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Celest Mech Dyn Astr DOI 10.1007/s10569-015-9608-5 ORIGIN AL AR TICLE

Figures of equilibrium of an inhomogeneous self-gravitating fluid
Ivan A. Bizyaev · Alexey V. Borisov · Ivan S. Mamaev

Received: 22 May 2014 / Revised: 19 December 2014 / Accepted: 10 February 2015 © Springer Science+Business Media Dordrecht 2015

Abstract This paper is concerned with the figures of equilibrium of a self-gravitating ideal fluid with a stratified density and a steady-state velocity field. As in the classical formulation of the problem, it is assumed that the figures, or their layers, uniformly rotate about an axis fixed in space. It is shown that the ellipsoid of revolution (spheroid) with confocal stratification, in which each layer rotates with a constant angular velocity, is at equilibrium. Expressions are obtained for the gravitational potential, change in the angular velocity and pressure, and the conclusion is drawn that the angular velocity on the outer surface is the same as that of the corresponding Maclaurin spheroid. We note that the solution found generalizes a previously known solution for piecewise constant density distribution. For comparison, we also present a solution, due to Chaplygin, for a homothetic density stratification. We conclude by considering a homogeneous spheroid in the space of constant positive curvature. We show that in this case the spheroid cannot rotate as a rigid body, since the angular velocity distribution of fluid particles depends on the distance to the symmetry axis. Keywords Self-gravitating fluid · Confocal stratification · Homothetic stratification · Chaplygin problem · Axisymmetric equilibrium figures · Space of constant curvature Mathematics Subject Classification 76U05

This is a revised version of the paper (Bizyaev et al. 2014), previously published in Russian. I. A. Bizyaev · I. S. Mamaev Udmurt State University, Universitetskaya 1, Izhevsk 426034, Russia A. V. Borisov Institute of Mathematics and Mechanics of the Ural Branch of RAS, S. Kovalevskaja Str. 16, Ekaterinburg 620990, Russia A. V. Borisov (B) National Research Nuclear University "MEPhI", Kashirskoye Shosse 31, Moscow 115409, Russia e-mail: borisov@rcd.ru

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I. A. Bizyaev et al.

1 Introduction This paper is concerned with exact solutions to the problem of (axisymmetric) figures of equilibrium of a self-gravitating ideal fluid with density stratification. First of all, we briefly recall the well-known results: For homogeneous fluid, the following ellipsoidal equilibrium figures for which the entire mass uniformly rotates as a rigid body about a fixed axis are well known: the Maclaurin spheroid (1742), the Jacobi ellipsoid (1834), In addition, in the case of a homogeneous fluid there also exist figures of equilibrium with internal flows: the Dedekind ellipsoid (1861), the Riemann ellipsoids (1861). Remark The discovery of the Dedekind and Riemann ellipsoids was inspired by the work of Dirichlet (1861) where the dynamical equations for a liquid homogeneous self-gravitating ellipsoid were obtained (for this system all the above-mentioned figures of equilibrium are fixed points). For a recent review of dynamical aspects concerning liquid and gaseous selfgravitating ellipsoids and a detailed list of references (see Borisov et al. 2009). We also note the integrability cases found in a related problem of gaseous ellipsoids (Gaffet 2001). While an enormous amount of research was devoted in the nineteenth and twentieth centuries to asymmetric figures of equilibrium [see, e.g., references in Appell (1921), Liouville (1834), Lyttleton (1953), Borisov et al. (2009) and Chandrasekhar (1969)], the Maclaurin spheroid remains the most important for applications to the theory of the figures of planets. However, it is well known that for all planets of the solar system the actual compression is different from the compression of the corresponding Maclaurin spheroid obtained from the characteristics of the planet.1 Usually this difference is attributed to the density stratification of the planet and it necessitates investigating inhomogeneous figures of equilibrium. For a stratified fluid mass rotating as a rigid body with small angular velocity , Clairaut (1743)2 obtained the equation of a spheroid which is an equilibrium figure to first order in 2 . Subsequently investigations of such figures were continued in the work of Laplace, Legendre and Lyapunov. Lyapunov obtained a final solution to this problem in the form of a power series in the small parameter 2 and found their convergence.3 On the other hand, Hamy (1889), Volterra (1903) and Pizzetti (1913, Chapter 12) showed that for a stratified fluid mass rotating as a rigid body there exist no figures of equilibrium in the class of ellipsoids. We present a modern formulation of a theorem which was proven in these works: Theorem Suppose the body consists of a self-gravitating, ideal, stratified fluid and the density is not constant along the volume. Assume that
1 Relevant calculations can be easily performed using the formulae of Sect. 3.3 and astronomical data available

from the Internet.
2 A. Clairaut took part in the first expeditions which confirmed I. Newton's viewpoint that the Earth is com-

pressed from the poles.
3 The proof of convergence was not published by Lyapunov. Problems of convergence for the Maclau-

rin and HuygensRoche figures are solved in Kholshevnikov and Elkin (2002), Kholshevnikov (2003) and Kholshevnikov and Kurdubov (2004).

123

Inhomogeneous self-gravitating fluid

the free surface of the fluid is an ellipsoid (it can be both triaxial and a spheroid), the density distribution (r) is such that the level surfaces (r) = const are ellipsoids coaxial with the outer surface. Then such a fluid mass configuration cannot be the figure of equilibrium rotating as a rigid body about one of the principal axes. Hamy proved this theorem for the case of a finite number of ellipsoidal layers with constant density, Volterra generalized this result to the case of continuous density distribution for a homothetic stratification of ellipsoids, and Pizzetti gave the simplest and most rigorous proof in the general case for both continuous and piecewise constant density distribution. We also mention the paper (Kong et al. 2010), which proposes higher-order corrections for finding the figures of equilibrium with stratified density. Such publications show that there is still no complete understanding regarding the equilibrium figures of celestial bodies with stratified density. We also note that VИronnet (1912) also attempted to prove this theorem for the case of continuous density distribution but made some errors. If one admits the possibility that the angular velocity of fluid particles is not constant for the entire fluid mass, then equilibrium figures for an arbitrary axisymmetric form of the surface and density stratification (Pizzetti 1913, Chapter 9) are possible. For example, Chaplygin (1948)4 explicitly showed a spheroidal equilibrium figure with a nonuniform distribution of angular velocities for the case of homothetic density stratification. It turns out that the surfaces with equal density (r) = const. do not coincide with the surfaces of equal angular velocity (r) = const . S. A. Chaplygin tried to use the resulting solution to explain the dependence of the angular velocity of rotation of the outer layers of the Sun on the latitude. In Montalvo et al. (1982) an explicit solution of another kind was found for which the equilibrium figure is a spheroid consisting of two fluid masses of different density 1 = 2 separated by the spheroidal boundary confocal to the outer surface, with each layer rotating at constant angular velocity such that 1 = 2 . A generalization of this solution to the case of an arbitrary finite number of "confocal layers" was obtained by Esteban and Vasquez (2001). In this paper we obtain a generalization of this solution to the case of an arbitrary confocal (both continuous and piecewise constant) density stratification. For comparison, we also present Chaplygin's solution for the homothetic stratification. In addition, we show that in the case of a space with constant curvature the homogeneous (curvilinear) spheroid is a figure of equilibrium only under the condition of a nonuniform distribution of the angular velocities of fluid particles (r) = const . In this case the solution can be represented as a power series in the space curvature.

