Astron. Astrophys. 341, 902-911 (1999)

3. Tests of the computer code

We have performed two tests to check the computer code. The first one, discussed in Sect. 3.1, consists of calculating a special case, namely that of a perfectly scattering atmosphere. As a second test, the theoretical and observed CLV of the continuum intensity are compared in Sect. 3.2.

3.1. Purely scattering atmosphere

Chandrasekhar (1960) has derived the exact solution of the transfer equation for a purely scattering atmosphere in radiative equilibrium, in which the angular dependence of the scattering is controlled by the Rayleigh phase matrix (8). Pure scattering refers to a conservative atmosphere with constant net flux, in which the whole opacity is due to scattering, so no pure absorption occurs. The Stokes , where denotes the intensity at disk center, and components of the continuum radiation field at the top of the atmosphere turn out to be independent of frequency and of all thermodynamic properties because of the lack of thermal coupling between the radiation field and the gas.

We have obtained a purely scattering atmosphere in our calculations in the following way: The scattering coefficient was artificially redefined as the sum of the original and , while the absorption coefficient was set equal to zero. The atmosphere is no longer self-consistent after these redefinitions. Nevertheless, Chandrasekhar's solution should be unrelated to the depth dependence of the temperature, density, and pressure.

All solar model atmospheres do indeed render identical center-to-limb variations of the polarization and the intensity for all wavelengths considered, from 4000 Å to 8000 Å. Moreover, these curves reproduce precisely the exact solution, as seen in Fig. 2. This verifies that scattering has been correctly implemented in the code.

Fig. 2. For the case of pure scattering a comparison is shown between the exact solution (filled circles), given by Chandrasekhar (1960), and our calculations (solid line). The left panel shows the CLV of the polarization and the right panel the limb darkening. All model atmospheres give the same solid lines for all wavelengths from 4000 Å to 8000 Å.

3.2. Comparison with observed limb darkening

Many observers have measured the solar limb darkening. The CLV curves of the intensity so obtained are then fitted to suitable analytical functions or limb-darkening laws, usually containing up to five fit parameters. In general these parameters depend on wavelength.

For the comparison of our calculations with observed CLVs we have chosen the analytical limb-darkening law given in Neckel (1996). It is not claimed that the function is representative of the Sun, but it is expected to best describe the average quiet Sun. Any new measurement will differ somewhat from this expression due to the limb-darkening variability. Likewise, it is unlikely that our calculations will perfectly reproduce it. We may however expect the calculated limb darkening of the quiet Sun models to agree reasonably well with the empirical data.

Fig. 3 shows a comparison between the observed (solid line) and the computed limb darkening of FALC₅ for two different wavelengths (note that MACKKL₆ renders the same results as FALC₅). The diagram to the right is representative of the worst case within the spectral range considered. Taking the natural variations in the Sun's actual CLV around the Neckel law into account we can conclude that the limb darkening of the quiet Sun is well reproduced with our code.

Fig. 3. Comparison of observed limb darkening (Neckel 1996; solid lines) and our calculations (dotted lines) for the average quiet Sun model FALC5 for two different wavelengths.

© European Southern Observatory (ESO) 1999

Online publication: December 16, 1998
helpdesk.link@springer.de