GENERALIZED AVERAGING PRINCIPLE AND THE SECULAR
EVOLUTION OF PLANET CROSSING ORBITS
GIOVANNI FEDERICO GRONCHI
Dept. of Mathematics, University of Pisa
Abstract. Planet crossing orbits give rise to mathematical singularities that make not possible to
apply the classical averaging principle to study the qualitative evolution of Near Earth Asteroids
(NEAs). Recently this principle has been generalized to deal with crossings in a mathematical model
with the planets on circular coplanar orbits. More accuracy is needed to compute the averaged
evolution of planet crossing orbits for different purposes: computing reliable crossing times for the
averaged motion, writing more precise proper elements and frequencies for NEAs, etc. In this paper
we present the generalization of the averaging principle using a model where the eccentricity and the
inclination of the planets are taken into account.
1. Introduction
The averaging principle is a powerful tool to study the qualitative behavior of the
solutions of Ordinary Differential Equations.
It consists in solving averaged equations, obtained by an integral average of the
original equations over some angular variables; if some conditions are satisfied,
the solutions of the averaged equations remain close to the solutions of the original
equations for a long time span. A review of the classical results on averaging
methods in perturbation theory can be found in (Arnold, Kozlov and Neishtadt
1997).
These methods have been successfully applied to compute the secular evolution
of the orbits of Main Belt Asteroids (Williams 1969). In this case the average is
done with respect to the fast angles representing the mean anomalies of the asteroid
and the planets present in the model.
An analytic study of the restricted circular twice-averaged three body problem
Sun-planet-asteroid can be found in (Lidov and Ziglin 1974) in the case in which
the orbit of the asteroid is uniformly close to the orbit of the planet.
We cannot apply the classical averaging principle if we want to compute the
secular evolution of a Near Earth Asteroid (NEA) because of the crossings that
may happen between its orbit and the orbits of the planets (especially the Earth);
an intersection of the orbits generates a second order polar singularity that makes
divergent the integrals in the right hand sides of the averaged equations.
The study of the dynamics of NEAs requires the development of theories that
can give statistical informations on the evolution of these objects, because the
evolution starting from a single initial condition is not representative for long time
Celestial Mechanics and Dynamical Astronomy 00: 1–23, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.1
2
spans due to the chaoticity of the orbits; so it is desirable to have an averaging
method also in this case.
Recently we have modified the classical averaging principle (Gronchi and Milani 1998) in order to compute the secular evolution of NEAs. We have shown
that also in the planet crossing case we can define piecewise smooth solutions
of equations involving the averaged perturbing function R, that correspond to the
classical averaged equations if there are no crossings. We have also used this theory
to compute proper elements and frequencies for all the known NEAs (Gronchi and
Milani 2001).
In the model used for these works, the planets have been considered on circular
coplanar orbits, following the example of (Kozai 1962): this simplification gives
rise to interesting properties of the phase space, such as the periodicity of the
averaged eccentricity and inclination (e I) as functions of the perihelion argument
ω and the existence of stable states, in a neighborhood of which there are possible
stable motions avoiding crossings (Gronchi and Milani 1999).
The results of this modified averaging principle, applied to NEAs, have been
compared with the results of numerical integrations of the full equations using
initial conditions of circular coplanar orbits for the planets. The comparison appears quite satisfying (Gronchi and Michel 2001) and the requirements to apply
the principle seems to be the same of the classical averaging: no mean motion
resonances between the asteroid and a planet nor close approaches between them.
We have also done a comparison of the previous results with full numerical
integrations starting from real initial conditions of the planets, that is taking into
account their eccentricities and inclinations. Even if the discrepancies concerning
the proper elements are not dramatic, the need of an averaging theory with the
planets on elliptic/inclined orbits is evident: such a theory would permit to obtain
more realistic times of crossing between the averaged orbits of NEAs and the orbits
of the planets, especially of the Earth, and then to do more reliable predictions
concerning the possibility of collisions; the improvement of the accuracy of the
crossing times would be useful also in the search for parent bodies of meteor
streams, using appropriate variables like the ones in (Valsecchi et al. 1999).
The generalization of the averaging principle including the eccentricities and
the inclinations of the planets is the subject of this paper. We have not included
here some computational details that would make longer the explanation of the
results: the reader interested can found them in (Gronchi 2002).
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.2
3
2. The averaged equations
Let us consider a restricted three body problem, Sun-planet-asteroid: we use, in a
heliocentric reference frame, the Delaunay variables L G Z g z , defined by
L k a
G k a 1 e2 Z k a 1 e2 cos I
g
z
nt
ω
Ω
t0 for the asteroid, where a e I ω Ω is the set of the Keplerian elements, k is
Gauss’s constant, n is the mean motion and t 0 is the time of passage at perihelion;
we also use the corresponding variables with a prime L G Z g z for the
planet.
The orbit of the planet is completely determined: it is the solution of the two
body problem Sun-planet. If we set E D G Z g z , the averaged equations of
motion for the asteroid can be written in the following form:
E˙ D
where E D
J ∇E D R
t
(1)
G Z g̃ z̃ are averaged Delaunay variables, J is the 4 4 matrix
J
O 2 I2
I2 O 2 t
(O2 and I2 are respectively the 2 2 zero and identity matrixes) and ∇ E D R is the
transposed vector of the integral average over of the partial derivatives of the
perturbing function R with respect to E D
∇E D R 1
2π π
π
2 π π
∇E D R d d ;
∇E D R ∂R
∂E D
The solutions of equations (1) are representative of the solutions of the full
equations of motion if there are no mean motion resonances between the asteroid
and the planet and if no close approaches occur between them.
