1
Evolution of the Earth-Moon system
Course Code:
Title of Work:
FYGC04
Course name:
Analytical Mechanics
Evolution of the Earth-Moon system
Date of submission:
2014-05-09
Family name:
First name:
Personal ID- number:
Abatte
Leif
880909-0734
Name of teacher:
Name of administrator:
Jürgen Fuchs & Igor Buchberger
Jürgen Fuchs
2
Abstract
This article reexamines the tidal evolution of the Earth-Moon system [5]. It is mostly based on
the work of Goldreich [1] and assumes that the reader knows a fair amount of astronomy and
astrophysics. A Hamiltonian formalism for the dynamics of the tidal friction will be derived.
A number of tidal friction models will be introduced and compared and towards the end of the
paper these will be integrated into the tidal evolution of the Earth-Moon system.
Table of content
Abstract ..................................................................................................................................... 2
Introduction .............................................................................................................................. 3
Developing the system .............................................................................................................. 4
Theories and assumptions ................................................................................................................. 4
Equations of motion ........................................................................................................................... 4
Simplifications .................................................................................................................................... 5
Jacobi coordinates ............................................................................................................................ 5
Delaunay variables........................................................................................................................... 6
Andoyer variables ............................................................................................................................ 6
Simplified equations of motion ......................................................................................................... 6
Earth-Moon Hamiltonian ................................................................................................................. 9
Tidal expressions and models ................................................................................................ 10
Tidal expressions .............................................................................................................................. 10
Tide models....................................................................................................................................... 10
Expressing the tides according to the Tidal models ............................................................ 13
Expression of the MacDonald tide model ...................................................................................... 13
Expression of the Darwin-Mignard tide model............................................................................. 13
Expressing the Darwin-Kaula-Goldreich tide model ................................................................... 14
Conclusion ............................................................................................................................... 17
Reference ................................................................................................................................. 17
3
Introduction
The Earth-Moon system has been a very important topic in astrophysics and astronomy. There
are several hypothesis to the origin of the Moon and one of them, being the more accepted
one; is that a proto-planet, the size of Mars, collided with Earth and the leftover debris after
the collision later formed the Moon. This is illustrated in figure 1. The Earth-Moon system
describes a very old system (4.5 billion years), and there have been many studies one of
which is done by Goldreich, and this paper reexamines his model. The Moon is a big satellite.
Its actually about one fourth the size of the Earth, which is indeed a big satellite. In
astrophysics moons are perceived as satellites when they orbit around a planet, and the fact
that our moon has this enormous size means that Earth has the biggest natural satellite in our
solar system. Most satellites are much smaller and their presence leads to smaller torque
effect.
Figure 1. Illustration of the formation of the Moon
In his studies Goldreich makes several strong assumptions with the intensions to maintain the
problem in a manageable size. What Touma & Wisdom (hereafter referred to as TW) have
done in the paper that is the basis of this report is to recheck the dynamics of the Earth-Moon
system, but under much less severe assumptions.
TW develop a Hamiltonian, in contrast to a Lagrangian, which is based on the framework of
Goldreich, but obtained in a way different from that of Goldreich’s. There are several tidal
models that were developed, one of which is examined here, is one derived by George
Darwin, the son of Charles Darwin. The Earth-Moon system developed by TW is carefully
examined when applied with the different tidal models; one will also see the differences of the
dynamics between them. Goldreich also shows a very important conclusion, which is that the
cross terms that were assumed negligible in similar former studies, are not so negligible after
all.
When incorporating these models, Darwin’s in particular, one can see the evolution of the
Earth-Moon system with tidal acceleration. Afterwards the system is fed with real observable
data and our system is viewed as it has evolved over the past few million years.
4
Developing the system
Theories and assumptions
The Goldreich theory is a relative old one, developed in the sixties, but still very relevant as
we shall see. The framework of his theory is reexamined, but as mentioned before, the
Hamiltonians developed by Goldreich and TW are not the same, despite similar assumptions.
This will lead to different results.
When developing the dynamics of the system one has to implement various assumptions,
them being;
•
•
•
•
•
•
•
•
•
Planetary perturbations.
The orbit of the Earth-Moon system, when revolving around the sun, is a circle.
The Moon is in a circular orbit around the barycenter of the Earth-Moon system
(Barycenter is, in astrophysics; the center of mass).
