3 The distribution of the S Dor variables on the HR-diagram

3.1 The HR-diagram

Figure 20 shows the position of the four categories of objects on the HR-diagram in four separate panels. The error bars are based on estimates (Sect. 2.2). The bars at the left represent the relative errors for the objects within the same stellar system. There are two bars at the right labeled "extra''. The first one with the sublabel "dist'' is due to the error in the distances of the three stellar systems relative to each other. The second one with the sublabel "BC hot'' is the effect of an error in temperature on the luminosity via the BC. It is only appropriate for the hottest S Dor variables since the BC is negligible for the cooler ones. Both errors should be incorporated to the three at the left by adding them quadratically and taking the square root, if objects of one stellar system are compared with those in another one. The dotted vertical line in each panel is the calculated threshold for dynamical instability when going from low to high temperatures according to Stothers & Chin (1996). Indeed, only a few objects lie on the left of it and within the estimated error.

Figure 20: The position of the four categories of S Dor variables on the theoretical HR-diagram. The fat dashed line in the four panels represents the SD-minimum strip. See for further explanation Sect. 3.2

3.2 Discussion on the HR-diagram

3.2.1 The s-a S Dor variables

The upper left panel of Fig. 20 shows the s-a variables: 4 galactic, 2 SMC and 8 LMC objects (Tables 1, 2 and 3). To prevent crowding, not all designations are given. The minimum state (dots) and the maximum state (circles) are connected by a line. Most of these lines, representing schematically the displacement during SD-phases, run more or less horizontal as expected. The more luminous the star, the larger the shift in temperature and the visual amplitude. For HDE269216 (LMC) only the average position is shown (open square), since no precise data for its extrema are known. For HDE269582 (LMC) only the minimum position could be derived (dot). The thick dashed line sketched through the minima (3 galactic, 1 SMC and 7 LMC objects) represents the "s-a SD-minimum strip'', which will be called from now on the "SD-minimum strip''. Such a relation has been first pointed out by Wolf (1989). The dashed line has been copied into the three other panels, which will be discussed hereafter. WRA751 (uncertain distance) and R4 (SMC; see below) have been ignored. The equation for the strip is:

$$\log L/L_{\odot} = 1.37\,\log T_{\rm eff} - 0.03.$$

It must be stressed that this strip is not necessarily the same as where the SD-eruptions originate. However, see the discussion on P Cyg in Sect. 3.2.2. The small scatter is surprising. It can partly be explained by the fact that errors in temperature cause a change in luminosity via the BC, which runs along almost the same inclination as the strip. If the small scatter is not accidental and intrinsic to the strip, then the distances are apparently rather reliable. The SMC object R4 (including an A-type companion) is much too faint: about 2 $.\!\!^{\rm m}$5 (bolometric) compared to other S Dor variables with the same temperature. This must be intrinsic. Whether this has anything to do with the fact that it also is a B[e] star (Tables 1 and 2; see also van Genderen & Sterken 1999) is unknown. B[e] stars are likely rapid rotators and according to Langer (1999) such stars should have a lower luminosity. On the other hand Langer & Heger (1998) concluded from the morphology and chemical enrichment of the nebula that R4 was originally composed of a close binary and a third star (the A-type companion). The close binary merged into a single star: the present B[e]/S Dor companion. In this model, supported by observations of the nebula by Pasquali et al. (2000), the merger star would be expected to add a large amount of He to the interior of the combined object, giving it a very large L/M ratio. However, how should this be reconciled with the unusual low luminosity of R4? The evolutionary tracks of stars on the verge of becoming a WR star leave the SD-area by bending down to the left (the most massive ones), or leaving it horizontally to the left (the less massive ones). Thus, R4 could be such a case. The second SMC object in this group, R40, lies only slightly above the SD-minimum strip. Clear systematic differences between the objects of the three stellar systems due to differences in metal content, are obviously absent. Perhaps R40 is the only one that to some extent responds to the expectation, since with low Z L goes up and $T_{\rm eff}$ drops (Stothers & Chin 1996) and the location of the photospheric Eddington limit goes up (Lamers & Noordhoek 1993). However, the observed deviation is not significant. According to the opaque-wind model of Davidson (1987) the maximum visual brightness cannot surpass the temperature limit to the red between 7000K and 8000K. Although an SD-phase is basically caused by an expanding radius and a decreasing temperature, rather than by an expanding pseudo-photosphere due to an increasing wind density, this group (and the other ones to be discussed below) seems to have a limit at $\sim$7000K. Perhaps the SD-phase is not only caused by a radius and temperature variation alone, but might at times be a mixture of an expanding radius/decreasing temperature and an expanding pseudo-photosphere, see van Genderen et al. (1998b, Sect. 4.5) The SD-minimum strip lies roughly at the same location as that of the dynamical unstable blue models of Stothers & Chin (1996), viz. between $\log L/L_{\odot}$ $\sim$6.1 and $\log T_{\rm eff}$ $\sim$4.5, and $\log L/L_{\odot}$ $\sim$5.4 and $\log T_{\rm eff}$ $\sim$3.9. Further, it should be noted that the red side of the SD-area overlaps the "Yellow Void'', a region of atmospheric instability of yellow hypergiants, investigated by Nieuwenhuijzen & de Jager (1995, 2000) and de Jager (1998), see Sect. 3.2.5 of the present paper.

