# The tully fisher relationship is used to determine distances

### The Local Tully-Fisher Relation

The Tully-Fisher relation is a correlation between the luminosity and the HI 21cm It is used to derive galaxy distances in the interval 7 to The candidates should have an inclination (i) between 90 and 45 degrees in order to measure. knowing the distances to galaxies is fundamental to a lot of problems, so measure v and use H0 is ≈ 70 km s-1 Mpc – NB this only (Tully-Fisher relation). Independent of the value of the Hubble Constant, the Tully-Fisher relation can be used to measure peculiar velocities.

The Faber-Jackson relation has a somewhat larger scatter but in this case surface brightness as a third parameter significantly improves the correlation. The result is a formulation called the "fundamental plane" [5,6]. It is interesting that the fundamental plane transforms considerably more closely into the virial theorem than the Tully-Fisher relation.

Distance Measurements In the s the situation regarding the extragalactic distance scale was in a sorry state. There was a debate over the value of this constant at the level of a factor two.

At its core was a critical issue.

## Distant galaxies: geometry, Tully-Fisher, and the fundamental plane

The standard cosmological model at the time held that the primary constituents of the universe were particles of matter that had been acting since the Big Bang to slow the cosmic expansion.

The "theory of inflation" anticipated that the density of matter amounted to the "critical value" required to give a flat topology. This model implies a specific link between the age of the universe and the expansion scale. The age of the universe was reasonably constrained by the age-dating of stellar populations in globular clusters. The theoretically preferred model required that the Hubble constant be at the very lowest of the range being seriously discussed at the end of the 20th Century.

Other methodologies emerged, such as the use of the bright end cutoff in the luminosities of planetary nebulae [9] and "surface brightness fluctuations " caused by the distribution of the brightest stars in galaxies dominated by old populations [10]. A paradigm shift came with the evidence from observations of supernovae of type Ia that the universe appears to be accelerating [11,12]. It seems that a repulsive dark energy is dynamically dominant.

The relationship between ages and expansion rate is altered. New methods to measure distance become available. The use of supernovae is particularly accurate.

If the brightest red giant branch stars are resolved in the image of a galaxy, the known luminosities of these stars gives a good distance [13]. However each of the various methods has a weakness: There is still an important role for the Tully-Fisher relation. It is no longer the most accurate method on a single case basis but since the application is to normal disk systems, with little restriction in range, it enables the determination of distances to many thousands of galaxies in all the environments that galaxies are found [14,15].

### Tully-Fisher and Fundamental plane relations as standard candles

Independent of the value of the Hubble Constant, the Tully-Fisher relation can be used to measure peculiar velocities, motions of galaxies that are deviations from the linear Hubble expansion. These motions are thought to be due to the gravitational influence of over- and under-densities of matter.

**The Tully-Fisher and Mass-Size Relations from Halo Abundance Matching - Harry Desmond**

There is an extensive literature discussing the relationship between the peculiar velocities of galaxies measured via the Tully-Fisher relation and the large-scale distribution of galaxies. Constraints on Galaxy Evolution The other uses of the Tully-Fisher relation derive from the constraints imposed on ideas of galaxy formation.

Although the general correlation between luminosity and rotation rate was anticipated, it was a surprise to find it to be so tight, with so little spread caused by additional parameters.

The dependence need not have been a power law. What defines the slope? It is somewhat different from what the virial theorem would give and what is seen with bulge dominated systems.

Especially, although there is an obvious linkage between the luminosity in stars and the stellar mass, why is the correlation so tight if most of the mass is non-baryonic? An empirical approach to these issues finds investigators pursuing the challenging task of observing spiral galaxies at large redshifts in order to look for the effects of evolution. This is difficult because the galaxies are faint, and have an angular size not much larger than the typical seeing at modern telescopes.

Galaxies with given rotation properties, thought to be specified by the mass of the host halos, were brighter at earlier epochs. However a correlation very similar to that seen today between luminosity and rotation rate was already in place when the universe was only half its present age [16]. Look at the pattern of the masers.

Does it look like the simple Keplerian motion of clouds in a simple ring? If all the clouds were in a ring, at a single distance from the central black hole, then we ought to see the largest speeds in the outermost blobs; but the observations show the largest speeds in the INNERMOST blobs.

It turns out that the geometry is pretty complex, something like this: The accretion disk is warped, and the masers we see are scattered throughout the disk, some much closer to the center than others.

But we can be pretty sure of several things: Of course, we can't directly observe all these quantities; the only things we measure directly are the angular positions of the masers, relative to the central source, and their velocities. IF we knew the distance d, then we could compute the orbital radius r and the mass of the central object M.

## One Universe at a Time

Let's play a little game. I'll choose one maser from the observed set, with the following properties: It seems that there is no unique solution: That should be no surprise. Both the orbital radius r and the central mass M depend linearly on the distance to the galaxy d.

Suppose we could measure the acceleration a of this little cloud of gas? If we could observe the acceleration, we could break the degeneracy and solve for the distance to the galaxy.

Where should we look? Here are the locations of the masing clouds again, as observed by radio interferometers: Which of these clouds might be the best choice to detect the acceleration?

### Tully-Fisher King - One Universe at a Time

Just how big is this acceleration? Can we really measure it? Well, let's find out. Use your values to fill in the acceleration column of your table. How could we measure such tiny accelerations? Well, the good thing about accelerations is that they can accumulate over time into large changes in velocity.