The gravitational potential is considered to be of second-order.
Earth is considered to be an axisymmetric body.
The finite size of the moon is ignored.
The perturbation from the sun is limited to a second-order potential.
Orbital periods are neglected, and
The motion of the nodal regression is also neglected.
Equations of motion
One starts off with determining the degrees of freedom this system has. There are three
related to the translational motion and three for the rotational movement for each body.
Applying the given assumptions, as stated above, such as; the circular orbit of the moon, the
axisymmetry and also averaging over the precession time gives one degree of freedom. This
degree of freedom is related to the obliquity and precession of the equinoxes of the Earth, this
in turn is related to the orbital inclination of the moon. Examining the dynamics of the system
we first derive the full equations of motion. Later we reduce the expression with the help of
implementing the assumptions, as stated above. Thus we start off with the description of the
Hamiltonian in its usual form:
1 p2
H = ∑ i + ∑Vij
(1)
2 i m i< j
The potential energy is expressed in a truncated Taylor expansion and Legendre polynomial,
both to the second order, where it describes the interaction between two axisymmetrical
bodies:
%
!
Gm1m2 " J 2 Re2
V ( x, ŝ) = −
$1− 2 P2 ( cos(θ ))'
r #
r
&
The general potential
energy expression
5
€
Where mi is the mass of the body, P2 (cos(θ )) is the Legendre polynomial of second degree,
!
Re is the equatorial radius of the extended body, r is the magnitude of the position vector x
from the center of mass of the extended body to the other mass, sˆ is the unit vector of the
symmetry axis and the angle between the these vectors is θ . Finally, J e is the oblateness
€
(which is basically the flatness) of the Earth, this is because Earth can’t be perceived
as a
point mass (because it is not a spherical body), so instead
€ it is better perceived as an oblate
body. This will result in a precession; this is what the second€term in the square brackets
describes. The index notations of the masses indicate the celestial body, 0 meaning the Earth,
1 the Moon and 2 being the Sun.
The potential becomes after the assumptions mentioned above:
V =−
% Gm0 m2 " J 2 Re2
% Gm1m2
Gm0 m1 " J 2 Re2
1−
P
cos(
θ
)
−
1−
P
cos(
θ
)
(
)
(
)
$
'
$
'−
2
01
2
02
r01 #
r012
r02 #
r022
r12
&
&
Simplifications
Jacobi coordinates
In order to simplify the Hamiltonian we need to express it in Jacobi coordinates. The Jacobi
coordinates are ideal for N-body problems because they facilitate the expression of the
Hamiltonian of the system; they separate the center of mass motion with the relative motion
without affecting the expression of the kinetic energy. They can be described in the following
4-praticle system example [9]: the first Jacobi coordinate is the difference between the first
! ! !
two position vectors, called R1 = r2 − r1 . The second coordinate is the absolute vector between
!
the third particle and the center of mass of the first two particles, R2 . The third Jacobi
!
coordinate is an absolute vector between the forth particle and the center of mass of R2 . This
can be illustrated with figure 2 [8]. In general, this procedure goes on depending on how
many bodies one is considering.
Figure 2. Jacobi coordinates on a 4-particle system [9]
6
Delaunay variables
Because the orbital motion of all bodies are governed by Kepler’s laws a body will follow a
Keplerian motion. The Delaunay variables are the canonical coordinates that solve the Kepler
problem and the Hamiltonian describing the Keplerian motion in the space reference-frame of
the total motion. In the present situation these coordinates are the following be; H: The
angular momentum projected on the z-space axis, h: the conjugate position, G: the magnitude
of the angular momentum, g: the conjugate coordinate, and some other ones which are
unspecified.
Andoyer variables
Andoyer variables are introduced to describe the rotational motion of the system. They are a
set of angles, which are based of sets of Euler angles that consists of three angular variables
and their conjugate momentum [3].
Simplified equations of motion
The first Jacobi coordinate in the a genera system is
! ! !
x"i = x i − X i−1 , where 0 < i < n
(2)
€ of the Jacobi coordinates above, the x!i! term is the barycenter of the
Just as the description
!
system i-bodies, where Xi−1 describes the barycenter of the i-1:th barycenter including the j:th.
The second Jacobi coordinate is
!
