**9. Planetary satellite dynamics**

86 Space Science

How did the planet of G186A survive the violent changing phases of the white dwarf, post main sequence evolution of star? A white dwarf is a spent out main sequence star which expands into Red Giant and then shrinks into a White Dwarf.

Five short period planets in multiple star system cannot be explained in a classical fashion. [Eggenberger et al 2003]

solar system Description Reason for enigma

secondary around each other is 25.7y;

Primary containing the exoplanet is as

Stars are orbiting each other at

massive as 3 times or more as compared to the secondary

Stars are orbiting each other at

Secondary star is a white dwarf; Primary containing the exoplanet is as

massive as 3 times or more as compared to the secondary

Stars are orbiting each other at distances of 20AU & an orbital period

massive as 3 times or more as compared to the secondary; Companion planet is MSini=1.7MJ

Massive short period planets are found in multiple star system

nothing beyond in the colder region that could nucleate and grow into a giant planet.

heliocentric distance just as hot-jupiters do in scorchingly tight orbits.

Mugrauer 2005, Hatzes et al 2003]

\* Initially it was thought that Giant planets must have formed in colder region far from their parent stars. Icy nuggets act as seeds that accumulate enough dust to build up to a critical mass where by runaway accretion it is enveloped by a large mass of gas giving birth to gas giants. These icy nuggets can form only beyond snowline[Sasselov & Lecar 2000]. But in HD 188753 this could not have happened. This is because the secondary system of star pair would truncate the disk to 1.3AU leaving

"Giant planets in circumstellar disks can migrate inward from their initial formation positions. Radial migration is caused by inward torques between the planet and disk, by outward torque between the planet and the spinning star and by outward torques due to Roche lobe overflow and consequent mass loss from the planet." [Trilling, Benz et al 1998]. Through numerical solutions it has been shown that taking all the torques into consideration, Jupiter-mass planets can stably arrive and survive at small

Table 7. The exoplanets which are conundrum.[Konacki 2005, Hatzes & Wuchterl 2005,

Orbital radius= 2.13AU; Orbital period=906d;

Primary containing the exoplanet is as

distances of 20AU

distances of 20AU;

of 56y;

Name of the extra-

HD 41004 (binary

GI 186(binary

γ Cephei(binary

19 binary or multiple star systems are inhabitated by a

planet

stars)

stars)

stars)

On 21st July 1994, the Silver Jubilee Celebration Year of Man's landing on Moon, NASA gave a press release stating that Moon has receded by 1 meter in 25 years from 1969 to 1994. Using this piece of data, the first Author redid the analysis of Earth-Moon System [Sharma 1995]. In a subsequent paper the Authors [Sharma, B. K. and Ishwar, B " Basic Mechanics of Planet-Satellite Interaction with special reference to Earth-Moon System", 2004, http://arXiv.org/abs/0805.0100 ] found that Satellites-Planet Systems have a characteristic lom(length of month)/lod(length of day) equation:

LOM/LOD = E×a3/2 – F×a2 [The proof is given in SOM\_Appendix A]

Where l.o.m. = length of month ( sidereal period of orbital rotation of the natural satellite around host planet which in case of our satellite Moon is 27.3 days);

l.o.d. = length of day ( spin period of the host planet which in our case is 24 hours or 1 solar day);

a = semi- major axis of the elliptical orbit of the satellite ( for Moon it is 3,84,400 Km);

E = JT/(BC);

JT = total angular momentum of the Satellite- Planet System,

= (Jspin )planet + (Jorbital)system + (Jspin)satellite ;

B = √[G(M + m)];

G = Gravitational Constant = 6.67 × 10-11 N-m2/Kg2 ;

M = mass of the host planet;

m = mass of the satellite;

C = Principal Moment of Inertia around the spin axis of the Planet;

F = m/[C(1+m/M)];

When lom/lod = 1 we have geosynchronous orbit.

$$\mathbf{E} \times \mathbf{a}^{3/2} - \mathbf{F} \times \mathbf{a}^2 = \mathbf{1} \tag{1}$$

Equation (1) has two roots and hence planet -satellite systems have two geo-synchronous. Only at these two Geo-synchronous orbits the system is in equilibrium because the orbits are non-dissipative. Elsewhere the system is dissipative hence in non-equilibrum either spiraling out to the outer geo-synchronous orbit or spiraling inward to its certain doom. The inner Geo-synchronous orbit lies at energy maxima whereas the outer Geosynchronous orbit is at energy minima. Therefore the inner geo-orbit is an unstable equilibrium orbit and the outer geo-orbit is a stable equilibrium orbit.

