Theory of mean motion resonances.

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1 Theory of mean motion resonances. Mean motion resonances are ubiquitous in space. They can be found between planets and asteroids, planets and rings in gaseous disks or satellites and planetary rings. They are also common in galaxies where star formation can be enhanced and may proceed either as a starburst or continuously over a long time period. In our solar system a macroscopic effect of MM resonances is represented by Kirkwood gaps in the orbital distribution of Main belt asteroids. In Figure the semimajor axis distribution of asteroids in the Main Belt is shown. Gaps of low density (Kirkwood gaps) are observed in correspondence to commensurabilities between the orbital period of Jupiter and that of the asteroid. This commensurability leads to a MM resonance configuration and, in some cases, to chaotic behaviour and escape from the Solar System. Trojans are also in a MM resonance (: between the orbital periods of the asteroid and Jupiter). There are about 2000 known Trojans of Juptier, 4 for Neptune and 3 for Mars.

2 The Kuiper Belt extends beyond the orbit of Neptune and it is populated by large bodies like Pluto and its satellite Charon. Pluto is in a 2:3 resonance with Neptune like other smaller bodies named Plutinos in the Kuiper Belt. Planets embedded in a disk excites the gas at MM resonances (Lindblad resonances) and exchange angular momentum with the disk migrating. For planets with a mass of a few Earth masses outer resonances win in terms of torque and the planet drifts towards the star. Largest planets (Jovian type) create a gap and evolve following the viscous infall of the disk towards the star. This evolution explains the large number of extrasolar planets on orbits very close to their parent star.

3 Hamiltonian of the 2-body problem. The equation of motion of the 2-body problem, represented by a mass m and a mass m 2 under the mutual gravitational attraction, is d 2 r dt 2 = G m m 2 r r 3 We now define =G m m 2 and the reduced mass = m m 2 m m 2 system is p= v. The Hamiltonian of the 2-body is so that the momentum of the H= 2 p 2 r and the Hamilton equations are ṙ= p H ṗ= r H We can make a coordinate change to Delunay canonical variables which are defined from the orbital elements in the following way l=m g= h= L= a G= a e 2 H = a e 2 cos i In these coordinates the Hamiltonian is H= L 2 It is easy to show that this is indeed the Hamiltonian of the 2-body problem. Since H does not depend on G and H the Hamilton equations for g and h are ġ=0 and ḣ=0. The only non-constant variable is the mean anomaly l whose equation is l= H L = 2 3 L 3 = a 3/2 = a 3 =n This equation tells us that the mean anomaly l has a fixed frequency equal to the mean motion n.

4 3-body problem in Hamiltonian form. Let's define the mass of the central body as M o, m the mass of the perturber and m=0 the mass of the test body. We consider the planar circular case with m on a circular orbit and all bodies moving on the same plane. The equation of motion is d 2 r dt = GM r 0 G m r r 2 r 3 r r 3 r r 3 where U r = GM 0 r r G m r r r r The Hamiltonian of the system is H= v2 r 3 2 GM 0 G m r r r r r r 3 = U r = v2 2 GM 0 R r,r r The Delunay variables (action-angle variables) are now defined as l=m g= h= L= G a G= L e 2 = G M o a e 2 H =G cosi= G M o a e 2 cosi and the reduced mass does not appear since the test body has 0 mass. In these new variables the Hamiltonian becomes H= G2 2 M o R L,G,H,l, g,h,t 2L 2 where the first term on the right is the Keplerian term and gives a 2-body motion around M o with constant frequency n, while the disturbing function R accounts for the changes in the other orbital elements and it can be thought as a perturbation of the 2-body motion. R can be expanded in a multiple Fourier series R=G m i=0 K i j G, L cos i l j l' g where the primes refer to the orbital elements of the body m. The reason why the index I ranges from [0, ] and not from [-, ] is due to the antisymmetry of the cos function cos(x) = cos(-x). As a consequence we have cos x y =cos x y cos x y =cos x y

5 so that cos i l j l' g =cos il j l ' g cos i l j l' g =cos il j l ' g and these terms can be collected in a single one. We talk of resonance when the following condition is met i l L,G j l' ġ L,G =0 i.e. if the frequency of the angle l is in a commensurability with the frequency of the angle l'-g. Often in literature a different index notation is used where i= p+q and j = -p. In this context the resonance condition becomes p q l L,G p l ' ġ L,G =0 When this condition is satisfied, the corresponding cos term in the Hamiltonian is constant or it changes slowly. As a consequence, its perturbative effects is stronger respect to the other terms and the Hamiltonian can be approximated as H= G2 2 M o G m 2L 2 K 00 Gm K i j cos where ψ is the slowly varying angle called critical argument =il j l ' g. The Hamiltonian can be further re-arranged but first we need to know what a canonical transformation is. Short tutorial on canonical transformation Definitions: A coordinate transformation from (p,q) to (w,z), where p and w are the momenta, is canonical if ) it preserves the Hamiltonian form of the equations of motion q i = H p i H p,q =k w, z ż i = H w i ṗ i = H q i ẇ i = H z i 2) It preserves the area, in other words the Jacobian of the transformation has determinant equal to w det p z p w q = z q

