Disturbance of the Keplerian movement

The Keplerian Mouvement is often disturbed by a third body (mass m3): the problem becomes not-integrable.

The following notation will be taken:

$\ P (X) = (x-m_1) (x-m_2) (x-m_3) = x^3- \ sigma_1 x^2 + \ sigma_2 X - \ sigma_3$ ,

who puts forward the symmetry of role of the three masses.

Two essential cases arise to the astronomer:

the movement of the Moon; " inégalités" Jupiter Saturn.

the general problem is still prone active research in mathematics: cf scholarpedia Three_Body_Problem. In particular the problem of double star planets is far from being concluded with precision (no general analytical solution exists, this shown by Poincaré).

the problem of the disturbance of a satellite by not-rotundity of the Earth is treated in space Géodésie.

The Moon

The problem of the movement of the Moon has three centuries of existence, since the " head-ache" of Newton (1687, Principia), while passing through Laplace, Jacobi, Hill, Delaunay, Brown, POINCARE, Birkhoff, KAM (1954 +…),… Chapront.

In experiments, LR (lunar-laser-ranging) gives, since the demounting of the mirrors on the Moon, a rather large precision (on average, the Moon moves away from the Earth of 3.84cm per annum: cf Veillet). The empirical semi theory of Chapront (IMCCE) gives éphémérides except for the meter. The search of precision continues of course.

100 pages would not be enough to explain this movement of the Moon. But one can include/understand simplified problems.

the problem restricts 2+1

One calls problem restricts the plane problem, where Sun and Ground have a Keplerian Mouvement circular, and where the Moon is regarded as a negligible mass. In this case, in the turning reference frame, R', where the line Sun_G_ Earth is motionless, the problem is brought back to the movement in the potentiel_de_Jacobi, on the condition of adding the Force of Coriolis.

The figure of the potentiel_de_Jacobi is very easy to trace, like here:

in the problem with two bodies, the effective potential of Leibniz (1689) is:

the hole of potential energy of the center - m.Gm1/r and
L1^2/2mr^2 with L orbital kinetic moment representing the centrifugal barrier which prevents from falling in-on this side circle pericentric. So energy being preserved, the satellite cannot move away beyond from a circle apocentric It [[theorem of Bertrand], special with the Newtonian case, involves the closing of the orbit by degeneration between the period of the radial oscillations and the period of the angular oscillations].

There is another manner of noting " formellement" this potential:

energy-effective: - m m1 a/r + r^2/2a^2, maximum in R = has

The potential _de_Jacobi is simply the bicentric sum of 2 energies, one compared to the Sun (of mass m1) and the other compared to the second body, the Earth, of mass m2, for a distance ST = a:

$(m_1+m_2) \ cdot E_ {jacobi} (r_ 1,r _2): = M \ cdot J (r_ 1,r _2)$ and

$M \ cdot J = = - m_1 \ cdot (\ frac {has} {r_1} + \ frac {r_1^2} {2a^2}) - m_2 \ cdot (\ frac {has} {r_2} + \ frac {r_2^2} {2a^2})$

The symmetry of role of m1 and m2 becomes apparent: the problem is thus rather easy to include/understand in bicentric Coordonnée S: SM = r1 and EM= r2.

The representation of the figure (in scilab, cf for example Murray-Dermott, p81) immediately highlights 5 points:

three aligned points of balance of Euler (1769): L1 between Sun and Earth, close to the Earth, then quasi-symmetrical L2 of L1 (: EL1 = EL2), and L3 (almost SL3 = ST = a),
the famous points of Lagrange (1773), inevitably at the exact distance r1=r2=a (symmetry of role in the expression of the potentiel_de_Jacobi), is L4 ahead and L5 behind movement.

The points of Euler are not stable.

