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Astron. Astrophys. 346, 626-632 (1999)

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3. Two methods for deriving the SMMF

3.1. Measuring the Stokes parameters I [FORMULA] V

If we assume that the solar magnetic field is weak ([FORMULA] 500 G) throughout the atmosphere where the sodium doublet is formed (low chromosphere), we can use the so called weak field approximation. In this case, the displacement due to the magnetic field, [FORMULA] is much smaller than the Doppler width of the original line profile (for a general description see e.g. Stix 1989). In this approximation, we can treat the magnetic field as a perturbation and make a Taylor expansion of the Stokes parameters in [FORMULA], up to first order:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] is the Zeeman split wavelength due to the permanent magnet in the field where the vapor cell is placed.

Adding and subtracting the non zero Stokes parameters, i.e. I and V:

[EQUATION]

GOLF makes 8 independent measurements with different combinations of the Stokes parameters (see appendix A and Fig. A1). To refer to them we have used a compact notation [FORMULA], where i refers to the polarization state of the incident solar light ([FORMULA]), j is the magnetic modulation ([FORMULA]) and k refers to the line wing (b, r). For simplicity, we are going to consider only the positions in the blue wings of the profiles and, at the end, we shall give the general solution including the measurements on the red wing.

[FORMULA] and [FORMULA] measurements of the nominal operating cycle of GOLF correspond to [FORMULA], while [FORMULA] and [FORMULA] correspond to [FORMULA] (see appendix and Fig. A1). Therefore, subtracting the two Eq. (5) and using, for example [FORMULA] and [FORMULA], we obtain two expressions:

[EQUATION]

[EQUATION]

and matching the two right hand sides of both equations:

[EQUATION]

In order to calculate this expression, we have to write the right hand side as a function of the GOLF measurements. [FORMULA] is the slope of the line at [FORMULA]. Using the measurement of the two polarization states of the incident solar light, and considering that the displacements are small, we can make a linear approximation:

[EQUATION]

where [FORMULA] is the displacement due to the magnetic modulation around the sodium cell ([FORMULA]). Both [FORMULA] and [FORMULA] can be written as a function of the magnetic field (the constant is computed for B in Gauss and [FORMULA] in Å):

[EQUATION]

where [FORMULA] is the longitudinal component of the solar magnetic field we are looking for, and [FORMULA] is the effective Landé factor of the transition. Strictly speaking, the wavelength and the [FORMULA] values, are the weighted sum for each of the sodium doublet components ([FORMULA], [FORMULA] and [FORMULA]). In the rest of this section, we shall only consider one of them, and demonstrate that the resultant expression is independent of the number of lines taken into account. Substituting expressions (9) and (10) in Eq. (8) we have:

[EQUATION]

and isolating the magnetic field, given in the same units as [FORMULA], we obtain:

[EQUATION]

To first order, [FORMULA] and [FORMULA] are equivalent to [FORMULA] and [FORMULA], thus to reduce the statistical noise, we can take their mean value to reach the final equation for one-wing only:

[EQUATION]

This new formula is valid only for one line (e.g. [FORMULA]). GOLF measures the 3 components of the sodium doublet and each one has three hyperfine components. Thus, the 8 independent measurements, [FORMULA], can be considered as the sum of the contributions, [FORMULA], of each independent component. Mathematically:

[EQUATION]

Eq. (12), valid only for each individual line, can be written as follows:

[EQUATION]

Assuming that the longitudinal magnetic field of the Sun [FORMULA] and that in the experiment are the same for each of the sodium doublet components, we can sum the 3 expressions (15):

[EQUATION]

Expanding this formula and introducing the definitions (14) we obtain again the general solution for the blue wing (13).

Also, as the red wing measurements are equivalent to the blue ones, a corresponding expression can be derived for them:

[EQUATION]

Combining the solutions for the blue (13) and red (17) wings we obtain the general expression of the longitudinal magnetic field of the Sun as a function of the 8 independent GOLF observables (García 1996):

[EQUATION]

3.2. Subtracting two velocities

Combining the velocity measurements from each of the two circular polarization states of the incident light, we can obtain, by another method, a magnitude which is proportional to the SMMF.

There are several methods to compute the velocity (Boumier et al. 1994; García et al. 1995). Using the so called "crossed ratios" method, defined for the first 4 measurements as follows:

[EQUATION]

where [FORMULA] is the calibration expression (for a complete description see Boumier 1991; García 1996):

[EQUATION]

The SMMF can be deduced calculating the difference between the velocities measured for each circular solar polarization state:

[EQUATION]

where the magnitudes at time ([FORMULA]) mean the same measurements as at time t but using the other circular polarization state of the incident light and [FORMULA] is measured in Gauss.

The results given by this method agree, within the errors, with the first one thus, in the rest of the paper, we will only refer to former one.

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© European Southern Observatory (ESO) 1999

Online publication: May 21, 1999
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