We find that the mass ratio q in quiescent black hole ( BH ) X-ray transients is tightly correlated with the ratio of the double peak separation ( DP ) to the full-width-half maximum ( FWHM ) of the H _ { \alpha } emission line , \log q = -6.88 ~ { } -23.2 ~ { } \log ( DP / FWHM ) .
The correlation is explained through the efficient truncation of the outer disc radius by the 3:1 resonance with the companion star .
This is the dominant tidal interaction for extreme mass ratios q = M _ { 2 } / M _ { 1 } \lesssim 0.25 , the realm of BH ( and some neutron star ) X-ray transients .
Mass ratios can thus be estimated with a typical uncertainty of \approx 32 % , provided that the H _ { \alpha } profile used to measure DP / FWHM is an orbital phase average .
We apply the DP / FWHM - q relation to the three faint BH transients XTE J1650-500 , XTE J1859+226 and Swift J1357-0933 and predict q = 0.026 ^ { +0.038 } _ { -0.007 } , 0.049 ^ { +0.023 } _ { -0.012 } and 0.040 ^ { +0.003 } _ { -0.005 } , respectively .
This new relation , together with the FWHM - K _ { 2 } correlation presented in Paper I ( 4 ) allows the extraction of fundamental parameters from very faint targets and , therefore , the extension of dynamical BH studies to much deeper limits than was previously possible .
As an example , we combine our mass ratio determination for Swift J1357-0933 with previous results to yield a BH mass of 12.4 \pm 3.6 M _ { \odot } .
This confirms Swift J1357-0933 as one of the most massive BH low-mass X-ray binaries in the Galaxy .