Logarithmic Spiral arms in the Milky Way I am trying to model the magnetic field of the inner 'disk' of the Milky Way Galaxy. I am following the model of Jansson and Farrar 2012.
In section 5.1.1 of this paper they describe the Milky Way as 8 logarithmic spiral regions with opening angle $i = 11.5^{\circ}$.
In this case, in cylindrical coordinates $(r,\phi,z)$ the dividing lines between these spiral regions is given by
$$ r = r_x  \, \text{exp}(\phi \, \text{tan}(90 - i))$$
where $r_x$ is a constant for each spiral arm.
I am confused as to how this equation can describe a full rotation of a spiral arm. For a whole rotation $\phi = 2 \pi$ and so $\text{exp}(\phi \, \text{tan}(90 - i)) \approx e^{30} \approx 10^{13}$. But surely the radius of a spiral arm in the MW does not change by such a huge factor?!
I feel that I am missing something here, and any guidance for what it is would be greatly appreciated.
Thanks
 A: Look at the model in context. In the sentence before giving the values for $r_{-x}$ (which you've written as $r_x$), the authors write

Between radii 5 kpc and 20 kpc there are eight logarithmic spiral regions with opening angle $i = 11.5^{\circ}$.

In other words, the model really only has physical meaning for 5 kpc < $r$ <20 kpc.
Some other things to take into account:


*

*This model is only valid for a certain region. While logarithmic spirals can indeed describe spiral arms well, they aren't successful for all values of $r$. For example, they fail to describe the inner region of the disk (i.e. 3-5 kpc).

*Spiral arms aren't always regular.

*We don't have a fantastic understanding of the spiral arms of the Milky Way. The 2-arm and 4-arm models are constantly competing with one another, and some minor arms (e.g. the Far 3 kpc Arm; see Dame & Thaddeus (2008)) have been discovered recently.


The important message here, though, is that the spiral arms are limited to main part of the galactic disk, which does not extend to such great distances.
