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2 Mon. Not. R. Astron. Soc. 000, 1-16 (0000) Printed 2 February 2008 (MN KTe X style file v1.4) Six detached w hite-dw arf close binaries* L. M o r a l e s - R u e d a 1,2, T. R. M a r s h 3,2, P. F. L. M a x t e d 4,2, G. N e l e m a n s 5,1 C. K a r l 6 arxiv:astro-ph/ v! 21 Feb 2005 R. N a p i w o t z k i 7, C. K. J. M o r a n 2 1 D epartm ent of Astrophysics, University of Nijm egen, P.O. Box 9010, 6500 GL Nijm egen, The N etherlands (lmr@ astro.kun.nl) 2 D epartm ent of Physics and A stronom y, University of Southam pton, Southam pton SO 17 1BJ, UK 3 D epartm ent of Physics, University of Warwick, Coventry, CV4 7AL, UK ( T.R.M arsh@ warwick.ac.uk) 4School of Chem istry and Physics, Keele University, Staffordshire S T 5 5BG, UK astro.keele.ac.uk) 5 In stitute of Astronom y, M adingley Rd, University of Cambridge, Cambridge CB3 0HA, UK 6D r R em eis-sternw arte, A stronom isches In stitu t der U niversitat Erlangen-Nürnberg, Sternw arstrasse 7, Bamberg, G erm any 7 D epartm ent of Physics & Astronom y, University of Leicester, University Road, Leicester L E I 7RH, UK Accepted ; Received ; in original form A B ST R A C T We determine the orbits of four double degenerate systems (DDs), composed of two white dwarfs, and of two white d w arf- M dwarf binaries. The four DDs, WD , WD , WD , and WD , show orbital periods of (5) d, (2) d, (6) d and (3) d respectively. These periods combined with estimates for the masses of the brighter component, based on their effective temperatures, allow us to constrain the masses of the unseen companions. We estimate that the upper limit for the contribution of the unseen companions to the total luminosity in the four DDs ranges between 10 and 20 per cent. In the case of the two white dwarf - M dwarf binaries, W D and WD , we calculate the orbital parameters by fitting simultaneously the absorption line from the white dwarf and the emission core from the M-dwarf. Their orbital periods are (1) d and (2) d respectively. We find signatures of irradiation on the inner face of WD s companion. We calculate the masses of both components from the gravitational redshift and the mass-radius relationship for white dwarfs and find masses of M0 and M0 for W D and WD respectively. This indicates that the stars probably reached the asymptotic giant branch in their evolution before entering a common envelope phase. These two white dwarf - M dwarf binaries will become cataclysmic variables, although not within a Hubble time, with orbital periods below the period gap. Key words: binaries: close - binaries: spectroscopic - white dwarfs 1 IN T R O D U C T IO N A b o u t 10 per cent of w hite dw arfs reside in close binary system s (N apiw otzki et al. 2003). T he form ation of w hitedw arf close-binary system s com m ences w ith a p air of m ain * The Isaac Newton and William Herschel telescopes are operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. Based on observations collected at the Centro Astronómico Hispano Alemán (CAHA) at Calar Alto, operated jointly by the Max-Planck Institut für Astronomie and the Instituto de Astrofísica de Andalucía (CSIC). Based on observations collected with ESO telescopes at the Paranal Observatory under programme IDs 165.H and 167.D sequence stars orbiting in a wide binary. T he m ost m assive sta r will evolve faster becom ing a giant and transferring m ass to its com panion. If m ass tran sfer is sufficiently rapid, th e result will be th e form ation of a com m on envelope (CE) m ade of th e o u ter layers of th e giant. T he o rb ital energy of th e binary can th e n be used to eject th e envelope, th e result being a binary com posed of a w hite dw arf and a m ain sequence com panion. If th e m ass ratio of th e initial com pon ents was close to unity, th e bin ary form ed after C E ejection can still be wide, w hereas in th e case of extrem e m ass ra tios, th e tw o com ponents will spiral in to eject th e envelope form ing a tig h ter binary. T his binary, called a p o st com m on envelope binary (P C E B ), if close enough, m ight becom e a cataclysm ic variable (CV) if m ass is transferred stab ly from 0000 RAS

3 2 L. Morales-Rueda et al. th e m ain sequence sta r to th e w hite dwarf. If, on th e other han d, th e bin ary is too w ide to becom e a CV, th e m ain sequence sta r will evolve into a giant and th e system will undergo an o th er C E phase. W hen th e envelope is ejected, th e resulting bin ary will be com posed of two w hite dw arfs a few solar radii ap a rt - called a double degenerate (D D ). For details on th e evolution and form ation of D D s see N elem ans et al. s (2001) recent po p u latio n synthesis studies and references therein. T he stu d y of w hite-dw arf close-binary stars can help us u n d erstand th e elusive C E phase th a t th ey m ust have gone th ro u g h, a t least once, during th eir evolution. T his phase is very difficult to stu d y in any o th er m anner as it lasts only of th e order of 1 to 100 years. For recent calculations on C E ejection efficiencies see Soker & H arpaz (2003) and references therein. T he subjects of th is paper, w hite dw arf - M dw arf binaries and D D s, are, as m entioned above, th e progenitors of CVs, and p o ten tial progenitors of T ype Ia supernova respectively. A short period D D w here th e com bined m ass of th e w hite dw arfs exceeds th e C h an d rasek h ar m ass m ight becom e a T ype Ia supernova (Iben & T utukov 1984), b u t no candidates have yet been found. W e should m ention th a t th is idea is still q u ite controversial w ith som e th eo rists claim ing th a t a DD does not becom e a T ype Ia supernova (Saio & N om oto 1998) and others claim ing th a t th is can be th e case if one includes ro ta tio n in th e calculations (P iersanti et al. 2003). T he ESO Supernova Ia P rogenitor Survey (SPY ) has as its m ain goal to search system atically for m assive, sh o rt period D D s in th e galaxy to establish th eir direct link w ith T ype Ia supernova and it is th e source of m ore th a n 100 recently discovered D D system s (N apiw otzki et al. 2003). E f forts are also been carried out to system atically find w hite - M dw arf binaries by using th e Sloan D igital Sky Survey (SDSS) resulting in m ore th a n 400 possible candidates (Silvestri, Hawley & Szkody2003). Schreiber & G ansicke (2003) review th e evolution of th e know n sam ple of w hite-dw arf close binaries w ith m ain sequence com panions, th e P C E B s, and conclude th a t, due to selection effects, th e (until th en) know n populatio n of P C E B s is d om inated by young system s w ith hot w hite dw arfs th a t will evolve into short period C V s (P < 3 h). T he P C E B sam ple is biased tow ard low m ass com panions as well w hich is th e reason w hy th ey evolve into short period CVs. In co n trast, th e P C E B s found in th e SDSS (R aym ond et al. 2003), published later on, show an average w hite dw arf te m p eratu re significantly lower, d em o n stratin g th a t to sam ple th e full p aram eter space b e tte r selection criteria have to be devised. In th is p ap er we stu d y 6 w hite dw arf close-binary stars. Four of th em are found to be D D s, th e o ther tw o are com posed of a w hite dw arf and an M dw arf star. T he separation of b o th com ponents in these binaries is of th e order of a few solar radii, as determ ined from th eir sh o rt o rb ital p eriods, so th ey are detached system s w here no m ass transfer betw een th em takes place. We o b tain th eir o rb ital solutions and com pare th e results o b tain ed w ith predictions draw n from p o p ulation synthesis studies. T able 1. Journal of observations. A description of the setup used with each telescope is given in the text. N indicates the number of spectra taken with each setup. * indicates high resolution spectra covering only the emission core of Ha. The brightness in B magnitudes of each target is also given. Object N Date Setup Inst. WD /2-3/3/96 AAT RGO (B = 14.37) 3 19/3/97 AAT RGO /2/98 INTa IDS 2 5/6/98 AAT RGO 2 15/4/03 INTb IDS 2 17/4/03 INTb IDS WD /2/97 INTb IDS (B = 14.9) /2/98 INTa IDS 9 3-5/3/99 INTa IDS 8 23/7/00 W HT ISIS 2 23/1/03 W HT ISIS WD /6/95 INTa IDS (B = 14.00) 13 29/2-3/3/96 AAT RGO /3/97 AAT RGO /6/97 INTa IDS /5/00 VLT UVES 5 4/5-18/8/01 VLT UVES /10/01 INTa IDS /2/02 3.5m TWIN WD /6/93 W HT ISIS (B = 15.30) /8/93 W HT ISIS /6/95 W HT ISIS 1 6/11/97 INTa IDS 1 22/11/97 W HT ISIS 2 15/4/03 INTb IDS W D /2-3/3/96 AAT RGO (B = 13.05) /3/97 AAT RGO WD * 10/6/96 W HT UES (B = 15.00) /6/95 INTa IDS 6 11/7/98 W HT ISIS 8 14/7/98 W HT ISIS 7 6-7/10/98 W HT ISIS 2 OBSERVATIONS A N D R E D U C T IO N T he d a ta used in th is stu d y were tak en over m any years (since 1993) using five different telescopes and seven different setups. T able 1 gives a list of th e num ber of sp ectra taken for each ta rg e t a t each observing cam paign indicating also w hich telescope was used in each case. AAT: denotes d a ta tak en w ith th e R oyal G reenw ich O b servatory (R G O ) sp ectrograph a t th e 4 m A nglo-a ustralian Telescope (A AT). T he setup consisted of th e 82 cm cam era w ith th e R 1200R grating centred in H a. T he CCD used was an M IT-LL (3kx1k) in fast read o u t m ode. T his com bination gives a dispersion of 0.23 A pixel- 1. IN Ta: denotes d a ta tak en w ith th e In term ediate D ispersion S pectrograph (IDS) a t th e 2.5m Isaac N ew ton Telescope (IN T) on th e island of L a Palm a. For these d ata, th e setup consisted of th e 500 m m cam era w ith th e R 1200R grating centred in H a and th e Tek (1kx1k) chip. T his com bination results in a dispersion of 0.39 A pixel- 1. IN T b: denotes d a ta tak en also w ith th e IDS at th e IN T b u t w ith a setup th a t consisted of th e 235 m m cam era w ith th e R1200B g rating and th e E EV 10 (2kx4k) CCD. T he

4 Six detached white-dwarf close binaries 3 w avelength range covered in th is case included H y and H,3. T his com bination results in a dispersion of 0.48 A pixel- 1. W H T: denotes sp ectra tak en w ith th e 4.2m W illiam H erschel Telescope (W H T ) on L a P alm a. M ost of th e spectr a were obtained using th e double arm sp ectrograph ISIS. O nly th e red sp ectra were used in our study. E xcept for th e sp ectra tak en in Jan u ary 2003, th e setu p for th e red arm consisted of th e 500 m m cam era w ith th e R 1200R grating and a Tek C CD (1kx1k) giving a dispersion of 0.40 A pixel- 1. For th e sp ectra of W D tak en in Jan u ary 2003, a M A R C O N I CCD (2kx4.7k) was used giving a dispersion of 0.23 A pixel- 1. In th e case of th e 4 high resolution spectr a tak en of W D (m arked w ith a sta r in T able 1), th e U trecht Echelle sp ectrograph (UES) was used w ith th e 35 cross disperser and a Tek (1kx1k) C CD giving a typical dispersion of 0.07 A pixel- 1. VLT: denotes d a ta tak en w ith th e U V -V isual Echelle S pectrograph (U V ES) a t th e U T 2 (K ueyen) V LT 8.2 m telescope located a t P a ra n a l O bservatory. T he setup used consisted of Dichroic 1 (central w avelengths 3900 A and 5640 A) w ith a E E V C CD (2kx4k) for th e blue arm and tw o CCD s, a E E V (2kx4k) and a M IT-LL (2kx4k) for th e red arm. T his setu p allows us to achieve alm ost com plete sp ectral coverage from 3200 A to 6650A w ith only tw o ~ 8 0 A wide gaps a t 4580 A and 5640 A. A slit w idth of 2.1 was used to m inim ise slit losses and th e C CD s were binned 2 x 2 to reduce read o u t noise. T his setup results in a sp ectral resolution of 0.36 A (or b e tte r if th e seeing disk is sm aller th a n th e slit w idth) a t H a. E xposure tim es were 5 min. 3.5 m: denotes d a ta taken w ith th e double beam T W IN sp ectro g rap h a t th e 3.5 m telescope in th e C alar A lto O b servatory. O nly th e red sp ectra were used in th is paper. T he setu p for th e red arm consisted of th e 230 m m cam era w ith th e T06 grating (1200 grooves m m - 1 ) and a SITe-CCD (2kx0.8k) giving a dispersion of 0.55 A pixel- 1. T he slit w id th was set to values betw een 0.8 and 1 arcsec depending on th e seeing. W e m ade sure in every case th a t th e sta r filled th e slit to avoid system atic errors in th e radial velocities caused by th e sta r w andering in th e slit during an exposure. For th e AAT, IN T, and W H T runs we obtained C ua r plus C un e fram es to calibrate th e sp e ctra in w avelength. In th e case of th e 3.5 m and V LT observations th e wavelength calibration arc used was a T ha r. A ll ta rg e t sp ectra were bracketed by arc sp ectra tak en w ithin one hour and th e w avelength scale interp o lated to th e tim e of m id-exposure. W e su b tra c te d from each im age a co n stan t bias level d e te r m ined from th e m ean value in its over-scan region. T ungsten flatfield fram es were obtained each night to correct for th e pixel to pixel response variations of th e chip. Sky flatfields were also o b tained to correct for th e pixel to pixel variations of th e chip along th e slit. A fter debiasing and flatfielding th e fram es, sp ectral ex tractio n proceeded according to th e optim al algorithm of M arsh (1989). T he arcs were ex tracted using th e profile associated w ith th eir corresponding targ et to avoid system atic errors caused by th e sp e ctra being tilted. U ncertainties on every point were p ropag ated th ro u g h every stage of th e d a ta reduction. W e did not a tte m p t to correct for light losses in th e slit. For th e V LT sp ectra a special procedure was applied to correct for a quasi-periodic rip ple p a tte rn appearing in m any of th e uncorrected m erged sp ectra (N apiw otzki et al. 2005). T he resulting V LT sp ectra were th e n divided by a sm oothed sp ectru m of a D C w hite dw arf, w hich by definition shows no spectral features a t all and therefore provides an excellent m eans of correcting for th e in stru m en tal response. 