Celestial Mechanics and Dynamical Astronomy manuscript No.(will be inserted by the editor)

Konstantin V. Kholshevnikov

Metric Spaces of Keplerian Orbits

Received: date / Revised version: date

Abstract Several metric spaces of Keplerian orbits and a set of their mostimportant subspaces, as well as a factor space (not distinguishing orbits withthe same longitudes of nodes and pericentres) are constructed. Topologicaland metric properties of them are established. Simple formulae to calculatethe distance are deduced. Applications to a number of problems of CelestialMechanics are discussed.

Keywords topological space, metric space, space of Keplerian orbits,manifold, embedding and imbedding in Euclidean space

1 Introduction

Many problems of Celestial Mechanics require an examination of topologi-cal and metric properties of spaces of Keplerian orbits. For example: SpaceDebris (identification of fragments of the same object); Space Guard (identi-fication of satellites launched together by the same rocket without detailed orany announcement); finding parent bodies for meteor streams; finding parentbodies for asteroids or comets having similar orbits.

Nevertheless for many years this topic did not attract sufficient attention.For the first time Gyorgyi (1968) has considered the connection of the Ke-plerian phase flow on an isoenergetic surface (in case of negative energy h)with a geodesic flow on a two-dimensional sphere, relying on symmetry prop-erties pointed out by Fock (1936). This result was independently obtainedby Moser in his more detailed paper (Moser, 1970). The generalization to anarbitrary value of h was made by Osipov: a short description of his resultscan be found in (Osipov, 1972), and a detailed exposition in (Osipov, 1977).

Astronomical Institute, St. Petersburg State University, Universitetsky pr. 28,St. Petersburg, Stary Peterhof, 198504 Russia, andInstitute for Applied Astronomy, nab. Kutuzova 10, 191187 RussiaE-mail: kvk@astro.spbu.ru

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The flow on a sphere must be replaced with the flow on an Euclidean planeor on a Lobachevskian plane in case of h = 0 and h > 0, respectively. Thisresult was repeated by Belbruno (1977) using a simpler representation viathe flow on one sheet of a two-sheeted hyperboloid embedded in a Lorentzspace (inducing a positive definite Riemannian metric on the hyperboloid).

A consequence important to the purpose of this paper was pointed out byMoser (1970): the 4-dimensional space of Keplerian orbits with fixed negativeenergy has the topological structure of the product of two two-dimensionalspheres. A simple proof without complicated considerations of flows is seen in(Stiefel and Scheifele, 1971, Part 3) (compare with Section 5 of the presentpaper).

In this article we deal with several 5-dimensional spaces of Keplerianorbits and their subspaces. Recently (Kholshevnikov and Vassiliev, 2004) weattempted to metrize the 5-dimensional space of bounded Keplerian orbits(elliptical and rectilinear). This Holder–type metric is natural in the sensethat it is independent of the set of orbital elements.

Earlier only artificial metrics, so called D-criterion (Southworth and Haw-kins, 1963), and its modifications (Porubcan, 1977; Drummond, 1981; Jopek,1993; Jopek and Froeschle, 1997; Valsecchi et al., 1999), were used in suchkinds of problems. They represent a sum of weighted squared differences (orincreasing positive functions of differences) between elements of two ellipsesand depend strongly on the concrete system of the orbital elements. RecentlyKlacka (1996) has performed a detailed analysis of D-criterion properties. Inparticular, he has shown that D-criterion is not a metric in the mathematicalsense due to the violation of triangle inequality, and has given examples ofcorrect metrics, unfortunately depending on the system of elements.

The metric introduced in (Kholshevnikov and Vassiliev, 2004) works per-fectly in case of not very elongated orbits. But it cannot be generalizedto unbounded orbits. So it implies the incompleteness of the correspondingspace, and it is of little use for near-parabolic cometary orbits.