2 Equations of motion and axisymmetric equilibrium figures 2.1 Equations of motion in curvilinear coordinates In this case, to solve specific problems, it is convenient to use special curvilinear (nonorthogonal) coordinates, which we denote by q = (q1 , q2 , q3 ). Therefore, we first represent the equations describing this system in an appropriate form. Suppose that an element of the fluid has coordinates q at a given time t .Let q = (q1 , q2 , q3 ) denote the rates of change of its coordinates during the motion. They depend on both the
4 This work was not published during the life-time of S. A. Chaplygin and appeared for the first time in his

posthumous collected works prepared by L. N. Sretenskii.

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coordinates q of the chosen element and time t : qi = qi (q, t ) and the total derivative of any function f of q, and t is calculated from the formula df f = + dt t f qi . qi (1)

i

Let G = gij denote the metric tensor corresponding to these coordinates. In the case of orthogonal coordinates G = diag(h 2 , h 2 , h 2 ), where h i are the LamИ coordinates. 123 As is well known (Kochin et al. 1963), the equations of motion for a fluid in a potential field can be represented as d dt T qi - T U 1 p =- - , qi qi qi (2)

where is the density, p is the pressure, U is the specific potential of external forces, and T is the specific kinetic energy of the fluid calculated from the formula T= 1 2 gij qi q j .
i, j

The continuity equation written in this notation becomes 1 + t g ( g qi ) = 0, g = det G. qi (3)

i

In the case of a self-gravitating fluid the gravitational potential U (q, t ) can be calculated from the equation U = 4 G(q, t ), (4) where G is the universal gravitational constant and the Laplacian is given by the well-known relation = 1 g qi gg
ij

q

j

,

g

ij

= G-1 ,

assuming that outside the liquid body the density vanishes: = 0. In the absence of external influences at the free boundary B of the fluid mass the pressure vanishes: p
B

= 0, Uin n Uout n

and the gravitational potential and its normal derivative are continuous: U
in B

=U

out B

,

B

=

B

,

(5)

where the indices in and out denote the quantities inside and outside the body, respectively (note that not only normal, but all first derivatives are continuous, even if the density is discontinuous). 2.2 Steady-state axisymmetric flows To explore possible figures of equilibrium, we choose curvilinear coordinates q = (r,, ), which are related to the Cartesian coordinates as follows x = r cos , y = r sin , z = Z (r,).

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Inhomogeneous self-gravitating fluid

Here the function Z (r,) is chosen so as to obtain a free surface of the fluid mass for one of the values = 0 . Its specific form will be defined by an appropriate problem statement. The metric tensor is given by 2 1 + Zr Zr Z 0 2 Z 0 , g = det G = rZ , G = Zr Z 0 0 r2 where Z r =
Z r

, Z =

Z

.

Remark Pizzetti (1913) used standard cylindrical coordinates (i.e., he set = z ), with the equation for the free surface being F (r, z ) = 0. From a practical point of view, this approach is inconvenient in searching for specific equilibrium figures of a stratified fluid. We shall seek a steady-state solution of (2), for which the velocity distribution has the form r = 0, = 0, = (r,), (6) and the functions U , p ,and do not depend on . Then, substituting (6)into(2)and (4), we obtain the system of equations U 1 + r r U = 4 r = p U 1 p = r 2 , + = 0, r G(r,),

2 1 1 1 + Zr rZ + rZ r r Z Z 1 - rZ r + rZ r , rZ r r =0

p (r,)

= 0.

(7)

Note that the continuity equation (3) holds identically. We choose the function Z (r,) defining the curvilinear coordinates in such a way that all coordinate surfaces = const are compact, and choose a value of = 0 which corresponds to the boundary of the fluid and defines the distribution of density (r,). Then, according to (7), after solving the equation for the potential one can always choose a distribution of pressure and of the squared angular velocity, which satisfy the first pair of equations:

p (r,) = -
0

U d ,

1 U 0 (r,0 ) + 2 (r,) = r r

U U - r r d , 0 = (r,0 ).

0

A possible obstruction to the existence of such equilibrium figures is that 2 (r,), defined from these equations, may turn out to be negative. The problem of equilibrium figures becomes more nontrivial when we impose some restrictions on the distribution of angular velocity. L. Lichtenstein and R. Wavre found sufficient conditions under which a fluid-filled region obviously possesses a plane of symmetry, see, for example, Lichtenstein (1933).

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Theorem Assume that for an inhomogeneous self-gravitating mass of perfect fluid the following is satisfied: 1. the fluid is at relative equilibrium where all particles rotate about the fixed axis O z , and their angular velocity depends only on the distance to the axis of rotation: = (r 2 ), 2. the density is a piecewise continuous function, 3. the fluid-filled region consists of a finite number of disjoint bounded (homogeneous or inhomogeneous) fluid volume whose boundaries are homeomorphic to spheres or tori. Then the fluid-filled region possesses a plane of symmetry perpendicular to the axis O z . It is also obvious that the center of mass lies on the intersection of the symmetry plane with the axis of rotation Oz .