6 3) A generatrix function of mixed variables exists so that, ad example, p= F q,w q F q,w z= w A simple example is F=k q w so that w= p/k and z=kq. The harmonic oscillator equations of motion can be solved by using a canonical transformation. We start from its Hamiltonian H= p2 2m m 2 q 2 2 A suited canonical transformation is generated by the function F q, z = m 2 q2 cotg z The new variables w,z are related to p,q in the following way p= F q w= F z =m qcotg z q=± =m q2 2 sin 2 z 2w m sin z p=± 2 m w cos z The new Hamiltonian is obtained by substituting the expressions for p,q in the old Hamiltonian K w,z =H p,q = w cos 2 w sin 2 z= w The Hamilton equation in the new coordinates are ż= K w = ẇ= K z =0 and the solution is z= t z 0 q t =± 2w 0 m sin t z 0 w=w 0 p t =± 2 m w cos t z 0 Moving back to the old coordinates we have found a solution of the equations of motion. The philosophy is that using a suited canonical transformation we move to coordinates where the Hamiltonian is much simpler (ad example it depends only on momenta) and the Hamilton equation are

7 easy to solve. Pendulum model for the resonance Hamiltonian The Hamiltonian H= G2 M o 2 2 L 2 G m K 00 L,G Gm K i j L,G cos depends on a single angle which is a combination of Delunay variables. The mean anomaly of the planet l' is simply time because l'=n' t where n' is the constant mean motion (or orbital frequency) of the mass m on a circular orbit. The goal is at this point to find a canonical transformation that changes the original Hamiltonia in a simpler one possibly with less degrees of freedom (attention, G is for the gravity constant and for the Delunay variable conjugate of g). We introduce the resonance variables of Poincare' =l l' g = p q l p l' g = p q G p L /q = L G /q The generatrix of this transformation is the following function f t,l, g,, = p q l p l ' g l g l ' = The dependence on t is again over l'=n' t. It is easy to show that = f = f L= f l G= f g The Hamiltonian in the new canonical variables (which are still action-angle variables) becomes K=H f t = G 2 2 M o 2 p q pn' n' Gm K 00 2, Gm K p q, p, cos R f The presence of is due to the time dependence of f on t t f l ' l ' = p n ' n ' t In R all the remaining fast changing terms of the perturbative function are collected. We will concentrate hereinafter on first order resonances for which q=± (see Winter and Murray, A&A 39, 290, 997). If q= the orbit of m is internal to that of m as in the case of Main Belt asteroids. The opposite if q= which is the case of Pluto and Neptune. When dealing with these resonances we will introduce a single index j for the resonance so that j= p q for the massless body and j = p for m. With this new index the Hamiltonian and the Poincare' variabels become

8 K= G2 M o 2 2 j 2 j n' n' Gm K, cos = GM 0 a j e 2 j = GM 0 a e 2 In the Hamiltonian we have neglected R and K 0 0 as small terms. The resonance condition becomes G2 2 M o = K = j n'=0 3 2 j This condition has been derived assuming that the term periodic function. The resonance condition translates to Gm K, cos averages to 0 being a j j n' n = j a 3/2 a 3/2 =0 Ad example, the 2: resonance between an asteroid and Jupiter occurs when the semimajro axis of the asteroid is about AU. To get the pendulum model, we need an additional effort to reduce the Hamiltonian to a single degree of freedom. Let's call r the exact value for which the above equation is satisfied (resonance condition). We can develop the Hamiltonian K in a Taylor series around r in the action variabel up to the second order. K= G2 2 M o M 2 o j 2 j r 2 G2 j r 3 G 2 M 2 o j 2 3 r 2 j r 4 r 2... j n' r j n' r... Combining the terms with the same terms in r we get to a simplified form K= A 0 A r A 2 r 2 B 0 cos where A 0 = G2 M o 2 2 j r 2 j n' r A = G2 M 2 o j j n'=0 3 j r A 2 = 3 G 2 M 2 o j 2 2 j r 4 The coefficient A is = 0 because it has the same form of the resonance condition, so it is 0 at the

9 resonance. The Hamiltonian then reads K= A 0 A 2 r 2 B 0 cos The action Φ is a constant of motion since K does not depend on the angle ϕ. The level curves for fixed energy are shown in figure. The red curve corresponds to the separatrix, which separates the libration motion from circulation. On the x-axis we must imagine to have ψ and on the y-axis r. We can derive the equation of the separatrix and in this way determines the amplitude of the resonance in the action space. We start from the level curves equation which can be derived assuming that the energy (the Hamiltonian K) is constant. Under this assumption we get = r ± K A B cos /2 0 0 A 0 For the separatrix, we know that when =0 then = r K= A 0 B 0. The equation for the separatrix is then and this leads to the equation = r ± B / 2 0 = A 2 cos r± 2B 0 sin A 2 2 The semi-amplitude of the resonance is ± 2B 0 A 2 Transforming this expression back to orbital elements lead to the resonance amplitude in semimajor axis (or mean motion). This can be found in Murray and Dermott (Nature, 30, 983) and it is n=± 2n2 e m M 0 a a ' f a a ' a= 2a 3n n

10 The amplitude depends also on the eccentricity of the orbit. In figure all main resonances in the asteroid belt are shown with their amplitudes. This grows with the eccentricity and when the resonance borders overlap the dynamical evolution is chaotic. This is seen in the following figure outlining the phase space. When the eccentricity (or mass ratio as in the case) grows the resonances overlap, the libration regions shrink and the evolution is chaotic.