There_{ } are 3 "observable" physical properties of_{ }
galaxies, R, L,_{ } and M, as there are in stars._{ } These
are given by the relations_{ }

The_{ } distance *d* enters into the small-angle_{ }
relation and the_{ } flux-luminosity relation._{ } We can
eliminate this between_{ } the two using the surface
brightness_{ }

which_{ } is independent of the distance, as_{ } we
discussed in Lecture 3._{ } We can rearrange
this to find_{ }

where_{ } we have introduced the mass-to-light ratio_{ }
*M*/*L*._{ } If we can assume that for a given_{ }
class of galaxies that_{ } the central surface brightness_{ }
I (in Lsun/pc^2)_{ } and the mass-to-light ratio_{ }
*M*/*L* *are constant*,_{ } then we find the
relation_{ }

which_{ } is the basis of some of the most useful_{ }
distance indicators in_{ } cosmology! For instance,_{ } if we
are observing_{ } spiral galaxies, the appropriate_{ }
velocity_{ } to use is the maximum velocity *v_max*_{ }
in the rotation curve which can be_{ } determined from H I
spectra._{ } Then, we have_{ }

which_{ } is known as the **Tully-Fisher relation**._{ }
On the other hand,_{ } if we are concerned with_{ } elliptical
galaxies, the appropriate velocity is the_{ } central *velocity
dispersion*_{ }

which_{ } is known as the **Faber-Jackson relation**.
Both_{ } of these relations were discovered empirically_{ }
and are justified through_{ } the arguments given above._{ }
The proportionality_{ } must be calibrated using galaxies_{ }
with known distances,_{ } then the relations can be used_{ }
to find the luminosity,_{ } and thus the distance, given_{ }
an observation of the apparent_{ } magnitude and velocity
width._{ } For example, a typical relation for ellipticals
is_{ }

where *M _{B}* is the absolute galaxy magnitude

Note_{ } that it is still under debate whether these_{ } relations are *universal*, that is, whether *I* and _{ }
*M*/*L* are dependent only on_{ }
the galaxy type,_{ } and do not depend on other things such as location
and environment._{ }

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*smyers@nrao.edu*
*Steven T. Myers*