U Gem Rediscussion of Radial Velocities and System Parameters. Józef I. S m a k

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1 ACTA ASTRONOMICA Vol. 51 (2001) pp U Gem Rediscussion of Radial Velocities and System Parameters by Józef I. S m a k N. Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, Warsaw, Poland jis@camk.edu.pl Received June 25, 2001 ABSTRACT Radial velocities measured from peaks of the emission lines (Smak 1976, Stover 1981), analyzed within the three-body approximation, give K 1 consistent with that measured directly from the absorption lines of the white dwarf (Long and Gilliland 1999). This implies that the three-body approximation is essential for correct description of the outer parts of the disk. Wings of the emission lines profiles are likely to be contaminated by contributions from parts of the stream which are overflowing the disk close to the white dwarf. Radial velocities measured from wings (Kraft 1962, Stover 1981), analyzed only in phase intervals free of contamination, give K 1 consistent with other determinations. New analysis of the spot eclipses gives i 69 Æ 2 Æ and shows that during outburst the disk expands up to about r d r Roche, while during quiescence it contracts from r d 0 75r Roche to r d 0 65 r Roche. However, the radius of the disk during quiescence obtained from V d sini appears larger: r d r Roche. System parameters are: M M, M M, R cm, R R, and distance d 96 4 pc. The radius of the secondary component, when compared with its mass, shows that the secondary is a normal main sequence star. Key words: binaries: eclipsing novae, cataclysmic variables Stars: individual: U Gem 1. Introduction U Gem, since its discovery as a spectroscopic (Kraft 1962) and eclipsing (Krzemiński 1965) binary, has become one of the most extensively observed dwarf novae. In spite of this there still exist significant discrepancies between individual determinations of some of the system parameters, notably in the case of K 1 and, consequently, the masses and radii of the components. The most recent determination of K 1 (Long and Gilliland 1999) was based on radial velocity variations measured for the first time directly from the absorption lines of the white dwarf, giving K km/s. Together with the well

2 280 A. A. determined K km/s (the mean value from Wade 1981 and Friend et al. 1990) this gives q or µ , where q M 2 M 1 and µ M 2 M 1 M 2 1µ This new value of K 1 is significantly lower than the original K km/s published by Kraft (1962). It is also lower than two other determinations: K km/s, measured from peaks of the emission lines (Smak 1976) and K km/s, obtained from their wings, and a nearly identical value from peaks (Stover 1981). Both of them, when combined with K 2, give µ Regarding the first of these two determinations one may recall here (as it was done already by Long and Gilliland) that the most probable solution (Smak 1976, Section 5), based on the three-body approximation (see Section 4 for details), gave also µ On the other hand, however, Stover (1981, Section III-d, Fig. 7) found no residuals in his peak velocities which would confirm the presence of three-body effects. This conclusion appeared also to be supported by the fact that the values of K 1 obtained from peaks and from wings were nearly identical. It should be added here that the value of µ 0 32 was used earlier (Smak 1984, 1996, Zhang and Robinson 1987) in the analysis of the hot spot eclipses resulting in the determination of the orbital inclination and variations of the disk radius. It is obvious that, should the correct value of µ be different, the analysis should be repeated, with the new, revised value of inclination to be used for the determination of masses and radii. Taking into account those problems it was decided to rediscuss the radial velocity data from Smak (1976) and Stover (1981) and to see whether (and how) the discrepancies mentioned above can be removed, and whether a self-consistent set of system parameters can be obtained. Section 2 will summarize the evidence concerning the γ-velocity and the phase shift φ φ spectr φ phot. In Section 3 we will repeat the spot eclipse analysis using µ Section 4 will be devoted to the analysis of radial velocities measured from peaks of the emission lines. It will be shown that when interpreted within the three-body approximation they give µ and K 1 consistent with values obtained by Long and Gilliland. In Section 5 we will demonstrate that the wings of the emission lines are likely to be affected by emission from parts of the stream which are overflowing the disk close to the white dwarf. It will be shown that the values of K 1, determined using only those phase intervals which are free of the expected contamination, become consistent with other determinations. System parameters and the distance will be redetermined in Sections 6 and 7. Finally, in Section 8 we will discuss our results and problems related to U Gem and other cataclysmic variables.

