3 The long-term behaviour: Secular theory

3.1 Restricted three-body case

We now recall Sect. 2.1.2 where we obtained the Hamiltonian of the averaged problem (Eq. (23)) and then apply Eq. (13) in order to write its first-order part (Eq. (24)) as

 

$$\sqrt{\mu_1} H_1 \mu_1 n_1^2 a_1^2 F(\phi) \frac{\delta a}{a_1} - X$$ = (42)

where $F(\phi)$ is a function of $\phi$,

$$\mu_1 n_1 [ g_1(\phi) 2 M +g_4(\phi) 2 N$$ X =

$$+g_2(\phi) \sqrt{n} a (2 M)^{1/2} e_1 \cos(\varpi-\varpi_1)$$

$$+g_3(\phi) \sqrt{n} a (2 M)^{1/2} e_1 \sin(\varpi-\varpi_1)$$

$$+g_5(\phi) \sqrt{n} a (2 N)^{1/2} I_1 \cos(\Omega-\Omega_1)$$

$$+g_6(\phi) \sqrt{n} a (2 N)^{1/2} I_1 \sin(\Omega-\Omega_1) ]$$ (43)

and

$$\begin{array}{l @{\hspace{2cm}} l} -\varpi & M=\frac{1}{2} n a^2 e^2 \\ [2mm] -\Omega & N=\frac{1}{2} n a^2 I^2 \end{array}$$ (44)

is an approximate canonical set of coordinates and conjugate momenta for the secular Hamiltonian (i.e. the Hamiltonian obtained after the elimination of the $\phi$-dependence).

3.1.1 Adiabatic approximation

By construction of the Hamiltonian of the averaged problem (Eq. (23)) the variables $(-\varpi,M)$ and $(-\Omega,N)$ vary on a timescale ${\cal O}(\mu_{1}^{-1/2} T_{1:1})$ and therefore are much slower than $\theta$, the angle variable of the integrable part H₀. The general procedure for dealing with this type of quasi-integrable systems involves a canonical transformation of variables such that the new Hamiltonian does not depend on the fast angle and thus the associated action is an adiabatic invariant (Lichtenberg & Lieberman 1983). To first-order in the small parameter $\sqrt{\mu_1}$, the transformed Hamiltonian is

$$\bar{H}=H_0+\sqrt{\mu_1}<H_1>_{\theta}$$ (45)

where

$$<H_1>_{\theta}\,=\frac{1}{2\pi} \int_{0}^{2\pi} H_1 \, {\rm ... ...c{1}{T_{1:1}} \oint H_1 \, \frac{{\rm d}\phi}{\dot{\phi}}\cdot$$ (46)

Now from Eq. (8)

$$\frac{1}{T_{1:1}} \oint \frac{\delta a}{a_1} \frac{{\rm d}\phi}{\dot{\phi}}={\cal O}(\mu_1)$$ (47)

hence we obtain the transformed Hamiltonian

 

$$\bar{H} H_0 - \bar{X}$$ = (48)

with

$$\bar{X} \mu_1 n_1 [ \bar{g}_1[l] (h^2+k^2)+\bar{g}_4[l] (p^2+q^2)$$ =

$$+\bar{g}_2[l] \sqrt{n_1} a_1 e_1 (h \sin{\varpi_1} + k \cos{\varpi_1})$$

$$+\bar{g}_3[l] \sqrt{n_1} a_1 e_1 (h \cos{\varpi_1} - k \sin{\varpi_1})$$

$$+\bar{g}_5[l] \sqrt{n_1} a_1 I_1 (p \sin{\Omega_1} + q \cos{\Omega_1})$$

$$+\bar{g}_6[l] \sqrt{n_1} a_1 I_1 (p \cos{\Omega_1} - q \sin{\Omega_1})]$$ (49)

where (k,h) and (q,p) are defined as

$$\begin{array}{l @{\hspace{2cm}} l} k= \sqrt{n_1} a_1 e \cos{\... ... I \cos{\Omega} & p= \sqrt{n_1} a_1 I \sin{\Omega}. \end{array}$$ (50)

Moreover,

$$g_i[l]=\frac{1}{T_{1:1}} \oint g_i(\phi) \frac{{\rm d}\, \phi}{\dot{\phi}}$$ (51)

where l is a parameter that characterises the size of the tadpole or horseshoe orbit which we define as

$$l=\pi/3-\phi_{\rm min}$$ (52)

and $\phi_{\rm min}$ is the minimal distance to m₁ obtained by solving Eq. (12) with $\dot{\phi}=0$.