3. Weak averaged solutions
The idea of the generalization of the averaging principle in (Gronchi and Milani
1998) starts from the fact that if there are no crossings between the orbits, by the
theorem of differentiation under the integral sign (Fleming 1964), the averaged
equations of motion (1) are equivalent to Hamilton’s equations with the averaged
perturbing function R:
E˙ D
J ∇E D R t
(2)
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.3
4
where
R
1
2π π
π
2 π π
R d d
(3)
We shall prove that it is possible to define piecewise smooth solutions of
equations (2) and the loss of regularity corresponds exactly to the intersection
configurations of the orbits: in fact we can give a twofold meaning to the right
hand sides of equations (2) at the node crossing, corresponding to two limit values
of the time derivatives (left and right) of the solutions at crossings.
We define as weak averaged solutions the solutions of equations (2). We observe
that they correspond to the classical averaged solutions as far as their trajectories in
the phase space do not pass through node crossing conditions between the orbits.
We also observe that the exchange of the differential and integral operators in
(2) is not essential for a theoretical definition of the weak solutions (they could
anyway be defined as the limits of the solutions of (1) coming from both sides of
the node crossing hyper-surfaces, defined later) but, as we shall see, this operation
is necessary to obtain analytic formulas for the discontinuity of the average of
the derivatives of R, that are not defined at the node crossings, and to define the
semianalytic procedure to compute the weak solutions.
In the next sections we shall see that it is possible to write R as a function
of a certain set of variables, called mutual elements; these elements are almost
everywhere regular functions of the Delaunay variables and are defined by the
mutual position of the osculating orbits of the asteroid and the planet. Using the
chain rule we shall see that we can write the equations of motion for the asteroid
in the more explicit form
˙
G
Z˙
∂R ∂E M
∂E M ∂g
∂R ∂E M
∂E M ∂z
g̃˙ z̃˙ ∂R ∂E M
∂E M ∂G
∂R ∂E M
∂E M ∂Z
(4)
where E M is a suitable subset of the mutual elements.
REMARK 1. In (Gronchi and Milani 1998) the variable Z is an integral of the
motion, corresponding to the third component of the angular momentum; here this
property is not valid any more.
The advantages coming from the use of the mutual elements will be clear in
the next sections: they will allow us to use the same formalism as in the circular
coplanar case for the planets. We shall skip the ‘tilde’ over the averaged variables
in the following to avoid the use of heavy notations.
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.4
5
4. The mutual reference frame
Given two closed non-coplanar osculating orbits of an asteroid and a planet in
the three body problem Sun-planet-asteroid we recall that the mutual nodal line is
the intersection between the two orbital planes. Furthermore we define as mutual
node each pair of points on the mutual nodal line, one belonging to the orbit of the
asteroid and the other to the one of the planet, that lie on the same side of the mutual
nodal line with respect to the common focus of the two conics. In this framework
there are two mutual nodes, the ascending and the descending one: they differ in
the change of sign of the z component along the asteroid orbit (negative to positive
in the first case and vice-versa in the second).
The mutual reference frame x y z is a heliocentric reference system such that
the x-axis is along the mutual nodal line, directed towards the mutual ascending
node; the y axis lies on the planet orbital plane, so that the orbit of the planet
lies on the x y plane (see Figure 1). We shall use the further convention that the
positive z-axis is oriented as the angular momentum of the planet.
z
mutual nodal
line
IM
ωM
*
y
ωM’
*
x
Figure 1. The mutual reference frame: the orbit of the planet lies on the x y plane; the x axis is along
the mutual nodal line and is oriented towards the ascending mutual node, marked with asterisks in
the figure.
Let ωM ωM
be the mutual pericenter arguments (the counter-clockwise angles
between the x axis and the pericenters) of the orbit of the asteroid and of the planet
respectively, and let IM be the mutual inclination between the two conics. We define
as mutual elements the set of variables a e a e ωM ωM
IM .
We can express ωM ωM
IM as functions of the Keplerian elements
ω Ω I ω Ω I , defined using a fixed heliocentric reference frame. Let us consider the unit vectors N ast and N pl , corresponding to the direction of the angular
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.5
6
momentum of the asteroid and the planet; each of them is orthogonal to one of the
osculating orbits and they are functions of the Keplerian elements:
N ast
sin Ω sin I
cos Ω sin I
cos I
N pl
;
sin Ω sin I cos Ω sin I cos I The unit vector pointing to the mutual ascending node is orthogonal to both
N ast and N pl , so it is defined by
Anod
NN pl
pl
N ast
N ast
where is the symbol of the vector product in
0 π is defined by
cos IM
N
(5)
N ast
pl
3.
The mutual inclination IM
where
is the Euclidean scalar product.
