondary star is tidally locked to the orbital period. Since the
secondary fills its Roche lobe (demonstrated by the mass transfer), the ratio of its rotational
velocity, Vrot sin i, to KK yields the mass ratio (Figure 1 shows how this works):
 Vrot sin i 

q  f 
 KK 
a2
Figure 4 – The dots are the mean V-band orbital light curve of SS Cyg from Voloshina &
Khruzina (2000). The basic double-humped variation is produced by ellipsoidal
variations of the K star. The extra amplitude and asymmetry of the hump at phase 0.75 is
produced by a bright spot on the outer edge of the disk. The solid line is a synthetic light
curve fitted to the data.
We modeled the light curve with 1) the ellipsoidal variations of the K star, 2) constant flux from
the accretion disk, and 3) variable flux from a spot on the outer edge of the disk. Because q is well
determined from spectroscopy, the fit yields a relation between orbital inclination and the amount
of flux coming from the disk. The relation is shown in Figure 5.
r2
Figure 3 - Top: Portion of the average spectrum of SS Cyg in the rest frame of the K star.
Bottom: Synthetic spectrum for a single, non-rotating star with the same spectral type.
The width of the absorption lines in the spectrum of SS Cyg is caused by a combination
of rotation, orbital motion, and instrumental resolution.
Figure 1 - If the secondary star in a binary fills its Roche lobe and is rotating
synchronously, the ratio of its rotational velocity to its projected orbital velocity is a
strong function of mass ratio because Vrot sin i K2  Vrot Vorb  r2 a 2  f (q)
3) The distorted secondary star shows ellipsoidal variations. In principle the ellipsoidal variations
give a relation between the mass ratio and the orbital inclination, although in practice systematic
errors must be carefully considered because of contamination by light from the accretion disk.
2. OBSERVATIONS
We obtained 23 spectrograms of SS Cyg with the High Resolution Spectrograph (HRS) on the
Hobby-Eberly Telescope between JD 2452084 and JD 2452114 (June and July 2001). SS Cyg
was in quiescence at the time (see Figure 2).
The spectrograms covered 5300 Å to 7000 Å at a resolution R = 30,000. The exposure times were
all 600 seconds, long enough to achieve the necessary signal-to-noise ratio without excessive
spectral smearing from the changing radial velocity of the secondary star as it progressed around
its orbit. A portion of the mean spectrum is shown in Figure 3.
3. ANALYSIS OF THE SPECTRUM
We used our “LinBrod” program to analyze the spectrum of SS Cyg (Bitner & Robinson 2006).
This program calculates the spectrum of a star that fills its Roche lobe in a close binary star by
summing wavelength-dependent specific intensities over the visible surface of the star. The
surface of the star is approximated by many (10,000 – 50,000) flat “tiles.” Specific intensities as a
function of wavelength are calculated for each tile using ATLAS9 and a modified version of the
spectrum synthesis program MOOG 2002 (Sneden 2002). Then for each orbital phase, LinBrod
calculates the emergent intensities, shifts the intensities to the radial velocity of the tile, and adds
Doppler-shifted intensities for all the visible tiles together to give the synthetic spectrum. The
spectrum is smeared to account for orbital motion and instrumental resolution.
We fit the synthetic spectra to all the individual spectrograms of SS Cyg simultaneously (not to the
mean spectrum) by χ2 minimization. The fits yield KK, q, and the fraction of the flux f coming
from the accretion disk and white dwarf. The results are shown in the following table.
Our Results
Martinez-Pais et al. (1994)
Figure 2 - The eruption light
curve of SS Cyg in 1981 and
2001. The horizontal bars
mark the dates when Hessman
et al. (1984) measured the
radial velocity curve of SS Cyg
in 1981 and when we obtained
our data in 2001. Both sets of
data were obtained during
quiescence but SS Cyg was
~0.6 magnitudes fainter when
we obtained our data.
North et al. (2002)
Weighted Mean
K (km/s)
q
f @ wavelength
162.5 ± 1.0
0.685 ± 0.015
0.54 ± 0.08 @ 5500 Å
162.5 ± 3
0.45 @ 6550 Å
165 ± 1
0.68 ± 0.02
163.7 ± 0.7
0.683 ± 0.012
0.315 ± 0.004 @ 6400 Å
Martinez-Pais et al. (1994) and North et al. (2002) used techniques similar to ours to measure KK
and q, and their results are also shown in the table. As we see no reason to prefer one result over
another, we have calculated the weighted means of the three measurements and recommend them
as the best estimates of these parameters for SS Cyg.
Figure 5 – The orbital inclination as a function of V-band flux contributed by the disk.
The solid line from lower left to upper right is measured from the amplitude of the
ellipsoidal variations; the parallel dashed lines are the 1-σ confidence limits. The vertical
lines are measured from the dilution of the absorption lines in the spectrum. The region
enclosed by the dashed lines gives the allowed range of inclinations.
5. THE ORBITAL INCLINATION AND MASSES
From the fits to the spectrum the fraction of the V-band flux contributed by the accretion disk and
white dwarf is f = 0.54 ± 0.08. With this value the fits to the orbital light curve yield an orbital
inclination 45° < i < 56.
The derived masses of the K star and white dwarf are MK = 0.55 ± 0.13 M and MWD = 0.81 ±
0.19 M, where the uncertainties are dominated by systematic errors in the orbital inclination.
6. THE EVOLUTIONARY STATE OF THE K STAR
Since the K star fills its Roche lobe, its radius is known. The ratio of its radius to its mass is
1.07 ≤ RK/MK ≤ 1.47 in solar units. The K star is 10% - 50% larger than the theoretical models for
single zero-age main-sequence stars with the same masses calculated by Chabrier & Baraffe
(1997). Thus, the K star does not obey the ZAMS mass-radius relation.
Kolb et al. (2001) have calculated theoretical evolutionary models for the secondary stars in masstransfer binaries. Their models, and also the models of Howell et al. (2001), predict that the
secondary star in a binary with the orbital period of SS Cyg should have a mass ~0.2 M greater
than the observed mass of the K star in SS Cyg -- if the secondary is unevolved. The Kolb et al.
models do, however, predict lower masses for K4-5 stars in which core hydrogen is depleted but
not exhausted. If the mixing length is maintained at α = 1.0, the depletion must be extreme to
match SS Cyg, Xc = 4  10-4; but if the mixing length is increased to α = 1.9, the depletion can be
more moderate. We interpret our results, therefore, as showing that the K star in SS Cyg has
significantly depleted the hydrogen in its core.
REFERENCES
Bitner, M. A., & Robinson, E. L. 2006, AJ, 131, 1712
Cannizzo, J. K., & Mattei, J. A. 1992, ApJ, 401, 642
Casares, J. 2005, arXiv:astro-ph/050307
Chabrier, G., & Baraffe, I. 1997, A&A, 327, 1039
Hessman, F. V. et al. 1984, ApJ, 286, 747
Howell et al. 2001, ApJ, 550, 897
Kolb et al. 2001, MNRAS, 321, 544
North, R. C. et al. 2002, MNRAS, 337, 1215
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