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

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5. Conversion of the raw signal to a velocity scale

5.1. General considerations

In this section we describe an analysis of the GOLF signal based on the direct photometric signals. In this approach it must be recognized that we are using the signal in an absolute photometric mode - one of the most difficult tasks in observational astronomy. Other approaches such as that described by Robillot et al. (1998) utilize a ratio of signals so that most of the effects only enter in second order (note that all corrections are applied in that approach even though the deduced velocity is less dependent on the corrections than is the case with this method). For the direct photometric method, any instrumental drift will find its way into the signal and can masquerade as a solar velocity. The only reason we can attempt this approach is the great stability of the observing platform and instrument. We are also aided by the separation in frequency between most drifts which occur over periods longer than one day and the solar oscillation modes of interest which have periods in the 0.05 to 3.0 hour range. However, there are also some gain fluctuations mostly from PM1 which produce abrupt jumps in this photomultiplier's counting rate and if uncorrected would add noise to the data in a frequency band overlapping that of interest. These events are mostly discrete and small in number and have been removed by applying a compensating discontinuity which is adjusted to assure smoothness across the jump.

In the face of these obstacles it may seen undesireable to use a direct photometric approach for the conversion of the GOLF signal into a quantity having dimensions of velocity. Our motivation is as follows: a prime objective of the GOLF project is to identify for the first time coherent solar oscillations in the period range 0.5 to 3.0 hours. Our ability to make this identification is improved if we understand the sources of non-coherent velocities and develop methods of modeling them from other data. The photometric signal is an integral of the intensity over the visible solar disk so that the full disk intensity can be considered to be a linear combination of the intensities produced by an array of pixel-sized sub-elements. If we can deduce the pixel by pixel signal due to non-coherent processes through the use of data from the MDI instrument on SOHO combined with ground-based observatory data, we can form the linear combination of these effects with those due to velocities and possibly correct for the non-coherent processes. The linear dependence present in treating the signal as photometric eases the task of isolating these effects. At shorter oscillation periods we can compare GOLF to other oscillation observations while at longer periods we can identify and possibly study the effect of solar active regions and supergranulation on the GOLF measurements. In this way we may be able to work our way into the [FORMULA]mode frequency range from above and below.

In the above context conversion to a velocity scale takes the form of a function of the observed signal as follows:


where [FORMULA] is an intensity-like signal that is a very slowly changing function of time including all identifiable properties of the instrument and orbit as well as the solar line profiles. The observed intensity [FORMULA] is a similarly corrected quantity derived from the moment by moment observation of the solar output. The sensitivity function [FORMULA] is well-known as a critical parameter for resonance scattering instruments and is related to [FORMULA] by:


It is important to note that even a perfect photometer system would not function as indicated in Eq. (25) because of the very large variations caused by the sun-spacecraft velocity variations and the consequent migration of the GOLF scattered wavelength relative to the solar spectral line. This subsection describes a method for calculating this effect.

The transformation from the observed count rate which we designate as [FORMULA] to an intensity-like variable allowing a velocity calculation of the form given in Eq. (25) requires a series of steps having varying degrees of reliability. The reduction sequence starts with the removal of instrumental effects along with the compensation for the inverse square effect of the variable distance. These steps will be described in García et al. (2000). These corrections for deadtime, stem temperature, the inverse square law and PM aging are all included here in deriving P, the photometric counting rate. Note, however, that the known correction for PM aging does not account for the full effects instrumental aging and this process is modelled in a more flexible fashion below. Correction for the effect of the variable distance and stem temperature can either increase or decrease the signal. The aging effects monotonically decrease the signal following a function which is probably roughly an exponential decay with and unknown decay rate. The first step in the reduction is then to correct the raw count rate [FORMULA] to a photometer rate [FORMULA] through the application of the deadtime correction, the inverse square distance correction and the stem temperature correction.

5.2. Use of the magnetic modulation

The corrections above are due to instrumental effects and the geometric inverse square effect of the variable distance. In addition there is the effect of the variable sun-spacecraft velocity on the solar line intensity at the working point of the GOLF instrument. This effect produces the instrument's sensitivity to velocity and is essential to the instrument operation. However, the large amplitude of velocity variation through the orbital path maps out a significant portion of the solar line profile and complicates the interpretation of the absolute photometry through the non-linearity of the functional relationship between velocity and intensity. Fortunately due to the magnetic modulation we have a method available to determine to a good approximation the combined solar line profile as seen by GOLF. At positive and negative magnetic modulation states we obtain photometric rates [FORMULA] and [FORMULA] which correspond to a velocity difference [FORMULA] which is an effective velocity offset between the two states of magnetic modulation. In essence we can use the difference in signal [FORMULA] to estimate the derivative [FORMULA]. By tracking the data over a large range in v, we can numerically integrate [FORMULA] to get [FORMULA] itself. This task suffers from the same difficulty of any absolute photometry in that [FORMULA] includes all the system gain uncertainties. We can overcome this problem by utilizing the ratio [FORMULA] as a function of time and orbital velocity since all gain uncertainties cancel in the ratio. Consequently, we have


where we can express the ratio of the two intensities in terms of an effective velocity offset given by:


with [FORMULA] in m s-1. We thus find that the orbital velocity plus the offset velocity is:


where [FORMULA]. Using the value of [FORMULA] from Eq. 24, we have [FORMULA] m s-1, [FORMULA], and [FORMULA] m s-1.

