PULSATION PROPERTIES

Although pulsations in EHes appear to be ubiquitous, it is not possible to summarize their properties with a single definition. Three groups may be identified; future observations will no doubt add to these.

V652Her variables - ``Z-bump'' pulsators. The first discovery of pulsation in an EHe was made by Landolt (1975), who discovered a 0.1day photometric period in V652Her, and by Hill et al. (1981), who measured the radial velocity curve and demonstrated the variations were due to radial pulsation. The pulsation is strictly periodic with regular light and radial-velocity curves. The discovery enabled Lynas-Gray et al. (1984) to deduce a direct mass $M_{\rm D}=0.7^{+0.4}_{-0.3}\mbox{\,$\rm M_{\odot}$}$ , whilst Kilkenny & Lynas-Gray (1982) discovered that the pulsation period was shrinking in a manner consistent with a secular contraction. These properties will be examined later.

Radial pulsations are mostly driven by the $\kappa$ mechanism. This occurs in a zone which gains thermal energy as it is compressed and loses thermal energy as it expands. A zone gains thermal energy if the incoming radiation flux at the lower boundary exceeds the outgoing flux at the upper boundary, i.e. the radiation flux is blocked. This occurs if the increase in opacity caused by the compression increases outwards, i.e. ${\rm d}[\delta \kappa]/{\rm d}r>0$ . The opacity variation due to a nearly adiabatic pulsation is given as

$\displaystyle \delta \kappa=\frac{\partial\kappa}{\partial\rho} \delta\rho +\f... ... d} T}\right)_{\rm ad} + \frac{\partial\kappa}{\partial T} \right] \delta T .$

Prior to 1990, all attempts to model the pulsation in V652Her found the star to be stable. While stars hotter than the classical Cepheid instability strip could show radial pulsations, this was only true if they were considerably more luminous than V652Her (Saio & Jeffery 1988, see below). This problem was overcome with the calculation of stellar opacities which more correctly included the contribution of iron-group elements at temperatures around $2 \times 10^5 \,$ K (Rogers & Iglesias 1992, Seaton et al. 1994). The opacity peak due to iron-group elements, often referred to as the ``Z-bump'', can have a similar effect to the Heii opacity peak at lower temperatures, particularly if the hydrogen-abundance is low. Saio (1993) showed that a `finger of instability' exists for helium stars with $T\sim20\,000\,$ K and which also have a sufficiently high metallicity and luminosity, such as V652Her.

V652Her lies right in the middle of this finger of instability, as do two other stars: HD144941 and LSS3184. If ``Z-bump'' instability was responsible for pulsations in V652Her, then these other stars should also pulsate with similar periods $\sim 0.1 \,$ day. Observations of HD144941 failed to find any evidence of variations (Jeffery & Hill 1996), but this was easily explained by its very low metallicity

(Harrison & Jeffery 1997, Jeffery & Harrison 1997). Prompted by Saio's (1995) prediction, Kilkenny & Koen (1995) discovered a 0.1day photometric period in LSS3184=BXCir and with radial velocities the radial pulsations have been more fully characterized by Kilkenny et al. (1999). The metallicities of V652Her and BXCir have been measured as

(Jeffery et al. 1999) and

(Drilling et al. 1998) respectively.

Jeffery & Saio (1999) have explored the extent of the Z-bump instability finger for radial and non-radial pulsations in terms of mass, metallicity and hydrogen abundance and have shown that it is principally quenched if metallicity is too low ( $Z\mathrel{\raise1.16pt\hbox{$<$}\kern-7.0pt \lower3.06pt\hbox{{$\scriptstyle \sim$}}}0.002$ ) or if the hydrogen abundance is too high ( $X\mathrel{\raise1.16pt\hbox{$>$}\kern-7.0pt \lower3.06pt\hbox{{$\scriptstyle \sim$}}}0.5$ ), for masses in the range 0.3-0.9 $\rm M_{\odot}$ . Since other hot helium-rich subdwarfs lying close to this Z-bump finger are known, it is possible that more V652Her variables remain to be discovered.

