Theory of the figures of balance

The theory of the figures of balance considered here results from many studies treating of the problem of the form of balance of the Earth, by supposing that this one is caused by the only force of Pesanteur, other than internal forces of cohesion or electric and magnetic forces. The theory presented rests entirely on the concept of Equipotential surface, and thus requires the existence of a potential of gravity inside the body which one studies. That excludes the presence from a differential rotation except if this one corresponds to a differential rotation in cylindrical layers coaxiales.
Although this problem is old of more than three centuries ( cf ellipsoidal Modèle of the Earth), it does not continue any less to be a current problem which has interesting applications in geodesy and geophysics. It is firmly enraciné in the general theory of the cosmic figures of body in rotation, which constitutes itself a field of active research in astrophysics and planetary physics. However, for cosmic bodies in slow rotation - as it is the case of the Earth - and in so far as we can admit that there is hydrostatic balance, the theory of the figures seems to have reached an acceptable level of completion. The study of the hydrostatic figure of the Earth did not only exert one deep influence on the birth and the subsequent development of the physical geodesy and the Géophysique, but it also largely contributed to the foundations of the physical of Newton, of the Hydrostatique, the Analytical mechanics , of the analyzes mathematical and of the Mathematical physics during XVIIIe and XIXe centuries, and is associated with the majority with the names with the mathematicians and physicists the most distinguished from this time.

Fundamental equations

To study the hydrostatic figure of balance of the Earth, one places oneself in the tallies general of the mechanics of the continuous mediums. The basic equations of a continuum are, in eulérienne formulation:

the equation of continuity , which expresses the conservation of the mass:

∂ρ/∂t + Σ_k=1→3 ∂ (ρ v_k)/∂x_k = 0,

the equation of motion , which expresses the conservation of the momentum, or impusion:

ρ ∂v_i/∂t + ρ Σ _k=1→3 v_k ∂v_i/∂x_k = f_i + Σ_k=1→3 ∂T_ik/∂x_k.

In these equations, ρ (x₁, x₂, x₃) indicates the density in a space point P located by its Cartesian coordinates (x₁, x₂, x₃), (v₁, v₂, v₃) the field speeds at the same point, (f₁, f₂, f₃) the voluminal field of force and T_ik with I, k=1,2,3 the tensor of the tensions in P. In the usual circumstances, that we will suppose realized here, the equation of rotation who expresses the conservation of the quantity of rotation, or kinetic moment, implies that the tensor of the tensions is symmetrical

T_ik = T_ki

and is thus reduced to six independent components instead of nine: T₁₁, T₂₂, T₃₃, T₁₂ = T₂₁, T₁₃ = T₃₁, T₂₃ = T₃₂.

Constitutive relation

In order to return the problem of a manageable figure of balance, it is necessary to give oneself a rheological law (or constitutive relation ) making it possible to express the field of the tensions in terms of fields derived from the field speeds or displacements. In general, it is supposed that the rheological behavior with the long-term of the studied continuous medium is that of a true fluid . Then, the tensor of the T_ij tensions is isotropic and can describe by means of a scalar field of pressure p, that is to say

T_ij = - p δ_ij,

where δ_ij represents the symbol of substitution of Kronecker: δ_ij is worth 1 if I = J and 0 if I ≠ J.

Permanent rotator

Definition of a permanent rotator

One can still more simplify the problem by supposing that the figure de' balance constitutes a permanent rotatory . One defines a body in permanent rotation as follows:

it is insulated in space and turns around a fixed axis;
its movement is stationary in a system of inertia, and the density of each individual mass point remains constant during the movement;
one can neglect the dissipative effects, in particular the frictions;
no force electromagnetic acts on the body.

Field speed of a permanent rotator

Let us note that one did not specify the angular velocity yet ω. Let us admit now that the body turns around the axis Ox₃ and take the origin O of the system of axes in the center of mass of the body. Let us consider cylindrical coordinates (H, λ, Z) defined in terms of the Cartesian coordinates (x₁, x₂, x₃) by the law of transformation

H = (x₁ ² + x₂ ²) ^1/2, λ = arctg (x₂/x₁), Z = x₃.

Thus, in a system of inertia, the components speed take the form

v_h = 0, v_λ = ω H, v_z = 0.

Equation of continuity for a permanent rotator

As the density of a mass point pertaining to a permanent rotator does not vary along its trajectory, we have

dρ/dt = ∂ρ/∂t + v_j ∂ρ/∂x_j = 0.

