Download A detached white dwarf/M dwarf binary with an

1996MNRAS.278..565M
Mon. Not. R. Astron. Soc. 278, 565-576 (1996)
A detached white dwarf/M dwarf binary with an orbital period of 2.47 h
T. R. Marsh 1 and S. R. Duck2
I
University of Southampton, Department of Physics, Highfield, Southampton SOl 7 IBJ
of Oxford, Department of Physics, Nuclear Physics Laboratory, Keble Road, Oxford OXI 3RH
2 University
Accepted 1995 August 15. Received 1995 August 4, in originaiform 1995 June 26
ABSTRACT
We find that the white dwarf GD 448 is a detached white dwarf/M dwarf binary with
an orbital period of 2.47 h. This is the shortest period known for such a binary,
placing it in the centre of the 'period gap' from 2 to 3 h in which few cataclysmic
variable stars (mass-transferring white dwarf/main-sequence star systems) are found.
The cooling age of the white dwarf (5 x 107 yr) shows that GD 448 was born in the
gap and has never been a cataclysmic variable star. We measure radial velocity semiamplitudes of K w =31.2±1.5 km S-l and K E =122.2±1.1 km S-l from Ha
absorption from the white dwarf and Ha and Ca II triplet emission from the M star.
The white dwarf absorption shows a gravitational redshift of 16.8 ± 1.6 km s-1,
leading to a mass of 0.44 ± 0.03 M0 for the white dwarf. The Ha and Ca II emissionline fluxes are modulated by a factor of 4, and are strongest when the M dwarf is
furthest from us. The flux variation is consistent with emission proportional to the
irradiating flux from the white dwarf, and yet the latter is not hot enough to have
produced the emission by photoionization alone. The asymmetric distribution means
that K E is less than the true K velocity of the M star, K M' From models of the emission
we find that 138 < KM < 168 km S-l and 0.08 < MM < 0.10 M 0 . Our models
independently predict light curves close to those observed. If the M dwarf is close to
its main-sequence radius, GD 448 will begin to transfer mass when its orbital period
is "" 1 h. A great surprise is that the width of the Ha emission from the M dwarf can
only be matched with the addition of 90 km s - 1 FWHM broadening. The excess
broadening does not affect the Ca II emission, and this suggests that it may arise from
thermal or pressure broadening.
Key words: binaries: close - binaries: spectroscopic - stars: individual: GD 448 stars: low-mass, brown dwarfs - novae, cataclysmic variables - white dwarfs.
1
INTRODUCTION
Over the past few years many detached but unresolved white
dwarf/main-sequence star binaries have been discovered.
Some, such as RE 2013 + 400 (Barstow et al. 1995), have
been found from X-ray emission from a hot white dwarf,
while others have been found from Balmer emission from the
main-sequence companion (Saffer et al. 1993). Such stars
may be the progenitor systems of the cataclysmic variable
stars (CVs). They can become CVs if angular momentum loss
can drive the stars together sufficiently rapidly. The distribution of orbital periods of such systems is thus a useful constraint upon models of the CV population, and in reverse
upon the evolution required to generate such close binary
stars (de Kool & Ritter 1993).
In this paper we present observations of the white dwarf
GD 448 (=WD 0710+ 741, LP034-185) which show that
it is a white dwarf/M dwarf binary with a period of only
2.4 7 h. With such a short period, GD 448 must one day start
to transfer mass once gravitational radiation and possibly
magnetic stellar braking have brought it into contact. However, a period of 2.47 h also raises the intriguing possibility
that GD 448 has already been a CV, because it is in a period
range through which CVs are thought to become detached.
The reason for this is that very few CVs are seen with periods
between 2 and 3 h, even though the angular momentum loss
needed to drive mass transfer continuously alters their
periods. This gap has been explained by supposing that the
angular momentum loss rate, which drives the mass donor
out of thermal equilibrium so that it becomes oversized for
©1996 RAS
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.278..565M
566
T. R. Marsh and S. R. Duck
its mass, decreases at orbital periods close to 3 h, which
allows the mass donor to shrink inside its Roche lobe
(Rappaport, Joss & Verbunt 1983; Spruit & Ritter 1983).
Mass transfer then ceases and the system coasts through the
gap only to start transferring mass again at P = 2 h. There
should therefore be many detached systems that were once
CVs in this period range. No such system has ever been
found, which is not surprising since there are many more
white dwarfs and M dwarfs than there are CVs and a
detached CV would be hard to distinguish from them.
GD 448 is thus an especially interesting system; however, we
shall see that there are good reasons to believe that it is
unlikely that GD 448 was ever a CV, and that it has instead
been born in the gap.
GD 448 was known to have a companion from the work
of Zuckerman & Becklin (1992), who found that it showed
an infrared excess. However, we chose to observe it because
of its low spectroscopic mass (0.35 M 0 ) determined in the
survey of Bergeron, Saffer & Liebert (1992); thus GD448
provides a further confirmation of the link between close
binary evolution and low-mass white dwarfs (Marsh, Dhillon
& Duck 1995).
The weather was clear for all the observations of GD 448,
although the seeing on the first night was poor (2 to 4 arcsec)
compared to the 1-arcsec seeing we enjoyed on the other
nights. Most of the exposures were 500 s long, with a few
longer integrations during periods of poor seeing. The
spectra were extracted with weights to give the maximum
signal-to-noise ratio. For each object spectrum the arc
spectra were extracted at the same position on the detector,
and then the wavelength scale derived from the arc pair was
interpolated in time for the object spectrum. The fits to the
arc calibration had rms scatters of about 1/30th of a pixel.
With a slit width of 1 arcsec our data are not photometric,
but observations of HD 19445 (Oke 1983) were used to
remove the sensitivity dependence on wavelength. (The slit
was held vertical during the observations.) There are telluric
features near 8200 A and beyond 8900 A in the red spectra,
and we attempted to remove these using the relatively
featureless star BD + 26 2606 (Oke 1983), following the
techniques of Wade & Horne (1988). BD+262606 does
have significant Paschen absorption, leading to the appearance of spurious Paschen emission lines in Fig. 1.