From Paper I we recall that $\bar{g}_5[l]=-2 \bar{g}_4[l]$ and $\bar{g}_6[l]=0$; then we write

$$2 \bar{g}_4[l] \mu_1 n_1 \Gamma[l]$$ = (53)

$$2 \bar{g}_1[l] \mu_1 n_1 \gamma[l]$$ = (54)

$$\frac{\bar{g}_2[l]}{2 \bar{g}_1[l]} - c[l] \cos{b[l]}$$ = (55)

$$\frac{\bar{g}_3[l]}{2 \bar{g}_1[l]} - c[l] \sin{b[l]}.$$ = (56)

We also note that from Eq. (14) we have

$$a_0[l]=\sqrt{\frac{8}{3} \mu_1 \left( f(\pi/3-l)-\frac{3}{2} \right)}$$ (57)

and we refer to Paper I for plots of $\Gamma[l]$, $\gamma[l]$, c[l] and b[l] as functions of a₀[l].

3.1.2 Secular solution

We now write Hamilton's equations for (k,h) as

$$\dot{k}+{\rm i} \dot{h} -{\partial \bar{X} \over \partial h}+{\rm i}{\partial \bar{X} \over \partial k}$$ =

$${\rm i} \gamma[l] (k+{\rm i} h)- {\rm i} \gamma[l] \sqrt{n_1} a_1 c[l] e_1 \exp[{\rm i}(\varpi_1+b[l])]$$ = (58)

which imply the solution for $(e,\varpi)$

 

$$e \exp[{\rm i} \varpi] e_{\rm p} \exp[{\rm i}(\gamma[l] t+\chi)]$$ =

$$+c[l] e_1 \exp[{\rm i} (\varpi_1+b[l])]$$ (59)

i.e. composed of a proper term (proper eccentricity $e_{\rm p}$ and proper precession frequency $\gamma[l]$) and a forced term (forced eccentricity c[l] e₁ and forced periapse b[l]).

Similarly, we write Hamilton's equations for (q,p) as

$$\dot{q}+{\rm i} \dot{p} -{\partial \bar{X} \over \partial p}+{\rm i}{\partial \bar{X} \over \partial q}$$ =

$${\rm i} \Gamma[l] (q+{\rm i} p)- {\rm i} \Gamma[l] \sqrt{n_1} a_1 I_1 \exp[{\rm i}\Omega_1]$$ = (60)

which imply the solution for $(I,\Omega)$

$$I \exp[{\rm i} \Omega]=I_{\rm p} \exp[{\rm i}(\Gamma[l] t+\Xi)]+ I_1 \exp[{\rm i} \Omega_1]$$ (61)

i.e. composed of a proper term (proper inclination $I_{\rm p}$ and proper precession frequency $\Gamma[l]$) and a forced term I₁ (note, however, that if we choose the orbital plane of $m_{\rm c}$ and m₁ as reference, then this forced term disappears).

3.2 The effect of an oblate planet

The Hamiltonian of the averaged problem has now six degrees of freedom (cf. Sect. 2.2) i.e. two more than in the restricted three-body case. However, we can eliminate these extra degrees of freedom by making a canonical transformation to the following variables

$$\begin{array}{l @{\hspace{2cm}} l} -\varpi_{\rm r}=-\varpi-\l... ... \tilde{\Lambda}_{{\rm f} 1}=N+\Lambda_{{\rm f} 1}. \end{array}$$ (62)

The transformed first-order part of the Hamiltonian (Eq. (32)) is then

 

$$\sqrt{\mu_1} \tilde{H}_1 -\beta_1 (\tilde{\Lambda}_{{\rm g}1}- \tilde{\Lambda}_{{\rm f} 1}) +\mu_1 n_1^2 a_1^2 F(\phi) \frac{\delta a}{a_1}$$ =