The equation of continuity thus implies that the movement is indivergentiel , i.e.

∂v_j/∂x_j = 0.

Symmetry of the field speed of a permanent rotator

In cylindrical coordinates, this equation becomes

∂ω/∂λ = 0,

what implies that a permanent rotation is characterized by a field speed which does not depend on the variable λ , that is to say

ω = ω (H, Z)

A particular case of a permanent rotator is a rigid rotatory , for which the angular velocity is constant:

ω = constant.

The body concerned rotates then in block: it is said that the corresponding figure of balance is in relative balance .

However, the assumption of a permanent rotation does not require only dρ/dt = 0, but also that ∂ρ/∂t = 0, from where one draws by means of the equation from continuity that

v_j ∂ρ/∂x_j = 0.

This last relation shows that for a permanent rotation the threads of current are orthogonal with the local gradient of density. In the cylindrical frame of reference that we adopted, this relation is written

ω ∂ρ/∂λ = 0.

Symmetry of the distribution of mass of a permanent rotator

By supposing that the body turns indeed (ω ≠ 0) like a permanent rotator compared to a reference frame of inertia, we thus find that

ρ = ρ (H, Z).

One from of deduced that the distribution of density in a permanent rotator has a axial symmetry , and the same conclusion is essential (see above) for the field angular velocity. Thus, this field ω = ω (H, Z) does not have a swirl in the Ox₁x₂ plan, therefore

x₂ ∂ω/∂x₁ - x₁ ∂ω/∂x₂ = 0.

Equation of motion of a permanent rotator

In terms of Cartesian coordinates, the field speed can thus be written

v₁ = - ω x₂, v₂ = ω x₁, v₃ = 0.

Under the terms of the relations ∂v_j/∂x_j = 0 and v_j ∂ρ/∂x_j = 0, the quantity ∂ (ρ v_j v_i)/∂x_j becomes simply ρ v_j ∂v_i/∂x_j. Consequently, the equation of motion takes the particular form

- ρ ω ² (x_i - δ_i3 x₃) = f_i - ∂p/∂x_i.

Potential gravific of a permanent rotator

According to the definition of a permanent rotator stated higher, it is obvious that the voluminal field of force f_i is simply the field of force generated by the gravitative autoattraction, i.e.

f_i = ρ X_i.

It is well-known that the specific gravific force X_i in a point P is provided by the expression

X_i (P) = G ∫_B ρ (Q) d^-2 (P, Q) e_i (P, Q) dτ (Q), (i=1,2,3)

where D (P, Q) represents the distance between the point attracted P and any not attracting Q of the body B. the e_i quantity indicates the IE component of an unit vector carried by the segment of right-hand side which joint P with Q, and directed P with Q. the element of volume in Q is indicated by dτ (Q). One knows just as the voluminal force X_i derives in each point from space from a scalar function V, called “potential of gravific autoattraction” or, more simply, potential gravific . Thus,

X_i = - ∂V/∂x_i, (i=1,2,3).

We define this potential V here, in agreement with the convention generally adopted by the physicists, as the potential energy per unit of mass of an material element located at the point P. This energy potential results from the gravitational interaction of the material element out of P with all the other material elements composing the body B, so that

V (P) = - G ∫_B ρ (Q) d^-1 (P, Q) dτ (Q).

According to this definition, the potential gravific is always negative , and corresponds physically to a energy binding . It should be noticed that the astronomers and the geodesists, and also of many geophysicists, adopt a different definition for the potential gravific, namely they compare it to the work which an material element of unit mass achieves in the field of gravific force X_i while passing from a point located ad infinitum at the point P. This work represents the potential energy per unit of changed mass of sign, and is thus positive. The convention of the physicists adopted here has the advantage, by retaining the direct correspondence between potential and potential energy, to make so that a mass point always tends to drive towards areas where the potential is lower. Constant G above is the Constante of gravitation, whose currently recommended numerical value is worth, in the International Système ,

G = (6.6742 ± 0.001) X 10^-11 m ³ s^-2 kg^-1.