3 Vol The γ-velocity 2. The γ-velocity and the Phase Shift There have been several determinations of the systemic velocity based on radial velocities measured either from the emission lines or from the absorption lines of the secondary component. Those which gave consistent results include: γ 42 km/s (Kraft 1962, wings, all lines), γ 40 6 km/s (Smak 1976, peaks, all lines), γ km/s (Smak 1976, S-wave, HeI λ4471), and γ 43 6 km/s (Friend et al. 1990, secondary, all lines). Those which gave different values were: γ 11 6 km/s (Stover 1981, wings, Hβ+Hγ), γ km/s (Stover 1981, S-wave, Hβ+Hγ), and γ km/s (Wade 1981, secondary, all lines). While no obvious explanation can be offered for those three deviating values, it appears fairly safe to adopt: γ 42 6 km/s The phase shift: φ φ spectr φ phot A phase shift: φ φ spectr φ phot between the true (spectroscopic) phases and phases defined by the eclipse of the hot spot is a natural consequence of the asymmetric location of the spot causing its eclipse to occur before the spectroscopic conjunction. To avoid possible confusion one should mention that when the radial velocity curves are fitted with an expression: K 1 sin φ φ 0 µ, then our definition of φ implies: φ φ 0. Values of φ determined earlier from the spot eclipse analysis with µ 0 32 (Smak 1984, 1996, Zhang and Robinson 1987) were all between φ and The new determination, with µ 0 27 (next Section) gives φ Turning to spectroscopic determinations of φ we first note that those based on radial velocity data measured from emission lines appear to give discordant, most likely spurious results (but see Section 5). On the other hand, those based on radial velocity data measured from absorption lines of the secondary component give consistently: φ (Wade 1981) and φ (Friend et al. 1990). Finally, we have an independent determination from the displacement of the infrared light curves (Berriman et al. 1983) giving φ Combining these results we can adopt: φ φ spectr φ phot Spot Eclipse Analysis The analysis of the hot spot eclipses, as described in details in Smak (1996), is repeated here with µ 0 27 (q 0 36) using Models 2a and 2b. It may be worth to recall here that those two models, involving the upper and lower limit estimates of the cross-section of the stream, result in the upper and lower estimates for the radius of the disk. The results are listed in Table 1, where s a is the effective "radius" of the spot in the orbital plane, while r 0 and dr dt describe variations of the disk

4 282 A. A. radius: r d r 0 dr t 2µ dt T a b l e 1 Spot Eclipse Solutions. Assumed: µ 0 27 (q 0 36 ) Mod. φ r 0 dr dt i s a 2a b As compared with earlier results (cf. Smak 1996, Table 1), all parameters (except for the disk radius; see below) turn out to practically the same. In particular, we obtain i 69 Æ 2 Æ. The radius of the disk (as could be expected) is systematically larger. During outburst the disk expands up to about r d r Roche, which is approximately equal to the commonly accepted value of the tidal radius r tid. During quiescence, between 20 and 80 days after outburst, the disk contracts from about r d 0 75 r Roche to about r d 0 65 r Roche. 4. K 1 and V d sin i from Peaks 4.1. k 1 and v d sini from Peaks in the Three-Body Approximation Following earlier papers (Paczyński et al. 1968, Piotrowski and Ziołkowski 1970, Smak 1976, Paczyński 1977) we use the three-body approximation to calculate, for a series of values of µ, families of periodic orbits around the primary component. This is done in standard dimensionless variables (hereafter denoted with small letters). General properties of such orbits were adequately described in earlier papers (cf. Smak 1976, Paczyński 1977) and therefore we will mention here only a few relevant points. Each orbit is parametrized by its "dimension"; in practice it is the distance from the primary of the arbitrarily chosen starting point. In further applications, however, we will prefer to use the mean radius normalized with the mean radius of the Roche lobe, i.e., r r Roche. As in previous investigations, our calculations within each family are terminated with the largest non-intersecting orbit. The dimensions of such a critical orbit depend strongly on µ (see Fig. 2 in Paczyński 1977) but, after normalization, we have r max r Roche (with little dependence on µ).