- X - Y (63)

with

$$\mu_1 n [ g_1(\phi) 2 M +g_2(\phi) \sqrt{n} a (2 M)^{1/2} e_1 \cos{\varpi_{\rm r}}$$ X =

$$+g_3(\phi) \sqrt{n} a (2 M)^{1/2} e_1 \sin{\varpi_{\rm r}}$$

$$+g_4(\phi) 2 N +g_5(\phi) \sqrt{n} a (2 N)^{1/2} I_1 \cos{\Omega_{\rm r}}$$

$$+g_6(\phi) \sqrt{n} a (2 N)^{1/2} I_1 \sin{\Omega_{\rm r}} ]$$ (64)

and

$$Y=\beta_1 (M-N) \left(1+\frac{\delta a}{a_1}\right)^{-7/2}- \beta_1 (M-N).$$ (65)

3.2.1 Adiabatic approximation

The first term on the right hand side of Eq. (63) can be dropped as there is no explicit dependence on $\lambda_{{\rm g} 1}$or $\lambda_{{\rm f} 1}$. Hence, Eq. (63) has the same form as Eq. (42), except for the term Y. This introduces the relative precession frequencies

$${\partial Y \over \partial M}=-{\partial Y \over \partial N} ... ...}\beta_1 \frac{\delta a}{a_1}= {\cal O}(\beta_1 \sqrt{\mu_1}).$$ (66)

As in general $\beta_1\ll n_1$, the variables $(-\varpi_{\rm r},M)$ and $(-\Omega_{\rm r},N)$still vary on a time-scale much longer than the co-orbital period $T_{1:1}={\cal O} (n_1 \sqrt{\mu_1})$. We can then follow the procedure described previously, obtaining an Hamiltonian as defined by Eqs. (45) and (46), i.e.

$$\bar{H} = H_0 -\bar{X} -\bar{Y}$$ (67)

with

$$\bar{X} \mu_1 n_1 [ \bar{g}_1[l] (h_{\rm r}^2+k_{\rm r}^2) +\bar{g}_4[l] (p_{\rm r}^2+q_{\rm r}^2)$$ =

$$+\bar{g}_2[l] \sqrt{n_1} a_1 e_1 k_{\rm r} + \bar{g}_3[l] \sqrt{n_1} a_1 e_1 h_{\rm r}$$

$$+\bar{g}_5[l] \sqrt{n_1} a_1 I_1 q_{\rm r} + \bar{g}_6[l] \sqrt{n_1} a_1 I_1 p_{\rm r} ]$$ (68)

and

$$\bar{Y}= \frac{1}{2} \Delta\bar{\beta} [(h_{\rm r}^2+k_{\rm r}^2)-(p_{\rm r}^2+q_{\rm r}^2)]$$ (69)

where $(k_{\rm r},h_{\rm r})$ and $(q_{\rm r},p_{\rm r})$ are defined as

$$\begin{array}{l @{\hspace{2cm}} l} k_{\rm r}= \sqrt{n_1} a_1 ... ...& p_{\rm r}= \sqrt{n_1} a_1 I \sin{\Omega_{\rm r}} \end{array}$$ (70)

and from Eq. (30)

$$\Delta\bar{\beta}= -\beta_1+\frac{\beta_1}{T_{1:1}} \oint \le... ...} \, \frac{{\rm d}\phi}{\dot{\phi}} ={\cal O}(\beta_1 \mu_1).$$ (71)

3.2.2 Secular solution

Hamilton's equations for $(k_{\rm r},h_{\rm r})$ are

 

$$\dot{k}_{\rm r}+{\rm i} \dot{h}_{\rm r} = \left(-{\partial \over ... ...\rm i} h_{\rm r}) -{\rm i} \gamma[l] \sqrt{n_1} a_1 c[l] e_1 \exp[{\rm i} b[l]]$$ (72)

and therefore the solution for $(e,\varpi_{\rm r})$ is

 

$$e \exp[{\rm i} \varpi_{\rm r}] = e_{\rm p} \exp[{\rm i}((\gamma[l... ...i)] +\frac{\gamma[l] c[l] e_1 \exp[{\rm i} b[l]]}{\gamma[l]+\Delta\bar{\beta}}.$$ (73)

