4. The magnetograph velocities and center-to-limb intensity curves
In order to evaluate surface integrals of Eqs. (6) and (7) we need to know the velocity function and we need to know how the intensity depends on center-to-limb position. The magnetograph system provides a direct way to obtain both these quantities. The essential feature of the system is that it automatically tracks the bisector position. Thus the magnetogram velocity map gives us the shift of the bisector and the limb darkening curve gives us the appropriate intensity of this bisector. The parameter defining the bisector as measured by the magnetograph is the separation between the wavelengths where the intensity is balanced. There are six values of we need to use, one for each phase of magnetic modulation for each of the three scattering components. These values are: pm.
We must define a velocity and intensity system relative to the above bisectors. At the bisector, the quantity of Eqs. (11) to (14) is zero. The value of then is the desired long-time average offset velocity of a pixel at position on the solar image. The average intensity of this pixel can then be expressed as the product of the bisector intensity
The q functions are shown in Fig. 3.
The task of calculating the intensity now consists of two parts: a determination of and a determination of the limb darkening function. Ideally we would use observations in both the D lines to obtain these functions. In fact, the difficulty with telluric contamination prevents the use of any magnetogram data for the D2 component. For this line the D1 limb darkening law has been adopted by rescaling to the level indicated by disk center line profiles. In addition, the extensive database at the Mt. Wilson 150-foot solar tower of observations of solar rotation and large scale velocities using the FeI line at nm makes it desireable to base the surface velocity patterns on this system.
Turning first to the reference velocity, we can identify four parts: the gravitational redshift , the sun-spacecraft velocity , a convective shift and a relative surface velocity . Although the last two of these velocities may in fact depend on the bisector choice, at the moment we neglect this effect since we do not have adequate data to determine these shifts. The disk center reference velocity defines a time dependent offset velocity which is the primary cause of the shift of the GOLF working points across the solar profile:
where is the central meridian angle, b is the solar latitude, and is the observed solar polar axis tilt. The rotation rate parameters , , and are related to the traditional coefficients used by the Mt. Wilson synoptic program. As can be seen from Eqs. (24) and (25) the rotational velocity is expressed in terms of for spatial dimensions and for inverse time. Here is a scattered light corrected synodic equatorial rotation rate which we take to be 2.84radian/s and the term with is mostly related to scattered light but might also include the effects of a vertical gradient in rotation rate. The and parameters are similar to the B and C quantities used by Howard et al. (1980), LaBonte & Howard (1982), Ulrich et al. (1988). However, these parameters are now given different designations to indicate that they are no longer variable and that the structure of the representation equations is now slightly different. We now leave these parameters fixed at radians/s. We also find that and are correlated with the measured scattered light. However, when the linear relations are extrapolated to zero scattered light, the value of remains non-zero with a typical value being -0.02radians/sec. The limb shift velocity is represented with a power series having the form:
and the values of are m/s. The meridional circulation velocity is given by:
Although the above detailed velocity functions have been derived from observations of nm, the magnetograms available using the D1 are consistent with these functions.
The limb darkening function is
where the normalization of the disk center intensity can be left somewhat arbitrary for one of the three scattering components. At disk center the ratios and are approximately 1.13 and 0.54 respectively. The limb darkening function for D1 is fit to a form similar to that used above for the limb shift in Eq. (26):
where and X is defined above in Eq. (27).
© European Southern Observatory (ESO) 2000
Online publication: January 29, 2001