3 RESULTS 3.1 A verage spectra Fig. 1 presents th e average red sp ectra for th e six system s discussed in th is paper. T he sp ectra of four of th e system s, W D , W D , W D and W D , are very sim ilar showing only very broad ab sorption a t H a. W D and W D , on th e o th er hand, show a double line stru c tu re in th eir H a line profile com posed of broad absorption com ing from th e w hite dw arf and narrow em ission from th e heated surface of th e M dw arf com panion. As th is narrow em ission moves w ithin th e absorption profile w ith th e o rb ital period, an average sp ectru m w ould show th e em ission broadened and for th a t reason for these tw o system s we present th e sp ectru m at a p articu lar o rb ital phase instead of an average of th e spectr a during an orbit. T he narrow em ission in W D is, on average, significantly stronger th a n for W D For W D , th e stren g th of th e narrow em ission core changes w ith o rb ital phase and only a t o rb ital phase 0.5 (w hen we are looking directly into th e heated face of th e M dw arf) reaches sim ilar stre n g th to th a t in W D See Section for details. 3.2 Four double degenerate system s To m easure th e radial velocities of th e four double degenerates: W D , W D , W D and W D , we used least squares fitting of a m odel line profile. T he m odel line profile is th e sum m ation of three G aussian profiles w ith different w idths and depths. For any given star, th e w idths and d ep th s of th e G aussians are optim ised and th e n held fixed while th eir velocity offsets from th e rest w avelengths of th e lines in question are fitted separately for each spectrum ; see M axted, M arsh & M oran (2000c) for fu rth er details of th is procedure. O nce th e radial velocities for each system were know n (see T able 2) we used a floating m ean periodogram to determ ine th e periods of our targ ets (e.g. C um m ing, M arcy & B u tler 1999). T he m eth o d consists in fitting th e d a ta w ith a m odel com posed of a sinusoid plus a co n stan t of th e form: Y + K s in ( 2 n f (t - to)), w here f is th e frequency and t is th e observation tim e. T he key point is th a t th e system ic velocity is fitted at th e sam e tim e as K and t 0. T his corrects a failing of th e well-known Lom b-scargle (Lom b 1976; Scargle 1982) periodogram w hich s ta rts by su b tractin g th e m ean of th e d a ta and th e n fits a plain sinusoid; th is is not th e b est approach for sm all num bers of points. W e obtained th e x 2 of th e fit as a function of f and th e n identified m inim a in th is function. Table 3 gives a list of th e o rb ital p aram eters derived for each D D binary star. T he o rb ital period of th e second best alias is also given, along w ith th e difference in x 2 betw een

5 4 L. Morales-Rueda et al. T able 2. Radial velocities for the 5 systems. In the case of the 2 white dwarf-m dwarf systems the values given are measured from the M dwarf RV (km s- 1 ) HJD RV (km s- 1 ) HJD RV (km s- 1 ) WD WD WD ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±1.60 WD ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±0.54 WD ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±2.44 WD ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±1.73 WD ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 3.47 WD ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±2.76

6 Six detached white-dwarf close binaries Wavelength (A) F ig u re 1. Average spectra for the six systems studied in this paper. T able 2. Continued. HJD RV (km s- 1 ) HJD RV (km s- 1 ) HJD RV (km s- 1 ) WD ± ± ± ± ± ± ± ± ±3.58 T able 3. List of the orbital periods measured for the four double degenerate systems studied. To, the systemic velocity, y, the radial velocity semi-amplitude, K, the reduced x 2 achieved for the best alias, the 2nd best alias and the x 2 difference between the 1st and 2nd aliases are also presented. When calculating the x 2 for both aliases we have added in quadrature a systematic error th at results in a reduced x 2 ~1 (see text for details). The number of data points used to calculate the orbital period is given in the final column under N. Object HJD (To) Period (d) Y (km/s) K (km/s) xleduced 2nd best alias (d) A x 2 N WD (5) (5) 39.05± ± (4) WD (1) (2) ± ± (1) WD (9) (5) 47.95± ± (5) WD (5) (3) 35.11± ± (6) 45 25

7 6 L. Morales-Rueda et al.. r'* \. ' \ ' \ ' ll I l( 1 iw"1) 1 N 1 l 1 ill ill H i ill ill H i - - K ' *\ / *\ -,*».. / X / X. \ / \ i f V / X /. \ / \ >. / \ / ". \ / \ / ' \ \* */ \ x. v *.»3/* v _ A \. s < \.. * W D ' W D W D W D Orbital phase Orbital phase Orbital phase Orbital phase F ig u re 2. Radial velocity curves for the four double degenerate systems. The data have been folded on the orbital period in each case. See Table 3 for the list of periods, radial velocity semiamplitudes and systemic velocities. Included in each panel is a plot of the residuals to the fit. The vertical scale on which the residuals have been plotted is twice the scale on which the radial velocities are plotted " x s o n n = 25 1 p n = n = n = W D W D : W D W D Frequency (cycles/d) Frequency (cycles/d) Frequency (cycles/d) Frequency (cycles/d) F ig u re 3. Each panel presents x 2 versus cycles/day obtained after the period search was carried out. The frequency with the smallest x 2 corresponds to the orbital frequency of the system. For clarity we have also included an inset showing a blow up of the region where the best period is. The number of radial velocity measurements used for the period search calculations, n, is shown in each panel. th e tw o best periods found. T he large difference in x 2 indicates th a t th e second best aliases are n o t plausible. T he resulting radial velocity curves (folded in th e o rb ital period) are presented in Fig. 2 and th e corresponding periodogram s (x 2 versus o rb ital frequency) in Fig. 3. E ach panel in th e periodogram includes a blow up of th e region in frequency w here th e m inim um x 2 is found. In each case, we com pute th e level of system atic u n certain ty th a t w hen added in q u a d ra tu re to our raw error estim ates gives a reduced x 2 ~ 1. By doing th is we are considering th e un-accounted sources of error such as tru e variability of th e sta r or slit-filling errors th a t cause th e poor fits of a few stars. Such errors are unlikely to be correlated w ith either th e o rbit or w ith th e statistical errors we estim ate, and therefore we add a fixed q u an tity in q u ad ratu re w ith our statistical errors as opposed to applying a sim ple m ultiplicative scaling to them. In all cases we use a m inim um value of 2 k m s -1 corresponding to 1/ 10th of a pixel w hich we believe to be a fair estim ate of th e tru e lim its of our d ata. T he last colum n of T able 4 gives th e value of system atic u n certain ty th a t we have added in q u a d ra tu re in each case. We th e n calculate th e probability of th e tru e o rbital periods being fu rth er th a n 1 and 10 p er cent from th e values we obtained - see M orales-r ueda et al. (2003a) and M arsh, D hillon & D uck (1995) for an explanation of th e m ethod used to calculate these probabilities - and present th em in T able 4. In all cases, th e probabilities of th e quoted periods being w rong are very low and we are certain th a t th e values given in Table 3 correspond to th e tru e o rb ital solution. In th e cases w here th e probability of th e o rb ital period being fu rth er th a n 1 and 10 per cent from our favoured value is th e sam e, th e significant probability lies w ithin a very sm all range around th e b est period, w ith all th e significant