3
2
OBSERVA TIONS
RESULTS
Fig. 1 shows a continuous sequence of spectra from January
21/22 in which an emission feature in Ha can be seen
crossing the absorption core of the white dwarf. We identify
the emission as coming from an M dwarf companion. Further
to the red, TiO bands from the companion can be seen, as
can Ca II near-infrared triplet emission, which again comes
from the M dwarf.
Compared to similar systems (Saffer et al. 1993; Schmidt
et al. 1995) the Ha emission in GD 448 is weak, and we had
to adopt a rather involved procedure to measure its radial
velocities. We first concentrated upon the red spectra, as the
NaI and Call lines are free of any features from the white
We used the double-beam spectrograph ISIS on the 4.2-m
William Herschel Telescope on the island of La Palma in the
Canary Islands. On the nights of 1995 January 20/21,21/22
and 24/25, we took three, 11 and 13 spectra respectively. On
the blue arm of ISIS we covered 6420 to 6820 A at 0.398 A
pixel- 1, offset to the red of H a to avoid poor charge transfer
columns. On the red arm we covered 7850 to 9360 A at
1.48 A pixel- 1 to search for molecular bands as a sign of
low-mass main-sequence or brown dwarf companions. The
full width half maximum (FWHM) was about 2 pixels for
each arm.
co
NaI
Call
I
III
GD 448
><
;j
r;:: ....
L--
L--
TiO
TiO
CIl
M6 dwarf
-500
o
500
Velocity (km s-l)
8000
8500
9000
Wavelength (.It)
Figure 1. The left-hand panel shows 11 spectra of GD448 centred on Ha taken on 1995 January 21/22; these show an emission component
in Ha crossing the narrow absorption core of the white dwarf. The right-hand panel shows the mean red spectrum which displays absorption
features from a late-type star and Ca II emission. An M6 standard spectrum (Gl65A + B) is shown for comparison scaled to match the contribution from the red star. The emission lines beyond 9000 A are artefacts of the removal of telluric lines.
© 1996 RAS, MNRAS 278, 565-576
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.278..565M
A short-period, detached white dwarf/M dwarfbinary
dwarf. We measured the Na I velocities by cross-correlation
with the spectrum of Gl65AB plotted in Fig. 1. Our spectrum is an average of the two stars' spectra, as they were too
close to be separated during reduction. This should not
matter, since they are both M6 (Kirkpatrick, Henry &
McCarthy 1991). A circular-orbit fit to the velocities then
allowed us to subtract the M dwarf from the spectra to leave
the Ca II region largely free of late-type features. We found
that the M dwarf contributed 15 ± 2 per cent of the flux at
8200 A.
Next we fitted Gaussian profiles to each of the three Ca II
lines in each spectrum. The widths of the Gaussians were
taken to be the same for all the lines, and the peak heights
were held in a fixed ratio for each spectrum. The width and
mean peak heights (four parameters) were adjusted by X2
minimization, holding the velocities, which were initially set
by eye, fixed. Then, holding the width, the velocities and
relative peak heights fixed, a peak height multiplier was fitted
for each spectrum to account for the marked change in
emission-line strength with phase. Finally, we held everything
fixed except the velocities. The latter were then fitted with
circular orbits, and the fitted velocities were used as inputs
for the next cycle of fits.
Although the Ca II lines are free from absorption, the
spectra are of lower resolution than the Ha data and so we
expected more precise values from Ha. The Ca II fits were
-'-400
-200
o
200
Velocity (km/s)
400
567
therefore used to set initial estimates for the Ha velocities in
order to allow a fit to be made to the Ha core from the white
dwarf. The white dwarf line was fitted in a similar manner
with multiple Gaussians, although we started by masking the
centre of the line where the Ha emission was. Once the fits
improved, we could model the sharp Ha absorption core,
and finally we were able to obtain fits to both emission and
absorption lines. As an additional constraint during this
process we found that the peak height variation was well
described by a sinusoid in phase, and therefore we fixed the
peak heights to follow the fitted sinusoid while the velocities
were being fitted. We required five Gaussians to fit the
absorption line, and their parameters (relative to a continuum normalized to 1) are listed in Table 2.
The final fits to Ha are displayed as a trailed spectrum in
Fig. 2, and the velocities are listed in Table 1. The circular
orbits fitted to the Ca II IR triplet emission, the Ha emission
and absorption lines, and the NaI absorption were of the
form
v = y + K sin 2Jt
(Tp
To),
where y is the systemic velocity, and K the radial velocity.
They are listed in Table 3. The NaI fits have been corrected
by + 30.5 km S-1 to account for the mean heliocentric
-400
-200
o
200
400
Velocity (km/s)
Figure 2. In the left panel we plot the Ha spectra as a trailed spectrum with time ascending, and in the right panel the spectra calculated from
the multicomponent fits made in order to derive the velocities. Gaps of hours or days between different groups of spectra cause the peculiar
pattern.
~ 1996
RAS, MNRAS 278, 565-576
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.278..565M
568 T. R. Marsh and S. R. Duck
Table 1. Heliocentric radial velocities.
