3 Upper limits on circumpulsar masses

We used a simple model to derive upper limits for the amount of circumstellar material in the form of grains. In the absence of indications on the dust composition provided by an accurate infrared spectrum of the dust, we assumed that the dust is composed of interstellar grains as described by Draine & Lee (1984). From the optical constants for this material (a mixture of silicates and graphite with a ratio of $\sim$1.1 by particle number), one computes the mean absorption coefficients $Q_{\rm abs}$ of spherical particles as a function of the wavelength and the particle size using calculations based on the Mie theory (Bohren & Huffman 1983). Using the standard collisional size distribution, i.e. n(a) = A a^-3.5 (Mathis et al. 1977), where the constant A ensures the proper normalization of the distribution, the flux radiated by a set of N particles at temperature $T_{\rm g}$ can be written as

$$% F_{\nu}(\lambda) = N \int_{a_{\rm min}}^{a_{\rm max}} 4 \pi... ...}(\lambda,a) \pi B_{\nu} (\lambda,T_{{\rm g}})~ n(a)~ {\rm d}a$$ (1)

where $a_{\rm min}$ and $a_{\rm max}$ are the minimum and maximum sizes of the grains set to 0.01 $\mu$m and 1000 $\mu$m, respectively, $B_{\nu}$ is the Planck function for blackbody emission per unit frequency. A lower cut-off size of 0.01 $\mu$m corresponds to the minimum size considered in dust emission models by Lazio et al. (2001); this minimum size is also comparable to the minimum grain size inferred for the interstellar medium dust grains (Mathis & Whiffen 1989). The maximum size is arbitrarily fixed to 1 mm, a size above which the integrated emission of the dust over the wavelength of interest (roughly 5 $\mu$m to 3 mm) is 10^-13 times lower than the integrated emission of the particles with sizes in the range 0.01 to 1000 $\mu$m (which means that we have currently no constraints on the mass of particles bigger than 1 mm). The influence of the minimum cut-off size is studied in Fig. 1 in which upper limits on the dust mass are plotted for the case of PSR B1534+12, parametrized by the minimum cut-off size.

Figure 1: Influence of the minimum cut-off in the grain size distribution on the upper limits of dust masses derived for PSR B1534+12 in the temperature range of 10-1500 K; cut-off values: a = 0.01 $\mu$m, b = 0.05 $\mu$m, c = 0.1 $\mu$m, d = 0.5 $\mu$m, e = 1 $\mu$m, f = 5 $\mu$m, g = 10 $\mu$m, h = 50 $\mu$m, i = 100 $\mu$m.

The range of circumpulsar mass limits allowed in the above model by our ISO data and the IRAS/Scanpi data, and for B1534+12 and B1257+12 by the published sub-mm and mm data are shown in Fig. 2.

Figure 2: Allowed range of upper limits of circumpulsar masses for dust temperature between 10 K and 1500 K; continuous line: our model; dashed line: the Foster & Fisher model (1996).

Each point in the plots of Fig. 2 represents an upper limit on the mass for a given temperature, the range of temperatures being chosen between 10 K (typical lower temperature of interstellar cold dust) and 1500 K (sublimation temperature of silicate dust). For each temperature, a probability density (coded by a grey-level on the left bars in the plots, the color of the points being reported on the bar) is computed by combining the partial probability density functions for each data point. A data point with a true value is assumed to follow a Gaussian partial probability density function with a standard deviation deduced from the error on each data point. A data point which corresponds to a lower limit is assumed to follow a half Gaussian-like partial probability density function for values greater than the data point value and an uniform probability density function for lower values. Most probable values for the temperatures are shown in the plots as the brightest points. The error bars overplotted correspond for each temperature point to 1/1000 of the maximum density of probability. Although the detections obtained in the IRAS beam at the position of pulsars are probably chance coincidences (van Buren & Tereby 1993), their influence on the most probable temperature of the grains is clearly seen in Fig. 2. The extrema of dust mass upper limits corresponding to temperatures 10 K and 1500 K are shown in Table 3.

We have also tested the global dust heating model used by Foster & Fischer (1996) which assumes that a fraction of the pulsar's spin-down luminosity is heating a dust disc and gives a relation between the total dust mass in the disc and the temperature. This dependance is shown in Fig. 2 as a dashed line overplotted. The pulsars' spin-down luminosities are shown in Table 1; for each pulsar, the parameter f expressing the fraction of spin-down luminosity converted into dust thermal energy is taken as 1 percent (Foster & Fischer 1996). Figure 2 shows that there is a temperature $T_{\rm cr}$ corresponding to the same upper limit of circumpulsar mass in the two models, if we allow f to increase slightly above 1 percent. These temperatures $T_{\rm cr}$ are shown in Table 3 together with the corresponding upper limits of circumpulsar masses $M_{\rm cr}$ in solar mass units.

 

 

Table 3: Upper limits on mass of emitting dust around pulsars computed at temperatures T_g = 10 K and 1500 K; and upper limits on mass at a temperature $T_{\rm cr}$ deduced from the model of Foster & Fischer (1996).

Pulsar	$M_{\rm 10K}$	$M_{\rm 1500K}$	$T_{\rm cr}$	$M_{\rm cr}$	$M_{\rm cr}$/$M_{\odot}$
	(kg)	(kg)	(K)	(kg)
B1534+12	<1027	<1020	10	<1027	< $5 \times 10^{-4}$
J2322+2057	<1030	<1020	30	<1026	< $5 \times 10^{-5}$
J2019+2425	<2 1030	<1020	30	<1026	< $5 \times 10^{-5}$
B0149-16	<1030	<1020	30	<1025	< $5 \times 10^{-6}$
B1604-00	<1030	<1020	30	<1025	< $5 \times 10^{-6}$
J0108-1431	<1028	< $2\;10^{18}$	30	<1023	< $5 \times 10^{-8}$
B1257+12	<1024	<1019	60	<1023	< $5 \times 10^{-8}$

We note that the latter upper limit of circumpulsar mass for PSR B1534+12 is 30 times smaller than the upper limit of $1.6\times 10^{-2}$ $M_{\odot}$ obtained by Phillips & Chandler (1994) in the sub-mm and mm ranges, using the Beckwith et al. (1990) results on circumstellar discs around T Tauri stars. Greaves & Holland (2000) using their upper limits of flux at 850 $\mu$m for B1534+12 and B1257+12, and the Foster & Fisher (1996) model with grain size 100 $\mu$m and a spin-down luminosity set at $2\times 10^{34}$ erg/s for both pulsars, deduced upper limits to disc masses typically lower than 10 Earth masses i.e. < $3\times 10^{-5}~M_{\odot}$.