8 Six detached white-dwarf close binaries 7 T able 4. List of probabilities th at the true orbital period of a system lies further than 1 and 10 per cent from our favoured value given in Table 3. Numbers quoted are the logarithms in base 10 of the probabilities. Column number 4 gives the value of the systematic uncertainty th at has been added in quadrature to the raw error to give a reduced x 2 ~ 1. Object 1% 10% systematic error (km s- 1 ) WD WD WD WD T able 5. The mass functions, f m, of the unseen components together with the larger lower limits obtained by assuming i = 90 and by substituting in the mass function equation our determination of the mass of the brighter component, M 1. Object fm (M 0 ) M 1(M0 ) M 2(M0 ) lower limit WD WD WD WD com p etition (i.e. n ex t best alias) placed outside th e 10 per cent region around th e best alias The unseen component of the binary B y know ing th e radial velocity sem iam plitude of one of th e com ponents of th e binary (the observable com ponent), K, and th e o rb ital period of th e system, we can th e n calculate th e m ass function of th e unseen com ponent by using: fm = M 23 sin3 i P K 1 (M 1 + M 2)2 2nG w here th e subscripts 1 and 2 refer to th e b righter and th e dim m er com ponents respectively. T he m ass function is th e lower lim it of th e m ass of th e unseen com ponent. T a ble 5 gives th e m ass functions of th e unseen com ponents for th e four D D s studied. In tw o cases (W D and W D ) th e com panion s m ass function is greater th a n 0.1M 0 w hich corresponds to th e m ass of a late M dw arf if it is a m ain sequence star. T he m asses of th e b righter com ponents of th e system s have been m easured by fitting th eir hydrogen line profiles to stellar atm osphere m odels using th e track s by A lthaus & B envenuto (1997) and can be su b stitu te d, to gether w ith th e assum ption of th e o rb ital inclinatio n of th e system being 90, in th e m ass function equation to give a larger lower lim it for th e m asses of th e unseen com ponents. T hese revised lower lim its (also presented in T able 5) are all g reater th a n 0.1M 0 w hich indicates th a t th e unseen com panions cannot be m ain sequence stars b e cause if th ey were we should be able to d etect th em (M arsh et al. 1995). T he unseen com panions m ust therefore be also com pact objects, probably w hite dwarfs. We searched for th e signature of th e faint com panions by shifting out th e fitted radial velocity for each binary (1) and th e n looking for differences in th e line profiles a t th e q u a d ra tu re phases (M arsh et al. 1995), i.e and T he sp ectra a t q u a d ra tu re phases were o b tained by averaging th e sp ectra contained in two separate phase ranges, i.e. th e sp ectra in th e range from 0.1 to 0.4 were averaged to o b tain th e phase 0.25 spectrum, and th e sp ectra in th e range from 0.6 to 0.9 to o b tain th e phase 0.75 spectrum. T he results are p lo tted in Fig. 4. A ny co n trib u tio n from th e com panion w hite dw arf should be seen as an asym m etry in th e line profile a t phase 0.25 th a t is m irrored a t phase 0.75 w ith respect to th e rest w avelength (M arsh et al. 1995). N one of th e four system s show a clear asym m etry of th is ty p e in th e line profiles. A second test th a t can be carried o u t to look for th e faint com panion consists in shifting o u t th e fitted radial velocity off th e sp ectra and creating a m ean sp ectru m by com bining all th e shifted spectra, su b tractin g th is m ean spectru m from th e individual shifted ones and p lo ttin g th e resulting sp ectra in a stack or trail. T his aids th e eye to identify any leftover absorption m oving w ith th e binary orbit. A D oppler m ap (M arsh & H orne 1988) can also be com puted from these stack of spectra. A ny o rb ital m otion leftover in th e sp ectra w ould appear in th e m aps as a sm all absorption region located in th e Vx = 0 axis of th e velocity m ap. We do not find any indication of th e presence of th e unseen com ponent in either trails or D oppler m aps in any of th e four system s studied How fa in t is the unseen companion? We explore th e question of how sm all th e co n trib u tio n of th e faint com ponent of th e system has to be so as n o t to be detected using th e m ethods discussed in th e previous section. To answ er th is question we create synthetic sp ectra th a t include th e absorption corresponding to th e brig h ter com ponent of th e system (in th e form of th ree G aussians), scaled to th e m easured value for each system, plus som e ex tra ab sorption m oving opposite to it (also represented by three G aussians) and determ ine for w hat percentage of brightness, relative to th e brig h t com ponent, we should be able to detect th e faint com ponent by looking at th e sp ectra around th e q u a d ra tu re phases (M arsh et al. 1995). T his m eth o d assum es th a t th e com panion stars have a sp ectru m sim ilar to th a t of th e b righter com ponent. A lthough th e fainter w hite dw arf is cooler and its H a line will be less deep in its spectrum, this seem s like a reasonable assum ption as th e com panions are also w hite dw arfs, unless of course th ey are n o t DA w hite dwarfs. In th e case of W D we find th a t th e b rig h t ness of th e com panion m ust be less th a n 10 per cent th e brightness of th e brig h t com ponent for us n o t to d etect it. T he values we find for W D , W D , and W D are 9 p er cent, 23 per cent and 17 p er cent respectively. T hese values tra n sla te into m agnitudes for th e com panions, in th e A region, th a t are respectively 2.5, 2.6, 1.6 and 1.9 fainter th a n th a t of th e brig h t com ponents. U sing th e calculated absolute V m agnitudes for th e bright com ponents (10.68, 10.35, and 10.24, see th e D iscussion Section) and th e b ro ad b an d colour indices for pure hydrogen and log g = 8 stellar atm osphere m odels com puted by B ergeron, W esem ael & B eaucham p (1995), we

