<h1>Yes.</h1>
<p><strong><a href="https://arxiv.org/abs/1106.1170" rel="nofollow noreferrer">Antoja et al. (2011)</a></strong> created two models: The Tight-Winding Approximation model (TWA) and the PERLAS model. They decided to use just the <span class="math-container">$m=2$</span> term for their TWA, based on data from <a href="https://en.wikipedia.org/wiki/Spitzer_Space_Telescope" rel="nofollow noreferrer">the Spitzer Space Telescope</a>.</p>
<p>The amplitude <span class="math-container">$A(R)$</span> was
<span class="math-container">$$A(R)=A_{sp}Re^{-R/R_{\Sigma}}$$</span>
Their <span class="math-container">$\phi_2(R)$</span> (denoted <span class="math-container">$g(R)$</span> in the paper) was of the form
<span class="math-container">$$g(R)=\left(\frac{2}{N\tan i}\right)\ln\left(1+\left(\frac{R}{R_{sp}}\right)^N\right)$$</span>
where <span class="math-container">$N$</span> measures, in their words,</p>
<blockquote>
<p>how sharply the change from a bar-like to spiral-like occurs in the inner regions.</p>
</blockquote>
<p>They approximated using <span class="math-container">$N=100$</span>, although <span class="math-container">$g(R)$</span> should properly use the limit <span class="math-container">$N\to\infty$</span>. <span class="math-container">$i$</span> is the pitch angle of the arms, which they measured to be <span class="math-container">$15.5$</span> to <span class="math-container">$12.8^{\circ}$</span>. <span class="math-container">$R_{sp}$</span>, the beginning of the spiral, was found to be <span class="math-container">$2.6$</span> to <span class="math-container">$3.6\text{ kpc}$</span>. For the other constants, <span class="math-container">$A_{sp}$</span> was taken to be <span class="math-container">$650$</span>-<span class="math-container">$1100$</span> or <span class="math-container">$850$</span>-<span class="math-container">$1300[\text{km s}^{-1}]^2\text{ kpc}^{-1}$</span>, depending on the exact arm shape, and the scale length <span class="math-container">$R_{\Sigma}$</span> was set to <span class="math-container">$2.5\text{ kpc}$</span>.</p>
<p>The pattern speed <span class="math-container">$\Omega_p$</span> was set in the range <span class="math-container">$15$</span>-<span class="math-container">$30\text{ km/s}$</span>, which fits nicely with <a href="https://astronomy.stackexchange.com/a/19177/2153">other values I've seen</a>. <span class="math-container">$\theta_0$</span> (denoted <span class="math-container">$\phi_0$</span> in the paper) was set to <span class="math-container">$88$</span> to <span class="math-container">$60^{\circ}$</span> - a little bigger than the spiral arm pitch angle. putting all this together, we get a potential of the form
<span class="math-container">$$\Phi_\text{spiral}(R,\theta,t)=A_{sp}Re^{-R/(2.5\text{ kpc})}\times\cos\left(2(\theta-88\text{ to }60)+\left(\frac{2}{100\tan (15.5\text{ to }12.8^{\circ})}\right)\ln\left(1+\left(\frac{R}{2.6\text{ to }3.6\text{ kpc}}\right)^{100}\right)-\Omega_{p}t\right)$$</span>
A <span class="math-container">$z$</span>-component could be multiplied to get a three-dimensional potential, likely exponentially decaying with height.</p>
<p><strong><a href="https://academic.oup.com/mnras/article/433/3/2511/1235707/Fitting-the-Lin-Shu-type-density-wave-theory-for" rel="nofollow noreferrer">Griv et al. (2013)</a></strong> also used density-wave theory. Their potential was
<span class="math-container">$$\Phi(R,\theta,t)=\Phi_{\text{mean}}(R)+\tilde\Phi\sqrt{\frac{R_0}{R}}\cos\phi$$</span>
where
<span class="math-container">$$\phi=\phi_0+m\left[\Omega_pt-\varphi+(1/\tan p)\ln(R/R_0)\right]$$</span>
which is essentially the same potential as discussed in the question, with
<span class="math-container">$$A(R)=\tilde{\Phi}\sqrt{\frac{R_0}{R}}$$</span>
The authors chose <span class="math-container">$m=1,2,3$</span> components, with different parameters for each. <span class="math-container">$p$</span> (in degrees) ranged from <span class="math-container">$-6$</span> to <span class="math-container">$-20$</span>, while <span class="math-container">$\phi_0$</span> (also in degrees) ranged from <span class="math-container">$239$</span> to <span class="math-container">$247$</span>. <span class="math-container">$\Omega_p$</span> was from <span class="math-container">$32.4$</span> to <span class="math-container">$37.6$</span> (<span class="math-container">$\text{ km/s}$</span>), also consistent with other models. <span class="math-container">$\tilde{\Phi}$</span> saw the most variation, ranging from <span class="math-container">$-65$</span> to <span class="math-container">$-390$</span> (in <span class="math-container">$\text{kpc}^{-2}\text{s}^{-2}$</span>). <span class="math-container">$R_0$</span> was set to <span class="math-container">$8\text{ kpc}$</span>, I believe. Again, an exponentially decaying function should account for the <span class="math-container">$z$</span>-dependence.</p>

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2020-02-04T13:49:32.997Z
I've just asked [Can the gravitational potential energy in a spiral galaxy be positive?](https://astronomy.stackexchange.com/q/34951/7982)
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2017-01-17T20:46:12.550Z
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2020-06-17T09:47:40.653Z
HDE 226868
<p>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.</p>

<p>This model describes the disk component of a galaxy with two main spiral arms. The <a href="https://en.wikipedia.org/wiki/Milky_Way#Spiral_arms" rel="nofollow noreferrer">spiral arms of the Milky Way</a> 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.</p>

<p>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)?</p>

<milky-way><spiral-arms>
Have parameters been determined for the Fourier components of a Milky Way spiral density-potential model?
HDE 226868
# Have parameters been determined for the Fourier components of a Milky Way spiral density-potential model?

(Asked by: username_0 on 2017-01-17)

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](https://en.wikipedia.org/wiki/Milky_Way#Spiral_arms) 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)?

**Tags:** milky-way, spiral-arms


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## 1 answers

### Answer by username_0 on 2017-01-17. Score: 2

# Yes.

**[Antoja et al. (2011)](https://arxiv.org/abs/1106.1170)** 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](https://en.wikipedia.org/wiki/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](https://astronomy.stackexchange.com/a/19177/2153). $\theta\_0$ (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)](https://academic.oup.com/mnras/article/433/3/2511/1235707/Fitting-the-Lin-Shu-type-density-wave-theory-for)** 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.

> **Comments:**
> **username_1**: I've just asked [Can the gravitational potential energy in a spiral galaxy be positive?](https://astronomy.stackexchange.com/q/34951/7982)  

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