3 Inhomogeneous figures with isodensity distribution of the angular velocity of layers 3.1 General equations for locally nonconstant and locally constant density distributions We now consider the case where the level surfaces of stratification of density coincide with the level surfaces of angular velocity (i.e., the fluids of equal density move with equal angular velocity); choosing them as coordinate lines = const, we represent this condition as = (), = (). (8) Eliminating the pressure from the first pair of equations of the system (7) (multiplying them by and differentiating the first one with respect to and the second one with respect to r and subtracting one from the other), we obtain () U (r,) = r ()2 () , r (9)

where the prime denotes the derivative with respect to . Let us consider the main consequences of this equation (expressing the restrictions to the gravitational potential inside the figure), which result from the conditions of mechanical equilibrium. We see that according to (9) it is necessary to analyze two cases separately. In the first of the cases we assume that () vanishes only at isolated points, in the second case we have to consider a situation where on the whole interval () 0, (1 ,2 ).Let us consider them in succession. 1. The case of locally nonconstant density If we assume that in some interval (1 ,2 ) () = 0, inside. then, according to (9), the potential U in this volume of fluid can be written as U (r,) = 1 u ()r 2 + v(). 2 (10)

If ( ) = 0 at some isolated point , then on the left and right of the potential is represented in the form (10), and due to continuity of U (r,) the limits of the functions u () and v() in on the left and right coincide. In this case, if ( ) = 0, then the equation d 2 () = 0 holds for the angular velocity of this layer. From the first pair of Eq. (7), d we obtain the unknowns p (r,) and () in the form
=

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Inhomogeneous self-gravitating fluid Fig. 1 Distribution with a layer on which the density takes a constant value

1 P () , p = - P ()r 2 - Q (), 2 () = u () - 2 ()

P () =
0

u ( ) ( ) d ,

Q () =
0

v ( ) ( ) d .

(11)

Obviously, p (r,)
=0

= 0,

d 2 d

=0

= 0.

Hence, it follows that the figure of equilibrium of a fluid with density stratification and angular velocity of the form (8) exists if and only if there exist functions Z (r,) and u (), v () satisfying the equation r, 1 u ()r 2 + v() = 4 G(), 2 (12)

and the potential inside the fluid mass has the form (10). 2. The case of locally constant density We now consider a situation where in some layer the density takes a constant value: () = 0 = const, (1 ,2 ), then, according to (9), we conclude that the angular velocity of the entire layer is also constant: () = 0 = const, (1 ,2 ). Taking this result into account, we integrate the first pair of Eq. (7) and obtain the following relation for the function U + p0 in the layer: U+ p 12 = 0 r 2 + 0 , 0 = const. 0 2 (13)

Furthermore, at all points at the boundaries of the layer = i , i = 1, 2 (see Fig. 1) the pressure inside and outside must be the same: pin (r,)
=i

=p

out

(r,)

=i

.

(14)

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I. A. Bizyaev et al. Fig. 2 Meridional sections of the surfaces = const

The potential in the layer also satisfies the Laplace equation r Uin (r,) = 4 G0 , and at the boundaries conditions (5) hold. In principle, in the general case an inhomogeneous figure of equilibrium can consist of parts, with both locally constant and locally nonconstant density. 3.2 The family of confocal spheroids Consider a particular case in which the sought-for solution exists. We shall show that in the case of confocal stratification of the density of a spheroid the gravitational potential is written as (10). Choose the parameterization of confocal stratification in R3 as follows z2 x 2 + y2 + 2 2 = 1, [0, +), d 2 (1 + 2 ) d where d is the focal distance of the meridianal section (see Fig. 2). Thus, the parameter defines the ratio between the small semiaxis of the spheroid and the focal distance, and the eccentricity e is expressed by the formula e= Expressing z , we find Z (r,) = ± d 2 2 - r
2

1 1 + 2

.

(15)

2 . 2 + 1

(16)

If the boundary of the spheroid filled with a fluid has semiaxes a and b (see Fig. 2), then the focal distance d and the coordinate of the boundary 0 are defined by d= a 2 - b2 , 0 = b a -b
2 2

.

(17)
r2 d 2 2

Remark It can be shown that for a prolate spheroidal stratification (i. e., for
d

+

of the layers (2 () < 0), therefore, we will not consider it.

z2 2 (2

+1)

= 1) this solution leads to a negative square of the angular velocity of rotation

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Inhomogeneous self-gravitating fluid

Proposition 1 The gravitational potential for a spheroid with confocal stratification has the form k 1 r 2 u () U= + d 2 v() , k = 4 G . (18) 2 2 1 + 2 For the internal points u in = I0 ()((1 + 32 )arcctg() - 3) - I1 ()(1 + 32 ) v = - I0 ()((1 + 2 )arcctg() - ) + I1 ()(1 + 2 ) + 2 I2 ()
in

I0 () =
0

( )(1 + 3 ) d , I1 () =
2 0

( )((1 + 3 2 )arcctg( ) - 3) d ,

I2 () =
0

( ) d .

(19)

For the external points u
out

= I0 (0 )((1 + 32 )arcctg() - 3), v

out

= I0 (0 )( - (1 + 2 )arcctg()). (20)

Proof In the case of confocal stratification (16) the Laplacian (7) has the form r, = 1 r r r r + 1 + 2 d
2

1+

2 2

-r

2

1 + 2

+ 2r

2 r

We shall search for a potential in the form (18). We show that then Eq. (12) reduces to two linear equations for the functions u () and v(). Indeed, applying the Laplacian r, to each term in (18) separately, we find k r, 2 1 r 2 u () 2 1 + 2 = r2 2(d (1 + 2 )2 - r 2 )
2

d du (1 + 2 ) d d

- 6u

2d 2 (1 + 2 )u kd 2 r, v() d 2 (1 + 2 )2 - r 2 2 d dv 1 + 2 =2 (1 + 2 ) . d (1 + 2 )2 - r 2 d d - Further, using the Poisson equation (12), we obtain: r2 2 d d
2

1 + 2

du d

- 6u + d
2

2 2

1 + 2 -r
2

d d

1 + 2

dv d

- 2d

1 + 2 u = 2 d

1 + 2

(),

whence, equating the coefficients with degrees r , we find the sought-for equations for the unknowns u () and v(): dv +2u -2 1 + 2 () = 0. d (21) As is well known, the solution (21) is represented as the superposition d d 1 + 2 du -6u +4() = 0, d d d 1 + 2 u () = u 0 () + u p (), v() = v0 () + v p (), (22)

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I. A. Bizyaev et al.

