# Tully-Fisher Relation

Written by User 1

Last Updated: January 17, 2011, 10:39 pm (UTC)
Originally created on January 17, 2011

Spiral galaxies are rotating bodies. The velocity of their rotation is related to their size. The Tully Fisher Relation approximately models this for type Sa, Sb and Sc galaxies. Using this information, it becomes possible to find its approximate distance from Earth.

There are three relations, one for each type of galaxy:

Sa: M = -9.95 * log(v) + 3.15
Sb: M = -10.2 * log(v) + 2.71
Sc: M = -11 * log(v) + 3.31

Where M is absolute magnitude and v is rotational velocity in km/s.

Rotational velocity is generally determined by redshift. For an edge-on galaxy, the redshift can be used directly. For other galaxies, the math must correct for the tilt of the galaxy. Using the H-alpha line:

z1 = (n1 - 6563) / 6563
z2 = (n2 - 6563) / 6563
v = c * (n1 + n2) / 2

Where c is the speed in light in km/s, z1 and z2 are the redshifts on the two sides of the galactic core (where rotation is fastest), n1 and n2 are the locations of the H-alpha lines on the sides (in nanometer) and v is rotational velocity in km/s.

Remember that the above is for edge-on galaxies only and will require manipulation with other galaxies. Face-on galaxies will not be able to use this as they will show no rotational redshift.

Taking this value and inserting it into any of the original formulas will yield the absolute magnitude of the galaxy. This can then be used to calculate the distance using the distance modulus.

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