The structure and dark matter halo core properties of dwarf spheroidal galaxies ( dSphs ) are investigated .
A double-isothermal ( DIS ) model of an isothermal , non self-gravitating stellar system , in gravitational equilibrium and embedded in an isothermal dark halo core provides an excellent fit to the various observed stellar surface density distributions \Sigma _ { * } ( r ) .
Despite its constant velocity dispersion , the stellar system can be well characterised by King profiles ( 42 ) with a broad distribution of concentration parameters c = \log ( r _ { *,t } / r _ { *,c } ) , with r _ { *,t } and r _ { *,c } the stellar tidal and core radius , respectively .
The DIS model confirms the suggestion of Kormendy & Freeman ( 44 ) that the core scale length of the stellar system , defined as a _ { * } = - ( d \ln \Sigma _ { * } / dr ^ { 2 } ) ^ { -1 / 2 } , is sensitive to the central dark matter density \rho _ { 0 } .
In contrast to single-component systems , r _ { *,t } however does not trace the tidal radius of the galaxy but the core radius r _ { c } of its dark matter halo . c is therefore sensitive to the ratio \sigma _ { * } / \sigma _ { 0 } with \sigma _ { * } and \sigma _ { 0 } the stellar and dark matter velocity dispersion , respectively .
Simple empirical relationships are derived that allow to calculate the dark halo core parameters \rho _ { 0 } , r _ { c } and \sigma _ { 0 } , given the observable quantities \sigma _ { * } , a _ { * } and c .
The DIS model is applied to the Milky Way ’ s dSphs .
Their halo velocity dispersions lie in a narrow range of 10 km/s \leq \sigma _ { 0 } \leq 18 km/s with halo core radii of 280 pc \leq r _ { c } \leq 1.3 kpc and r _ { c } \approx 2 a _ { * } .
All dSphs follow closely the same universal scaling relations \langle \rho _ { 0 } r _ { c } \rangle \equiv \rho _ { 0 } \times r _ { c } = 75 _ { -45 } ^ { +85 } M _ { \odot } pc ^ { -2 } and \sigma _ { 0 } ^ { 2 } \times r _ { c } ^ { -1 } = 0.45 _ { -0.27 } ^ { +0.51 } ( km/s ) ^ { 2 } pc ^ { -1 } that characterise the cores of more massive galaxies over a range of 18 magnitudes in blue magnitude M _ { B } .
For given \langle \rho _ { 0 } r _ { c } \rangle the core mass is a strong function of core radius , M _ { c } \sim r _ { c } ^ { 2 } .
Inside a fixed radius r _ { u } , with r _ { u } the logarithmic mean of the dSph ’ s core radii , the total mass M _ { u } = 2.17 \langle \rho _ { 0 } r _ { c } \rangle r _ { u } ^ { 2 } is however roughly constant .
Outliers with smaller masses are expected for dSphs with core radii that are much larger or smaller than r _ { u } .
For the Milky Way ’ s dSphs we find r _ { u } = 400 \pm 100 pc and M _ { u } = 2.6 \pm 1.4 \times 10 ^ { 7 } M _ { \odot } , in agreement with Strigari et al .
( 78 ) .
Due to their small r _ { c } , the core densities of the Galaxy ’ s dSphs are very higher , with \rho _ { 0 } = 0.03 - 0.3 M _ { \odot } pc ^ { -3 } .
The dSphs would have to be on galactic orbits with pericenters smaller than a few kpc in order for their stellar systems to be affected by Galactic tides which is very unlikely .
dSphs should therefore be tidally undisturbed .
Observational evidence for tidal effects might then provide a serious challenge for the cold dark matter scenario .