!
x!0 = X n−1
Combining these expressions the general kinetic energy can be expressed in the following
manner:
2
1 p"
T= ∑ i
2 i m"i
Where m"i = mi
momentum.
€
ηi−1
m! are the Jacobi reduced masses and p"i = mi x˙ "i are the Jacobi canonical
ηi i
€
Here ηi is the c.m. up to particle i
€
j
€
η j = ∑mj
i=0
7
Note, all Jacobi variables are represented with a prime symbol.
Now, the kinetic energy is expressed in term of Jacobi coordinates, its time for the potential
energy to also be expressed in Jacobi coordinates. The first step is to express the distance
! !
between the i:th and the j:th particle/body, xi − x j .
We start with
! !
#m ! &
!
! !
!
!
! − xi = xi+1 − xi − Xi − Xi−1 = xi+1 − xi − % i xi! (
xi+1
$ ηi '
(
)
After some simplifications the distance can be expressed as follow:
! ! ! i−1 m !
xi − x0 = xi" + ∑ j x"j
j=1 η j
Taking this to our three-body system, Earth-Moon-Sun, the Jacobi-coordinates are the
following:
! ! !
x1 − x0 = x1" ,
! ! ! m !
x2 − x0 = x"2 + 1 x1" and
η1
! ! ! ! m ! ! m !
x2 − x1 = x"2 − x1" + 1 x1" = x"2 + 0 x1"
η1
η1
Here the indices 0,1 and 2 represent the Earth, Moon and the Sun, respectively.
The distances between each body is expressed by expressing them with Legendre polynomial
truncated to the second order and inverted. The inverted distance between the Earth and the
Sun is:
! !
! !
1 1 * m1 x1! ⋅ x!2 m12 r1!2 $ x1! ⋅ x!2 '
= ,1−
+
P
+...
/
&
)
2
r02 r2! + η1 r2!2
η12 r2!2 % r1!r2! ( .
The distance between the Moon and the Sun is, after simplifications:
! !
! !
,
1 1 ) m0 x1! ⋅ x!2 m02 r1!2 # x1! ⋅ x!2 &
= +1+
+
P
+...
.
%
(
2
r12 r2! * η1 r2!2
η12 r2!2 $ r1!r2! ' The potential energy can now be expressed with the help of Jacobi coordinates:
! !
! !
&
m0 m1 # J 2 Re2
m0 m2 # J 2 Re2
m1 x" ⋅ x"2 m12 r1"2 * x1" ⋅ x"2 -&
V = −G
+ 2 2 P2 ,
%1− 2 P2 ( cos(θ 2" )) −
/(
%1− 2 P2 ( cos(θ1"))( − G
r1" $
r1"
r2" $
r2"
η1 r2"2
η1 r2" + r1"r2" .'
'
8
! !
! !
m1m2 , m0 x1# ⋅ x#2 m02 r1#2 & x1# ⋅ x#2 ) /
−G 2 .1+
+ 2 2 P2 (
+1
r2# - η1 r2#2
η1 r2# ' r1#r2# * 0
After some cancellations and simplifications the potential, which corresponds to the space
reference-frame, is:
€
! !
m0 m1 J 2 Re2
m0 m2 J 2 Re2
m1m2 m02 r1!2 $ x1! ⋅ x!2 '
V =G
P2 ( cos(θ1!)) + G
P2 ( cos(θ 2! )) − G 2
P2 &
)
r1! r1!2
r2! r2!2
r2! η12 r2!2 % r1!r2! (
The kinetic energy of the motion of the space reference-frame is the Kepler Hamiltonian:
H Kepler =
1 p2 µ
µ
mµ 2
− =−
=− 2
2 m r
2a
2L
Where µ = Gm0 m1 , L2 = maµ and a is the is the semi-major axis of the orbit. The entire space
reference-frame Hamiltonian is:
€
H Space
€
mµ 2
= H Kepler + V = − 2 + V
2L
This is essentially the Hamiltonian that describes the space reference-frame of the system,
after this to completely describe the entire system one needs to describe the dynamics of the
€
body with a body-frame.
The rotatonal kinetic energy of Earth is described with the help of Andoyer variables.
