Spherical polytropic balls cannot mimic black holes
  The so-called black hole shadow is a dark region which is expected to appear
in a fine image of optical observation of black holes. It is essentially an
absorption cross section of black hole, and the boundary of shadow is
determined by unstable circular orbits of photons (UCOP). If there exists a
compact object possessing UCOP but no black hole horizon, it can provide us
with the same shadow image with black holes, and a detection of shadow image
cannot be a direct evidence of black hole existence. Then, this paper examine
whether or not such compact objects can exist under some suitable conditions.
We investigate thoroughly the static spherical polytropic ball of perfect fluid
with single polytrope index, and then investigate a representative example of
the piecewise polytropic ball. Our result is that the spherical polytropic ball
which we have investigated cannot possess UCOP, if the sound speed at center is
subluminal (slower-than-light). This means that, if the polytrope treated in
this paper is a good model of stellar matter in compact objects, the detection
of shadow image is regarded as a good evidence of black hole existence. As a
by-product, we have found the upper bound of the mass-to-radius radio (M/R) of
polytropic ball with single index, M/R < 0.281, under the
subluminal-sound-speed condition.
