  Polytropic transonic solutions of spherically symmetric and steady galactic
winds in the gravitational potential of a dark matter halo (DMH) with a
supermassive black hole (SMBH) are studied. The solutions are classified in
terms of their topological features, and the gravitational potential of the
SMBH adds a new branch to the transonic solutions generated by the gravity of
the DMH. The topological types of the transonic solutions depend on the mass
distribution, the amount of supplied energy, the polytropic index $\gamma$, and
the slope $\alpha$ of the DMH mass distribution. When $\alpha$ becomes larger
than a critical value $\alpha_\mathrm{c}$, the transonic solution types change
dramatically. Further, our model predicts that it is possible for a slowly
accelerating outflow to exist, even in quiescent galaxies with small $\gamma$.
This slowly accelerating outflow differs from those considered in many of the
previous studies focusing on supersonic outflows in active star-forming
galaxies. In addition, our model indicates that outflows in active star-forming
galaxies have only one transonic point in the inner region ($\sim$ 0.01 kpc).
The locus of this transonic point does not strongly depend on $\gamma$. We
apply the polytropic model incorporating mass flux supplied by stellar
components to the Sombrero Galaxy, and conclude that it can reproduce the
observed gas density and the temperature distribution well. This result differs
significantly from the isothermal model, which requires an unrealistically
large mass flux (Igarashi et al. 2014). Thus, we conclude that the polytropic
model is more realistic than the isothermal model, and that the Sombrero Galaxy
can have a slowly accelerating outflow.