HJD
-2449700
38.58823
38.60250
38.61926
39048273
39049331
39.50547
39.51428
39.52138
39.53256
39.53967
39.54676
39.55571
39.56279
39.56987
42040623
42041228
42041819
42.50958
42.51551
42.52144
42.60755
42.61347
42.61940
42.62531
42.68122
42.68713
42.69306
Ha
emission
kms- l
absorption
kms- l
-150.1±9.0
-47.7±12.7
64.5±5.8
-67.2±2.2
-131.1±204
-156.5±3.5
-151.5±6.3
-10004±9.5
-48.0±18.5
24.4±9.6
62.0±704
84.7±4.9
73.9±3.3
45.7±2.5
-116.0±8.8
-78.0±10.2
5.4±12.1
-100.4±8.0
-70.4±9.1
-35.9±15.9
-120.8±6.9
-106.9±9.0
-58.0±1O.2
-13.0±14.1
-93.3±3.2
-131.9±3.9
-151.5±3.7
3.1±5.1
-28.8±4.5
-55.1±304
-19.1±304
-0.1±2.9
8.2±2.9
13.9±3.5
-1.6±4.0
-36.7±6.6
-40.3±4.2
-45.2±4.2
-52.1±4.0
-51.2±3.5
-39.2±3.3
-204±3.7
-16.0±3.7
-19.8±404
-15.2±304
-18.0±3.3
-31.9±5.7
-2.5±3.6
-13.6±3.6
-16.3±3.7
-29.1±5.2
-6.3±4.5
4.7±4.8
5.1±3.9
Table 2. Parameters of Gaussian fits to
Ha from the white dwarf.
Component
1
2
3
4
5
FWHM
(A~
1.20±0.05
5.9±004
28.0±1.0
78.1±204
238±19
Height
-0.186±0.007
-0.077±0.005
-0.124±0.004
-0.140±0.004
-0.043±0.003
velocity of Gl65AB and, because of the poor signal-to-noise
ratio of the Na I measurements, we fixed the zero crossing
time To and orbital period P to the weighted mean of the Ha
and Ca II fits
HJD= 244 9739.530 67(19)+ 0.103 0420(1O)E,
where the figures in brackets are the 1 a uncertainties in the
last two digits of their respective constants. This is our best
ephemeris for GD 448.
Figs 3 and 4 show the radial velocities of Ha and of the
Call triplet and NaI plotted against phase. Fig. 5 shows the
equivalent widths (EW) of Ha and Call triplet (scaled to
match the strongest line at 8542 A) versus phase along with
sinusoidal fits of the form
EW=A +Bsin2:n:(~-0.25)A,
HJD
-2449700
38.58840
38.60264
38.61932
39048278
39049332
39.50552
39.51434
39.52145
39.53264
39.53970
39.54680
39.55578
39.56287
39.56993
42040630
42041232
42041822
42.50965
42.51556
42.52147
42.60772
42.61362
42.61956
42.62549
42.68135
42.68726
42.69316
Call
NaI
kms- l
kms- l
-1I604±12o4
-66.1±14.6
63.5±7.3
-72.8±3.0
-134.8±3.2
-169.0±4.8
-136.1±9.0
-108.9±13.9
-33.8±18.5
11.3±14.3
43.8±1O.0
74.6±6.7
7204±4.7
4204±3.5
-1I0.8±12.8
-68.2±1504
-16.9±15.7
-114.1±11.8
-90.9±1404
8.7±15.0
-115.5±1O.1
-89.2±13.7
-66.8±1604
-34.3±16.1
-9504±4.5
-138.2±5.9
-16004±5.5
-133.6±94.2
-123.9±48.1
52.5±102.0
-18004±201.8
-138.0±115.1
-240.6±90.1
-158.0±44.7
-43.2±75.0
-95.8±52.6
0.3±59.7
1804±49.6
90.8±52.9
45.6±49.1
127.7±63.6
-98.9±5404
-37.2±62.5
-61.3±34.6
-51.6± 110.2
-206.2±74.2
-20.2±47.0
-141.2±47.1
-106.8±48.1
-25.1±59.1
-55.0±5304
-269.6±87.7
-373.1± 108.0
-251.9±84.3
which were used to provide extra stability during the radial
velocity measurements. The best-fitting values of the constants A and B, the duty cycle B /A, and the FWHM of the
emission lines are listed in Table 4. The equivalent width of
Ha is referred to the continuum interpolated from regions
away from the absorption line. The best-fitting Ca II line flux
ratios were /(8662)//(8542) = 0.837 ± 0.042 and /(8498)/
/(8542)=0.675 ±0.035. Finally, Fig. 6 shows the equivalent
widths of the Na I 8200 line, and provides some indication
that the N a I is strongest at phase 0, in antiphase to the
emission line. Thus NaI 8200 is weakest on the side facing
the white dwarf, as has been seen in CVs (Wade & Horne
1988).
4
4.1
DISCUSSION
The gravitational redshift of the white dwarf
The fits listed in Table 3 show an offset between the systemic
velocity of the white dwarf and its companion which is presumably caused by the gravitational redshift of the white
dwarf. To determine the red shift, we must first correct for the
redshift of the M dwarf. Caillault & Patterson (1990) fit a
mass-radius relation of the form
logR/R0 =0.79610gM/M0 -0.037,
and for a mass near 0.1 M0 this translates to a redshift,
GM/Rc, of 0.4 km s-I, a value which is relatively insensitive
©1996 RAS,MNRAS 278, 565-576
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.278..565M
A short-period, detached white dwarf/M dwarf binary
569
Table 3. Circular orbit fits.
Ha emission
Ha absorption"
Call triplet
NaI8200 b
x2
23.5
38.6
28.2
20.8
'Y (km S-I)
-39.0±1.1
-22.7±1.1
-42.7±1.5
-34±11
K (km S-I)
122.3±1.4
-31.2±1.5
122.1±1.9
132±l9
P (d)
0.103042 ± 0.000012
0.103027 ± 0.000044
0.103042 ± 0.000017
To - 2449739
0.53072 ± 0.00024
0.53026 ± 0.00097
0.53059 ± 0.00032
"The uncertainties on the absorption fit have been scaled by (38.6/23)1/2 to account approximately for the poor fit. There were 27 points with four parameters giving 23 degrees of
freedom.
bFor the fit to Na I, To and P were fixed at the mean values of the Ha and Ca II fits.
o
o
r----.----~--_.----.---_,r_--_r----r_--_r----._--~----._--_,
~
~
I
rn
S
C
.....:>,
'0
0
0
Ql
>
«i
:a
al
1-0
CJ
'E::
.....