9 8 L. Morales-Rueda et al. >> CO CD X 3 X Z >> CO QJ T Velocity (k m / s ) Velocity (k m / s ) F ig u re 4. The spectra averaged around the quadrature phases for the four systems studied. In each case, the lower spectrum corresponds to quadrature phase 0.25 and the top one to phase There is no clear asymmetry in the line profile at phase 0.25 th at gets mirrored in phase 0.75 for any of the systems. This indicates th at we cannot detect the faint companion of the systems. o b tain u p p er lim its for th e absolute R m agnitudes of th e faint com ponents of 13.3, 13.0, 12.5 and 12.2 respectively. 3.3 Two w hite dw arf/m dwarf binaries In th e case of W D (aka B P M 6502) and W D th e sp ectra are com posed of absorption lines th a t have th eir origin in th e w hite dw arf plus em ission cores th a t have th eir origin in th e M dw arf. T his e x tra em ission com ponent m akes th e m easuring of th e radial velocities m ore com plicated, as th e absorption com ing from th e w hite dw arf has its core filled by th e M dw arf em ission. A way to m easure sim ultaneously th e radial velocities of b o th com ponents is to use least squares fitting of a m odel line profile as in th e previous case b u t th is tim e using a m odel line profile th a t is th e sum of four G aussian profiles. T hree of th e G aussians fit th e absorption com ponent and one fits th e emission. T he steps followed to carry out these com plex fits consisted of: 1) fitting only th e em ission lines w ith a single G aussian function and obtaining th e radial velocities associated to th e em ission line for each spectrum, 2) calculating th e o rbital solution for th e em ission lines by m eans of obtaining a periodogram from th e radial velocities m easured, 3) using this o rb ital solution to fix th e o rbit of th e th ree G aussians th a t will fit th e absorption com ponent of th e lines and 4) o b tain ing th e radial velocity sem iam plitude and system ic velocity for th e w hite dw arf by fitting all th e sp ectra sim ultaneously w ith one em ission and th ree absorption G aussians. T his m eth o d was easily applicable to W D as th e em ission com ing from th e M dw arf is com parable to th e H a absorption core (see Fig. 1). In th e case of W D th e em ission com ponent is very strong com pared w ith th e absorption core m aking th e fitting of th e d a ta m ore difficult and th e results o b tained less accurate. For W D we find 7 very close aliases w ith very sim ilar values of x 2. T a ble 6 gives a list of th e o rb ital solutions for th e 7 aliases. T he solutions for K 2, y 2, K i and 71 are consistent w ithin th e errors for all aliases. Previous studies of W D (K awka et al. 2000) result in an orb it solution in w hich th e o rb ital period is consistent w ith our alias num ber 3 and th e values for K 2, y 2, K 1 and y 1 are consistent w ith those from our 7 aliases. In T able 7 we present th e orb ital solutions for W D and W D T he results presented for W D correspond to th e first alias show n in T able 6. In Fig. 5 we plot th e radial velocities m easured for th e em ission and absorption com ponents to g eth er w ith th e b est fits

10 Six detached white-dwarf close binaries 9 Orbital phase F ig u re 5. Orbital solution for W D and WD Included in each panel is a plot of the residuals to the fit to the emission line component. The vertical scale on which the residuals have been plotted is larger than the scale on which the radial velocities are plotted. T able 6. O rbital solution for the 7 aliases found for W D P (d) To Y2 K 2 x 2 A2red T able 7. List of the orbital periods measured for the two white dwarf-m dwarf systems studied. To, the systemic velocity, y, the radial velocity semi-amplitude, K, for both the white dwarf and the M dwarf, and the reduced x 2 achieved for the best alias are given The number of data points used to calculate the orbital period is also given (1) (7) 7.93± ± (1) (7) 7.83± ± (1) (7) 8.03± ± (1) (7) 7.73± ± (1) (7) 8.14± ± (1) (7) 7.64± ± (1) (7) 8.26± ± N p (d) T o (d) - K w d (k m s 1) YWD (k m s- 1 ) Km (k m s- 1 ) YM (k m s- 1 ) x 2r ed q= M M /M w d W D WD / (1) (2) (7) (1) ± ± ± ± ± ± ± ± ± ±0.017 given in T able 7. T he error bars in th e radial velocities are sm aller th a n th e size of th e sym bols used to plot th e d ata. N otice th a t th ere are 4 ex tra points in th e fit to th e radial velocity of th e M dw arf for W D T his accounts for th e 4 high resolution sp ectra tak en w ith U ES th a t only covered th e core of th e H a line The M dw arf companions We binned th e sp ectra for b o th binary system s into 10 phase bins and p lo tted th em in Fig. 6 as a stack of sp ectra w ith o rb ital phase in th e vertical axis. T he advantage of p resen t ing th e sp ectra in th is way is th a t we can explore th e line flux variations during an orbit. In th e case of W D

11 1 0 L. M o r a le s - R u e d a e t al. WD Ha WD Ha Velocity (k m /s) Velocity (k m /s) F ig u re 6. Trailed phase binned spectra for W D and WD The orbit is plotted twice. The variable brightness in the emission line component, coming from the M dwarf companion, is clear in WD WD i \ + - -t + * " * - i : Orbital phase Orbital phase W D < * O rb ita l p h a s e O rb ita l p h a s e O rb ita l p h a s e O rb ital p h a s e F ig u re 7. Flux modulation seen on the emission from the M dwarf component ofw D (top) and WD (bottom). th e stren g th of th e em ission line does not vary significantly w ith o rb ital phase. T his is n o t th e case for W D w here th e line flux increases a t o rb ital phases around 0.5. T his phase corresponds to th e w hite dw arf and th e M dw arf being aligned w ith th e line of sight, th e w hite dw arf being closer to us. A t th is phase we are looking to th e h eated face of th e M dwarf. To d eterm ine how th e line flux varies w ith phase, we fitted th e line profiles once m ore w ith four G aussians, th ree F ig u re 8. Flux modulation seen on the emission from the M dwarf component ofw D (top) and WD (bottom). This tim e the data has not been folded in the orbital period. for th e absorption com ponent and one for th e em ission com ponent b u t th is tim e we allowed th e height of th e em ission com ponent to vary. T he results are displayed in Fig. 7. T he variation in flux is very significant in W D and as we m entioned earlier it can be explained w ith em ission com ing from th e irrad iated face of th e M dw arf, th e side facing th e w hite dw arf. T his has im p o rtan t consequences for th e calculation of th e m ass of th e w hite dw arf carried o u t in th e following sections as we have to account for distortions on th e M dw arf to correct its radial velocity sem iam plitude. W hen th e com panion is heavily irrad ia ted th e em ission line

