Magnatohydrostatic Clouds

Consider here the effect of the magnetic field. In the magnetized medium, the Lorentz force

$$\mbox{\boldmath${F}$}=\frac{1}{4\pi}(\mbox{\boldmath${\nabl... ...}{4\pi}(\mbox{\boldmath${B\cdot\nabla}$})\mbox{\boldmath${B}$}$$ (4.27)

works in the ionized medium. The first term of equation (4.27), which is called the magnetic pressure, has an effect to support the cloud against the self-gravity.

The virial analysis is also applicable to the magnetohydrostatic clouds. The terms related to the magnetic fields are

$$M = \int\frac{B^2}{8\pi}dV+ \int_S (\mbox{\boldmath${r \cdot B}$})\mb... ...ldmath${B \cdot n}$} dS-\int_S \frac{B^2}{8\pi}\mbox{\boldmath${r \cdot n}$} dS$$

$$\simeq \int \frac{B^2-B_0^2}{8\pi}dV\simeq \frac{1}{6\pi^2}\left(\frac{\Phi_B^2}{R}-\frac{\Phi_B^2}{R_0}\right),$$ (4.28)

where $\Phi_B$ represents a magnetic flux and it is assumed to be conserved if we change the radius, $R$, that is $\Phi_B=\pi B_0R_0^2=\pi BR^2$. Equation (4.23) becomes

$$4\pi \bar{p}R^3 - 4\pi p_0 R^3 -\frac{3}{5}\frac{GM^2}{R}+\frac{1}{6\pi^2}\frac{\Phi_B^2}{R}=0,$$ (4.29)

where we ignored the term $\frac{1}{6\pi^2}\frac{\Phi_B^2}{R_0}$. The last two terms are rewritten as

$$\frac{3}{5}\frac{G}{R}\left(M^2-M_\Phi^2\right),$$ (4.30)

where $M_\Phi$ is defined as $3GM_\Phi^2/5=\Phi_B^2/6\pi^2$.

This shows the effects of the magnetic fields:

1. B-fields effectively reduce the gravitational mass as $M^2-M_\Phi^2=M^2-5\Phi_B^2/(18\pi^2G)$. This plays a part to support a cloud.
2. However, even a cloud contracts (decreasing its radius from $R_0$ to $R$), the ratio of the gravitational to the magnetic terms keeps constant since these two terms are proportional to $\propto R^{-1}$. Thus, if the magnetic term does not work initially, the gravitational term continues to predominate over the magnetic term.

If $M < M_\Phi$, a sum of last two terms in equation (4.29) is positive. Since the second term of rhs of equation (4.25) is positive, there is one $R$ which satisfies equation (4.29) irrespective of the external pressure $p_0$. While, if $M>M_\Phi$, there is a maximum allowable external pressure $p_0$. Therefore, $M=M_\Phi$ gives a criterion whether the magnetic fields work to support the cloud or not. More realistic calculation [Mouschovias (1976a,1976b), Tomisaka et al (1988)] gives us a criterion

$$G^{1/2}\frac{d m}{d \Phi_B}=\frac{G^{1/2}\sigma}{B}=0.17\simeq \frac{1}{2\pi},$$ (4.31)

where $\sigma$ and $B$ means the column density and the magnetic flux density. A cloud with a mass

$$M > \frac{\Phi_B}{2\pi G^{1/2}}$$ (4.32)

is sometimes called magnetically supercritical, while that with

$$M < \frac{\Phi_B}{2\pi G^{1/2}}$$ (4.33)

is magnetically subcritical.

More precisely speaking, the criterion showed in equations (4.32) and (4.33) should be applied for a cloud which has a much larger mass than the Bonnor-Ebert mass. That is, even without magnetic fields, the cloud less-massive than the Bonnor-Ebert mass has a hydrostatic configuration shown in Figure 4.2 (left). The cloud with central density of $\rho_c=10$ has a stable density distribution. To fit the numerical results, Tomisaka et al (1988) obtained an expression for the critical mass when the cloud has a mass-to-flux ratio $d m/d \Phi_B$, the isothermal sound speed $c_{\rm is}$, and the external pressure $p_0$ as

$$M_{cr}=1.3\left\{ 1-\left[\frac{1/2\pi}{G^{1/2}d m/d \Phi_B ... ..._{r=0}}\right]^2\right\}^{-3/2}\frac{c_s^4}{p_0^{1/2}G^{3/2}}.$$ (4.34)

This shows that the critical mass is a decreasing function of the mass-to-flux ratio or increasing function of the magnetic flux. And the critical mass becomes much larger than the Bonnor-Ebert mass $\simeq c_s^4/(p_0^{1/2}G^{3/2})$ only when the mass-to-flux ratio at the center of the cloud is reaching $1/2\pi$ at which the term in the curry bracket goes to zero. Hereafter, we call here the cloud/cloud core with mass larger than the critical mass $M_{\rm cr}$ a supercritical cloud/cloud core. The cloud/cloud core less-massive than the critical mass is subcritical.