Restoration of the Tully-Fisher Relation by Statistical Rectification

Abstract

I employ the Lucy rectification algorithm to recover the inclination-corrected distribution of local disk galaxies in the plane of absolute magnitude ($M_i$) and HI velocity width ($W_{20}$). By considering the inclination angle as a random variable with a known probability distribution, the novel approach eliminates one major source of uncertainty in studies of the Tully-Fisher relation: inclination angle estimation from axial ratio. Leveraging the statistical strength derived from the entire sample of 28,264 HI-selected disk galaxies at $z < 0.06$ from the Arecibo Legacy Fast ALFA (ALFALFA) survey, I show that the restored distribution follows a sharp correlation that is approximately a power law between $-16 > M_i > -22$: $M_i = M_0 - 2.5\beta \ [\log(W_{\rm 20}/250 {\rm km/s})]$, with $M_0 = -19.77\pm0.04$ and $\beta = 4.39\pm0.06$. At the brighter end ($M_i < -22$), the slope of the correlation decreases to $\beta \approx 3.3$, confirming previous results. Because the method accounts for measurement errors, the intrinsic dispersion of the correlation is directly measured: $\sigma(\log W_{20}) \approx 0.06$ dex between $-17 > M_i > -23$, while $\sigma(M_i)$ decreases from $\sim$0.8 in slow rotators to $\sim$0.4 in fast rotators. The statistical rectification method holds significant potential, especially in the studies of intermediate-to-high-redshift samples, where limited spatial resolution hinders precise measurements of inclination angles.