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.