Next: Secular evolution of asteroid
Up: Secular perturbation theory
Previous: Evolution equations for twoplanet
Secular evolution of planetary orbits
As a specific example of the use of orbital perturbation theory,
let us determine the evolution of the osculating orbital elements of the two planets in our model solar system
due to the secular terms in their disturbing functions (i.e., the terms that are independent of the
mean longitudes
and
). This is equivalent to averaging the osculating
elements over the relatively short timescales associated with the periodic terms in the disturbing functions
(i.e., the terms that depend on
and
, and, therefore, oscillate on timescales similar
to the orbital periods of the planets).
From Equations (10.16)(10.18),
the secular part of the first planet's disturbing function takes the form

(10.29) 
where
because, as can be demonstrated (Brouwer and Clemence 1961),
Evaluating the righthand sides of Equations (10.10)(10.15)
to
(it is assumed that
,
,
,
,
,
,
,
,
,
,
), we find that




(10.34) 




(10.35) 




(10.36) 




(10.37) 




(10.38) 
and 





(10.39) 
because
. It follows that
, and




(10.40) 




(10.41) 




(10.42) 




(10.43) 
and 





(10.44) 
Note that the first planet's mean angular velocity is slightly modified
in the presence of the second planet, but that its major radius remains the same.
From Equations (10.26)(10.28),
the secular part of the second planet's disturbing function takes the form

(10.45) 
where
Evaluating the righthand sides of Equations (10.20)(10.25)
to
, we find that




(10.48) 




(10.49) 




(10.50) 




(10.51) 




(10.52) 
and 





(10.53) 
It follows that
, and




(10.54) 




(10.55) 




(10.56) 




(10.57) 
and 





(10.58) 
The second planet's mean angular velocity is also slightly modified in the
presence of the first planet, but its major radius remains the same.
Let us now generalize the preceding analysis to take all eight of the major planets in the solar system into account.
Let planet
(where
runs from
to
) have mass
, major radius
, eccentricity
, longitude of the
perihelion
, inclination
, and longitude of the ascending node
.
As before, it is convenient to introduce the alternative orbital elements
,
,
, and
. It is also helpful to define the following parameters:

(10.59) 
and

(10.60) 
as well as
where
is the mass of the Sun.
It then follows, from the preceding analysis, that the secular terms in the planetary disturbing functions cause the
,
,
, and
to vary in time as
where




(10.67) 




(10.68) 




(10.69) 
and 





(10.70) 
for
.
Here, Mercury is planet 1, Venus is planet 2, and so on, and Neptune is planet 8. Note that the
time evolution of the
and the
, which determine the eccentricities of the planetary orbits, is decoupled from that of the
and the
, which determine the inclinations.
Let us search for normal mode
solutions to Equations (10.63)(10.66) of the form
It follows that
At this stage, we have effectively reduced the problem of determining the secular evolution of the planetary orbits to a pair of matrix eigenvalue equations (Gradshteyn and Ryzhik 1980c)
that can be solved via standard numerical techniques (Press et al. 1992). Once we have determined the eigenfrequencies,
and
,
and the corresponding eigenvectors,
and
, we can find the phase angles
and
by demanding that, at
, Equations (10.71)(10.74) lead to the values of
,
,
, and
given in Table 4.1.
Table 10.1:
Eigenfrequencies and phase angles obtained from LaplaceLagrange secular evolution theory.















1 
5.462 
5.201 
89.65 
20.23 
2 
7.346 
6.570 
195.0 
318.3 
3 
17.33 
18.74 
336.1 
255.6 
4 
18.00 
17.64 
319.0 
296.9 
5 
3.724 
0.000 
30.12 
107.5 
6 
22.44 
25.90 
131.0 
127.3 
7 
2.708 
2.911 
109.9 
315.6 
8 
0.6345 
0.6788 
67.98 
202.8 

