# About the hump on galaxy rotation curves

The past days I have been studying the rotation curves of disk galaxies and I am currently trying to understand how we can extract information about the dark matter of a galaxy by looking its rotation curve. A typical and well behaved rotation curve of a disk galaxy rises fast at the start, then it peaks and produces a "hump"/"bump" and then it remains constant (or slightly declining). I would like to ask what exactly causes the "hump"/"bump" on the rotation curve and what it can tell us about the galaxy.

## 1 Answer

The first hump that you see is caused by the galactic disk. The gravitational potential for this is given by* $$\Phi_{disk}=-\frac{GM_{disk}}{\left(r^2+(a+\sqrt{b^2+z^2})^2\right)^{1/2}}$$ where $a$ is the scale radius of the disk, $b$ the scale height, and all other variables take their usual meaning. The rotational velocity is then the square root of the $r$-derivative of $\Phi$ times r: $$v_{rot,disk}=\left(\frac{GM_{disk}r^2}{\left(r^2+(a+\sqrt{b^2+z^2})^2\right)^{3/2}}\right)^{1/2}$$ If you plot this (with the appropriate units), you should get that first bump you ask about.

This particular bump can tell us two things:

1. The scale radius
2. The mass of the disk

A smaller value of $a$ will lead to a peak at shorter $r$ while also increasing the rotational velocity; a larger $a$ will lead to the opposite change. A larger $M_{disk}$ will produce a larger peak rotational velocity (assuming you fixed $a$, your model's peak would occur at the same point).

*There are other models of potentials, see Section 7.4 of this link; the model I am using is called the Miyamoto-Nagai potential.

• I always thought that the first hump is due to the bulge-disk transition, not only to the disk. – user115376 Nov 13 '16 at 20:55
• @user115376: The bulge is certainly decreasing there and the disk increasing, so I guess that's a more accurate description than saying it is caused by the disk itself (though I'd still argue that my statement may be slightly misleading but not incorrect). – Kyle Kanos Nov 13 '16 at 21:07
• I think it is due to both, not only the disk because there are many high resolution rotation curves of spiral galaxies that the bulge is negligible and no bump is observed, while the disk surely contributes, plus the bulge length scale is always before the disk length scale – user115376 Nov 13 '16 at 21:11