ASTM001/MAS423 SOLAR SYSTEM

1. ASTM001/MAS423 SOLAR SYSTEM Carl Murray Lecture 9: Secular Perturbations

2. Introduction Although we know that we cannot solve the three-body problem, is there any way of obtaining an approximate analytical solution for the three- and N-body problems? The answer is “Yes” provided we make some approximations about the orbits of the masses involved. This is called secular perturbation theory. The subsequent solution can then be used to find the behaviour of test particles moving under the gravitational effect of the orbiting masses. This gives rise to the concept of proper elements. The starting point is to consider two masses orbiting a central mass with the assumptions that (i) the orbits of the masses do not intersect, (ii) the eccentricities and inclinations are small and (iii) no resonances are involved.

3. Secular Theory for Two Planets (1) We saw in the last lecture how the disturbing function is made up of terms that can be classified as secular, resonant or short-period. Secular terms are those that are independent of the mean longitudes. They can only arise from terms with zeroth order arguments; to second order in e and I we have: There are no indirect terms! where j=0

4. Secular Theory for Two Planets (2) This gives, for the basic direct part of the disturbing function: where subscript 1 is for the inner body and subscript 2 for the outer body. The individual disturbing functions are: We can write the Laplace coefficients as:

5. Secular Theory for Two Planets (3) From Kepler’s Third Law: Hence:

6. Secular Theory for Two Planets (4) The perturbation equations for each planet can be combined by writing: external perturber internal perturber Note that:

7. Secular Theory for Two Planets (5) We can write the 8 coefficients as the elements of two matrices: To lowest order in the eccentricities and inclinations the equations of motion are now: Note: (i) the symmetrical nature of the equations and (ii) the singularities that occur for circular or planar orbits.

8. Secular Theory for Two Planets (6) We can overcome the singularities if we introduce new variables: In these variables the (common) disturbing function is now: and we can transform the equations of motion using:

9. Secular Theory for Two Planets (7) We have: The equations of motion become: Explicitly we have:

10. Secular Theory for Two Planets (8) Note that the equations are now separable. We have a system of coupled, first order, linear differential equations with constant coefficients. Their solution can be reduced to an eigenvalue problem and can be written as: eigenvalues of matrix A corresponding eigenvectors eigenvalues of matrix B corresponding eigenvectors The b and g are phases; they and the amplitudes are determined from starting conditions.

11. Secular Theory for Two Planets (9) We have derived the classic Laplace-Lagrange solution for secular perturbations of (two) interacting bodies orbiting a central mass. Note that the characteristic equation for the matrix B is: where This reduces to: This has the obvious root f = 0 and there is a singularity in the solution. This is where the inclination-node solution differs from the eccentricity-pericentre one. Note too that the Laplace-Lagrange solution is independent of mean longitudes (naturally, because we used only secular terms!) and therefore we have no information on the positions of the planets.

12. Secular Theory for Two Planets (10) Our solution implies that the orbits of the two planets are stable for all time. However, it is important to remember the assumptions under which the solution was derived (non-intersection of orbits, small e and Iand no resonances). Now consider the application of the above theory to the motion of Jupiter and Saturn orbiting the Sun.

13. Jupiter and Saturn (1) Using the following values: We obtain:

14. Jupiter and Saturn (2) The matrices are now: With characteristic equations:

15. Jupiter and Saturn (3) The resulting eigenfrequencies are: The corresponding eigenvectors are: with phases:

16. Jupiter and Saturn (4) The full solution gives: Note the coupling between Jupiter and Saturn – when Jupiter is at maximum eccentricity Saturn is at minimum eccentricity, etc. Remember that this solution assumes that there are no resonant terms between Jupiter and Saturn.

17. Free and Forced Elements (1) The disturbing function for a test particle (of negligible mass) perturbed by two planets is: where:

18. Free and Forced Elements (2) Using the same substitutions as before, the equations of motion are: Where we already know the solutions for Jupiter and Saturn. Substituting these we obtain:

19. Free and Forced Elements (3) Taking another derivative of each equation gives: Where the four frequencies are given by:

20. Free and Forced Elements (4) The solutions take the form: where and the two phases are constants determined from the boundary conditions. We also have: Note that the forced elements are time varying and that the free elements are fixed by location.

21. Free and Forced Elements (5) Geometrically we can represent each solution as circular motion around a centre that is moving on a predetermined path: We can illustrate this with a numerical integration of particles started with the same a, the same free e (and I) and randomised free pericentres (and nodes).

22. Free and Forced Elements (6) Recall that the values of the forced eccentricity and inclination will depend on the equations: Here A and B are just functions of the semi-major axes and the perturbing masses. Also the eigenfrequencies are just functions of semi-major axes. What happens when or ??

23. Free and Forced Elements (7) These divisors can be zero at certain values of semi-major axis of the particle. At such locations the free precession (or regression) rate of the pericentre (or node) is equal to one of the eigenfrequencies of the two-planet system. The theory suggests that at these locations there will be singularities in the behaviour of the eccentricity or inclination. What happens to the forced eccentricity (and inclination) as we approach the orbit of a perturber given that the Laplace coefficients   as a  1 In fact (see book for details): Where the subscript l denotes the value for the perturber.

