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Astron. Astrophys. 332, 1123-1132 (1998)

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2. Beaming physics

The thermal models outlined in the Papers I-III divides the surface of the asteroid into typically a few thousand planar facets. The facets are heated by the Sun and scattered visual and thermal radiation from neighbouring facets. As the asteroid rotates, much of the heat is absorbed during the daytime and is later re-emitted on the night side, due to the heat conduction into the material below the surface of the facets. The predicted flux at a given wavelength [FORMULA] and distance D from the asteroid, is computed by integrating the contribution from each facet, such that

[EQUATION]

where [FORMULA] is the intensity from a point S on the surface, in the direction towards the observer, and µ the directional cosine of the facet dS. If dS is out of sight, [FORMULA] by definition. The intensity [FORMULA] is essentially the Planck function, multiplied by the wavelength and directional dependent emissivity.

The problem with this, however, is that there will be substantial temperature variations even on small scales due to the rough terrain. The emission from a facet is not that of a single black body radiator at a given temperature.

Here the assumption is that the beaming effects are caused by the surface roughness on a very small scale, but still large enough for the geometric optics approximation to be valid. On a scale too small, the temperature variations will be smeared by the heat conduction. Therefore the methods outlined in Paper III can be applied here, but now on a much smaller scale.

Roughness is added to a small part of the planar facet. The coordinate system used is to let the xy -plane coincide with the facet, and have the z -axis pointing outwards from the surface. The rough surface is described by [FORMULA]. The emission from this rough patch is then compared to that of a flat surface with the same projected area, which gives a correction factor for the whole facet.

Both approaches to the surface roughness discussed here are isotropic, i.e. no preferred azimuth direction. In the first case are craters shaped as hemispherical segments covering a given fraction of the flat surface. The second is a Gaussian random surface, completely described by the r.m.s. slope [FORMULA].

As in Paper III, the spectrum is divided into the visual (V) where the solar irradiation dominates, and the infrared (IR) where the thermal emission takes place. The temperature variations over the rough surface is determined by solving the multiple scattering problem in the V and IR bands. Both internal and external shadows have to be considered. The heat conduction into the regolith is taken into account, but only in one dimension. The observed flux in different directions is compared to the flux from a smooth surface with the same physical properties.

2.1. Notation

The notation used here is given in Table 1. Since it is preferable to work in normalised units:


[TABLE]

Table 1. The notation.


[EQUATION]

With this the Planck function becomes

[EQUATION]

2.2. Radiative transfer

View factors have been used several times before in order to solve the radiative heat transfer problem on rough surfaces in the solar system (Hansen, 1977; Spencer, 1990; Vogler et al., 1991; Johnson et al., 1993; Lagerros, 1996a). A more general introduction is given by Modest (1993). The view factor from a facet da at a point [FORMULA] to [FORMULA] at [FORMULA] is defined as the fraction of the radiative energy leaving the first and directly striking the latter. Assuming Lambertian facets, the view factor is

[EQUATION]

where [FORMULA] and [FORMULA] are the corresponding surface normals, and [FORMULA] is 1 if the two facets are visibile to each other, otherwise it is taken to be 0.

The radiosity [FORMULA] is the total flux of visual light reflected from a point [FORMULA]. This is given by multiplying the total irradiation at [FORMULA] by the reflectance [FORMULA]. Since the irradiation equals the insolation plus the radiosity from other parts of the surface, it follows that

[EQUATION]

For the IR the source term of the thermal emission is [FORMULA], by which

[EQUATION]

The temperature u is determined by the energy balance between the energy thermally emitted, the heat conducted into the regolith, the radiation from the neighbouring facets and the Sun, such that

[EQUATION]

where the gradient [FORMULA] is for Cartesian coordinates normalised by the thermal skin depth [FORMULA]. Note the dependency on the thermal history - as the asteroid rotates - of u in Eqs. (5) and (6), if [FORMULA]. On the other hand, if heat conduction can be neglected due to the slow spin rate of the body, or the low conductivity of the regolith, the thermal parameter is [FORMULA]. By combining Eq. (5) and Eq. (6) it follows that

[EQUATION]

These Fredholm integral equations of the second kind have to be solved starting with [FORMULA] in Eq. (4). If there is no heat conduction, [FORMULA] is given by Eq. (7). Otherwise, the IR radiosity has to be derived by iterating between Eq. (6) and Eq. (5), as described in Paper III. Only in a few special cases is it possible to find analytical solutions. In general the rough surface has to be divided into a finite number of facets. By this the kernel [FORMULA] is replaced by the view factor matrix [FORMULA], and the integral equations are turned into matrix equations. A high precision method for calculating the view factor between finite triangular facets is given in Sect  A.1.

