Semi-analytical models of galaxy formation can be used to predict the evolution of the number density of early-type galaxies as a function of the circular velocity at the virial radius , v _ { c, vir } .
Gravitational lensing probability and separation distribution on the other hand are sensitive to the velocity dispersion ( or circular velocity ) at about the effective radius .
We adopt the singular isothermal ellipsoid ( SIE ) lens model to estimate the velocity dispersion at the effective radius .
The velocity dispersion from strong lensing based on the SIE , \sigma _ { SIE } is then closely related to the observational central stellar velocity dispersion , \sigma _ { cent } ; we have empirically \sigma _ { SIE } \approx \sigma _ { cent } .
We use radio lenses from the Cosmic Lens All-Sky Survey and the PMN-NVSS Extragalactic Lens Survey to study how the velocity dispersions , \sigma _ { SIE } , are related to v _ { c, vir } ; if the galaxy were a singular isothermal sphere up to the virial radius , v _ { c, vir } = \sqrt { 2 } \sigma _ { SIE } .
When we include both the lensing probability and separation distribution as our lensing constraints , we find \sigma _ { SIE } / ( 200 { { km~ { } s ^ { -1 } } } ) = [ ( 1.17 _ { -0.26 } ^ { +0.40 } ) v _ { c, vir% } / ( 200 { { km~ { } s ^ { -1 } } } ) ] ^ { 0.22 ^ { +0.05 } _ { -0.04 } } for 200 { { km~ { } s ^ { -1 } } } \la \sigma _ { SIE } \la 260 { { km~ { } s ^ { -1 } } } ; at \sigma _ { SIE } = 200 { { km~ { } s ^ { -1 } } } , the ratio \sqrt { 2 } \sigma _ { SIE } / v _ { c, vir } is about 1.65 ^ { +0.57 } _ { -0.37 } ( 68 % CL ) but decreases to 0.65 _ { -0.12 } ^ { +0.15 } ( 68 % CL ) for \sigma _ { SIE } = 260 { { km~ { } s ^ { -1 } } } .
These results are consistent with those of Seljak ( 44 ) obtained from galaxy-galaxy weak lensing for galaxies of around L _ { * } .
However , our results clearly suggest that the ratio must vary significantly as \sigma _ { SIE } is varied and are marginally discrepant with the Seljak results at \sigma _ { SIE } = 260 { { km~ { } s ^ { -1 } } } .
The scaling \sigma _ { SIE } \propto v _ { c, vir } ^ { 0.22 \pm 0.05 } is broadly consistent with those from galaxy occupation statistics studies and the most recent galaxy-galaxy weak lensing study .
We discuss briefly the implications of our results for galaxy formations and structures .
These constraints can be significantly strengthened when larger lens samples become available and the accuracy of semi-analytical model predictions improves .