Here we represent the spaces of Keplerian orbits and of non-rectilinearKeplerian orbits as smooth manifolds embedded in an Euclidean space. Itallows us to metrize the manifolds using an Euclidean distance, and to deducethe topological structure of orbital spaces and their subspaces in a very simplemanner.

Hence the problem of representation of Keplerian orbits as points in ametric space is solved from the mathematical point of view. It is not so fromthe astronomical point of view. As J.Klacka notes, there exists an arbitrari-ness in choosing weight factors and the degree p defining Holder–type metric.Most probably the best choice of these parameters depends on the astronom-ical problem, and one set of them fits well to cometary and meteor orbits andthe other does for asteroidal ones.

Finally, we stress that we deal with unperturbed Keplerian orbits only.Our approach may be used for real celestial bodies during a time-span en-suring sufficiently small perturbations (days for Earth satellites, centuries forcomets and asteroids in case of avoiding close encounters with planets). Forlonger times our approach is valid under additional conditions. For regularperturbations one may use proper elements instead of osculating ones (Roig

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and Beauge, 2005). In case of small variations of eccentricities and inclina-tions (great perturbations of longitudes of pericentres and nodes are admissi-ble) one may use a 3-dimensional factor space instead of 5-dimensional spaceof orbits. But in case of great perturbations the Keplerian approximation oforbits does not work.

2 Orbits

In Astronomy the notion Keplerian orbit has several meanings (which areusually clear from the context). Let us list the main ones.

1. Solution to the equation

r + κ2

r

r3= 0 (1)

as a curve parametrized by time, and embedded in an appropriate space(configuration, phase, extended phase, and other ones). Here r ∈ D1 =R3 \ 0 is the position vector, r = |r|, κ2 is the gravitational parameter.The orbit may be thought of as a wire with a bead sliding along it.The orbit may be identified with a 6-dimensional initial vector of positionand velocity, so the space of orbits coincides with the phase space D =D1 × D2, the last being the velocity space D2 = R3. The space D can beeasily metrized by using an arbitrary metric of R

6 ⊃ D.2. Class of all parametrizations of the solutions of equation (1) not changing

the direction of time, i.e. a set of points of the orbit in the previous senseembedded in a suitable space with the order of their passage.The orbit may be thought of as a wire with a marked direction.The orbit is determined uniquely by means of 5 elements.

3. Set of points of the orbit in the first sense. The surrounding space playsno role here.The orbit may be thought of as a wire.

For different problems we need different definitions of orbits. For orbitdetermination we use the orbit in the first sense. For asteroid identification,search for a parent comet of a meteor stream, we use the orbit in the secondsense. For determination of the minimal distance and possibility of intersec-tion we use the orbit in the third sense. In the present paper we deal withorbits in the second sense.

3 Space of Curvilinear Orbits H(b)

The main difficulty arises from the degenerate case of rectilinear orbits. Atthe same time they are of little practical use. So we begin with a simplercase, that of curvilinear orbits.

Each Keplerian non-rectilinear orbit E may be uniquely determined bya pair (c, e) = (cx, cy, cz; ex, ey, ez) of the area-vector c = r × r and theLaplace–Runge–Lenz vector

e =r × c

κ2− r

r.

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Let us remind that e = |e| is the numerical eccentricity of E , and if e 6= 0then e is directed from the focus to the pericentre of the orbit E .

Conversely, each pair (c, e) ∈ R6 satisfying the relations ce = 0, c = |c| >0 represents a non-rectilinear orbit.

Hence we can introduce the space H(b), b > 0 of curvilinear orbits E(c, e)having c > b as a 5-dimensional manifold embedded in R6 by relations

ce = cxex + cyey + czez = 0, (2)

c > b. (3)

In other words, H(b) is a part of the 5-dimensional cone (2) of the seconddegree embedded in R6, that is situated outside the Cartesian product of the3-dimensional space of vectors e and the 3-dimensional ball c2

x + c2y + c2

z 6 b2

of the space of vectors c. For b = 0 the mentioned ball degenerates into thepoint cx = cy = cz = 0.

