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Astron. Astrophys. 321, 643-651 (1997)

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3. Data analysis

The first steps of analysis included flat fielding, dark current correction and the consideration of the effects of stray light. We took into account spectrally undispersed and dispersed scattered light. The former originates from e.g. the scattering of sunlight at randomly distributed dust particles inside the spectrograph which reaches the detector everywhere with the same intensity. The latter is related to light which is scattered into the umbra from outside the spot region.

We assume that the observed and the real intensity profile inside the umbra, [FORMULA] and [FORMULA] respectively, are related by


where [FORMULA] denotes the intensity profile of the adjacent quiet sun and [FORMULA] the contribution of atmospheric stray light to the umbral line profile. A similar relation can be formulated between the observed ([FORMULA]) and the real ([FORMULA]) continuum intensity ratio umbra/photosphere:


Assuming that [FORMULA] is more or less constant inside the umbra, the knowledge of either the real umbral intensity contrast [FORMULA] or the amount of stray light [FORMULA] enables to reconstruct the real umbral intensity profiles.

Owing to the lack of additional data (e.g. aureoles) that could enable us to determine the actual amount of stray light, we adopted a literature value (Maltby et al. 1986) for the umbral continuum contrast [FORMULA] of 0.21 at [FORMULA]  846.85 nm.

In the next step we selected scans that show well defined UDs: two series of the [FORMULA]  846.85 nm line, each consisting of 27 spectra taken between 8:44 and 9:23 UT and one series of the [FORMULA]  630.25 nm line consisting of 25 spectra recorded between 11:06 and 11:40 UT. There are various UDs scattered over the umbra observed. The isolated UDs in one of the [FORMULA]  846.85 nm sequences (CUD1) and in the [FORMULA]  630.25 nm sequence (CUD2) are located well within the umbra. The UD pair in the second [FORMULA]  846.85 nm sequence is found close to the umbra-penumbra boundary (PUDs), where the intensity gradient is large and the magnetic field lines are much more inclined against the surface normal than in the central part of the umbra. In the following PUD1 denotes the UD closer to the penumbra and PUD2 the UD closer to the central part of the umbra. Besides this, the slices belonging to the single dot series pass through the darkest parts of the umbra, whereas those of the double dot series cut through the periphery of the umbra.

3.1. Magnetic field strength and brightness temperature

Spectroscopic determinations of solar magnetic field strength make use of the Zeeman effect. For the simple case of a triplet the equation


specifies the relationship between the wavelength shift of the [FORMULA] -components [FORMULA] relative to the wavelength of the unshifted line position [FORMULA] given in nm and the magnetic field strength [FORMULA] in Gauss. Hence the Zeeman equation can be used to infer the field strength from the observed broadening of the Stokes-I profile if the line is sufficiently split. This direct method is limited to the case, where the line-forming layer is permeated by a strong magnetic field and the inclination to the line-of-sight is small (Balthasar & Schmidt 1993).

Since the blend of the [FORMULA]  846.85 nm line deforms the [FORMULA] -component and may also affect the position of the [FORMULA] -component the direct method seems not to be appropriate to determine the field strength. In order to obtain more reliable results, we calculated synthetic line profiles and fitted them to the observed ones (see Sect. 4).

The brightness temperature T was derived by converting the continuum intensity I into temperature via the Planck law and assuming local thermal equilibrium :


where [FORMULA] and [FORMULA] is the quiet sun continuum intensity and temperature respectively.

3.2. Determination of the local background

We considered the UD brightness under the following aspect (Koutchmy & Adjabshirzadeh 1981): the central intensity of the UD profile [FORMULA] is measured above a local pseudo-background level [FORMULA] obtained by interpolating between two footpoints as seen from each side of the UD. A similar interpolation method was used to derive the ratio of magnetic field strength inside the UD to that of the surrounding dark umbral material. The knowledge of the variation of the field strength inside the umbra enables to attribute a field strength value to the footpoints. Between these values we interpolate linearly on the local magnetic background. The magnetic profile between the two footpoints is approximated through a second order polynomial fit. Together with the position of the UD in the spectra a value of magnetic field strength can be attached to the UD and the umbral pseudo-background (see Fig. 2). This technique works out well for an isolated UD located centrally in the umbra. For the case of two adjacent UDs - as in sequence PUD1/2 - a slightly different method was used. Since the two PUDs are magnetically unresolved, the interpolation on the local magnetic background was done only between the outer footpoints.

[FIGURE] Fig. 2. Determination of the local intensity and magnetic background. Shown are the variation of the magnetic field strength (thin line) and the intensity (thick line) along the slit. Magnetic field values below 1000 G are not reliable.
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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998