When the natural satellite is at the inner geo-orbit it is easily perturbed by solar wind or cosmic particles or solar insolation. It tumbles out on an expanding outward spiral path or it falls short of the inner geo-orbit on inward collapsing spiral path. Inward collapsing spiral path is entirely a runaway path. The outward spiral path, because of energy conservation, is initially an impulsive gravitational runaway phase which quickly terminates because of tidal dissipation in the central host body due to tidal streching and squeezing . This runaway phase is the gravitational sling shot phase. After the gravitational sling shot phase, the natural satellite coasts on its own towards the outer geo orbit. Our Moon is on a midway course in its journey towards the outer geo-orbit. Charon, a satellite of Pluto, has already arrived at the outer geo-orbit. The satellite may remain stay put in the outer geo-orbit as Charon is doing or it may be deflected as our Moon will be.

Enigma of the Birth and Evolution of Solar Systems

reactions in the corresponding moon.

2004, http://arXiv.org/abs/0805.0100].

generality. This will be done in a future paper.

have essentially analyzed Sun-planet as a two body problem.

bodies are analyzed.

secondary system.

velocity profile.

May Be Solved by Invoking Planetary-Satellite Dynamics 89

energy decreases. Traveling "above" and "below" a moon alters the direction modifying only the orientation (and angular momentum magnitude). Intermediate flyby orientation change both energy and angular momentum. Accompanying these actions there are reciprocal

The above slingshot effect is in a three body problem. In a three body problem , the heaviest body is the primary body. With respect to the primary body the secondary system of two

In case of planet flyby, planet is the primary body and the moon- spacecraft constitute the

While analyzing the planetary satellites, Sun is the primary body and planet-satellite is the secondary system. But in our Keplerian approximate analysis, Sun has been neglected without any loss of generality and without any loss of accuracy. In fact the general trend of evolution of our Moon has been correctly analyzed [Sharma, B. K. and Ishwar, B " Basic Mechanics of Planet-Satellite Interaction with special reference to Earth-Moon System",

While analyzing the Sun-planet system, galactic center is the primary body and Sun-planet is the secondary system. But in our analysis the galactic center has been neglected and we

In a similar fashion in the analysis of Planet Flyby-Gravity Assist Maneuvers, Planet is the primary body. The planet can be neglected and moon-spacecraft can be treated as a two body problem and the same results can be obtained without any loss of accuracy or

The gravitational sling shot becomes clearer if we look at the radial acceleration and radial

Fig. 2. Radial Acceleration Profile of Moon (Within aG1 the Moon is accelerated inward. Beyond aG1 the Moon is rapidly accelerated outward under the influence of an impulsive gravitational torque due to rapid transfer of spin rotational energy. The maxima of the outward radial acceleration occurs at a1. (This is the peak of the impulsive sling shot torque.)

#### **10. The new hypothesis- gravitational sling shot model of planet-satellite system**

The Authors did the Keplerian-approximated analysis of Earth-Moon, Mars-Deimos-Phobos and Pluto-Charon [Sharma & Ishwar 2004A, Sharma, Rangesh & Ishwar 2009]. The Authors were able to generate the outward expanding spiral path of Moon as shown in Figure 1.

In a sequel paper on the New Perspective of the Solar System[Sharma & Ishwar 2004B, Sharma 2011], it was established that Planets experience a similar kind of impulsive slingshot phase due to Sun as our Moon does due to Earth. This leads to new paradigm on the birth and evolution of our as well as extra Solar Systems.

#### **10.1 THe phenomena of gravitational slingshot**

Planet fly-by, gravity assist is routinely used to boost the mission spacecrafts to explore the far reaches of our solar system[Dukla, Cacioppo & Gangopadhyaya 2004, Jones 2005, Epstein 2005, Cook 2005]. Voyager I and II used the boost provided by Jupiter to reach Uranus and Neptune. Cassini has utilized 4 such assists to reach Saturn.