12 Six detached white-dwarf close binaries 11 used to m easure its radial velocity will give us th e value associated w ith th e irradia ted face, not th e value for th e centre of m ass of th e com panion. M axted et al. (1998) find th a t p ro b ably as a result of optical d e p th effects, B alm er em ission lines induced by irrad iatio n are significantly broadened ren dering m easurem ents of radial velocities from fitting these lines m ore inaccurate. In th e case of W D , th ere is also som e flux m odulation present a t a fainter level b u t peaking a t phase 0.6 ra th e r th a n phase 0.5. T his is probably associated to chrom ospheric activity of th e M dw arf ra th e r th a n irrad iation. To m ake sure th a t th is variability is not o rb ital we have also p lo tted th e flux of th e em ission com ponent versus u n folded o rb ital phase in Fig. 8. W ee see th a t for W D th e variability observed is not phase dep en d en t im plying th a t is intrinsic to th e M dw arf, probably chrom ospheric. In th e case of W D th e variability observed is larger and it peaks a t phase 0.5 confirm ing th a t it is due to irrad i ation of th e inner face of th e M dw arf by th e w hite dwarf The masses o f the components O nce th e radial velocity sem iam plitudes for b o th com ponents have been m easured, together w ith th e o rb ital period we can use Eq. 1 to o b tain th e m ass function for th e w hite dw arf and th e M dw arf in each case. If we com bine Eq. 1 w ith q = M m /M w d = K w d / K m we o b tain larger lower lim its for th e m asses of b o th com ponents: M w d = P K m (K w d + K m )2 27tG sin3 i M m = P K w d (K w d + K m ) 2 _ (2) 2tvG sin i V ; T he actu al m ass of th e w hite dw arf can be determ ined from th e g rav itatio n al redshift and th e m ass-radius relationship for w hite dw arfs (A lthaus & B envenuto 1997). F irst we calculate th e difference in system ic velocities for b o th system com ponents (yw d y m ). We m ust th e n add corrections for (i) th e redshift of th e M dw arf, G M m / R m c, w here th e radius of th e M dw arf has been calculated by using th e m assradius relationship given by C aillault & P a tte rso n (1990): log R /R q = log M /M q 0.037, (ii) th e difference in transverse D oppler shifts of b o th com ponents: (K m K W d ) / 2c sin 2 i, (iii) th e p o ten tial a t th e M dw arf produced by th e w hite dwarf: G M w d / ac, (iv) and th e p o ten tial a t th e w hite dw arf due to th e M dwarf: G M m / ac, w here a is th e distance betw een b o th stars. T he m ass of th e w hite dw arf is th e n calculated by com paring th e resu lting g rav itatio n al redshift w ith m odels for low m ass helium w hite dw arfs (A lthaus & B envenuto 1997; B envenuto & A l th a u s 1998). T he inclination of th e system can th e n be calculated by using Eq. 2. In order to calculate th e corrections to th e w hite d w arf s T able 8. Summary of the param eters for W D and WD BA indicates Benvenuto & Althaus (1998). See text for explanations on the different values given. Param eter white dwarf M dwarf W D fm (Mq ) (2) (1) M(Mq ) from Eq (9) (4) M(Mq ) from q 0.75(5)/0.78(5) (5)/0.1735(5) M(Mq ) from BA 0.75(7)/0.78(7) m (Mq ) if CO WD 0.72 i ( ) 16 WD fm ( M Q ) 0.233(4) (5) M(Mq ) from Eq (3) 0.12(2) M(Mq ) from q 0.61(3)/0.64(3) (5)/0.1925(5) M(Mq ) from BA 0.61(3)/0.64(3) m (m q ) if CO WD 0.59 i( ) 60/59 g rav itatio n al redshift we h ad to assum e a value for th e m ass of th e M dw arf. W e perform ed several iteratio n s of these calculations u n til we o b tained consistent values for all th e param eters. T his m ethod has been used previously (M arsh & D uck 1996; M axted et al. 1998) to o b tain th e m asses of b o th com ponents in pre-c V system s. T he results obtained after these iteratio n s are presented in T able 8. We chose as th e b est estim ates for th e corrections described above and th e final p aram eters of th e iterations, those th a t resulted in a m ass for th e w hite dw arf th a t was closer to th a t calculated by B envenuto & A lthaus (1998). A m inim um value for th e m asses of th e com ponents was found w hen we assum ed a helium core w hite dw arf, w ith m etallicity Z = and an o u ter hydrogen envelope of fractional m ass (i.e. m ass of en v elo p e /to tal m ass of th e star) We assum ed a T eff = K and K for W D and W D respectively (B ragaglia, R enzini & B ergeron 1995; B ergeron, Saffer & L iebert. 1992). A m axim um value was found if instead we assum ed th a t th e fractional m ass of th e o u ter hydrogen envelope was 4 x 10-4 in th e case of W D and 2 x in th e case of W D T he m inim um and m axim um values found for th e m asses are given in T able 8. For W D , j WD j M is 36.16±2.72 k m s - 1. Afte r iteratin g for different values for th e m ass of th e M dw arf we find th a t if M m is M q th e corrections i, ii, iii and iv are respectively 0.48, 0.09, 0.24 and 0.05 k m s - 1, giving a value for th e w hite dw arf redshift of k m s - 1. If, on th e o th er han d, M m is M q th e corrections are respectively 0.48, 0.10, 0.25, 0.05 k m s - 1, giving a value for th e w hite dw arf redshift of k m s - 1. For W D , j WD j M is ± k m s- 1. For a m ass for th e M m = M q or betw een M q, th e values for i, ii, iii, and iv are 0.49, 0.04, 0.12 and 0.04 k m s - 1, giving k m s -1 for th e w hite dw arf redshift. If a carbon-oxygen core w hite dw arf w ith m etallicity Z = 0 and a hydrogen envelope of fractional m ass 10-4 is assum ed instead, th e values o b tained for th e m asses of th e w hite dw arf are 0.72 and 0.59 M q for W D and W D respectively.

13 12 L. Morales-Rueda et al. T able 9. New values for the M dwarf param eters depending on its filling factor ƒ for WD R m cp is the radius calculated from the equation of Caillault & Patterson (1990) given in Section A range of values for several param eters are given. These are the result of assuming two different masses for the outer hydrogen envelope of the white dwarf. See text for details. f k m q m m i a r m r m cp km s 1 M O R R R T he m asses o b tained for b o th w hite dw arfs are u n usually high indicating th a t th e sta r probably reached th e asym ptotic giant bran ch (AGB) in its evolution. T he initial binaries m ust have been very w ide in order for th is to happen. W e notice th a t th e m asses calculated here differ significantly from those given in Section. 