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Measuring the Virial Factor in SDSS DR5 Quasars with Redshifted H$β$ and Fe ii Broad Emission Lines

Under the hypothesis of gravitational redshift induced by the central supermassive black hole, and based on line widths and shifts of redward shifted H$β$ and Fe ii broad emission lines for a sample of 1973 $z<0.8$ SDSS DR5 quasars, we measured the virial factor in determining supermassive black hole masses, usually estimated by the reverberation mapping (RM) method or the relevant secondary methods. The virial factor had been believed to be from the geometric effect of broad-line region. The measured virial factor of Fe ii is larger than that of H$β$ for 98% of these quasars. The virial factor is very different from object to object and for different emission lines. For most of these quasars, the virial factor of H$β$ is larger than these averages that were usually used in determining the masses of black holes. There are three positive correlations among the measured virial factor of H$β$, dimensionless accretion rate and Fe ii/H$β$ line ratio. A positive three-dimensional correlation is found among these three quantities, and this correlation indicates that the virial factor is likely dominated by the dimensionless accretion rate and metallicity. A negative correlation is found between the redward shift of H$β$ and the scaled size of broad-line region radius in units of the gravitational radius of black hole. This negative correlation will be expected naturally if the redward shift of H$β$ is mainly from the gravity of black hole. Radiation pressure from accretion disk may be a significant contributor to the virial factor.