Symmetry of the figures of balance

When one studies the theory of the figures for cosmic bodies like the Earth or planets, i.e. for heterogeneous bodies in rotation around a polar axis, the main difficulty comes owing to the fact that one cannot admit a priori a form determined for a Equipotential surface, as in the case from the theory from the figures from balance from the homogeneous masses (for example, the spheroid of Maclaurin or the ellipsoidal of Jacobi ) considered in astronomy or from the ellipsoidal from reference discussed in physical geodesy. Consequently, the examination of the elements of symmetry of a figure of balance makes it possible to simplify the theory appréciablement. In the continuation of this article, we will consider the case of the Earth, but the general theory also applies to the other planetary bodies like with simple stars.

First of all, we suppose that the Earth is in permanent rotation, thus excluding from our study the phenomena of tide or the electromagnetic phenomena, for example. One saw higher than under these conditions, the angular field velocity and the field of density have both a symmetry of revolution. It follows that the potential gravific of a permanent rotator has him-also to it even axial symmetry. This property of symmetry of V (P) = - G ∫_B ρ (Q) d^-1 (P, Q) dτ (Q) can be established mathematically as follows: are (H, λ, Z) and (h', λ', z') the cylindrical coordinates of the point potentié P and a point potentiant unspecified Q, respectively. The corresponding Cartesian coordinates of P and Q, i.e. (x₁, x₂, x₃) respectively (x'₁, x'₂, x'₃), are expressed by means of the relations

x₁ = H cos λ, x₂ = H sin λ, x₃ = Z,
x'₁ = h' cos λ', x'₂ = h' sin λ', x'₃ = z'.

The element of volume in Q is dτ (Q) = dx'₁ dx'₂ dx'₃ = h' dh' dz' dλ', and outdistances it between P and Q is

D (P, Q) = + h' ² + (z' - Z) ² - 2:00 h' cos (λ - λ') ^1/2.

Like ρ (Q) = ρ (h', z') does not depend on the angle λ', integration on λ' in the expression of V (P) can be replaced by an integration on the difference λ'-λ with the same limits in integration 0 and 2π that for λ'. This way, the variable λ disappears from the integral, and

V = V (H, Z)

together with

ρ = ρ (H, Z)

and

ω = ω (H, Z).

Principle of symmetry of Pierre Curie

In fact, the axial symmetry of V could have been deduced from the axial symmetry of ρ while making use of the principle of symmetry established by the French physicist Pierre Curie (1859 - 1906). This principle can state in a concise way as follows: Any effect produced by a given cause must have at least the same elements of symmetry but can be more symmetrical than the cause. Any dissymmetry met in a given effect must also exist in the cause which produced the effect. In corollary, if there exists a relation of reciprocity between a cause and an effect, in the sense that the cause can just as easily be regarded as the effect, and the effect as the cause, both - causes and effect - must have exactly the same elements of symmetry . An example of this case is provided by a relation between the tensions and the deformations, namely what we higher called a “constitutive relation”. It seems that one did not find yet until now of experiment nor of observation referring to phenomena macrophysic which contradicts the principle of Curie, and this last should be regarded as a universal law of Nature as well as the laws of thermodynamics, for example. We will adopt this point of view to deduce certain properties here from the figures of balance.

First of all, we note that the gravific attraction force is central, and thus has more the high degree of symmetry, that of the sphere. The principle of Curie requires whereas the arrangement of the masses produces by the only action of gravific attraction must correspond to spherical symmetry: any less symmetrical configuration is excluded if one wants to have balance. A mathematical proof owing to the fact that the sphere is the single figure of possible balance of a fluid body isolated at rest was given by Lyapunov in 1884.

Then, let us consider the combined action of a gravific attraction force and a force axifuge. The latter introduces a certain asymmetry compared to the sphere by the fact that the force axifuge has the symmetry of a cylinder of cross-section. The principle of Curie thus requires that the distribution of balance of the masses has same axial symmetry, since there exists a relation of reciprocity between the distribution of density and the potential. It is the case studied higher in mathematical terms.

Equation of state: barotropes and baroclines

A particular point must hold all our attention at this stage. Indeed, in order to supplement the formulation of the problem, we must include a suitable equation of state. In general form, this one can be written

p = p (ρ, T, φ₁, φ₂,…, φ_N).

The letter p indicates the pressure, ρ is the density, T is the temperature, and the whole of the variables φ₁, φ₂,…, φ_N describes the chemical composition and mineralogical. By definition, a baroclinic structure (or, simply, a barocline ) is a system for which a physical relation of this form exists. However, in certain special circumstances which are roughly filled in the case of the interior of the Earth, the pressure can simply be connected to the density, namely

p = p (ρ).

