# spiral arms

Another interesting structure revealed in protoplanetary disks with state-of-the-art telescopes is spiral arms. On the left in the above figure is a scattered light image of the SAO 206462 disk (Garufi et al. 2013). We are looking at the photons which are originally emitted by the central star but then scattered off the surface of the disk toward us. On the right is my attempt to reproduce the observation with a planet in the disk (Bae et al. 2016a). The planet excites two spiral arms (S1 and S2) interior to its orbit. A vortex (V) forms at the inner edge of the gap opened by the planet. At this snapshot, one of the spiral arms (S1) passes through the vortex (V), and this is how we interpret the bright blob V in the observation.

Now, how does a planet excite multiple spiral arms? To make things simple, let's consider a very small mass planet, say a few Earth-mass planet around a solar-mass star, for which we can safely ignore non-linear effects. The gravitational potential of the planet can be decomposed into a Fourier series, a sum of individual azimuthal modes having different azimuthal wavenumbers $m$ $=1$, $...$, $\infty$. Each individual azimuthal mode excites $m$ wave modes at its Lindblad resonance, which is known for many decades since a series of seminal papers by Goldreich & Tremaine (1978a,b, 1979).

Before I move forward, I have some examples in the figure below to help you visualize how this works. In the figure the potential of the planet is centered at $(X,Y)=(1,0)$ and what I've plotted here is the surface density distributions perturbed by individual azimuthal modes of the Fourier decomposed potential with $m=2$, $3$, $4$, and $5$. For the left panel, for example, I'm showing a result from a simulation in which I included only $m=2$ azimuthal mode of planet's potential. As shown, an $m$th azimuthal mode excites $m$ spiral wave modes, which are indicated with $n=0$, $1$, ..., $m-1$. I show only four different $m$s but you can imagine the same happens for any given $m$.

When non-linear effects can be ignored we can simply add up the perturbations driven by individual modes to reconstruct the perturbation driven by the full potential of the planet. When we do this, you will see three spiral arms emerge. The primary spiral arm, the one directly attached to the planet, forms from superposition of the $n=0$ components in the above Fourier representation. Similarly, the secondary and tertiary arms form from superposition of the $n=1$ and $n=2$ components, respectively.  See also an animated version here, in which I add the perturbations driven by individual modes one at a time from $m=1$ to $m=30$.

Some of you might have noticed that non-zero $n$ components do not excite at the same location in the disk. For example, if you compare the $n=2$ component in $m=3$, $4$, and $5$ Fourier mode, they indeed excite out of phase. However, they can be in phase as they propagate because the propagation of wave modes depend on their wavenumber $m$.

I understand that the three spiral arms are very tightly wound and do not really look like the ones we have observed so far. That's true and it's probably because the planets responsible for the observed spiral arms are much more massive than the one used in this example. Non-linear effects become increasingly important as the planet mass increases. For a larger mass planet spiral arms are more opened and the separation between the spirals increases. See the snapshot below for two-armed spirals excited by a 3 Jupiter-mass planet. Doesn't this look more similar to the spiral arms you know of?