3.2.2 The w-a S Dor variables

The upper right panel shows the w-a members: 4 galactic and 8 LMC objects (Tables 4, 5 and 6). For most of these stars the parameters have been plotted as averages, since the extrema are not precisely known. HD38489 is the only one with a relative large range: minimum (dot) and maximum (circle) are connected by a line, although the light variation is poorly known. R85 has a small range, minimum (dot) and maximum (circle) are connected by a line. That the long-term low-amplitude variation represents an SD-phase indeed is obvious from the three spectral type determinations indicated in Fig. 13: at minimum the star is hotter than at the maximum. Two famous objects: $\eta$ Car (Fig. 11) and P Cyg, belong to this group. The S Dor variable connected to the first object is hidden in a lot of circumstellar material, so its brightness and temperature are only roughly known (Table 4). According to the principle of a "Chinese lantern'' any light variation of the variable is imitated by the bipolar reflection nebula (mainly by the dominant SE part more or less pointing to us, van Genderen et al. 1999). It appears that the light variation in the visual during the interval 1992-1994 showed strong similarities with that of R85 during 1984-1986: both S Dor variables showed a cycle of an S-SD phase with microvariations superimposed (van Genderen et al. 1999). This proves that the S Dor variable hidden in $\eta$ Car must also be a normal S Dor variable. This is supported by the spectrum which looks like that of P Cyg (Ebbets et al. 1997). Its most probable position is outlined by the dashed-dotted error box. The cause of the large uncertainty is explained in the notes to Table 4. The present status of $\eta$ Car is discussed in Sect. 3.2.6. After the SD-eruptions in the 17th century, P Cyg's total displacement in $\log T_{\rm eff}$ to the right on the HR-diagram amounts to 0.046, assuming constant luminosity (Lamers & de Groot 1992; de Groot & Lamers 1992). This is indicated by the short horizontal line in Fig. 20. If this is not much in error, the eruptions occurred close to its present position and according to the model of Langer et al. (1994), P Cyg ought to be at the beginning of an S Dor stage. It is of interest to note that the age of the oldest shell of P Cyg, possibly a relic of an eruption, amounts to some 2 10⁴yr (Meaburn et al. 1999). This suggests that P Cyg has been that long an S Dor variable. According to Stothers's (1999b) prediction, P Cyg is now exactly in a state of marginal dynamical instability, as is indeed observed by de Groot et al. (2001) (see Sect. 1.6 and Table 6 in the present paper). Only microvariations are present and a possible very low-amplitude SD-cycle. The Stothers & Chin (1995) models do not predict much variation in the luminosity, either during or directly after the mass-loss episodes. In the course of the "blue loop'' after a mass-loss episode, the star, in this case P Cyg, should be (nearly) inactive, as is observed. Thus, the observations of P Cyg support my view that S Dor variables are a sub-class of the $\alpha$ Cyg variables (Sect. 1.6). It seems obvious that the microvariations are hardly influenced by the various SD-instabilities: they are present when the star is not, or at most an extremely weak-active S Dor variable, or they are imperturbably present during high-amplitude SD-phases. Presumably they should be considered as small-scale instabilities of the outer layers, typical for $\alpha$ Cyg variables in general. If the $M_{\rm bol}$ of P Cyg was a magnitude brighter during the eruption of the 17th century, which is not certain at all (Lamers & de Groot 1992), the displacement followed an inclination steeper than the SD- minimum strip. The model of Langer et al. (1994) does not predict such a large variation in the luminosity. With a few exceptions, the w-a members obey the SD-minimum strip, but with more scatter. Most of them are Be stars and/or are suspected of having a gaseous disk e.g. R99 and R123 in the LMC (Table 4). R149 (Be+neb, LMC) is like R4 (B[e], SMC) in the previous panel, much too faint, which could be intrinsic (Sect. 3.2.1). I suspect that most of the w-a variables should fit the SD-minimum strip because they are supposed to be in a minimum of both types of SD-phases. The s-a variable HR Car experienced such a minimum lasting about 20yr and the s-a variable R71 has now (1999) been in such a state for 15yr, only showing weak-active SD phases (Fig. 4; van Genderen et al. 1997b; Sterken et al. 1997b). Such minima might even last much longer. Note that R81 (B2.5eqIa⁺, LMC) and R123 (Bpec, LMC), were s-a variables in the previous century (Figs. 12 and 15, respectively).