5 Vol Fig. 1. Disk radius variations obtained with µ 0 27 and two models for the spot geometry. Smaller symbols are of lower weight. Crosses correspond to eclipses observed at the onset of an outburst, when the disk just begins to expand; they are plotted twice: at their proper t and at t 1 and 2. Bars at left mark r d r Roche 1 0, 0.9, and 0.8. Lines represent Eq. (2) with parameters listed in Table 1. Radial velocity variations are then calculated assuming that the disk extends up to r d r max. In the simpler, "outer-ring" approximation (cf. Smak 1976, Paczyński 1977) we consider only the outermost ring (note that at larger radii such a "ring" is no longer circular). In the second, presumably more realistic, "full-disk" approximation we determine radial velocities from the "blue" and "red" peaks of the line profiles. These are calculated by integrating contributions from the surface of the disk, assuming that the emitting atoms are distributed as f rµ r 1 5 (Smak 1981, Stover 1981) and including a moderate amount of broadening (to make the resulting profiles resemble more closely the observed ones). As a result of calculations described above we have a two-parameter family of radial velocity curves v r φµ, the two parameters being µ and r d. Those model curves can now be analyzed in the same way as in the case of the observed curves i.e., by fitting the formula v r φµ γ k 1 sin φ φ 0 µ v d 3µ

6 284 A. A. where the sign corresponds to the "red" and "blue" peaks. At this point we must recall that the observed radial velocity curves measured from the "red" or "blue" peak of the emission lines are at certain phases contaminated by the S- wave component. Consequently, in order to avoid this contamination, the observed curves must be analyzed using only those phase intervals which are free of this contamination (see Section 4.2 below). Accordingly, the same procedure must be applied to model curves. The resulting values of k 1 differ, in general, from µ and, likewise, the values of v d differ from their Keplerian equivalent. Prior to their use in the analysis of the observational data (to be presented below) it will be more convenient, however, to convert them to their "observable" equivalents. First, using the well determined value of K 2, we introduce the true value of K 1 : K 1 µ 1 µ K 2 4µ The "observed" values of K 1 and V d sini, corresponding to k 1 and v d, are then 4.2. K 1 and V d sini from Peaks k K 1 K 1 1 µ 5µ V d sini K 1 k 1 v d 6µ There are only two sets of radial velocity data measured from peaks of the emission lines (Smak 1976, Stover 1981). Similar data could (and should) have been extracted from spectra collected by Honeycutt et al. (1987) for the purpose of Doppler tomography (Marsh et al. 1990, Kaitchuck et al. 1994) but, regrettably, this has not been done. The first set (Smak 1976), analyzed using phase intervals free of contamination by the S-wave component, i.e., φ phot for the red peak and φ phot and for the blue peak, gave: K km/s and V d sini km/s. The second set (Stover 1981) was re-analyzed for the purpose of the present investigation, with radial velocities from peaks of the Hβ and Hγ lines (Fig. 7 in Stover 1981) being analyzed in the same phase intervals as in the case of the first set, and with a fixed value of φ (corresponding to φ 0 025; see Section 2.2). The resulting parameters: K km/s and V d sini km/s, are practically identical with those obtained by Stover (1981). In further analysis we will also need information concerning the dimensions of the disk at the time the two sets of observations were made. Using Eq. (2), with parameters listed in Table 1, and with, respectively, t 52 days and 67 days, we get r d and for the first set and r d and for the second set, the two values for each set corresponding to Models 2a and 2b. Note that these

