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Astron. Astrophys. 343, 983-989 (1999)

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4. Temperature response functions

We aim at associating the intensity fluctuations at certain distances from D2 line centre to certain heights in the solar atmosphere. We do this by means of temperature response functions [FORMULA], in short RFs (Mein 1971, Beckers & Milkey 1975, Caccin et al. 1977, Kneer et al. 1980). First, we calculate the intensity [FORMULA] from the VAL C model atmosphere (Vernazza et al. 1981). The line parameters for D2 and the sodium atomic parameters and abundance as in Kneer & Nolte (1994) are used. The calculations are done in LTE since we are exclusively interested in the wing intensities. This is justified from the calculations of e.g. Bruls et al. (1992). Next, we compute the intensity change [FORMULA] if we change in the height range [FORMULA] the temperature by the factor [FORMULA]. We define the temperature response function as


where [FORMULA] is the continuum intensity. In this way [FORMULA] simply gives the reaction of the intensity to a temperature change at z.

We keep the pressure unchanged when modifying the temperature. It was shown by Kneer & Nolte (1994) that, for Na D2, pressure fluctuations have a much smaller influence on the wing intensities than temperature fluctuations. We may savely take the intensity pattern in the wings as proxy for the temperature structure.

Fig. 4a depicts some response functions calculated in the above manner with [FORMULA], which is small, i.e. in the linear regime of the reaction of [FORMULA] to temperature changes. The minimum of [FORMULA] for the continuum (leftmost solid curve in Fig. 4a) is due to the negative contributions of the H- opacity (cf. Kneer et al. 1980).

[FIGURE] Fig. 4a-c. Temperature response functions [FORMULA] in units of [FORMULA] km[FORMULA]; a  with infinite spectral resolution at distances [FORMULA] from line centre (in mÅ, from right to left ): 0, 100, 200,..., 700, and continuum (solid leftmost curve); b  dotted: continuum as in a ; solid: after averaging over an 8 Å wavelength band centred at D2; other curves: response functions after convolution with Airy's FPI function of 200 mÅ FWHM at line centre (dashed), at 250 mÅ (dot-dashed) and at 450 mÅ off line centre (dot-dot-dot-dashed); c  linear combinations of response functions in b , see text for details.

However, instead of using the RFs in Fig. 4a calculated for infinite spectral resolution and in order to mimic the actual resolution, we convolved [FORMULA] also with Airy's FPI function for 200 mÅ FWHM at fixed z. We call the results [FORMULA]. Some are given in Fig. 4b, at line centre, at 250 mÅ and at 450 mÅ off line centre. All RFs span larger height ranges than those of Fig. 4a. Besides, the speckle reconstructed image was obtained through a 10 Å FWHM filter centred at D2. Since we do not have the actual transmission curve we approximate the contributions of intensities within the filter pass-band to the fluctuations by averaging [FORMULA] for fixed z over 8 Å. The effect is also shown in Fig. 4b. One notices the difference between the dotted curve, the pure continuum RF, and the solid curve, the averaged RF, which we denote by [FORMULA]. The latter has more contributions from higher atmospheric layers. A number which will be needed later is the intensity [FORMULA] averaged over 8 Å relative to the continuum intensity: [FORMULA].

The RFs in Fig. 4b possess a substantial overlap of contributing atmospheric heights. We can reduce this by using linear combinations of the RFs in Fig. 4b. Suitable combinations are found by estimates and the restriction that the RF be not too negative. Some are shown in Fig. 4c: solid curve (1a): [FORMULA]; long-dashed (1b): [FORMULA]; short-dashed (2): [FORMULA]; dot-dashed (3): [FORMULA] .

We can now better discriminate between various formation heights of intensity fluctuations. The centres of gravity of the RFs 1a, 1b, 2, and 3 approximately are at heights of -10 km (without the tail above 70 km), 0 km, 125 km, and 165 km, respectively.

We must not expect that the temperature fluctuations are small, [FORMULA]5% at fixed heights. To study the non-linearity of the D2 wing intensities we calculated RFs with [FORMULA]T/T =[FORMULA]0.25. We restrict the discussion to the linear combinations as in Fig. 4c. They are presented in Fig. 5a for [FORMULA]T/T = +0.25 and in Fig. 5b for [FORMULA]T/T = -0.25, with the same coding as in Fig. 4c. Although the general behaviour of the RFs is similar to those of Fig. 4c, we notice the strong temperature dependence of the D2 wing intensities: Large temperature enhancements increase very much the ionization of sodium and weaken the wing strength of D2. In that case, the RFs of the wings move to somewhat lower heights compared to those of Fig. 4c. On the other hand, strong temperature reductions lead to highly increased neutral sodium densities and to increased wing strengths. In that case, low wing intensities are "formed" in high atmospheric layers. The wavelength integrated image, (and linear combinations of it with wing images) is less dramatically influenced by temperature decreases.

[FIGURE] Fig. 5a and b. Temperature response functions [FORMULA] after the same treatment leading to the RFs of Fig. 4c; a  [FORMULA]T/T(z) = +0.25; b  [FORMULA]T/T[FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999