The evolution of the Earth-Moon system based on the dark matter field
  fluid model
The evolution of the Earth-Moon system based on the dark fluid model
The evolution of the Earth-Moon system based on  
the dark matter field fluid model 
Hongjun  Pan 
Department of Chemistry 
University of North Texas, Denton, Texas 76203, U. S. A. 
 Abstract 
The evolution of Earth-Moon system is described by the dark matter field fluid 
model with a non-Newtonian approach proposed in the Meeting of Division of Particle 
and Field 2004, American Physical Society. The current behavior of the Earth-Moon 
system agrees with this model very well and the general pattern of the evolution of the 
Moon-Earth system described by this model agrees with geological and fossil evidence. 
The closest distance of the Moon to Earth was about 259000 km at 4.5 billion years ago, 
which is far beyond the Roche’s limit. The result suggests that the tidal friction may not 
be the primary cause for the evolution of the Earth-Moon system. The average dark 
matter field fluid constant derived from Earth-Moon system data is 4.39 × 10-22 s-1m-1. 
This model predicts that the Mars’s rotation is also slowing with the angular acceleration 
rate about -4.38 × 10-22 rad s-2. 
Key Words. dark matter, fluid, evolution, Earth, Moon, Mars 
1. Introduction 
The popularly accepted theory for the formation of the Earth-Moon system is that 
the Moon was formed from debris of a strong impact by a giant planetesimal with the 
Earth at the close of the planet-forming period (Hartmann and Davis 1975). Since the 
formation of the Earth-Moon system, it has been evolving at all time scale. It is well 
known that the Moon is receding from us and both the Earth’s rotation and Moon’s 
rotation are slowing. The popular theory is that the tidal friction causes all those changes 
based on the conservation of the angular momentum of the Earth-Moon system. The 
situation becomes complicated in describing the past evolution of the Earth-Moon 
system. Because the Moon is moving away from us and the Earth rotation is slowing, this 
means that the Moon was closer and the Earth rotation was faster in the past.  Creationists 
argue that based on the tidal friction theory, the tidal friction should be stronger and the 
recessional rate of the Moon should be greater in the past, the distance of the Moon 
would quickly fall inside the Roche's limit (for earth, 15500 km) in which the Moon 
would be torn apart by gravity in 1 to 2 billion years ago. However, geological evidence 
indicates that the recession of the Moon in the past was slower than the present rate, i. e., 
the recession has been accelerating with time. Therefore, it must be concluded that tidal 
friction was very much less in the remote past than we would deduce on the basis of 
present-day observations (Stacey 1977). This was called “geological time scale 
difficulty” or “Lunar crisis” and is one of the main arguments by creationists against the 
tidal friction theory (Brush 1983).  
But we have to consider the case carefully in various aspects. One possible 
scenario is that the Earth has been undergoing dynamic evolution at all time scale since 
its inception, the geological and physical conditions (such as the continent positions and 
drifting, the crust, surface temperature fluctuation like the glacial/snowball effect, etc) at 
remote past could be substantially different from currently, in which the tidal friction 
could be much less; therefore, the receding rate of the Moon could be slower. Various 
tidal friction models were proposed in the past to describe the evolution of the Earth-
Moon system to avoid such difficulty or crisis and put the Moon at quite a comfortable 
distance from Earth at 4.5 billion years ago (Hansen 1982, Kagan and Maslova 1994, Ray 
et al. 1999, Finch 1981, Slichter 1963). The tidal friction theories explain that the present 
rate of tidal dissipation is anomalously high because the tidal force is close to a resonance 
in the response function of ocean (Brush 1983). Kagan gave a detailed review about those 
tidal friction models (Kagan 1997). Those models are based on many assumptions about 
geological (continental position and drifting) and physical conditions in the past, and 
many parameters (such as phase lag angle, multi-mode approximation with time 
dependent frequencies of the resonance modes, etc.) have to be introduced and carefully 
adjusted to make their predictions close to the geological evidence. However, those 
assumptions and parameters are still challenged, to certain extent, as concoction.  