4, m easured by fitting th e line profiles to stellar atm osphere m odels. T his discrepancy suggests th a t th e redshift m easurem ents m ay n o t be reliable, p erh ap s not surprising given th e difficulty of separatin g th e M sta r em ission from th e w hite dw arf absorption. M easurem ents of th e w hite dw arf a t U V w avelengths w ould be helpful. As m entioned in Section 3.3.1, th e flux m odulation seen in Fig. 7 for W D indicates th a t th e M dw arf com panion is strongly irrad ia ted and therefore th e value m easured for K M is probably a lower lim it for th e tru e radial velocity sem iam plitude. T his im plies th a t th e M dw arf m ass and inclination given in T able 8 are u p p er and lower lim its respectively. To calculate how d isto rted th e M dw arf is, we calculate how m uch th e radial velocity sem iam plitude of th e com panion changes as a function of its radius and how its m ass and inclination are affected. W e present th e results in Table 9. T he radius of th e M dw arf is given in term s of a linear filling fraction, f, defined as th e ratio of th e stellar radius m easured from th e centre of m ass to th e inner L agrangian point. A value of f = 1 im plies th a t th e M dw arf fills its Roche lobe. For these calculations we have taken M w d = 0.61 and 0.64 M q (the m axim um and m inim um values calculated above) and K WD = k m s - 1. If th e M sta r does n o t deviate too far from th e m ain sequence we expect it to fill a t least 0.4 of its R oche lobe w hich tran slates into a tru e radial velocity sem iam plitude in th e range 168 < K M < 222 k m s - 1, and a m ass betw een < M m < M q The masses o f the M dwarf companions A n independent estim ate of th e m asses of th e M ty p e com panions can be done from th eir absolute infrared m agnitudes. A lthough th e w hite dw arfs dom inate th e flux in th e optical range in b o th system s, th ey produce only a m inor fraction of th e infrared lum inosity. T he distances of th e system s were com puted from th e p aram eters of th e w hite dw arfs given in Section 4. Since th e flux co n trib u tio n of th e M dw arfs in th e blue p a rt of th e spectra, used for th e m odel atm osphere fits, is very sm all, we do not expect system atic effects caused by spectral contam ination. J, H, and K m agnitudes were retrieved from th e 2MASS point source catalogue. We com puted th e w hite d w arf s contrib u tio n using th e colour calibration of B ergeron et al. (1995) and su b tracte d it from th e observed fluxes. C orrections are sm all for W D and do not exceed 25% (10%) for th e J (K) b an d flux of W D Finally, th e M dw arf m asses were com puted from th e calibration of H enry & M cc arthy (1993). R esults are listed in T a ble 10. O ur error estim ates include photom etric errors and distance uncertain ties resulting from th e sp ectral analysis of th e w hite dw arf. W e adopted N apiw otzki, G reen & Saffer s (1999) estim ates for th e external fit errors. For b o th system s, th e value obtained for th e m ass of th e M dw arf com panion from its infrared m agnitudes is in agreem ent w ith th a t obtained from th e sp ectral analysis in section In th e case of W D , th e revised m ass after taking into account heating by th e w hite dw arf is considered. 4 D ISC U SSIO N T able 11 gives a list of all th e detached w hite-dw arf binaries w ith know n o rbits know n up to now. M ost values have been tak en from R itte r & K olb (2003). T he 6 system s discussed in th is p ap er are also included. L iebert, B ergeron & H olberg (2005), B ergeron et al. (1992) and B ragaglia et al. (1995) o b tain th e tem p eratu res and gravities for th e system s studied in th is paper. T heir values, to g eth er w ith our determ in atio n for th e m asses of th e w hite dw arf, are given in Table. 12. Fig. 9 presents th e theoretical m ass versus o rb ital p e riod d istrib u tio n for D D s from th e populatio n m odel as described in N elem ans et al. (2004). T he four D D s discussed in th is p ap er are p lo tted over th e theoretical d istrib u tio n and fall right in th e expected range of values according to N elem ans et al. (2004). T he direct progenitors of th e double w hite dw arf b in a ries (giant plus w hite dw arf) could have h ad a w ide variety of m asses and periods, leading to inferred efficiencies of th e C E th a t are poorly constrained (see N elem ans & T out 2004, fig. 5). We calculated th e possible progenitor system s of th e

14 Six detached white-dwarf close binaries 13 T able 10. Infrared properties and mass estimates for the M dwarf companions. dist (pc) Mj Mh Mk M /M q W D ± ± ± ± ±0.010 WD ±9 9.28± ± ± ±0.009 T able 11. List of all the detached white dwarf binaries with known orbital periods (given in days). The type of binary is also given where WD = white dwarf; M = M dwarf; sdo /sdb = O /B subdwarf;? = uncertain. * indicates periods measured in this paper. References for the orbital periods not measured in this paper are (a) Bragaglia, Greggio & Renzini 1990, (b) Koen, Orosz & Wade 1998, (c) Orosz & Wade 1999, (d) Maxted et al. 2000a, (e) Morales-Rueda et al. 2003a, (f) Maxted et al. 2000b, (g) Marsh 1995, (h) Marsh et al. 1995, (i) Moran et al. 1999, (j) Saffer, Livio & Yungelson 1998, (k) Napiwotzki et al. 2002, (m) Maxted, Marsh & Moran 2002, (n) Holberg et al. 1995, (o) Drechsel et al. 2001, (p) Kilkenny et al. 1998, (q) Maxted et al. 1998, (r) Wood & Saffer 1999, (s) Orosz et al (t) Bruch & Diaz 1998, (u) Gizis 1998, (v) Wood, Harmer & Lockley 1999 (w) Delfosse et al (x) Napiwotzki et al. 2001, (y) Maxted et al. 2000c, (z) Saffer, Liebert & Olszewski 1988, (aa) Heber et al. 2003, (ab) Karl et al. 2003, (ac) O Donoghue et al. 2003, (ad) Maxted et al. 2002, (ae) Hillwig et al. 2002, (af) Maxted et al. 2004, (ag) Kawka et al. 2000, (ah) Kawka et al. 2002, (ai) Saffer et al. 1993, (aj) O Brien, Bond & Sion 2001, (ak) Sing et al. 2004, (al) Moran, Marsh & Bragaglia 1997, (am) Morales-Rueda et al. 2003b, (an) Edelmann, Heber & Napiwotzki 2002, (ao) Napiwotzki et al. 2004, (ap) Fuhrmeister & Schmitt 2003, (aq) O Toole, Heber & Benjamin 2004, (ar) Robb & Greimel 1997, (as) Heber et al. 2004, (at) Raymond et al. 2003, (au) Gansicke et al. 2004, (av) Wood, Robinson & Zhang 1995, (aw) Bruch, Vaz & Diaz 2001, (ax) Pigulski & Michalska 2002, (ay) Shimansky, Borisov & Shimanskaya 2003, (az) Rauch & Werner 2003, (ba) Chen et al. 1995, (bb) Green, Richstone & Schmidt 1978, (bc) Landolt & Drilling 1986, (bd) Bell, Pollacco & Hilditch 1994, (be) Pollacco & Bell 1994, (bf) Lanning & Pesch 1981, (bg) Bleach et al. 2002, (bh) Vennes & Thorstensen WD, sdob + WD Object P orb Type Ref. WD, sdob + M Object P orb Type Ref. sdob +? Object P orb Ref. W D W D /W D a, al PG sdb/m m H E ao KPD sdb/w D b, c HS W D/M o PG am KPD sdb/w D d PG sdb/m p KPD e PG sdb/w D e GD w d / m q HE ao PG w d / w d g MT Ser sdo/m aw HE ao WD w d / w d f HW Vir sdb/m r PG e WD w d / w d h HS w d / m au PG an PG sdb/w D i NN Ser w d / m ax PG am PG sdb/w D i EC w d / m ac HE ao HE w d / w d ab J w d / m at PG e PG sdb/w D j, i HS sdb/m as PG d Feige sdb/w D aq GD w d / m ay HD aa HE w d / w d k BPM w d / m ah PG e PG sdb/w D i PG sdb/m e PG am PG sdb/w D e PG w d / m s HE ao PG sdb/w D e AA Dor sdo/m az PG e, an PG sdb/w D e WD w d / m ae PG e PG sdb/w D e CC Cet w d / m ai UVO an WD w d / w d h RR Cae w d / m t HE ao WD w d / w d * TW Crv sdo/m ba KPD am WD w d / w d * W D w d / m *, ag HD e HE sdb/w D x GK Vir w d / m bb PG e WD w d / w d m KV Vel w d / m bc PG e Feige w d / w d n RXJ w d / m ar KPD e L w d / w d z UU Sge w d / m bd PG am WD w d / w d m V447 Lyr sdo/m be PG e PG sdb/w D j V1513 Cyg w d / m u PG am W D w d / w d h V471 Tau W D /K aj HE ao WD w d / w d h RXJ w d / m af PG e WD w d / w d * HZ w d / m bf PG e WD w d / w d * PG w d / m v W D ao PG w d / w d ad EG UMa w d / m bg WD d REJ w d / m v PG e WD w d / m * PG e REJ w d / m v PG e HS w d / k ak IN CMa w d / m ah BE UMa w d / k av REJ w d / m ap Feige w d / m bh G ab w d / m w