7 Vol radii are measured at a specific position angle defining the location of the hot spot. When converted to the mean radius (in the sense defined in Section 4.1) and normalized with the mean radius of the Roche lobe they become: r d r Roche 0 76 and 0.64 for the first set and r d r Roche 0 74 and 0.62 for the second set The Three-Body Model Analysis We now compare the observed values of K 1, V d sini, and r d r Roche with the three-body model predictions. The first comparison, involving K 1 and V d sin i is shown in Fig. 2. The two main conclusions resulting from this comparison are as follows. The disk is large, its dimensions being, roughly, comparable to those of the largest non-intersecting orbit. In fact, the radius of the disk obtained from the first set using the "full disk" approximation (which is presumably more realistic; see Section 4.1), turns out to be r d r Roche 0 95, and this is significantly larger than the "critical" value of r max r Roche Regarding the mass ratio parameter µ we can conclude that its values obtained from the two sets, using two model approximations, turn out to be within errors practically identical. We can adopt: µ Fig. 2. Observed values of K 1 and V d sini from Smak (1976) filled squares and Stover (1981) filled triangles are compared with a grid of three-body models, with µ and r d r Roche , corresponding to the "outer ring" (left) and "full disk" (right) approximation. The second comparison, involving K 1 and r d r Roche is shown in Fig. 3. The pairs of values of r d r Roche plotted in this comparison are lower and upper limits based on two models of the spot eclipse (see Section 3); unfortunately, they differ considerably. Taking into account this uncertainty, together with formal errors of K 1, we obtain: µ

8 286 A. A. Fig. 3. Observed values of K 1 from Smak (1976) filled squares and Stover (1981) filled triangles are plotted against r d r Roche, with the two sets of symbols representing the upper and lower limits for the disk radii (see text for details). Theoretical lines from three-body models are labelled with µ and correspond to the "outer ring" (left) and "full disk" (right) approximation. We now conclude that the two, partly independent determinations give µ and µ , i.e., values which within errors are practically identical. More importantly, they agree almost perfectly with µ obtained by Long and Gilliland (1999). Using the two values of µ we can determine the true values of K 1 (see Eq. 4), getting: K km/s and K km/s. Obviously, they also agree with K km/s obtained by Long and Gilliland A Test for the Presence of Three-Body Effects We now return to the problem of applicability of the three-body approximation to the description of motion of the outer parts of the disk. As mentioned in the Introduction, Stover (1981) found no residuals in his peak velocities which would confirm the presence of three-body effects. Nevertheless we decided to perform such a test again. As described in Section 4.2 we re-analyzed the peak velocities measured by Stover (1981) from the Hβ and Hγ lines using phase intervals free of contamination by the S-wave component. The resulting solution is now used to calculate the residuals from the best-fit sine-wave curves. The same procedure is applied to the radial velocity curves calculated within the three-body model with µ 0 27 and r d r Roche 0 85, using the "outer ring" and "full disk" approximations. The resulting residuals are converted into km/s. A comparison between the observed and theoretical residuals is shown in Fig. 4. In spite of considerable observational scatter, it can be seen that in all three segments there appears to be a good agreement

9 Vol between the general trend shown by the observed and predicted residuals. It is clear, however, that more accurate observational data would be needed to make such a test more decisive. Fig. 4. Residuals from the best-fit sine-wave curves in the phase intervals used in the solutions. Crosses represent residuals corresponding to the observed peak velocities from Stover (1981). Solid lines show residuals obtained from radial velocity curves calculated within the "full disk" approximation, while broken lines within the "outer ring" approximation. 5. K 1 from Wings As already mentioned in the Introduction, Stover (1981) measured radial velocities from the wings of the emission lines, with the resulting value of K km/s, which is considerably larger than the new value obtained by Long and Gilliland (1999). In what follows we propose a possible explanation for this discrepancy. The wings of the line profiles are produced in the inner parts of the disk and are supposedly free of contaminations (such as the S-wave) which affect the peaks. This, however, may not necessarily be true. One of the effects to be considered is

10 288 A. A. connected with the possibility that only part of the stream collides with the outer edge of the disk, forming the hot spot, while its other parts may continue to overflow the disk (Kunze et al. 2001); the standard S-wave component appears, in fact, to originate in the overflowing parts of the stream just above (and below) the hot spot (Smak 1985). Fig. 5. Dimensionless radial velocity curves from points in the stream (calculated with µ 0 27). Solid line corresponds to the location of the hot spot. See text for details. Fig. 5 shows radial velocity curves (the "S-waves") for a sequence of points in the stream starting with a point where the hot spot (and the standard S-wave) is formed up to and beyond its closest approach to the white dwarf. As can be seen there are two phase intervals where velocities from the stream, as it passes close to the primary component, become very high. To check whether they are sufficiently high to affect the "red" or the "blue" wing of the line profile produced in the disk we use the profiles published by Stover (1981, Fig. 5) and find that the half-width of the emission lines, measured at intensity level equal to 1/2 of the peak intensity, is V 1100 km/s, while at 1/4 of the peak intensity it is V 1600 km/s. In dimensionless units these two values correspond to v 2 7 and 4 0. Comparing them with the largest amplitude "S-waves" shown in Fig. 5 we conclude that, indeed, at certain phase intervals the contribution from the stream is likely to affect