The second possible scenario is that another mechanism could dominate the 
evolution of the Earth-Moon system and the role of the tidal friction is not significant. In 
the Meeting of Division of Particle and Field 2004, American Physical Society, 
University of California at Riverside, the author proposed a dark matter field fluid model 
(Pan 2005) with a non-Newtonian approach, the current Moon and Earth data agree with 
this model very well. This paper will demonstrate that the past evolution of Moon-Earth 
system can be described by the dark matter field fluid model without any assumptions 
about past geological and physical conditions. Although the subject of the evolution of 
the Earth-Moon system has been extensively studied analytically or numerically, to the 
author’s knowledge, there are no theories similar or equivalent to this model.  
2.  Invisible matter 
In modern cosmology, it was proposed that the visible matter in the universe is 
about 2 ~ 10 % of the total matter and about 90 ~ 98% of total matter is currently 
invisible which is called dark matter and dark energy, such invisible matter has an anti-
gravity property to make the universe expanding faster and faster.   
If the ratio of the matter components of the universe is close to this hypothesis, 
then, the evolution of the universe should be dominated by the physical mechanism of 
such invisible matter, such physical mechanism could be far beyond the current 
Newtonian physics and Einsteinian physics, and the Newtonian physics and Einsteinian 
physics could reflect only a corner of the iceberg of the greater physics.  
If the ratio of the matter components of the universe is close to this hypothesis, 
then, it should be more reasonable to think that such dominant invisible matter spreads in 
everywhere of the universe (the density of the invisible matter may vary from place to 
place); in other words, all visible matter objects should be surrounded by such invisible 
matter and the motion of the visible matter objects should be affected by the invisible 
matter if there are  interactions between the visible matter and the invisible matter.  
If the ratio of the matter components of the universe is close to this hypothesis, 
then, the size of the particles of the invisible matter should be very small and below the 
detection limit of the current technology; otherwise, it would be detected long time ago 
with such dominant amount. 
With such invisible matter in mind, we move to the next section to develop the 
dark matter field fluid model with non-Newtonian approach. For simplicity, all invisible 
matter (dark matter, dark energy and possible other terms) is called dark matter here. 
3. The dark matter field fluid model 
In this proposed model, it is assumed that: 
1.  A celestial body rotates and moves in the space, which, for simplicity, is uniformly 
filled with the dark matter which is in quiescent state relative to the motion of the 
celestial body. The dark matter possesses a field property and a fluid property; it can 
interact with the celestial body with its fluid and field properties; therefore, it can have 
energy exchange with the celestial body, and affect the motion of the celestial body. 
2.  The fluid property follows the general principle of fluid mechanics. The dark matter 
field fluid particles may be so small that they can easily permeate into ordinary 
“baryonic” matter;  i. e., ordinary matter objects could be saturated with such dark matter 
field fluid. Thus, the whole celestial body interacts with the dark matter field fluid, in the 
manner of a sponge moving thru water. The nature of the field property of the dark matter 
field fluid is unknown. It is here assumed that the interaction of the field associated with 
the dark matter field fluid with the celestial body is proportional to the mass of the 
celestial body. The dark matter field fluid is assumed to have a repulsive force against the 
gravitational force towards baryonic matter. The nature and mechanism of such repulsive 
force is unknown. 
With the assumptions above, one can study how the dark matter field fluid may 
influence the motion of a celestial body and compare the results with observations. The 
common shape of celestial bodies is spherical. According to Stokes's law, a rigid non-
permeable sphere moving through a quiescent fluid with a sufficiently low Reynolds 
number experiences a resistance force F 
rvF πμ6−=       (1) 
where v is the moving velocity, r is the radius of the sphere, and μ is the fluid viscosity 
constant. The direction of the resistance force F in Eq. 1 is opposite to the direction of the 
velocity v. For a rigid sphere moving through the dark matter field fluid, due to the dual 
properties of the dark matter field fluid and its permeation into the sphere, the force F 
may not be proportional to the radius of the sphere. Also, F may be proportional to the 
mass of the sphere due to the field interaction. Therefore, with the combined effects of 
both fluid and field, the force exerted on the sphere by the dark matter field fluid is 
assumed to be of the scaled form 
        (2) mvrF n−−= 16πη
where n is a parameter arising from saturation by dark matter field fluid, the r1-n can be 
viewed as the effective radius with the same unit as r, m is the mass of the sphere, and η 
is the dark matter field fluid constant, which is equivalent to μ. The direction of the 
resistance force F in Eq. 2 is opposite to the direction of the velocity v. The force 
described by Eq. 2 is velocity-dependent and causes negative acceleration. According to 
Newton's second law of motion, the equation of motion for the sphere is 
   mvr
m n−−= 16πη      (3) 
Then  
       (4) )6exp( 10 vtrvv
n−−= πη
where v0 is the initial velocity (t = 0) of the sphere. If the sphere revolves around a 
massive gravitational center, there are three forces in the line between the sphere and the 
gravitational center: (1) the gravitational force, (2) the centripetal acceleration force; and 
(3) the repulsive force of the dark matter field fluid.  The drag force in Eq. 3 reduces the 
orbital velocity and causes the sphere to move inward to the gravitational center. 