The space H(b) can be easily metrized. The natural and simultaneouslysimple metric corresponds to the one of the surrounding space R6, for examplethe Euclidean one. Namely, we define the distance between two orbits Ek =E(ck, ek) ∈ H(b) as

(E1, E2) =

√1

κ2L(c1 − c2)2 + (e1 − e2)2, (4)

L > 0 being a scale factor having the physical dimension of length. Thedistance (4) is dimensionless. If needed we may change by ∗ = L havingthe dimension of length.

Remark 1. The right-hand side of (4) contains a weight quantity L. Itmust be chosen properly when dealing with an astronomical problem.

Remark 2. One may use another metric in R6, for example the Holder

one. But it seems too cumbersome except in the limiting Laplace case relatedto the L1–norm or norm ‖ · ‖, namely

˜(E1, E2) =|c1 − c2|

κ√

L+ |e1 − e2|. (5)

The results below do not depend on the type of metric.The manifold H(b) is non-compact. First, it is unbounded. Second, its

complement to the cone (2) contains a boundary c2x + c2

y + c2z = b2. If b > 0

the boundary may be added to H(b) which results in a manifold H(b) withboundary. If b = 0 one cannot do it. In fact, no orbit corresponds to thepoint (c, e) = (0, 0, 0; 2, 0, 0) (rectilinear orbits have a unit vector e) while acontinuum of rectilinear orbits having arbitrary energy constant

h =r2

2− κ2

r

corresponds to a point (0, 0, 0; 1, 0, 0).

Theorem 1The space H(b), b > 0 is arcwise–connected.

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Proof. Let Ek = E(ck, ek), k = 1, 2 be two points lying in the space H(b).Let us connect them by a continuous curve situated in H(b) in two steps:1. rotate the pair of vectors c1, e1 as a rigid body so that their directionscoincide with those of c2, e2.2. not altering the directions, change their lengths so that the lenghts of c1, e1

coincide with those of c2, e2, respectively.Both operations preserve the relations (2, 3). The theorem is prooved.

Theorem 2The space H(b), b > 0 is a 5-dimensional algebraic open manifold without

singular points.

Only the last statement needs a proof. Singular points of H(b) satisfy theequation gradF = 0, F being the left-hand side of (2). Hence, c = e = 0which is incompatible with (3).

The metric space H(b), b > 0 is not complete: for cn → c0, en = e0 and|c0| = b the sequence E(cn, en) has no limit in H(b). For b > 0 the metricspace may be completed adding the boundary c = b. We have seen that ifb = 0 such a completion has no useful sense.

4 Space of Orbits H

Now we construct the space of all orbits, including rectilinear ones withc = 0, e = 1. If c = 0 and h < 0, then e, h determine E . But if the energyis nonnegative there are two separate orbits: the ascending one and the de-scending one, having r > 0 and r < 0, respectively. Nevertheless let us agreeto consider these two orbits as one orbit. Otherwise we cannot metrize thespace of orbits. For example, under fixed h > 0 and e → 1, c → 0 the hy-perbola tends to the union of two branches of rectilinear orbit. If we assumethe branches to be two different orbits then the continuity breaks and themetrization is impossible.

Under this assumption an orbit E may be uniquely determined by a triplet(c, e, h). Conversely, each triplet (c, e, h) ∈ R7 satisfying the relations

ce = 0, 2hc2 − κ4(e2 − 1) = 0 (6)

represents an orbit.Hence we may introduce the space H of orbits E(c, e, h) as a 5-dimensional

algebraic manifold embedded in R7 by equations (6) of the second and thirddegree, respectively.

A non-dimensional metric in H is induced by the Euclidean metric in R7:

1(E1, E2) =

√1

κ2L(c1 − c2)2 + (e1 − e2)2 +

L2

1

κ4(h1 − h2)2 , (7)

L and L1 being scale factors having the physical dimension of length. Ifdesired one may change 1 by ∗1 = L1 having the dimension of length.

