Have parameters been determined for the Fourier components of a Milky Way spiral density-potential model?

Question

Spiral arms in galaxies can be modeled by potentials of the form $$\Phi_{\text{spiral}}(R,\theta,z,t)=\sum_mA_m(R,z)\cos(m\theta-m\theta_0-\phi_m(R)-\Omega_pt)$$ where $A_m(R,z)$ is the $m$th-order coefficient based on some potential - such as a Miyamoto-Nagai potential - $\phi_m(R)$ is a function typically of the form $$\phi_m(R)=\alpha_m\ln (R/R_{0,m})$$ for some characteristic radius $R$, and $\Omega_p$ is some pattern speed. I've read that typically the $m=0$ component dominates, followed by the $m=2$ term. There may also be an $m=4$ term, though it is typically weaker.

This model describes the disk component of a galaxy with two main spiral arms. The spiral arms of the Milky Way are more complicated; some consider it to have two major arms and two more subordinate arms, while others consider it to have four major arms.

Have any potential or density models of the Milky Way been created using these Fourier components, and if so, what are the parameters used (the $m$s, $A_m$s, $\phi_m$s, and $R_{0,m}$s)?

Answer 1

Yes.

Antoja et al. (2011) created two models: The Tight-Winding Approximation model (TWA) and the PERLAS model. They decided to use just the $m=2$ term for their TWA, based on data from the Spitzer Space Telescope.

The amplitude $A(R)$ was $$A(R)=A_{sp}Re^{-R/R_{\Sigma}}$$ Their $\phi_2(R)$ (denoted $g(R)$ in the paper) was of the form $$g(R)=\left(\frac{2}{N\tan i}\right)\ln\left(1+\left(\frac{R}{R_{sp}}\right)^N\right)$$ where $N$ measures, in their words,

how sharply the change from a bar-like to spiral-like occurs in the inner regions.

They approximated using $N=100$, although $g(R)$ should properly use the limit $N\to\infty$. $i$ is the pitch angle of the arms, which they measured to be $15.5$ to $12.8^{\circ}$. $R_{sp}$, the beginning of the spiral, was found to be $2.6$ to $3.6\text{ kpc}$. For the other constants, $A_{sp}$ was taken to be $650$-$1100$ or $850$-$1300[\text{km s}^{-1}]^2\text{ kpc}^{-1}$, depending on the exact arm shape, and the scale length $R_{\Sigma}$ was set to $2.5\text{ kpc}$.

The pattern speed $\Omega_p$ was set in the range $15$-$30\text{ km/s}$, which fits nicely with other values I've seen. $\theta_0$ (denoted $\phi_0$ in the paper) was set to $88$ to $60^{\circ}$ - a little bigger than the spiral arm pitch angle. putting all this together, we get a potential of the form $$\Phi_\text{spiral}(R,\theta,t)=A_{sp}Re^{-R/(2.5\text{ kpc})}\times\cos\left(2(\theta-88\text{ to }60)+\left(\frac{2}{100\tan (15.5\text{ to }12.8^{\circ})}\right)\ln\left(1+\left(\frac{R}{2.6\text{ to }3.6\text{ kpc}}\right)^{100}\right)-\Omega_{p}t\right)$$ A $z$-component could be multiplied to get a three-dimensional potential, likely exponentially decaying with height.

Griv et al. (2013) also used density-wave theory. Their potential was $$\Phi(R,\theta,t)=\Phi_{\text{mean}}(R)+\tilde\Phi\sqrt{\frac{R_0}{R}}\cos\phi$$ where $$\phi=\phi_0+m\left[\Omega_pt-\varphi+(1/\tan p)\ln(R/R_0)\right]$$ which is essentially the same potential as discussed in the question, with $$A(R)=\tilde{\Phi}\sqrt{\frac{R_0}{R}}$$ The authors chose $m=1,2,3$ components, with different parameters for each. $p$ (in degrees) ranged from $-6$ to $-20$, while $\phi_0$ (also in degrees) ranged from $239$ to $247$. $\Omega_p$ was from $32.4$ to $37.6$ ($\text{ km/s}$), also consistent with other models. $\tilde{\Phi}$ saw the most variation, ranging from $-65$ to $-390$ (in $\text{kpc}^{-2}\text{s}^{-2}$). $R_0$ was set to $8\text{ kpc}$, I believe. Again, an exponentially decaying function should account for the $z$-dependence.