3.2.3 The ex-/dormant S Dor variables

The bottom panel at the left of Fig. 20, shows the ex-/dormant variables: 5 galactic and 3 LMC objects (Tables 8, 9 and 10). Most of them lie below the SD-minimum strip. Whether this is accidental or intrinsic (e.g. due to further evolution to the blue, see Sect. 3.2.1) is unknown. Again, there is one object with a much too low luminosity, the galactic object HDE326823. Lopes et al. (1992) and Sterken et al. (1995b) suspect that it is on its way to the WN stage. One could speculate that the same is true for R4 and R149 discussed in Sects. 3.2.1 and 3.2.2, respectively. According to Weis (1999) S119 has a shell typical for an S Dor variable, although the age may amount up to $\sim$10⁵yr, which is relatively high.

3.2.4 The candidate and former candidate S Dor variables

The bottom panel at the right of Fig. 20 shows the candidate S Dor variables, at least those for which luminosity and temperature are known: 6 positive candidates (+ sign, bracketed if the position is very uncertain) and 4 negative candidates (- sign) (Tables 11, 12 and 13). The Pistol Star (PS) is represented by two models. Their positions far from the SD-minimum strip could point to a large uncertainty in the distance and/or the reddening, just like for most of the other candidates. Further, 3 non-candidates (dots) are also plotted. The positive candidate IRC+10420 is the only object which was situated on the left side of the 7000-8000K border (discussed in Sect. 3.2.1) some decades ago, but which evolved quickly to about 7900K in 1994 (Oudmaijer 1995; Nieuwenhuijzen & de Jager 2000). The star has been plotted in Fig. 20 with the last-mentioned temperature. This object might still be a yellow hypergiant. See for a further discussion on the possible relationship between the two types of variables Sect. 3.2.5.

