Volume 544, August 2012
|Number of page(s)||15|
|Published online||25 July 2012|
Average Ca ii H profiles of observations No. 1 (top), 2 (3rd row), and 4 (2nd row). Bottom: average Ca ii IR profile of observation No. 4. Same layout as Fig. 2.
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For completeness, Fig. A.1 shows the average profiles of the remaining observations used in this study. The average and minimal intensity profiles differ only slightly for both the Ca ii IR line at 854.2 nm and Ca ii H. The maximum observed intensity of Ca ii H varies significantly from observation to observation, in contrast to the minimum intensity, implying that extreme excursions towards higher intensity happen more frequently than those towards lower intensity.
Table B.1 lists the complete statistics at three (four) wavelengths in the Ca ii H (Ca ii IR) line. The skewness of the magnetic locations is up to three to five times larger than that of the field-free locations for the intermediate wavelengths (396.632 nm and 854.131 nm).
Intensity statistics at selected wavelengths.
Conversion from wavelength to geometrical height for the Ca ii IR line at 854.2 nm. Top: intensity response function. The red line denotes the centre of the response function at each wavelength. Bottom: geometrical height corresponding to the centre of the response function.
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To determine the formation height of a given wavelength in the Ca ii lines, we synthesized the spectral lines with the SIR code (Ruiz Cobo & del Toro Iniesta 1992) in LTE using a modified version of the HSRA model (Gingerich et al. 1971) without any chromospheric temperature rise (cf. BE09). Using the original HSRA model yielded comparable results (see Fig. 9 for the case of Ca ii H). The two atmosphere models are fully identical up to log τ = −4. We added a temperature perturbation of 1 K to each of the 75 points in optical depth, one after the other, and synthesized the corresponding spectra. The difference to the unperturbed profile ΔI(λ,τ) = I(λ,T + ΔT(τ)) − I(λ,T) yields the intensity response function with wavelength for perturbations at a given optical depth (upper panel of Fig. C.1). For each λ, we fitted a Gaussian to ΔI(λ,τ), where the centre of the Gaussian yields the optical depth attributed to that wavelength, τ(λ). With the tabulated values of τ and z in the original HSRA model, one can then obtain the corresponding geometrical height, z(λ) (lower panel of Fig. C.1). The respective curve for Ca ii H was derived in a fully analogous way (see, e.g., BE09). The attributed formation heights inferred from the intensity response function closely match those derived from phase differences (BE08, BE09, their Fig. 1).
Appendix D: Conversion between intensity and brightness temperature fluctuations using the Planck curve
Calculation of brightness temperature variation corresponding to intensity fluctuations of ± σI/Imean. Top: for λ = 854.213 nm in the core of the Ca ii IR line. Bottom: for λ = 396.847 nm in the core of the Ca ii H line.
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Parameters and temperature fluctuations at selected wavelengths in the Ca ii H line.
For the conversion from relative intensity fluctuations to brightness temperature fluctuations, one first needs to attribute a characteristic temperature to the average intensity at each wavelength. To this extent, we used the results of the previous section. The centre of the intensity response function at each wavelength λ yields the optical depth τ(λ) corresponding to that wavelength (e.g., third column of Table D.1). The reference atmosphere model then provides the temperature T0(τ) at that optical depth, where we assume that the average intensity Imean(λ) at the wavelength is directly related to the average temperature T0(τ(λ)). In the LTE assumption, this relation between intensity and temperature is given by (D.1)For a given intensity variation, e.g., the rms fluctuation σI, one finds that (D.2)For , one obtains (D.3)Owing to the non-linearity of Eq. (D.2), ΔT is different when adding or subtracting the rms fluctuation from Imean. We therefore used the average value of ΔT for the two signs as the temperature variation that corresponds to the rms intensity fluctuation. To obtain the brightness temperature difference corresponding to the increase/decrease in I by one standard deviation σI, the variation in the emergent intensity around the average value T0 is calculated with Eq. (D.1). The relative change in I/I0(λ,T0) is displayed in Fig. D.1 for two wavelengths in the line core of Ca ii H (bottom panel of Fig. D.1) and Ca ii IR (top panel of Fig. D.1). The temperatures corresponding to 1 ± σI/Imean(λ) are then read off from the intersections with the curve of relative intensity variation. The use of the modified HSRA model reduces the temperature rms (Fig. 9) because of the dependence on T-1 in Eq. (D.3). We used the original HSRA as a temperature reference because the majority of the solar atmosphere should presumably be closer to the original than the modified HSRA model. Table D.1 lists the relevant parameters and resulting temperature values for several wavelengths in the Ca ii H line, sorted in order of monotonically increasing wavelength and formation heights. We point out that the values of T0 and z were taken from the original HSRA model and were not derived from the observed spectra themselves.
© ESO, 2012
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