TBody
1
1 # L2A L2B L2C &
2
2
2
= ( Aω A + Bω B + Cω C ) = % +
+ (
2
2$ A
B C'
Where A, B and C are the main moments, the variables LA , LB and LC are the angular
momenta that corresponds to each main moment of inertia and ω A , ω B and ω C are the
€
projections of the spin vectors on the relative principle axis.
€ €
€
The Hamiltonian for the body reference-frame is then, after some simplifications *:
€
€
€2
2
2
2
" G − L0 %" sin (l0 ) cos (l0 ) % L20
H Body = $ 0
+
'$
'+
B & 2C
# 2 &# A
where l is the mean anomaly, G now represents the magnitude of the angular momentum (not
the gravitational constant).
9
Earth-Moon Hamiltonian
Now we can put together the two Hamiltonians, to express the entire system with a single
Hamiltonian, which is called the Earth-Moon Hamiltonian:
H Lunar ≡ H Space + H Body +V
mµ 2 # G02 − L20 &# sin 2 (l0 ) cos2 (l0 ) & L20
= − 2 +%
+
+V
(%
(+
2L $ 2 '$ A
B ' 2C
∴ H Lunar = −
€
mµ 2 # G02 − L20 &# sin 2 (l0 ) cos2 (l0 ) & L20
+%
+
(%
(+
2L2 $ 2 '$ A
B ' 2C
+G
m0 m1 J 2 Re2
P2 ( cos(θ1!))
r1! r1!2
m m J R2
+G 0 2 2 2 e P2 ( cos(θ 2! ))
r2! r2!
! !
m1m2 m02 r1"2 $ x1" ⋅ x"2 '
−G 2
P2 &
)
r2" η12 r2"2 % r1"r2" (
(3)
As described before the first term in equation (3) is the undisturbed orbital motion of the
body, the second and third term describe the rotational motion of the body, the fourth term
describe the Moon’s torque on the Earth and here we can also see the effect of Earth’s
oblateness on the Earth-Moon orbit. The firth term describe the sun’s torque on the Earth, and
the effect of Earth’s oblateness on the orbit that the Earth has around the Sun, but it is
unvarying. Finally, the last (sixed) term describes the Sun’s torque on the Earth-Moon orbit
and the effects they give. But as stated in the assumptions, the Earth-Moon orbit is regarded to
be in a fixed orbit around the Sun, so it will be discarded.
____________________________
* The simplifications are that the body axes are aligned with the principal axes:
LC = LZ = L = G cos(J) , LA = LX = G sin(J )sin(l) and LB = Lγ = G sin(J )cos(l)
10
Tidal expressions and models
Tidal expressions
The Hamiltonian is now simplified by incorporating the assumption such as that the EarthMoon orbit is circular about the Sun and the Earth-Moon orbit is circular etc. The
Hamiltonian in the end is expressed as:
H Lunar = C1 cos2 (ε ) + C2 cos2 (I ) + C3 cos2 (i)
Where
C1 = −G
m0 m2 J 2 Re2 3
m1m2 m0 a12 3
m0 m1 J 2 Re2 3
C
=
−G
C
=
−G
2
a2
a22 4 , 3
a2 η1 a22 8
a1 a12 4 ,
is the mutual obliquity; which is the angle between the equatorial plane and the plane of the
earth rotation around the sun, I is the angle between the angular momentum and the z space
€
€
axis and i is the orbital inclination.
ε
€
Tide models
Now we will express the torques that the Moon and the Sun have on the Earth, which result
the tides. The Earth cannot be perceived as a rigid body because of the oceans, thus as the
Moon orbits, it gravitationally attract the oceans toward itself and creates a bulge of water that
follows it wherever it goes. As the Moon orbits, and this bulge follows, it will collide with the
coast and create friction. This effect would not be so significant if the Moon weren’t so big, as
mentioned before, it is actually about one fourth the size of the Earth, and this creates a
significant effect on the conditions of Earth.
This water bulge that follows the Moon around, due to the Moon’s gravitational pull, isn’t
directly “under” the moon. That is, if you would stand on the Moon’s surface at the point
closest to Earth and you would look up, you would see a bulge of water precisely above you,
right? Well, not exactly; the bulge of water would actually be a bit shifted towards the
opposite of the rotational direction of the Earth. This shift is called “tidal phase lag” and has
to be incorporated to the equations later by the introduction of the different tidal models that
will be presented. Tides are also affected by the Sun, but they are not as significant as the
ones made from the Moon.