where u 0 and v0 are a general solution of the homogeneous system [when () = 0], while u p and v p are a particular solution of the inhomogeneous system. In this case one can choose u 0 () = A
1

1 + 32 + A
2

2

1 + 32 arcctg - 3 , (23)

v0 () = - A1 2 + A

1 - 2 arcctg + + A3 arcctg + A4 ,

where A1 , A2 , A3 and A4 are arbitrary constants of integration. Using a modification of the method of variation of constants, the particular solution can be represented in terms of single integrals:

u p () =

1 + 3

2

arcctg - 3
s

1 + 3

2

( ) d

- 1 + 32
s

1 + 3

2

arcctg - 3 ( ) d ,

v p () =
s

(arcctg) S ( ) d - arcctg
s

S ( ) d , (24)

2 S () = 2 + 1 () - u p (),

In the general case, for each of the integrals an arbitrary constant can be chosen as the lower bound s in (24). The conditions which must be satisfied by the potential have the form 1. Away from the spheroid, the potential must tend to zero:

lim

u out () = 0, 1 + 2
out

lim v

out

() = 0.

(25)

2. At the boundary of the spheroid = 0 , the potential must be a smooth function: u in (0 ) = u u (0 ) = u
in

(0 ), v in (0 ) = v (0 ), v (0 ) = v
in

out

(0 ), (0 ). (26)

out

out

3. As 0, the potential on the section z = 0, r (0, d ) must be a smooth function, i.e., the values of its derivatives must be the same at the points z + and z - as 0 (see Fig. 1). This yields the condition u
in =0

= 0, vin

=0

= 0.

(27)

Let us satisfy the first condition (25). To do so, we express the potential outside as a power 1 series in : u out () = 3A 1 + 2
out 1

+O

1 ,v

out

= -A

out 1

2 + A

out 4

+O

1

to give Aout = Aout = 0. 1 4 Next, we satisfy the condition (26). To simplify the system (25), we choose a particular solution in such a way that it vanishes on the surface. It is easily seen that this can be achieved by choosing s = 0 . Moreover, in this case Eq. (26)are satisfiedifweset Ain = Ain = 0, 1 4 Aout = Ain , Aout = Ain . 2 2 3 3

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Inhomogeneous self-gravitating fluid

From Eq. (27) we find two remaining constants Ain and Ain : 2 3
0 0

A=
in 2 0

1 + 3

2

( ) d ,

A = -2
in 3 0

S ( ) d .

Now, in order to obtain the relations (19), we only need to simplify the expression for Ain : 3
0 0

A = -2
in 3

1+
0

2

() d + 2
0

u p () d ,

1 ( ) ( ) d ,

2u p () = 2 1 ()
0

2 ( ) ( ) d - 2 ()
0 2

1 () = 1 + 3

arcctg - 3, 2 () = 1 + 32 .
in 3

We take the second integral in the expression for A primitives

by parts. To do this, we define the

1 () = (1 + 2 )arcctg - , 1 () = 1 (), 2 () = ( + 1), 2 () = 2 (), to give
0 0 0

2u p () d =
0 0

1 ()

2

() - 2 ()

1

() () d = -2
0

2 () d .

Thus, we finally obtain
0

Ain = -2 2
0

1 + 3

2

( ) d .

Writing the solution (24) with the known integration constants, taking the iterated integrals in v p () by parts, as was done above, and reducing similar terms, we obtain (19)and (20). Remark If we make a change of the variable = ix in Eq. (21), they take the form of inhomogeneous Legendre equations with n = 2 and n = 1. We note that no assumptions about differentiability of the distribution of () were used in proving this proposition. Therefore, this result holds for any (including piecewise constant) distribution for which the integrals (19)converge.5 Now, using the representations (19) and (18) for the potential inside the figure, we can find () and p (r,) from the equilibrium conditions [see the first pair of equations in (7)]. Finally, integrating by parts, we obtain the following theorem. Theorem 1 Suppose that an inhomogeneous perfect fluid fills an oblate spheroid with the 1 semiaxes a0 = d (1 + 2 ) 2 , b0 = d 0 and that the surfaces of constant density of the fluid 0 coincide with the family of confocal ellipsoidal surfaces with the semiaxes a = d (1 + 2 )
1 2

5 In the case of piecewise constant distribution the function () is a combination of Dirac delta functions.

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I. A. Bizyaev et al.

and b = d , [0,0 ]. Then, for each density distribution : [0,0 ] R, there is a stationary motion such that each ellipsoidal surface of the family moves as a rigid body around the common axes with the angular velocity ()2 (0 ) 1 + 32 arcctg(0 ) - 30 0 = I0 (0 ) 2 G () 1 + 2 0 - 2 ()
0

( )

I0 ( ) 1 + 3

2

arcctg( ) - 3 - I1 ( ) 1 + 3 1 + 2

2

d , (28)

and with the pressure distribution p (r,) = 4 G
0

( )

r2 -d (1 + 2 )2

2

( I0 ( )(1 - arcctg( )) + I1 ( ) ) d . (29)

3.3 The homogeneous Maclaurin spheroid Let the density be constant everywhere inside some spheroid: () = 0, 0 , 0 < , 0 < 0 ,

where 0 is defined by (44). In this case we find the gravitational potential from Proposition 1. Inside the spheroid it can be represented as U = 2 G 1 r 2 u in () + d 2 v in () , 2 1 + 2 1 + 2 arcctg0 - 0 - 22 , 0 2 - 0 1 + 2 arcctg0 .

u in () = 0 0 1 + 32 v in () = 0 1 + 2 0

Comparing this expression with (18) and (20) as a consequence of this representation of the potential, we obtain the well-known Maclaurin theorem (Chandrasekhar 1969) in the case of a spheroid. Theorem 2 The gravitational potential that is produced by an inhomogeneous spheroid with confocal stratification and density () at the external point is the same as the potential of a homogeneous spheroid with the density 1 0 = = 0 (1 + 2 ) 0
0

1 + 3
0

2

( ) d .

(30)

Next, from (28) and taking into account the relationship (15) between 0 and the eccentricity, we obtain the well-known expression for the angular velocity 0 of the Maclaurin spheroid 2 0 1 - e2 2 3 - 2e2 arcsin e - 3e 1 - e2 . = 0 1 + 30 arcctg0 - 30 = 2 G0 e3 (31) Using (29), we find the pressure for the Maclaurin spheroid: 2 - 2 (1 - 0 arcctg0 ) p =0 d 2 1 + 2 2 G0
2

1 + 2

1 + 2 - r 0

2

.