Let us consider the unit vectors X 0 X 0 , pointing to the position of the
perihelion of the orbit of the asteroid and the planet respectively; they also are
functions of the Keplerian elements:
X 0
cos Ω cos ω sinΩ sin ω cos I
sin Ω cos ω cosΩ sin ω cos I
sin ω sin I
;X 0
cos Ω cos ω sinΩ sin ω cosI
sin Ω cos ω cosΩ sin ω cosI
sin ω sin I
and the mutual perihelion arguments ω M ωM
are given by
A X 0 A X 0
cos ωM
sin ωM
nod
nod
N ast
A X 0 A X 0 N cos ωM
sin ωM
nod
nod
pl
REMARK 2. Let E K a e I ω
and E K be the subset
of the Keplerian elements of the asteroid and the planet defining the averaged
perturbing function. The transformation
Ω
E
K
E K
a e I ω Ω a e a e ωM ωM
IM from Keplerian to mutual elements is not 1-1; in fact you can rigidly rotate both
orbits together around the origin of the fixed reference system and the mutual
elements do not change, while there is a change in the Keplerian ones. Notice
that equation (5) is not defined if the mutual inclination between the two orbits is
zero.
We consider the subset E M a e ωM ωM
IM of the mutual elements de
pending on the asteroid Keplerian elements. The derivatives of the mutual elements
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.6
7
with respect to the Delaunay variables appearing in equations (4) can be easily
computed by the chain rule, passing through the Keplerian elements of the asteroid:
∂E M
∂E D
in fact we observe that
O ∂E M
∂E K
O2 3
I2
∂E M ∂E K
;
∂E K ∂E D
IM ∂ ω M ωM
∂ I ω Ω
t
23
and
∂E K
∂E D
M O2t 3
O2 I2
(6)
where I2 is the 2 2 identity matrix, O2 and O2 3 are respectively the 2
2 3 zero matrixes, and
M
with β 1
k a
0
β e
cotanI β
0
0
1 β sin I 2 and
1 e2 .
REMARK 3. Note that some of the coefficients of the matrix M becomes singular
for I 0. This is a fictitious singularity that can be removed by a rotation of the
fixed reference frame.
More generally, given a planetary model with the Sun, N planets (N 1) and
an asteroid, where the eccentricities and the inclinations of the orbits of the planets
are taken into account, we can define N mutual reference frames using the orbits of
the asteroid and one planet at a time, provided that each of the mutual inclinations
is different from zero.
In Figure 2 we show the two mutual reference frames y 1 y2 y3 and z1 z2 z3 in
the case of four bodies: the Sun, two planets and an asteroid; we also draw the fixed
reference frame x1 x2 x3 .
When we consider a restricted N 2 body problem with N 1, we suppose
the motion of the planets as known a priori: this motion can actually be computed,
for example, by means of a synthetic theory like in (Carpino et al. 1987). We can
take advantage of the simplified description of the perturbing function R in the
mutual reference frames to use a formalism very close to the one of the coplanar
case for the planets. The perturbing function is a sum of terms R i (with i 1 N)
of
and in each Ri the dependence on the variables of the asteroids and the ones
the only planet i appears. We can study the case of only one perturbing planet: the
perturbation of all the planets that we want to include in the model, up to the first
order in the perturbing masses µi (the ratio between the mass of the planet i and the
mass of the Sun), will be obtained by the sum of the contribution of each planet.
From now on we shall consider a system of three bodies: Sun, planet, asteroid.
The quantities related to the planet will be distinguished by a prime from the same
quantities for the asteroid.
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.7
8
x3
y3
z3
mutual nodal line
with planet 2
mutual nodal line
with planet 1
y2
planet 2
planet 1
Sun
z2
x1
x2
y1
z1
asteroid
Figure 2. Two mutual reference frames y1 y2 y3 and z1 z2 z3 are defined by the pairs of orbits
asteroid, planet 1 and asteroid, planet 2 . Also the fixed reference frame x1 x2 x3 is drawn in
the figure.
5. Geometry of the osculating orbits
In the mutual reference frame, as described in Figure 1, we can write the equations
of the two osculating orbits of the asteroid and the planet
P u :
P u :
x a cos ωM cos u e y a sin ωM cos u e z a sin ωM cos u e x
y
z
β sin ωM sin u
β cos ωM sin u cos IM
β cos ωM sin u sin IM
a cos ωM
cos u e a sin ωM
cos u e 0
β sin ωM
sin u β cos ωM
sin u (7)
(8)
where β 1 e2 , β 1 e 2 and u u are the eccentric anomalies of the
asteroid and the planet respectively.
The positive distance between a point on the orbit of the asteroid and a point on
the orbit of the planet is defined by its square
2
u u P u P u a2 1 e cos u 2 a 2 1 e cos u 2 β sin u sin ωM
2aa cos u e cos ωM β sin u sin ωM cos u e cos ωM
cos IM cos u e sin ωM β sin u cos ωM cos u e sin ωM
β sin u cos ωM
2
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.8
9
For later use we define the function
D
u
u
5.1. C ROSSING
CONDITIONS AND VALUES OF THE ANOMALIES AT THE
ASCENDING NODE
If the mutual inclination between the two orbits is different from zero, the node
crossing conditions between the asteroid and the planet are represented, in terms
of the mutual variables, by
dnod
dnod
1
1
aβ2
e cos ωM
aβ2
e cos ωM
1
1
a β 2
e cos ωM
a β 2
e cos ωM
0
(ascending node crossing) ;
0
(descending node crossing)
As the theory in both cases is formally equivalent, we shall discuss only the
ascending node crossing case.
The relations
a 1 e cos u 1
aβ2
;
e cos ωM
a 1 e cos u 1
a β 2
e cos ωM
(9)
define the value of the eccentric anomalies u u at the ascending node (even if there
are no crossings): from (9) we obtain
cos u ∂u
∂e
cos ωM e
;
1 e cos ωM
sin ωM
;
β 1 e cos ωM ∂u
∂ωM
β sin ωM
;
1 e cos ωM
sin u 1
β
e cos ωM
We also define
u e sin u
that is the value of the mean anomaly of the asteroid at crossing. These formulas
are valid also for the planet by formally replacing u e ω M β with u e ωM
β .