Because there is no air friction in space and there is no direct contact between the celestial
bodies, this friction effect that the Moon creates on the Earth has to be counteracted in order
for energy to be conserved. Thus this torque slows down the rotation of the Earth and
increases the orbit of the Moon, resulting in both celestial bodies separating over time. As
times passes (millions of years) Earth’s rotation will slow down to the point were it is equal to
the orbital velocity of the Moon. This has already happened with the Moon, because it doesn’t
spin, it revolves around earth showing us only one side of its surface, this is called Tidal
locking [7].
11
TW utilize Goldreich’s notation, which are; aˆ is the unit vector along the spin axis of the
earth, bˆ is the unit vector normal to the plane of the orbit of the Moon, and cˆ is the unit
vector of the normal to the ecliptic, see figure 3.
€
€
€
Figure 3. Unit vectors defined by Goldreich
Goldreich also uses the following torque notations:
!
!
T0 , which is the torque made on Earth, T1 is the torque made on the Moon, and the torque that
€
the Sun does on Earth is negligible, so in order to conserve energy:
!
!
T0 = −T1
Their tidal torques affect the time derivatives of their corresponding angular momentum:
dG0 !
dG1 ! ˆ
dH1 !
€dH 0 !
= T0 ⋅ aˆ ,
= T0 ⋅ cˆ ,
= T1 ⋅ b and
= T1 ⋅ cˆ
dt
dt
dt
dt
The time derivative of the Hamiltonian is:
!
!
% T! ⋅ €
% T! ⋅ cˆ
€ T! ⋅ bˆ (
€ % T! ⋅ cˆ (
ˆ T ⋅ aˆ
dH Lunar €
T1 ⋅ bˆ (
0 b
1
1
0
1
* + 2C2 cos(I)'
*
= 2C1 cos(ε )'
+
− 4 cos(ε )
+ cos(i)
* + 2C3 cos(i)'
dt
G1
G1 )
G1 )
& G0 )
& G0
& G1
€
Now that the Hamiltonian is expressed and its time derivatives incorporate the tidal equations,
let’s examine the different tidal models that exist. In this paper only four tidal models are
examined, they are: The MacDonald tide model, the Darwin tide model with constant phase
lags and with phase lags proportional to frequency, and the Darwin-Mignard tide model.
1. In the MacDonald tide model the distortion of the body is presented as a second order
harmonics distortion. It is the friction effect from the Moon, and it is delayed with a
constant phase lag. This phase lag is only adapted to circular orbits; it does not take
into account eccentric orbits. This model does not consider any Solar torque or SolarMoon cross torques.
2. The Darwin tide model involves the introduction of a phase lag for each of the Fourier
expanded tide-raising potential terms to represent the friction effect, but this model is
12
not so convenient, because there are numerous terms to implement this phase lag and
they are essentially unconstrained by observations.
3. The Darwin-Mignard tide mode involves, as the MacDonald tide model, the
description of the distortion of the body in terms of a second harmonic distortion, but
here it is delayed with respect to the tide-rising potential with a constant time lag. This
model has a simple analytical form.
TW inform us that Goldreich provides an important contribution, which is the recognition that
there can be average tidal effects on one body due to the tide-raised by another body. As the
earth spins around its axis if it isn’t aligned with its orbital plane normal (the ecliptic plane
normal) then the spin can carry the tidal bulge out of the Moon’s orbital plane. This can in
turn produce average in-plane torques on a third body.
These tidal torques are expressed in terms of components of two sets of coordinate systems.
One corresponding to the ecliptic plane, where its direction is denoted as fˆi , which means:
fˆ = cˆ . The other one corresponding to the Moon’s orbital plane and it is denoted with eˆ ,
i
3
€
which also means: eˆ3 = bˆ , see figure 3. This also means eˆ1 = aˆ × bˆ and fˆ1 = aˆ × cˆ .
€
Thus, once that is established the torque can be expressed in terms of these coordinate
€
systems:
€
€
€
!
!
T = T1eˆ1 + T2eˆ2 + T3eˆ3 or T = T1"fˆ1 + T2" fˆ2 + T3" fˆ3
Their corresponding projections are:
!
