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 H i 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 H i -selected disk galaxies at z < 0.06 from the Arecibo Legacy Fast ALFA 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 β [ log ( W 20 / 250 km / s ) ] , with M 0 = −19.77± 0.04 and β = 4.39 ± 0.06. At the brighter end ( M i < −22), the slope of the correlation decreases to β ≈ 3.3, confirming previous results. Because the method accounts for measurement errors, the intrinsic dispersion of the correlation is directly measured: σ ( log W 20 ) ≈ 0.06 dex between −17 > M i > −23, while σ ( M i ) decreases from ∼0.8 in slow rotators to ∼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.