If such is the case, it will be said that it is about a barotropic equation of state and the corresponding structure is a barotrope . At first approximation, we can regard the Earth as a barotrope.

The fundamental distinction between models barotropic and model baroclinic is in their respective stratifications. It is clear that surfaces of equal density (isopycnic surfaces ) and surfaces of equal pressure (isobar surfaces ) coincide in a barotrope. On the other hand, in the baroclines surfaces of equal density cut surfaces of equal pressure, unless very specific conditions are not met.

Taking account of the axial symmetry of ρ and V, the equation of motion of a permanent rotator becomes in cylindrical coordinates

ρ^-1∂p/∂h = - ∂V/∂h + ω ² H, ρ^-1∂p/∂z = - ∂V/∂z, (ρ H) ^-1∂p/∂λ = 0.

The fact that ∂p/∂λ = 0 establishes the symmetry of revolution of the field of pressure, i.e.

p = p (H, Z).

Relative hydrostatic balance

One defines the components of the Pesanteur in cylindrical coordinates by the relations

g_h = - ∂V/∂h + ω ² H, g_z = - ∂V/∂z, g_λ = 0.

Consequently, in a body in permanent rotation, gravity is orthogonal everywhere with the isobars, as one notes it while remembering the definition of the gradient. Let us note that this property is very general: it is valid just as easily for the baroclines that for the barotropes, and that it is valid for a uniform rotation as for a differential rotation, in so far as the angular field velocity preserves an axial symmetry. In general, one defines the potential axifuge like

Z = - ½ ω ² H ²,

but it is obvious that this definition is valid only for a uniform rotation . If such is the case, we can employ a system of axes fixed in the body and turning with him. If each particle of the body is at rest compared to this reference frame which is not inertia, it is said that the body is in relative hydrostatic balance or, more simply, in hydrostatic balance. Generally, in the geodetic theories or geophysics, one admits that such is the case for the Earth.

On the contrary, if the angular velocity is not constant from one point to another, it can exist only a hydrodynamic balance . A differential rotation of this type probably plays a big role in giant stars and planets, but seems excluded in telluric planets because of a too strong viscous or electromagnetic coupling between the various layers. The studies of the general figures of balance often give the most importance to zonal rotation, than one names meridian circulation or currents of von Zeipel, or currents of Eddington, or currents of Jardetzky. However, the study of the currents of convection generally does not form part of the theory of the figures of balance.

Theorem of Wavre-Poincaré

The relations above defining the components of gravity show that a potential axifuge Z, and thus a Equipotential surface

U = V + Z = const

can exist if and only if ω does not depend on Z, i.e. if the angular velocity is constant on the cylindrical surfaces centered on the axis of rotation, in other words if ω = ω (H). In this case more general than uniform rotation, one defines the potential axifuge by

Z = - ∫_0→h ω ² (h') h' dh'.

We note, in particular, that for a uniform rotation, the preceding relation is reduced well to the formula Z = - ½ ω ² H ², as it should be.

An important conclusion that one can draw from the existence of a potential of rotation, and thus of the existence of a potential of gravity, is

U (H, Z) = V (H, Z) - ∫_0→h ω ² (h') h' dh',

so that

g_h (H, Z) = - ∂U/∂h, g_z (H, Z) = - ∂U/∂z, g_λ (H, Z) = 0.

One often gives to this conclusion the name of theorem of Wavre-Poincaré , and one can state it in the following form: For a permanent rotator, any of the properties which follow involve the three others automatically, namely

the angular velocity is constant on cylinders centered on the axis of rotation;
gravity derives from a potential;
gravity is normal on the surfaces of equal density;
surfaces of equal density coincide with surfaces of equal pressure. Indeed, according to the equations for the gradient of the pressure and the components of gravity provided higher, one can write for the total differential

dp = (∂p/∂h) dh + (∂p/∂z) dz + (∂p/dλ) dλ

the following equation:

dp = ρ (g_h dh + g_z dz),

thus

= - ρ^-1 dp.