6. The Wetherill function
Let us consider the two straight lines r r tangent to both ellipses in the ascending
nodal points P u P u and parametrized with the mean anomalies so that
P u t and r t have the same velocities (derivatives with respect to t) in P u Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.9
10
and P u t and r
are
t have the same velocities in P u . The equations for r r x x F y y G cos IM z z G sin IM r:
x
y
z
r :
x
y
0
F
G
where
F
G
ae sin ωM β 1
a 1 e cos ωM β
F
G
1
1
a e sin ωM
β
a 1 e cos ωM
β
1
and
x
1
aβ2
;
e cos ωM y z 0;
x
1
a β 2
;
e cos ωM
y
z
0
REMARK 4. The condition on the velocity at the tangency points can be easily checked using a reparametrization of the straight lines by the parameters
u u , that are related to the mean anomalies by Kepler’s equations
u e sin u
u
e sin u
(10)
DEFINITION 1. We call Wetherill function (Wetherill 1967) the positive distance
d between a point on r and a point on r . The square of this function is
d2
where k k
F F x x 2kk 2
z2
k2 F 2
GG cos IM 2
y y and k k
G F F
F 2
2dnod
k 2
k
G 2
2
k dnod 2
.
Note that d2 is a quadratic form in the variables k k : it is homogeneous when
there is a crossing at the ascending node. We can write it in the more concise form
d2
d2 κ
κtAκ
Btκ
dnod 2
(11)
where
κ
k k ;
F
A21
2
with components
dnod F dnod F
B1
B2
For later use we define
A
B 2 B 1 B2 ;
2
A11
A12
A22
G F F F G
2
2
2
A11 A12
A21 A22
;
GG cos IM as the squared Wetherill function in coordinates u u :
2
u u d2
u
u The geometry of Wetherill’s approximation of the orbits is strictly connected
with the degeneracy of the matrix A :
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.10
11
PROPOSITION 1. The matrix A is always positive definite if IM 0. If IM 0 we
have degeneracy of A if and only if the straight lines r r are parallel: in this case
A is positive semi-definite.
Proof. A is a symmetric 2 2 matrix and it is positive definite if and only if its
principal invariants, the trace tr A and the determinant det A , are positive. By a
direct computation we have
tr A det A a2
a 2
2
1
2e
cos
ω
e
1 2e cos ωM
M
β2
β 2
a2 a 2
e sin ωM cos IM 1 e cos ωM
β2 β 2
e sin ωM
1
e cos ωM 2
sin2 IM 1
e cos ωM
From the above expressions we deduce that tr A det A 0
IM 0
e sin ωM 1
e cos ωM
2
e 2 1
2e cos ωM
e2 0 and that
e sin ωM
1
e cos ωM 0
that corresponds to the straight lines defined by r r being parallel.
DEFINITION 2. We call tangent crossings the crossing orbital configurations for
which det A 0.
In the development of this theory we shall assume that the mutual inclination
IM is different from zero during the whole evolution, or at least in a neighborhood
of each crossing between the orbits; this implies that no tangent crossings occur.
7. Kantorovich’s method
The inverse of the Wetherill function 1 d can be used to extract the principal part
from the perturbing function, whose direct term is proportional to 1 D. The average
of the indirect term is zero, so it does not appear in this theory.
The distance D is a 2π-periodic function in both variables and this property
can be used to shift the integration domain 2 : π π π π
in a suitable way, according to the node where the orbits cross each other, in such
a way that the crossing values will be always internal points of this domain.
We shall use Kantorovich’s method (Demidovich 1966) to study the regularity
properties of the averaged perturbing function R defined in (3). When the orbits intersect each other this function is defined by a convergent integral of an unbounded
function, in fact the integrand has a first order polar singularity in the points ,
that correspond to collisions.
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.11
12
We shall prove that in computing the derivatives of R with respect to the mutual
variables, for instance the e-derivatives, we can use the decomposition
2π 2 ∂
R µk2 ∂e
∂ 1
2 ∂e
D
1
d d
d
∂
∂e 1
d d
2 d
(12)
where µ is the perturbative parameter corresponding to the ratio between the mass
of the planet and the mass of the Sun; that is we shall prove the validity of the
hypotheses of the theorem of differentiation under the integral sign to exchange the
symbols of integral and derivative in front of the remainder function 1 D 1 d.
Therefore we shall need only to study the regularity properties of the last term in
(12), which is easier to handle.
REMARK 5. We use Kantorovich’s method of singularity extraction in a wider
extent: the derivatives of the remainder function still have a polar singularity, but
it is of order one, so that the integrals over of these derivatives are convergent.
Actually the decomposition (12) allows us to extract the main singular term from
the derivative of R.
8. Integration of 1 d
Let 2 crossing point
:
π π π π . First we move the ascending node
to the origin of the reference system by the translation
τ :
k k κ
(13)
2
with k k ; we call
the translated domain. Then we perform a
coordinate change to eliminate the linear terms in the squared Wetherill function
d 2 κ defined by equation (11). We define the inverse of the transformation used
for this purpose
Ξ
1
:ψ
κ T ψ
S
(14)
2 ψ
2 are the new variables, and T is a 2 2
where S S1 S2 y y real-valued matrix.
