  The Petermann factor and the phase rigidity are convenient measures for
various aspects of open quantum and wave systems, such as the sensitivity of
energy eigenvalues to perturbations or the magnitude of quantum excess noise in
lasers. We discuss the behavior of these two important quantities near
non-Hermitian degeneracies, so-called exceptional points. For small generic
perturbations, we derive analytically explicit formulas which reveal a relation
to the spectral response strength of the exceptional point. These formulas shed
light on the possibilities for enhanced sensing in passive systems. The
predictions of the general theory are successfully compared to analytical
solutions of a toy model. Moreover, it is demonstrated that the connection
between the Petermann factor and the spectral response strength provides the
basis for an efficient numerical scheme to calculate the latter. Our theory is
also important in the presence of the unavoidable imperfections in the
fabrication of exceptional points as it allows to determine of what is left of
the sensitivity for such imperfect exceptional points studied in experiments.