However, if the sum of the centripetal acceleration force and the repulsive force is 
stronger than the gravitational force, then, the sphere will move outward and recede from 
the gravitational center. This is the case of interest here. If the velocity change in Eq. 3 is 
sufficiently slow and the repulsive force is small compared to the gravitational force and 
centripetal acceleration force, then the rate of receding will be accordingly relatively 
slow. Therefore, the gravitational force and the centripetal acceleration force can be 
approximately treated in equilibrium at any time. The pseudo equilibrium equation is 
GMm 2
2 =       (5) 
where G is the gravitational constant, M is the mass of the gravitational center, and R is 
the radius of the orbit.  Inserting v of Eq.  4 into Eq. 5 yields  
   )12exp( 1
R n−= πη     (6) 
       (7) )12exp( 10 trRR
n−= πη
where 
R =       (8) 
R0 is the initial distance to the gravitational center.  Note that R exponentially increases 
with time. The increase of orbital energy with the receding comes from the repulsive 
force of dark matter field fluid.  The recessional rate of the sphere is 
dR n−= 112πη      (9) 
The acceleration of the recession is 
   ( Rr
Rd n 21
12 −= πη ) .     (10) 
The recessional acceleration is positive and proportional to its distance to the 
gravitational center, so the recession is faster and faster. 
According to the mechanics of fluids, for a rigid non-permeable sphere rotating 
about its central axis in the quiescent fluid, the torque T exerted by the fluid on the sphere 
ωπμ 38 rT −=       (11) 
where ω is the angular velocity of the sphere.  The direction of the torque in Eq. 11 is 
opposite to the direction of the rotation. In the case of a sphere rotating in the quiescent 
dark matter field fluid with angular velocity ω, similar to Eq. 2, the proposed T exerted 
on the sphere is 
( ) ωπη mrT n 318 −−=      (12) 
The direction of the torque in Eq. 12 is opposite to the direction of the rotation. The 
torque causes the negative angular acceleration 
=       (13) 
where I is the moment of inertia of the sphere in the dark matter field fluid 
   ( )21
2 nrmI −=       (14) 
Therefore, the equation of rotation for the sphere in the dark matter field fluid is 
   ωπη
d −−= 120      (15) 
Solving this equation yields 
       (16) )20exp( 10 tr
n−−= πηωω
where ω0 is the initial angular velocity.  One can see that the angular velocity of the 
sphere exponentially decreases with time and the angular deceleration is proportional to 
its angular velocity.   
For the same celestial sphere, combining Eq. 9 and Eq. 15 yields 
     (17) 
The significance of Eq. 17 is that it contains only observed data without assumptions and 
undetermined parameters; therefore, it is a critical test for this model. 
For two different celestial spheres in the same system, combining Eq. 9 and Eq. 
15 yields 
   67.1
1 −=−=⎟⎟
   (18) 
This is another critical test for this model. 
4. The current behavior of the Earth-Moon system agrees with the model 
 The Moon-Earth system is the simplest gravitational system. The solar system is 
complex, the Earth and the Moon experience not only the interaction of the Sun but also 
interactions of other planets. Let us consider the local Earth-Moon gravitational system as 
an isolated local gravitational system, i.e., the influence from the Sun and other planets 
on the rotation and orbital motion of the Moon and on the rotation of the Earth is 
assumed negligible compared to the forces exerted by the moon and earth on each other. 
In addition, the eccentricity of the Moon's orbit is small enough to be ignored. The data 
about the Moon and the Earth from references (Dickey et .al., 1994,  and Lang, 1992) are 
listed below for the readers' convenience to verify the calculation because the data may 
vary slightly with different data sources. 
