The Faint Stellar Object SDSS J1257+3419 is a Dark Matter Dominated System

A recent study has revealed SDSS J1257+3419 is either a faint and small dwarf galaxy or a faint and widely extended globular cluster. In this Letter, the author suggests this stellar system is a dwarf spheroidal (dSph). Adopting an observational relation between binding energy and mass of old stellar systems, we derive a relation between mass and size of dSphs by assuming that they are dark matter dominated and virialized objects. Letting half-light radius represent size of SDSS J1257+3419, we find that its mass is $\sim 7\times 10^6$ solar mass. This indicates mass-to-light ratio ($M/L$) of SDSS J1257+3419 is about 1000 in the solar unit. This large $M/L$ is expected from a Mateo plot of dSphs. Thus, we insist SDSS J1257+3419 is a dSph.