A space-craft which passes " behind" the moon gets an increase in its velocity(and orbital energy) relative to the primary body. In effect the primary body launches the space craft on an outward spiral path. If the spacecraft flies "infront" of a moon, the speed and the orbital

Fig. 1. Lunar Orbital Radius expanding spiral trajectory obtained from the simulation for the age of Moon (i.e. from the time of Giant Impact to the present times covering a time span of 4.5Gyrs).

88 Space Science

The Authors did the Keplerian-approximated analysis of Earth-Moon, Mars-Deimos-Phobos and Pluto-Charon [Sharma & Ishwar 2004A, Sharma, Rangesh & Ishwar 2009]. The Authors were able to generate the outward expanding spiral path of Moon as shown in Figure 1.

In a sequel paper on the New Perspective of the Solar System[Sharma & Ishwar 2004B, Sharma 2011], it was established that Planets experience a similar kind of impulsive slingshot phase due to Sun as our Moon does due to Earth. This leads to new paradigm on the

Planet fly-by, gravity assist is routinely used to boost the mission spacecrafts to explore the far reaches of our solar system[Dukla, Cacioppo & Gangopadhyaya 2004, Jones 2005, Epstein 2005, Cook 2005]. Voyager I and II used the boost provided by Jupiter to reach

A space-craft which passes " behind" the moon gets an increase in its velocity(and orbital energy) relative to the primary body. In effect the primary body launches the space craft on an outward spiral path. If the spacecraft flies "infront" of a moon, the speed and the orbital

Fig. 1. Lunar Orbital Radius expanding spiral trajectory obtained from the simulation for the age of Moon (i.e. from the time of Giant Impact to the present times covering a time span

of 4.5Gyrs).

**10. The new hypothesis- gravitational sling shot model of planet-satellite system** 

birth and evolution of our as well as extra Solar Systems.

Uranus and Neptune. Cassini has utilized 4 such assists to reach Saturn.

**10.1 THe phenomena of gravitational slingshot** 

energy decreases. Traveling "above" and "below" a moon alters the direction modifying only the orientation (and angular momentum magnitude). Intermediate flyby orientation change both energy and angular momentum. Accompanying these actions there are reciprocal reactions in the corresponding moon.

The above slingshot effect is in a three body problem. In a three body problem , the heaviest body is the primary body. With respect to the primary body the secondary system of two bodies are analyzed.

In case of planet flyby, planet is the primary body and the moon- spacecraft constitute the secondary system.

While analyzing the planetary satellites, Sun is the primary body and planet-satellite is the secondary system. But in our Keplerian approximate analysis, Sun has been neglected without any loss of generality and without any loss of accuracy. In fact the general trend of evolution of our Moon has been correctly analyzed [Sharma, B. K. and Ishwar, B " Basic Mechanics of Planet-Satellite Interaction with special reference to Earth-Moon System", 2004, http://arXiv.org/abs/0805.0100].

While analyzing the Sun-planet system, galactic center is the primary body and Sun-planet is the secondary system. But in our analysis the galactic center has been neglected and we have essentially analyzed Sun-planet as a two body problem.

In a similar fashion in the analysis of Planet Flyby-Gravity Assist Maneuvers, Planet is the primary body. The planet can be neglected and moon-spacecraft can be treated as a two body problem and the same results can be obtained without any loss of accuracy or generality. This will be done in a future paper.

The gravitational sling shot becomes clearer if we look at the radial acceleration and radial velocity profile.

Fig. 2. Radial Acceleration Profile of Moon (Within aG1 the Moon is accelerated inward. Beyond aG1 the Moon is rapidly accelerated outward under the influence of an impulsive gravitational torque due to rapid transfer of spin rotational energy. The maxima of the outward radial acceleration occurs at a1. (This is the peak of the impulsive sling shot torque.)

Enigma of the Birth and Evolution of Solar Systems

the first derivative of Eq.2 to zero we get:

*Iap*

*v a*

of the system.

value of K.

From Eq.4, structure exponent 'M' is calculated.