Moon: 
    Mean radius:    r  = 1738.0 km 
    Mass:       m = 7.3483 × 1025 gram 
    Rotation period = 27.321661 days 
    Angular velocity of Moon = 2.6617 × 10-6 rad s-1 
    Mean distance to Earth Rm= 384400 km 
    Mean orbital velocity v = 1.023 km s-1 
    Orbit eccentricity    e = 0.0549 
    Angular rotation acceleration rate = -25.88 ±  0.5 arcsec century-2 
       = (-1.255 ± 0.024) × 10-4 rad century-2
       = (-1.260 ± 0.024) × 10-23 rad s-2
    Receding rate from Earth = 3.82 ±  0.07 cm year-1 = (1.21 ±  0.02) × 10-9 m s-1 
Earth: 
    Mean radius:    r  = 6371.0 km 
    Mass:       m = 5.9742 × 1027 gram 
    Rotation period = 23 h 56m 04.098904s = 86164.098904s 
    Angular velocity of rotation = 7.292115 × 10-5 rad s-1 
    Mean distance to the Sun Rm= 149,597,870.61 km 
    Mean orbital velocity v = 29.78 km s-1 
    Angular acceleration of Earth = (-5.5 ± 0.5) × 10-22 rad s-2
The Moon's angular rotation acceleration rate and increase in mean distance to the Earth 
(receding rate) were obtained from the lunar laser ranging of the Apollo Program (Dickey 
et .al., 1994). By inserting the data of the Moon's rotation and recession into Eq.  17, the 
result is 
 039.054.1
10662.21021.1
1092509.31026.1
   (19) 
The distance R in Eq. 19 is from the Moon's center to the Earth's center and the number 
384400 km is assumed to be the distance from the Moon's surface to the Earth's surface. 
Eq. 19 is in good agreement with the theoretical value of -1.67. The result is in accord 
with the model used here. The difference (about 7.8%) between the values of -1.54 and -
1.67 may come from several sources: 
1. Moon's orbital is not a perfect circle 
2. Moon is not a perfect rigid sphere. 
3. The effect from Sun and other planets. 
4. Errors in data. 
5. Possible other unknown reasons. 
The two parameters n and η in Eq. 9 and Eq. 15 can be determined with two data 
sets.  The third data set can be used to further test the model.  If this model correctly 
describes the situation at hand, it should give consistent results for different motions. The 
values of n and η calculated from three different data sets are listed below (Note, the 
mean distance of the Moon to the Earth and mean radii of the Moon and the Earth are 
used in the calculation). 
    The value of n:          n = 0.64 
    From the Moon's rotation:   η  = 4.27 ×  10-22  s-1 m-1
    From the Earth's rotation:     η  = 4.26 × 10-22  s-1 m-1
    From the Moon's recession:  η = 4.64 × 10-22  s-1 m-1
One can see that the three values of η are consistent within the range of error in the data. 
    The average value of  η:   η = (4.39 ± 0.22) × 10-22 s-1 m-1
By inserting the data of the Earth's rotation, the Moon’s recession and the value of n into 
Eq. 18, the result is 
14.053.1
6371000
1738000
1021.11029.7
1092509.3105.5
)64.01(
  (20) 
This is also in accord with the model used here. 
The dragging force exerted on the Moon's orbital motion by the dark matter field 
fluid is -1.11 × 108 N, this is negligibly small compared to the gravitational force between 
the Moon and the Earth ~ 1.90 × 1020 N;  and the torque exerted by the dark matter field 
fluid on the Earth’s and the Moon's rotations is T = -5.49 × 1016 Nm and -1.15 × 1012 Nm, 
respectively.  
5. The evolution of Earth-Moon system 
Sonett  et al. found that the length of the terrestrial day 900 million years ago was 
about 19.2 hours based on the laminated tidal sediments on the Earth (Sonett et al., 
1996). According to the model presented here, back in that time, the length of the day 
was about 19.2 hours, this agrees very well with Sonett et al.'s result.   
Another critical aspect of modeling the evolution of the Earth-Moon system is to  
give a reasonable estimate of the closest distance of the Moon to the Earth when the 
system was established at 4.5 billion years ago. Based on the dark matter field fluid 
model, and the above result, the closest distance of the Moon to the Earth was about 
259000 km (center to center) or 250900 km (surface to surface) at 4.5 billion years ago, 
this is far beyond the Roche's limit. In the modern astronomy textbook by Chaisson and 
McMillan (Chaisson and McMillan, 1993, p.173), the estimated distance at 4.5 billion 
years ago was 250000 km, this is probably the most reasonable number that most 
astronomers believe and it agrees excellently with the result of this model. The closest 
distance of the Moon to the Earth by Hansen’s models was about 38 Earth radii or 
242000 km (Hansen, 1982).  