0
0
....
I
i:1
Q)
CJ
.S
Ql
:I::
o
~
L -__- L____~__~____~__~____~____L __ _~_ _ _ _~_ _~_ _ _ _J __ _~
o
I
0.2
0.4
0.6
0.8
Orbital phase
Figure 3. The phase-folded heliocentric radial velocities and circular orbit fits to GD 448 from the Ha line.
0
0
I
(\I
~
~
I
............:...... - .. -
rn
S
C
./ _,"f
0
Call IR triplet
.--- -. ----........
..
----__
,-'-" ,
:>,
.....
/"
------.--.. __
_. ___ .... ____ •______ ,!!, ,-f
'0
0
Ql
>
«i
:a
al
1-0
0
0
(\I
I
0
0
(\I
CJ
'E::
.....
i:1
0
Q)
CJ
.S
0
0
:I::
I
Ql
,_,'If{/
(\I
---
.
l-----l--- ___ t
NaI 8200
_/Jt",+---r
t ----------____ j_
1 jJ ,/""
jll··l···f~.. I
f
$f-, f
---
0
0
"<i'
I
o
0.2
0.4
0.6
0.8
Orbital phase
Figure 4. The phase-folded heliocentric radial velocities and circular orbit fits to GD 448 from the Ca II emission and Na I absorption. The
vertical scale of the Na I plot covers a wider range than the Ca II plot in order to include all the data.
©1996 RAS, MNRAS 278, 565-576
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.278..565M
570
T. R. Marsh and S. R. Duck
g
C\l
..c:
....,
'0
.~
....,
lD
i::
Q)
«l
.;:;>
0"
Q)
Q)
;§
i::
.!3
'"'"
·s
lD
a
r£I
a
o
0.2
0.6
0.4
0.8
Orbital phase
Figure 5. The emission-line equivalent widths are plotted versus orbital phase along with sinusoidal fits fixed to peak at phase 0.5. Solid circles
represent the Ha values, and crosses the Ca II 8542-A values.
Table 4. Fits to the line emission.
Ha
x
2
A (A)
B (A)
BfA
FWHM (A)
37.0
0.629 ± 0.010
0.401 ± 0.012
0.638 ± 0.022
2.23 ± 0.05
Call
18.9
1.082 ± 0.021
0.709 ± 0.027
0.655 ± 0.028
3.21 ± 0.08
to the exact mass because of the form of the mass-radius
relation, although it could be up to a factor of 2 smaller if the
star is oversized for its mass, with the maximum size set by
the Roche lobe. A further (K~ - K?-vv )/2c sin2 i = 0.1 km S-1
comes from the difference in transverse Doppler shift of the
two stars, and so the white dwarf's redshift is 16.8 ± 1.6 km
S-I. We use only the Ha emission systemic velocity in order
to avoid any systematic offsets (although we have not
detected any) between velocities of the red and blue arm
data.
The uncertainty on this measurement is purely statistical
and takes no account of any systematic effects. It is as low as
it is because of the long total exposure on the target
(17600 s). Some caution should be exercised in accepting
this measurement because of the presence of Ha emission
overlapping the central absorption core. If its shape is not
Gaussian, then a systematic error is possible. It is unfortunately extremely difficult to assess the likely size of any such
error, and we merely note the possibility.
A redshift of 16.8 kIn s -1 corresponds to a mass for a
zero-temperature helium white dwarf of 0.402 ± 0.027 MG
(Hamada & Salpeter 1961). Although this is consistent with
the spectroscopic mass of 0.35 ± 0.03 MG measured by
Bergeron et al. (1992), their value includes a correction for
the finite temperature of the white dwarf derived from the
evolution sequences of Wood (1990), and we must do the
same. Without the correction, Bergeron et al.'s gravity gives a
mass of 0.30 MG' The correction of 0.05 MG is larger than
white dwarfs of a similar effective temperature in Bergeron et
al.'s sample, because GD 448 has such a low gravity. Since
the spectroscopic method is based upon gravity which scales
as R-2, whereas the redshift depends upon the gravitational
potential, the change to the redshift-based estimate will be
different. First, assume that close to Mw= 0.4 M G , the radius
scales as M~. From Nauenberg's (1972) analytic massradius relation we find
1 (Mw/1.44r2/3+(Mw/1.44)2/3
a- - -
-
3 (Mw/1.44) 2/3_(Mw/1.44)2/3'
for Mw in solar masses. This gives a = - 0.48 for Mw= 0.4
MG' Therefore the gravity scales as g oc M~;f6, whereas the
redshift scales as g oc M{v48. A fractional change in the radius
of e then leads to a 2e fractional change in g and therefore a
1.02e change in the mass based upon a spectroscopic
analysis; similarly, it leads to a 0.68e change in the red shiftbased estimate. Therefore any revision of the radius has
about 1.5 times less effect on the mass deduced from the
redshift compared to the mass from model atmospheres. We
deduce a corrected mass of 0.435 M G , assuming that heliumcore models suffer a similar correction to carbon-core
models.
A further revision is needed if, as is now coming to be
thought, there is a thick (10 - 4 M G ) layer of hydrogen present
(Bragaglia, Renzini & Bergeron 1995). The spectroscopic
mass then becomes 0.38 (P. Bergeron, private communication). The red shift-based estimate should therefore be
increased by another 0.03/1.5 = 0.02 M G , raising it to 0.45
MG' The existence of a thick hydrogen layer is not certain, so
we will take as our final estimate a half-way choice of
Mw= 0.44 ± 0.03 M G , where the uncertainty has been
increased to reflect our uncertainty over the hydrogen layer.
