Astron. Astrophys. 338, 340-352 (1998)

2. The thermophysical model

The thermophysical model of asteroids used here has been discussed in detail by Lagerros (1996, 1997, 1998). In essence, the surface temperature is calculated from the energy balance between absorbed solar radiation, the thermal emission, and heat conduction into the surface material. The disk integrated model flux at the wavelength is

where is the distance to the observer, the Planck function, and the direction cosine µ projects the surface element dS towards the observer. The "beaming function" , and the wavelength- and direction-dependent emissivity are discussed below.

The most important aspects of the model parameters are outlined in the following:

2.1. Shape, size and spin vector

An estimate of the size and shape of the asteroid is required for the surface integral in Eq. (1). Essentially any shape can be used in the model, from perfectly spherical to highly irregular. In this case (except for 4 Vesta as discussed below) the shapes are assumed to be ellipsoids. For the shape and spin vector the following input parameters are used by the model:

For asteroids of the sizes and rotational periods considered here, the damping time scale for tumbling asteroids to principal-axis rotation is short, compared to the collision time scale (Burns & Safronov 1973). Thus, the spin vector is assumed to be given in a right-handed system and parallel to the semi-minor axis c.

The absolute size is a key physical property, since from Eq. (1) the disk integrated flux is proportional to the projected area () at all wavelengths.

In order to predict the flux at any given time it is necessary to know the orientation and position of the asteroid relative to the Sun and the observer. The positional data is given from celestial mechanics at a very high precision.

The orientation can be computed for any time if the spin vector and the absolute rotational phase at an epoch are known. Together with the axis ratios it is possible to predict the projected area of the ellipsoid at any given time, and hence the light curve.

2.2. Albedo and emissivity

The temperature at a given point on the surface is determined by several factors. The albedo gives the fraction of reflected solar radiation, while the remaining fraction is absorbed and heats the surface. The cooling is determined by the wavelength dependent hemispherical emissivity. The model parameters are:

where G is the slope parameter for the phase curve in the HG-magnitude system (Bowell et al. 1989). Together with the geometric albedo this gives the Bond albedo, which is assumed to be close to the bolometric albedo.

The emissivity is used for two purposes. First, the hemispherical emissivity averaged over wavelength (weighted by the Planck function) gives the "bolometric emissivity". Secondly, the is tied to the volume single scattering albedo of the regolith particles. By this it is possible to derive the directional emissivity used in Eq. (1).

2.3. Heat conduction

Heat conduction has the effect of lowering the amplitude of the diurnal temperature variations. In general, by increasing the thermal conductivity, more energy is emitted on the night side at the expense of the day side temperature. There is also a morning / afternoon asymmetry in the flux due to the thermal inertia. The model parameter for the heat conduction is

where , with being the thermal conductivity, the density and the heat capacity of the surface material.

The heat conduction is much more important at the shorter IR wavelengths. The importance for the far-IR calibration is that mid-IR ground-based photometry is used, for which the thermal inertia have to be taken into account.

2.4. The thermal IR beaming

The thermal emission from atmosphere less solar system bodies like the Moon or the asteroids has a tendency to be "beamed" into the solar direction. This behaviour is probably due to the large surface roughness and porosity. The beaming function corrects the temperature for this effect in Eq. (1). The beaming function is calculated by adding spherical segment craters (Lagerros 1998) to a smooth surface. The roughness is described by the model parameters

The beaming is a very noticeable effect, and can be studied by investigating the non-Lambertian phase curves of asteroids.

2.5. Multiple scattering

Many asteroids are probably rather irregular, which means that there can be multiple scattering and mutual heating between well separated parts of the surface. Multiple scattering can be taken into account (Lagerros 1997) by adding a term for the scattered field to Eq. (1). This option is not used here, since in general it makes only a minor effect, and the shapes of the asteroids are not known well enough to model it in detail.

© European Southern Observatory (ESO) 1998

Online publication: September 8, 1998
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