# Accuracy and assumptions in deriving the Tully-Fisher relation

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?

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). 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.

In agreement with the above answer, a good zero order "motivation" can be found in:
http://www2.astro.psu.edu/~caryl/a480/lecture7_10.pdf. Since Tully-Fischer boils down to the mutual interaction between the baryonic mass of galaxies and the dark mass of their halos (star formation, angular momentum exchange, ...) there is no "derivation" in a textbook sense (yet), because too much of that picture remains unknown. Rather, one tries to draw conclusions about such interactions from Tully-Fischer type data. It's a subject of current research.

• Avoid posting links without at least summarizing their content, in case the link goes dead someday. This is especially important for a link to course material on a personal page - it's likely this will change from semester to semester, or the person may move and their website may disappear. – Kyle Oman Nov 2 '15 at 23:04
• @KyleOman in this case it may be easiest for someone to edit in the relevant information. (I can do it, but not now) – David Z Nov 4 '15 at 16:05