In addition to the above uncertainties, the use of standard
cooling sequences for either the red shift or spectroscopic
masses is questionable because of the unusual prior evolution of the white dwarf in this system, and it should be
©1996 RAS, MNRAS 278, 565-576
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.278..565M
A short-period, detached white dwarf/M dwarf binary
571
C\I
~
....,
""
~
'"0
.~
....,
"
Q)
til
:>
';3
Q)
co
til
j
j
0"
0
0
C\I
lJ'l
0
z
0
o
1.5
0.5
2
Orbital phase
Figure 6. The equivalent widths of Na I 8200 A are plotted 'versus phase and have been repeated over two cycles.
stressed that the white dwarf here is a helium, not a carboni
oxygen, white dwarf - the difference this makes to the
evolution is not clear. The above is a rather long-winded way
of saying that GD 448 is not a suitable object for testing the
spectroscopic versus gravitational masses (see Bergeron,
Liebert & Fulbright 1995 for cases which are reliable). Even
without these complications, direct comparison between the
spectroscopic and redshift estimates is dangerous, since we
selected GD 448 because of its low spectroscopic mass and
this could lead to a statistical bias towards low values. The
latter effect leads us to use our red shift-based mass only in
what follows, but no qualitative difference would occur if we
used the lower spectroscopic mass, except that the M dwarf
would move into the brown dwarf mass range.
4.2
Orbital parameters and the M star's line emission
Knowing the mass of the white dwarf, M w , and the radial
velocity semi-amplitudes of the white dwarf and its companion M star, Kw= 31.2 ± 1.5 km S-I and K M= 122.2± 1.1
km s - I (a weighted average of Ca II and Ha), we can immediately obtain the mass of the red starfrom MM=(KwIKM)Mw
= 0.112 ± 0.009 Mo. The orbital inclination follows from
. 3'
1=
sm
PKM(Kw+ KM)2
2'Jt GMw
(1)
where P is the orbital period. We find i = 24~3 ± 0~7.
Unfortunately, it is clear from the variation in emissionline flux seen in Fig. 5 that the semi-amplitudes derived from
the emission lines may not provide an accurate estimate of
K M . The emission lines peak in strength at phase 0.5, when
we see the side of the M star which faces the white dwarf.
Presumably, then, the emission is a result of irradiation by
the white dwarf. Assuming that the red star co-rotates with
the binary, its centre of mass will move faster than the side
which faces the white dwarf, and therefore the measured
values, K E , will be less than the true K M • Thus the mass and
inclination estimates above are really upper and lower limits
respectively.
The size of the distortion depends upon the size of the M
star, with the maximum occurring if the M star is close to
filling its Roche lobe. If, as Figs 5 and 6 seem to imply, Na I
8200 A is distorted in the opposite sense to the emission
lines, it gives an upper limit to K M • Unfortunately, we do not
have the signal-to-noise ratio in these data to make use of this
constraint. Therefore we now consider models of emission
over the M star in order to estimate the likely distortion. We
do not distinguish between Ha and Call, since the ratios
BIA listed in Table 4 are consistent with each other.
Our procedure was as follows. We assumed that the
emission-line strength was proportional to the flux from the
white dwarf incident per unit area on the M star. The models
were parametrized in terms of a linear filling fraction, f,
which we define as the ratio of the stellar radius measured
from the centre of mass towards the inner Lagrangian point
divided by the radius to the inner Lagrangian point. A value
of f = 1 corresponds to a star that fills its Roche lobe, and in
general f5,1. We fixed Kw=31.2 km s-1, Mw =0,44, and
then for eachfwe chose a value of the mass ratio q = KwlKM
which then fixes KM and i from above. We then calculated
the predicted radial velocity curve, taking account of the
distorted surface of the M star. The predicted curve was
fitted with a sinusoid to give a predicted K E' If this was less
(more) than the observed value of 122.2 km S-I, the value of
q was reduced (raised) and the calculation repeated. Finally,
a value of q was found for which the predicted value of KE
matched the observed value to within 0.05 km S-I.
We had to specify how the emission-line flux F from each
point varied with viewing angle. If we imagine a thin layer of
constant source function lying over a dark background (the
M star), then integration of the radiative transfer equation
gives
Foccos O(l-exp- rlcos 0),
where r is the vertical optical depth through the emission
layer, and 0 is the angle between our line of sight and the
©1996 RAS, MNRAS 278, 565-576
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.278..565M
572
T. R. Marsh and S. R. Duck
normal to the surface. The extremes of this model are
T» 1, and F constant for
T « 1. In reality the optically thick case can never be exactly
achieved, because the line profile must become optically thin
in its wings, but the Lambert's law case is still a useful
approximation. When the source function is not constant,
then limb darkening or brightening can occur in the optically
thick case. However, we do not include any limb darkening,
because, as we shall see, the zero-limb-darkened models only
just fit the light curves and limb darkening only makes
matters worse. Limb brightening, on the other hand, could
allow the optically thick model to appear the same as the
optically thin case. However, we have not considered this
possibility further.
Table 5 lists the parameters of our models. The table
shows that, allowing for systematic effects only, the mass of
the red starfalls in the range 0.081 < MM/M0 < 0.112, which
is in the very late M dwarf range. The light curves provide an
independent check on our models. Fig. 7 shows the predicted light curves scaled to match the peak of the Ha equivalent widths. The cos fJ dependence of the optically thick
models means that their light curves are always more sharply
peaked than the equivalent optically thin case. Given that the
models were entirely based upon the radial velocities, the
predicted light curves match the data well, and show that
there are no serious errors in our assumptions. Unfortunately, the light curves tell us nothing new, and in particular
they do not constrain the filling factor since the data lie
between the extreme cases for the entire range of f. The
degree of modulation of the light curve depends mainly upon
the orbital inclination, although it could decrease if there was
any emission on the dark side of the M star. This does not
appear to be significant in GD 448, but may affect GD 245
which, although it is of a higher inclination than GD 448, has
a very similar emission-line light curve (Schmidt et al. 1995).