11 Vol the wings. There are two possible tests to verify this hypothesis. The most direct test would involve checking whether there is any asymmetry, due to the expected contribution from the stream, between the "red" and "blue" wings at those critical phases. This would require high quality spectrophotometric data and it can only be hoped that someone with access to such data will eventually perform such a test. We can perform another test here, which consists of re-analyzing radial velocity data from Stover (1981, Table 1), using only those phase intervals in which both wings remain unaffected. Upon inspection of Fig. 5 we adopt φ and This solution, illustrated in Fig. 6, gives: γ 10 5 km/s, K km/s, φ , i.e., φ Considering the simplicity of our assumptions it is almost surprising to find that the value of K 1 agrees so well with that obtained by Long and Gilliland and with our revised values obtained from peaks. Even more surprising is the value of φ which is consistent with that adopted in Section 2.2. Finally we note that points in the phase intervals not used in the solution (open symbols in Fig. 6) do, indeed, show deviations which are qualitatively consistent with predictions which could be made from Fig. 5. As already mentioned in the Introduction the value of K 1 obtained by Kraft (1962) from radial velocities measured from wings of the emission lines was K km/s, i.e., considerably larger than all other determinations. Could this also be due to the contributions from the stream? To answer this question we re-analyze Kraft s data in essentially the same way as it was done above for Stover s data. Since there are only seven data points in the phase intervals (see above) used in the solution we are forced to adopt φ and limit our solution to γ and K 1. The solution, illustrated in Fig. 6, gives: γ km/s, K km/s, this value of K 1 being within 1σ limits consistent with other determinations! 6. System Parameters The values of K 1 determined or re-determined in previous Sections agree within 1σ limits with that published by Long and Gilliland (1999). Their formal accuracy, however, is much lower. As a result, when the weighted mean is calculated, it comes out nearly identical with K km/s from Long and Gilliland. Adopting this value, together with K km/s from Wade (1981) and Friend et al. (1990), and i 69 Æ 2 Æ from Section 3, we now determine the main parameters of the system (Table 2). Note that due to our higher value of inclination they differ slightly from those obtained by Long and Gilliland (1999, Table 5). The radius of the secondary component was obtained under the usual assumption that it fills its Roche lobe. When compared with its mass it shows that the secondary is a normal main sequence star. The radius of the primary was calculated from its mass using the mass-radius relations for the carbon-oxygen white dwarfs from Hamada and Salpeter (1961) and Nauenberg (1972); in the case of a

12 290 A. A. Fig. 6. Top: Radial velocities from wings of the Hβ (squares) and Hγ (triangles) lines measured by Stover (1981). Bottom: Radial velocities from wings measured by Kraft (1962). Filled symbols denote radial velocities used in the solution. Solid lines are the best fit sine-wave curve resulting from the solution. T a b l e 2 System Parameters i Æ 69 2 M 1 M M 2 M A R R cm R 2 R d pc 96 4 helium white dwarf R 1 would be slightly smaller, but well within the formal errors given in Table 2. Using M 1 and R 1 we calculate the gravitational redshift: γ grav km/s. The observed value, obtained with γ WD km/s

