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Astron. Astrophys. 364, 799-815 (2000)

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3. Determination of the magnetic modulation from two-wing observations

Although the data between January 19, 1996 and April 1, 1996 are perturbed by a variety of commissioning activities, it represents the only period during which both the blue and red wings of the Na D lines were observed from space with the GOLF system which includes the magnetic modulation. The classic resonance scattering helioseismometer yields an observable designated as R:

[EQUATION]

The use of the modulated electromagnet permits a monitoring of the net solar line profile as averaged by the instrument. The amplitude of magnetic modulation can be found in addition through the use of the orbital changes in the sun-spacecraft velocity. The magnetic modulation combined with the wing selection by the rotating polarizers provides four measurable quantities: [FORMULA]. These can be combined to yield a number of different expressions for R like that of Eq. (5). First, if we take [FORMULA] and [FORMULA], we recover Eq. (5) exactly. Second following Boumier (1991), Garca (1996) and Boumier et al. (1991) we can define red and blue analogues of Eq. (5) as follows: 1

[EQUATION]

If the scattering process included only one term instead of three, the difference between [FORMULA] and [FORMULA] would be the same as would result from a velocity shift equivalent to that implied by the separation between the wavelengths of the two states of magnetic modulation. The asymmetrically placed set of three scattering wavelengths causes the centroid of the combination to be dependent on the scattering gas temperature. We can estimate the effective value of the wavelength separation by smoothing each R function in time and then considering each to be a function of the orbital velocity plus an offset. If we represent these smoothed functions of velocity v by the notation: [FORMULA] and [FORMULA] then we should have [FORMULA] for some value of [FORMULA]. This suggests the strategy of plotting [FORMULA] versus [FORMULA] and [FORMULA] versus [FORMULA] so that if [FORMULA] is properly chosen the two curves should coincide. Fig. 2 shows this comparison with [FORMULA] m s-1. This figure shows the full range of R values returned over this span of orbital velocity.

[FIGURE] Fig. 2. Comparison of [FORMULA] and [FORMULA] as functions of orbital velocity [FORMULA] m s-1. The curve for [FORMULA] is shown as dashed.

The value of [FORMULA] which produces the most exact agreement between the [FORMULA] and [FORMULA] is uncertain due to irregularities in the longer term trends of the two functions. The magnitude of this uncertainty can be estimated by examining [FORMULA] as a function of [FORMULA] with [FORMULA] as a parameter. This comparison is shown in Fig. 3. The value of [FORMULA] determines the average value of [FORMULA] so that if [FORMULA] is too small, [FORMULA] is positive and if [FORMULA] is too large, [FORMULA] is negative. The correct [FORMULA] leaves the average of [FORMULA] near zero. A figure of the merit for any choice of [FORMULA] is [FORMULA], the rms variation in [FORMULA] considered as a function of [FORMULA]. The best value of [FORMULA] is that which minimizes [FORMULA]. Due to the irregularities in the [FORMULA] function as shown in Fig. 3 this minimum is not precisely defined. The behavior of [FORMULA] versus [FORMULA] is shown in Fig. 4. The breadth of the minimum in Fig. 4 provides an estimate for the uncertainty of [FORMULA] which is about 0.5 m s-1. Although this value is well determined and is based on signals which have been corrected for the known effects such as dead-time and resonance cell stem temperature variations, the observations were made during a period of the experiment when a variety of other parameters were undergoing adjustment. Therefore, there could be systematic effects which cause the above formal error to be an underestimate of the actual uncertainty in the [FORMULA] parameter.

[FIGURE] Fig. 3. The difference between [FORMULA] and [FORMULA] as functions of orbital velocity.

[FIGURE] Fig. 4. The behavior of [FORMULA] as a function of [FORMULA].

In order to convert [FORMULA] into an effective magnetic field modulation amplitude we use the derivative of Eq. (5) to obtain a relationship between the line intensity slopes and [FORMULA]:

[EQUATION]

Next we need to relate the amplitude of magnetic modulation [FORMULA] to [FORMULA] and this requires consideration of the interaction between the solar line profile and the sodium scattering. Each scattering component has a wavelength [FORMULA] according to Eq. (1). The sun-spacecraft velocity determines [FORMULA] where v is the sum of orbital velocity, the Einstein shift velocity [FORMULA] m s-1, convective correlation shifts [FORMULA], and the velocity signal we wish to measure [FORMULA]. The convective correlation shift is poorly determined theoretically and depends on the definition of the position of the solar line. Each solar line profile j can be considered as a function [FORMULA] where w is an input parameter having dimensions of wavelength as indicated by the subscript [FORMULA] and the bracket notation indicates a functional relationship. In our case w is the wavelength difference between the scattering component and a centroid of the solar line. Each solar line profile [FORMULA] can be taken as a constant when averaged over long time periods. These represent the D1 and D2 integrated sunlight profiles. The observed intensities are then:

[EQUATION]

which yield:

[EQUATION]

Eqs. (11) and (12) require knowledge of the line profiles over some range in v near each of the working points. For the full velocity range of the orbital motion of SOHO, the line profiles are significantly non-linear. For the purpose of the determination of the magnetic modulation, we may consider a restricted range as illustrated in Fig. 2 of [FORMULA] m s-1 in which case the profile is nearly linear. We may then define a reference velocity [FORMULA] and consider [FORMULA] to be small along with the magnetic modulation velocity [FORMULA]. We may also consider the line profile to be a function of velocity instead of wavelength: [FORMULA]. After adopting the following compact notation for the derivative with respect to velocity:

[EQUATION]

and defining:

[EQUATION]

and using a linear expansion about the reference velocity we obtain:

[EQUATION]

Two properties of these definitions are worth emphasizing: first, even though we have used a velocity scale for the argument of the line profile function, we have reversed the sign in order to account for the fact that the velocity appears with a negative sign in Eqs. (14 - 17); and second, the slopes [FORMULA] are positive while the slopes [FORMULA] are negative.

We can now express the intensity change due to the magnetic modulation as

[EQUATION]

Utilizing the fact that [FORMULA] we can write [FORMULA] as

[EQUATION]

giving [FORMULA] which can be inserted into Eq. (8) to obtain:

[EQUATION]

Due to the temperature dependence of the scattering strengths [FORMULA], the value of [FORMULA] also depends on the temperature as well as the offset velocity. The total range in temperature as indicated by the platinum probe is from 169.9o C to 171.6o C during the entire first year of operation. During the two wing period which is of interest the range was restricted to 169.9o C to 170.2o C. This corresponds to a gas temperature between approximately 154.9o and 155.1o C where there is an uncertain offset to the true gas temperature. The [FORMULA] functions as derived from the Delbouille et al. (1973) integrated sunlight profiles are shown in Fig. 5. Since the offset velocity at the central time of the sequence was 315 m s-1, we can derive [FORMULA]. From this result we may derive:

[EQUATION]

The derived [FORMULA] corresponds to a magnetic modulation of [FORMULA] gauss. Eq. (24) provides a precise determination of the magnetic modulation amplitude and is adopted for the remainder of this paper.

[FIGURE] Fig. 5. The function [FORMULA] as a function of [FORMULA]. The derivative is dimensionless while the velocity differs from the orbital velocity by the convective velocity offset. The cell gas temperature is shown for each of the two lines. The values chosen bracket the range observed during the first year of operation. For the purpose of finding the magnetic modulation, we need the gas temperature during the two-wing period which ranged between 154.9o C and 155.1o C.

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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