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3 Models

The emission of asteroids beyond $\sim$$\mu $m is dominated by radiation thermally emitted from their surface. A blackbody fit and the Standard Thermal Model (STM) have been applied to the obtained infrared data to determine the range of surface (blackbody and sub-solar) temperatures.

To determine the black-body temperature of the observed asteroids we fitted the infrared data with a Planck function multiplied by the solid angle of the objects. Both the solid angle and the black-body temperature were treated as free parameters.

The sub-solar temperature of the observed objects was computed by applying the STM (Lebofsky & Spencer 1989), already used by Tedesco et al. (1992) in the IRAS asteroid survey. It assumes a non-rotating spherical asteroid, in instantaneous equilibrium with solar insolation, observed at 0$^{\circ}$solar phase angle. In this ideal situation, in which the thermal inertia is neglected, and the asteroid nightside emission is thus not taken into account, the sub-solar temperature is given as:

\begin{displaymath}T_{\rm SS} = \left[ \frac{(1-A) S} {\eta \epsilon \sigma} \ri...
... ^{1/4}
{\rm and~~~} T(\Omega) = T_{\rm SS} \cos^{1/4}(\Omega)
\end{displaymath} (1)

where $\Omega$ is the solar zenith angle, A is the bolometric Bond albedo, S is the solar flux at the distance of the asteroid, $\eta$ (infrared beaming) is an empirical factor adjusted so that the model matches the integrated flux of the object at a given wavelength, $\epsilon$ is the wavelength-independent emissivity and $\sigma$ is the Stefan-Boltzmann constant. Moreover, the infrared beaming and the phase angle geometry are empirically corrected, generally with correction factors of $\eta = 0.756$ and 0.01 mag deg-1, respectively (see Lebofsky & Spencer 1989).

In Table 2 the sub-solar and black-body temperatures, and the albedo and diameter values estimated by applying STM to our data are reported. The table shows also the absolute magnitude H and the slope parameter G used as input, and diameters and albedos obtained by Tedesco et al. (1992) on the basis of IRAS data. Our estimations of diameters and albedos are in agreement with the values given by Tedesco et al. (1992), with the exception of the albedos of 511 Davida and 914 Palisana which deviate from the IRAS results. This can be due to differences in the viewing geometry between the IRAS and ISO observations. In the simplified STM the real shape of the asteroid is not taken into account: this can influence the albedo and diameter calculations. To model the thermal continuum of previous ISO observations, Dotto et al. (2000) and Barucci et al. (2002) used the advanced thermophysical model (TPM) developed by Lagerros (1996, 1997, 1998). To properly apply this advanced model we need the knowledge of several physical parameters of the analysed asteroids. Unfortunately, we do not have a good estimation of the pole direction, shape, infrared beaming and thermal inertia of the five asteroids here discussed. A comparison between the diameter and albedo values obtained by STM and TPM has been possible only for 77 Frigga, for which an estimation of the rotational state is available (Erikson 2000). We applied TPM, considering the wavelength dependent emissivity as stated by Müller & Lagerros (1998), and default values for thermal inertia and beaming parameters $\rho$ and f, and we obtained results similar to those computed by STM. Since the rotational and physical parameters of the asteroids here discussed are not sufficiently well known, we preferred to apply the simplest STM, with the minimum number of free parameters, without introducing new possible sources of error.

Using STM we computed the expected flux at the time of ISO observations. Then we divided the observed spectra for the STM expected flux, obtaining the "Relative Obs/Mod''.

 

 
Table 2: Absolute magnitude H, slope parameter G, computed temperatures and diameter and albedo values, IRAS diameters and albedos (D and pH) with their uncertainties ($\sigma _D$ and  $\sigma p_H$).

Object
H G Diameter Albedo Black-body Sub-solar IRAS IRAS IRAS IRAS
          temp. temp. D $\sigma _D$ pH $\sigma p_H$
  (mag)   (km)   (K) (K) (km) (km)    

77 Frigga
8.52 0.16 $70 \pm 4$ $0.146 \pm 0.005$ 223 259 69.25 2.1 0.144 0.009
114 Kassandra 8.26 0.15 $103 \pm 4$ $0.084 \pm 0.005$ 228 265 99.64 1.9 0.0884 0.003
308 Polyxo 8.17 0.21 $ 151\pm 7$ $0.043 \pm0.002$ 232 265 140.69 3.8 0.0482 0.003
511 Davida 6.22 0.16 $303 \pm 8$ $0.064 \pm 0.003$ 232 268 326.07 5.3 0.054 0.0023
914 Palisana 8.76 0.15 $71 \pm 7$ $ 0.113 \pm 0.004$ 220 257 76.61 1.7 0.0943 0.004



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