3.2.5 Discussion on the possible evolutionary connection between S Dor variables and yellow hypergiants

According to de Jager (1998) IRC+10420 is a yellow hypergiant, like $\rho$ Cas and HR8752. HR8752 appeared to have evolved quickly to the blue during the 20th century (Zsoldos 1986), like IRC+10420 (e.g. Kastner et al. 1995; Oudmaijer 1995; Nieuwenhuijzen & de Jager 2000). $\rho$ Cas may be doing the same, although its behaviour is more chaotic. The observations and calculations of Nieuwenhuijzen & de Jager (1995, 2000) and de Jager (1998) indicate that HR8752 and $\rho$ Cas are now bouncing against the cool border ($\sim$7500K) of the Yellow Void, a region of atmospheric instability, which overlaps the cool side of the SD-area (although no S Dor variables reside in the Void, just at its lower boundary, see Fig. 21). This is accompanied by enhanced mass ejections. For HR8752 the bouncing occurs in possible cycles of $\sim$10yr. It is unknown how and when these objects will cross this border to enter the Yellow Void. With other words: are these fast evolving yellow hypergiants in fact proto-S Dor variables? The opposite question has been posed by de Jager (1998: Sect. 2). According to Garcia-Segura et al.'s (1996b) evolutionary model for a 35$M_\odot$ star: possibly not. Their model predicts an extremely rapid evolution from the yellow hypergiant stage to the WR stage: about 100yr! However, as de Jager & Nieuwenhuijzen (see above) have demonstrated, the evolution of the yellow hypergiants to the blue will be stopped by the Yellow Void for some time, but it is unknown for how long. Once they have passed, or avoided this barrier somehow, there is perhaps not much time left for a (blue) S Dor stage. On the other hand, note that the S Dor variables like the w-a variables HD168607, R85, R74, the s-a variable R110 and the ex-/dormant variable HD168625 in Fig. 20, lie just at the lower edge of the Yellow Void (Fig. 21) and that not one S Dor variable is situated right inside the Void, (only the s-a variables cross the Void during SD-phases and are obviously not in a normal state then). Therefore, we may assume that once the envelopes of the yellow hypergiants have lost so much mass that it results in more atmospheric stability, they become low-luminous S Dor variables after all and for a time long enough to be observed. Note that both types of variables are generally blueward evolving stars and that the masses of the yellow hypergiants (Nieuwenhuijzen & de Jager 2000) are most likely of the same order as that of the S Dor variables like R110: 10$M_\odot$ (Stahl et al. 1990), HD160529: 13$M_\odot$ (Sterken et al. 1991) and HD168625: 10$M_\odot$ (Robberto & Herbst 1998). Consequently, in some respects, yellow hypergiants could be very well proto-S Dor variables, consistent with de Jager's (1998) supposition. Besides, the cycles of mass ejections of the yellow hypergiants and the SD-phases of S Dor variables may be related instability mechanisms. It should be noted that yellow hypergiants also belong to the $\alpha$ Cyg variables, viz. the cool ones, showing microvariations on a time scale of hundreds of days.

Figure 21: The structure of the S Dor area on the HR-diagram. It shows amongst others the SD-triangle in which the s-a S Dor variables move to-and-fro during the SD-phases, the SD-minimum strip (fat dashed line) with the thinly dashed lines as its bandwidth, and the Yellow Void