24. Jupiter, Saturn + Particle (1) Consider the perturbations on a test particle (with arbitrary semi-major axis) due to the secular effect of Jupiter and Saturn. The first step is to calculate the secular solution for Jupiter and Saturn. This gives the following eigenfrequencies: Then we calculate A as a function of semi-major; A can be thought of as the free precession of the test particle at each semi-major axis arising from the “ring” terms in the expansion of the disturbing function. Since the masses and semi-major axes are fixed, the value of A is a constant for any semi-major axis of a test particle. At the orbit of a perturber (Jupiter or Saturn) we have singularities:

25. Jupiter, Saturn + Particle (2)

26. Jupiter, Saturn + Particle (3) Therefore we expect to see singularities in the values of the forced eccentricity and inclination where the curve of A intersects the eigenfrequencies. This is what happens:

27. Jupiter, Saturn + Particle (4) We have already shown that the forced eccentricity (and inclination) tends to the value of the eccentricity (and inclination) of the perturber at the orbit of the perturber. Given that the eccentricity (and inclination) of the perturber change because of the secular interactions between the two planets, the previous curves will fluctuate but the singularities will stay at the same locations. This work suggests that particles with the same semi-major axis and same free eccentricity (and inclination) will have the same forced eccentricity (and inclination) at any instant. This means that in (h,k) space they will move along circles with the same radius (the free eccentricity), with a similar result for inclination.

28. Jupiter, Saturn + Particle (5) We can see this in the results of a numerical simulation of particles: Note how the circle is maintained over the 30,000 years of the integration, but the centre shifts as the planets interact.

29. Jupiter, Saturn + Particle (6)

30. Jupiter, Saturn + Particle (7)

31. General Secular Perturbations (1) The theory of two perturbing planets can be extended to N planets orbiting a central mass. It is also possible to include other effects such as oblateness of the central body (this removes the degeneracy in the eigenfrequencies). The general secular solution is: cf. Two planets:

32. General Secular Perturbations (2) This means that we can now derive a secular theory for the long-term behaviour of the planets in the solar system. This was first done by Laplace in the 18th century. However, he had to make some allowance for the 5:2 near-resonance between Jupiter and Saturn (la grande inegalité). Recall that the secular theory we have derived makes no allowance for resonances (or near-resonances) between planets. One of the classic secular theories from the last century is by Brouwer & van Woerkom (1950). Their inclusion of the Jupiter-Saturn near 5:2 resonance led to the addition of several eigenfrequencies.

33. General Secular Perturbations (3) Eigenfrequencies from Brouwer & van Woerkom’s theory: We can use their theory to plot the evolution of planetary elements over long time periods:

34. General Secular Perturbations (4)

35. General Secular Perturbations (5)

36. General Secular Perturbations (6)

37. General Secular Perturbations (7)

38. General Secular Perturbations (8) 8-planet model: 2-planet model:

39. Inner Solar System

40. Outer Solar System

41. General Free + Forced Elements (1) We can also extend the theory of free and forced elements to include N planets (and the effects of oblateness):

42. General Free + Forced Elements (2) Again we can expect singularities in the forced elements where A and B match the eigenfrequencies of the system. Note the three intersections that occur in the asteroid belt.

43. Hirayama Families & Dust Bands (1) We have shown that the eccentricity (and inclination) of a dust particle can be thought of as being the vector sum of two distinct parts: The proper or free eccentricity that reflects the “inherent” eccentricity of the particle. The forced eccentricity that is a function of where the particle is relative to the perturbing masses. In 1918 Hirayama derived free elements for the known population of asteroids and showed that there were specific concentrations of asteroids:

44. Hirayama Families & Dust Bands (2) Osculating Proper

45. Hirayama Families & Dust Bands (3) By removing the short-period perturbations due to the planets, Hirayama had identified various asteroid families which are believed to be the collision products of impact events. Note that the families are clusterings in semi-major axis, eccentricity and inclination.

46. Hirayama Families & Dust Bands (4) If we plot the (h,k) and (p,q) of the asteroid families we clearly see the circles associated with asteroids having the same free and forced eccentricity (and inclination) but randomised free pericentres (and nodes).

47. Hirayama Families & Dust Bands (5)

48. Hirayama Families & Dust Bands (6) In 1984 the IRAS satellite discovered that the zodiacal cloud had distinct bands:

49. Hirayama Families & Dust Bands (7) The IRAS dust bands were subsequently explained as being the dust formed by the collisions that produced the major asteroid families Themis, Eos and Koronis. Background flux at 25mm. Smoothed residual data (solid line) with Themis, Eos and Koronis dust model (dashed line).

50. Hirayama Families & Dust Bands (8) The cumulative effect of the dust at the same eccentricity and inclination but different pericentres and nodes leads to the appearance of bands.