5 Infrared radiation

The IRAS satellite detected infrared radiation from all stars of our sample with the exception of RZ Psc. This radiation is unquestionably thermal emission from circumstellar dust and opens a way to estimate the total mass of circumstellar dust by modelling the spectral energy distribution.

We tried to reproduce the observed spectral energy distributions in the infrared by simple spherically symmetric models. The shell is defined by the inner and outer radii of the shell, $r_{\rm i}$ and $r_{\rm o}$, and the density within it is approximated by a power law $\rho(r) \propto r^{-d}$. The nature and optical properties of the dust grains are not known very well. We assume the dust to be a mixture of bare silicate and graphite spheres with the optical constants as given by Draine (1985) and a size distribution $n(a) \propto a^{-3.7}$, resembling the distribution function of Mathis et al. (1977). The stellar radiation is approximated by a black body of the effective temperature of the star.

The relatively small optical depth of the shell is an a-posteriori justification for assuming a smooth density distribution in the model calculations. In the infrared we observe the emission from the whole shell, and since the average optical depth is small, we can expect that geometrical effects due to the cloudy structure are rather small. A related argument can be used to justify the assumption of spherical symmetry. There is growing evidence that circumstellar matter also around Herbig Ae/Be stars is distributed in the form of disks (Waters & Waelkens 1998). However, as long as the matter is optically thin, the actual geometry does not really matter because the differing geometries can be accounted for by changing the exponent d of the density distribution. For instance, an exponent d found by fitting a spherically symmetric model is equivalent to an exponent $d_{\rm disk}=d-1$ of a disk of constant width.

   

Table 5: Characteristics of the best fitting models

	VX Cas	RZ Psca	WW Vul	BH Cep	BO Cepb	SV Cep
Star:
$T_{\rm eff}$ (K)	10000	5250	9520	6400	6900	10500
Luminosity L ( $L_{\hbox{$\odot$ }}$)	53.8	5.5	54.0	7.4	3.85 (7.7)	93.8
Distance D (pc)	850	480	550	400	400 (566)	530
Circumstellar shell:
Inner boundary $r_{\rm i}$ (AU)	1.67	(4.0)	2.7	11.7	250.7 (355)	13.4
Outer boundary $r_{\rm o}$ (AU)	8.42102	(4.4103)	4.7103	$2.48\,10^3$	$2.93\,10^3$ ( $4.14\,10^3$)	5.9103
Exponent of
density distribution d	1.0	(1.5)	1.45	1.35	1.5	1.6
Total mass M ( $M_{\hbox{$\odot$ }}$)	2.910-4	(3.5 $\,10^{-4})$	6.3$\,10^{-4}$	6.210-4	1.210-3 (2.410-3)	7.310-4
Lower limit of
minima duration $\tau_{\rm min}$ (d)	1.9	2.1	2.5	3.9	19.6 (16.5)	5.8

$\parbox{18cm}{\noindent \\ $^a$\space RZ Psc was not detected by IRA... ... in brackets are for the hypothetical central binary of 2 identical F2 stars. }$

Figure 10: Comparison of the observed spectral energy distributions with the model fits. Bars denote upper limits

Table 5 lists parameters of our best spherically symmetric models. For BO Cep we also give the parameters of the model that is based on the binary hypothesis outlined in Sect. 4.2. Figure 10 shows the comparison of the models for VX Cas, BH Cep, and BO Cep with the observations. For similar comparisons for WW Vul, RZ Psc, and SV Cep see Friedemann et al. (1994a).

In an earlier paper we argued that the density distribution may be reflected in the minimum duration distribution if the circumstellar matter is primarily concentrated in the obscuring clouds (Friedemann et al. 1995). The duration of a minimum depends on the orbital velocity of the cloud and, therefore, on its distance from the star. If we approximate distribution of the minimum durations $\tau$, $N(\tau)$, by

$$N(\tau) \propto \tau^{-\alpha},$$ (1)

then the number of clouds per unit volume, $n_{\rm c}(r)$, should follow the power law

$$n_{\rm c}(r) \propto r^{-(\frac{\alpha}{2}+1)}.$$ (2)

Denoting the mean dust mass of a cloud at distance r by $\bar{m}_{\rm c}(r)$, the radial density distribution of the dusty matter concentrated in the clouds is

$$\rho_{\rm c}(r) \propto \bar{m}_{\rm c}(r)\cdot r^{-(\frac{\alpha}{2}+1)}.$$ (3)

From the modelling of the infrared excess by means of spherically symmetric dust shells, we most often found an exponent of $d \approx 1.5$ for the density distribution (see Table 5), implying a power law for the minimum duration distribution with $\alpha \approx 1$, if most of the circumstellar dust is concentrated in clouds and the dust masses within the clouds are independent of their distance from the star. A look at Fig. 7 shows that a power law is rather a bad representation of the distribution function. With the possible exception of SV Cep and RZ Psc, it would predict much more minima of short duration than observed. As we have mentioned above, we could have underestimated the number of shorter minima systematically. Because of gaps in the time series we cannot detect a certain fraction of the shortest minima and we cannot decide whether the data points belong to, say, two short minima or a longer one. If we take the duration distributions at face value, two possible explanations come to mind. First, the dust masses of the innermost clouds could be systematically larger than the masses of the clouds lying farther out. While a larger mass make the cloud more stable against tidal disruption and enlarges its lifetime, the extinction by the cloud and thus the depth of the minimum would also increase. However, there is no indication in the data that the shortest minima have larger amplitudes than the longer ones. Second, the proportion of dust that is not concentrated in clouds but diffusely distributed through the circumstellar shell could be larger in the innermost region of the shell, e.g., as a consequence of a greater instability of the clouds near the star. Here we would like to mention the work by Krivova (1997) on WW Vul, who constructed a model where the infrared radiation is primarily emitted by a diffusely distributed shell with some absorbing clouds producing the Algol-like minima.

The inner boundary $r_{\rm i}$ of the circumstellar shells and the duration of the shortest minima $\tau_{\rm min}$ offer a further connection between the models based on the infrared emission and the properties derived for the cloud ensembles. Assuming circular orbits, the duration of a central occultation by a cloud at distance r from the star is

$$\tau (r) = 4 R_* \sqrt{r/{GM_*}},$$ (4)

where R_* and M_* are the radius and the mass of the star, respectively, and the cloud radius is assumed to be equal R_*. The shortest minima should occur for $r=r_{\rm i}$. The resulting values $\tau_{\rm min}$ are listed in the last line of Table 5. Stellar masses and radii are adopted as appropriate for main sequence stars of the respective spectral class.

Comparing the lower limits $\tau_{\rm min}$ in Table 5 with the histograms of Fig. 7, a large discrepancy occurs in the case of BO Cep. There are smaller differences for BH and SV Cep, too, but we do not think that they are serious enough to argue against the basic assumption, i.e., that cloud ensemble and infrared emitting circumstellar dust shell occupy the same space and are identical. Obviously, this basic assumption has to be abandoned for BO Cep. The infrared emitting shell is located much farther from the star than the objects responsible for the minima. It remains an open question whether this spatial separation is a key to understand the synchronization between the periodic component of the light variations and occasionally occurring deeper minima.