Those of Lagrange are LINEAIREMENT_stables if $\ sigma_1^2 < 27 \ sigma_2$ ()

The points L1 and L2 were largely studied for placements of satellites in orbit (Halation-orbit of Michalodimitrakis (1958), study resumed by Simo, etc). The points L4 and L5 for other types of satellites.

the problem of Hill: 1+1+1

m2/m1 << 1: moon example - Ground: 0.0123; Jupiter-Sun: 0.001; Earth_Sun: 0.000 003

The surface of Jacobi becomes of " pseudo-révolution" around the Sun. One can calculate the dimensions of Jacobi of the points (L1, L2), (L3, L4):

$\ 2J (L_1) = (9m_2/M) ^ {2/3} - 10m_2/3M$
$\ 2J (L_2) = (9m_2/M) ^ {2/3} - 14m_2/3M$
$\ 2J (L_3) = m_2/M$
$\ 2J (L_4) = - m_2/M$ .

Hill considered the case: m2/M negligible in front of (m2/M) ^ (2/3) (approximately 1/100 in the case Earth-Sun)

Then the problem is still simplified. So it is often given to study, because simple but nonsimplistic.

The system " in general is taken; cartésien" of origin E of axis Ey towards L2 and perpendicular Ex. The hill of Jacobi becomes symmetrical compared to y' Ey (what was already true), but also compared to x' Ex, and presents an infinite hole-well in E, of which the " séparatrice" is to say it quickly two symmetrical chains intersecting in L1 and L2; but attention the level lines will have " cornes" in L1 and L2. AND ATTENTION: space is " polarisé" by the Force of Coriolis, the trajectories will be deviated on the right. AND the system of equations presents a scaling in (m2/M) ^ (1/3) (help for interpretation!): there is no free parameter!

The Problème of Hill is thus attractive by its apparent simplicity: a simple symmetrical hill with rectilinear ridge and then a very deep hole in E (origin); more the " effect; dahu" caused by the Force of Coriolis.

1. If M (Moon) has an energy lower than L1 , it is trapped in the equipotential singular one passing by L1 and L2 and will not be able to escape from this lobe of Hill (let us recall TL1 = A. (m2/3M) ^ (1/3)).

BUT its trajectory will not be simple for as much! In this type of situation, the good idea is to take coordinates of Jacobi: Ground E like origin, EM = $\ vec {R}$ , G barycentre the Ground-Moon, and GS = $\ vec {R}$ . Note of history : the first to have had this fantastic idea of a point G without geometrical significance is Jeremiah Horrocks (1637!): immediately, the tables rodolphines were improved. Then Newton was caught the head with the problem of the influence of the Force of tide of the Sun on the movement of the Moon (note: one is often surprised by the fact that the Sun attracts more the Moon than the Earth does it! but, seen origin Ground, it is only the influence differential, known as of tide, which intervenes).

2. If M has an energy higher than that of L1 (we will still call it M, but it is not any more the Moon!), the situation becomes completely interesting: the " shock-rebond" , M against Ground, is completely surprising in its diversity:

One could think that it is about a simple rebound out of hairpin (a horseshoe is trop-trop broad in such a case): yes, the intuition is good: the trajectory (quasi-circular) coming from right-hand side, along X = B >0 remains quasi parallel to the Iso-Jacobi located in B/2 (approximately the excess of slope of the hill and the force of Coriolis causes a " drift transistor of Hall" effect; from quasi constant speed and thus M will turn IN FRONT OF the Earth (and not like an ordinary Keplerian movement around the Earth, essential difference). But this intuition overall right is to rehabilitate in the vicinity of separating which leads to the " corne" L2 (not singular of the hill of Jacobi and thus along concave arc L2L1 towards the hole: speed |dy/dt| brutally will decrease (approximately the particle falls into the well), the force of Coriolis will increase terribly and to make it go up, whipped , but the return will be " hoquetant" in the shapes of arches more or less trochoidal. To acquire intuition on such an example is very formative: it is advisable to trace is-even all these beautiful (?) trajectories, then to recall them in the reference frame galiléen (That gives a presentiment of complexity to come).