Similarly, Hamilton's equations for $(q_{\rm r},p_{\rm r})$ are

 

$$\dot{q}_{\rm r}+{\rm i} \dot{p}_{\rm r} \left(-{\partial \over \partial p_{\rm r}}+ {\rm i} {\partial \over \partial q_{\rm r}} \right) (\bar{X}+\bar{Y})$$ =

$${\rm i} (\Gamma[l]-\Delta\bar{\beta}) (q_{\rm r}+{\rm i} p_{\rm r})-{\rm i} \Gamma[l] \sqrt{n_1} a_1 I_1$$ = (74)

and therefore the solution for $(I,\Omega_{\rm r})$ is

 

$$I \exp[{\rm i}\Omega_{\rm r}] = I_{\rm p} \exp[{\rm i} ((\Gamma[l... ...lta\bar{\beta}) t+\Xi)] +\frac{\Gamma[l] I_1}{\Gamma[l]-\Delta\bar{\beta}}\cdot$$ (75)

From Eqs. (73) and (75) we see that the oblateness term, $\Delta\bar{\beta}>0$, leads to a decrease in the secular precession periods and the magnitude of the forced terms. However, this effect is only significant if $\Delta\bar{\beta} \sim \gamma$(or $-\Gamma$). As $\gamma \ge 3.375 n_1 \mu_1$ (see Fig. 1a in Paper I) and in general $\beta_1\ll n_1$, one can largely neglect the term $\Delta\bar{\beta}$ in that which concerns the evolution of the eccentricity; in this case the secular solution reduces to Eq. (59). On the other hand, very small amplitude tadpole orbits can have $\Gamma\sim -\beta_1 \mu_1$ (see Fig. 1d in Paper I) and thus the effect of the term $\Delta\bar{\beta}$ on the evolution of the inclination can be visible.

3.3 The effect of additional massive bodies

The Hamiltonian of the averaged problem has $2\times N +4$ degrees of freedom (cf. Sect. 2.3). In order to reduce the number of degrees of freedom we perform a canonical transformation to the following variables

$$\begin{array}{l @{\hspace{2cm}} l} -\varpi_{{\rm r}}=-\varpi-... ... \tilde{\Lambda}_{{\rm f} k}=N+\Lambda_{{\rm f} k}. \end{array}$$ (76)

We also perform N-1 canonical transformations of the same type to the following variables

$$\begin{array}{l @{\hspace{2cm}} l} \check{\lambda}_{{\rm g} i... ...k} \Lambda_{{\rm f} i}+\tilde{\Lambda}_{{\rm f} k}. \end{array}$$ (77)

Then, the transformed first-order part of the Hamiltonian (Eq. (40)) is

 

$$\sqrt{\mu_k} \check{H}_1 -\check{\Lambda}_{{\rm g} k} g_k-\check{\Lambda}_{{\rm f} k} f_k$$ =

$$-\sum_{i \neq k} \Lambda_{{\rm g} i} (g_i-g_k) -\sum_{i \neq k} \Lambda_{{\rm f} i} (f_i-f_k)$$

$$+ \mu_k n_k^2 a_k^2 F(\phi) \frac{\delta a}{a_k} - X -Y$$ (78)

with

$$\mu_k n [ g_1(\phi) 2 M + g_4(\phi) 2 N$$ X =

$$+g_2(\phi) \sqrt{n} a (2 M)^{1/2} \sum_{i} e_{k,i} \cos(\varpi_{\rm r}+\check{\lambda}_{{\rm g} i})$$

$$+g_3(\phi) \sqrt{n} a (2 M)^{1/2} \sum_{i} e_{k,i} \sin(\varpi_{\rm r}+\check{\lambda}_{{\rm g} i})$$

$$+g_5(\phi) \sqrt{n} a (2 N)^{1/2} \sum_{i} I_{k,i} \cos(\Omega_{\rm r}+\check{\lambda}_{{\rm f} i})$$

$$+g_6(\phi) \sqrt{n} a (2 N)^{1/2} \sum_{i} I_{k,i} \sin(\Omega_{\rm r}+\check{\lambda}_{{\rm f} i})]$$ (79)

and

Y = (A_k-g_k) M +(B_k-f_k) N

$$+ \sum_{j \neq k} A_j \sqrt{n} a (2 M)^{1/2} \sum_{i} e_{j,i} \cos(\varpi_{\rm r}+\check{\lambda}_{{\rm g} i})$$

$$+ \sum_{j \neq k} B_j \sqrt{n} a (2 N)^{1/2} \sum_{i} I_{j,i} \cos(\Omega_{\rm r}+\check{\lambda}_{{\rm f} i}).$$ (80)

And the first two terms on the right hand side of Eq. (78) can be dropped as there is no explicit dependence on $\lambda_{{\rm g} k}$ or $\lambda_{{\rm f} k}$.

