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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. REFERENCES 4.6 The nature ofthe M star Apart from the strong line emission, the M star appears to be little affected by the white dwarf. The M star contributes 575 Barstow M. A. et ai., 1995, MNRAS, 272, 531 Bergeron P., Saffer R. 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