Table Of ContentQuantum effects on Lagrangian points and displaced periodic
orbits in the Earth-Moon system
Emmanuele Battista∗
Dipartimento di Fisica, Complesso Universitario di Monte S. Angelo,
Via Cintia Edificio 6, 80126 Napoli, Italy
5
Istituto Nazionale di Fisica Nucleare, Sezione di Napoli,
1
0
Complesso Universitario di Monte S. Angelo,
2
r Via Cintia Edificio 6, 80126 Napoli, Italy
a
M
1 Simone Dell’Agnello†
3
Istituto Nazionale di Fisica Nucleare,
]
c
Laboratori Nazionali di Frascati, 00044 Frascati, Italy
q
-
r
g
Giampiero Esposito‡
[
2 Istituto Nazionale di Fisica Nucleare, Sezione di Napoli,
v
3 Complesso Universitario di Monte S. Angelo,
2
7
Via Cintia Edificio 6, 80126 Napoli, Italy
2
0
.
1 Jules Simo§
0
5
Department of Mechanical and Aerospace Engineering,
1
:
v University of Strathclyde, Glasgow, G1 1XJ, United Kingdom
i
X
(Dated: April 1, 2015)
r
a
1
Abstract
Recent work in the literature has shown that the one-loop long distance quantum corrections to
the Newtonian potential imply tiny but observable effects in the restricted three-body problem of
celestial mechanics, i.e., at the Lagrangian libration points of stable equilibrium the planetoid is
not exactly at equal distance from the two bodies of large mass, but the Newtonian values of its
coordinates are changed by a few millimeters in the Earth-Moon system. First, we assess such a
theoretical calculation by exploiting the full theory of the quintic equation, i.e., its reduction to
Bring-Jerrard form and the resulting expression of roots in terms of generalized hypergeometric
functions. By performing the numerical analysis of the exact formulas for the roots, we confirm
and slightly improve the theoretical evaluation of quantum corrected coordinates of Lagrangian
libration points of stable equilibrium. Second, we prove in detail that also for collinear Lagrangian
points the quantum corrections are of the same order of magnitude in the Earth-Moon system.
Third, we discuss the prospects to measure, with the help of laser ranging, the above departure
from the equilateral triangle picture, which is a challenging task. On the other hand, a modern
version of the planetoid is the solar sail, and much progress has been made, in recent years, on
the displaced periodic orbits of solar sails at all libration points, both stable and unstable. The
present paper investigates therefore, eventually, a restricted three-body problem involving Earth,
Moon and a solar sail. By taking into account the one-loop quantum corrections to the Newtonian
potential, displaced periodic orbits of the solar sail at libration points are again found to exist.
PACS numbers: 04.60.Ds, 95.10.Ce
∗E-mail: [email protected]
†E-mail: [email protected]
‡E-mail: [email protected]
§E-mail: [email protected]
2
I. INTRODUCTION
From the point of view of modern theoretical physics, the logical need for a quantum
theoryofgravityissuggested bytheEinsteinequationsthemselves, which tellusthatgravity
couples to T , the energy-momentum tensor of matter, in a diffeomorphism-invariant way,
µν
by virtue of the tensor equations [1, 2]
1 8πG
R g R = T . (1.1)
µν µν µν
− 2 c4
When Einstein arrived at these equations, although he had already understood that the
classical Maxwell theory of electromagnetic phenomena is not valid in all circumstances, the
only known forms of T were classical, e.g., the energy-momentum tensor of a relativistic
µν
fluid, or even just the case of vacuum Einstein equations, for which T vanishes. In due
µν
course, it was realized that matter fields are quantum fields in the first place (e.g., a massive
Dirac field, or spinor electrodynamics). The quantum fields are operator-valued distribu-
tions [3], for which a regularization and renormalization procedure is necessary and even
fruitful. However, the mere replacement of T by its regularized and renormalized form
µν
T on the right-hand side of Eq. (1.1) leads to a hybrid scheme, because the classical
µν
h i
Einstein tensor R 1g R is affected by the coupling to T . The question then arises
µν − 2 µν h µνi
whether the appropriate, full quantum theory of gravity should have field-theoretical nature
or should involve, instead, other structures (e.g., strings [4] or loops [5] or twistors [6]),
and at least 16 respectable approaches [7] have been developed so far in the literature. To
make such theories truly physical, their predictions should be checked against observations.