3.3.1 Secular resonances

We now assume that the perturbers move on circular and co-planar orbits, in which case the terms in Eq. (78) depending on $(\check{\lambda}_{{\rm g} i},\Lambda_{{\rm g} i})$ and $(\check{\lambda}_{{\rm f} i},\Lambda_{{\rm f} i})$ also disappear and the Hamiltonian reduces to

 

$$H_0+ \mu_k n_k^2 a_k^2 F(\phi) \frac{\delta a}{a_k}$$ H =

$$-(2 g_1(\phi) \mu_k n +A_k-g_k) M$$

$$-(2 g_4(\phi) \mu_k n +B_k-f_k) N.$$ (81)

Equation (81) represents a one-degree of freedom (hence integrable) system depending on the parameters M and N. Therefore, we can perform a canonical transformation to action-angle variables which eliminates $\phi$ from the Hamiltonian (Lichtenberg & Lieberman 1983). To first-order in the small parameter $\sqrt{\mu_k}$, this is accomplished by averaging over the angle variable of the zero-order term H₀, which is equivalent to averaging over the co-orbital period T_1:1. The transformed Hamiltonian is

$$\bar{H}=H_0-(\gamma_k[l] +\bar{A}_k-g_k) M -(\Gamma_k[l]+\bar{B}_k-f_k) N$$ (82)

which has proper frequencies

  

$$\dot{\varpi}_{\rm r} -{\partial \bar{H} \over \partial M}=\gamma_k[l]+\bar{A}_k-g_k$$ = (83)

$$\dot{\Omega}_{\rm r} -{\partial \bar{H} \over \partial N}=\Gamma_k[l]+\bar{B}_k-f_k$$ = (84)

where $\gamma_k[l]=2 \bar{g}_1[l] \mu_k n_k$, $\Gamma_k[l]=2 \bar{g}_4[l] \mu_k n_k$; and from Eq. (38)

$$\bar{A}_k \frac{1}{T_{1:1}} \oint A_k \frac{{\rm d} \phi}{\dot{\phi}} =A_{k,k} (1+{\cal O}(\mu_k))$$ = (85)

$$\bar{B}_k \frac{1}{T_{1:1}} \oint B_k \frac{{\rm d} \phi}{\dot{\phi}} = B_{k,k} (1+{\cal O}(\mu_k)).$$ = (86)

Secular resonances involving the pericentres (or nodes) occur when $\dot{\varpi}_{\rm r}$ (or $\dot{\Omega}_{\rm r}$) is equal to a forcing frequency g_i-g_k (or f_i-f_k), i.e. when

  

$$\gamma_k[l]+\bar{A}_k=g_i$$ (87)

$$\Gamma_k[l]+\bar{B}_k=f_i.$$ (88)

Note that the proper frequencies of precession of the Trojan orbit are in fact $\dot{\varpi}=\dot{\varpi}_{\rm r}+g_k$ and $\dot{\Omega}=\dot{\Omega}_{\rm r}+f_k$ which coincide respectively with the left hand side of Eq. (87) and Eq. (88). Moreover, the terms $\gamma _k$ (and $\Gamma _k$) and the terms $\bar{A}_k$ (and $\bar{B}_k$) are respectively the contribution from the mass m_k and the contribution from the additional massive bodies m_j (where $j\neq k$).

