What is the relation between and for a cosmic string?

We examine the relation between the deficit angle and the mass per unit length of a cosmic string. In the weak-field approximation. In general we define a correction term by ()-1. We show that depends on the gravitational strength of the string, but is independent of the thickness of the string. Taking the zero-thickness limit we obtain an infinitely thin string for which. The implications of this result for the treatment of strings as singular line sources in general relativity is discussed.