The theory outlined here is generally referred to as LaplaceLagrange secular evolution theory. The
eigenfrequencies, eigenvectors, and phase angles obtained from this theory are listed in Tables 10.110.3.
Note that the largest eigenfrequency is of magnitude
arc seconds per year, which translates to an
oscillation period of about
years. In other words, the typical timescale over which the secular evolution of the solar system predicted by LaplaceLagrange theory takes place is at least
years, and is, therefore, much
longer than the orbital period of any planet.
Figure:
The full circles show the observed planetary perihelion precession rates (lefthand panel) and ascending node
precession rates (righthand panel) at J2000. All ascending nodes are measured relative to the mean ecliptic at J2000. The empty circles show the theoretical
precession rates calculated from LaplaceLagrange secular evolution theory.
Source (for observational data): Standish and Williams (1992).

Figure 10.2 compares the observed perihelion and ascending node precession rates of the planets at
(which
corresponds to the epoch J2000) with those calculated from the theory described previously. It can be seen that, generally, there
is good agreement between the theoretical and observed precession rates, which gives us some degree of confidence in the
theory. On the whole, the degree of agreement exhibited in the lefthand panel of Figure 10.2 is
better than that exhibited in Figure 5.1, indicating that the LaplaceLagrange secular evolution theory
described in this chapter is an improvement on the (highly simplified) Gaussian secular evolution theory
outlined in Section 5.4.
Observe that one of the inclination eigenfrequencies,
, takes the value zero. This is a consequence of the conservation
of angular momentum. Because the solar system is effectively an isolated dynamical system, its net angular momentum
vector,
, is constant in both magnitude and direction. The plane normal to
that passes through the
center of mass of the solar system (which lies very close to the Sun) is known as the invariable plane. If all
of the planetary orbits were to lie in the invariable plane then the net angular velocity vector of the solar system would be
parallel to its fixed net angular momentum vector. Moreover, the angular momentum vector would be parallel to one of
the solar system's principal axes of rotation. In this situation, we would expect the solar system to remain in
the invariable plane. (See Chapter 8.) In other words, we would not expect any time evolution of the planetary
inclinations. (Of course, lack of time variation implies an eigenfrequency of zero.) According to Equations (10.73) and (10.74), and the
data shown in Tables 10.1 and 10.3, if the solar system were in the inclination eigenstate associated with
the null eigenfrequency,
, then we would have
for
.
Because
and
, it follows that all of the
planetary orbits would lie in the same plane, and this planewhich is, of course, the invariable planeis
inclined at
to the ecliptic plane. Furthermore, the
longitude of the ascending node of the invariable plane,
with respect to the ecliptic plane, is
. Actually, it is generally more convenient to measure
the inclinations of the planetary orbits with respect to the invariable plane, rather than the ecliptic plane, because the
inclination of the latter plane varies in time. We can achieve this goal by simply omitting the fifth inclination
eigenstate when calculating orbital inclinations from Equations (10.73) and (10.74).
8pt
Table:
Components of eccentricity eigenvectors
obtained from LaplaceLagrange secular evolution theory.
All components multiplied by
.











2 
3 
4 
5 
6 
7 
8 










18128 
629 
404 
66 
0 
0 
0 
0 
2 
2331 
1919 
1497 
265 
1 
1 
0 
0 
3 
154 
1262 
1046 
2979 
0 
0 
0 
0 
4 
169 
1489 
1485 
7281 
0 
0 
0 
0 
5 
2446 
1636 
1634 
1878 
4331 
3416 
4388 
159 
6 
10 
51 
242 
1562 
1560 
4830 
180 
13 
7 
59 
58 
62 
82 
207 
189 
2999 
322 
8 
0 
1 
1 
2 
6 
6 
144 
954 

Consider the
th planet. Suppose one of the
coefficients
, sayis sufficiently large that

(10.79) 
This is known as the Lagrange condition (Hagihara 1971). As can be demonstrated, if the Lagrange condition is satisfied then the eccentricity of the
th planet's orbit varies between the minimum value