For example, applications of the covariant theory lead to detailed predictions for the cross
sections of various scattering processes [8], but such phenomena (if any) occur at energy
scales inaccessible to observations, and also the effects of Planck-scale physics on cosmology,
e.g., the cosmic microwave background radiation and its anisotropy spectrum [9–13], are not
yet easily accessible to observations, although cosmology offers possibly the best chances for
testing quantum gravity [14].
Recently, inspired by the work in Refs. [15–22] on effective field theories of gravity, where
it is shown that the leading (i.e., one-loop) long distance quantum corrections to the New-
tonian potential are entirely ruled by the Einstein-Hilbert part of the full action functional,
some of us [23, 24] have assumed that such a theoretical analysis can be applied to the
3
long distances and macroscopic bodies occurring in celestial mechanics [25–27]. More pre-
cisely, the Newtonian potential between two bodies of masses m and m receives quantum
A B
corrections leading to [22]
Gm m k k
V (r) = A B 1+ 1 + 2 +O(G2), (1.2)
Q
− r r r2
(cid:18) (cid:19)
where [23]
G(m +m )
A B
k κ , (1.3)
1 1
≡ c2
G~
k κ = κ (l )2. (1.4)
2 ≡ 2 c3 2 P
Equation (1.2) implies that, ε > 0, there exists an r value of r such that
0
∀
Gm m k k
V (r)+ A B 1+ 1 + 2 < ε, r > r . (1.5)
Q r r r2 ∀ 0
(cid:12) (cid:18) (cid:19)(cid:12)
(cid:12) (cid:12)
This feature will play(cid:12)an important role in our concludi(cid:12)ng remarks in Sec. V.
(cid:12) (cid:12)
Wealso stress that the dimensionless parameter κ depends onthedimensionless parame-
1
ter κ . In other words, k is a post-Newtonian term which only depends on classical physical
2 1
constants, but its weight, expressed by the real number κ , is affected by the calculational
1
procedure leading to the fully quantum term k , where the real number κ weighs the Planck
2 2
length squared. More precisely, the perturbative expansion involves only integer powers of
Newton’s constant G:
∞ ∞
Gm m Gm m k
V (r) A B 1+ f (r)Gp A B 1+ n , (1.6)
Q ∼ − r p ∼ − r rn
! !
p=1 n=1
X X
where, upon denoting by L and L the gravitational radii L GmA,L GmB, one has
A B A ≡ c2 B ≡ c2
the coefficients k = k (L +L ,(l )2). At one loop, i.e., to linear order in G, where
n n A B P
(m +m )1 ~ 1
A B
f (r) = κ +κ , (1.7)
1 1 c2 r 2c3r2
we can only have the contribution (LA+LB) with weight equal to the real number κ , and
r 1
the contribution (lP)2 with weight equal to the real number κ . Although the term (lP)2 is
r2 2 r2
overwhelmed by the term (LA+LB), the two are inextricably intertwined because κ is not a
r 1
free real parameter but depends on κ : both κ and κ result from loop diagrams. Thus,
2 1 2
the one-loop long distance quantum correction is the whole term
k k (L +L ) (l )2
f (r)G = 1 + 2 = κ A B +κ P , (1.8)
1 r r2 1 r 2 r2
4
where κ takes a certain value because there exists a nonvanishing value of κ . The authors
1 2
of Ref. [22] found the numerical values
41
κ = 3, κ = . (1.9)
1 2 10π
The work in Ref. [23] considered the application of Eqs. (1.2)-(1.4) and (1.9) to the circular
restricted three-body problem of celestial mechanics, in which two bodies A and B of masses
α and β, respectively, with α > β, move in such a way that the orbit of B relative to A is
circular, and hence both A and B move along circular orbits around their center of mass C
which moves in a straight line or is at rest, while a third body, the planetoid P, of mass m
much smaller than α and β, is subject to their gravitational attraction, and one wants to
evaluate the motion of the planetoid. On taking rotating axes1 with center of mass C as
origin, distance AB denoted by l, and angular velocity ω given by
G(α+β)
ω = , (1.10)
l3
r
one has, with the notation in Fig. 1, that the quantum corrected effective potential for the
circular restricted three-body problem is given by GU, where [23]
1(α+β) α k k β k k
U = (x2 +y2)+ 1+ 1 + 2 + 1+ 3 + 2 , (1.11)
2 l3 r r r2 s s s2
(cid:18) (cid:19) (cid:18) (cid:19)
where r and s are the distances AP and BP, respectively, while here
G(m+α) G(m+β)
k κ , k κ ,κ = κ = 3. (1.12)
1 ≡ 1 c2 3 ≡ 3 c2 1 3
The equilibrium points are found by studying the gradient of U and evaluating its zeros.
There exist indeed five zeros of gradU [23]. Three of them correspond to collinear libration
points L ,L ,L of unstable equilibrium, while the remaining two describe configurations of
1 2 3
stable equilibrium at the points denoted by L ,L . The simple but nontrivial idea in Refs.
4 5
[23, 24] was that, even though the quantum corrections in (1.2) involve small quantities,
the analysis of stable equilibrium (to linear order in perturbations) might lead to testable
departures from Newtonian theory, being related to the gradient of U, and to the second
derivatives of U evaluated at the zeros of gradU. The quantum corrected Lagrange points
L and L have coordinates (x(l),y (l)) and (x(l),y (l)), respectively, where
4 5 + −
(r2(l) s2(l)+b2 a2)
x(l) = − − , (1.13)
2(a+b)
1 To be self-consistent, some minor repetition of the text in Ref. [23] is unavoidable.
5
y
P
s
x
r B
b
C
a
A
FIG. 1: The figure shows the two bodies of large mass, A and B, the center of mass C, and the
planetoid at P.
y (l) = r2(l) x2(l) 2ax(l) a2, (1.14)
±
± − − −
p
where r(l) 1 , s(l) 1 , w(l) ans u(l) being the real solutions of an algebraic equation
≡ w(l) ≡ u(l)
of fifth degree (see Sec. II). Interestingly, r(l) = s(l), and hence to the equilateral libration
6
points of Newtonian celestial mechanics there correspond points no longer exactly at vertices
of an equilateral triangle. For the Earth-Moon-satellite system, the work in Ref. [24] has
found
x x 8.8 mm, y y 5.2 mm, (1.15)
Q C Q C
− ≈ | |−| | ≈
where x (resp. x ) is the quantum corrected (resp. classical) value of x(l) in (1.13), and
Q C
the same for y and y obtainable from (1.14). Remarkably, the values in (1.15) are well
Q C
accessible to the modern astrometric techniques [28].
6
On the other hand, much progress has been made along the years on modern models
of planetoids and their displaced periodic orbits at all Lagrange points L ,L ,L ,L ,L ,
1 2 3 4 5
to linear order in the variational equations for Newtonian theory. In particular, a mod-
ern version of planetoid is a solar sail, which is propelled by reflecting solar photons and
therefore can transform the momentum of photons into a propulsive force. Solar sailing
technology appears as a promising form of advanced spacecraft propulsion [29–33], which
can enable exciting new space-science mission concepts such as solar system exploration and
deep space observation. Although solar sailing has been considered as a practical means
of spacecraft propulsion only relatively recently, the fundamental ideas had been already
developed towards the end of the previous century [34].
Solar sails can also be used for highly nonKeplerian orbits, such as closed orbits displaced
high above the ecliptic plane [35]. Solar sails are especially suited for such nonKeplerian
orbits, since they can apply a propulsive force continuously. This makes it possible to
consider some exciting and unique trajectories. In such trajectories, a sail can be used as
a communication satellite for high latitudes. For example, the orbital plane of the sail can
be displaced above the orbital plane of the Earth, so that the sail can stay fixed above
the Earth at some distance, if the orbital periods are equal. Orbits around the collinear
points of the Earth-Moon system are also of great interest because their unique positions
are advantageous for several important applications in space mission design [36–41].
Over the last few dacades, several authors have tried to determine more accurate approx-
imations of such equilibrium points [42]. Such (quasi-)Halo orbits were first studied in Refs.
[42–47]. Halo orbits near the collinear libration points in the Earth-Moon system are of
great interest, in particular around the L and L points, because of their unique positions.
1 2
However, a linear analysis shows that the collinear libration points L ,L and L are of the
1 2 3
type saddle center center, leading toaninstability intheir vicinity, whereas theequilateral
× ×
equilibrium points L and L are stable, in that they are of the type center center center.
4 5
× ×
AlthoughthelibrationpointsL andL arenaturallystableandrequire asmallacceleration,
4 5
the disadvantage is the longer communication path length from the lunar pole to the sail.
If the orbit maintains visibility from Earth, a spacecraft on it (near the L point) can be
2
used to provide communications between the equatorial regions of the Earth and the polar
regions of the Moon. The establishment of a bridge for radio communications is crucial
for forthcoming space missions, which plan to use the lunar poles. Displaced nonKeplerian
7
orbits near the Earth-Moon libration points have been investigated in Refs. [29–33, 48].
This brief outline shows therefore that the analysis of libration points does not belong just
to the history of celestial mechanics, but plays a crucial role in modern investigations of
space mission design.
Section II studies in detail the algebraic equation of fifth degree for the evaluation of
noncollinear libration points L and L , by first passing to dimensionless units and then
4 5
exploiting the rich mathematical theory of quintic equations and their roots. Section III
derives and solves the algebraic equation of ninth degree for the evaluation of quantum
corrections to collinear Lagrangian points L ,L ,L . Section IV outlines the prospects to
1 2 3
measurethequantumcorrectedcoordinatesobtainedinSec. IIwiththehelpoflaserranging.
Section V evaluates displaced periodic orbits at the quantum-corrected Lagrange points L
4
and L , and a detailed comparison with the results of Newtonian celestial mechanics is
5
also made. Concluding remarks and open problems are presented in Sec. VI, while the
Appendices describe relevant background material on the theory of algebraic equations.
II. ALGEBRAIC EQUATIONS OF FIFTH DEGREE FOR w(l) AND u(l)
In Ref. [23] it has been shown that the gradU = 0 condition at noncollinear libration
points leads to the algebraic equations of fifth degree
w5 +ζ w4 +ζ w3 +ζ = 0, (2.1)
4 3 0
u5 +ζ˜ u4 +ζ˜u3 +ζ˜ = 0, (2.2)
4 3 0
where
2κ G(m+α) 1 1 1 1
ζ = 1 , ζ = , ζ = , (2.3)
4 3κ c2l2 3 3κ l2 0 −3κ l2l3
2 P 2 P 2 P
2κ G(m+β)
ζ˜ = 1 , ζ˜ = ζ , ζ˜ = ζ . (2.4)
4 3κ c2l2 3 3 0 0
2 P
Such formulas tell us that it is enough to focus on Eq. (2.1), say, where, to exploit the
mathematical theory of quintic equations, we pass to dimensionless units by defining
1 γ
w = , (2.5)
r ≡ l
P
where γ is a real number to be determined. The quintic equation obeyed by γ is therefore
γ5 +ρ γ4 +ρ γ3 +ρ = 0, (2.6)
4 3 0
8
where ρ ,ρ ,ρ are all dimensionless and read as
4 3 0
2κ G(m+α)
ρ ζ l = 1 , (2.7)
4 ≡ 4 P 3κ c2l
2 P
1
ρ ζ l2 = , (2.8)
3 ≡ 3 P 3κ
2
1 l 3 l 3
ρ ζ l5 = P = ρ P . (2.9)
0 ≡ 0 P −3κ l − 3 l
2 (cid:18) (cid:19) (cid:18) (cid:19)
At this stage, we can exploit the results of Appendix A, by virtue of which Eq. (2.6) can be
brought into the Bring-Jerrard [49, 50] form of quintic equations
γ5 +d γ +d = 0. (2.10)
1 0
Since we are going to need the roots of the quintic up to the ninth or tenth decimal digit, the
form (2.10) of the quintic will turn out to be very useful, because it leads to exact formulas
for the roots which are then evaluated numerically, which is possibly better than solving
numerically the quintic from the beginning. Hermite [51] proved that this equation can be
solved in terms of elliptic functions, but we use the even more manageable formulas for the
roots displayed in Ref. [52]. For this purpose, the crucial role is played by the number
3125( d )4
σ − 0 . (2.11)
≡ 256 ( d )5
1
−
We can further simplify Eq. (2.10) by rescaling γ according to
γ = χγ˜. (2.12)
The quintic for γ˜ is then
d d
γ˜5 + 1γ˜ + 0 = 0. (2.13)
χ4 χ5
One can choose χ in such a way that
d
1 = 1 = χ = χ(d ) = ( d )14, (2.14)
− χ4 ⇒ 1 − 1
and the corresponding σ of (2.11) reads as
3125 d 4
σ˜ = 0 = σ. (2.15)
256 −(χ(d ))5
(cid:18) 1 (cid:19)
If σ˜ < 1, which is the case that holds in the Earth-Moon system by virtue of the numerical
| |
results for d and d obtained from the algorithm of Appendix A, then the analysis of Ref.
0 1
9
[52] shows that the five roots of the quintic (2.13), here written as γ˜5 γ˜ β˜ = 0, are
− −
obtained from the parameter
d
β˜ 0 (2.16)
≡ −(χ(d ))5
1
occurring in σ˜, and from hypergeometric functions of order 4, according to [52]
γ˜ i β˜ 5 iβ˜2 5 β˜3 F (σ˜)
1 4 32 −32 0
γ˜ 1 β˜ 5 β˜2 5 β˜3 F (σ˜)
2 = − 4 32 32 1 , (2.17)
γ˜ i β˜ 5 iβ˜2 5 β˜3F (σ˜)
3 − 4 −32 −32 2
γ˜ 1 β˜ 5 β˜2 5 β˜3 F (σ˜)
4 4 −32 32 3
γ˜ = β˜F (σ˜), (2.18)
5 1
−
where, having defined the higher hypergeometric function
a , a , ..., a , a ∞
F : σ˜ F(σ˜) F 1 2 n−2 n−1 = C σ˜s, (2.19)
s
→ ≡ b , b , ..., b , σ˜
1 2 n−2 s=0
X
with the coefficients evaluated according to the rules
(a ,s)(a ,s)...(a ,s)(a ,s)
C 1, C 1 2 n−2 n−1 , (2.20)
0 s
≡ ≡ (1,s)(b ,s)...(b ,s)(b ,s)
1 n−3 n−2
(λ,µ) λ(λ+1)(λ+2)...(λ+µ 1), (2.21)
≡ −
one has
1 , 3 , 7 , 11
F (σ˜) F −20 20 20 20 , (2.22)
0
≡ 1, 1, 3, σ˜
4 2 4
1, 2, 3, 4
F (σ˜) F 5 5 5 5 , (2.23)
1
≡ 1, 3, 5, σ˜
2 4 4
9 , 13, 17, 21
F (σ˜) F 20 20 20 20 , (2.24)
2
≡ 3, 5, 3, σ˜
4 4 2
7 , 9 , 11, 13
F (σ˜) F 10 10 10 10 . (2.25)
3
≡ 5, 3, 7, σ˜
4 2 4
This representation of the roots γ˜ is discovered by pointing out that Eqs. (2.13) and (2.14)
i
suggest considering such roots as functions of σ˜ = σ. By taking derivatives of Eq. (2.13)
10