We obtain the vector S by setting to zero the coefficients of the linear terms in
the quadratic form in the new variables ψ: the equation to solve is
2AS
B A
B 0
whose solutions are
S1
1 22 B2 A12 det A (15)
;
S2
B1 A12 B2 A11 det A Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.12
13
Furthermore we can determine the non-degenerate matrix T in order to obtain
T t AT
I2
where I2 is the 2 2 identity matrix: we set
T
1 τ
0
σ τρ
1 ρ with
τ
det A ;
A11
ρ
A11 ;
The transformation
Ξ:κ
ψ R κ S
1
T
1
A11
where
R
σ A12 τ
0
(16)
σ
ρ
brings the quadratic form d 2 κ into a simpler form in the new variables ψ, and the
2
domain
is transformed into a parallelogram with two sides parallel to the y axis
(see Figure 3); after performing the change of variables (16) we have
d2 Ξ
where
d dmin
nod
1
1
ψ y 2
y2
dmin 2
1
2 2
F G 2F F GG cos IM
det A F 2G 2
1 2
(17)
is the minimum distance between the two straight lines r and r .
REMARK 6. dmin is in general smaller than the absolute value of the nodal distance dnod , and it is zero only when dnod 0. It can be used as an analytic
approximation for the MOID (Minimum Orbital Intersection Distance) (Bonanno
1999; Valsecchi et al. 2002) when the minimum distance between the two orbits is
reached in a neighborhood of the nodal points (usually when the mutual inclination
is not small).
Using the variable changes (13), (16) and the transformation to polar coordinates, whose inverse is
Π
1
:
r
θ
y
y
r cos θ
r sin θ
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.13
14
τ -l’,l-
l
l
Ξ
k
l’
y
v3 r 2
k’
l’
v2
r3
r1
v4
2
2
y’
v5
r4
2
Ξ[
]
2
after the two coordinate changes that
Figure 3. Description of the transformations of the domain
2
are used to bring the squared Wetherill function into the form y2 y 2 dmin in the new variables
y y . In the parallelogram on the right side we show the decomposition of the domain used for the
integration.
we have
1
d d
2 d
1
det A 1
where Ξ 1 Π 1 the r variable we obtain
2
Ξ
where ri , with i 1
by
2
4
∑
y2
r
r2
1
θi dmin 1
y 2
2
dmin dy dy
dr dθ
dmin i 1 θi
2
2
:
2
. Integrating in
ri2 θ dθ 2πdmin 4, are the lines delimiting the parallelogram Ξ
r1 θ r3 θ 2
and
1
det A det A 1
d d
2 d
ρτ π S1 ;
ρ cos θ σ sin θ
ρτ π S1 ; r4 θ ρ cos θ σ sin θ
r2 θ ρ π
S2 sin θ
(18)
2
, defined
;
ρ π S2 ;
sin θ
θ1 θ5 2π and θ j , with j 2 5, are the counter-clockwise angles between
the y -axis and the vertex v j seen from
the origin of the axes (see Figure 3). These
angles are defined by the following relations:
0 θ2 θ3 π θ4 θ5 2π ;
tan θ2
ρ π
σ π
S2 S2 τ π
S1 ;
tan θ3
ρ π
σ π
S2 S2 τπ
S1 Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.14
;
15
tan θ4
σπ
ρπ
S2 S2 τπ
S1 ; tan θ5
ρπ
σπ
S2 S2 τ π
S1 REMARK 7. The integrals in formula (18) are elliptic and the integrand functions
are bounded. These integrals are differentiable functions of the mutual orbital elements E M , therefore we have only to control the term d min in order to study the
differentiability of the integral average of 1 d.
9. Boundedness of the remainder function
Let κ k k . If there is a crossing at the ascending node, from
the equations of the orbits (7), (8) and from Kepler’s equations (10) we deduce that
Taylor’s development of D 2 κ D2 in a neighborhood of κ 0 0 is given
by
D 2 κ
d 2 κ
O κ 3
where O κ 3 is an infinitesimal of the same order as κ
the following
3
for κ
(19)
0. We prove
LEMMA 1. If there is an ascending node crossing between the orbits, there exist
a neighborhood U0 of κ 0 0 and two positive constants B1 B2 such that
B1 d 2 κ D 2 κ
B2 d 2 κ κ
U0
Proof. First we notice that d 2 κ κt A κ is a homogeneous quadratic form in κ
because the two orbits intersect each other. A is positive definite, hence there exist
two positive constants C1 C2 such that
C1 κ 2
κt A κ
C2 κ 2
κ
From the relations (19) and (20) we can easily prove that
lim
κ
0
D 2 κ
d2 κ
2
(20)
1
that directly implies the existence of U 0 and of the constants B1 B2 .
We can now prove this result:
PROPOSITION 2. The remainder function 1 D 1 d is bounded even if there is
an ascending node crossing between the orbits.
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.15
16
Proof. If there are no crossings the remainder function is trivially bounded, in fact
2 and the minimum value of d is d
D 0 for each min that,
from equation (17) and for IM 0, can be zero only if dnod 0.
