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I understand the mathematical derivation of the Tully Fisher relation from basic physics formulas, as shown on this site. However, after using the physics equations, it seems that several assumptions are made from this point on.

First are statistical assumptions. There are statistical errors because the observable, luminous mass of the galaxy is less than the actual mass of the galaxy, and the mass of the galaxy is assumed to be only the observed mass, not the actual mass. Second, this relationship seems to assume that all galaxies are perfectly circular (with negligible thickness).

So, with all these assumptions necessary for the Tully-Fisher relation, how is the relationship derived, how can corrections be made for the assumptions when attempting calculations using this relationship, and why is the Tully-Fisher relation generally accepted by the astrophysics community?

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The Tully-Fisher relation is first and foremost, and historically, an observational relation. The 'derivation' of the relation, is really more of a 'motivation'. The relation is based on a myriad details of galaxy and star formation, and dynamics---an accurate 'derivation' could only be based on numerical simulations (which is the basis for most modern fits to the observed relationships).

For example, a large contribution to the velocity distribution of a galaxy will be its history of interactions with nearby objects (e.g. other galaxies). This has nothing to do with the intrinsic properties of the galaxy itself, and thus there isn't really a way to 'account' for it in the analytical equation.

If you look at these plots (and note that its actually plotted opposite of the usual way), you'll see that there is incredible intrinsic scatter, not only in each plot, but also between the figures (which are each for a different observed color, but the same sample of galaxies). enter image description here Plots are from this paper: http://adsabs.harvard.edu/abs/2010A&A...521A..27F

While the Tully-Fisher relation is very important, and kind of a staple of galactic astronomy, its reflective of a useful tendency---not a tight, intrinsic scaling.

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