Contribution of gravitational self-interaction to Δphi and μ for a cosmic string

We consider the contribution of gravitational self-interaction to the mass per unit length μ and the deficit angle Δphi of a long, static, straight cosmic string. The equations describing the structure and gravitational field of the string are expanded in powers of the gravitational strength of the string. To lowest order these equations yield the usual results for the Nielsen-Olesen string including Vilenkin's formula Δphi = 8πμ. In the next order a numerical treatment of the equations gives the gravitational corrections to Δphi and μ