UDC 521.11+533.9

HYDROMECHANICS

SOME PARTICULAR SOLUTIONS OF THE EQUATIONS DESCRIBING AXISYMMETRIC MOTIONS OF A GRAVITATING RAREFIED PLASMA

E. V. RYAZANOV

(Presented by Academician L. I. Sedov on 1 X 1966)

Consider axisymmetric nonstationary motions of a rarefied plasma in the presence of a magnetic field and self-gravitation. We shall assume here that the magnetic-field intensity vector \(\mathbf H\) is perpendicular to the trajectories of the particles. Then the magnetic lines of force may be: 1) either straight lines parallel to the axis of symmetry; 2) or concentric circles with centers on the axis of symmetry; 3) or helical lines.

If dissipative processes are absent, then the equations describing such motions (with allowance for pressure anisotropy) may be written in the form

\[ \rho \frac{dv_r}{dt} +\frac{\partial}{\partial r}(p_\perp+h) +\frac{1}{r}\left[(p_\perp-p_\parallel)\frac{h_\varphi}{h} +2(h_\varphi+Gm\rho)\right]=0, \]
\[ \frac{d\rho}{dt} +\rho\left(\frac{\partial v_r}{\partial r}+\frac{v_r}{r}\right)=0, \qquad \frac{\partial m}{\partial r}-2\pi\rho r=0, \tag{1} \]
\[ \frac{d}{dt}\left(\frac{h_z}{\rho^2}\right)=0, \qquad \frac{d}{dt}\left(\frac{h_\varphi}{\rho^2 r}\right)=0, \qquad \frac{d}{dt}\left(\frac{p_\perp}{\rho h^{1/2}}\right)=0, \qquad \frac{d}{dt}\left(\frac{p_\parallel h}{\rho^3}\right)=0. \]

Here \(r\) is the distance from the particle to the axis of symmetry; \(t\) is time; \(v_r\) is the radial velocity; \(\rho\) is the density; \(m\) is the mass; \(h=h_z+h_\varphi\); \(h_z=H_z^2/8\pi\); \(h_\varphi=H_\varphi^2/8\pi\); \(H_z\) and \(H_\varphi\) are the axial and transverse components of the magnetic-field intensity vector; \(G\) is the gravitational constant; \(p_\perp\) and \(p_\parallel\) are the pressures, respectively perpendicular and parallel to the magnetic field (transverse and longitudinal pressures).

In case 1) \(h_z\ne0,\ h_\varphi=0\); in case 2) \(h_z=0,\ h_\varphi\ne0\); in case 3) \(h_z\ne0,\ h_\varphi\ne0\).

In the absence of gravitational forces, equations analogous to equations (1) were obtained earlier in works \((^{1,2})\). In work \((^2)\) some exact solutions of system (1) for \(G=0\) were found.

If we assume that the dependence of the radial velocity of a particle on \(r\) and \(t\) has the form

\[ v_r=\frac{r}{\mu(t)}\frac{d\mu}{dt}, \tag{2} \]

where \(\mu(t)\) is a function of time (to be determined), then by direct verification it is easy to confirm the existence of the following particular exact solutions of system (1).

I.

\[ \rho=\rho_0\mu^{-2},\qquad h_z=\left[{}^{1}/{}_{2}b_3\rho_0\xi^2+b_2-P(\xi)\right]\mu^{-4}, \]
\[ h_\varphi=0,\qquad p_\perp=P(\xi)\mu^{-4},\qquad p_\parallel=Q(\xi)\mu^{-2}. \tag{3} \]

Here \(P(\xi)\) and \(Q(\xi)\) are arbitrary functions of the Lagrangian coordinate \(\xi=r/\mu\). The dependence \(\mu(t)\) is found from the differential equation:

\[ \left(\frac{d\mu}{dt}\right)^2 =b_3\mu^{-2}-2b_1\ln\mu+b_4=f_1(\mu); \tag{3'} \]

\(b_1,\ldots,b_4\) are arbitrary constants, \(\rho_0=b_1/2\pi G\).

II.
\[ \rho=\frac{R'}{r\mu}=\frac{T'}{r^2},\quad h_z=0,\quad h_\varphi=\frac{1}{r^2}\left[a_4+a_5(\xi T-\Pi_1)-2\pi G R^2\right], \]
\[ p_\perp=\frac{1}{r\mu^2}(a_2+a_3T),\quad p_\parallel=\frac{a_1}{\mu^4}T', \]
\[ \left(\frac{d\mu}{dt}\right)^2 =2a_3\mu^{-1}-a_1\mu^{-2}-2a_5\ln\mu+a_6=f_2(\mu). \tag{4} \]

Here \(R(\xi)\) is an arbitrary function; \(a_1,\ldots,a_6\) are arbitrary constants; the functions \(\Pi_1(\xi)\) and \(T(\xi)\) are related to the function \(R(\xi)\) by the relation
\[ T'=\xi R'=\Pi_1'' \]
(the prime denotes differentiation with respect to \(\xi\)).