We list the predicted main-sequence radii based upon
Caillault & Patterson's (1990) mass-radius relation in Table
5. The main-sequence radius and the true radius coincide for
I"" 0.53. However, allowing for uncertainty in Caillault &
Patterson's relation which we are extrapolating from their
lowest mass datum of 0.13 M 0 , and for the possibility of
mass transfer which leads to an oversized star, or for the M
star being too young to have achieved its main-sequence
radius, we claim only that I> 0.4. In the range favoured for
main-sequence radii, 0.4<1<0.7, the optically thin models
are marginally superior in Fig. 7.
We conclude this section with a summary. We have found
that the strong irradiation effects are well described by
emission in proportion to the flux incident from the white
dwarf. If it does not deviate too far from the main sequence,
the M star must be at least 40 per cent the size of its Roche
lobe. Although the measured radial velocity semi-amplitude
KE = 122.2 km S-I, the true semi-amplitude KM is in the
range 138<KM <168 km S-I for a filling factor range
0.4 <I < 1.0. Similarly, the mass of the M star is in the range
0.08 to 0.10 M 0 , with another 0.01 M0 of statistical
uncertainty.
F ex cos fJ (Lambert's law) for
4.3
Emission-line widths
Once fitted to the radial velocities, the geometrical model for
the line emission also fits the variation of the emission line
Table 5. Models of the M star's emission.
f
0.0
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
0.9
1.0
1.0
Optically q
thin/thick
Either
0.2553
Thin
0.2473
Thick
0.2482
Thin
0.2392
Thick
0.2408
Thin
0.2309
Thick
0.2332
Thin
0.2226
Thick
0.2255
Thin
0.2144
Thick
0.2177
Thin
0.2065
Thick
0.2102
Thin
0.1992
Thick
0.2030
Thin
0.1928
Thick
0.1965
Thin
0.1880
Thick
0.1913
Thin
0.1860
Thick
0.1886
J{M'
MM
kms- I
M0
122.2
126.2
125.7
130.4
129.6
135.1
133.8
140.2
138.4
145.5
143.3
151.1
148.4
156.6
153.7
161.8
158.8
166.0
163.1
167.7
165.4
0.112
0.109
0.109
0.105
0.106
0.102
0.103
0.098
0.099
0.094
0.096
0.091
0.092
0.088
0.089
0.085
0.086
0.083
0.084
0.082
0.083
iO
24.3
25.0
25.0
25.8
25.7
26.7
26.5
27.6
27.3
28.7
28.2
29.7
29.2
30.8
30.2
31.8
31.2
32.6
32.1
33.0
32.5
Q
a
RM
RCpb
R0
R0
R0
A
0.759
0.757
0.757
0.755
0.756
0.754
0.754
0.752
0.753
0.750
0.751
0.749
0.749
0.747
0.748
0.746
0.746
0.745
0.745
0.744
0.745
0.000
0.027
0.027
0.054
0.054
0.080
0.080
0.106
0.106
0.130
0.131
0.155
0.156
0.178
0.179
0.202
0.203
0.225
0.226
0.249
0.250
0.161
0.157
0.158
0.153
0.154
0.149
0.150
0.144
0.146
0.140
0.142
0.136
0.138
0.132
0.134
0.129
0.131
0.126
0.128
0.125
0.127
00
4264
1066
479
277
186
140
115
103
100
103
'We have taken Mw=0.44 M0 and Kw=31.2 km S-1 for all ofthese models.
bThe main-sequence radius based upon MM according to the relation of Caillault &
Patterson (1990).
©1996 RAS, MNRAS 278, 565-576
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.278..565M
A short-period, detached white dwarf/M dwarf binary
o
o
0.5
573
0.5
Orbital phase
Figure 7. The predicted light curves of the models for different filling factOrs f. The dashed lines represent the optically thick models, and the
dotted lines represent the optically thin models.
a
m
~
'I
r----,----~---,----~----r_--_r----._--~----,_--~----_.--_.
o
a
.-<
'"
a
L __ _- L_ _ _ _
o
~
__
~
____
0.2
~
_ _ _ _L __ _
0.4
~
____
~
0.6
__
~
_ _ _ _J __ _
~
_ _ _ _ _ L_ _
~
0.8
Orbital phase
Figure 8. The FWHM of the Ha emission are broader than expected (the dotted-line model). The dashed lines represent the predicted widths
for exposure times of 600 and 1200 s (the uppermost line) when a further blurring of FWHM = 90 km S-I is added. Circles, triangles, open
diamonds and stars represent 500-, 600-, 900- and 1200-s exposures.
flux, but how does it match the line profiles? The projected
equatorial rotation velocity of the M star is equal to
(Kw+ KM)RM/a, which is less than 66 km S-1 from Table 5.
Moreover, only half the star produces any emission, so we
expect widths of order "" 60 km s - 1. In the I band, our
resolution of "" 105 km S-1 FWHM is not high enough, but
the Ha data have a resolution of 40 km S-1 and should
resolve the profiles. That they do is clear from Fig. 1, since
the 40 km s -1 resolution is equivalent to 2 pixels and yet the
emission lines are several pixels in width. In fact, the lines are
much broader than we expect: when fitting the radial velocities we found a best-fitting FWHM of 2.22 ± 0.05 A for
H a, equivalent to 100 ± 2 km s - 1. We will show in this
section that, as our approximate calculation implies, the
geometrical model alone cannot fit the observed widths, and
instead an additional broadening mechanism must blur the
profiles by a further 90 km s - 1.
We first fitted the FWHM individually while keeping all
the fit parameters described earlier constant and forcing the
areas of the emission lines to match the fitted sinusoidal
variation. The results are plotted against phase in Fig. 8.
Given the model described in the previous section, it is
simple to calculate line profiles. In addition to the model
previously defined, we added 40 km s - 1 of instrumental
© 1996 RAS, MNRAS 278, 565-576
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.278..565M
574 T. R. Marsh and S. R. Duck
broadening and we calculated the effect of finite exposure
lengths by· trapezoidal averaging over the width of a bin.