€
T ⋅ aˆ = T2 sin(ε )€+ T3 cos(ε ) = T2$ sin(I) + T3$ cos(I)
!
T ⋅ bˆ = T1#sin(h0 − Ω)sin(i) + T2# cos(h0 − Ω) + T3# cos(i)
€
€
€!
& sin(i)sin(I)sin(h0 − Ω) )
& cos(I) − cos(i)cos(ε ) )
T ⋅ cˆ = T1( −
+ + T2 (
+ + T3,
sin(ε )
sin(ε )
'
*
'
*
13
Expressing the tides according to the Tidal models
Here we express the tidal models in these two coordinate systems; we start off with the MacDonald
tides:
Expression of the MacDonald tide model
€
•
T1 = 0
•
2Am E(q) − q$2K(q)
T2 =
sin(2δ )
πa16
q
•
€
T3 =
2Am
q'K(q)sin(2δ )
πa16
Where:
3
A = GmRe5 k 2
2
q"2 = 1 − q 2 , q 2 =
€
1 − cos 2 (ε )
%n
(2 % n
(
1+ ' 1 Ω * − 2' 1 Ω * cos(ε )
&
&
0)
0)
€
Here n1 is the mean motion of the lunar orbit, Ω0 is the angular rotation rate of the Earth, k2
€ is the potential Love number (Love numbers are parameters
€
of the both the rigidness of a
planet and the susceptibility of its shape caused by the tidal potential) [8], K and E are
€
elliptical integrals of the first and second
rad and
€ kind. The phase lag was taken to be 0.04635
€
this cannot be more precise because the exact position of the tide bulge is unknown.
Expression of the Darwin-Mignard tide model
Lunar tides created by average lunar torques are:
•
•
•
€
€
In these models the tides caused by the solar torques are considered; their expression are:
•
•
€
€
€
!
(
k Gm 2 R 5 % 3
T ⋅ â = Δt 2 61 e & Ω0 sin 2 (ε ) + 3cos (ε [Ω0 cos(ε − n1 )]))
'2
*
a1
!
k Gm 2 R 5
T ⋅ bˆ = Δt 2 61 e {3(Ω0 cos(ε − n1 )}
a1
!
*
k Gm 2 R 5 ' 3
T ⋅ cˆ = Δt 2 61 e ( Ω0 [cos(I) − cos(i)cos(ε )] + 3cos(i)[Ω0 cos(ε − n1 )] +
)2
,
a1
•
!
)
k Gm 2 R 5 & 3
T ⋅ aˆ = Δt 2 6 2 e ( Ω0 sin 2 (I) + 3cos(I)[Ω0 cos(I − n 2 )] +
'2
*
a2
!
*
k Gm 2 R 5 ' 3
T ⋅ bˆ = Δt 2 6 2 e ( Ω0 [cos(ε ) − cos(I)cos(i)] + 3cos(i)[Ω0 cos(I − n 2 )] +
)2
,
a2
2
5
!
k Gm R
T ⋅ cˆ = Δt 2 6 2 e 3[Ω0 cos(I − n 2 )]
a2
{
}
14
There are two cross terms between the average sun torques and average lunar torques in every
model (except the MacDonald model).
The cross torques associated with Average Lunar torques due to solar tides are:
•
€
€
€
•
•
!
k 2Gm1m2 Re5 3 2
9
3
ˆ
T ⋅ a = Δt
[ sin (i)sin 2 (I)cos(2(h0 − Ω0 )) − sin 2 (i)sin 2 (I) − cos(I)sin(I)cos(i)sin(i)
6
a2
8
8
4
3
⋅ cos(h0 − Ω) + sin 2 (I)]
4
! ˆ
T⋅ b = 0
!
(
k Gm m R 5 % 3
T ⋅ ĉ = Ω0 Δt 2 16 2 e '− cos(i)sin(i)sin(I )cos(h0 − Ω)*
& 4
)
a2
The average torque, due to lunar tides are:
•
!
k Gm m R 5 3
9
T ⋅ â = Ω0 Δt 2 31 3 2 e [ sin 2 (i)sin 2 (I )cos(2(h0 − Ω) − sin 2 (i)sin 2 (I ) −
a1 a2
8
8
3
3
− cos(I )sin(I )cos(i)sin(i)cos(h0 − Ω) + sin 2 (I )]
4
4
•
•
!
