Thread by @DudeDarkmatter

Here follows a thread on the use and misuse of the exponential disk approximation in extragalactic astronomy.

Exponential disk is a common and useful way to quantitively approximate the light distribution of rotating galaxies. By azimuthally averaging around ellipses, we can plot the surface brightness as a function of radius. Here is an example for the LSB galaxy UGC 1230.

The image at left has been reduced to the graph at right (blue points). The light falls off exponentially. This is typical of galaxy disks. All the complexity of the image can - as an approximation - be described by 2 parameters: the central surface brightness and scale length.

This is a handy approximation for many purposes. It gives a shorthand for the appearance of a galaxy, whether it is large or small, high or low surface brightness. We can talk about properties across the population of disks, their relative numbers, selection effects, etc.

Galaxies exist over a wide range of both size and surface brightness, up to some maximum limit. We have yet to identify any minimum limit, as our knowledge is circumscribed by selection effects.

This plot from xxx.lanl.gov/abs/astro-ph/9…

Another nifty application of the exponential disk is in dynamics. As first shown by Freeman (1970; see also Binney & Tremaine) one can write down an analytic expression for the rotation curve of a thin, exponential disk. This makes a lot of calculations easy.

Too easy, if we care about the details of dynamics. It was shown already in the ‘80s that the exponential disk approximation fails to capture important aspects of the dynamics. To do that right, one has to solve the Poisson equation numerically. That can be done, but is more work

So often people persist in using the exponential disk approximation in situations where they should not. This was rife enough that Jerry Sellwood felt obliged to remind people of this already ancient result at a conference in 1998. arxiv.org/abs/astro-ph/9…

Here is the illustration from Sellwood. Three disks of the same mass but slight different luminosity profiles (left). These lead to rather different rotation curves (right). This makes a huge difference to any discussion of mass modeling & dark matter halos.

At the time, Jerry was concerned with the use of V(2.2 scale lengths) as the radius of maximum contribution of the stars to the total rotation curve. That is only correct for pure exponentials (dashed line). It can be and often is very wrong for real galaxies. Yet usage persists.

If one wishes to describe the entire shape of the rotation curve, one cannot use the exponential disk. Sorry. I love exp disk. But all approximations have their limits of applicability. It just ain’t good enough for dynamics.

Here are a few examples done right, from arxiv.org/abs/1609.05917 It is only when you do it right by solving the Poisson equation numerically that you find the correct radial acceleration relation. If instead you pretend disks are exponential, you will screw it up.

I see LOTS of papers, both theoretical and observational, that make this elementary mistake. It might be a reasonable place to start, but if you claim to explain the RAR with purely exponential disks, you’ve just self-identified as a poser who doesn’t know WTF he’s talking about.

I will refrain from making list of arxiv papers that suffer from this failing. (I remain too polite for twitter. Seriously. This is the mild version.) There are lots of claims out there. At most one can be correct. You are safe to discard those that don’t get past exponential dik