Remark 1. The right-hand side of (7) contains two weight quantitiesL, L1. They must be chosen properly when dealing with an astronomicalproblem.

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Remark 2. As above, one may use another metric in R7. The results belowdo not depend on the type of metric.

The space H(b) for an arbitrary b > 0 is a subset of the space H. Evidently,

6 1. (8)

On the other hand, for any two orbits lying in H(b)

h1 − h2 =κ4

2c2

1c2

2

[c2

2(e2

1 − e2

2) + (1 − e2

2)(c2

1 − c2

2)].

Hence for any bounded part of H(b), b > 0, and for any compact part of H(0)there exists a constant A such as

1 6 A. (9)

Consequently, and 1 define identical topology in H(b), b > 0. In particular,if a sequence En ∈ H(0) converges to E0 ∈ H(0) in the metric , then it alsoconverges to E0 in the metric 1. Conversely, convergence in the metric 1

implies convergence in the metric .The situation is more complicated in case of Cauchy (fundamental) se-

quences. Remind the definition: a sequence xn in a metric space is called aCauchy sequence if lim (xn, xk) = 0 when n, k tend to infinity. In a metricspace every convergent sequence is a fundamental sequence. The conversestatement is valid in a complete metric space.

In the space H(b), b > 0 the notions of fundamental sequences in themetrics and 1 coincide, though it is not true for b = 0. Indeed, the sequenceEn = E(cn, en, hn) ∈ H(0) where

cn =c1

n, en = e1

√n + 1

2n, hn = nh1 ,

|c1| = κ√

L , |e1| =√

2 , c1e1 = 0, h1 =κ2

2L1

is a Cauchy sequence in the metric , whereas En diverges in the metric 1.Roughly speaking all rectilinear orbits lying on the same ray are identical forthe metric , whereas the metric 1 distinguishes them.

Theorem 3The space H is arcwise–connected.

Proof. Let Ek = E(ck, ek, hk), k = 1, 2 be two points lying in the space H.Let us now consider the curve

h(t) = h1 , c(t) = (1 − t)c1, e(t) =e1

e1

√1 +

2h1

κ4c2

1(1 − t)2 , 0 6 t 6 1.

For e1 = 0 one should use an arbitrary unit vector e0 orthogonal to c1 insteadof e1/e1 in the above relation. It is easy to verify that E(t) = E(c(t), e(t), h(t))belongs to H, i.e. the equalities (6) hold true. Our curve connects the pointsE(0) = E1 and E(1), equal to E(0, e1/e1, h1) or E(0, e0, h1). Hence we cansuppose c1 = c2 = 0. Connectedness of the subspace H extracted by theequation c = 0 is evident.

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Theorem 4The space H is a 5-dimensional algebraic open manifold without singularpoints.

Only the last statement needs a proof. Consider the Jacobian matrix offunctions representing the left-hand sides of equations (6)

J =

(ex ey ez cx cy cz 0

4hcx 4hcy 4hcz −2κ4ex −2κ4ey −2κ4ez 2c2

).

The rank of J is equal to 2 at regular points of H, and 1 or 0 at singularones. Let us find all solutions to the inequality rankJ (c, e) 6 1.

The last minor of J ∣∣∣∣cz 0

−2κ4ez 2c2

∣∣∣∣

is equal to 0 if and only if cz = 0. Similarly, cx = cy = 0, so c = 0. Then wefind that e = 0 which is incompatible with the second equation of (6).

Theorem 5The space H is complete.

The proof follows from the completeness of R7 and the continuity of theleft-hand sides of (6).

5 Subspaces of H

Besides the whole space H its subspaces play an important role in CelestialMechanics. Let us describe the most important cases.