(
Gk m m R 5 % 3
3
T ⋅ b̂ = Ω0 Δt 2 31 3 2 e ' cos(h0 − Ω)sin(i)sin(I )cos2 (i) − cos(I )cos(i)sin 2 (i)*
&4
)
a1 a2
4
!
T ⋅ ĉ = 0
The reaction torques on the Earth are opposite of the torques on exterior bodies.
Expressing the Darwin-Kaula-Goldreich tide model
Here we concentrate on the average torques as before, but now with an equal phase lag, which
is chosen to be:
sin(εt ) = 0.0927
(4)
So, the average lunar torques due to lunar tides are:
•
•
•
!
&
Gk m 2 R 5 # 3 9
T ⋅ â = sin(εt ) 2 61 e % − sin 4 (ε )(
$ 2 16
'
a1
!
%
Gk m 2 R 5 " 3
3
T ⋅ b̂ = sin(εt ) 2 61 e $ cos(ε ) + cos(ε )sin 2 (ε )'
#2
&
a1
4
2
5
!
%
Gk m R " 3
9
3
3
T ⋅ ĉ = sin(εt ) 2 61 e $ cos(i)cos(ε ) + cos(i)cos(ε )sin 2 (ε ) + cos(I ) + cos(I )sin 2 (ε )'
#4
&
a1
16
4
16
15
The average solar torques due to solar tides are:
•
•
•
!
&
Gk m 2 R 5 # 3 9
T ⋅ â = sin(εt ) 2 61 e % − sin 4 (I )(
$ 2 16
'
a1
!
%
Gk m 2 R 5 " 3
9
3
T ⋅ b̂ = sin(εt ) 2 61 e $ cos(i)cos(I ) + cos(i)cos(I )sin 2 (I ) + cos(ε )sin 2 (I )'
#4
&
a1
16
4
!
%
Gk m 2 R 5 " 3
3
T ⋅ ĉ = sin(εt ) 2 62 e $ cos(I ) + cos(I )sin 2 (I )'
#2
&
a2
4
The average Solar-Lunar cross-torque tides are:
•
!
Gk m m R 5 3
3
9
9
T ⋅ â = sin(εt ) 2 31 3 2 e [ sin 2 (I ) − sin 4 (I ) − sin 2 (i)sin 2 (I ) + sin 2 (i)sin 4 (I ) −
a1 a2
4
8
8
16
3
− cos(i)cos3 (I )sin(i)sin(I )cos(h0 − Ω)]
4
•
•
!
T ⋅ b̂ = 0
! !
Gk m m R 5 3
T ⋅ c = sin(εt ) 2 31 3 2 e [− cos(I )sin 2 (i)sin 2 (I )cos(2(h0 − Ω) −
a1 a2
8
"3
%
3
3
$ cos(i)sin(i)sin(I ) − cos(i)sin(i)sin (I )' cos(h0 − Ω) ]
#4
&
8
The average solar torques due to lunar tides:
•
•
•
!
Gk2 m1m2 Re5 3 2
3
9
9
T ⋅ â = sin(εt )
[ sin (I ) − sin 4 (I ) − sin 2 (i)sin 2 (I ) + sin 2 (i)sin 4 (I ) −
3 3
a1 a2
4
8
8
16
3
3
− cos(i)cos3 (I )sin(i)sin(I )cos(h0 − Ω) + sin 2 (i)sin 4 (I )cos(2(h0 − Ω))]
4
16
5
!
Gk m m R 3
3
T ⋅ b̂ = sin(εt ) 2 31 3 2 e [ sin(i)sin(I )cos(h0 − Ω) − cos(i)cos(I )sin 2 (i) −
a1 a2
4
4
9
27
− sin 3 (I )sin(i)cos(h0 − Ω) + sin 3 (i)sin 3 (I )cos(h0 − Ω) +
8
32
3
3
+ cos(i)cos(I )sin 2 (i)sin 2 (I ) − sin 3 (i)sin 3 (I )cos(3(h0 − Ω))]
4
32
!
