  In this paper, we derive a precise estimate of the mean of the extinction
time of the contact process with a fixed infection rate on a star graph with
$N$ leaves. Specifically, we determine not only the exponential main factor but
also the sharp sub-exponential prefactor of the asymptotic formula for the mean
extinction time as $N\to\infty$. Previously, such detailed asymptotic
information on the mean extinction time of the contact process was known only
for complete graphs. To achieve these results, we first provide an accurate
estimation of the quasi-stationary distribution on non-extinction of the
contact process, utilizing special function theory and refined Laplace's
method. Subsequently, we employ the recently developed potential theoretic
approach to metastability of non-reversible Markov processes, enabling us to
deduce these results. The integration of these methodologies represents a novel
approach developed in this paper, which has not previously been used in the
study of the contact process.