(32)

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Inhomogeneous self-gravitating fluid

It can be shown that the level surfaces (32) are homothetic spheroids. Using (16) and a relation defining the homothetic stratification, which in our case takes the form d we find r= d
2 2

r2 z2 + 2 2 = m, d 0 1 + 2 0 1 + 2 2 - 2 0 m 2 - 2 0

1 + 2 0

.

Then, substituting r into (32), we obtain p = d 2 2 1 + 2 (1 - 0 arcctg0 ) (1 - m ). 0 0 2 2 G0 If we compare the expressions (31)and (28)for = 0 , then we obtain the following result: Theorem 3 For an arbitrary confocal stratification the angular velocity on the outer surface of the inhomogeneous spheroid is the same as the angular velocity 0 of the Maclaurin spheroid with density 0 = :
2 0 = 0 2 G

1 + 32 arcctg(0 ) - 30 , 0

(33)

where is the average density of the spheroid (30). Thus, we see that for any confocal stratification the rotation of the body (planet) does not differ visually from from that of the Maclaurin spheroid. 3.4 A spheroid with piecewise constant density distribution As mentioned above, the relations (28) and (29) hold for any stratification of density (). Nevertheless, for the case of piecewise constant density distribution we present a derivation, which is different from the previous one, for angular velocity and pressure distributions to demonstrate how the equilibrium conditions can be used in the case of locally constant density (see Sect. 3.1). We now consider a spheroid with piecewise constant density, i.e., consisting of a sequence of embedded homogeneous layers with different densities. We will label the outer layer, as before, by the index 0 and the last internal layer by the index n . Thus, we obtain a spheroid consisting of n + 1 layers: 0 < , 0, 0 , 1 < < 0 , () = 1 , 2 < < 1 , ..., ... n , 0 < < n . The case of two layers of different density (in our notation n = 1) is considered in Montalvo et al. (1982), and the generalization of this case to an arbitrary number of layers is found in Esteban and Vasquez (2001). Interestingly, almost all calculations presented below are contained in Hamy (1889), although he used them not to search for new figures of equilibrium but to prove the absence of inhomogeneous figures of equilibrium with rigid body rotation (see Sect. 1).

123

I. A. Bizyaev et al.

From (13) we find that the pressure inside the k th layer is given by p (k ) = Gr i
2 2 k u in () - 2 G 1 + 2

+ 2 Gd 2 vin () + k , k = 0, 1,..., n .

where k < < k +1 . Further, taking into account that the pressure at the outer boundary is zero and the potential and the pressure at the boundary between the layers change continuously, we obtain the following relations for unknown angular velocities:
2 0 0 u in (0 ) = 0 ,..., 2 G 1 + 2 0 2 2 n -1 n -1 u in (n ) n n = + n , 2 G 2 G 1 + 2 n 0 = 0 , 1 = 1 - 0 ,...,n = n - n

-1

.

This yields the angular velocity for the k th layer in the form
2 k k = 2 G k

i
i =0

u in (i ) . 2 1 + i

We obtain the expression for u in (i ) from (19):
2 2 u in (i ) = I0 (i ) 1 + 3i arcctgi - 3i - I1 (i ) 1 + 3i .

To calculate I0 (i ) and I1 (i ), we use the Heaviside function (x ) = 0, 1, x < 0, x 0,

and represent the density of the spheroid under consideration as
n

() =
i =0

i (i - ).

Integrating, we find that
n

I0 (i ) =
j =i +1 j 2 2i 2 1 + 3i

j

j

1 + 2 j .

I1 (i ) =
i =0

j

-

j

1 + 2 arcctg j - j

j

As a result, we obtain an expression for the angular velocity of the k -th layer in the form k 2 2i 2 2i 1 + 3i k k = i j j 1 + 2 arcctg j - j - j 2 2 2 G 1 + i j =0 1 + 3i i =0 n 2 1 + 3i arcctgi - 3i (34) j j 1 + 2 . + j 2 1 + i j =i +1

123

Inhomogeneous self-gravitating fluid Fig. 3 a A graph showing the dependence of the relation on the layer . b A graph showing the dependence of the angular velocity on the layer . c The period of revolution T depending on the layer

(a)

(b)

(c)

3.5 A spheroid with continuous density distribution To keep track of the dependence of the angular velocity of the layers on the change in density, we consider an inhomogeneous spheroid with different functions of density distribution of the following form: ( () = n0) 1 - n n , n = 2, 4, 6, (35) where n and n are some constants (n has the meaning of density at the center of the spheroid). We will determine their values from the given average density of the body dV = dV and the given ratio between the density on the surface and the average density of the body =
(0 ) (0) (0)

, (1 + n )(3 + n ) 1 + 2 (1 - )-n 0 0 (3 + n ) 1 - (1 + n ) 1 + 2 0 (3 + n ) (1 + n ) 1 n (1 + + 2 0 n )2 0 + 3(1 + n )2 0 - 1 - 3(1 + n )2 0 +3+n .

=

0 =

As an example, assume that the eccentricity e0 and the quantity , which are the same as the data of the Earth (Williams 2004): e0 = 0.08181, = 2.5.
Figure 3a shows the dependences of on the coordinate of the layer for (35). As we can see, the density increases most sharply at the center of the spheroid for n = 2 and then, as n increases, the density decreases. To find the angular velocity, we substitute the density distributions (35) into (14) and obtain the dependence of the angular velocity on the layer. A graph of this dependence is shown in Fig. 3b. (Since the explicit formulae for () are unwieldy, we do not present them here.) For the angular velocity with density distribution (35) one may draw the following conclusion from Fig. 3b: the closer the center of the spheroid, the larger the angular velocity; specifically, the larger the value of density at the center of the spheroid (with n = 2), the larger the increase in the angular velocity.

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Next, we calculate the numerical value of the dependence of the period of revolution for each layer. If we assume the average density to be the same as that of the Earth = 5.51 g/cm3 , then we obtain the dependences of T () presented in Fig. 3c.

4 The Chaplygin problem: a spheroid with homothetic density distribution As is well known, the homothetic stratification is given by z2 r2 + 2 = , [0, +), b2 a where, assuming that a and b are the principal semiaxes of a spheroid filled with a fluid (see Fig. 4), we obtain 0 = 1, Again we set = ( ) (does not depend on r ), 0,
=1

Z (r, ) = ±b -

r2 . a2 1, > 1.

