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Astron. Astrophys. 364, 816-828 (2000)

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5. The sensitivity function

The planned GOLF signal would have involved the ratio [FORMULA]. In this case the normalization factor [FORMULA] would have cancelled out. Due to the poor operation of the rotating polarization mechanisms, the GOLF instrument now operates in a single wing and the normalization factor is found by comparison to a temporal average of the returned integrated-sunlight signal. For the purpose of the simulation of the GOLF signal this average can also be calculated by integrating [FORMULA] or [FORMULA] over the visible solar disk since we use line profile functions which represent an average of the time-variable profiles. To avoid duplication of formulae which are essentially identical, we replace the subscript b or r with w as a generalized wing designation where w can take on the values b and r. The average intensity is then:


The apparent velocity signal detected by GOLF is then the change in intensity due to a mode distributed over the surface according to [FORMULA] relative to the average intensity multiplied by a conversion factor [FORMULA]:


We define a sensitivity S to be a function of x and y which when multiplied by the surface velocity [FORMULA] and integrated over the visible disk yields the observed change in velocity. Consequently, the integral in Eq. (32) shows that the appropriate definition of the sensitivity function is:


Due to the presence of three scattering components, the character of the sensitivity functions depends on the resonance cell temperature through the [FORMULA] functions. If a single scattering component were present, the response function would simply be [FORMULA]. Consequently, it is useful to examine these three functions independently to help anticipate the character of a change which could be caused by a change in the balance between the three scattering strengths. These functions are shown in Fig. 4 in a format similar to that used for Fig. 3. The range of velocity shown in these figures is slightly larger than is seen by the GOLF instrument including the effects of orbital velocity variation and solar rotation. The solar disk center curves for [FORMULA] have a restricted range since they do not involve solar rotation. However, the significant non-linearity shown for the 60o case and which is present at 75o as well, indicates that the representation of the solar line profiles as triangles or trapezoids is not a good approximation. Due to the fact that the signal comes from all three components, some of the slope variation seen in this figure in fact cancels out and the net non-linearity is not as extreme as is shown in this figure.

[FIGURE] Fig. 4. The sensitivity functions for each of the scattering components. Each of these curves is the local line slope normalized by the local bisector intensity. The velocity is relative to the wavelength of the local line bisector.

The velocity conversion factor can be estimated in two ways. The variation in the orbital velocity corresponds to a uniform shift of the whole solar surface, i.e. it is equivalent to picking [FORMULA] to be a constant which may arbitrarily be taken as unity. Eq. (32) then yields:


The second method is by tracking the observed GOLF output as a function of [FORMULA] and applying the analysis of Ulrich et al. (1998). The comparison is shown in Fig. 5 for two values of cell stem temperature. There are four points of comparison relevant on this figure: the value of [FORMULA], the slope of [FORMULA] as a function of [FORMULA], the separation between the values of [FORMULA] and [FORMULA] parameterized as [FORMULA]) and the curvature of [FORMULA] as a function of [FORMULA]. The excellent agreement between all these parameters represents a strong validation of the model. Of these four points of comparison, only the curvature is not well reproduced from the ground-based profile model.

[FIGURE] Fig. 5. The conversion factor [FORMULA] derived from the GOLF measurements on SOHO shown as the solid lines to the same parameter derived from the ground-based line profiles for two stem temperatures: 146oK (short-dashed lines) and 155oK (long-dashed lines).

While values for many parameters go into this model, these parameters are all determined from data other than that shown in Fig. 5. Furthermore, we are primarily interested in using spatially resolved data to reproduce the GOLF signal which means that we do not need [FORMULA] separately but rather use [FORMULA] to convert the GOLF signal to a velocity and at the same time use [FORMULA] to convert the spatially resolved data to an equivalent GOLF velocity. As long as the model is consistent between these two applications, most uncertainties cancel. The largest uncertainty in the model concerns the treatment of the spectral resolution of the Mt. Wilson system and the effect of line smearing due to spatially unresolved line-of-sight velocities. Either of these effects produce changes in the [FORMULA] functions at the 5% level and we believe they cancel so that it is best to adopt the directly observed profiles as we have done here. The scattered light corrections to both GOLF and the Mt. Wilson profiles are constrained respectively by GOLF data taken during cool cell periods and the fit of the core intensity of the Mt. Wilson profiles to those measured by the double-pass system of Delbouille et al. (1973). The values of [FORMULA] are changed by 2% when this correction is dropped. The amplitude of magnetic modulation almost exclusively governs the separation between the [FORMULA] curves for the two states of modulation. Note that the amplitude of magnetic modulation was adjusted in the paper describing the instrument sensitivity Ulrich et al. (2000) in order to account for the two wing data. The results here also depend on the value adopted in that paper but no further adjustment was made to achieve the agreement shown here.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001