  We consider a nonlinear Schr\"{o}dinger (NLS) equation with any positive
power nonlinearity on a star graph $\Gamma$ ($N$ half-lines glued at the common
vertex) with a $\delta$ interaction at the vertex. The strength of the
interaction is defined by a fixed value $\alpha \in \mathbb{R}$. In the recent
works of Adami {\it et al.}, it was shown that for $\alpha \neq 0$ the NLS
equation on $\Gamma$ admits the unique symmetric (with respect to permutation
of edges) standing wave and that all other possible standing waves are
nonsymmetric. Also, it was proved for $\alpha<0$ that, in the NLS equation with
a subcritical power-type nonlinearity, the unique symmetric standing wave is
orbitally stable.
  In this paper, we analyze stability of standing waves for both $\alpha<0$ and
$\alpha>0$. By extending the Sturm theory to Schr\"{o}dinger operators on the
star graph, we give the explicit count of the Morse and degeneracy indices for
each standing wave. For $\alpha<0$, we prove that all nonsymmetric standing
waves in the NLS equation with any positive power nonlinearity are orbitally
unstable. For $\alpha>0$, we prove the orbital instability of all standing
waves.