Using the second of Eq. (7) and noting that p be represented as

= 0, we obtain the pressure, which can d , 1 = (1).

p (r, ) = 1 U (r, 1) - ( )U (r, ) +
1

U (r, )

In a similar manner, substituting the pressure from the first of Eq. (7), we obtain 1 U 1 (r, 1) + (r, ) = r( ) r
2

1

U (r, ) d . r

(36)

Thus, to complete the solution, we only need to find the potential from the equation r, U (r, ) = 4 G( ). In Ferrers (1875) a convenient integral representation of the potential for a (three-axial) ellipsoid with homothetic density stratification is obtained. Applying it to the case of the spheroid = 1 gives
Fig. 4 Meridional sections of the surfaces = const with homothetic stratification

123

Inhomogeneous self-gravitating fluid Fig. 5 Meridional sections of the surfaces 2 = const with G equal spacings
2

U (r, z ) = Ga b
in

22 0

f (1) - f

r2 a 2 +s

+

z2 b2 +s

(s ) f (1) - f
r2 a 2 +s

ds ,

U

out

(r, z ) = Ga 2 b

2 s
0

+

z2 b2 +s

(s )

ds , (37)

(s ) = (a 2 + s ) b2 + s , where the function f ( ) is related with the density of the fluid by ( ) = df ( ) , d

and the quantity s0 for given (r, z ), which correspond to a point outside the liquid spheroid, is defined as the root of the equation z2 r2 +2 = 1. a + s0 b + s0
2

As an example, we consider the density distribution of the form ( ) = 0 (1 - n ), n = 1, 2, 3. (38)

Given the average density of the body and the ratio between the densities at the center and on the surface = 0 , we now define the constants 0 and : 1 = Set = 5, b 1 =. a 2 -1 (3 + 2n ) , 0 = . 3 + 2n (39)

1, 2, 3 are shown in Fig. 5. The graphs of change in the relation 2 along the small G semiaxis b is shown in Fig. 6. For the densities from Figs. 5 and 6 one can draw the following conclusions:
2

Further, we find the potential from (37) and obtain the angular velocity from (36). The 2 meridional sections of the surfaces 2 = const with equal spacings for different n = G

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I. A. Bizyaev et al. Fig. 6 The change of 2 G along the small semiaxis b for different n
2

1. The closer the center of the spheroid, the slower the change in the angular velocity. 2. For n = 1 the level surfaces near the center of the spheroid are concentric spheres. Further, as n increases, the region in which the level lines are closed surfaces increases. For n > 1 these closed surfaces are no longer surfaces of the second order. Let us consider in more detail the angular velocity at the boundary of the spheroid at the equator with densities of the form (38), but now with an arbitrary n .From(36), changing the variable s = a 2 (t - 1), we obtain the angular velocity on the surface:
2 n (r, 1) = 0 e 2 G 2

1-e

2 1

t -1 t
2 2

t -e

2 3/2 n 2

1- that is, for r = a we have

t

-n 2n

t -e

(t - 1)e

r2 +t 1-e a2

dt ,

2 n (a , 1) = 0 e 2 G

2

1-e

2 1

(t - 1) 1 - t t
2

-n

t -e
2

2 3/2

dt .

Explicitly integrating gives
2 n (a , 1) 20 e2 2 = 0 m + 2 G 3 + 2n

1-e

2

2n 1 - e 2 + 3 - 2e 5 + 2n

F

3 5 7 ,n + ,n + ,e 2 2 2

2

-1 ,

(40) 2 where m is the dimensionless angular velocity of the Maclaurin spheroid: 1 - e2 2 3 - 2e2 arcsin e - 3e 1 - e2 . m = e3 Substituting the expression (40) into the relation for the angular velocity, we obtain for two values of n
2 0 (a , 1) 2 (a , 1) 2 = = m . 2 G0 (1 - ) 2 G0

Further, we shall define 0 and from various known data for the Earth: We are given the average density of the body = 5.51 g/cm3 and the ratio between the densities on the surface and at the center 0 = 5. In this case 0 and are defined by (40), 1 and the dependence of the period of revolution at the equator T on n is shown in Fig. 7a.

123

Inhomogeneous self-gravitating fluid Fig. 7 The dependence of period T on n at the equator a for
= 5.51g/cm3 and 0 = 5, 1 b for = 5.51g/cm3 and

(a)

= 2 .5

(b)

Fig. 8 The dependence of period T on the polar radius on the surface of the inhomogeneous spheroid = 5.51 g/cm3 and = 2.16 for n = 1, n = 2 and n=3

As can be seen in Fig. 7a, T (n ) reaches the minimum at the point T (0.8675) = 24.1610 h. We are given the average density of the body, = 5.51 g/cm3 , and the ratio between the density on the surface and the average density, 1 = = 2.5. The dependence of the period of revolution at the equator T on n is shown in Fig. 7b. The dependence of the period of revolution T on the polar radius r on the surface is shown in Fig. 8.
3

5 Figures of equilibrium in S

One of the generalizations of the above results is that they are carried over to the spaces of constant curvature S 3 and L 3 , by analogy with Celestial Mechanics of point masses (Borisov and Mamaev 1999, 2006; Killing 1885; Kozlov and Harin 1992; SchrЖdinger 1940). There is a vast classical and recent literature on the dynamics of gravitating point masses (see Albouy 2013; Borisov et al. 2004; Borisov and Mamaev 2006, 2007; Bizyaev et al. 2014), in which, for example, the well-known analogs of the Kepler law and those of the three-body problem were studied. However, a particular generalization of the theorems of Newtonian potential to S 3 and L 3 was performed only in Kozlov (2000). As will be shown below, in this case the problem of equilibrium figures becomes considerably more complex. In particular, even in the case of homogeneous ellipsoids the rigid body rotation of a fluid mass is impossible (we recall that an ellipsoid in curved space is said to be a body obtained by the intersection of the

