§18 Isothermal cloud

Let us consider now the case of an isothermal cloud, for which the density in the cloud $\rho_{\mathrm c}$ varies as a function of the position $r$ in the following way (Binney & Tremaine (1997)):

$$\rho_{\mathrm c}= \frac{ \xi }{ r^2 } .$$ (18.1)

where $\xi$ is a constant of proportionality. In other words, because the jet and cloud are maintained in pressure balance, the pressure acting on the jet is given by:

$$\frac{p}{p_0} = \left( \frac{r_0}{r} \right )^2.$$ (18.2)

For this isothermal case, it is easy to verify that the factors in the expansions for the gravitational potential $\phi$ and the pressure $P$ as defined by eqs.(17.12)-(17.13) are given by:

$$\alpha = \frac{ 1 - \Gamma \left( 3 - 2\Gamma \right) }{ \sin^{2\Gamma} \negthickspace \varphi_0}, \qquad \tilde\alpha = \xi \ln(\sin\varphi_0) + \frac{3}{2}\xi,$$

$$\beta = \frac{ 4 \Gamma \left( 1 - \Gamma \right) }{ \sin^{ 2 \Gamma } \negthinspace \varphi_0 }, \qquad \tilde\beta = 2 \xi,$$ (18.3)

$$\zeta = \frac{ \Gamma \left( 2\Gamma - 1 \right) }{ \sin^{ 2 \Gamma } \negthinspace \varphi_0 }, \qquad \tilde\zeta = \frac{ \xi }{ 2 }.$$

This solution corresponds to that found by Raga & Cantó (1996) for the case in which no gravity is present, i.e. $\tilde\alpha \! = \! \tilde\beta \! = \! \tilde\zeta \! = 0$. From the solutions obtained above in eq.(17.14) and eq.(18.3) it follows that the dimensionless parameter $\Lambda$ defined as:

$$\Lambda \equiv G \frac{ \xi }{ M_0^2 c_0^2 } = G \frac{ \rho_0 r_0^2 }{ M_0^2 c_0^2 } .$$ (18.4)

is a number that parametrises the required solution.

The deflection of jets in isothermal clouds may be important for interstellar molecular clouds and the jets associated with Herbig-Haro objects. For this case we can obtain a value for the parameter $\Lambda$ . If we adopt a particle number density of $n_\mathrm{H} \! \sim \! \unit{ \power{10}{2} }{ \centi \meter \rpcubed }$, and a temperature $T \! \sim \unit{ 10 }{ \kelvin }$ for a cloud with radius $r_0 \! \sim \unit{ 1 }{ pc }$ (Hartmann, 1998; Spitzer, 1998), then

$$\Lambda \sim \frac{ 10^ {-2} }{ M_0^2 } \left( \frac{ r_0 }{ 1 ... ... \mathrm{cm}^{-3} } \right) \left( \frac{ T }{ 10 \mathrm{K} } \right)^{-1}.$$ (18.5)

The same calculation can be made for the cases of radio jets interacting with the gas inside a cluster of galaxies. For this case, typical values are $n_\mathrm{H} \sim \unit{ \power{10}{-2} } { \centi \metre \rpcubed }, T \sim \unit{ \power{10}{7} }{ \kelvin } \text{ and } r_0 \sim \unit{ 100 }{ \kilo pc }$ (Longair, 1992; Longair, 1998). With these values, the parameter $\Lambda \sim 10^{-2} / M_0^2$, exactly as eq.(18.5).

The fact that jets are formed in various environments such as giant molecular clouds and the gaseous haloes of clusters of galaxies with the same values of the dimensionless parameter $\Lambda$ provides a clue as to why the jets look the same in such widely different environments.

From its definition, the parameter $\Lambda$ can be rewritten as $\Lambda = ( 3 / 4 \pi ) ( G { \mathsf{M} } / r ) ( 1 / \ensuremath{v}_0^2 )$, where ${ \mathsf{M} }$ is the mass within a sphere of radius $r_0$. This quantity is roughly the ratio of the gravitational potential energy from the cloud acting on a fluid element of the jet, to its kinetic energy at the initial position $r_0$. The parameter $\Lambda$ is thus an indicator of how large the deflections due to gravity are going to affect the trajectory of the jet. The bigger the number $\Lambda$, the more important the deflection caused by gravity will be. In other words, when the parameter $\Lambda \gg 1$ the jet becomes ballistic and bends towards the centre of the cloud. When $\Lambda \ll 1$ the deflections are dominated by the pressure gradients in the cloud and the jets bend away from the centre of the cloud.

Fig.(III.3) shows plots for three different values of $\Lambda$ with initial Mach numbers of $M_0 \! = \! 5$ and $M_0 \! = 10$. A comparison with a numerical integration of eq.(17.14) using a fourth-order Runge-Kuta method is also presented in the figures by dashed lines. This comparison shows that as long as the deflections are sufficiently small, or as long as the Mach number of the flow in the jet is sufficiently large, the analytic approximations discussed above are a good approximation to the exact solution.

Figure III.3: Deflection produced in a jet due to the collision with an isothermal cloud (semicircle) of radius $r_0$. The jet penetrates the cloud from the right, parallel to the $x$ axis. Different trajectories are shown in each diagram for different initial heights of $y/r_0\!=\!0.05$, $0.15$,...,$0.95$ as measured from the $x/r_0$ axis. In each figure the top diagram corresponds to the case in which gravitational effects are not considered (Raga & Cantó, 1996). The middle and bottom diagrams represent trajectories for which gravitational effects are taken into account and the parameter $\Lambda$ has values of $10^{-6}$, $0.01$ respectively in units of the square of the initial Mach number $M_0^{2}$ of the jet (see text). For the plots, a polytropic index $\kappa \! = \! 5/3$ for the flow in the jet was assumed. The diagrams at the top and bottom correspond to initial Mach numbers for the jet flow $M_0 \! = \! 5$ and $M_0 \! = \! 10$ respectively. The dashed lines in the graphs represent the direct numerical integration of the equation of motion. The continuous lines are the analytic approximations discussed in the text.

Footnotes

... as

The parameter $\Lambda$ is an important number which can be obtained by dimensional analysis. For, the problem in question is characterised by the gravitational constant $G$, a ``characteristic length'' $r_0$ and the values of the velocity of the jet and the density at this point which are $\ensuremath{v}_0$ and $\rho_0$ respectively. Three independent dimensions (length, time and mass) describe the whole hydrodynamical problem. Since four independent physical quantities ( $G, \rho_0, \ensuremath{v}_0,$and$r_0$) are fundamental for the problem we are interested, the Buckingham $\Pi$-Theorem (Sedov, 1993) of dimensional analysis demands the existence of only one dimensionless parameter $\Lambda$, which is given by eq.(18.4).