According to this model, the length of day of the Earth was about 8 hours at 4.5 
billion years ago. Fig. 1 shows the evolution of the distance of Moon to the Earth and the 
length of day of the Earth with the age of the Earth-Moon system described by this model 
along with data from Kvale et al. (1999), Sonett et al. (1996) and Scrutton (1978). One 
can see that those data fit this model very well in their time range.  
Fig. 2 shows the geological data of solar days year-1 from Wells (1963) and from 
Sonett et al. (1996) and the description (solid line) by this dark matter field fluid model 
for past 900 million years. One can see that the model agrees with the geological and 
fossil data beautifully.  
The important difference of this model with early models in describing the early 
evolution of the Earth-Moon system is that this model is based only on current data of the 
Moon-Earth system and there are no assumptions about the conditions of earlier Earth 
rotation and continental drifting. Based on this model, the Earth-Moon system has been 
smoothly evolving to the current position since it was established and the recessional rate 
of the Moon has been gradually increasing, however, this description does not take it into 
account that there might be special events happened in the past to cause the suddenly 
significant changes in the motions of the Earth and the Moon, such as strong impacts by 
giant asteroids and comets, etc, because those impacts are very common in the universe. 
The general pattern of the evolution of the Moon-Earth system described by this model 
agrees with geological evidence. Based on Eq. 9, the recessional rate exponentially 
increases with time. One may then imagine that the recessional rate will quickly become 
very large.  The increase is in fact extremely slow. The Moon's recessional rate will be 
3.04 × 10-9 m s-1 after 10 billion years and 7.64 × 10-9 m s-1 after 20 billion years. 
However, whether the Moon's recession will continue or at some time later another 
mechanism will take over is not known. It should be understood that the tidal friction 
does affect the evolution of the Earth itself such as the surface crust structure, continental 
drifting and evolution of bio-system, etc; it may also play a role in slowing the Earth’s 
rotation, however, such role is not a dominant mechanism.  
Unfortunately, there is no data available for the changes of the Earth's orbital 
motion and all other members of solar system. According to this model and above results, 
the recessional rate of the Earth should be 6.86 × 10-7 m s-1 = 21.6 m year-1 = 2.16 km 
century-1, the length of a year increases about 6.8 ms and the change of the temperature is 
-1.8 × 10-8 K year-1 with constant radiation level of the Sun and the stable environment on 
the Earth. The length of a year at 1 billion years ago would be 80% of the current length 
of the year. However, much evidence (growth-bands of corals and shellfish as well as 
some other evidences) suggest that there has been no apparent change in the length of the 
year over the billion years and the Earth's orbital motion is more stable than its rotation.  
This suggests that dark matter field fluid is circulating around Sun with the same 
direction and similar speed of Earth (at least in the Earth's orbital range). Therefore, the 
Earth's orbital motion experiences very little or no dragging force from the dark matter 
field fluid. However, this is a conjecture, extensive research has to be conducted to verify 
if this is the case.  
6. Skeptical description of the evolution of the Mars 
The Moon does not have liquid fluid on its surface, even there is no air, therefore, 
there is no ocean-like tidal friction force to slow its rotation; however, the rotation of the 
Moon is still slowing at significant rate of (-1.260 ± 0.024) × 10-23 rad s-2,  which agrees 
with the model very well. Based on this, one may reasonably think that the Mars’s 
rotation should be slowing also. 
The Mars is our nearest neighbor which has attracted human’s great attention 
since ancient time. The exploration of the Mars has been heating up in recent decades. 
NASA, Russian and Europe Space Agency sent many space crafts to the Mars to collect 
data and study this mysterious planet. So far there is still not enough data about the 
history of this planet to describe its evolution. Same as the Earth, the Mars rotates about 
its central axis and revolves around the Sun, however, the Mars does not have a massive 
moon circulating it (Mars has two small satellites: Phobos and Deimos) and there is no 
liquid fluid on its surface, therefore, there is no apparent ocean-like tidal friction force to 
slow its rotation by tidal friction theories. Based on the above result and current the 
Mars's data, this model predicts that the angular acceleration of the Mars should be about 
-4.38 × 10-22 rad s-2. Figure 3 describes the possible evolution of the length of day and the 
solar days/Mars year, the vertical dash line marks the current age of the Mars with 
assumption that the Mars was formed in a similar time period of the Earth formation. 
