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Consider a hydrostatic balance of isothermal cloud.
By the gas density,
, the isothermal sound speed,
, and the gravitational potential,
, the force balance is written as
![\begin{displaymath}
-\frac{c_{\rm is}^2}{\rho}\frac{d \rho}{d r}-\frac{d \Phi}{d r}=0,
\end{displaymath}](img852.png) |
(4.1) |
and the gravity is calculated from a density distribution as
![\begin{displaymath}
-\frac{d \Phi}{d r}=-\frac{GM_r}{r^2}=-\frac{4\pi G}{r^2}\int_0^r \rho r^2 dr,
\end{displaymath}](img853.png) |
(4.2) |
for a spherical symmetric cloud, where
represents the mass contained inside the radius
.
The expression for a cylindrical cloud is
![\begin{displaymath}
-\frac{d \Phi}{d r}=-\frac{G\lambda_r}{r}=-\frac{2\pi G}{r}\int_0^r \rho r dr,
\end{displaymath}](img854.png) |
(4.3) |
where
represents the mass per unit length within a cylinder of radius being
.
For the spherical symmetric case, the equation becomes the Lane-Emden equation with the polytropic index of
(see Appendix C.1).
This has no analytic solutions.
However, the numerical integration gives us a solution shown in Figure 4.1 (left).
Only in a limiting case with the infinite central density, the solution is expressed as
![\begin{displaymath}
\rho(r)=\frac{c_{\rm is}^2}{2\pi G}r^{-2}.
\end{displaymath}](img857.png) |
(4.4) |
Increasing the central density, the solution reaches the above Singular Isothermal Sphere (SIS) solution.
On the other hand, a cylindrical cloud has an analytic solution (Ostriker 1964) as
![\begin{displaymath}
\rho(r)=\rho_c \left( 1+ \frac{r^2}{8H^2} \right)^{-2},
\end{displaymath}](img858.png) |
(4.5) |
where
![\begin{displaymath}
H^2={c_{\rm is}^2}/{4\pi G \rho_c}.
\end{displaymath}](img859.png) |
(4.6) |
Far from the cloud symmetric axis, the distribution of equation (4.5) gives
![\begin{displaymath}
\rho(r)\propto r^{-4},
\end{displaymath}](img860.png) |
(4.7) |
while the spherical symmetric cloud has
![\begin{displaymath}
\rho(r)\propto r^{-2}
\end{displaymath}](img861.png) |
(4.8) |
distribution.
Subsections
Next: Problem 1
Up: Local Star Formation Process
Previous: Local Star Formation Process
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Kohji Tomisaka
2007-07-08