3.2.6 The present status of $\eta$ Car

The fact that all S Dor variables with reasonable reliable distances and reddenings lie on a relatively narrow band on the HR-diagram, can have consequences for an object like $\eta$ Car. Its distance (2.3kpc), foreground reddening (0 $.\!\!^{\rm m}$50) and total luminosity ( $\log L/L_{\odot}$ = 6.65, or $M_{\rm bol}$ = -11.9) seem to be reliable as well. Nevertheless, its position if plotted in Fig. 20 is too far above the S Dor area, whatever temperature is used. Note that more than 90% of the luminosity is due to IR radiation. Thus, apart from the S Dor variable, more IR sources, or sources leading to heating of dust and gas complexes (such like explosive events in the past, accreting mass processes, etc.), might be present inside $\eta$ Car (van Genderen et al. 1994, 1995, 1999, 2001). Note that the radial velocity curve based on lines of the Paschen series is useless and that according to Davidson et al. (2000) the binary hypothesis is weak, in contrast with the view of Damineli et al. (2000). The justification of serious doubt on the radial velocity curve of Damineli et al. (1997) is certainly not new: two years before the study of Davidson et al. (2000), de Groot & Henderson (1998, quoted by van Genderen et al. 1999) already expressed their serious concern. They found that the HeI 5875Å  and 6678Å  lines (presumably originating deeper in the wind) gave radial velocities completely different from those on which the excisting binary models were based. The "spectroscopic events'', the X-ray observations and the spectroscopy suggest a colliding-wind binary (Damineli et al. 1997, 2000; Pittard 2000). Another indication that $\eta$ Car hides a secret is based on the detection of peculiar light variations of non-stellar origin, sometimes only clearly present in the near-UV, sometimes also detectable in V and B and deviating from the normal SD-variability. Most striking are the periodicities in the near-UV, detected by our multi-photometric monitoring campaigns. They could point to the presence of a luminous accretion disk which can occur in semi-detached binaries (van Genderen et al. 1994, 1995, 1999, 2001). Calculations based on formulae derived from energy balance arguments (Bath 1979; Bath & Pringle 1985) show that variations in such a disk can be measurable. This would imply the presence of a semi-detached binary, but other scenarios without a semi-detached binary are not impossible as well (van Genderen et al. in prep.). Our photometric analysis revealed that the nett rising trend of the brightness from the near-IR (see also Whitelock et al. 1994) to the near-UV between 1974 and 1992 is not due to a star's photosphere, but due to variations in the amount of circumstellar hot gas and dust (van Genderen et al. 1994). This is supported by a near-IR study of Smith & Gehrz (2000). They conclude that the near-IR variations are mainly due to morphological changes of the ejecta in the core and their variable illumination. In this respect it is of interest to mention that according to Viotti & Rossi (1999) the core is likely enshrouded by a $1\hbox{$^{\prime\prime}$ }$- $2\hbox{$^{\prime\prime}$ }$ dense circumstellar cloud. They believe also that in the immediate surroundings of the core, dust is at present condensating from the stellar wind at a very high rate. See for a bipolar nebula older than the Homunculus, Sect. 1.8.

3.3 Discussion on the structure of the SD-area

Figure 21 summarizes the structure of the SD-area. It shows the HD-limit, the triangle in which the s-a variables move to and fro during the SD-phases, the SD-minimum strip with its limits (thinly dashed lines) in which most of the S Dor variables reside, and the Yellow Void for yellow hypergiants. The SD-area between the thinly dashed lines is 0.6 wide in $\log L/L_{\odot}$, or 1 $.\!\!^{\rm m}$5 in $M_{\rm bol}$ and confined to the right of the instability boundary of Stothers & Chin (1996) (vertical dotted line). The close similarity between the slopes of the SD-area and the oblique part of the empirical HD-limit indicates that the SD-phenomenon is somehow related to the Eddington limit as expected (the locus where $g_{\rm eff}$ = 0, a situation favourable for instability and increasing mass loss, e.g. Lamers & Noordhoek 1993; de Koter 1993; Humphreys & Davidson 1994). The SD-upper limit slopes down more deeply to the red than the HD-limit. Figure 21 also shows the atmospheric Eddington limits (AEL) for post-mainsequence stars (PMS) for constant Eddington factors $\Gamma$ = 0.90 and 0.95 (Z = 0.02). Those for Z = 0.008 are not very different (Lamers 1997). Between and above these lines, the atmospheres are expected to become losely bound, thus, for  0.9. These lines fit the upper part of the SD-minimum strip very well (contrary to the lower part). Lamers (1997) speculated that the objects with low luminosity do not come directly from the MS, but should be post-RSG, as opposed to those belonging to the upper part. According to the chemical abundances of the ejecta, with the exception of that of P Cyg, all could be post-RSG (Sect. 1.7). The calculations of Stothers & Chin (1996) predict that most S Dor variables are in their second (thus hot/blue) SD-instability stage, thus, post-RSG, supporting many observational findings. Thus, the reason why the high luminous variables do fit the AEL for PMS stars, while they are possibly post-RSG, is not clear.