(10.80) 
and the maximum value

(10.81) 
Moreover, on average, the
th planet's perihelion point precesses at the associated eigenfrequency,
. The precession
is prograde (i.e., in the same direction as the orbital motion) if the frequency is positive,
and retrograde (i.e., in the opposite direction) if the frequency is negative.
If the Lagrange condition is not satisfied then all we can say is that the maximum eccentricity is given by Equation (10.81),
and there is no minimum eccentricity (i.e., the eccentricity can vary all the way down to zero). Furthermore,
no mean precession rate of the perihelion point can be identified.
It can be seen from Table 10.2 that the Lagrange condition for the orbital eccentricities is satisfied for all planets except
for Venus and Earth. The maximum and minimum eccentricities, and mean perihelion precession rates, of the
planets (when they exist) are given in Table 10.4. Note that Jupiter and Uranus have the same mean
perihelion precession rates, and that all planets that possess mean precession rates exhibit prograde precession.
Table:
Components of inclination eigenvectors
obtained from LaplaceLagrange secular evolution theory.
All components multiplied by
.











2 
3 
4 
5 
6 
7 
8 










12548 
1180 
850 
180 
2 
2 
2 
0 
2 
3548 
1006 
811 
180 
1 
1 
0 
0 
3 
409 
2684 
2446 
3595 
0 
0 
0 
0 
4 
116 
685 
451 
5021 
0 
1 
0 
0 
5 
2751 
2751 
2751 
2751 
2751 
2751 
2751 
2751 
6 
27 
14 
279 
954 
636 
1587 
69 
7 
7 
333 
191 
173 
125 
95 
77 
1757 
206 
8 
144 
132 
129 
122 
116 
112 
109 
1181 

There is also a Lagrange condition associated with the inclinations of the planetary orbits (Hagihara 1971).
This condition is satisfied for the
th planet if one of the

, sayis sufficiently large that

(10.82) 
The fifth inclination eigenmode is omitted from this summation because we are now
measuring inclinations relative to the invariable plane.
If the Lagrange condition is satisfied then the inclination of the
th planet's orbit with respect to the invariable plane varies between the minimum value

(10.83) 
and the maximum value

(10.84) 
Moreover, on average, the ascending node precesses at the associated eigenfrequency,
. The precession
is prograde (i.e., in the same direction as the orbital motion) if the frequency is positive,
and retrograde (i.e., in the opposite direction) if the frequency is negative.
If the Lagrange condition is not satisfied then all we can say is that the maximum inclination is given by Equation (10.84),
and there is no minimum inclination (i.e., the inclination can vary all the way down to zero). Furthermore,
no mean precession rate of the ascending node can be identified.
It can be seen from Table 10.3 that the Lagrange condition for the orbital inclinations is satisfied for all planets except
for Venus, Earth, and Mars. The maximum and minimum inclinations, and mean nodal precession rates, of the
planets (when they exist) are given in Table 10.4. The four outer planets, which
possess most of the mass of the solar system, all have orbits whose inclinations to the invariable plane remain small.
On the other hand, the four relatively light inner planets have orbits whose inclinations to the invariable
plane can become relatively large. Observe that Jupiter and Saturn have the same mean
nodal precession rates, and that all planets that possess mean precession rates exhibit retrograde precession.
Table 10.4:
Maximum/minimum eccentricities and inclinations of planetary orbits, and mean perihelion/nodal precession rates,
from LaplaceLagrange secular evolution theory. All inclinations relative to invariable plane.







Planet 













Mercury 
0.130 
0.233 

4.57 
9.86 

Venus 
0.000 
0.0705 

0.000 
3.38 

Earth 
0.000 
0.0638 

0.000 
2.95 

Mars 
0.0444 
0.141 

0.000 
5.84 

Jupiter 
0.0256 
0.0611 

0.241 
0.489 

Saturn 
0.0121 
0.0845 

0.797 
1.02 

Uranus 
0.0106 
0.0771 

0.902 
1.11 

Neptune 
0.00460 
0.0145 

0.554 
0.800 


Figure 10.3:
Time variation of the eccentricity (top left), inclination (top right), perihelion precession
rate (bottom left), and ascending node precession rate (bottom right), of Mercury, as predicted by LaplaceLagrange
secular perturbation theory. Time is measured in millions of years relative to J2000. All inclinations
are relative to the invariable plane. The horizontal dashed lines in the top panels
indicate the predicted minimum and maximum eccentricities and inclinations from
Table 10.4. The horizontal dashed lines in the bottom panels
indicate the predicted mean perihelion and ascending node precession rates from same
table.

