Tully-Fisher Distance Estimate

• Tully-Fisher Distance Estimate is an extragalactic method that correlates spiral galaxy luminosity with rotational velocity to compute redshift-independent distances.
• It employs precise photometric and color corrections alongside HI linewidth measurements to derive calibrated absolute magnitudes and reduce scatter.
• The methodology underpins key cosmological analyses, including cosmic flow mapping, mass distribution studies, and independent Hubble constant measurements.

The Tully-Fisher (TF) Distance Estimate is a fundamental extragalactic methodology exploiting the empirical correlation between the luminosity of spiral galaxies and their rotational velocity to derive redshift-independent distances. Precision and calibration in the TF relation underpin cosmological analyses, including measurement of the Hubble constant, cosmic flow mapping, and mass distribution studies. The method is applicable across photometric bands (optical, near-IR, mid-IR), in both luminosity-TF and baryonic-TF formulations, and is supported by rigorous calibration protocols utilizing geometric and standard-candle distances.

1. Mathematical Formulation and Calibration of the Tully-Fisher Relation

The canonical form of the TF relation in a passband $\lambda$ is: $$M_\lambda = a_\lambda\,\bigl[\log W^i_{mx} - 2.5\bigr] + b_\lambda$$ where $M_\lambda$ is absolute magnitude, $W^i_{mx}$ is the inclination-corrected maximum HI linewidth (proxy for $2V_{max}$), with $a_\lambda$ and $b_\lambda$ the slope and zero point, respectively. For mid-infrared ([3.6] μm) calibration (Sorce et al., 2013):

• Uncorrected relation:

$$M^{b,i,k,a}_{[3.6]} = -20.34 \pm 0.10 - (9.74 \pm 0.22)\bigl(\log W^i_{mx}-2.5\bigr)$$

• Color-corrected, reduced-scatter form:

$$M_{C_{[3.6]}} = -20.34 \pm 0.08 - (9.13 \pm 0.22)\bigl(\log W^i_{mx}-2.5\bigr)$$

Corrections applied to apparent magnitudes include Milky Way foreground extinction ($A_b$), internal extinction in the disk ($A_i$), k-correction for bandpass shift ($A_k$), and an aperture correction for extended-source PSF wings ($A_a$). For color corrections (when $I - [3.6]$ photometry is available), a pseudo-magnitude is constructed: $$C_{[3.6]} = [3.6]^{b,i,k,a} - \Delta[3.6]^{\rm color}$$ where

$$\Delta[3.6]^{\rm color} = -0.47\,\left[\left(I - [3.6]\right) + 0.77\right]$$

The slope is empirically constrained via large cluster samples; the zero point is set by nearby galaxies with independently measured distances (Cepheids, TRGB).

2. Workflow: Practical Distance Estimation Procedure

Given observations of $[3.6]_{\rm obs}$, $W_{m50}$, inclination $i$, and if available $I-[3.6]$, the workflow is:

1. Photometric Corrections: $$[3.6]^{b,i,k,a} = [3.6]_{\rm obs} - A_b^{[3.6]} - A_i^{[3.6]} - A_k^{[3.6]} + A_a^{[3.6]}$$
2. Color Correction (optional): $$C_{[3.6]} = [3.6]^{b,i,k,a} - \Delta[3.6]^{\rm color}$$
3. HI width correction: $W^i_{mx}$ is derived from $W_{m50}$ after instrumental, relativistic, and inclination corrections.
4. Absolute Magnitude Prediction: Use either the raw TF relation or color-corrected form as appropriate.
5. Distance Modulus and Physical Distance:

$$\mu = m_{\rm corr} - M; \quad D = 10^{(\mu+5)/5}\,\mathrm{Mpc}$$

1. Residual Malmquist Bias: Apply

$\mu^c = \mu + 0.0065(\mu - 31)^2$

An individual galaxy’s distance uncertainty is typically $\sigma_{\mathrm{TF}}\approx0.44$ mag (color-corrected), corresponding to $\sim22\%$ fractional error; raw scatter is marginally higher ($0.49$ mag, $\sim25\%$).

3. Calibration Samples, Error Budget, and Propagation

Template slopes are set by fitting to 213 galaxies in 13 clusters spanning $1000-10000$ km s$^{-1}$. Zero point calibration employs 26 galaxies with robust Cepheid or TRGB distances. The derived Hubble constant from calibrator clusters at $V_{\rm CMB} > 4000$ km s$^{-1}$ is: $$H_0 = 73.8\pm1.1_{\rm stat}\pm3.8_{\rm sys} \,\mathrm{km\,s}^{-1}\,\mathrm{Mpc}^{-1}$$ finalized as $$H_0 = 74\pm5\,\mathrm{km\,s}^{-1}\,\mathrm{Mpc}^{-1}$$ (Sorce et al., 2013).

Principal sources of uncertainty:

• Photometry ($\pm0.05$ mag)
• Inclination ($\lesssim8\%$ in $W^i_{mx}$ for $i$ near $45^\circ$ cutoff)
• Linewidth measurement ($\lesssim20$ km s$^{-1}$)
• Extinction, k-correction ($\sim0.02$ mag)
• Cosmic scatter: cluster depth, asymmetries, and population variation

Cluster distances benefit from error reduction $\propto N^{-1/2}$ for $N$ members.

4. Advantages, Bandpass Considerations, and Color Corrections

Mid-infrared (3.6 μm) TF estimation offers several distinct advantages:

• Uniform Spitzer photometry across the sky; zero-point stability $<1\%$
• Negligible Galactic and internal extinction coefficients
• Old stellar populations dominating the flux; tight mass–luminosity coupling
• Low background, short exposures ($\sim4$ min) suffice for all-sky coverage

Despite slightly increased scatter relative to $I$-band when color information is unavailable, a simple additive color correction recovers full accuracy. The color-corrected mid-IR TF achieves comparable precision to $I$-band ($\sim0.2$ mag scatter, $\sim20\%$ distance error), while maintaining the extinction and PSF advantages intrinsic to the mid-IR regime.

5. Limiting Factors, Applicability, and Best Practice Recommendations

The method is valid for spiral galaxies up to $\sim200$ Mpc, beyond which peculiar velocity noise becomes limiting. Mid-IR TF is robust against photometry and extinction errors but subject to the adopted calibration sample’s reliability. Optimal application requires:

• Strict consistency in photometric correction recipes
• High S/N HI profiles and accurate inclination measurements
• Exclusion of irregular, strongly star-forming, or interacting systems which elevate TF scatter
• Preferential use of color corrections where possible
• Comprehensive propagation of calibration, measurement, and cosmic variances in the uncertainty model

For cluster/group applications, joint distance estimates reduce intrinsic scatter, enabling precision peculiar-velocity and $H_0$ analysis.

6. Contextual Significance in the Distance Ladder and Cosmological Applications

The mid-infrared Tully-Fisher calibration (Sorce et al., 2013) provides a robust independent rung in the extragalactic distance ladder, complementary to Cepheid and TRGB standard candles, and supported by geometric anchors such as maser distances. Its calibration stability and intrinsic accuracy are critical for precision $H_0$ measurements and for systematic mapping of cosmic flows and mass distribution. As galaxy samples, ancillary photometry, and HI databases expand, the mid-IR TF method's reach and impact are correspondingly amplified.

This methodology remains central to the Cosmicflows project, large-scale velocity field inference, and the contemporary debate over Hubble constant determination—providing independent, well-characterized distances out to significant cosmological scales.