Most of our exposures are 500 or 600 s long, but there were
three each of 900- and 1200-s duration taken before we
were aware of the short orbital period. Although the profiles
are not precisely Gaussian, they are close enough that their
FWHM should be comparable to the FWHM we derived
from the data. In any case, the model for 600-s exposures
and f= 1 (plotted as the dotted line in Fig. 8) is so far from
the observations that the precise definition of FWHM is a
detail. The FWHM of the model profiles vary markedly with
phase, since at quadrature only one side of the star has
emission and therefore the profile is narrow compared with
the conjunction phases. This effect is accentuated by the
rapid orbital motion at the conjunction phases which smears
the profiles still more. The data show signs of such a variation, but the observed FWHM are always larger than predicted.
Since the geometrical model seems otherwise well
founded, and because there is little room for manoeuvre (the
f= 1 model gives the broadest possible lines), we conclude
that there must be a broadening mechanism intrinsic to the
line-emitting region. The upper dashed lines in Fig. 8 represent models with 90 km s - I of extra broadening added for
exposure times of 600 and 1200 s. Interestingly, the width of
the Ca II lines, at 3.21 ± 0.08 A or 113 ± 3 km s - I, is comparable to the resolution of '" 1 05 km S-I. This suggests that
whatever broadens the Ha emission does not affect Ca II in
the same way. This rules out kinematic explanations of the
broadening, such as turbulence or flares, in favour of
pressure or thermal broadening. If thermal broadening
provides the solution, the emission must come from material
at 180 000 K. High-resolution observations of the Ca II
emission would provide a very useful additional constraint.
4.4
Production of the emission-line flux on the M star
The distribution of emission over the M star is strongly
suggestive of irradiation, and yet the effective temperature of
the white dwarf (19 000 K from Bergeron et al. 1992) is not
very high. For photoionization leading to optically thin Ha
emission and for a specific geometry, one number is needed
to predict the equivalent width. This number is the product
of (i) the fraction of recombinations which lead to an Ha
photon and (ii) the ratio of the number of ionizing photon S-I
emitted to the photon density in photon s - 1 A-I at the
wavelength of Ha. Since the white dwarf's photosphere
dominates at Ha, and assuming that it does so shortward of
911 A, the second quantity is a function of the shape of its
spectrum only. In Table 5 we list the value that this product
(denoted by Q) must have in order for our models to match
the peak equivalent width ofHa of 1 A (for the optically thin
models only). The value of Q, which can be thOUght of as the
equivalent width that would be observed if the M star intercepted all the radiation from the white dwarf, is given
approximately by 4:n:j(xQ ) times the peak equivalent width,
where Q is the solid angle subtended by the M star at the
white dwarf, and x is the fraction of the irradiated surface
visible to us at phase 0.5. The fraction visible does not vary
very much, and so Q is dominated by the solid angle factor.
A blackbody spectrum with T = 19 000 K matching the
temperature fitted by Bergeron et al. (1992) gives
Q = 200 A; accounting for 0.3 Ha photons per recombination for case B (Hummer & Storey 1987), this drops to 60 A,
which is not enough to produce the observed emission. The
discrepancy is far worse for a stellar atmosphere, since at this
effective temperature a stellar atmosphere produces far
fewer photons short of the Lyman limit than does a blackbody. For example, a Te = 20 000 K, log g = 7 atmosphere
(Wesemael et al. 1980) and a recombination factor of 0.3
gives Q '" 0.2 A, which is 500 times too small to match the
observed equivalent width.
Rapidly rotating M dwarfs often show chromo spheric Ha
emission. Accounting for dilution by the white dwarf, the
equivalent width of the Ha emission with respect to the M
dwarf is about 50 A. This is large compared with most active
M dwarfs, which show equivalent widths of order 5 A, but
there is one known with an equivalent width of over 200 A
(PC 0025 +0447, Sclmeider et al. 1991). A better means of
comparison is through the ratio of Ha flux, L Ha , to the
bolometric luminosity, L bol' We estimate log L Haj L bol =
- 3.2 for GD 448 (see Section 4.6 for a discussion of the
luminosity of the M star). Consistent with the comparison of
equivalent widths, this value is large compared with most
active stars, although it is not the largest value known. Therefore a purely chromo spheric origin is at least possible and,
given the extremely rapid rotation of the M star, perhaps not
surprising. However, to accept this requires dismissing the
phasing of the light curve of emission-line flux as a coincidence. We consider this to be very unlikely, especially as
there are two other white dwarfjM dwarf binary stars which
share the problem of too cool a white dwarf and Ha
emission which peaks at phase 0.5 (PG 1026 + 002, Saffer et
al. 1993; GD245, Schmidt et al. 1995). Instead, it may be
that the combination of a chromosphere and hot photospheric irradiation can produce more emission than either
can alone.
Schmidt et al. (1995) suggest that photoionization from
the n = 2 level could provide the solution, and perhaps it also
requires some chromo spheric heating to maintain a sufficient
population of hydrogen atoms in this state.
4.5
Is GD 448 crossing or was it born in the period gap?
The orbital period of GD 448 is in the middle of the period
gap of cataclysmic variables, and it appears to have many of
the hallmarks of the detached systems that cataclysmic
variables are thought to become while traversing the gap.
However, we will now show that the time-scales involved
make it very unlikely that GD 448 can have become
detached at a period of 3 h, and we believe instead that it
must have emerged from the common-envelope phase close
to its present orbital period.
