## 4. The magnetograph velocities and center-to-limb intensity curvesIn 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
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 D 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, 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 D 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 D where and © European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |