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We explore the stability of different galaxy light concentration indices as afunction of the outermost observed galaxy radius. With a series of analyticallight-profile models, we show mathematically how varying the radial extent towhich one measures a galaxy's light can strongly affect the derived galaxyconcentration. The "mean concentration index", often used for parameterizinghigh-redshift galaxies, is shown to be horribly unstable, even when modelingone-component systems such as elliptical, dwarf elliptical and pure exponentialdisk galaxies. The C_31 concentration index performs considerably better but isalso heavily dependent on the radial extent, and hence exposure depth, of anygiven galaxy. We show that the recently defined central concentration index isremarkably stable against changes to the outer radius, providing a meaningfuland reliable estimate of galaxy concentration. The index n from the r^(1/n)models is shown to be monotonically related with the central concentration oflight, giving the index n a second and perhaps more tangible meaning. With asample of elliptical and dwarf elliptical galaxies, we present correlationsbetween the central light concentration and the global parameters: luminosity(Pearson's r = -0.82), effective radius (r = 0.67), central surface brightness(r = -0.88), and velocity dispersion (r = 0.80). The more massive ellipticalgalaxies are shown to be more centrally concentrated. We speculate that thephysical mechanism behind the recently observed correlation between the centralvelocity dispersion (mass) of a galaxy and the mass of its central supermassiveblack hole may be connected with the central galaxy concentration. That is, wehypothesize that it may not simply be the amount of mass in a galaxy but ratherhow that mass is distributed that controls the mass of the central black hole.Comment: (aastex, 18 pages including 13 figures

Galaxy Light Concentration. I. Index Stability and the Connection with Galaxy Structure, Dynamics, and Supermassive Black Holes