If there is a crossing at the ascending node we have to investigate the local
behavior of the remainder function in a neighborhood of , where
both D and d can vanish. The boundedness of the remainder function can be shown
using the previous lemma, for which there exists a neighborhood U 0 and a positive
constant B1 such that the relations
d κ
D κ
D κ
B1 d κ ;
1
B1 d κ hold for each κ U0 . It follows that the local behavior of the remainder function
can be bounded in this way:
1
D κ
1
d κ
d
We observe that d
κ D 2 κ
d κ D κ d κ D κ
κ
constant C1 such that d 2
2
2
B 1
B1 2
κ
κ
D 2 κ
3
κ
2
1
1
D κ
3
1
2
1
d κ
L
κ
U0
REMARK 8. Although the remainder function 1 D 1 d is bounded, it is not
continuous in when there is a crossing at the ascending node; it can
be seen, for instance, by computing the limits of this function along the straight
lines k λk λ
0.
as k
10. Singularities of the EM -derivatives of R
Kantorovich’s method is used to describe the singularities of the derivatives of
the averaged perturbing function with respect to the mutual variables appearing in
equations (4).
First we prove that the derivatives of the remainder function are always
Lebesgue integrable over 2 , so that the average of the remainder function is
differentiable: indeed its derivatives can be computed by exchanging the position
of the integral and differential operators as in (12). Then we shall see that, if the
two orbits intersect each other, there is a discontinuous term in the derivatives of
the average of 1 d that is responsible of the discontinuity of the derivatives of R.
These derivatives admit two limit values at crossing (coming from d nod 0 and
from dnod 0).
3
d3 κ
D κ O κ and that by (20) there is a positive
κ C κ ; hence there exists a constant L 0 such that
d
1
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.16
17
10.1. S INGULARITIES
e ω M ωM
- DERIVATIVES
OF THE OF THE REMAINDER
FUNCTION
Let us set υ u u and ν v v u u u u . We can apply Taylor’s formula
with the integral remainder to the vector functions P u P u and write
P u
P u
P u
Pu u v
P u u
u s Pss s ds
u
Pu u v
u
u
u
t Ptt t dt
The vector functions defining the straight lines r u r u and r u u have the same Taylor’s development, up to the first order in ν v2 v 2 ,
as P u and P u respectively, so that we can write
r r u
P u
r u P u Pu u v
u
Pu u v
u s rss s ds
u
u
u
u
t rtt t dt
We prove the following
THEOREM 1. If there is an ascending node crossing between the orbits at
u u u u , the derivatives of the remainder function 1 1 with respect
to the mutual elements e ωM ωM
, computed at the node crossing, can be bounded
by functions with a first order polar singularity in u u , so they are Lebesgue
integrable over 2 .
Proof. We shall consider only the derivatives with respect to e: the proof for the
other derivatives is similar. First we note that
∂
∂e
1
υ
Let us write
where
2
e0 2
e1 2
e2 1
2
3
∂
υ ∂e
∂
∂e
2
υ
2
∂
∂e
;
1
υ
for the Euclidean scalar product. We have
∂
P u
∂e
∂
P u
2
∂e
∂
P u
2
∂e
2
υ
2
e0
2
e1
P u 2
2
e2
P u P u
P u Pu u v Pu u v P u υ
(21)
;
;
u
u
u
2
∂
υ ∂e
1
3
u s Pss s ds u
u
t Ptt t dt Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.17
18
and
∂
∂e
υ
2
2
e0
2
e1
2
e2
(22)
where
∂
r u
∂e
∂
r u
2
∂e
∂
2
r u
∂e
2
e0 2
2
e1 2
e2 r u P u
r u Pu u v Pu u v r
;
;
u
u
u u s rss s ds u
If we set the crossing conditions P u u
u
t r tt t dt P u , we have
2
e0 P u 2
e0 0
and, in particular, the constant terms in Taylor’s developments of ∂ 2 ∂e and
∂ 2 ∂e vanish.
The terms defined by 2e 2 and 2e 2 are at least infinitesimal of the second order
with respect to ν as υ
u u , so that the first order terms in ν at crossing can
2
be given only by e 1 and 2e 1 .
Using the theorems on the integrals depending on a parameter we obtain
∂
∂e
u
u
u
u s Pss s ds u
u
t Ptt t dt
u
∂Pss
∂u
∂P
u s
s ds u t tt t dt
Puu u v ∂e
∂e
∂e
u
u
u
u
∂
s rss s ds u
u t r tt t dt ∂e u
u
u
u
∂rss
∂u
∂r tt
u s
s ds u t t dt
ruu u v ∂e
∂e
∂e
u
u
u
∂u
P u v ;
∂e u u
∂u
r u u u v ∂e
so that these two expressions are at least infinitesimal of the first order with respect
to ν . As this terms are multiplied by first order terms in the expressions of 2e 1
and 2e 1 , they give rise to at least second order terms.
We can conclude that the first order terms in the expressions (21) and (22) are
equal and they are given by
2
∂
P u
∂e
P u ∂u
Pu u ∂e
∂u
P u ∂e u
Pu u v Pu u v ;
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.18
19
therefore the asymptotic developments of the e-derivatives of
a neighborhood of υ u u are
∂
∂e
υ
2
αv
β v
∂
∂e
υ ;
υ
2
where α β are independent on u u and
υ and
second order with respect to ν as υ
u u .
Using the decomposition
1
1
3
3
1
1
1
2
β v
2
υ in
υ
υ are infinitesimal of the
1
αv
υ and
2
1
2
the boundedness of the remainder function 1 1 and Lemma 1 (that do not
depend on the choice of the coordinates used for the proof), we conclude that there
exist two constants L1 L2 0 such that
∂
∂e
1
υ
∂
∂e
in a neighborhood of υ
theorem.
