  We consider the high spin expansion for the null geodesics in the Kerr
spacetime. We expand the null geodesic equation successively to higher orders
in deviation from extremity. Via the method of matched asymptotic expansion,
the radial integrals are obtained analytically. It turns out that the analytic
expressions are very sensitive to the value of the shifted Carter constant $q$.
We show that for a large $q$, the analytic expressions can be used to study
observational electromagnetic signatures for astrophysical black holes like
M87*. However, for a small $q$, the high spin expansion method can only be
applied to (near-) extreme black holes.