3.3.2 Adiabatic approximation

We will now assume that not only the variables $(-\varpi_{\rm r},M)$ and $(-\Omega_{\rm r},N)$ but also the angles $\check{\lambda}_{{\rm g} i}$ and $\check{\lambda}_{{\rm f} i}$vary on a time-scale much longer than the co-orbital period T_1:1. Note that while the first assumption will in general be true, the same does not necessarily apply to the second assumption. When the co-orbital frequency is comparable to one of the forcing frequencies, a low order resonance can occur in which case the adiabatic approximation does not provide a good description of the system. Nevertheless, if we ignore this possibility then we can apply Eqs. (45) and (46) to obtain the transformed Hamiltonian

$$\bar{H} = H_0- \sum_{i \neq k} \Lambda_{{\rm g} i} (g_i-g_k) -\sum_{i \neq k} \Lambda_{{\rm f} i} (f_i-f_k) - \bar{X} -\bar{Y}$$ (89)

with

$$\bar{X} \mu_k n_k [ \bar{g}_1[l] (h_{\rm r}^2+k_{\rm r}^2) +\bar{g}_4[l] (p_{\rm r}^2 +q_{\rm r}^2)$$ =

$$+\bar{g}_2[l] \sqrt{n_k} a_k \sum_{i} e_{k,i} (k_{\rm r} \cos{\check{\lambda}_{{\rm g} i}} -h_{\rm r} \sin{\check{\lambda}_{{\rm g} i}})$$

$$+\bar{g}_3[l] \sqrt{n_k} a_k \sum_{i} e_{k,i} (h_{\rm r} \cos{\check{\lambda}_{{\rm g} i}} +k_{\rm r} \sin{\check{\lambda}_{{\rm g} i}})$$

$$+\bar{g}_5[l] \sqrt{n_k} a_k \sum_{i} I_{k,i} (q_{\rm r} \cos{\check{\lambda}_{{\rm f} i}} -p_{\rm r} \sin{\check{\lambda}_{{\rm f} i}})$$

$$+\bar{g}_6[l] \sqrt{n_k} a_k \sum_{i} I_{k,i} (p_{\rm r} \cos{\check{\lambda}_{{\rm f} i}} +q_{\rm r} \sin{\check{\lambda}_{{\rm f} i}} )]$$ (90)

and

$$\bar{Y} \frac{1}{2}(\bar{A}_{k}-g_k) (h_{\rm r}^2+k_{\rm r}^2) +\frac{1}{2}(\bar{B}_{k}-f_k) (p_{\rm r}^2+q_{\rm r}^2)$$ =

$$+\sum_{j \neq k} \bar{A}_{j} \sqrt{n_j} a_j \sum_{i} e_{j,i} (k_{... ...\cos{\check{\lambda}_{{\rm g} i}} -h_{\rm r} \sin{\check{\lambda}_{{\rm g} i}})$$

$$+\sum_{j \neq k} \bar{B}_{j} \sqrt{n_j} a_j \sum_{i} I_{j,i} (q_{... ...\cos{\check{\lambda}_{{\rm f} i}} -p_{\rm r} \sin{\check{\lambda}_{{\rm f} i}})$$ (91)

where $(k_{\rm r},h_{\rm r})$ and $(q_{\rm r},p_{\rm r})$ are defined as

$$k_{\rm r}+{\rm i} h_{\rm r} \sqrt{n_k} a_k e \exp[{\rm i} \varpi_{\rm r}]$$ = (92)

$$q_{\rm r}+{\rm i} p_{\rm r} \sqrt{n_k} a_k I \exp[{\rm i} \Omega_{\rm r}]$$ = (93)

and from Eq. (38)

$$\bar{A}_j \frac{1}{T_{1:1}} \oint A_j \frac{{\rm d} \phi}{\dot{\phi}} =A_{k,j} (1+{\cal O}(\mu_k))$$ = (94)

$$\bar{B}_j \frac{1}{T_{1:1}} \oint B_j \frac{{\rm d} \phi}{\dot{\phi}} =B_{k,j} (1+{\cal O}(\mu_k)).$$ = (95)

3.3.3 Secular solution

The evolution of $z=\sqrt{n_k} a_k e \exp[{\rm i} \varpi]$ is described by

 

$$(\dot{z} - {\rm i} g_k z) \exp[{\rm i} \lambda_{{\rm g} k}] = \le... ...\rm r}}+ {\rm i} {\partial \over \partial k_{\rm r}} \right) (\bar{X}+\bar{Y})=$$

$${\rm i} \bigg( (\gamma_k[l]+\bar{A}_k-g_k) z - \gamma_k[l] \sqrt{n_k} a_k c[l] e_k \exp[{\rm i}(\varpi_k+b[l])]$$

$$+ \sum_{j \neq k} \bar{A}_j \sqrt{n_j} a_j e_j \exp[{\rm i} \varpi_j] \bigg) \exp[{\rm i} \lambda_{{\rm g} k}]$$ (96)

which is the equation of a forced harmonic oscillator with proper frequency $\gamma_k[l]+\bar{A}_{k}$ and forcing frequencies g_i; hence the solution for $(e,\varpi)$ is