3.4 The maximum light amplitude due to the SD-phases

In Fig. 22, the relation between the observed maximum light amplitude ($\triangle$ $V_{\rm max}$), largely based on the schematic light curves presented here, see also Table 16 to be discussed in Sect. 3.5) and the luminosity at minimum apparent brightness ( $\log L/L_{\odot}{\rm min}$) is investigated. Note that such a relation has already been investigated by Wolf (1989) and called the "amplitude-luminosity'' relation. The figure shows 12 s-a- (dots), 3 w-a- (circles) and 1 ex-/dormant (square) S Dor variables. Since most of the observed light amplitudes are lower limits (unless the objects were observed for a century or so, in which case the amplitude can be considered as statistically reliable upper limits), the arrows point to the right. Objects with unreliable parameters have not been plotted. The curve represents the maximum excursions in the s-a SD-triangle from the SD-minimum strip to the boundary at $\log T_{\rm eff}$ = 3.84 at constant luminosity. Thus, from the maximum observed shift in temperature follows $\triangle$BC $_{\rm max}$, which is therefore equal to $\triangle$ $V_{\rm max}$. The higher the luminosity, the larger the expected light amplitude (which agrees more or less with the observations), since the radiation pressure is an important agent of the SD-instability.

Figure 22: The maximum observed light amplitude as a function of the luminosity at minimum apparent brightness. The relation between maximum light amplitudes in the s-a SD-triangle for genuine horizontal displacements on the HR-diagram (thus, then $\triangle$ $V_{\rm max}$ = $\triangle BC_{\rm max}$) is represented by the curved line

Most of the objects lie close to the curve. Part of the scatter must be caused by the fact that $\triangle$ $V_{\rm max}$ is not always equal to $\triangle$BC $_{\rm max}$ (thus, no precise horizontal displacement in the HR-diagram). Another part is due to errors in the physical parameters and the fact that the genuine maximum amplitude has not been observed. Evidently, this maximum amplitude due to the sum of the two SD-phases (S-SD and L-SD phases) can amount to about 2 $.\!\!^{\rm m}$5. Note that this is only slightly less than that for the SD-eruptions of $\eta$ Car and P Cyg!