A trajectory quasi-circular compared to the Sun can take eccentricity: there is a " effect; sling-déviation" who in other circumstances is made profitable for probes. This effect is as well studied in the situation of the Anneau of the planets and the Satellite shepherd S.

Obviously, there does not exist trajectory coming from the right-hand side with x= B negative: the dahus M cannot run along the datum line, because of the Force of Coriolis. The dissymmetry of the figure is a example particularly striking effect of this force and often quoted of this fact.

- - -

All this was described within a conservative purely Hamiltonian framework. Reality is sometimes different bus from the friction effects (Effet Poynting-Robertson) can appear. And the problem becomes more severe to study.

The problem Jupiter-Saturn

outline:

One puts the problem in the usual form Ho + small disturbance $\ epsilon^2$ .H1, in notation of combined variables Action-angles.

Here Ho is simply the sum of the Hamiltonians H (Jupiter) + H (Saturn) in the heliocentric coordinates (O, sun like origin) of Poincaré or Delaunay. One can optimize still a little by cancelling the average of the disturbance by the method of Moyennisation of Bogoliubov-Krylov, modern version of that of Gauss.

This method then makes it possible to develop the éphémérides in power of $\ epsilon$ ; but one knows, since Poincaré that any situation is not possible: in the majority of the cases, the series are divergent; it is advisable to stop the series at the good place.

The critical problem is that of the small denominators: if an interior planet has its period equal to that of Jupiter. p/q p/q rationnel>1 with p weak, then there is Résonance. In the same way for an external planet time with p/q <1: the case of Saturn is close to 2:5). Sometimes these resonances are destabilizing, and sometimes stabilizing (cf Sicardy, course m2 on line).

The LARGE theoretical opening was that of theorem KAM (1954).

The calculators type lambda-calculation (calculation symbolic system) made enormous progress also, making it possible to calculate exhibitors of Stabilité of Lyapunov. Wisdom, then Laskar thus established that the telluric planets can be strongly disturbed by the planets joviennes, as well in their eccentricity, as their slope, even their obliqueness. The case of Pluto was re-examined recently and led to the déstatuer: the solar system has nothing any more but 8 planets.

Research is still very active on the rings and the " petits" body. Who more is, the precision led to look at the relativistic effects and all (or almost) is still to make Pioneer.

The general problem: double star planet

Being approached without a culture in rather large topology it would not know: cf scholarpedia: Three_Body_Problem.

What one can say is at least this:

there is transition between the preceding problems and the general problem unresolved from continuity: the variety of situations is simply larger.

the problem of the homographic configurations is basic and fundamental.

the theorem of the virial D ² I/dt ² = 4 E_cinétique +2 E_potentielle and the Inequality of Sundman > L ² /2I + (dI/dt) ²/2I play a great part.

the destiny eschatologic of the 3 masses is related to the possibility of extracting as much from energy than one wants it of a double pseudo-collision, the third mass while profiting to have a speed of escape, all this at kinetic time given: the case of the Problème to three bodies pythagorician is exemplary. Chazy, then Alexeev classified these destinies.

As POINCARE had strongly underlined, the periodic trajectories play a key role. Theorem KAM will come to confirm this brilliant intuition.

All periodic curiosities are explored with rapture. And the computers allow studies more and more.

the cases of Intégrabilité become more accessible via Zyglin and Ramis-Morals, which utilize the theory of the Groupe of differential Welshman.

To look at how becomes deformed the triangle S, E, M is capital (homothety, it, is easily taken into account, like had seen it Newton (1666)) (cf Bill Casselman). The Dynamique symbolic system becomes an important component, as well as the algebraic Analyze.

See too

Keplerian Movement
Problem with NR body + its excellent references
Theory of the disturbances
Inequality of Sundman

References

Murray and Dermott, Solar system Dynamics, ED CUP, 1999, ISBN 0-521-57597-4

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