13 Vol (Long and Gilliland 1999) and γ 42 6 km/s (Section 2.1) is: γ grav γ WD γ km/s. While not exactly equal the two values agree within the 1σ limits and this confirms earlier conclusion by Long and Gilliland. 7. The Distance 7.1. Distance from the K-Magnitude of the Secondary This is the most commonly used method of determining the distance to a cataclysmic binary. The basic equation of the method is (Bailey 1981) m M 5log d 5 K 5log R 2 R µ S K 7µ where d is the distance (in parsecs), K is the observed magnitude, and S K is the surface brightness in the K-band. We now apply this method to U Gem. To find S K we proceed as follows. From M 2, using logt e vs. M M relation based on Popper s (1980) data, we find logt e From R 2, using logt e vs. R R relation based on data from Reid and Gillmore (1984), we find log T e With logt e , using log T e vs. V Kµ relation from Reid and Gillmore (1984), we get V Kµ This value of V Kµ implies spectral type M2 (Bessell 1991, Table 2) or M2-3 (Berriman 1987, Fig. 1), i.e., considerably earlier than directly estimated M4 (cf. Harrison et al. 2000). Finally, using the S K vs. V Kµ relation from Berriman (1987) we find S K , where the error is largely due to scatter shown by points defining this relation (Fig. 1 in Berriman 1987). With K (Harrison et al. 2000) and R 2 from Table 2 we get from Eq. (7): m M or d 91 9 pc Distance from the Ultraviolet Flux of the Primary To determine distance from the observed ultraviolet flux we reverse the procedure used by Cheng et al. (1997). Using their Eq. (7), together with mean values of the "scale factors" S wd and S belt obtained by them from the two-temperature (white dwarf + accretion belt) solution and our value of R 1 (Table 2) we get d pc. This value corresponds to the full visibility of the white dwarf. One should note, however, that the bottom hemisphere of the white dwarf is likely to be obscured by the disk (cf. Smak 1992). In such a case R WD in Eq. (7) of Cheng Ô et al. should be replaced with R WD x, where x is the visible fraction of the flux from the white dwarf; at i 69 Æ we have x Taking this into account we get d pc Summary of Distance Determinations In addition to the two photometric determinations presented above we have the very accurate trigonometric parallax of π mas measured by Harrison et al. (2000), giving d pc. The weighted mean values of the three

14 292 A. A. determinations are: d pc and pc for, respectively, the full visibility and partial obscuration of the white dwarf. As a final result we adopt: d 96 4 pc, or m M Discussion Our re-discussion of radial velocities of U Gem measured from the emission lines satisfactorily explains the origin of discrepancies between values of K 1 obtained from wings and from peaks and K 1 measured directly from the absorption lines of the white dwarf. We have shown, in particular, that K 1 and V d sin i measured from peaks (avoiding phases where contamination by the S-wave component is present) and analyzed within three-body approximation, give the correct value of µ (or q) and, consequently, with K 2 being known, the true, correct value of K 1. This, together with the results of a test shown in Fig. 4, implies that the three-body effects in the outer parts of the disk are indeed important. We have also shown that radial velocities measured from wings are likely to be contaminated by emission components from parts of the stream which are overflowing the disk close to the white dwarf and, as a result, produce K 1 which is systematically too high. Correct value of K 1, consistent with other determinations, can be obtained, however, when only those phase intervals which are free of this hypothetical contamination are used. This implies that such stream overflow effects are indeed present. In a broader context our results open the problem of systematic accuracy of values of K 1 determined earlier for all other cataclysmic variables, including those measured from peaks as well those from wings which, until now, were considered "safer" and more reliable. It is strongly recommended that K 1 from peaks be determined using the three-body approximation while that from wings be based on phase intervals free of contamination by the stream effects. Returning to U Gem we should note one problem which requires further studies. The radius of the disk obtained via three-body analysis (Section 4.3; see Fig. 2) from V d sini is r d , i.e., comparable, or even larger that the "critical" value of r max 0 85 r Roche ; note that this refers to the time when spectroscopic observations were made, i.e., 52 and 67 days after outburst. On the other hand, the corresponding mean radius of the disk obtained from the spot eclipse analysis (Sections 3 and 4.2) is significantly lower: r d r Roche. Only during outburst it becomes comparable to the tidal radius r tid 0 9 r Roche. The new value of the distance to U Gem (Section 7), based on three independent and highly consistent determinations, is significantly larger than most values published over the last two decades (e.g., Berriman 1987, Warner 1995).

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