Tightly Wound Spirals

Figure 3.7: Tightly wound (left) vs loosely wound (right) spirals.

We assume the wave driven by the self-gravity has a form of tightly-wound spiral [Fig.3.7(left)]. When we move radially, the density $\delta \Sigma$ varies rapidly. While, it changes its amplitude slowly in the azimuthal direction. In a mathematical expression, if we write the density perturbation $\delta \Sigma$ as

$$\delta \Sigma =A(R,t)\exp[i m\phi+i f(R,t)],$$ (3.22)

where the amplitude of spiral $A(R,t)$ is a slowly varing function of $R$, a tightly wound spiral means the shape function varies fast (the radial wavenumber $k\simeq d f/d R$ is large enough). We consider the gravitational force from the vicinity of $(R_0,\phi_0)$, since the $\delta \Sigma$ oscillates and cancels even if we integrate over large region. Thus,

$$\delta \Sigma (R,\phi,t)\simeq \Sigma_a \exp [ik(R_0,t)(R-R_0)],$$ (3.23)

where

$$\Sigma_a=A(R_0,t)\exp[im\phi_0+f(R_0,t)].$$ (3.24)

Notice that the density perturbation [eq.(3.23)] is similar to that studied in $\S$2.6. The potential should be expressed in a similar form to equation (2.64) as

$$\delta \Phi \simeq -\frac{2\pi G \Sigma_a}{\vert k\vert}\exp[ik(R_0,t)(R-R_0)],$$ (3.25)

which simply means

$$\Phi_a=-\frac{2\pi G \Sigma_a}{\vert k\vert}.$$ (3.26)

If we set $R=R_0$, we obtain our final result for the potential due to the surface density perturbation

$$\delta \Phi (R, \phi, t) \simeq -\frac{2 \pi G}{\vert k\vert}A(R,t)\exp[im\phi+f(R,t)].$$ (3.27)

Differentiating this equation with $R$ and ignoring the term $dA(R,t)/dR$ compared to that of $df(R,t)/dR=k$, we obtain

$$\delta \Sigma(R,\phi,t)=i \frac{{\rm sign}(k)}{2\pi G}\frac{d \delta \Phi(R,\phi,t)}{d R},$$ (3.28)

Neglecting the terms $\propto 1/R$ compared to the terms containing $\partial /\partial R$, equations (3.19), (3.20), and (3.21) are rewritten as

$$i(m\Omega - \omega) \Sigma_a + i k \Sigma_0 u_a = 0,$$ (3.29)

$$u_a [\kappa^2 -(m\Omega -\omega)^2] = (m\Omega - \omega) k (\Phi_a + h_a),$$ (3.30)

and

$$v_a [\kappa^2 -(m\Omega -\omega)^2] = i \frac{\kappa^2}{2\Omega} k (\Phi_a + h_a),$$ (3.31)

Using these equations [(3.29), (3.30), and (3.31)], $\Phi_a=-2\pi G \Sigma_a/\vert k\vert$, and $h_a=c_s^2\Sigma_a/\Sigma_0$, we obtain the dispersion relation for the self-gravitating instability of the rotating gaseous thin disk

$$\left(m\Omega -\omega\right)^2=k^2c_s^2 - 2\pi G \Sigma_0 \vert k\vert + \kappa^2.$$ (3.32)

Generally speaking, the epicyclic frequency depends on the rotation law but is in the range of $\Omega \mbox{\raisebox{0.3ex}{$<$}\hspace{-1.1em} \raisebox{-0.7ex}{$\sim$}} ... ...\mbox{\raisebox{0.3ex}{$<$}\hspace{-1.1em} \raisebox{-0.7ex}{$\sim$}} 2 \Omega$ (see Table 3.1 for $\kappa$ for typical rotation laws). It is shown that the system is stabilized due to the epicyclic frequency compared with a nonrotating thin disk [eq.(2.65)].

Table 3.1: Epicyclic frequency vs rotation law.

Rotation	$\kappa$
Rigid-body rotation $\Omega=$const.	$2\Omega$
Flat rotation $v_\phi=$const.	$\sqrt{2}\Omega$
Kepler rotation $v_\phi \propto r^{-1/2}$	$\Omega$