1. H1 = H ∩ h = const, h < 0. Changing variables κ2c = c√−2h we

represent H1 as intersection of a unit sphere S5 and a cone K

e2

1 + e2

2 + e2

3 + c2

1 + c2

2 + c2

3 = 1,

c1e1 + c2e2 + c3e3 = 0. (10)

In this section we denote the components of vectors c, e with subscripts1, 2, 3.As it is noted in (Stiefel and Scheifele, 1971) the substitution

p = c + e, 2c = p + q,

q = c − e, 2e = p − q

shows that H1 may be thought of as a Cartesian product of two two-dimensional spheres

H1 = S2 × S

2.

Indeed, (c, e) belongs to H1 (i.e. the equations (10) hold true) if and onlyif vectors p,q are unit vectors.Subspaces of H1 corresponding to c = 0 ⇒ e = 1 or e = 0 ⇒ c = 1 areboth the two-dimensional unit spheres S2.

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2. H2 = H ∩ h = const, h = 0, represents an intersection of a cylinder K1

and the cone K

e2

1+ e2

2+ e2

3= 1,

c1e1 + c2e2 + c3e3 = 0. (11)

Evidently H2 may be thought of as a tangent bundle of a two-dimensionalunit sphere TS2.A subspace of H2 corresponding to c = 0 ⇒ e = 1 is the two-dimensionalunit sphere S2.

3. H3 = H ∩ h = const, h > 0. Changing variables κ2c = c√

2h werepresent H3 as intersection of a hyperboloid K2 and the cone K

e2

1+ e2

2+ e2

3− c2

1− c2

2− c2

3= 1,

c1e1 + c2e2 + c3e3 = 0. (12)

Substitution

p = e/e , e = p√

1 + q2 ,

q = c, c = q

shows that H3 may be thought of as a tangent bundle of a two-dimensionalunit sphere TS2. Indeed, (c, e) belongs to H3 (i.e. the equations (12) holdtrue) if and only if vector p is a unit one, and vector q is orthogonal top.A subspace of H3 corresponding to c = 0 ⇒ e = 1 is the two-dimensionalunit sphere S2.

4. H4 = H ∩ c = const, c 6= 0, represents a plane R2. Indeed, e is anarbitrary vector lying in the plane orthogonal to c, and h = κ4(e2−1)/2c2

is uniquely determined by c, e.5. H5 = H ∩ c = 0 represents a Cartesian product of a unit sphere and

a straight line S2 × R. Indeed, e is an arbitrary unit vector, and h isan arbitrary real number. The space H5 is topologically equivalent toR3 \ 0.

6. H6 = H ∩ c = const, c 6= 0, represents a tangent bundle of a two-dimensional unit sphere TS2. Indeed, for each vector c/c describing theunit sphere S2 one can find an arbitrary vector e lying in the plane or-thogonal to c, and h = κ4(e2 − 1)/2c2 is uniquely determined.

7. H7 = H ∩ e = const, e 6= 0, e 6= 1, represents a punctured planeR2 \ 0. Indeed, c is an arbitrary non-zero vector orthogonal to e, andh = κ

4(e2 − 1)/2c2 is uniquely determined.8. H8 = H∩e = const, e = 1, represents a union of a plane and a straight

line orthogonal to it R2 ∪ R embedded in R

3 with coordinates (c1, c2, h).Indeed, c may be an arbitrary non-zero vector orthogonal to e, and thenh = 0, or c may be a zero-vector and then h is an arbitrary real number.

9. H9 = H ∩ e = 0 represents a punctured 3-dimensional space R3 \ 0.Indeed, c is an arbitrary non-zero vector, and h = −κ

4/2c2 is uniquelydetermined by c.

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10. H10 = H ∩ e = const, e 6= 0, e 6= 1, represents a punctured tangentbundle of a two-dimensional unit sphere TS2. Indeed, for each e one canfind an arbitrary non-zero vector c orthogonal to e, and h = κ4(e2−1)/2c2

is uniquely determined.11. H11 = H ∩ e = 1 represents a union of a tangent bundle of a two-

dimensional unit sphere and a Cartesian product of a unit sphere by astraight line TS2 ∪

(S2 × R

). Indeed, for each e one can find either an

arbitrary non-zero vector c orthogonal to e, and then h = 0, or c = 0independently of e, and h is an arbitrary real number.