An arbitrary displacement on an equipotential surface provides = 0, thus implying that dp = 0. Isobar surfaces thus coincide with the p and equipotential surfaces, = p (U) or U = U (p). We thus find because of = - ρ^-1 dp that the density is also constant on equipotential surface. In conclusion, the isobars, the isopycniques ones and the equipotential surfaces all coincide if there exists a potential of gravity, and forces it per unit of mass due to gravity is normal on these surfaces. In particular, it is normal with the isopycniques ones. Reciprocally, let us consider a body for which the isobars and the isopycniques ones coincide. One can then define a function U (p) = - ∫ dp/ρ (p), in terms of which the relation dp = (∂p/∂h) dh + (∂p/∂z) dz + (∂p/dλ) dλ becomes

= - g_h dh - g_z dz,

showing that is an exact total differential. Consequently, the relations g_h (H, Z) = - ∂U/∂h, g_z (H, Z) = - ∂U/∂z, g_λ (H, Z) = 0 must apply, and the field of gravity derives from a potential. Finally, let us suppose that gravity is normal everywhere on the surfaces of equal density, that is to say G x' ∇ ρ = 0. Like g' = - ∇ U = ρ^-1 ∇ p, the coincidence of isobar and isopycnic surfaces is established, and all the statements of the theorem of Wavre-Poincaré are shown.

Baroclines and pseudo-barotropes

It is important to realize that the condition ω = ω (H), or the equivalent condition ∂ω/∂z = 0, does not require any preliminary knowledge of the equation of state. According to the theorem of Wavre-Poincaré, the isopycniques ones of a barocline will thus coincide with its isobars if ∂ω/∂z = 0. However, it is obvious that of such baroclines particular should be called in a more suitable way of the pseudo-barotropes , because they share the majority of their properties with the true barotropes, and are distinguished clearly from the true baroclines.

This observation enables us to return to the considerations of symmetry. First of all, the principle of Curie teaches us that the barotropes are at least as symmetrical as the true baroclines, but can be more symmetrical.

Theorem of Lichtenstein

The principle of Curie (without broken symmetry) allows inférer another property of very important symmetry of the figures of balance: any rotatory permanent barotropic or pseudo-barotropique must have an additional element of symmetry, namely an equatorial plan, i.e. a normal plan with the axis of rotation which passes by the center of mass . This statement constitutes the theorem of Lichtenstein . But this physical proof of the theorem of Lichtenstein in all its general information would require the unicity of solution, and the rigorous mathematical demonstration is not commonplace whole.

The case of a true barocline is somewhat more complicated, but Dive showed in 1930 which a barocline in differential rotation is symmetrical compared to an equatorial plan if the angular velocity is everywhere a bi-univocal function of the density and distance to the axis of rotation.

Theory of the field interns of Clairaut, Laplace and Lyapunov

Let us regard the Earth as a barotrope turning at a uniform angular velocity ω around the axis Ox₃. We wish to find his form of balance for a distribution of density ρ specified along a tilted direction of a certain angle compared to the axis of rotation. Under the term “appears of balance” we do not include/understand only the shape of external surface and the external field of gravity, but also the shape of an unspecified internal layer, i.e. the form of all isopycnic, isobar and equipotential surfaces which all coincide in the case studied here. We will take the origin O of the frame of reference to the center of mass of the Earth, and we will indicate by R the radius vector which joint the origin at an arbitrary point located on an equipotential surface given

U (R, θ) = V (R, θ) + Z (R, θ) = constant.

As it is usual, the Greek letter θ ( theta ) indicates the geocentric colatitude, and R = | R | is the distance to the center of mass. In spherical terms of coordinates R, θ, λ the potential axifuge is written

Z (R, θ) = - ½ ω ² R ² sin ² θ

Z (R, θ) = - ⅓ ω ² R ² - P₂ (cos θ).

Functions of figure

For an unspecified barotrope in slow rotation, and in particular for the Ground models, the equipotential surfaces differ only slightly from the spherical form. It is thus natural, taking into account the elements of symmetry which these surfaces must have, to seek the equation describing the surface of an unspecified layer in the form

R (S, θ) = S + Σ_n=0→∞ s_2n (S) P_2n (cos θ).

The variable S indicates a judiciously selected parameter having to make it possible to recognize of which layer it acts - to simplify, one will say that S represents the name of the layer considered. Thus, one can choose S as equatorial ray has, or like polar ray C, of the layer in question. In the continuation, we choose S like average ray equivolumetric of the layer, in other words, as ray of the sphere which has same volume as that contained under the layer.