Figure 10.3 shows the time variation of the eccentricity, inclination, perihelion precession
rate, and ascending node precession rate, of Mercury, as predicted by the LaplaceLagrange
secular perturbation theory described earlier. It can be seen that the eccentricity and inclination
do indeed oscillate between the upper and lower bounds specified in Table 10.4.
Moreover, the perihelion and ascending node precession rates do appear to oscillate about the mean values (
and
,
respectively) specified in the same table.
Figure 10.4:
The eccentricity eigenfrequencies obtained from Brouwer and van Woerkom's
refinement of standard LaplaceLagrange secular evolution theory (filled circles), compared with the corresponding
eigenfrequencies from Table 10.1 (open circles).

According to LaplaceLagrange secular perturbation theory, the mutual gravitational interactions between the various planets in the
solar system cause their orbital eccentricities and inclinations to oscillate between fixed bounds on
timescales that are long compared to their orbital periods. Recall, however, that these results depend on a great many approximations; the neglect of all nonsecular terms in the
planetary disturbing functions, as well as the neglect of secular terms beyond first order in the planetary masses, and
beyond second order in the orbital eccentricities and inclinations. It turns out that, when the neglected
terms are included in the analysis, the largest correction
to standard LaplaceLagrange theory is a secondorder (in the planetary masses) effect caused by periodic
terms in the disturbing functions of Jupiter and Saturn that oscillate on a relatively long timescale (i.e., almost 900 years), due to
the fact that the orbital periods of these two planets are almost commensurable (i.e., five times the orbital period of
Jupiter is almost equal to two times the orbital period of Saturn). In 1950, Brouwer and van Woerkom worked out a modified version of LaplaceLagrange
secular perturbation theory that takes the aforementioned correction into account (Brouwer and van Woerkom 1950). This refined secular evolution theory is
described, in detail, in Murray and Dermott (1999). As is illustrated in Figure 10.4, the values of the
eccentricity eigenfrequencies
and
in the Brouwervan Woerkom theory differ somewhat from those specified in Table 10.1. The
corresponding eigenvectors are also somewhat modified. The Brouwervan Woerkom theory contains two additional, relatively smallamplitude, eccentricity eigenmodes that
oscillate at the eigenfrequencies
and
. On the other hand, the Brouwervan Woerkom theory does not give rise to any significant modifications to the inclination eigenmodes.
Figure 10.5 shows the maximum and minimum orbital eccentricities predicted by the Brouwervan Woerkom
theory, compared with the corresponding limits from Table 10.4. It can be seen that the refinements introduced
by Brouwer and van Woerkom modify the oscillation limits for the orbital eccentricities of Mercury and Uranus somewhat, but do
not significantly change the limits for the other planets. Of course, the oscillation limits for the orbital inclinations are
unaffected by these refinements (because the inclination eigenmodes are unaffected).
Figure 10.5:
The maximum and minimum orbital eccentricities predicted by Brouwer and van Woerkom's
refinement of standard LaplaceLagrange secular evolution theory (filled circles), compared with the corresponding
limits from Table 10.4 (open circles).

It must be emphasized that the Brouwervan Woerkom secular evolution theory is only approximate in nature. In fact,
the theory is capable of predicting the secular
evolution of the solar system with reasonable accuracy up to a million or so years into the future or the past.
However, over longer timescales, it becomes inaccurate because the true longterm dynamics
of the solar system contains chaotic elements. These elements originate from two secular resonances among the
planet:
, which is related to the gravitational interaction of Mars and the Earth; and
, which is related to the interaction of Mercury, Venus, and Jupiter (Laskar 1990).
Next: Secular evolution of asteroid
Up: Secular perturbation theory
Previous: Evolution equations for twoplanet
Richard Fitzpatrick
20160331