[FIGURE] Fig. 6. 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 spanned during operations.

As long as the photomultipliers respond linearly to solar radiation the ratio [FORMULA] should be a function of orbital velocity alone. Thus we can express this ratio as a power series expansion:


where [FORMULA] for our present data. We could also consider K to be larger than 2 with the coefficients [FORMULA] for [FORMULA]. In practice the data can be adequately fit with just [FORMULA] and [FORMULA] and inclusion of higher terms would raise the risk of fitting spurious features. Figs. 7a and b show the relationship between [FORMULA] and [FORMULA] for PM1 and PM2. These figures include all data between April 11, 1996 and Feb. 16, 1998 except for the 5 day interval in Nov, 1996 when the GOLF system experienced a brief cooldown where the missing data has been replaced with an interpolated fill. For PM2 the ratio remains a unique function of [FORMULA] whereas for PM1 the most recent data near [FORMULA] m s-1 deviates from the previous pattern. Generally the data from PM2 appears more reliable than that for PM1.

[FIGURE] Fig. 7a and b. The magnetic modulation ratio [FORMULA] for the two photomultipliers as a function of the orbital velocity [FORMULA]. The upper subfigure a shows the function for PM1 and the lower subfigure b shows the function for PM2.

Although not needed for the primary data set beginning April 11, 1996, the magnetic modulation on the red wings of the lines is also of interest as a diagnostic of instrument performance and as a tool for verification of the analysis method. Table 3 gives both the fitting coefficients for the correlations shown in Figs. 7a and b and for similar correlations obtained for the red wing data between Jan. 20, 1996 and April 1, 1996. The blue wing coefficients apply to the full data set while those for the red wing are available only for these early operations. The red wing data in particular have been contaminated by a variety of commissioning activities and are less reliable for this reason as well as having reduced reliability due to the smaller range in orbital velocity spanned by the sequence.


Table 3. Fitting coefficients for the magnetic modulation ratio

5.3. Correction for the working point wavelength variation due to sun-spacecraft velocity changes

As the velocity of the spacecraft relative to the sun varies through the orbit, the intensity of the solar lines should change due to the shift of the central wavelength of the solar line relative to the fixed wavelengths of the scattering points. We wish to correct the output of the GOLF instrument for this effect by deducing the effective shape of the solar line from the magnetically modulated position of the scattering points and the drift in relative velocity itself. The line profiles have been defined in Eqs. 9, 10, 18 and 19. Whereas most of the discussion has been carried out in terms of ratios of outputs in order to eliminate the need to use absolute photometry, the line profile functions required in above equations are not written in the form of a ratio. They can be converted to a ratio by the adoption of a reference intensity defined to be [FORMULA]. A convenient choice of [FORMULA] is that where the sun-spacecraft relative velocity is zero using the low field state of magnetic modulation.

To illustrate the method of obtaining the line intensity from the observed ratio of intensities in the two magnetic modulation states, we retain only [FORMULA] and represent the intensity with terms quadratic in the velocity. The representation of the line intensities are to this approximation:


We can also represent [FORMULA] as the product of [FORMULA] and [FORMULA] from Eq. 31:


We can equate these two expressions for [FORMULA] and equate coefficients of like powers of [FORMULA] to obtain two expressions for [FORMULA] and [FORMULA]:


which are readily solved for [FORMULA] and [FORMULA]. In the Appendix we show how to find the [FORMULA] for an arbitrary number of coefficients using as many empirically determined [FORMULA] as needed to represent the observed intensity ratio [FORMULA]. The signal functions [FORMULA] and [FORMULA] are then


The line intensity functions [FORMULA] are shown in Fig. 8. The derivatives [FORMULA] and [FORMULA] can be calculated straightforwardly from these expansions allowing a [FORMULA] function to be determined for [FORMULA] and [FORMULA] individually. These sensitivity functions are shown in Fig. 9 as functions of the orbital velocity.

[FIGURE] Fig. 8. The solar line intensity function [FORMULA] as a function of the orbital velocity [FORMULA]

[FIGURE] Fig. 9. The velocity sensitivity conversion factors [FORMULA] as functions of the orbital velocity [FORMULA].

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Online publication: January 29, 2001