In this paper , we measure the optical-to-virial velocity ratios V _ { opt } / V _ { 200 c } of disk galaxies in the Sloan Digital Sky Survey ( SDSS ) at a mean redshift of \langle z \rangle = 0.07 and with stellar masses 10 ^ { 9 } M _ { \odot } < M _ { * } < 10 ^ { 11 } M _ { \odot } . V _ { opt } / V _ { 200 c } , the ratio of the circular velocity measured at the virial radius of the dark matter halo ( \sim 150 kpc ) to that at the optical radius of the disk ( \sim 10 kpc ) , is a powerful observational constraint on disk galaxy formation .
It links galaxies to their dark matter haloes dynamically and constrains the total mass profile of disk galaxies over an order of magnitude in length scale .
For this measurement , we combine V _ { opt } derived from the Tully-Fisher relation ( TFR ) from Reyes et al .
with V _ { 200 c } derived from halo masses measured with galaxy-galaxy lensing .
In anticipation of this combination , we use similarly-selected galaxy samples for both the TFR and lensing analysis .
For three M _ { * } bins with lensing-weighted mean stellar masses of 0.6 , 2.7 , and 6.5 \times 10 ^ { 10 } M _ { \odot } , we find halo-to-stellar mass ratios M _ { 200 c } / M _ { * } = 41 , 23 , and 26 , with 1 \sigma statistical uncertainties of around 0.1 dex , and V _ { opt } / V _ { 200 c } = 1.27 \pm 0.08 , 1.39 \pm 0.06 , 1.27 \pm 0.08 ( 1 \sigma ) , respectively .
Our results suggest that the dark matter and baryonic contributions to the mass within the optical radius are comparable , if the dark matter halo profile has not been significantly modified by baryons .
The results obtained in this work will serve as inputs to and constraints on disk galaxy formation models , which will be explored in future work .
Finally , we note that this paper presents a new and improved galaxy shape catalogue for weak lensing that covers the full SDSS DR7 footprint .