The temperature of the white dwarf of 19 000 K gives a
cooling age of 5 x 10 7 yr (from cooling models kindly
supplied by Wood). Gravitational radiation takes almost 40
times longer - 1.8 x 10 9 yr - to change the orbital period of
GD 448 from 3 to 2.47 h. This problem cannot be avoided
by invoking an additional braking mechanism, because the
time-scale on which the M star can shrink is of order its
thermal time-scale,
tKH=3 x 10 7 (MjM0)2(RjR0t'
( L jL0 1 yr. Taking bolometric magnitudes from Bessell
(1991) and using Caillault & Patterson's mass-radius relation, we ffurl a thermal time-scale of about 3 x 109 yr for a
t
©1996 RAS, MNRAS 278, 565-576
© Royal Astronomical Society • Provided by the NASA Astrophysics Data System
1996MNRAS.278..565M
A short-period, detached white dwarf/M dwarf binary
0.1-M0 M dwarf. Therefore the orbital period cannot be
changed enough in the time available (5 x 10 7 yr) without the
Roche lobe shrinking faster than the M dwarf and therefore
keeping the system in contact. Although we have used carbon-core models to deduce the cooling age, it seems unlikely
that helium-core models will extend the cooling age by the
factor of 60 needed to match the M dwarf's thermal timescale.
GD 448 must start to transfer mass in the future. As the
mass ratio will remain constant until contact, the size of the
Roche lobe of the M star scales with orbital period as p2/3.
Therefore mass transfer will occur at a period of 2.5 j3/2 h.
On Caillault & Patterson's (1990) mass-radius relation,
f = 0.53, and mass transfer will occur at P ,.. 1 h, taking some
2.5 x 109 yr to get to this stage. Thus GD 448 may violate the
80-min orbital period minimum (Paczynski & Sienkiewicz
1981) by virtue of not transferring mass until a late stage,
rather in the way that the systems transferring mass in the
period gap may have been born directly into it. Our discovery of GD 448 in a relatively small sample of objects
suggests that similar systems may be common. However,
once it does start to transfer mass, GD 448 will not do so at a
particularly high rate and it may not be very obvious. In fact,
the parameters of GD 448 are such that its future evolution
is a little unusual for a CV, as it is launched immediately on to
a course that most CVs only reach at the end of their careers.
If the radius of the M star scales as R ex: M~, then one can
show that (e.g. King 1988)
ill
For low-mass main-sequence stars a ,.. 1; however, when, as
is the case here, the mass-loss time-scale is shorter than the
thermal time-scale of the M star, the latter responds adiabatically and expands with a approaching -1/3. Substituting
i
32 G 3 MWMM(Mw+MM)
a4
-J=57
where a is the separation of the two stars (Landau & Lifshiftz
1958), we find that the initial mass transfer rate lies between
10 -10 and 2 x 10 -10 M0 yr-l, with the higher rate corresponding to the adiabatic case.
The rates estimated above give mass-transfer time-scales
- MMIMM of order 10 9 yr, less than the M star's thermal
time-scale, as remarked above. More detailed calculations
are needed to establish whether the mass transfer time-scale
is short enough to make the M star expand immediately, but
eventually it must expand as its thermal time-scale will
increase still more as it loses mass. It will not be long before
the M star ceases to burn hydrogen at all, and it will start to
become degenerate. Degenerate stars follow the same
Rex: M- 1/ 3 relation that adiabatic fully convective stars track.
Therefore the orbital period will start to increase as, or very
soon after, mass transfer starts, and the system will fade into
obscurity along with other short-period CVs.
15 ± 2 per cent of the light at 8200 A and is negligible in the
visual band. The M dwarf is 0.15 mag brighter in / compared
to the white dwarf than it is at 8200 A (Marsh et al. 1995),
and Bergeron et al. (1992) list an absolute visual magnitude of 10.15. Finally, from Greenstein (1984), v-/=
- 0.7 ± 0.05 for white dwarfs of similar effective temperature
to GD 448, and therefore we obtain an absolute magnitude
at / of 12.76 ± 0.15 for the M dwarf.
Zuckerman & Becklin (1992) find M K =9.80±0.15 for
the M star in GD 448, and thus / - K = 2.96 ± 0.2. Old disc
M dwarfs with / - K = 2.96 ± 0.2 have an absolute / mag of
12.0~8:~ according to Bessell (1991), and therefore within the
uncertainties the M dwarf in GD 448 is not obviously different from main-sequence M dwarfs, although the main
sequence itself is not well known for such low masses.
4.7 Conclusions
We have found that GD 448 is a white dwarfJM dwarf binary
with a 2.47-h orbital period, which makes it the shortest
period example of such binaries found to date. We measure a
gravitational redshift of 16.8 ± 1.6 km s -I for the white
dwarf, which implies a mass of 0.44 ± 0.03 M 0 . Together
with the radial velocity semi-amplitudes, this fixes the masses
and inclination of the system. However, the semi-amplitude
of the M star comes from emission lines which are concentrated upon the side which faces the white dwarf, and it is
therefore an underestimate of the true K M • The light curve of
the line emission is well fitted by irradiation models, but the
results depend upon the unknown degree of filling of the
Roche lobe of the M star, f. For likely values of f, the mass of
the M star lies in the range 0.08 to 0.10 M 0 . The geometrical
model predicts too narrow a width for the Ha line emission
from the M star, and intrinsic broadening of 90 km s - 1 must
be invoked to match the data. The Ca II triplet emission does
not show the same broadening, which suggests that thermal
or pressure effects are responsible.
The cooling age of the white dwarf is much shorter than
the time required for gravitational radiation to reduce the
orbital period from 3 h to its present value, which suggests
that GD 448 is not a cataclysmic variable star traversing the
period gap. It will start to transfer mass once gravitational
radiation has shrunk the orbit to a period of about 1 h, after
which the period will increase and the system will fade as the
M star becomes degenerate.
ACKNOWLEDGMENTS
We thank the staff of the La Palma observatory for their
support during the observations. TRM was supported by a
PPARC Advanced Fellowship during the course of this work.
SRD was supported by a PPARC post-doctoral grant. We
thank the referee, P. Bergeron and M. Wood for their help on
the mass of the white dwarf. The analysis was carried out on
the Starlink network.
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The nature ofthe M star
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