Astron. Astrophys. 345, 635-642 (1999)

5. Flux variations

After investigating the limb behaviour of the faculae, we tested how well the flux spectrum, determined by adding together appropriately weighted intensity spectra, would agree with our previous simpler approach when fitting the spectral variation of the disk-integrated flux (Papers I and II). To this end we compared a three-component model with the irradiance variations as compiled by Lean (Lean 1997 and priv. comm) and with the measurements from VIRGO (see also Fligge et al. 1998). Finally, we also checked for agreement with the results of Mitchell & Livingston (1991), who estimated the contribution of the spectral lines to the irradiance variations.

5.1. Ultraviolet spectral irradiance variations

The irradiance variation between solar-activity maximum and minimum as a function of wavelength are shown in Fig. 7. The dotted line shows the UV data as compiled by Lean (1997) for wavelengths smaller than 400 nm, and beyond 400 nm her estimate of the flux variations. The solid line is our model fit assuming a global facular filling factor of 2.3% and a spot filling factor of 0.23%. These filling factors were obtained by minimising between the observations and the model between 200 and 400 nm while requiring the total irradiance variation to be 0.1%, according to the ACRIM (Willson & Hudson 1991) and ERB (Kyle et al. 1994) measurements. As no error estimates were available, and so as to avoid overemphasizing the spectral ranges with the largest variations, we arbitrarily set the errors to a fixed fraction of the measured variations.

Fig. 7. Relative flux (or equivalently irradiance) variations over the solar cycle vs. wavelength. The dotted curve represents observations for wavelengths shorter than 400 nm. The solid line shows the relative irradiance variations resulting from a 3-component model with a facular filling factor of 2.3% and a spot filling factor of 0.23%. The total irradiance variation predicted by the model is 0.1%.

The fit between 300 and 400 nm is not quite as good as in Paper I where we used a much simpler model. One of the reasons for this is that the flux is now formed over a larger height range, so that changes of individual temperature-depth points in the model atmosphere no longer affect relatively narrow and well-defined wavelength regions. In addition, the observations are also rather unreliable in this particular wavelength range (G. Rottman, priv. comm.). Furthermore, we cannot expect to obtain accurate results in the UV with a strict LTE approach. This problem becomes particularly acute for the shortest wavelengths, where the radiation comes from the highest layers. For these wavelengths the NLTE approach of Fontenla et al. (1999) is certainly to be preferred.

The ratio of facular to sunspot filling factor is around 10, which is only slightly less than what was measured by Chapman et al. (1997) around activity maximum.

5.2. VIRGO data

Following Paper II, we compare the output of the new model calculations with time series of total and spectral irradiance obtained with the VIRGO (Variability of Irradiance and Gravity Oscillations, Fr"ohlich et al. 1997) instrument onboard SOHO. The time variations of facular and sunspot filling factors are assumed to follow the Mg II k index and sunspot areas, respectively. The details are given in Paper II. We follow that paper exactly, except for the way that the quiet-sun, facular and sunspot flux spectra are constructed.

The resulting model time series of the total solar irradiance and of the irradiance at the three VIRGO wavelength bands look very similar to the time series plotted in Fig. 5 of Paper II and are not plotted again here. A quantitative comparison of the time series shows that due to the increased variations around 400 nm the fit for VIRGO's blue channel is improved. The total, green and blue channels can now be fitted consistently within less than 1% deviation from the RMS variations of the measured time series. Unfortunately, however, the same fit leads to variations of the red channel which are clearly too large (by more than 10%) compared to the measurements.

Therefore, if all four channels are weighted equally, the overall agreement between the reconstructed and the observed time series is of the order of 4-5% only and the reconstructed solar total and spectral irradiance variations are of the same quality as the ones presented in Paper II. The results are summarized in Table 1 which shows the ratios between the RMS variation of the different color channels for the observed and modelled time-series, respectively.

Table 1. Ratio between the RMS variation of the total (), red (), green () and blue () colour channels for the observed (by VIRGO) and modelled time-series, respectively.

We have also reconstructed the total irradiance over the whole time that solar irradiance has been measured and compared it with the composite put together by Fröhlich & Lean (1998). Once again, the results are similar to those presented in Paper II.

5.3. Line blanketing

The course of the solar cycle can also be tracked by changes in the line blanketing. The line-blanketing variations during solar cycle 21 have been measured by Mitchell & Livingston (1991) who found that in disk-integrated spectra the spectral lines in the 500 to 560 nm range are on average 1.4% shallower and have 0.8% smaller equivalent widths at solar maximum than at solar minimum. As expected, the blanketing effect was stronger at the blue end of their spectra than at the red end.

