SpringerLink
Forum Springer Astron. Astrophys.
Forum Whats New Search Orders


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

Previous Section Next Section Title Page Table of Contents

Appendix A: derivation of the solar line intensity as a function of [FORMULA]

In order to convert the photomultiplier rate ratio to a function which represents the dependence of [FORMULA] on the orbital velocity we consider that [FORMULA] can be expanded in a power series of the orbital velocity and that [FORMULA] and [FORMULA] are obtained by inserting [FORMULA] and [FORMULA] into this power series. According to Eq. (30) we can write:

[EQUATION]

The intensities can then be expressed in terms of the general power series:

[EQUATION]

where [FORMULA] is the [FORMULA] intensity when [FORMULA] and for compactness of the sums we have defined [FORMULA]. After utilizing the fact that [FORMULA], the difference in intensities can be expressed as

[EQUATION]

[EQUATION]

The quantity [FORMULA] can be expanded as a binomial to get

[EQUATION]

After inserting the expansions for [FORMULA] into Eq. (A.3) we get:

[EQUATION]

We seek to collect terms with like powers of [FORMULA] and solve for the coefficients which result in Eq. (31) being satisfied for all values of [FORMULA]. In the double sum contained in Eq. (A.6) any particular power of [FORMULA] can appear in more than one term. In order to collect all the terms having the same power of [FORMULA] let us reverse the order of summation. We have to correctly set the limits of summation in the new order so that we continue to include the same terms originally present. As written in Eq. (A.6 the terms present have [FORMULA] and [FORMULA] so that when we exchange the order of the summation the limits transform as follows:

[EQUATION]

which allows us to reexpress the intensity difference as:

[EQUATION]

where we have defined:

[EQUATION]

We can combine Eqs. (31) and (A.8) for [FORMULA] and express the result as a function of [FORMULA]:

[EQUATION]

The second half just comes from inserting the expansion for [FORMULA] into the denominator. This can be solved after multiplying by [FORMULA] on both sides to yield:

[EQUATION]

After collecting the coefficients of like powers of [FORMULA] we get the system of equations (see Eq. (A.12) on top of the page). This system can now be solved for all the [FORMULA] with [FORMULA] since [FORMULA] and all other quantities can be evaluated from the observations. Note that the [FORMULA] values need not differ from zero for more than [FORMULA]. In practice as indicated in Table 3 we have retained non-zero [FORMULA] for [FORMULA]. The equations also apply to the red wing data with the only change required being that of reversing the sign of [FORMULA]. We could have retained the positive sign of [FORMULA] and utilized a ratio [FORMULA] but this would have involved a different weighting of the three scattering components. The method suggested here of retaining [FORMULA] and reversing the sign of [FORMULA] leaves the three scattering components distributed nearly the same way relative to their average and provides a more symmetric treatment of the two wings. An IDL code to derive the [FORMULA] coefficients from the A coefficients is available from the UCLA GOLF website.

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
helpdesk.link@springer.de