Given the recently deduced relationship between X-ray temperatures and stellar velocity dispersions ( the “ T – \sigma relation ” ) in an optically complete sample of elliptical galaxies ( [ Davis & White 1996 ] ) , we demonstrate that L > L _ { * } ellipticals contain substantial amounts of dark matter in general .
We present constraints on the dark matter scale length and on the dark-to-luminous mass ratio within the optical half-light radius and within the entire galaxy .
For example , we find that minimum values of dark matter core radii scale as r _ { dm } > 4 ( L _ { V } / 3 L _ { * } ) ^ { 3 / 4 } h ^ { -1 } _ { 80 } kpc and that the minimum dark matter mass fraction is \gtrsim 20 % within one optical effective radius r _ { e } and is \gtrsim 39 – 85 % within 6 r _ { e } , depending on the stellar density profile and observed value of \beta _ { spec } .
We also confirm the prediction of Davis & White ( 1996 ) that the dark matter is characterized by velocity dispersions that are greater than those of the luminous stars : \sigma _ { dm } ^ { 2 } \approx 1.4 – 2 \sigma _ { * } ^ { 2 } .
The T – \sigma relation implies a nearly constant mass-to-light ratio within six half-light radii : M / L _ { V } \approx 25 h _ { 80 } M _ { \odot } / L _ { V _ { \odot } } .
This conflicts with the simplest extension of CDM theories of large scale structure formation to galactic scales ; we consider a couple of modifications which can better account for the observed T – \sigma relation .