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Assessment of Cowling approximation in computing ellipticity of a magnetized non-barotropic star

A deformation of a neutron star due to its own magnetic field is an important issue in gravitational wave astronomy, since a misaligned rotator with small ellipticity may emit continuous gravitational wave that may be observed by ground-based detectors. Recently Mastrano et al. (2011,2013) evaluated deformations induced by both poloidal and toroidal magnetic field in non-barotropic model stars by neglecting the gravitational field perturbation (Cowling approximation). Following their treatment in non-barotropic fluid and magnetic configurations, we here assess the effect of gravitational perturbation that they neglected. We show that the ellipticity computed with gravitational perturbation is roughly twice as large as that obtained by Cowling approximation. We should allow this amount of error in using the neat analytic treatment proposed by them.