H7 - the Tully-Fischer relation

image: Sparke and Gallagher (2007)

it is an empirical relation that correlates the width of the velocity profile with luminosity

L = 3 \times 10^{10} L_{⊙} {(\frac{v}{200 {km s}^{- 1}})}^{4}

where, $v$ is the width of the velocity profile

the width of a spectral line can be a distance-independent indicator of luminosity
combined with $m$ , it can be used to obtain the distance via the distance modulus
considering Keplerian motion:

v^{4} = G^{2} M \frac{M}{R^{2}}

considering the mass to light ratio: $M \propto L$ , constant for spirals
the surface brightness, $I \propto L / R^{2} \propto M / R^{2}$
therefore:

v^{4} \propto L I

for spirals, $I$ is constant, hence:

v^{4} \propto L