1
υ
1
3
1
2
1
υ
υ
υ
3
3
1
υ
3
1
υ
υ L1
β v υ1 L2
u u . This is sufficient to prove the statement of the
10.2. T HE e ωM ωM
- DERIVATIVES
OF THE AVERAGE OF
αv
1 d
2
As det A 0 and is in the interior part of 2 , we have dmin
ri2 θ 0 for
each θ θi θi 1 and for each i 1 4. Then we can use again the theorem of
differentiation under the integral sign and
compute, for instance, the derivative of
the average of 1 d with respect to e as
∂
∂e 1
d d
2 d
∂
∂e
1
det A 1
det A 4
1
2 i∑
1
θi θi
1
θi 4
∑
1
dmin i 1 θi
∂
dmin ∂e
dmin 2
2
ri2 θ 2
ri2 θ dθ 2πdmin ∂
d
∂e min
dθ 2π
ri2 θ (23)
We have similar formulas for the derivatives with respect to ω M ωM
, obtained
simply by substitution of the partial derivative operators.
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.19
20
The discontinuities present in the terms
∂
d ;
∂e min
∂
d ;
∂ωM min
∂
dmin
∂ωM
are responsible of the discontinuities in the derivatives of the averaged perturbing
function.
10.3. S INGULARITIES
R
OF THE IM - DERIVATIVE OF
We observe that we do not need to perform the splitting of Kantorovich’s method
to compute the derivative with respect to IM of the integral average of 1 , in fact
we have the following
PROPOSITION 3. If there is an ascending node crossing, the derivative of 1 with respect to the mutual inclination IM can be bounded by a function with a first
order polar singularity in u u , so it is Lebesgue integrable over 2 .
Proof. We have
∂
∂IM
1
υ
1
2
3
∂
υ ∂IM
2
υ
so we need to prove that ∂ 2 ∂IM is an infinitesimal of the second order with
respect to ν as υ
u u . For this purpose we only need to check the vanishing
of the term
2
∂
P u
∂IM
P u ∂u
Pu u ∂IM
∂u
P u ∂IM u
Pu u v Pu u v (24)
that formally represents the first order terms in the derivative of 2 υ with respect
to IM , as we can see from similar computations in Theorem 1. The expression in
(24) vanishes because P u P u u and u do not depend on IM .
REMARK 9. The previous proposition puts in evidence an aspect of the singularities of the derivatives of R that also appeared in the theory developed by (Gronchi
and Milani 1998): there is a variable such that the derivative of R with respect to it
is a continuous function.
Using a model with all the planets on circular coplanar orbits, the time derivative
of the averaged motion of the node does not present loss of continuity at node crossings because the derivative of the perturbing function with respect to the inclination
between the asteroid orbits and the common orbital plane of the planets has a first
order polar singularity, just like the derivative with respect to I M in Proposition 3.
On the other hand in this work the derivative with respect to the inclination I of
the asteroid orbit with respect to a plane of the fixed frame has not necessarily a
first order polar singularity because I I IM ωM ωM
, so that the derivatives with
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.20
21
respect to both ωM and ωM
, that have second order polar singularities, are also
involved:
˙
Ω t
1
∂R
kβ a sin I ∂I 1
∂R ∂IM
kβ a sin I ∂IM ∂I
∂R ∂ωM
∂ωM ∂I
∂R ∂ωM
∂ωM
∂I We conclude that, if we take into account the
eccentricities and the inclinations
of the planets, the time evolution of the node Ω t is not necessarily smooth.
10.4. D ISCONTINUITY
IN THE DERIVATIVES OF
R
The averaged perturbing function R can be regarded as a function of the mutual variables a e a e ωM ωM
IM ; the e ωM ωM
-derivatives of R are smooth
functions in each of the two regions d nod 0 and dnod 0 and they have a smooth
extension onto the boundary of each of these two regions. Correspondingly these
derivatives have a twofold definition on the (hyper-) surface d nod e e ωM ωM
0.
The results of the previous sections allow us to write formulas for the differences between the two values of the derivatives of R at the ascending node
crossing.
DEFINITION 3. Let
∂
∂
∂
;
;
∂e ∂ωM ∂ωM
and
∂
∂
∂
;
;
∂e ∂ωM ∂ωM
be the partial derivative operators applied in the regions of the space where d nod
0 and dnod 0 respectively.