3.5 The duration of maximum and minimum state

Variables currently in or close to minimum state, especially the w-a and ex-/dormant variables, may be on the verge of ending their S Dor-life, or may have just ended it. However, the probability that this is so, is low, as may be concluded from the following discussion. This seems to indicate that their relative quiescence is in most cases only temporary. Table 16 lists the variables, ordered according to category, of which the probable apparent brightness before 1950 could be traced. The header brightness/activity is divided into two columns: <1950 and 1950-2000, listing whether the objects were in a "high'' or "low'' state. The column "ampl'' lists the observed total light range of the SD-phases, which inevitably is in most cases a lower limit. The column " $t_{\rm trend}$'' lists the probable limit of the duration between "high'' and "low'' brightness of the SD-activity, or vice versa. The column " $t_{\rm low}$'' lists the duration of the "low'' state. The last column lists the reference numbers, or refers to a figure, or tables, in the present paper. It appears that at least 8 objects: viz. 5 s-a-, 2 w-a- and 1 ex-/dormant objects (uncertain cases are bracketed and are not included in this statistic), were presumably "high'' before, in some cases long before, 1950. At least 4 objects: viz. 2 s-a-, 1 w-a- and 1 ex-/dormant objects were presumably "low'' before, or long before, 1950. Of two specimens of the latter group, R40 and R127, together with R110 ("low'' state before 1950 uncertain) we witnessed their developing SD-phases in the last few decades. This means that the precise durations of the "high'' and "low'' states before 1950 are unknown because of the lack of observations. It appears that most of the time scales of the declines and rises ( $t_{\rm trend}$) lie between 20 and 50yr. The durations of the observed non-active stages ( $t_{\rm low}$) are of the same order. $\zeta^{1}$ Sco has already been low for 100yr (Sterken et al. 1997a). For HD160529 the "high'' state ( $t_{\rm high}$ not listed in Table 16) already has lasted 110yr (Fig. 2). To this sample P Cyg can be added, which seems to have been hardly active for perhaps 300yr (however, a very low amplitude oscillation with a time scale of 8yr has been detected by de Groot et al. 2001; this could be an S-SD cycle, see Table 6). Most of the objects have not been monitored photometrically for longer than a few decades (since 1950), thus, it is quite thinkable that most of the current (1950-2000) "low'' or "nearly low'' states are only temporary. It is therefore impossible to say which of the ex-/dormant variables are really "ex'' and on their way to become a WN-type star. If the numbers of variables are counted showing a low activity in the last two-three decades, say with light amplitudes 0 $.\!\!^{\rm m}$2, (thus, difficult to discover), one finds about 24 (5 s-a-, 10 w-a-, 8 ex-/dormant and 1 positive candidate, viz. HD80077). The total number of variables amount to 35 (excluding the other 7 positive candidates, because there is no information on any light variation: most of them are still shrouded by dense circumstellar material). On the other hand, there are only a few which turned out to have been highly, or moderately active for 110yr or more, viz. AG Car (van Genderen et al. 1997a), HR Car (but with a long lasting inactive interval, van Genderen et al. 1997b) and HD160529 (see above), while S Dor (not listed in Table 16) has been active for at least 165yr (to be more precise since at least $\sim$1834, see Thackeray 1974; van Genderen et al. 1997a). Some of the S Dor variables in M 31 and M 33 also show an activity lasting at least 100yr (AF And in M 31, Var.A in M 33), others are only active at intervals of 20yr (e.g. Var.C in M 33) in between periods of quiescence lasting a few decades (Hubble & Sandage 1953; Sharov 1975, 1990; Rosino & Bianchini 1973; Humphreys et al. 1988). Consequently, the numbers and examples above suggest that S Dor variables are for most of their lifetime ($\sim$70%) close to (= w-a), or in a minimum (= dormant) state. Long-lasting active stages (thus, with frequent SD-phases) are rare. Whether S Dor variables can be very active for most of their lifetime is unknown, but obviously such objects are rare too.

3.6 Total numbers of S Dor variables in the Galaxy and the Magellanic Clouds

If the numbers of the three categories (s-a, w-a and ex-/dormant) are complete for the volume of the galactic disc around the Sun (maximum radii 6kpc for the s-a- and 2.5kpc for the other two groups, excluding He3-519 with a possible distance of 8kpc, see Table 9), then the total numbers in the galactic disk can be estimated. Table 17 summarizes the results. For the determination of the multiplication factors ("mult.f.'') the radius of the galactic disk has been taken to be 12kpc with a negligible thickness. There should then be about 200 S Dor variables. If the number for the LMC is complete, which is not certain at all, there are 10 times more S Dor variables in the Galaxy than in the LMC. However, considering that the mass ratio between galactic disk and the LMC is presumably of the same order (80 and $10~10^{9}\,M_{\odot}$, respectively), there is not much difference in their frequency. Assuming that the incompleteness is the same for both Magellanic Clouds, the ratio found for the numbers of S Dor variables as well as for the masses of the two stellar systems one finds LMC/SMC $\sim$6 (the mass for the SMC amounts to $1.5~10^{9}\,M_\odot$). Due to the low number of variables in the SMC, this result is not very reliable.