The total irradiance variations can be seen as a combined effect of the flux variations from the continuum and those of the lines. In the following, we define the line blanketing as the ratio of the average absorbed irradiance to the average continuum irradiance in the unblanketed spectrum. In the wavelength range between 500 and 560 nm, Mitchell & Livingston (1991) measured the line blanketing to be 0.076 and the change in the line blanketing to be %. They then calculate that the irradiance change due to the change in line blanketing is -0.066 % in this wavelength range (see their Eq. 17). We use our calculations to proceed the opposite way and calculate the contribution of the line blanketing from the difference between the continuum variations and the variations of the emergent spectra that include the absorption lines via the ODFs. Our aim is twofold. On the one hand, we use the Mitchell & Livingston (1991) data to test our model. On the other hand, since our model covers a far larger wavelength range than their measurements, we can predict the contribution of line blanketing to the total irradiance variations more easily. In both cases the result is only approximate since we use LTE and ODFs to represent the spectral lines.

Table 2 lists the flux changes , where the subscripts f, s and q indicate the facular, spot flux and quiet-sun flux, respectively. The four rows show results for the wavelength ranges of 300 to 400 nm, 400 to 500 nm, 500 to 560 nm, i.e. the one investigated by Mitchell & Livingston (1991), and for the "total" spectral range (164-160 000 nm). The first column gives the wavelength range; Columns 2 and 3 list the flux changes produced by the facular and spot models for the continuum; Columns 4 and 5 the facular and spot changes for the emergent spectrum, and Columns 6 to 8 show the relative flux change produced by the combination of spots and faculae (for filling factors of 0.23 and 2.3% respectively) in the continuum (6), the emergent spectrum (7) and in the line blanketing (8).

Table 2. The relative flux changes (in percent) of the continuum and emergent spectrum (including lines) due to faculae and spots in different wavelength ranges. The columns and symbols are explained in the text.

Columns 9 and 10 show our calculations of how much the line blanketing and spectral variations in each wavelength range contribute to the total irradiance variations. Column 9 details the contribution of the line blanketing changes alone; Column 10 shows the contribution of the changes in the emergent spectrum (i.e. line plus continuum). The values in parentheses are the estimates by Mitchell & Livingston (1991) based on an assumed facular contrast of 0.02 at disk centre. Whereas Columns 2 to 5 are independent of the facular and spot filling factors, Columns 6 to 10 have been calculated for the filling factors derived from the fits in Sect. 5.1, i.e. for a facular filling factor of 0.023 and a spot filling factor of 0.0023. For the range between 500 to 560 nm, our line-blanketing contribution of -0.078% (see Column 8 in Table 2) agrees to better than 20% with the value of -0.066% measured by Mitchell & Livingston (1991).

Mitchell & Livingston (1991) also estimate the percentage contribution of the line blanketing and of the continuum variations to the total irradiance changes as measured by ACRIM. Their estimates for the continuum contribution are very sensitive to the assumed contrast function for the faculae, so that they list two sets of results, one for faculae that show no contrast at disk centre, and one where the contrast at disk centre is 0.02. The latter is the one our models (that have a contrast of about 0.04 at disk centre) should be compared to. (The continuum contributions are mainly due to spots and are hence negative. For higher facular contrasts the overall continuum contributions will consequently be smaller.)

The predictions of our models are listed in the last two columns of Table 2, the values in parentheses are the results of Mitchell & Livingston (1991), who adopted a facular filling factor of 0.033. Considering the approximative nature of the extrapolations by Mitchell & Livingston (1991) to shorter wavelength ranges and the uncertainties that are introduced into our calculations by considering ODFs only, the agreement between the two data sets is reasonably good.

Of particular interest is the line-blanketing contribution to the total irradiance variations. If our models are to be believed, then continuum variations are negligible and contribute only to a small amount to the total irradiance variations. We have to point out, however, that the exact contributions of the continuum are very model-dependent, particularly so at UV wavelengths, where they cannot be checked against observations due to the flux redistribution into the UV. The contributions of the line blanketing depend to a certain extent on the contrast and on the temperature stratification. If we choose a slightly different stratification (which still reproduces the observations almost as well), the contribution of the line blanketing to total irradiance variations is decreased to 90%. We cannot as yet rule out that line blanketing variations may contribute as little as 70 to 80% to total irradiance variations, but values much below these appear unlikely.

The reason for the dominance of the line blanketing is that most of the continuum-flux increase due to faculae is cancelled out by a corresponding decrease due to sunspots. If the sunspot and facular filling factors were of similar magnitude, the main variations would be due to changes in the continuum and the line changes would approximately cancel each other out. As the facular filling factor is about an order of magnitude larger, the continuum variations due to faculae and sunspots almost cancel out (see Columns 2 and 3) and the line-blanketing change introduced by the faculae becomes the dominant contribution to the total irradiance variation.

© European Southern Observatory (ESO) 1999

Online publication: April 19, 1999
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