Such description was not given before according to the author’s knowledge and is 
completely skeptical due to lack of reliable data.  However, with further expansion of the 
research and exploration on the Mars, we shall feel confident that the reliable data about 
the angular rotation acceleration of the Mars will be available in the near future which 
will provide a vital test for the prediction of this model. There are also other factors 
which may affect the Mars’s rotation rate such as mass redistribution due to season 
change, winds, possible volcano eruptions and Mars quakes. Therefore, the data has to be 
carefully analyzed. 
7.  Discussion about the model 
From the above results, one can see that the current Earth-Moon data and the 
geological and fossil data agree with the model very well and the past evolution of the 
Earth-Moon system can be described by the model without introducing any additional 
parameters; this model reveals the interesting relationship between the rotation and 
receding (Eq.  17 and Eq.  18) of the same celestial body or different celestial bodies in 
the same gravitational system, such relationship is not known before. Such success can 
not be explained by “coincidence” or “luck” because of so many data involved (current 
Earth’s and Moon’s data and geological and fossil data) if one thinks that this is just a 
“ad hoc” or a wrong model, although the chance for the natural happening of such 
“coincidence” or “luck” could be greater than wining a jackpot lottery; the future Mars’s 
data will clarify this; otherwise, a new theory from different approach can be developed 
to give the same or better description as this model does. It is certain that this model is 
not perfect and may have defects, further development may be conducted. 
James Clark Maxwell said in the 1873 “ The vast interplanetary and interstellar 
regions will no longer be regarded as waste places in the universe, which the Creator has 
not seen fit to fill with the symbols of the manifold order of His kingdom. We shall find 
them to be already full of this wonderful medium; so full, that no human power can 
remove it from the smallest portion of space, or produce the slightest flaw in its infinite 
continuity. It extends unbroken from star to star ….” The medium that Maxwell talked 
about is the aether which was proposed as the carrier of light wave propagation. The 
Michelson-Morley experiment only proved that the light wave propagation does not 
depend on such medium and did not reject the existence of the medium in the interstellar 
space. In fact, the concept of the interstellar medium has been developed dramatically 
recently such as the dark matter, dark energy, cosmic fluid, etc. The dark matter field 
fluid is just a part of such wonderful medium and “precisely” described by Maxwell. 
7. Conclusion 
The evolution of the Earth-Moon system can be described by the dark matter field 
fluid model with non-Newtonian approach and the current data of the Earth and the Moon 
fits this model very well. At 4.5 billion years ago, the closest distance of the Moon to the 
Earth could be about 259000 km, which is far beyond the Roche’s limit and the length of 
day was about 8 hours. The general pattern of the evolution of the Moon-Earth system 
described by this model agrees with geological and fossil evidence. The tidal friction may 
not be the primary cause for the evolution of the Earth-Moon system. The Mars’s rotation 
is also slowing with the angular acceleration rate about -4.38 × 10-22 rad s-2. 
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Caption 
Figure 1, the evolution of Moon’s distance and the length of day of the earth with 
the age of the Earth-Moon system. Solid lines are calculated according to the dark matter 
field fluid model. Data sources: the Moon distances are from Kvale and et al. and for the 
length of day: (a and b) are from Scrutton ( page 186, fig. 8), c is from Sonett and et al.  
The dash line marks the current age of the Earth-Moon system.  
Figure 2, the evolution of Solar days of year with the age of the Earth-Moon 
system. The solid line is calculated according to dark matter field fluid model. The data 
are from Wells (3.9 ~ 4.435 billion years range), Sonett (3.6 billion years) and current 
age (4.5 billion years). 
Figure 3, the skeptical description of the evolution of Mars’s length of day and the 
solar days/Mars year with the age of the Mars (assuming that the Mars’s age is about 4.5 
billion years). The vertical dash line marks the current age of Mars. 
Figure 1, Moon's distance and the length of day of Earth 
change with the age of Earth-Moon system
The age of Earth-Moon system (109 years)
0 1 2 3 4 5
Distance
Length of day
Roche's limit
Hansen's result
Figure 2,  the solar days / year vs. the age of the Earth
The age of the Earth (109 years)
3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6