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sphere S 3 or the Lobachevsky space L 3 , embedded in R4 , with a conical quadric). One of the difficulties is due to the fact that although some generalizations of Ivory's theorem on the potential of the elliptic layer (Kozlov (2000)) are possible, this and similar theorems cannot be completely extended to S 3 and L 3 (they are closely related to the homogeneity of plane space). Remark Generalizations of the problem of equilibrium figures to the relativistic case are also possible, see, e.g., the review (Meinel et al. 2008). Unfortunately, attempts to obtain explicit analytical exact solutions along these lines have yielded no results so far. This direction is a new research area. 5.1 Steady-state axisymmetric solutions in S
3

To explore possible figures of equilibrium in S 3 , we choose curvilinear coordinates, as was done for the plane space E 3 . For convenience, we assume S 3 to be embedded into E 4 , then the transition to the coordinates under consideration has the form x0 = ± R 2 - r 2 - Z 2 (r,), x1 = Z (r,), x2 = r cos( ), x3 = r sin( ), where Z (r,) is defined, as before, by the specific problem statement. The metric tensor can be represented as g11 g12 0 G = g12 g22 0 , 0 0 r2 where
2 g11 = 1 - Z r +

Z (r + ZZr )2 , g12 = R2 - r 2 - Z 2

rZ + R 2 - r R2 - r 2 - Z

2 2

Z

r

, g22 =

R2 - r
2 2

2

Z

R -r -

2 Z2

.

We shall seek a steady-state solution for which the velocity distribution of fluid particles has the form r = 0, = 0, = (r,). As above, assuming that the density depends only on and using the equations of Sect. 2.1, we obtain the system U U 1 p 1 p + = r 2 , + = 0, r r r U = 4 G(),
r

=

x0 rZ r

rZ x0 1 0 Z r x0

1-

r2 R2

r
r

+

x0 Z x x0 + rZ r r + r x0

( Z - rZ R2 r ( Z - rZ r ) Zr + R2 r ( Z - rZ r ) Zr + R2
2 1 + Zr -

)2 ,

(41)

123

Inhomogeneous self-gravitating fluid

where x0 = R 2 - r 2 - Z 2 (r,) and it is assumed that the density () vanishes everywhere outside the body (0 < ), and at the free boundary = 0 the pressure is zero as well: p (r,)|
=0

= 0.

As we can see, the hydrodynamical equations remain the same as in E 3 . Therefore, as in Sect. 4, their solution inside the region ( 0 ) filled with fluid can be represented as

p (r,) = 0 U (r,0 ) - ()U (r,) +
2 0

U (r,)

1 dU (r,) = 0 (r,0 ) + r() dr 5.2 A homogeneous spheroid in S
3

0

d() d , 0 = (0 ), d d() dU (42) (r,) d . dr d

We now consider in more detail the case of a homogeneous spheroid, when, for 0 ,the density () = 0 = const. The generalization of confocal stratification in S 3 is given as follows (cf. Sect. 3.2):
2 2 x0 x2 x 2 + x3 R . - 21 2 - 2 2 = 0, 0, 2 2 2 R -d d d d 1+ 2

Hence, we obtain Z (r,) = ± d 2 2 - r
2

R 2 + d 2 2 . R 2 1 + 2

As in the previous case (see Sect. 3.2), the parameter d and the boundary 0 of a liquid spheroid with the semiaxes a and b are given by d= b a 2 - b2 , 0 = . 2 - b2 a
d d

According to (42), in the case of a homogeneous spheroid velocity of the fluid depends only on r : 2 (r ) = 1 U (r,0 ). r r

= 0, therefore, the angular

(43)

We shall seek solutions to the equation for the potential (41) in the form of a power series d2 in the parameter R 2 :

U (r,) = 2 Gd

2 n =0

d R

2n

Un (r,).

As can be shown, all terms of this series are polynomials in r . It is convenient to represent them as

Un (r,) =
n =0

r d

2m

u n , () . 2m (1 + 2 )m

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The potential U0 (r,) is equal (up to a multiplier) to the potential of the Maclaurin spheroid (see Sect. 3.3): U0 (r,) = u inside the spheroid ( 0 ): u u
in 0 ,0 in 0 ,1 0 ,0

() +

r 2 u 0,1 () , d 2 2(1 + 2 )

() = 0 1 + 2 0

2 - 0 1 + 2 arcctg0 , 1 + 2 arcctg0 - 0 - 22 , 0

() = 0 0 1 + 32

outside the spheroid (0 < ): u u
out 0 ,0 out 0 ,1

() = 0 0 1 + 2 0 () = 0 0 1 + 2 0

- 1 + 2 arcctg , 1 + 32 arcctg - 3 .
2

We shall assume that the space curvature is very small ( R ourselves to calculating the first correction U1 (r,) = where the functions u
1,0

a 2 ) and, therefore, restrict

r 4 u 1,2 () r 2 u 1,1 () +2 +u 4 4(1 + 2 )2 d d 2(1 + 2 )
1,1

1,0

(),

(), u

(), and u du 1, d du 1, d
0 ,1 2

1,2

() satisfy the equations
1,2

d d d d d d The functions u ,u

1 + 2 1 + 2

- 20u - 6u
1,2

+ 16u

0 ,1

= 0, du 0, d
1

1

1,1

- 1 + 2

- 6(2 + 2 )u 1 + 2
0 ,1

+ 8u
0

+ 40 1 + 2 = 0,
1,1

du 1, d

- 2u

- 1 + 2

du 0, d

0

+ 22 u
1,0 1,1

+ 0 1 + 2

= 0.

(44)

, an d u

1,2

must also satisfy the following boundary conditions: = 0 , m = 0 , 1, 2 . = du
2 out 1,m =0

du

in 1,m

d u
in 1,m =0

=u

out 1,m

=0

,

=0 du in m 1,

d
=

=0
R d

d .

, m = 0 , 1, 2 .

U1 (r,)

=O R

Since the solution of the resulting system is rather unwieldy, we omit it here and confine ourselves to the expression for the angular velocity of the fluid, for which, according to (43), we find u in 1 (0 ) 1 2 (r ) 0, +2 = 2 G R 1 + 2 0 u
in 1,2

(0 )
2 2 0

1+

r2 +

u

in 1,1

(0 ) 2 0

1+

d

2

+O

d4 R4

.