12. H(b) = H ∩ c > b, b > 0. It is examined in §3.

Remark 1. Dimension of H(b), H is equal to 5, that of H1 , H2 , H3 , H6 , H10 , H11

is 4, that of H5 , H9 is 3, and that of H4 , H7 , H8 is 2.Remark 2. Spaces H1 ÷ H7, H9, H10 are arcwise–connected algebraic

open manifolds without singular points.

6 Spaces of Orbits, Coplanar Case

Each solution to the equation (1) is a plane curve. Fixing the orbital planewe obtain important subspaces of the aforementioned spaces. Without lossof generality we suppose x, y – plane as the orbital one, so c = (0, 0, c),e = (α, β, 0). Equation (2) holds true automatically. Assume that R3, R4 areEuclidean spaces with coordinates (α, β, c) and (α, β, c, h) respectively.

Considering the space of curvilinear orbits G(b), b > 0 it is natural to fixthe orientation of the x, y – plane, i.e. suppose c > 0. Hence, the space G(b)is now a part of R3 selected by the inequality

c > b. (13)

In other words, G(b) is an open half-space of R3, and its properties are wellknown.

On the contrary, considering the space G of orbits it is natural not tofix the orientation of the x, y – plane, i.e. suppose c to be an arbitrary realnumber. Hence, G is now a 3-dimensional algebraic manifold embedded inR

4 by equation

2hc2 − κ4(α2 + β2 − 1) = 0 (14)

of the third or degree.Metrics (4) and (7) operate in G(b) and G respectively.

Theorem 6The space G is a complete arcwise–connected 3-dimensional algebraic openmanifold without singular points.

The proof is much simpler than in case of 3-dimensional vector c and maybe omitted.

Let us describe the most important subspaces of G.

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1. G1 = G ∩ h = const, h < 0. Changing variables κ2c = c√−2h we

represent G1 as a unit sphere S2

α2 + β2 + c2 = 1. (15)

2. G2 = G ∩ h = const, h = 0, represents a cylinder S1 × R:

α2 + β2 = 1, (16)

while c remains an arbitrary real number.3. G3 = G ∩ h = const, h > 0. Changing variables κ2c = c

√2h we

represent G3 as a hyperboloid of one sheet

α2 + β2 − c2 = 1. (17)

It is curious that the phase flow in this case may be thought of as a flowon one sheet of a two-sheeted hyperboloid (Belbruno, 1977).

4. G4 = G ∩ c = const, c 6= 0, represents a paraboloid of revolution

h = κ4(α2 + β2 − 1)/2c2 . (18)

5. G5 = G ∩ c = 0 represents a cylinder S1 × R: circle (16) for α, β whileh remains an arbitrary real number.

6. G6 = G ∩ e = const, e 6= 1, represents a curve

hc2 = A, (19)

A = κ4(e2−1)/2 being a non-zero constant. Topologically G6 is equivalentto two separated straight lines.

7. G7 = G ∩ e = const, e = 1, represents a union of two orthogonalstraight lines R ∪ R embedded in R2 with coordinates (c, h). Indeed, cand h satisfy the equation (19) with A = 0.

8. G8 = G∩e = const, e 6= 0, e 6= 1, coincides with G6×S1. TopologicallyG8 is equivalent to two separated cylinders S1 × R.

9. G9 = G ∩ e = 1 coincides with G7 × S1. Topologically, G9 is equiv-alent to two 2-dimensional cylinders having one and only one commoncircumference. Such a surface can be imbedded (but not embedded) inR3.

10. G(b) = G ∩ c > b, b > 0. As it is shown, G(b) is an open half-space ofR3.

Remark 1. Dimension of G(b), G is equal to 3, that of G1 , G2 , G3 , G4 , G5 , G8 , G9

is 2, that of G6 , G7 is 1.Remark 2. Spaces G1 ÷ G5 are arcwise–connected algebraic open man-

ifolds without singular points. Spaces G6 and G8 consist of two separatedarcwise–connected algebraic open manifold without singular points.