T ⋅ ĉ = 0
16
Tide model comparison
The different tidal models can now be computed and plotted. Comparing these models we see,
in figures 4-7, that they don’t differ that much except for small lunar semi major axes. There
the Darwin-Kaula-Goldreich models show a smaller value than the other tide models. This is
due to the fact that in the past the lunar orbit did not look the same as it does now. As the
moon was forming, billions of years ago, the orbit looked a lot different so these models break
down if we go back far enough in time. This is a well-known time scale problem.
Figure 4: The graph shows the models where their maximum and minimum values plotted over a processional period. The
evolution of the tides is shown with the Darwin-Mignard model being the solid line, the Darwin-Kaula-Goldreich model
shown as the dotted line and the MacDonald model shown as dashed lines. [1, figure 3]
Figure 5: Here the different models are compared in terms
of their average torques. The graph shows the evolutions of
these models in terms of the orbital inclination, i, and the
semi-major axis of the lunar orbit. The solid lines represent
the Darwin-Mignard model, the dotted lines represent the
Darwin-Kaula-Goldreich model and the dashed lines
represent the MacDonald tide model. [1, figure 4]
Figure 6: Here the different average torques are compared
for the different tide models. Here the obliquity of the Earth
is plotted vs. the semi-major axis of the lunar orbit. Again,
the Darwin-Mignard tide model is represented with the solid
lines, the Darwin-Kaula-Goldreich model is represented
with the dotted lines and the MacDonald tide model are
represented with the dashed lines. [1, figure 5]
17
What we can see from these images is that the inclusion of solar torques and cross terms has
larger effect than the changing in the frequency dependence of the tidal torques in the
Darwin-Kaula-Goldreich and the Darwin-Mignard models. We can conclude that the mutual
obliquity of the lunar orbit to the Earth’s equator is large at small lunar semi-major axes.
Conclusion
After developing a suitable Hamiltonian that describes the Earth-Moon system and applying
and comparing different tidal models, we can conclude that Goldreich’s Evolution of the
Earth-Moon system does not depart from the re-examined results that TW provide. This is
also true when real-time data are analysed using this model, which is not included here
(Touma & Wisdom, 1994, [1]), extracted from the DE 102 paper written by Newhall [6].
They, yet again, show this result. This paper shows that Goldreich captured the essence of the
dynamics of the Earth-Moon system. This is not surprising if one knows the brilliancy of
Goldreich as seen in the Efroimsky-paper [4]; where he mentions that Goldreich’s version of
the Darwin-Kaula tidal model was actually written without any proof and decades later
proved [4]. This paper serves to enlighten the Earth-Moon interactions and its dynamics.
Reference
[1] Touma J & Wisdom J. Evolution of the Earth Moon system (1994). Volume 105, number 5.
Review of scientific paper. pp.1943-1961. http://adsabs.harvard.edu/full/1994AJ....108.1943T
[2] Wikipedia. Jacobi coordinates (13/12/23): http://en.wikipedia.org/wiki/Jacobi_coordinates
[3] Noyelles B, Lemaître A, and Vienne A. Titan’s rotation A 3-dimentional theory (2008).
Modelling of data. pp.959. http://www.aanda.org/articles/aa/pdf/2008/06/aa7991-07.pdf
[4] Efroimsky M. Can the tidal quality factors of terrestrial planets and moons scale as positive
powers of the tidal frequency? (2012). Review of models. pp.8.
http://arxiv.org/pdf/0712.1056.pdf
[5] Goldreich P. History of the lunar orbit (1966). Review of Geophysics and Space Physics.
Volume 4, pp.411-439.
http://onlinelibrary.wiley.com/doi/10.1029/RG004i004p00411/abstract
[6] Newhall X.X. DE 102 - A numerically integrated ephemeris of the moon and planets spanning
forty-four centuries (1983). Astronomy and Astrophysics (ISSN 0004-6361), vol. 125, no. 1,
Aug. 1983, pp. 150-167. NASA-supported research.
[7] Scharringhausen B. Is the Moon moving away from the Earth? When was this
discovered? (2002). Third paragraph.
http://curious.astro.cornell.edu/question.php?number=124
18
[8] Wikipedia. Love Number (2014-05-08). http://en.wikipedia.org/wiki/Love_number
[9] Cornille P. Advanced Electromagnetism and Vacuum Physics (2003). Volume 21.
Page 102.
http://books.google.se/books?id=y8sSFTDkQ20C&pg=PA102&redir_esc=y#v=onepa
ge&q&f=false