The final step is to compute the integrated flux from the surface towards the observer:

[EQUATION]

The model described here allows for inhomogeneous surface material since both the absorptivity and emissivity may vary over the surface (e.g mineral mixtures). This possibility is not explored any further. The [FORMULA] and [FORMULA] are taken to be constant over the surface, which also results in constant sub-solar temperature [FORMULA], and thermal parameter [FORMULA].

2.3. Spherical craters

As discussed above, spherical "craters" on a smooth surface have been used several times as a surface roughness model. Typically the hemispherical segments cover a fraction f of the surface. The shape of the craters is described by S, the depth of the segment divided by the diameter of the sphere. Without loss of generality, the radius of the sphere can be taken to be unity. The surface area of the segment is then [FORMULA], and the area of the reference surface or the projected area of the crater rim is [FORMULA].

Due to the symmetries of the sphere it is possible to solve the integral equations analytically. Buhl et al. (1968) made use of this, but assumed the emissivity to equal 1, and neglected the scattered visual radiation. The approach here is more general.

The view factors inside a sphere (Modest, 1993) are given by

[EQUATION]

if [FORMULA] in spherical coordinates, with [FORMULA] at the bottom of the crater. This means that the kernel [FORMULA] in the integral equations above is independent of [FORMULA], and thus also the integrals.

Thus Eqs. (4), (5) and (6) can be written in respective order:

[EQUATION]

where [FORMULA], [FORMULA] and [FORMULA] are constants. Again, if heat conduction can be neglected, Eq. (7) can be used, and is transformed into

[EQUATION]

where [FORMULA] is a constant. All these constant can easily be found by using the same procedure. For example, solving Eq. (10) for [FORMULA] and substituting it into the definition of the constant [FORMULA] gives

[EQUATION]

It is useful to note that

[EQUATION]

where the latter expression comes from the fact that the total energy entering the crater has to pass the projected area of the crater rim. From this it follows that

[EQUATION]

Thus the radiosity [FORMULA] equals the local directional cosine [FORMULA] plus a constant radiation field, regardless of the position in the hemispherical segment "crater". Again, a solution can be found by first calculating [FORMULA] and then iterating between Eqs. (11) and (12).

With no heat conduction

[EQUATION]

and from Eq. (13),

[EQUATION]

That is, the temperature is determined by the direct solar irradiation and a constant term, which is the same for the whole crater.

If [FORMULA] is the zenith distance ([FORMULA]) and [FORMULA] is the azimuth of the Sun, the local directional cosine is given by first calculating

[EQUATION]

and then taking care of the shadow from the crater rim

[EQUATION]

The same method of course applies to [FORMULA].

The final step is to compute the observed flux from Eq. (8)

[EQUATION]

This flux should then be compared to the flux from the reference surface

[EQUATION]

where [FORMULA] if heat conduction is neglected.

The procedure is then basically to calculate u from for example Eq. (18) and then numerically integrate the flux from Eq. (19). If the crater coverage is a fraction f, the flux to be compared with the reference flux [FORMULA] is simply [FORMULA].

An important issue is correcting the albedo for the surface roughness. For a spherical asteroid, the Bond albedo equals the bi-hemispherical reflectance. That is, the fraction of incoming light scattered in all directions from the surface, if isotropically illuminated. For the spherical crater this isotropic field [FORMULA] is passing through the crater rim. From Eq. (9) and basic view factor algebra, the source term [FORMULA] in Eq. (10) is replaced by [FORMULA]. Using view factor algebra, the local radiosity [FORMULA] adds up to the flux [FORMULA] passing out from the rim. From this the bi-hemispherical albedo of the spherical crater is

[EQUATION]

Taking the crater coverage f into account, the bi-hemispherical reflectance becomes

[EQUATION]

since geometric optics is assumed.

The roughness of the Gaussian random surfaces to be discussed next is measured by [FORMULA], the r.m.s. of the slopes on the surface. For the hemispherical segment model, the corresponding value is

[EQUATION]

which will be used when comparing the models. Note that this measure of the roughness fails when the segments becomes hemispheres, since [FORMULA] when [FORMULA].

2.4. Gaussian random surfaces

A real rough surface is the result of a number of stochastic processes, rather than perfect craters forming on a flat surface. A more realistic model is probably to assume some random distribution of surface slopes or heights relative to a mean reference surface. A method to generate stochastic surfaces is discussed in Sect. A.2.

The Gaussian random surfaces in Sect. A.2.are controlled by only one parameter, the r.m.s. slope [FORMULA]. The algorithm gives the height [FORMULA] above the mean plane in a square grid. Each side in the grid has a length [FORMULA] and consists of [FORMULA] points. As illustrated in Fig. 1 the surfaces are triangulated by simply dividing each square of neighbouring points in the mesh into two triangles. The result is a rough surface described by a discrete set of facets, with well defined surface areas and normals.