Substituting the solution for u in m (0 ) and expressing 0 in terms of the eccentricity of the 1, boundary using the formula e = 1 2 and d 2 = a 2 -b2 , we obtain an explicit representation
1+0

123

Inhomogeneous self-gravitating fluid Fig. 9 Dependences of 00 , 11 , and 10 on the eccentricity e

for the angular velocity in the form 2 (r ) = 2 G0
00

11

10

d2 1 + 2 11 r 2 + 10 a 2 + O , R R4 3 3 1 - e2 2 - 2 arcsin e - 2 1 - =- e e e 30 35 1 - e2 12 - 2 + 4 arcsin e + =- e e 2e 27 10 1 - e2 16 - 2 + 4 arcsin e - = e 2e e
00

e

2

, 1-e 1-e
2 2

4 55 35 - +4 3 3e 2 2e 41 1 10 - 2+ 4 3 6e e

,

.

The graphs of dependence of each of the corrections for the angular velocity on the eccentricity is presented in Fig. 9. Thus, in the space of constant (positive) curvature the homogeneous liquid self-gravitating spheroid cannot rotate as a rigid body, and the angular velocity distribution of fluid particles depends only on the distance to the symmetry axis: = (r ). Remark For completeness we also present the equations which describe axisymmetric figures of equilibrium in curvilinear orthogonal coordinates (, , ) and are defined as follows:
2 2 x0 x1 ( - )( + ) R2 , = = , = 2 d2 +1 d2 d 2 2 x2 x3 (1 + )(1 - ) (1 + )(1 - ) 2 = = cos2 , sin , 0 < < , 0 < < 1. d2 +1 d2 +1

In this case the system (41) takes the form U U 1 p d 2 1 p + =- (1 - )2 , + = () 2( + 1) () 2 U (, ) = 4 G(), 4 ( - ) R 2 = (1 + ) ( - ) + (1 - ) ( + ) . + ( + ) d 2 (1 + )2 , ( + 1)

This form of equations is preferable if it is necessary to obtain a solution in terms of quadratures (and not in the form of a power series).

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6 Discussion In this paper we have systematically analyzed the problem of inhomogeneous axisymmetric equilibrium figures of an ideal self-gravitating fluid. We have obtained the most general solution describing a stratified spheroid [the angular velocity of the fluid takes the same value on the layer with equal density, i.e., = ()]. This solution naturally yields the abovementioned spheroids with piecewise constant density distribution (Esteban and Vasquez 2001; Montalvo et al. 1982). It is shown that the angular velocity of the outer surface of the spheroid with confocal stratification of density is the same as that of the homogeneous Maclaurin spheroid with density . Therefore, this model cannot be used to explain the deviation of the compression of planets from the compression of the Maclaurin spheroids rotating with the same angular velocity. We have also presented a fairly detailed review (and a formulation in modern terms) of results in this vein. Of special note is Chaplygin's work (previously unpublished and found in archives) on spheroids with homofocal density stratification. In the last section we have considered the problem of the conditions for equilibrium of a homogeneous spheroid in the spaces of constant curvature S 3 and shown that in this case the fluid cannot rotate as a rigid body and that the angular velocity of fluid particles depends only on the distance to the symmetry axis: = (r ). We conclude by pointing out some open problems related to possible generalizations of the above results. 1. The stratified analogs of the Maclaurin spheroids raise the question of their stability. Of particular importance is here in all probability their secular stability, which was considered by Lyapunov (1959) for the case of homogeneous fluid density. In his analysis of the perturbation of the free surface by spherical harmonics he concluded that the higher the order of a harmonic, the larger the value of eccentricity at which the loss of secular stability occurs. In the general case Lyapunov arrived at the conclusion that the secular stability of the Maclaurin spheroids is lost under arbitrary deformations if the eccentricity becomes equal to 0.8126 [for the special case of ellipsoidal perturbations this result was obtained by Dirichlet (1861)]. As far as this problem is concerned, no finite-dimensional equations governing the dynamics of stratified ellipsoids have been obtained so far (see Dirichlet 1861; FassР and Lewis 2001; Riemann 1861). Because of this it is difficult to obtain all sufficient stability criteria determined by the finite dimensionality of the system (Lyapunov's theorem, KAM theory). 2. Historically, attempts to derive the first equations of stratified ellipsoids go back to Betti (1881), but, as Tedone (1895) noted, Betti made a mistake in his study. In this connection, the question of possible existence of three-axial inhomogeneous ellipsoids still awaits its solution. 3. The above solution for spheroids with confocal stratification is evidently the only solution possible, for which = (), but no proof of this fact has been found. 4. The problem of stability of the figures of equilibrium with respect to both ellipsoidal and arbitrary perturbations is also an open question. 5. Another interesting problem is that of obtaining an explicit solution (not in the form of a power series) for a homogeneous spheroid in curved space and the search for other possible figures of equilibrium in the spaces of constant curvature. 6. We recall that for the Maclaurin and Jacobi ellipsoids there exists a "dynamical" generalization, due to Dirichlet, where the self-gravitating liquid ellipsoid retains an ellipsoidal shape but changes the directions and dimensions of the semiaxes during its motion. It is unknown whether there exists such a dynamical generalization for inhomogeneous figures of equilibrium.

123

Inhomogeneous self-gravitating fluid

Remark A simple extension of Dirichlet's method, for example, to a family with confocal density stratification (see Sect. 3) is impossible since in Dirichlet's solution the same fluid particles move in ellipsoids forming at each instant of time a homothetic (and not confocal) foliation. 7. Another possible generalization involves finding the figures of equilibrium of a stratified gas cloud. In this case, in order to close the system (7), one uses, as a rule, thermodynamical equations (for applications to fluid mass dynamics see the review (Borisov et al. 2009) and references therein). In particular, one of the simplest assumptions used in Dyson (1968)isthat the temperature of the fluid/gas is constant along the entire volume T (r,) = T0 = const. In the case of an ideal gas this leads to a linear relation between density and pressure p = , = RT0 , where R is the universal gas constant. Assuming that = (), we obtain from (7)and (45) the system U () U = r 2 , = , r () r U = 4 G(). One of the unknowns in these equations is the function Z (r,) characterizing possible equilibrium figures of the cloud of an ideal gas. Remark To close the system, one can use, instead of the equation of state (45), the condition that the fluid flow be barotropic.
Acknowledgments The authors thank A. Albouy for useful advice and invaluable assistance in the course of work. The work of Alexey V. Borisov was carried out within the framework of the state assignment to the Udmurt State University "Regular and Chaotic Dynamics". The work of Ivan S. Mamaev was supported by the RFBR Grants 14-01-00395-a.

(45)

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