 

$$e \exp[{\rm i} \varpi]= e_{\rm p}\exp[{\rm i}((\gamma_k[l]+\bar{A... ...}\bar{A}_{j} e_{j,i}} {\gamma_k[l]+\bar{A}_{k}-g_i}\exp[{\rm i}(g_i t+\chi_i)].$$ (97)

The evolution of $Z=\sqrt{n_k} a_k I \exp[{\rm i} \Omega]$ is described by

 

$$(\dot{Z} - {\rm i} f_k Z) \exp[{\rm i} \lambda_{{\rm f} k}] = \le... ...{\rm r}}+ {\rm i} {\partial \over \partial q_{\rm r}} \right) (\bar{X}+\bar{Y})$$

$$= {\rm i} \bigg( (\Gamma_k[l]+\bar{B}_k-f_k) Z - \Gamma_k[l] \sqrt{n_k} a_k I_k \exp[{\rm i}\Omega_k]$$

$$+ \sum_{j \neq k} \bar{B}_j \sqrt{n_j} a_j I_j \exp[{\rm i} \Omega_j] \bigg) \exp[{\rm i} \lambda_{{\rm f} k}]$$ (98)

which is the equation of a forced harmonic oscillator with proper frequency $\Gamma_k[l]+\bar{B}_{k}$ and forcing frequencies f_i; hence the solution for $(I,\Omega)$ is

 

$$I \exp[{\rm i} \Omega]= I_{\rm p}\exp[{\rm i}((\Gamma_k[l]+\bar{B}_{k}) t+\Xi)]$$

$$+\sum_{i} \frac{\Gamma_k[l] I_{k,i}-\sum_{j\neq k}\bar{B}_{j} I_{j,i}} {\Gamma_k[l]+\bar{B}_{k}-f_i}\exp[{\rm i}(f_i t+\Xi_i)].$$ (99)

We now recall from Paper I that $\bar{B}_j=(1+{\cal O}(\mu_k)) B_{k,j}$, and that by definition of eigen-values and eigen-vectors, $B_{k,k} I_{k,i}+\sum_{j\neq k} B_{k,j} I_{j,i}=f_i I_{k,i}$, so that

 

$$I \exp[{\rm i} \Omega] I_{\rm p}\exp[{\rm i}((\Gamma_k[l]+\bar{B}_{k}) t+\Xi)]$$ =

$$+\sum_{i} I_{k,i} \exp[{\rm i}(f_i t+\Xi_i)]$$

$$+ \sum_{i} \frac{{\cal O}(\mu_k \mu_i) I_{k,i}}{\Gamma_k[l]+\bar{B}_{k}-f_i} \exp[{\rm i}(f_i t+\Xi_i)].$$ (100)

This is to say that when $\Gamma_k[l]+\bar{B}_{k}=f_{i}$, the forcing terms in the second-order differential equation that describes the evolution of $I \exp[{\rm i} \Omega]$ are smaller than in the non-resonant case by a mass ratio factor. Indeed, one can show that these forcing terms have amplitudes $I_{k,i} (\Gamma_k[l]+\bar{B}_{k}+f_{i}) [(\Gamma_k[l]+\bar{B}_{k}-f_{i})+ {\cal O}(\mu_k \mu_i)]$, which reduce to $2 f_i I_{k,i} {\cal O}(\mu_k \mu_i)$at the exact resonance.

Note that although the secular solution obtained here (Eqs. (97) and (99)) has essentially the same form as that obtained in Paper I, the terms $\bar{A}_j$ and $\bar{B}_j$ are now defined as averages over T_1:1 (which nonetheless coincide, to lowest order, with the quantities defined in Paper I). The basic improvement with respect to Paper I is the correct derivation of the thresholds of validity of the secular solution which as we have seen now simply depends on the validity of the adiabatic approximation.