## Appendix A: derivation of the solar line intensity as a function ofIn order to convert the photomultiplier rate ratio to a function which represents the dependence of on the orbital velocity we consider that can be expanded in a power series of the orbital velocity and that and are obtained by inserting and into this power series. According to Eq. (30) we can write: The intensities can then be expressed in terms of the general power series: where is the intensity when and for compactness of the sums we have defined . After utilizing the fact that , the difference in intensities can be expressed as The quantity can be expanded as a binomial to get After inserting the expansions for into Eq. (A.3) we get: We seek to collect terms with like powers of and solve for the coefficients which result in Eq. (31) being satisfied for all values of . In the double sum contained in Eq. (A.6) any particular power of can appear in more than one term. In order to collect all the terms having the same power of 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 and so that when we exchange the order of the summation the limits transform as follows: which allows us to reexpress the intensity difference as: where we have defined: We can combine Eqs. (31) and (A.8) for and express the result as a function of : The second half just comes from inserting the expansion for into the denominator. This can be solved after multiplying by on both sides to yield: After collecting the coefficients of like powers of
we get the system of equations (see
Eq. (A.12) on top of the page). This system can now be solved for
all the with
since
and all other quantities can be
evaluated from the observations. Note that the
values need not differ from zero
for more than . In practice as
indicated in Table 3 we have retained non-zero
for
. The equations also apply to the
red wing data with the only change required being that of reversing
the sign of . We could have retained
the positive sign of and utilized a
ratio but this would have involved
a different weighting of the three scattering components. The method
suggested here of retaining and
reversing the sign of 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
coefficients from the © European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |