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Mercury's capture into the 3/2 spin-orbit resonance as a result of its chaotic dynamics

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Mercury is locked into a 3/2 spin-orbit resonance where it rotates three times on its axis for every two orbits around the sun. The stability of this equilibrium state is well established, but our understanding of how this state initially arose remains unsatisfactory. Unless one uses an unrealistic tidal model with constant torques (which cannot account for the observed damping of the libration of the planet) the computed probability of capture into 3/2 resonance is very low (about 7 per cent). This led to the proposal that core-mantle friction may have increased the capture probability, but such a process requires very specific values of the core viscosity. Here we show that the chaotic evolution of Mercury's orbit can drive its eccentricity beyond 0.325 during the planet's history, which very efficiently leads to its capture into the 3/2 resonance. In our numerical integrations of 1,000 orbits of Mercury over 4 Gyr, capture into the 3/2 spin-orbit resonant state was the most probable final outcome of the planet's evolution, occurring 55.4 per cent of the time.
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..............................................................
Mercury’s capture into the 3/2
spin-orbit resonance as a
result of its chaotic dynamics
Alexandre C. M. Correia
1,2
& Jacques Laskar
2
1
Departamento de
´
sica da Universidade de Aveiro, Campus Universita
´
rio de
Santiago, 3810-193 Aveiro, Portugal
2
Astronomie et Syste
`
mes Dynamiques, IMCCE-CNRS UMR8028, Observatoire
de Paris, 77 Avenue Denfert-Rochereau, 75014 Paris, France
.............................................................................................................................................................................
Mercury is locked into a 3/2 spin-orbit resonance where it rotates
three times on its axis for every two orbits around the sun
1–3
. The
stability of this equilibrium state is well established
4–6
, but our
understanding of how this state initially arose remains unsatis-
factory. Unless one uses an unrealistic tidal model with constant
torques (which cannot account for the observed damping of the
libration of the planet) the computed probability of capture into
3/2 resonance is very low (about 7 per cent)
5
. This led to the
proposal that core–mantle friction may have increased the
capture probability, but such a process requires very specific
values of the core viscosity
7,8
. Here we show that the chaotic
evolution of Mercury’s orbit can drive its eccentricity beyond
0.325 during the planet’s history, which very efficiently leads to
its capture into the 3/2 resonance. In our numerical integrations
of 1,000 orbits of Mercury over 4 Gyr, capture into the 3/2 spin-
orbit resonant state was the most probable final outcome of the
planet’s evolution, occurring 55.4 per cent of the time.
Tidal dissipation will drive the rotation rate of the planet towards
a limit equilibrium value x
l
(e)n depending on the eccentricity e and
on the mean motion n (see Methods). In a circular orbit (e ¼ 0) this
equilibrium coincides with synchronization (x
l
(0) ¼ 1), but
x
l
(e
0
) ¼ 1.25685 for the present value of Mercury’s eccentricity
(e
0
¼ 0.206), while the equilibrium rotation rate 3n/2 is achieved
for e
3/2
¼ 0.284927. In their seminal work
5
, Goldreich and Peale
assumed that Mercury passed through the 3/2 resonance during its
initial spin-down. They derived an analytical estimate of the capture
probability into the 3/2 resonance and found P
3/2
¼ 6.7% for the
eccentricity e
0
. With the updated value of the momentum of inertia
9
ðB 2 AÞ=C . 1:2 £ 10
24
; this probability increases to 7.73%, and
our numerical simulations with the same setting give P
3/2
¼ 7.10%
with satisfactory agreement.
In fact, using the present value of the eccentricity of Mercury is
questionable, as the eccentricity undergoes strong variations in
time, owing to planetary secular perturbations. Assuming a random
date for the crossing of the 3/2 resonance for 2,000 orbits, we
found numerically P
BVW50
3=2
¼3:92% and P
BRE74
3=2
¼5:48% for these-
cular (averaged) solutions of Brouwer and Van Woerkom
10
and
Bretagnon
11
. It should be stressed that with the regular quasiper-
iodic solutions BVW50 or BRE74, as for the fixed value of the
eccentricity e
0
, the 3/2 resonance can be crossed only once, because
e , e
3/2
. This will no longer be the case with a complete solution for
Mercury’s orbit that takes into account its chaotic evolution
12,13
.In
this case, Mercury’s eccentricity can exceed the characteristic value
e
3/2
(Fig. 1), and additional capture into resonance can occur.
To check this new scenario, it is not possible to use a single orbital
solution because, owing to its chaotic behaviour, the motion cannot
be predicted precisely beyond a few tens of millions of years. We
have thus performed a statistical study of the past evolutions of
Mercury’s orbit, with the integration of 1,000 orbits over 4 Gyr in
the past, starting with very close initial conditions. This statistical
study was made possible by the use of the averaged equations for the
motion of the Solar System
12,13
that have recently been readjusted
and compared to recent numerical integrations
14
, with very good
agreement over nearly 35 Myr.
Owing to the chaotic evolution, the density function of the 1,000
solutions over 4 Gyr is a smooth function (Fig. 1), similar, but not
equal, to a gaussian curve
14
. The mean value of the eccentricity
e
LA04
is slightly higher than
e
BVW50
and
e
BRE74
; but the main difference is a
much wider range for the eccentricity variations, from nearly zero to
more than 0.45. The planet eccentricity can now increase beyond e
3/
2
during its history. Even if these episodes do not last for a long time,
they will allow additional capture into the 3/2 spin-orbit resonance.
For each of these 1,000 orbital motions of Mercury, we have
numerically integrated the rotational motion of the planet, taking
into account the resonant terms of equation (2), for p ¼ k/2 with
k ¼ 1,…,10, the tidal dissipation, and the planetary perturbations,
starting at t
0
¼ 24 Gyr, with a rotation period of 20 days. Because e
is not constant, the ratio x(t) of the rotation rate of the planet to its
mean motion n will tend towards a limit value
~
x
l
ðtÞ (see Methods)
that is similar to an averaged value of x
l
(e(t)), and capture into
resonance can now occur in various ways.
Type I is the classical case, where e , e
p
(Fig. 2a). It is only in this
case that the probability formula of Goldreich and Peale
5
will apply.
In type II, the eccentricity oscillates around e
p
at the time when the
spin rate x(t) decreases towards p. The tidal dissipation thus drives
x(t) several times across p, greatly increasing the probability of
capture (Fig. 2b). Types I and II can only occur in the first few Myr,
as the spin rate decreases from faster rotations. We distinguish these
cases from type III, where the planet is not initially captured into
Figure 1 Probability density function of Mercury’s eccentricity. Values are computed over
4 Gyr for the two quasiperiodic solutions BVW50 (ref. 10) (a) and BRE74 (ref. 11) (b) and
for the numerical integration of the secular equations of refs 12 and 14 for 1,000 close
initial conditions (LA04, c). The mean values of the eccentricity in these solutions are
respectively
e
BVW50
¼0:177;
e
BRE74
¼0:181; and
e
LA04
¼0:198: The vertical dotted
line is the characteristic value e
3/2
.
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resonance p ; but later on, as the orbital elements evolve, the
eccentricity increases beyond e
p
, and tidal dissipation accelerates
the spin rate beyond p, leading to additional capture (Fig. 2c).
Over 1,000 orbits, a few were initially trapped in high-order
resonances (one in 7/2, one in 4/1, two in 9/2 and three in 5/1), but
these were associated with high values of the planet’s eccentricity. As
the eccentricity decreased these resonances became unstable, and
none of these high-order resonances survived. They did eventually
get trapped for a long time into the 5/2 resonance, but even that did
not survive over the full history of the planet. Indeed, the stability of
the resonances depends on the eccentricity of Mercury, and except
for the 1/1 resonance, the resonances may become unstable for very
small values of the eccentricity (Table 1).
We followed all 1,000 solutions, starting from 24 Gyr, until they
reached the present date or were captured into the 2/1, 3/2 or 1/1
resonances. Unlike in previous studies, we found that capture into
the 1/1 resonance is possible, because the eccentricity of Mercury
may decrease to very low values at which capture can occur and the
resonance remain stable. Over 554 solutions that were captured into
the 3/2 resonance, a single one, initially captured at 23.995 Gyr,
escaped from resonance at about 22.396 Gyr. The solution then got
trapped into the 1/1 resonance at 22.290 Gyr, capture that was
favoured by the very low eccentricity required to destabilize the 3/2
resonance (Table 1). Out of the 56 solutions initially trapped into
the 2/1 resonance, ten were destabilized, and only two of them were
further captured, one into the 3/2 resonance, and one into the 1/1
resonance. Globally, we obtained a final capture probability:
P
1=1
¼ 2:2%; P
3=2
¼ 55:4%; P
2=1
¼ 3:6% ð1Þ
The remaining 38.8% non-resonant solutions end with nearly the
same final rotation rate of x
f
¼ 1.21315, because all the orbital
solutions are very close in the vicinity of the origin. Among the
solutions captured into the 3/2 resonance, we can distinguish 31
solutions of type I, 168 of type II and 355 of type III (Fig. 2).
With the consideration of the chaotic evolution of the eccentri-
city of Mercury, we thus show that with a realistic tidal dissipative
model that properly accounts for the damping of the libration of the
planet, and without the need for some additional core–mantle
friction, the present 3/2 resonant state is the most probable outcome
for the planet.
Additionally, from the present state of the planet, we can derive
an interesting constraint on its past evolution. Of all 554 orbits
trapped into the 3/2 resonance, for 521 of them (94.0%) Mercury’s
eccentricity exceeded 0.325 in the past 4 Gyr. The conditional
probability that Mercury’s eccentricity exceeded 0.325, given that
its rotation is trapped into the 3/2 resonance, is thus 94.0%. The 3/2
resonant state of Mercury thus becomes an observational clue that
the chaotic evolution of the planet orbit led its eccentricity beyond
0.325 over its history.
The largest unknown in this study remains the dissipation factor
k
2
/Q of K (equation (4)) (ref. 15). A stronger dissipation would
increase the probability of capture into the 3/2 resonance, because
x(t) would follow more closely x
l
(e(t)) (Fig. 2), whereas lower
dissipation would slightly decrease the capture probability. This study
should apply more generally to any extrasolar planet or satellite
whose eccentricity is forced by planetary perturbations. A
Methods
Tidal dissipation and core–mantle friction will drive Mercury’s obliquity (the angle
between the equator and the orbital plane) close to zero. For zero-degree obliquity, and in
the absence of dissipation, the averaged equation for the rotational motion near resonance
p (where p is a half-integer) is
4,5
:
_x ¼ 2
3
2
n
B 2 A
C
Hðp; eÞ sin2ð
2 pMÞð2Þ
where
is the rotational angle, x ¼
_
=n is the ratio of the rotation rate to the mean motion
n, M is the mean anomaly and H(p, e) are Hansen coefficients
5,16
. The moments of inertia
are A , B , C, with C ¼
y
mR
2
, where m and R are the mass and radius of the planet, and
y is a structure constant.
Tidal models independent of the frequency (constant-Q models) do not account for
the damping of the amplitude of libration that is at present observed on Mercury
5,17
.
Moreover, these models introduce discontinuities into the equations and can thus be
considered as unrealistic approximations for slow rotating bodies
18
. Therefore, we use here
for slow rotations a viscous tidal model, with a linear dependence on the tidal frequency.
Its contribution to the rotation rate is given by
5,18,19,20
:
_x ¼ 2K½
Q
ðeÞx 2 NðeÞ ð3Þ
with
Q
ðeÞ¼ð1 þ 3e
2
þ 3e
4
=8Þ=ð1 2 e
2
Þ
9=2
; NðeÞ¼ð1 þ 15e
2
=2 þ 45e
4
=8 þ 5e
6
=16Þ=ð1 2
e
2
Þ
6
; and
K ¼ 3n
k
2
yQ
R
a

3
m
0
m

ð4Þ
where k
2
and Q are the second Love number and the quality factor, while a, m and m
0
are
the semi-major axis, the mass of the planet and the solar mass, respectively. Equilibrium is
achieved when _x ¼ 0; that is, for constant e, when x ¼ x
l
(e) ¼ N(e)/Q(e).
For a non-constant eccentricity e(t), the limit solution of equation (3) is no longer
x
l
(e), but more generally:
~
x
l
ðtÞ¼ xð0ÞþK
ð
t
0
NðeðtÞÞgðtÞdt

=gðtÞð5Þ
where gðtÞ¼expðK
Ð
t
0
Q
ðeðtÞÞdtÞ:
Figure 2 Typical cases of capture into the 3/2 resonance. The rotation rate x(t ) (bold
curve) and limit value x
l
(e(t )) (dotted curve) are plotted versus time (Gyr). a, Type I is the
classical case
5
:Ase , e
3/2
, the limit value x
l
is always lower than 3/2. b, In Type II, at
the time when x reaches the resonant value 3/2, e is oscillating around e
3/2
, leading to
multiple crossings of the resonance, with ultimately a capture. c, Type III corresponds to
solutions that have not been captured during the initial crossing of the resonance, but later
on, as the eccentricity increases beyond e
3/2
.
Table 1 Critical eccentricity e
c
(p) for the resonance p
p e
c
(p)
.............................................................................................................................................................................
1/1
3/2 0.000026
2/1 0.004602
5/2 0.024877
3/1 0.057675
7/2 0.095959
4/1 0.135506
9/2 0.174269
5/1 0.211334
.............................................................................................................................................................................
If e , e
c
(p), the resonance p becomes unstable, and the solution may escape the resonance.
The critical eccentric ity e
c
(p) is obtained by the resolution of ½
Q
ðeÞp 2 NðeÞ=Hðp;eÞ¼
ð3=2KÞ n½ðB 2 AÞ=C:
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Using y ¼ 0.3333, k
2
¼ 0.4 and Q ¼ 50 (refs 15, 21), we have
K ¼ 8.45324 £ 10
27
yr
21
. Assuming an initial rotation period of Mercury of 10 h, we
estimated that the time needed to despin the planet to the slow rotations would be about
300 million years. This is why we started our integrations in the slow-rotation regime, with
a rotation period of 20 days (x < 4.4) and a starting time of 24 Gyr, although these values
are not critical.
Received 12 March; accepted 4 May 2004; doi:10.1038/nature02609.
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206, 1240 (1965).
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(1965).
3. Colombo, G. Rotation period of the planet Mercury. Nature 208, 575 (1965).
4. Colombo, G. & Shapiro, I. I. The rotation of the planet Mercury. Astrophys. J. 145, 296–307 (1966).
5. Goldreich, P. & Peale, S. J. Spin orbit coupling in the Solar System. Astron. J. 71, 425–438 (1966).
6. Counselman, C. C. & Shapiro, I. I. Spin-orbit resonance of Mercury. Symp. Math. 3, 121–169 (1970).
7. Goldreich, P. & Peale, S. J. Spin-orbit coupling in the solar system 2. The resonant rotation of Venus.
Astron. J. 72, 662–668 (1967).
8. Peale, S. J. & Boss, A. P. A spin-orbit constraint on the viscosity of a Mercurian liquid core. J. Geophys.
Res. 82, 743–749 (1977).
9. Anderson, J. D., Colombo, G., Espsitio, P. B., Lau, E. L. & Trager, G. B. The mass, gravity field, and
ephemeris of Mercury. Icarus 71, 337–349 (1987).
10. Brouwer, D. & Van Woerkom, A. J. J. The secular variations of the orbital elements of the principal
planets. Astron. Pap. Am. Ephem. XIII, part II, 81–107 (1950).
11. Bretagnon, P. Termes a
`
longue pe
´
riodes dans le syste
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me solaire. Astron. Astrophys. 30, 141–154 (1974).
12. Laskar, J. The chaotic motion of the solar system. Icarus 88, 266–291 (1990).
13. Laskar, J. Large-scale chaos in the Solar System. Astron. Astrophys. 287, L9–L12 (1994).
14. Laskar, J. et al. Long term evolution and chaotic diffusion of the insolation quantities of Mars. Icarus
(in the press).
15. Spohn, T., Sohl, F., Wieczerkowski, K. & Conzelmann, V. The interior structure of Mercury: what we
know, what we expect from BepiColombo. Planet. Space Sci. 49, 1561–1570 (2001).
16. Hansen, P. A. Entwickelung der products einer potenz des radius vectors mit dem sinus oder cosinus
eines vielfachen der wahren anomalie in reihen. Abhandl. K. S. Ges. Wissensch. IV, 182–281 (1855).
17. Murray, C. D. & Dermott, S. F. Solar System Dynamics (Cambridge Univ. Press, Cambridge, 1999).
18. Correia, A. C. M. & Laskar, J. Ne
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ron de Surgy, O. Long term evolution of the spin of Venus. I. Theory.
Icarus 163, 1–23 (2003).
19. Munk, W. H. & MacDonald, G. J. F. The Rotation of the Earth; A Geophysical Discussion (Cambridge
Univ. Press, Cambridge, 1960).
20. Kaula, W. Tidal dissipation by solid friction and the resulting orbital evolution. J. Geophys. Res. 2,
661–685 (1964).
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Handbook of Physical Constants 1–31 (American Geophysical Union, Washington DC, 1995).
Acknowledgements This work was supported by PNP-CNRS, Paris Observatory CS, and
Fundac¸a
˜
o para a Cie
ˆ
ncia e a Technologia, POCTI/FNU, Portugal. The numerical computations
were made at IDRIS-CNRS, and Paris Observatory. Authors are listed in alphabetic order.
Competing interests statement The authors declare that they have no competing financial
interests.
Correspondence and requests for materials should be sent to J.L. (Laskar@imcce.fr).
..............................................................
Laser-induced ultrafast spin
reorientation in the antiferromagnet
TmFeO
3
A. V. Kimel
1
, A. Kirilyuk
1
, A. Tsvetkov
1
, R. V. Pisarev
2
& Th. Rasing
1
1
NSRIM Institute, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen,
The Netherlands
2
Ioffe Physico-Technical Institute, 194021 St.-Petersburg, Russia
.............................................................................................................................................................................
All magnetically ordered materials can be divided into two
primary classes: ferromagnets
1,2
and antiferromagnets
3
. Since
ancient times, ferromagnetic materials have found vast appli-
cation areas
4
, from the compass to computer storage and more
recently to magnetic random access memory and spintronics
5
.In
contrast, antiferromagnetic (AFM) materials, though represent-
ing the overwhelming majority of magnetically ordered
materials, for a long time were of academic interest only. The
fundamental difference between the two types of magnetic
materials manifests itself in their reaction to an external mag-
netic field
in an antiferromagnet, the exchange interaction leads
to zero net magnetization. The related absence of a net angular
momentum should result in orders of magnitude faster AFM spin
dynamics
6,7
. Here we show that, using a short laser pulse, the
spins of the antiferromagnet TmFeO
3
can indeed be manipulated
on a timescale of a few picoseconds, in contrast to the hundreds of
picoseconds in a ferromagnet
8–12
. Because the ultrafast dynamics
of spins in antiferromagnets is a key issue for exchange-biased
devices
13
, this finding can expand the now limited set of appli-
cations for AFM materials.
To deflect the magnetization of a ferromagnet from its equili-
brium, a critical field H
FM
cr
< H
A
of the order of the effective
anisotropy field is required. In contrast, the response of an anti-
ferromagnet to an applied field remains very weak before the
exchanged-enhanced critical field H
AFM
cr
<
ffiffiffiffiffiffiffiffiffiffiffiffiffi
H
A
H
ex
p
is reached. In
most materials the exchange field H
ex
.. H
A
(H
A
, 1T,
H
ex
< 100 T) and thus H
AFM
cr
.. H
FM
cr
: This difference is related to
the fact that in an antiferromagnet, no angular momentum is
associated with the AFM moment. This large rigidity of an anti-
ferromagnet to an external field also shows up in the magnetic
resonance frequency
14
, where spin excitations start at q <
g
ffiffiffiffiffiffiffiffiffiffiffiffiffi
H
A
H
ex
p
: This is in contrast to q < gH
A
in a ferromagnet,
which can result in a difference of more than two orders of
magnitude.
Indeed, dynamical many-body theory calculations
15
show a
possibility of AFM dynamics with a time constant of a few
femtoseconds only. Experimentally, the ultrafast dynamics of an
antiferromagnet is still an intriguing question. The problem how-
ever is far from trivial, as there is no straightforward method
for the manipulation and detection of spins in AFM materials.
Therefore, an appropriate mechanism should be found that would
deflect the AFM moments on a timescale down to femtoseconds,
and this change should subsequently be detected on the same
timescale.
The solution to this problem can be found in the magnetocrystal-
line anisotropy. Indeed, a rapid change of this anisotropy can lead,
via the spin–lattice interaction, to a reorientation of the spins
11,12
.
Such anisotropy change, in turn, can be induced by a short
femtosecond laser pulse in a material with a strong temperature-
dependent anisotropy. The subsequent reorientation of the spins
can be detected with the help of time-resolved linear magnetic
birefringence
16
, which enables us to follow the change of the
direction of spins in antiferromagnets, similar to the Faraday and
Kerr effects in ferromagnets.
The rare-earth orthoferrites RFeO
3
(where R indicates a rare-
earth element) investigated here are known for a strong tempera-
ture-dependent anisotropy
17,18
. These materials crystallize in an
orthorhombically distorted perovskite structure, with a space-
group symmetry D
16
2h
(Pbnm). The iron moments order antiferro-
magnetically, as shown in Fig. 1, but with a small canting of the
spins on different sublattices. The temperature-dependent aniso-
tropy energy has the form
19,20
:
F
ðTÞ¼
F
0
þK
2
ðTÞsin
2
v þK
4
sin
4
v ð1Þ
where v is the angle in the xz plane between the x axis and the AFM
moment G, see Fig. 1, and K
2
and K
4
are the anisotropy constants of
second and fourth order, respectively. Applying equilibrium con-
ditions to equation (1) yields three temperature regions corre-
sponding to different spin orientations:
G
4
ðG
x
F
z
Þ: v ¼ 0; T $ T
2
G
2
ðG
z
F
x
Þ: v ¼ 1=2
p
; T # T
1
G
24
: sin
2
v ¼ K
2
ðTÞ=2K
4
; T
1
# T # T
2
ð2Þ
where T
1
and T
2
are determined by the conditions K
2
(T
1
) ¼ 22K
4
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... In the applied literature, it is very common to use a simplified version of the tidal torque that can be obtained by averaging the dissipation over time (Celletti and Chierchia 2009;Celletti and Lhotka 2014;Correia et al. 2004), so that the tidal torque becomes proportional to the derivative of the rotation angle. ...
... Consider the motion of a rigid body, say a satellite S, with a triaxial structure, rotating around an internal spin-axis and, at the same time, orbiting under the gravitational influence of a point-mass perturber, say a planet P. A simple model that describes the coupling between the rotation and the revolution of the satellite goes under the name of spin-orbit problem, which has been extensively studied in the literature in different contexts (see, e.g., Beletsky 2001;Celletti 1990b, a;Correia et al. 2004;Wisdom et al. May 1984). ...
... The expression for the tidal torque can be simplified by assuming (as in Peale (2005), Correia et al. (2004)) that the dynamics is essentially ruled by the average T d of the tidal torque over one orbital period, which can be written as ...
Article
Full-text available
We consider the dissipative spin–orbit problem in Celestial Mechanics, which describes the rotational motion of a triaxial satellite moving on a Keplerian orbit subject to tidal forcing and drift. Our goal is to construct quasi-periodic solutions with fixed frequency, satisfying appropriate conditions. With the goal of applying rigorous KAM theory, we compute such quasi-periodic solutions with very high precision. To this end, we have developed a very efficient algorithm. The first step is to compute very accurately the return map to a surface of section (using a high-order Taylor’s method with extended precision). Then, we find an invariant curve for the return map using recent algorithms that take advantage of the geometric features of the problem. This method is based on a rapidly convergent Newton’s method which is guaranteed to converge if the initial error is small enough. So, it is very suitable for a continuation algorithm. The resulting algorithm is quite efficient. We only need to deal with a one-dimensional function. If this function is discretized in N points, the algorithm requires \(O(N \log N) \) operations and O(N) storage. The most costly step (the numerical integration of the equation along a turn) is trivial to parallelize. The main goal of the paper is to present the algorithms, implementation details and several sample results of runs. We also present both a rigorous and a numerical comparison of the results of averaged and not averaged models.
... Consider the motion of a non-rigid satellite S that we assume to have a triaxial shape and principal moments of inertia A < B < C. We assume that the barycenter of the satellite S moves on an elliptic Keplerian orbit with semimajor axis a, eccentricity e, and with the planet P in one focus. The satellite rotates around the smallest physical axis, in such a way that the spin-axis is perpendicular to the orbit plane (see, e.g., [17][18][19][20][21]). We normalize the units of measure of time so that the orbital period T orb is equal to 2π , which implies that the mean motion is n = 2π /T orb = 1; we introduce the perturbative parameter ε, which measures the equatorial ellipticity of the satellite: ...
... 3 (20) and Γ denotes the gamma function. ...
... We remark that [26] provides a better estimate for the constant C 0 in the symplectic case. Its expression is more complicated than (20). However, for our parameter values, it seems that the estimate (20) suffices to reach the final result of getting analytic estimates close to the break-down. ...
We provide evidence of the existence of KAM quasi-periodic attractors for a dissipative model in Celestial Mechanics. We compute the attractors extremely close to the breakdown threshold. We consider the spin–orbit problem describing the motion of a triaxial satellite around a central planet under the simplifying assumption that the center of mass of the satellite moves on a Keplerian orbit, the spin-axis is perpendicular to the orbit plane and coincides with the shortest physical axis. We also assume that the satellite is non-rigid; as a consequence, the problem is affected by a dissipative tidal torque that can be modeled as a time-dependent friction, which depends linearly upon the velocity. Our goal is to fix a frequency and compute the embedding of a smooth attractor with this frequency. This task requires to adjust a drift parameter. We have shown in Calleja et al. (2020) [14] that it is numerically efficient to study Poincaré maps; the resulting spin–orbit map is conformally symplectic, namely it transforms the symplectic form into a multiple of itself. In Calleja et al. (2020) [14], we have developed an extremely efficient (quadratically convergent, low storage requirements and low operation count per step) algorithm to construct quasi-periodic solutions and we have implemented it in extended precision. Furthermore, in Calleja et al. (2020) [15] we have provided an “a-posteriori” KAM theorem that shows that if we have an embedding and a drift parameter that satisfy the invariance equation up to an error which is small enough with respect to some explicit condition numbers, then there is a true solution of the invariance equation. This a-posteriori result is based on a Nash-Moser hard implicit function theorem, since the Newton method incurs losses of derivatives. The goal of this paper is to provide numerical calculations of the condition numbers and verify that, when they are applied to the numerical solutions, they will lead to the existence of the torus for values of the parameters extremely close to the parameters of breakdown. Computing reliably close to the breakdown allows to discover several interesting phenomena, which we will report in Calleja et al. (2020) [28]. The numerical calculations of the condition numbers presented here are not completely rigorous, since we do not use interval arithmetic to estimate the round off error and we do not estimate rigorously the truncation error, but we implement the usual standards in numerical analysis (using extended precision, checking that the results are not affected by the level of precision, truncation, etc.). Hence, we do not claim a computer-assisted proof, but the verification is more convincing that standard numerics. We hope that our work could stimulate a computer-assisted proof.
... In our solar system, only Mercury experiences reversals, and just barely. The striking spin-orbit resonance and large eccentricity that make this possible may reflect a history (and future) of chaotic dynamics 5,17 . Also in our solar system, observers on Hyperion see Saturn and the sun rise and set chaotically 18 , while observers on Nix and Hydra see Pluto and the sun rise and set chaotically 19 . ...
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Earth’s rapid spin, modest tilt, and nearly circular orbit ensure that the sun always appears to move forward, rising in the east and setting in the west. However, for some exoplanets, solar motion can reverse causing alien suns to apparently move backward. Indeed, this dramatic motion marginally occurs for Mercury in our own solar system. For exoplanetary observers, we study the scope of solar motion as a function of eccentricity, spin–orbit ratio, obliquity, and nodal longitude, and we visualize the motion in spatial and spacetime plots. For zero obliquity, reversals occur when a planet’s spin angular speed is between its maximum and minimum orbital angular speeds, and we derive exact nonlinear equations for eccentricity and spin–orbit to bound reversing and non-reversing motion. We generalize the notion of solar day to gracefully handle the most common reversals.
... Mercury, making exactly three sidereal rotations per one orbital period (Pettengill & Dyce 1965), is the only planet in the solar system captured into a supersynchronous resonance. This resonance happens to be the most likely outcome of Mercury's − = = spin-orbit evolution even without the assistance of a liquid core friction, due to the relative proximity of the planet to the Sun and its considerable eccentricity, which could have reached even higher values in the past (Correia & Laskar 2004). Mercury also has rather short characteristic times of spin-down of the order of 10 7 yr (Noyelles et al. 2013), which indicate that the massive molten core was formed after the capture event. ...
Experiment Findings
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I investigate the spin-orbital evolution of the potentially habitable super-Earth GJ 667Cc in the multiple system of at least two exoplanets orbiting a nearby M dwarf. The published radial velocities for this star are re-analyzed and evidence is found for additional periodic signals, which could be taken for two additional planets on eccentric orbits making the system dynamically inviable. Limiting the scope to the two originally detected planets, we assess the dynamical stability of the system and find no evidence for bounded chaos in the orbital motion.
... The condition that these planets might not satisfy could be the optimal orbital and spin velocities. Mercury is known to be coupled to the Sun in a 3/2 spin-orbit resonance wherein the planet spins three times about its axis for every two orbits around the Sun [28,29]. The planet Venus has a sidereal rotation period which is longer than its orbital period by 18 days. ...
Preprint
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The existence of plate tectonics on the Earth is directly dependent on the internal viscosity contrast, mass of the planet, availability of liquid water and an internal heat source. However, the initial conditions of rotational velocity and revolutionary periodicity of the Earth around the Sun too must have been significant for the inception of plate tectonics. The initial orbital conditions of the Earth were significantly influenced by the diametrical processes of core segregation and Moon formation and that had probably led to the eventuality of initiation and persistence of plate tectonics. The change in the orbital conditions could have rendered the Earth to evolve in a near-linear trend so that the rotational periodicity of the planet (TP) could approach the time taken for the planet to travel one degree in its orbit around the Sun (T1degree), that is TP ~ T1degree. Such an optimal condition for the rotational and revolutionary periodicities could be essential for the development of plate tectonics on the Earth. This hypothesis has direct implications on the possibility of plate tectonics and life in extrasolar planets and potentially habitable solar planetary bodies such as Europa and Mars.
... which is known as the pseudo-synchronization (Correia and Laskar 2004). Indeed, this is not a solid rotation, because the equilibrium is slightly super-synchronous at nonzero eccentricities. ...
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Close-in co-orbital planets (in a 1:1 mean-motion resonance) can experience strong tidal interactions with the central star. Here, we develop an analytical model adapted to the study of the tidal evolution of those systems. We use a Hamiltonian version of the constant time-lag tidal model, which extends the Hamiltonian formalism developed for the point-mass case. We show that co-orbital systems undergoing tidal dissipation favour either the Lagrange or the anti-Lagrange configurations, depending on the system parameters. However, for all range of parameters and initial conditions, both configurations become unstable, although the timescale for the destruction of the system can be larger than the lifetime of the star. We provide an easy-to-use criterion to determine whether an already known close-in exoplanet may have an undetected co-orbital companion.
... Consider the motion of a non-rigid satellite S that we assume to have a triaxial shape and principal moments of inertia A ă B ă C. We assume that the barycenter of the satellite S moves on an elliptic Keplerian orbit with semimajor axis a, eccentricity e, and with the planet P in one focus. The satellite rotates around the smallest physical axis, in such a way that the spin-axis is perpendicular to the orbit plane (see, e.g., [Bel01,Cel90,Cel10,CL04,WPM84]). ...
Preprint
Full-text available
We provide evidence of the existence of KAM quasi-periodic attractors for a dissipative model in Celestial Mechanics. We compute the attractors extremely close to the breakdown threshold. We consider the spin-orbit problem describing the motion of a triaxial satellite around a central planet under the simplifying assumption that the center of mass of the satellite moves on a Keplerian orbit, the spin-axis is perpendicular to the orbit plane and coincides with the shortest physical axis. We also assume that the satellite is non-rigid; as a consequence, the problem is affected by a dissipative tidal torque that can be modeled as a time-dependent friction, which depends linearly upon the velocity. Our goal is to fix a frequency and compute the embedding of a smooth attractor with this frequency. This task requires to adjust a drift parameter. The goal of this paper is to provide numerical calculations of the condition numbers and verify that, when they are applied to the numerical solutions, they will lead to the existence of the torus for values of the parameters extremely close to the parameters of breakdown. Computing reliably close to the breakdown allows to discover several interesting phenomena, which we will report in [CCGdlL20a]. The numerical calculations of the condition numbers presented here are not completely rigorous, since we do not use interval arithmetic to estimate the round off error and we do not estimate rigorously the truncation error, but we implement the usual standards in numerical analysis (using extended precision, checking that the results are not affected by the level of precision, truncation, etc.). Hence, we do not claim a computer-assisted proof, but the verification is more convincing that standard numerics. We hope that our work could stimulate a computer-assisted proof.
... which is known as the pseudo-synchronization (Correia and Laskar, 2004). Indeed, this is not a solid rotation, because the equilibrium is slightly super-synchronous at non-zero eccentricities. ...
Preprint
Full-text available
Close-in co-orbital planets (in a 1:1 mean motion resonance) can experience strong tidal interactions with the central star. Here, we develop an analytical model adapted to the study of the tidal evolution of those systems. We use a Hamiltonian version of the constant time-lag tidal model, which extends the Hamiltonian formalism developed for the point-mass case. We show that co-orbital systems undergoing tidal dissipation either favour the Lagrange or the anti-Lagrange configurations, depending on the system parameters. However, for all range of parameters and initial conditions, both configurations become unstable, although the timescale for the destruction of the system can be larger than the lifetime of the star. We provide an easy-to-use criterion to determine if an already known close-in exoplanet may have an undetected co-orbital companion.
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BepiColombo is a joint mission between the European Space Agency, ESA, and the Japanese Aerospace Exploration Agency, JAXA, to perform a comprehensive exploration of Mercury. Launched on $20^{\mathrm{th}}$ 20 th October 2018 from the European spaceport in Kourou, French Guiana, the spacecraft is now en route to Mercury. Two orbiters have been sent to Mercury and will be put into dedicated, polar orbits around the planet to study the planet and its environment. One orbiter, Mio, is provided by JAXA, and one orbiter, MPO, is provided by ESA. The scientific payload of both spacecraft will provide detailed information necessary to understand the origin and evolution of the planet itself and its surrounding environment. Mercury is the planet closest to the Sun, the only terrestrial planet besides Earth with a self-sustained magnetic field, and the smallest planet in our Solar System. It is a key planet for understanding the evolutionary history of our Solar System and therefore also for the question of how the Earth and our Planetary System were formed. The scientific objectives focus on a global characterization of Mercury through the investigation of its interior, surface, exosphere, and magnetosphere. In addition, instrumentation onboard BepiColombo will be used to test Einstein’s theory of general relativity. Major effort was put into optimizing the scientific return of the mission by defining a payload such that individual measurements can be interrelated and complement each other.
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In a previous paper (Laskar, Nature, 338, 237-238), the chaotic nature of the solar system excluding Pluto was established by the numerical computation of the maximum Lyapunov exponent of its secular system over 200 Myr. In the present an explanation is given for the exponential divergence of the orbits: it is due to the transition from libration to circulation of the critical argument of the secular resonance 2(g 4 −g 3 )−(s 4 −s 3 ) related to the motions of perihelions and nodes of the Birth and Mars. An other important secular resonance is identified: (g 1 −g 5 )−(s 1 −s 2 ). Its critical argument stays in libration over 200 Myr with a period of about 10 Myr and amplitude from 85° to 135°. The main features of the solutions of the inner planets are now identified when taking these resonances into account. Estimates of the size of the chaotic regions are determined by a new numerical method using the evolution with time of the fundamental frequencies. The size of the chaotic regions in the inner solar system are large and correspond to variations of about 0.2 arcsec/year in the fundamental frequencies. The chaotic nature of the inner solar system can thus be considered as robust against small variations of the initial conditions or of the model. The chaotic regions related to the outer planets frequencies are very thin except for g 6 which present variations sufficiently large to be significant over the age of the solar system.
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Dissipation of tidal energy in the earth's mantle and the moon was calculated assuming a dissipation factor 1/Q constant throughout both bodies. In the mantle the dissipation varies from about 2 × 10−6/Q erg cm−3 sec−1 near the pole at the bottom of the mantle to about 0.02 × 10−6/Q erg cm−3 sec−1 near the surface. The effects of compressibility and inhomogeneity are less than 3%. In a homogeneous moon the dissipation varies from a maximum of about 0.03 × 10−6/Q erg cm−3 sec−1 near the center to a minimum of about 0.4 × 10−9/Q erg cm−3 sec−1 at the surface. A theory of orbital evolution is developed in which the disturbing function is expressed in a Fourier series with respect to time, so that the effects of variation of dissipation factor 1/Q, or lag angle ϵ, with amplitude and frequency can be examined. Comparisons with results of other authors are made.
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Due to planetary perturbations, there exists a large chaotic zone for the spin of the terrestrial planets , Nature 361, 608-612). The crossing of this zone in the past can lead Venus' spin to its present retrograde configuration for most initial conditions, but through two different processes , Nature 411, 767-770). Here, we present in full details the dissipative models used for this study of the spin evolution of Venus. The present state of Venus is an equilibrium between gravitational and thermal atmospheric tidal torques , Icarus 11, 356-366). We present here a revised model for the thermal atmospheric tides which does not suffer the singularity at synchronous states arising in previous studies. This new model should thus provide a more realistic description of the final stages of Venus' evolution. Assuming that the present spin of Venus is in a final state, we describe the resulting constraints on the various dissipative parameters. We show that the capture in the 1:1 spin orbit resonance during Venus' history is unlikely and becomes impossible when the dense atmosphere is present as this resonance becomes unstable. Our study is presented in a very general setting and should apply to any terrestrial planet with a dense atmosphere.
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Numerous integrations of the solar system have been conducted, with very close initial conditions, totaling an integration time exceeding 100 Gyr. The motion of the large planets is always very regular. The chaotic zone explored by Venus and the Earth is moderate in size. The chaotic zone accessible to Mars is large and can lead to eccentricities greater than 0.2. The chaotic diffusion of Mercury is so large that its eccentricity can potentially reach values very close to 1, and ejection of this planet out of the solar system resulting from close encounter with Venus is possible in less than 3.5 Gyr.