4 Discussion

4.1 The $\mathsfsl{\beta}$ Pic disk

For widely adopted parameters, the stellar disk subtends an angle in the sky of less than 1 mas and the photosphere generates a flux density at the Earth of less than 1 mJy at 1200 $\mu$m. The stellar contribution to our SEST measurements can therefore be safely ignored. Also, any line emission in this band pass is likely to be totally negligible (Liseau & Artymowicz 1998; Liseau 1999 and references therein).

Assuming an opacity law of the form $\kappa_{\nu}= \kappa_0~(\nu/\nu_0)^{\beta}$, the flux density at long wavelenghts from an optically thin source of a certain dust population can be expressed as

$$F_{\nu} = \frac {2 k \kappa_0 \Omega}{c^2 \nu^{\beta}_0}~\nu^{2+\beta} \int T_{\rm dust}(z)~{\rm d}z~~ .$$ (1)

If the particles dominating the emission at 850 $\mu$m and 1200 $\mu$m, respectively, can be assumed to yield the same integral, e.g. because they share the common temperature $T_{\rm dust}$ and/or occupy similar locations in space along the line of sight z, the average spectral index $\beta$ can be obtained from our 1200 $\mu$m and the 850 $\mu$m fluxes by Holland et al. (1998), using

$$\beta = - 2 + \frac{ {\rm d}\log {F_{\nu}} }{ {\rm d}\log {\n... ...frac { \log { (F_{850}/F_{1200}) } }{ \log {(1200/850) } }~~ ,$$ (2)

yielding $\beta = 0.5$ for a point source at the stellar location. The index becomes $\beta = 1$, if we use the fluxes for the extended source, viz. integrated over a radius of 40 $^{\prime \prime }$ centered on the star. This includes blob B, contributing some 20% to the 850 $\mu$m flux. A "correction'' for this would again indicate a lower $\beta$ value and we conclude that the dust in the $\beta ~ {\rm Pic}$ disk has a shallow opacity index, perhaps even below unity (Dent et al. 2000 suggest $\beta = 0.8$). This is illustrated in Fig. 3, which displays the long wavelength spectral energy distribution of $\beta ~ {\rm Pic}$, together with weighted Rayleigh-Jeans spectra for $\beta= 0,~1~{\rm and}~2$, and could mean that the grains dominating the millimeter-wave emission are different from those scattering most efficiently in the visual and the near infrared. This value of $\beta$ is significantly lower than those found in the interstellar medium (ISM), where typically $\beta \sim 2$ (Hildebrand 1983), but it is similar to that found in protostellar disks (e.g., Beckwith et al. 1990; Mannings & Emerson 1994; Dutrey et al. 1996), indicating significant differences between the dust particles in the ISM and those in the $\beta ~ {\rm Pic}$ disk. As was also already concluded by Chini et al. (1991), the presence of relatively large grains is suggested, with maximum radii in excess of 1 mm ( $\max{a}/\lambda$  $\stackrel {>}{_{\sim}}$ 1). Intriguing, however, is the existence of such large grains possibly as far away as 1000 AU from the star (see, e.g., Takeuchi & Artymowicz 2001; Lecavelier des Etangs et al. 1998).

 

 

Table 1: SIMBA 1200 $\mu$m flux densities of the $\beta ~ {\rm Pic}$ disk.

Feature	Relative	$F_{\nu}(1200$ $\mu$m)	Remarks
	Offset ( $^{\prime \prime }$)	(mJy/beam)
A	(0, 0)	$24.3 \pm 3.0$	$\beta$ Pic disk
		$35.9 \pm 9.7$	integrated over a radius of 40 $^{\prime \prime }$
B	(-21, -26)	$\ldots$	SW blob (Holland et al. 1998):
			contaminated by $\beta ~ {\rm Pic}$ disk
C	(-27, -44)	$9.7\pm 3.0$	SW blob (this paper)

4.2 Weak dust features in the 1.2 mm image

The asymmetric flux distribution displayed in Fig. 2 may be surprising. In agreement with our observations, Chini et al. (1991) and Dent et al. (2000) too were unable to detect any emission in the northeast part of the disk, where we place a $3 \sigma$-upper limit on the mass of 0.2  $M_{\oplus}$ (see below; the dust temperature at 500 AU $T_{\rm dust} = 45$ K, when $T_{\rm dust}(r) = 110~(r/26~{\rm AU})^{-0.3}$ K, see Liseau & Artymowicz 1998).

Given the low signal-to-noise ratio (S/N), the reality of this lopsidedness is difficult to assess, but asymmetries in the $\beta ~ {\rm Pic}$ disk have been noticed also at other wavelengths. For instance, in scattered light, the receding NE side of the disk extends much further and is much brighter than the SW disk. In contrast, the shorter, approaching SW disk seems much thicker (Kalas & Jewitt 1995; Larwood & Kalas 2001). The situation is reversed in the thermal infrared (albeit on smaller spatial scales), where the SW disk appears significantly brighter and more extended than the NE side (Lagage & Pantin 1994; Wahhaj et al. 2002; Weinberger et al. 2002). This could be due to a "Janus-effect'', i.e. the NE being dominated by "bright'' dust particles (high albedo, silicates), whereas in the SW, the majority of dust grains is "dark'' (high absorptivity, carbonaceous?). What would accomplish such uneven distribution in the disk is not clear, but large differences in albedo, by more than one order of magnitude, are not uncommon, for instance, in solar system material. Also, in order to understand the nature of feature C (and B) velocity information would be valuable.

Blob C would be situated in the disk midplane and on the second contour in the scattered light image of Larwood & Kalas (2001) ( 22 < R < 25 mag/arcsec², see also Kalas & Jewitt 1995). No obvious distinct feature is seen at its position. However, to be detectable with SIMBA at the SEST, any point source at mm-wavelengths would not be point-like at visual wavelengths, and its optical surface brightness could be very low.

To gain some quantitative insight we ran numerical models, exploiting Mie-theory, for a variety of plausible dust mixtures regarding the chemical composition and grain size distributions (for details, see Pantin et al. 1997). The equilibrium temperatures were found to be in the interval 16 K to 58 K and depending on the dust albedo and scattering phase function, the predicted integrated scattered light, to be consistent with our SEST observations, spans 8 magnitudes. The two most extreme cases considered were (1) bright cold dust (albedo = 1 at visual wavelengths, $T_{\rm dust}=16$ K) which is scattering isotropically and has albedo = 0.2 at thermal wavelengths, which results in a spatially integrated R-magnitude of 17.3 and (2) dark warm dust (albedo = 0.02 in the visual, $T_{\rm dust}=58$ K) which gives rise to "comet scattering'', i.e. 14% of isotropic at 90$^{\circ}$, and has zero albedo at thermal wavelengths, resulting in an integrated R-magnitude of 25.6. These extreme cases are felt to be either too optimistic or too conservative and an intermediate case might be more appropriate. Our adopted model includes isotropically scattering dust at $T_{\rm dust}=25$ K with albedo = 0.2 at visual and zero albedo at thermal wavelengths, yielding an integrated R-magnitude of 20. An about 10 $^{\prime \prime }$ source (R=25 mag/arcsec²) would thus be consistent with both the optical data and our SIMBA measurement and a deep R-band search might become successful. Similarly, the integrated 850 $\mu$m flux density is predicted to be slightly less than 18 mJy and should become readily detectable. However, blob C is situated outside the figure of Holland et al. 1998.

Feature C is perfectly aligned with the optical disk plane, and its (hypothetical) mass can be estimated from

$$M_{\rm dust} = \frac{D^2~F_{\nu}}{\kappa_{\nu}~B_{\nu}(T_{\rm dust})}~~ ,$$ (3)

where the distance D is assumed to be that of $\beta ~ {\rm Pic}$ and the adopted dust absorption coefficient $\kappa_{250~{\rm GHz}} \sim 1$ cm² g^-1. This estimate of $\kappa_{\nu}$ is probably correct within a factor of three (Beckwith et al. 2000 and references therein). For the dust temperature $T_{\rm dust}=25$ K, the dust mass is of the order of ten lunar masses (0.16  $M_{\oplus}$), which would be comparable to the dust mass of the disk proper, being overall much warmer. Another factor of about three uncertainty stems thus from the dust temperature, provided $10~{\rm K} < T_{\rm dust} < 60$ K.

Figure 3: The SED of the $\beta ~ {\rm Pic}$ disk from 10 to 1300 $\mu$m. Data up to 200 $\mu$m are from Heinrichsen et al. (1999) (52 $^{\prime \prime }$ to 180 $^{\prime \prime }$). The datum at 800 $\mu$m is from Zuckerman & Becklin (1993) (integrated over a radius = 25 $^{\prime \prime }$). The 850 $\mu$m data are from Holland et al. (1998) and refer to a 14 $^{\prime \prime }$ beam and to the integrated flux over a radius = 40 $^{\prime \prime }$. The latter should be comparable to our 1200 $\mu$m point for 40 $^{\prime \prime }$; the lower is for a 25 $^{\prime \prime }$ beam, and that at 1300 $\mu$m (24 $^{\prime \prime }$ beam) is from Chini et al. (1991). For reference, the spectral slopes longward of 100 $\mu$m for three values of the dust emissivity index $\beta$, discussed in the text, are shown by the straight lines.

The famous "SW-blob'' in the 850 $\mu$m image has received considerable interest by the debris disk community. According to Dent et al. (2000), this feature, labelled B in Fig. 2, is real. It is not readily apparent in our 1200 $\mu$m image, however, presumably due to the combination of low contrast and reduced angular resolution. To test this idea, we performed numerical experiments, i.e. convolving two point sources, at the appropriate positions of A and B and with varying flux ratios, with the SIMBA beam. Flux ratios, normalised to the peak value, which are consistent with our observations are in the range 0.25 - 0.45, with the most compelling being about 0.3 (comparable to that for blob C). In combination with feature C, blob B accounts for the southward bridge seen in Fig. 2.

For radiation mechanisms generating power law spectra and/or thermal dust emission from blob B, the spectral slope is given by $\alpha = \beta_{A} + 2 - \Delta \log R_{\lambda}/\Delta \log \lambda$, where $\beta_{A}$, as before, refers to A = (0 $^{\prime \prime }$, 0 $^{\prime \prime }$), i.e. the $\beta ~ {\rm Pic}$ disk, and where $R_{\lambda} = \left [ F_{\nu}(A)/F_{\nu}(B) \right ]_{\lambda}$ and $\lambda = 850$ $\mu$m or 1300 $\mu$m. Because of the relatively low S/N of the SCUBA and SIMBA data, the actual flux ratios are highly uncertain and, furthermore, calibration uncertainties and telescope beam effects could potentially introduce large errors. The combined observations of blob B suggest $\log { (R_{850}/R_{1200}) } \sim 0$, yielding $\beta_{B} \sim \beta_{A}$, i.e. consistent with the spectrum of A.