We define the operator ‘Diff’ to describe the differences in the right hand sides of
equations (2) at dnod 0, when we pass from a region where dnod 0 to a region
where dnod 0. We have
Diff
Diff
Diff
Diff
∂R
∂G ∂ R
∂G
∂ R
∂G
β
∂R
Diff
ke a
∂e cotanI ∂ωM
∂R
Diff
kβ a ∂I
∂ωM ∂R
∂Z ∂ R
∂Z
∂ R
∂Z
∂R
∂g ∂ R
∂g
∂ R
∂g
∂ωM
∂R
Diff
∂ω
∂ωM ∂R
∂z ∂ R
∂z
∂ R
∂z
∂ωM
∂R
Diff
∂Ω
∂ωM 1
∂ωM
∂R
Diff
kβ a sin I ∂I
∂ωM ∂ω∂I Diff ∂ω∂R ;
∂ω∂I Diff ∂ω∂R ;
M
M
M
∂ω∂ω Diff ∂ω∂R ∂ω∂Ω Diff ∂ω∂R M
M
;
M
M
;
M
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.21
22
where
Diff
∂R
∂e Diff
∂R
∂ωM Diff
∂R
∂ωM ∂ R
∂e
∂ωM
and
∂
d
∂e nod Diff
∂
d
∂ωM nod Diff
∂
d
∂ωM nod Diff
Γ EK E
∂ R
∂ωM k2 µ
2π 2
K
∂ R
∂e ∂ R
∂ωM ∂ R
with
Γ EK E
K
Γ EK E
Γ EK E
Diff
∂ R
∂ωM 2π
det A
1
∂
d
;
∂e nod K
Diff
∂
d
;
∂ωM nod K
Diff
∂
d
;
∂ωM nod 1
det A
F 2 G 2 2F F GG cosIM
F 2G 2
1 2
2a cosω 1 e 2e 1 e cosω ;
e sin ω ;
∂ ∂ω d ∂ ∂ω d 2a 1 1 eecosω
e sin ω ∂ ∂ω d ∂ ∂ω d 2a 1 1 ee cos
ω ∂ dnod
∂e
∂ dnod
∂e
nod
nod
M
M
nod
nod
M
M
2
M
M
2
2
M
M
2
2
M
M
2
11. Conclusions
We can use the formulas of Section 10 to define piecewise smooth solutions of
equations (2). Let us consider two constant values for the semimajor axis a a such
that there are no mean motion resonances between the asteroid and the planet, and
let us suppose that close approaches are avoided during the evolution of the asteroid. We also assume that the time evolution of the variables e I ω Ω is known
and consider equations (2). The derivatives of the averaged perturbing function
R are continuous functions (indeed smooth) of the mutual variables in each of
the connected regions where crossings are avoided; some of these derivatives are
subjected to a loss of regularity
on the node crossing surfaces.
When a solution G t Z t g̃ t z̃ t arrives at the ascending node crossing
surface dnod 0 at time t , coming from a region in the phase space such that
dnod 0, we can take as the new Cauchy problem, defining the future motion, the
one obtained by adding the terms described in Section 10.4 to the right hand sides
of equations (2).
Note that if a solution arrives at dnod 0 coming from a region where dnod 0
we can define the new Cauchy problem by subtracting the same terms.
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.22
23
For very peculiar choices of the initial conditions it is possible that this definition does not give rise to a unique solution, but there are two ways of continuing
the solution after the singularity. This happens when the derivatives of d nod with
respect to the mutual variables vanish at the node crossing.
This theory can be used in a completely analogous way for crossings at the descending node, and it is also suitable to deal with double crossings, that is crossings
at both ascending and descending node with the same planet.
12. Acknowledgements
The author is grateful to A. Milani for his several suggestions and comments during
the writing of this work.
References
Arnold, V.I., Kozlov, V.V. Neishtadt, A.I.: 1997. ‘Mathematical Aspects of Classical and Celestial
Mechanics’, Springer
Bonanno, C.: 1999. ‘An analytical approximation for the MOID and its consequences’, Astron.
Astrophys. 360, 411–416
Carpino, M., Milani, A., Nobili, A. M: 1987. ‘Long-term numerical integrations and synthetic
theories for the motion of the outer planets’, Astron. Astrophys, 181, 182–194
Demidovic, B.P. and Maron I.A.: 1966. ‘Foundations of Numerical Mathematics’, SNTL Praha.
Fleming, W. H.: 1964. ‘Functions of Several Variables’, Addison-Wesley
Gronchi, G.F.: 2002. ‘Theoretical and computational aspects of collision singularities in the N-body
problem’, PHD Thesis, University of Pisa, in preparation
Gronchi, G.F. and Michel, P.: 2001. ‘Secular orbital evolution, proper elements and proper frequencies for Near-Earth Asteroids: a comparison between semianalytic theory and numerical
integrations’, Icarus 152, 48–57
Gronchi, G.F. and Milani, A.: 1998. ‘Averaging on Earth-crossing orbits’, Cel. Mech. Dyn. Ast. 71/2,
109–136
Gronchi, G.F. and Milani, A.: 1999. ‘The stable Kozai state for asteroids and comets with arbitrary
semimajor axis and inclination’, Astron. Astrophys. 341, 928–935
Gronchi, G.F. and Milani, A.: 2001. ‘Proper elements for Earth crossing asteroids’, Icarus 152, 58–69
Kozai, Y.: 1962. ‘Secular Perturbations of Asteroids with High Inclination and Eccentricity’, Astron.
J. 67, 591–598
Lidov, M.L. and Ziglin, S.L.: 1974. ‘The analysis of restricted circular twice-averaged three body
problem in the case of closed orbits’, Cel. Mech. Dyn. Ast. 9, 151–173
Valsecchi, G. B., Jopek, T. J. and Froeschlè, Cl.: 1999, ‘Meteoroid streams identification: a new
approach’, MNRAS 304, 743–750
Valsecchi, G.B., Milani, A., Gronchi, G.F., Chesley, S.R.: 2001. ‘Resonant returns to close
approaches: analytical theory’, Astron. Astrophys., submitted
Wetherill, G. W.: 1967. ‘Collisions in the asteroid belt’, J. Geophys. Res. 72 2429–2444
Williams, J.G.: 1969. ‘Secular Perturbations in the Solar System’, PHD Thesis , University of
California, Los Angeles.
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.23
Celmecproc_gronchi.tex; 30/01/2002; 22:32; p.24