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7 Algorithms for Distances and 1 Computations

Distances and 1 can be easily calculated. For each orbit E there exists aset of elements p, e, i, Ω, ω as the semi-latus rectum, eccentricity, inclination,longitude of ascending node, and the argument of pericentre, respectively.

Let us express c, e via their lengths and directions

c = κ√

pc0 , e = ee0 , (20)

where orthogonal unit vectors c0 and e0 are

c0 = sin i sinΩ, − sin i cosΩ, cos i,e0 = cosω cosΩ − cos i sin ω sin Ω, cosω sin Ω + cos i sinω cosΩ, sin i sin ω.

Note that the representation (20) always exists, though it is not unique incases c = 0 and e = 0. Substituting (20) into (4) one gets

2(E1, E2) =1

L(p1 + p2 − 2

√p1p2 cos ξ) + (e2

1 + e2

2 − 2e1e2 cos η) (21)

with

cos ξ = cos i1 cos i2 + sin i1 sin i2 cos(Ω1 − Ω2), (22)

cos η = (cosω1 cosω2 + cos i1 cos i2 sinω1 sin ω2) cos(Ω1 − Ω2)+

(cos i2 cosω1 sinω2−cos i1 sinω1 cosω2) sin(Ω1−Ω2)+sin i1 sin i2 sinω1 sin ω2 .(23)

Finally, taking (7) into account we obtain

2

1(E1, E2) = 2 +L2

1

4

(1

a1

− 1

a2

)2

, (24)

a being a semi-major axis. For h = 0 we put 1/a = 0.

8 Factor Space

Orbits of a significant part of celestial bodies evolve almost-periodically. Theirsemi-major axes, eccentricities, and inclinations are almost-periodic functionsof time, while longitudes of nodes and pericentres have secular trends as well.So the important role is played by the factor space consisting of the orbitsidentified in case of equal a, e, i not taking into account Ω, ω.

The position of x, y – plane, as well as the orbital plane is important forinclination to be determined uniquely. So we must reject rectilinear orbitshaving a continuum of orbital planes.

Introduce the space F(b), b > 0 as the factor space of H(b) by the equiv-alence above. Elements of F(b) may be thought of as triplets (γ, δ, e), withγ = c cos i and δ = c sin i, where

c > b, 0 6 i 6 π, e > 0. (25)

12

Evidently F(b) is a part of a quarter-space R3, δ > 0, e > 0, lying outside acylinder γ2 + δ2 6 b2. If b = 0 the cylinder degenerates into a straight lineγ = δ = 0.

The space F(b) is neither open nor closed. If b > 0 one may construct

the closure F(b) defined by c > b in (25). Obviously F(b) is a 3-dimensionalmanifold with boundary. If b = 0 such a completion has no useful sense.

The natural metric in F(b) is the minimum of

2(E1, E2) = min (E1, E2) (26)

over all Ω1, Ω2, ω1, ω2.Relations (22 – 23) show that cos ξ and cos η attain simultaneously their

largest values cos i1 cos i2 + sin i1 sin i2, and 1, respectively, for ω1 = ω2 =Ω1 − Ω2 = 0. This fact can be demonstrated also by purely geometric con-siderations.

Finally,

2

2(E1, E2) =

1

L[p1 + p2 − 2

√p1p2 cos(i1 − i2)] + (e1 − e2)

2. (27)

Acknowledgements We are cordially grateful to Professor S.A.Klioner for fruit-ful discussions, and to the reviewers L.Florıa and G.F.Gronchi for detailed analysisof the paper and valuable remarks.

This work is supported by Russian Foundation for Basic Research, (Grants 05-02-17408, 06-02-16795), and Council by President of Russia for Support of LeadingScientific Schools (Grant NS-4929.2006.2).

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