PVTel variables - radial ``strange'' mode pulsators. One of the brightest EHes, PVTel was considered to show irregular brightness and radial velocity variations on timescales of weeks, months and years (Walker & Hill 1985). More systematic observations of another EHe, FQAqr, led to the discovery of small-amplitude ( $\sim$ $\mbox{$\rm0\hspace{-0.25em}\stackrel{m}{.} \hspace{-1.0mm}1$}$ ) photometric variations with an apparent period of about $21\,$ day (Jeffery & Malaney 1985). Subsequent observations confirmed the variations, but the period was ambiguous (Jeffery et al. 1986). More recently, five years worth of data demonstrated that variations persist on a characteristic timescale of $\sim21\,$ day, but with no long-lasting coherent period (Kilkenny et al. 1999). These variations are accompanied by small-amplitude velocity variations of a few km $\,$ s $^{-1}$ (Lawson et al. 1993). Similar properties have since been detected in a number of other EHes including PVTel, NOSer, V2244Oph, V354Nor and V1920Cyg (cf. Lawson et al. 1993). This group all have $8\,000\leq T/\,$ K $\leq 15\,000$ , low surface gravities and $7\leq\Pi/\,$ day $\leq 25$ , where $\Pi$ here represents the characteristic timescale.

Variability of similar character but longer $\Pi$ has been recorded in RCrB stars and associated with radial pulsations for some time. These pulsations are reviewed by Lawson & Kilkenny (1996). Whilst RCrB pulsators are relatively cool, EHes are considerably hotter than, for examples, classical Cepheids. Lying to the blue of the classical instability strip, their pulsations are a consequence of the extremely non-adiabatic conditions in the envelopes of stars with high

ratios. Dubbed `strange' modes, the pulsations are primarily associated with regions of density inversion, such as the Heii ionization zone (Saio et al. 1998). Strange modes are characteristically different to $\kappa$ -modes since their frequencies change rapidly with stellar parameters (e.g.

). For EHes and RCrBs, two consequences noted by Saio & Jeffery (1988) are that (i) the stability criterion is effectively provided by the

ratio, and (ii) $\Pi$ and

are related approximately linearly.

The extreme non-adiabacity of EHe envelopes provides a possible explanation for their quasi-periodic behaviour. If the start and end states for each pulsation cycle are not identical, each cycle will not resemble the previous cycle exactly in either amplitude or duration. Over time, the oscillation will forget its history or, effectively, lose phase coherence, even though the local characteristic timescale will be unchanged. The failure of nonlinear calculations of RCrB models to show limit cycles (Saio & Wheeler 1985) supports this proposal, whilst Fadeyev (1993) found considerable disagreement between the results of linear and non-linear calculations. Further non-linear calculations for PVTel and RCrB variables are required.

V2076Oph variables - non-radial ``strange'' mode pulsators. The most luminous EHes with $T>20\,000\,$ K are also small-amplitude variables. The light curves of V2076Oph and V2205Oph are considerably more complicated than those of the PVTel variables and have shorter characteristic timescales of $0.7-1.1\,$ day and $3-9\,$ day respectively (Lynas-Gray et al. 1987, Jeffery et al. 1985). It appears that the variations are multi-periodic, and that the characteristic timescales are longer than anticipated for radial fundamental or first harmonic pulsations. The conclusion is that both stars pulsate non-radially, possibly in a low-order

-mode. Radial velocity measurements support this conclusion, with line-profile variations in V2205Oph indicating

or 3 (Jeffery & Heber 1992).

Linear radial pulsation theory indicates that these stars should be unstable to strange-mode pulsations. However, the most unstable radial mode is no longer similar to the fundamental or first harmonic, but a much higher-order mode (Saio & Jeffery 1988). Glatzel & Gautschy (1992) investigated non-adiabatic non-radial pulsations in a limited helium star evolution sequence, and found strange-mode instabilities at temperatures up to the limit of their study at $T\sim20\,000\,$ K. The similar appearance of the instabilities for radial and non-radial pulsations suggests that non-radial strange-modes may be responsible for the variability in V2076Oph and V2205Oph. However the models used by Glatzel & Gautschy (1992) are less evolved than these stars are likely to be. An important experiment will be to perform linear non-radial pulsation analyses for any evolution models constructed to explain the origin of these EHes.

A major observational difficulty concerns both the multi-periodicity and the extreme non-adiabacity of the pulsations. Existing observations need to be substantially improved both in sampling rate and duration in order to fully resolve the frequency structure of the light curves. However, if the quasi-periodicity of radial pulsations in cooler stars extends to the hotter non-radial counterparts, frequency analyses of long data trains will be doomed from the outset. The observation and modelling of non-radial pulsations in extremely luminous stars (including EHes) presents a major challenge for astrophysics.

**Figure:** Illustration of the merged-binary white dwarf evolution model for V652Her (Saio & Jeffery 2000).
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