III.
\[ \rho=\frac{R'}{r\mu},\quad h_z=\frac{k^2}{r^2\mu^2}\Phi(\xi),\quad h_\varphi=\frac{1}{r^2}\Phi(\xi), \]
\[ p_\perp=\frac{c_9}{\alpha}\frac{(\mu^2+k^2)^{1/2}}{\mu^4}\,\xi R',\quad p_\parallel=\frac{c_3}{\mu^2(\mu^2+k^2)}\,\xi R', \]
\[ \left(\frac{d\mu}{dt}\right)^2 =c_1\mu^{-2}-c_3(\mu^2+k^2)^{-1}-2c_5\ln\mu +c_9\mu^{-2}(\mu^2+k^2)^{1/2}+ \tag{5} \]
\[ +\frac{c_9}{k}\frac{2+\alpha}{\alpha}\ln \frac{k+(\mu^2+k^2)^{1/2}}{\mu} +c_{11}, \]
\[ R(\xi)=b\xi^\alpha-\beta,\quad \Phi(\xi)=A-2\pi G R^2+c_5\int R'\xi^2\,d\xi. \]

Here \(A=2\pi G\beta^2\); \(\alpha\) can take only two values \((\alpha=1,\alpha=2)\). In this case \(\beta=0,\ 3c_1=k^2c_5\), if \(\alpha=1\), and \(2c_1=k^2(c_5-4\pi Gb)\), if \(\alpha=2\); \(c_1,c_3,c_5,c_9,c_{11},\beta,b,k^2\) are constants.

Let us note that, for the magnetohydrodynamic equations, solutions analogous to solutions I—III were found by A. G. Kulikovskii \((^3)\), I. M. Yavorskaya \((^4)\), McVittie \((^5)\), and the author \((^6)\).

Solutions I—III can be generalized to the case when the plasma particles have, in addition to the radial velocity \(v_r\), also a rotational velocity \(v_\varphi\), determined from the formula
\[ v_\varphi^2=\xi\chi'(\xi)\mu^{-2}. \tag{6} \]

A similar generalization was made earlier by Yu. P. Ladikov \((^7)\) for magnetic hydrodynamics with isotropic pressure and by V. P. Korobeinikov \((^2)\) for the hydrodynamics of a rarefied plasma without taking gravitational forces into account.

Then, in the case \(h_\varphi=0\), we shall have
\[ p_\perp=(P+\rho_0\chi)\mu^{-4}, \tag{7} \]
where \(\chi(\xi)\) is an arbitrary function; the remaining functions depend on \(r\) and \(t\) according to formulas (3), (3′).

In the case \(h_z=0\) we obtain:
\[ p_\parallel=(a_1-\chi'/\xi)T'\mu^{-4}, \tag{8} \]
where \(\chi(\xi)\) is an arbitrary function, and the functions \(\rho(r,t)\), \(p_\perp(r,t)\), \(h_\varphi(r,t)\), \(\mu(t)\) are determined by formulas (4).

For \(h_\varphi\ne0,\ h_z\ne0\), and \(v_\varphi\ne0\), solution (5) is valid, in which the constants \(\alpha\) and \(A\) are arbitrary, while the function \(\chi(\xi)\) is found from the formula
\[ \chi'=k^2\left[\left(c_5-\frac{c_1}{k^2}\right)\xi -4\pi G\frac{R}{\xi} -\frac{2\Phi(\xi)}{\xi^2R'}\right]. \]

Solutions I—III can be used in some specific problems.

The behavior of the functions \(f_1(\mu)\), \(f_2(\mu)\), which determine various types of gas motion, has been investigated in detail in works \((^3,^4,^6)\). From these

It follows from the investigations that in case I a complete expansion of the plasma in the presence of gravitational forces and a magnetic field is impossible. If \(b_3 < 0\), then collapse of the plasma toward the axis of symmetry cannot occur. In case II expansion is possible if \(a_5 < 0\), and impossible if \(a_5 > 0\). If \(a_1 > 0\), then the plasma particles cannot contract into an infinitely thin filament.

Received
20 IX 1966

REFERENCES

1. G. Chew, M. L. Goldberger, F. E. Low, Proc. Roy. Soc. A, 236, 442 (1956), Russian translation: Problems of Modern Physics, No. 7, 139 (1957).
2. V. P. Korobeinikov, Applied Mathematics and Technical Physics, No. 1, 153 (1962).
3. A. G. Kulikovskii, DAN, 114, No. 5, 984 (1957).
4. I. M. Yavorskaya, DAN, 114, No. 5, 988 (1957).
5. G. C. McVittie, Rev. Modern Phys., 30, No. 3, 1080 (1958).
6. E. V. Ryazanov, PMM, 23, issue 1, 187 (1957).
7. Yu. P. Ladikov, DAN, 137, No. 2, 303 (1961).