15 14 L. Morales-Rueda et al. log P(hr) log P(hr) F ig u re 9. Left panel: mass of the bright white dwarf for the four DDs discussed in this paper as measured by Bragaglia et al. (1995), Liebert et al. al. (2005) and Bergeron et al. (1992) (see Table 5) versus period distribution. In the grey scale we plot the mass to period distribution of DDs according to theory (for the model described in Nelemans et al. (2004)). Right panel: mass ratio (only upper limits taken from Table 5) for the four DDs studied versus period distribution. The theoretical distribution according to the Nelemans et al. (2004) is also plotted. T able 12. Temperatures and gravities measured for the bright component of the system by fitting the hydrogen line profiles to stellar atmosphere models. (a) Bragaglia et al. (1995), (b) Liebert et al. (2005) and (c) Bergeron et al. (1992). V represents the V magnitude taken from the literature and M y is the absolute magnitude. The masses given have been determined using cooling tracks by Althaus & Benvenuto (1997). WD V Teff(A') log g M /M q M v Ref a b a c a c tw o w hite dw arf plus M sta r binaries. If th e high m asses inferred from th e g rav itatio n al redshifts were right, th is m eans th e direct progenitors of th e w hite dw arfs m ust have been highly evolved giants. U sing th e equations in Hurley, Pols & T out (2000) in th e sam e way as described in N elem ans & T out (2004) we calculated th e possible progenitor system s. For W D we find possible progenitors w ith m asses typically in th e range M q, while for W D th e progenitors typically have m asses betw een M q. B ecause of th e ra th e r extrem e m ass ratio s and th e large radii of th e giants, th e C E is m ost likely caused by tid al interactio n, ra th e r th a n R oche-lobe overflow (H ut 1980) and th e binaries will generally not be synchronised. W e use th e form alism derived for sta r - p lan et interactions by Soker (1996) to calculate th e separation betw een th e tw o stars at w hich th e C E sets in. T he required C E efficiencies are b e tw een 0.1 and 1 for W D and betw een 0.1 and 1.6 for W D B o th system s can also be explained w ith th e gam m a-algorithm (N elem ans & T out 2004), w ith values of y around 1.5. If, on th e o th er h an d, th e lower w hite dw arf m asses presented in T able 12 are right, th e progenitor m asses inferred are in th e range M q for W D and M q for W D T he required C E efficiencies derived in th is case lie in th e sam e ranges as for th e m ore m assive w hite dw arf altern ativ e presented above. U sing th e binary results presented in Tables 7 and 8, and th e equations from Schreiber & G ansicke (2003), we have calculated th e w hite dw arf cooling age using m odels by W ood, R obinson & Zhang (1995), t cooi, th e o rb ital period of th e binary at th e end of th e C E phase, P CE, th e tim e it will tak e for th e binary to s ta rt m ass tran sfer (and becom e a CV) assum ing classical m agnetic braking (CM B) and assum ing reduced m agnetic braking (R M B ), t sd, and th e o rbital period w hen m ass tran sfer sta rts, P sd. T hese values are p resented in th e to p th ree rows of T able 13. T he in p u t m asses used for th e w hite dw arfs are those obtained assum ing th a t th ey are carbon-oxygen core w hite dwarfs. T he in p u t m asses for th e com panions are 0.17M q for W D and < M m < M q for W D (as calculated in Sectio n 3.3.2). T he value of P CE for b o th binaries is very close to th eir present o rb ital periods w hich indicates th a t they are very young P C E B s. T hese system s will n o t becom e CVs w ithin a H ubble tim e (except p erhaps W D if RM B takes place), assum ing T0 = 1.3 x yrs (Ferreras, Melchiorri & Silk 2001), and w hen they do, th eir o rb ital periods will place th em below th e CV period gap. T he evolutionary properties of W D and W D are also calculated by using th e lower w hite dw arf m asses presented in T able 12 and th e m asses of th e M dw arfs obtained in Section T hese are presented in th e b o tto m two rows of Table 13. T he values for all p aram eters in th is case are sim ilar to those o b tained for th e larger w hite dw arf m asses and th e conclusions reached are equivalent. 5 CONCLUSIONS W e have obtained th e o rb ital solution for four D D system s and tw o w hite dw arf - M dw arf binaries. W e find th a t th e w hite dw arf com panions for th e four D D s studied co n trib u te betw een 10 and 20 per cent of th e to ta l lum inosity and c 0000 RAS, MNRAS 000, 1-16

16 Six detached white-dwarf close binaries 15 T able 13. Evolutionary properties of W D and WD The top three rows present the results obtained assuming the white dwarf masses given in Table 8 (0.72 and 0.59 M q respectively). 1 and 2 indicate calculations obtained for WD by assuming Mm = 0.12 and M q respectively. The bottom two rows present the results obtained when the lower white dwarf masses given in Table 12 are assumed instead. In this case the M dwarf masses calculated in Section are used. The values of tcool and tsd given in the table are in fact the logs of tcool and tsd in years. WD cool P CE (d) t sd P sd CMB RMB CMB RMB (d) 1G42-69G 7.92 G.3381 G G.3G 1G.G8 G.G73 2GG G.7412 G G.96 G.G55 2GG G.7412 G G 1G.78 G.G rem ain undetected. T he m asses and periods obtained for these system s agree w ith theoretical m ass period d istrib u tions (based on N elem ans et al. (2004)). In th e case of th e w hite dw arf - M dw arf binaries we have been able to m easure th e m otion of b o th com ponents. We find th a t th ere are signatures of strong irrad iatio n of th e su r face of th e M dw arf com ponent in W D T h e w hite dw arf m asses calculated from th eir g rav itatio n al redshift are unusually high com pared to those m easured by fitting th eir hydrogen lines w ith stellar atm osphere m odels, and could be th e result of th e evolution of giant stars w ith m asses betw een 1.25 and 3.5 M q th a t w ent th ro u g h a C E phase as a result of tid al interaction. T hese tw o binaries are young P C E B s th a t will evolve into CVs, although not w ithin a H ubble tim e, w ith o rb ital periods below th e period gap. A C K N O W LEDGEM ENTS LM R was su p p o rted by a P P A R C post-d o cto ral g ran t and by N W O -V ID I g ran t to P.J. G root during th e course of th is research. GN is su p p o rted by P P A R C and an N W O -V E N I grant. 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