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.
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:
- The scale radius
- 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.
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