Volume 574, February 2015
|Number of page(s)||16|
|Section||Stellar structure and evolution|
|Published online||22 January 2015|
Appendix A: Demonstration of the error propagation formula for the non-linear extension of the acoustic radius inversion
Equation (26) is obtained with a little algebra. First, we treat the observed frequencies, νobs,i, and the inverted acoustic radius, τinv, as independent stochastic variables with ϵi being the individual noise realisations for each frequency, ϵτ the resultant deviation on τinv, and and the average of the stochastic variables νobs,i and τinv, respectively. Furthermore, we assume that: (A.3)Using the fact that τinv = qoptτref, with the definition of qopt given in Eq. (25), and the separation into stochastic and average contributions defined previously, we obtain: (A.4)where we assumed that ϵi is much smaller than 1, thereby allowing us to linearise the above equation. We now apply the formula for the variance of a linear combination of independent stochastic variables and obtain (A.5)where we used the following equivalences: (A.6)Equation (A.5)then leads directly to Eq. (26) when using the definition of qopt given in Eq. (25).
The following figures illustrate the quality of the kernel fits for some of the test cases we presented in the article. Although these plots are redundant from the visual point of view, we wish here again to stress that they are crucial to the understanding of the quality of a SOLA inversion and justify the accuracy of the results presented in the previous sections.
© ESO, 2015
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.