This way, the problem to find the form of balance is brought back to the problem to determine an infinite whole of functions s_2n (S), n=0,1,2,3,… that one calls the “ functions of figure ”. In order to be able to use the preceding formula in practice, it is necessary that the order of magnitude of the functions of figure decreases sufficiently quickly with order N to ensure a reasonably fast convergence of the series appearing in the member of right-hand side. For the Ground models in particular, such is well the case as it will be shown further.

If we truncate the series with n=1, we are in the presence of the first order approximation which corresponds to the traditional case of the theory of Clairaut . If we truncate the series with n=2, we obtain the approximation of the second order which corresponds to the theory of Darwin-in Sitter . For the Earth, the approximation of order 2 corresponds to neglect effects of approximately 10^-5; as the average radius of the Earth is of 6371 km, one thus makes errors which can amount typically with a few meters. Such errors are incompatible with the precision of geodetic measurements reached at present, which east provide the ray of the Earth with a margin of a few centimetres. Since the end of the year 1970, various authors considered in the case of the hydrostatic figure of the Earth of the approximations of order 3 and provided the suitable formulas of them. Actually, in so far as one does not wish to make effective calculations, it is hardly more difficult to develop the theory of the figures of balance with an general order n=N than to limit itself to the order n=2 or n=3, for example.

Theory of Clairaut-Radau

If we neglect the terms out of O (m ²) or smaller, the equations of figure are reduced to the only equation

F₂ = - s₂ S₀ + S₂ + T₂ - (1/3) m = 0

and flatness becomes

F = - (3/2) s₂.

Combining both, one obtains the equation intégro-differential of Clairaut

F S₀ + (3/2) (S₂ + T₂) - (1/2) m = 0.

The quantities S₂ and T₂ should be evaluated only in the order O (m), and S₀ with the order O (1). Consequently, with X = s/s₁ and δ = ρ/ρ, one finds

S₂ (X) = - (2/5) x^-5 ∫_y=0→x δ (there) D (y⁵ F),

T₂ (X) = - (2/5) ∫_y=x→1 δ (there) df.

Differential equation of Clairaut

In this form, the equation of Clairaut is a linear equation intégro-differential for the flatness of the internal layers of the Earth. It can be solved repeatedly by the general method indicated higher for the functions of figure. However, the traditional approach consists to derive the equation from Clairaut in the form intégro-differential compared to X and to eliminate the quantities m and T₂ by making use of the two last relations, thus obtaining - (5/2) S₂ = (F - ⅓ X df/dx) S₀, then to multiply this intermediate result by x⁵ and to derive once again compared to X. One then obtains the differential equation of Clairaut

D ² f/dx ² + 6 δ x^-1 S₀^-1 df/dx + 6 x^-2 (δ S₀^-1 - 1) F = 0.

One can numerically integrate this last ordinary differential equation while imposing that F remains finished in x=0 (in the center) and that F satisfies in x=1 (on surface) in the condition

df (1) /dx + 2 F (1) = (5/2) Mr.

One leads to this condition by multiplying the equation intégro-differential of Clairaut above by x⁵, by posing X then = 1 and by noting that S₀ (1) = 1.

Differential equation of Clairaut-Radau

Throughout the XIXe century, in the absence of the abundant data later by the seismology, the astronomers tried to determine the density of the internal layers of the Earth by imposing two constraints on various functions of test ρ = ρ (X): (1) the function of test, by integration on volume, was to provide the actual value M of the Masse of the Earth; (2) by integrating the equation of Clairaut, the function of test was to provide the actual value F of the geometrical flatness on the surface. However, before the advent of the computers and, especially, the personal microcomputers in second half of the XXe century, the solution of the differential equation of Clairaut was obtained by a numerical integration which was to be done with the hand and which was extremely tiresome.

For this reason, elegant method of solution suggested in 1885 by the French astronomer of Prussian origin Rodolphe Radau (1835 - 1911), which enormously simplifies the integration of the equation of Clairaut, became method of standard resolution and remains until now of a great value in theoretical work having recourse to the flatness calculated with the first order.

The method of Radau consists in transforming the equation of Clairaut by introducing the function without dimension known as “parameter (or variable) of Radau”

η (X) = X f^-1 df/dx,

by means of which one obtains the equation of Clairaut-Radau

D (1+η) ^1/2 S₀/dx = 5 x⁴ S₀ F (η),

with

F (η) = (1 + η/2 - η ² /10) (1+η) ^-1/2.

Equation of Radau

Theory of Darwin-in Sitter

Base of the theory of the figures

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