Doppler shift and speed of rotating objects in space I understand the concept of how we can use the doppler effect to know if an object is spinning, in the sense that the part of the object spinning towards us will exhibit a blueshift, and the part spinning away will exhibit a redshift.
However, how can we determine the rotation rate using doppler effect? My professor said to do so by "measuring the widths of spectral lines," but I would just like to know what I should be looking for. I would assume that the closer the spectral lines are together, the faster the object is spinning but I would like for somebody to either confirm that or help me understand what it actually means.
 A: The answer is that you can't. Other information and assumptions are necessary.
The angular velocity (which is what I assume is meant by the "rotation rate" here) is given by $v/r$, where $v$ is the tangential velocity around the centre of rotation and $r$ is a radial distance from the centre of motion.
When observing a distant, rotating object, then assuming that rotation axis is oriented optimally, then a spectral line will be broadened by rotational motion. The side coming towards you emits blue-shifted light and the side moving away emits redshifted light. Providing your spectrograph has the resolving power to discern these shifts then they are manifested either by a broadening of spectral features (if the object is spatially unresolved  i.e. is viewed as a point of light) or an obvious relationship between position on the object and doppler shift in the case of an object can be spatially resolved.
In the former case, the width of a spectral line yields $2v$, in the latter then we can directly measure the difference in velocity between opposite sides of the object and this is $2v$.
To turn this into a rotation rate requires an estimate of $r$. In general, this is not known exactly for an astronomical object, because it involves multiplying the angular size of an object by its estimated distance, and is therefore a source of uncertainty in the rotation rate. For spatially unresolved objects you can't even do this, so you need some other means of estimating $r$.
A final complication is the angle of inclination of the rotation axis. In general we observe a "projected rotation velocity" $2v \sin i$, where conventionally $i=90^{\circ}$ means the rotation axis is at right angles to the line of sight. Thus one also needs to know $i$ in order to estimate the rotation rate.
A: If you consider a rotating body, some parts of the object will be moving towards you and some parts away. These additional velocity components will give you a different Doppler shift, and a different observed frequency of light. When considered the whole visible surface of the rotating object, there will be a continuum of different Doppler shifts, and therefore a wider spectral line.
