  For the formation of a black hole in the gravitational collapse of a massless
scalar field, we calculate a critical exponent that governs the black hole
angular momentum for slightly non-spherical initial data near the black hole
threshold. We calculate the scaling law by second-order perturbation theory. We
then use the numerical results of a previous first-order perturbative analysis
to obtain the numerical value mu ~ 0.76 for the angular momentum critical
exponent. A quasi-periodic fine structure is superimposed on the overall power
law.