[FIGURE] Fig. 1. A sample Gaussian random surface with r.m.s. slope [FORMULA], described by 2 048 triangles. The solar radiation comes from a zenith angle of [FORMULA]. Multiple scattering in the visual and the IR is taken into account, and the surface temperature is indicated by the gray scale from black (cold) to white (hot).

All points within the spherical crater are visible to each other. The only shadows to consider are the "external", those created by the crater rim when viewed or illuminated from an external point. For the stochastic surfaces the visibility of a given point to other points on the surface or positions outside the surface is more complex. A given point is obviously not visible if its surface normal is oriented away from the other position in question. If the two facets are facing each other, it is necessary to check for other triangles obstructing the connecting line. This was achieved by using standard ray tracing methods from the computer literature (Glassner, 1990, Chap. 7).

Furthermore, to avoid unwanted effects close to the boundary of the square patch, the periodic nature of the Gaussian rough surface is used. All calculations involving shadows and radiative transfer are done by first shifting the facet under consideration to the centre position of the square patch. This is essentially achieved by performing the vector subtraction modulus [FORMULA], for points on the surface. An alternative approach would perhaps be to use only some inner region, cutting away the boundary, but that is very inefficient since the computational costs increases rapidly with N.

2.5. The constant background approximation

The integral equations in Sect. 2.2are difficult to solve in the general case. The solutions for the spherical craters suggests a possible zeroth order approximation: the radiosity is determined by the direct solar irradiation plus a constant diffuse field (Eq. 10). This field can be derived by using the concept of self heating, as introduced in Paper III.

Assume a constant radiosity over a rough surface. The fraction of the total energy emitted escaping into space is given by the ratio [FORMULA]. The self heating is the fraction not escaping, but heating other parts of the surface:

[EQUATION]

From Sect. A.2 in Paper III the self heating of the surface with spherical craters is

[EQUATION]

For a Gaussian random surface, the ratio between the ensemble surface area and its projected area on the xy -plane is given by

[EQUATION]

where [FORMULA] is the incomplete gamma function (Muinonen and Saarinen, 1997).

The constant background field approximation is then to assume that [FORMULA]. Taking the mean over the surface A in Eq. (10) results in

[EQUATION]

For the spherical craters with [FORMULA], this exactly reproduces the result in Eq. (14). This directly suggests a first comparison between the two roughness models, by numerically computing the mean and variance of [FORMULA] for a sample stochastic surface.

2.6. Heat conduction

The possibility of heat conduction into the regolith is formally introduced in Eq. (6). If the thermal parameter [FORMULA] is non-zero, the heat diffusion equation has to be added to the set of equations in Sect. 2.2, that is for the interior

[EQUATION]

Strictly speaking this is a 3-D problem that has to be solved together with the integral equations for the radiative heat transfer. There are, however, limits to the length scale where the surface roughness is causing the beaming effect. The upper limit comes from the simple fact that the roughness itself must start to decrease with increasing length scales at some point. The lower limit is determined by the smearing of the lateral temperature variations due to the heat conduction. This should occur on a scale comparable to the thermal skin depth, which typically is rather small (Spencer, 1990). Within these limits, it is assumed that the conduction of heat into the regolith is more important than across the surface.

The method for solving the 1-D heat conduction problem simultaneously with the radiative heat transfer equations is discussed in Paper III. For every point on the surface the heat diffusion equation into the regolith is solved along an axis parallel to the local normal [FORMULA].

2.7. The beaming function

There are several different ways to quantify the beaming. In Paper I and III a factor [FORMULA] was multiplied to the Planck function. Saari et al. (1972) corrected the brightness temperature by a "directional factor" D. Whichever approach, the correction is in principle a function of the viewing geometry, diurnal thermal history, and wavelength. The STM beaming correction, on the other hand, is primarily designed for disk integrated data of spherical asteroids. Thus the surface temperature is corrected by a factor [FORMULA] regardless of the position on the surface, and a linear phase correction is applied to the disk integrated flux.

The directional factors are used here, with the slight difference that the emissivity is not unity when computing the brightness temperature. The emissivity corrected brightness temperature equals the physical temperature of the surface if there is no beaming. The advantage of this approach is that the beaming correction is fairly insensitive to the effects of the heat conduction (see below). That is, if [FORMULA] is the brightness temperature of a flat surface with heat conduction ([FORMULA]), and [FORMULA] without, then

[EQUATION]

where [FORMULA] is the brightness temperature for the same surfaces with roughness added. The temperature [FORMULA] is given by the model in Paper I. If the approximation in Eq. (23) is valid, the beaming corrected [FORMULA] can be derived from [FORMULA]. This is more desirable since introducing heat conduction as described above requires much more computational efforts.

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© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998
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