Astronomy and Astrophysics – Astrophysics
Scientific paper
Aug 2010
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2010apj...719..985v&link_type=abstract
The Astrophysical Journal, Volume 719, Issue 1, pp. 985 (2010).
Astronomy and Astrophysics
Astrophysics
Scientific paper
Section 4.3 of the published article presents the erroneous conclusion that the incoherent sum of covariant orthogonally polarized modes causes the variance of the total intensity to increase and that of the major polarization to decrease. This incorrect result is derived in Appendix C from the unsubstantiated expression for the cross-covariance matrix presented in Equation (33). Although this ansatz yields results that are consistent with previous work, such as Equations (5) and (6) of McKinnon & Stinebring (1998) and Equation (5) of Cordes et al. (1978), it does not take account of the covariances between the Stokes parameters as expressed by our Equation (28).
The correct form of Equation (33) is derived by considering the incoherent superposition of two sources described by Stokes parameters A and B, the result of which is indistinguishable from a single stationary stochastic source of polarized radiation with mean Stokes parameters A + B. Substitution of S = A + B into Equation (28), yields where C A and C B are the covariance matrices of A and B as defined by Equation (28) and the rectified cross-covariance matrix is given by
Here, the intensity correlation coefficient rhov is reintroduced to generalize the result; it varies from zero (disjoint modes) to unity (superposed modes). To verify the above expression for the cross-covariance matrix, consider the superposition of orthogonally polarized modes; in this case, A \cdot B} = -|A||B| and, in the natural basis defined by A
If the intensities, degrees of polarization, and degrees of freedom of the modes are equal (A 0 = B 0 = S 0/2, |A|=|B|, and rhov A = rhov B = rhov), then the superposition of the modes will be unpolarized and the covariance matrix reduces to C = sigmav2 S 2 0 I, where I is the 4 × 4 identity matrix, as is easily confirmed using the above equations.
In the natural basis, it is readily seen that the cross-covariance terms due to the sum of orthogonally polarized modes contribute equally to the variances of the total intensity and the major polarization. That is, orthogonally polarized modes cannot explain the differences in the variances observed by McKinnon (2004) and summarized in Table 1. The differences in the variances can be explained by the coherence and/or obliquity of the modes. For example, the partially coherent sum of linearly polarized orthogonal modes would produce an additional elliptically polarized component that would increase the variances of the total intensity and the minor polarizations. Similarly, the superposition of non-orthogonal modes would add to the variances of the minor polarizations. In each case, the variance of the total intensity will exceed that of the major polarization, the distribution of the polarization vector will become ellipsoidal, and the major axis of the ellipsoid will no longer be parallel to the mean polarization vector.
The rectified cross-covariance matrix does not affect the main conclusion of Section 4.4; that is, the excess dispersion of the polarization vector, first observed as modal broadening by Stinebring et al. (1984), is completely due to self-noise intrinsic to the pulsar signal. However, the correction does impact on the proposal to develop a new technique for producing mode-separated profiles. All of the equations presented in Appendix C must be replaced by a more general expression that includes the partial coherence and/or non-orthogonality of the modes. As before, this development is left for future work.
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