 

**of our planet Earth by primary-centric analysis3**

Earth-Moon System", 2008, http://arXiv.org/abs/0805.0100].

receded from 15,000Km to the present Lunar Orbit of 384,400Km.

May Be Solved by Invoking Planetary-Satellite Dynamics 91

Fig. 3. Radial Velocity Profile of Moon. (Beyond aG1, Moon is rapidly accelerated to a

central body as well as it tidally heats up the central body by tidal deformations.

1.5 <sup>2</sup>

calculate the value of 'K' from Eq. 2 equated to Vmax at semi-major axis 'a2'.

1 1

maximum radial velocity,Vmax, at a2 where Sling-Shot Effect terminates and radial acceleration is zero. Then onward Moon coasts on it own towards the outer Geo-Synchronous Orbit aG2)

rotational energy is transferred to the primary. This rotational energy causes spin-up of the

Since Eq.2 has a maxima at a2 therefore the first derivative of Eq. 2 has a zero at a2. Equating

We donot yet know the structure constant K. We make an intelligent guess of Vmax and

Using these values of 'K' and 'M' the time integral equation is set up and tested for the age

This transit time should be of the order of 4.5Gy in the case of Iapetus because that is the age of Iapetus.[ Castillo- Rogez et al (2007)]. Through several iterations we arrive at the correct

**10.3 Theoretical verification of the experimentally observed 'lengthening of day' curve** 

Since the birth of Earth-Moon System, Earth's spin has been slowing down and Moon has been receding. Earth's spin has slowed down from 5 hours to 24 hours today and Moon has

3 [Sharma, B. K. and Ishwar, B " Basic Mechanics of Planet-Satellite Interaction with special reference to

<sup>1</sup> , , *G Iapresent <sup>G</sup> Iap*

*da a a transittime froma tothe present valueof a*

(5)

<sup>2</sup> 2 2.5 0.5 0 *E M a F M a M at a Iap Iap* (4)

#### **10.2 Setting up of the time integral equation.**

In setting up the time integral equation the first step is to set up the radial velocity expression which has been derived in SOM\_Appendix A.

The radial velocity expression is as follows:

$$\frac{da\_{\rm lap}}{dt} = \frac{K}{a\_{\rm loop}^M} \left(\frac{\alpha}{\Omega} - 1\right) \cdot \frac{2a^{1/2}}{m \cdot B} = \frac{K}{a\_{\rm loop}^M} \left(E \cdot a\_{\rm loop}^{3/2} - F \cdot a\_{\rm loop}^2 - 1\right) \cdot \frac{2a^{1/2}}{m \cdot B}$$

Or

$$\text{cov}\left(a\_{\text{lap}}\right) = \frac{da\_{\text{lap}}}{dt} = \frac{2K}{a\_{\text{lap}}^M} \cdot \frac{1}{m \cdot B} \cdot \left(E \cdot a\_{\text{lap}}^2 - F \cdot a\_{\text{lap}}^{2.5} - \sqrt{a\_{\text{lap}}}\right) \tag{2}$$

Where K is the structure constant and M is the structure exponent. All the other symbols are defined as before. Equation 2 gives the radial velocity of natural Satellite Iapetus with respect to Saturn.

Between aG1 and aG2 , ω/Ω is greater than Unity hence radial velocity is positive and recessive.

At less than aG1 , ω/Ω is less than Unity hence radial velocity is negative and secondary approaches primary.

At greater than aG2 , ω/Ω is negative which is physically not possible in a prograde system hence system is untenable and it is a forbidden state.

Spin to Orbital velocity equation yields a root when it is in second mean motion resonance (MMR) position. That is:

$$\frac{d\phi}{d\Omega} = E \cdot a\_{lap}^{3/2} - F \cdot a\_{lap}^2 = \mathbf{2} \tag{3}$$

This gives a root at a2 which is gravitation resonance point and I assume that after the secondary undergoes gravitational sling shot impetus, it attains maximum recession velocity at this point. After this maxima, recession velocity continuously decreases until it reaches zero magnitude at outer Clarke's Orbit as shown in Figure 3.

Thus as is evident from Eq.2, recession velocity is zero at aG1 and aG2. From aG1 to a2 , the system is in conservative phase and secondary experiences a powerful sling-shot impulsive torque which imparts sufficient rotational energy to the secondary by virtue of which the secondary coasts on its own from a2 to aG2 during which time the system is in dissipative phase, Secondary is exerting a tidal drag on the central body and all the rotational energy released by the central body as a result of *de-spinning* is lost as tidal heat, but not completely. This tidal heat is produced during tidal deformation of both the components of the binary if the secondary is not in synchronous orbit. Our Moon is presently in synchronous orbit hence it is not undergoing tidal heating but Earth is undergoing tidal heating.

When the secondary tumbles into sub-synchronous orbit it experiences a negative radial velocity which launches it on a collapsing spiral and the system is *spun-up* . In this collapsing phase, secondary exerts an accelerating tidal torque on the central body and 90 Space Science

In setting up the time integral equation the first step is to set up the radial velocity

 1/2 1/2 2 2 3/2 2 1 1 *Iap M M Iap Iap*

 *Iap* 2 1 2 2.5 *Iap M Iap Iap iap*

Where K is the structure constant and M is the structure exponent. All the other symbols are defined as before. Equation 2 gives the radial velocity of natural Satellite Iapetus with

Between aG1 and aG2 , ω/Ω is greater than Unity hence radial velocity is positive and

At less than aG1 , ω/Ω is less than Unity hence radial velocity is negative and secondary

At greater than aG2 , ω/Ω is negative which is physically not possible in a prograde system

Spin to Orbital velocity equation yields a root when it is in second mean motion resonance

3/2 2 2 *Ea Fa Iap Iap*

This gives a root at a2 which is gravitation resonance point and I assume that after the secondary undergoes gravitational sling shot impetus, it attains maximum recession velocity at this point. After this maxima, recession velocity continuously decreases until it reaches

Thus as is evident from Eq.2, recession velocity is zero at aG1 and aG2. From aG1 to a2 , the system is in conservative phase and secondary experiences a powerful sling-shot impulsive torque which imparts sufficient rotational energy to the secondary by virtue of which the secondary coasts on its own from a2 to aG2 during which time the system is in dissipative phase, Secondary is exerting a tidal drag on the central body and all the rotational energy released by the central body as a result of *de-spinning* is lost as tidal heat, but not completely. This tidal heat is produced during tidal deformation of both the components of the binary if the secondary is not in synchronous orbit. Our Moon is presently in synchronous orbit

When the secondary tumbles into sub-synchronous orbit it experiences a negative radial velocity which launches it on a collapsing spiral and the system is *spun-up* . In this collapsing phase, secondary exerts an accelerating tidal torque on the central body and

*v a Ea Fa a*

*dt m B <sup>a</sup>* (2)

(3)

*da K aK <sup>a</sup> Ea Fa dt a a m B m B*

**10.2 Setting up of the time integral equation.** 

The radial velocity expression is as follows:

Or

respect to Saturn.

approaches primary.

(MMR) position. That is:

recessive.

expression which has been derived in SOM\_Appendix A.

hence system is untenable and it is a forbidden state.

 

*Iap Iap*

*da K*

hence it is not undergoing tidal heating but Earth is undergoing tidal heating.

zero magnitude at outer Clarke's Orbit as shown in Figure 3.

*Iap*

Fig. 3. Radial Velocity Profile of Moon. (Beyond aG1, Moon is rapidly accelerated to a maximum radial velocity,Vmax, at a2 where Sling-Shot Effect terminates and radial acceleration is zero. Then onward Moon coasts on it own towards the outer Geo-Synchronous Orbit aG2)

rotational energy is transferred to the primary. This rotational energy causes spin-up of the central body as well as it tidally heats up the central body by tidal deformations.

Since Eq.2 has a maxima at a2 therefore the first derivative of Eq. 2 has a zero at a2. Equating the first derivative of Eq.2 to zero we get:

$$E(2-M)a\_{lap}^{1.5} - F(2.5-M)a\_{lap}^2 - (0.5-M) = 0 \text{ at } a\_2 \tag{4}$$

From Eq.4, structure exponent 'M' is calculated.

We donot yet know the structure constant K. We make an intelligent guess of Vmax and calculate the value of 'K' from Eq. 2 equated to Vmax at semi-major axis 'a2'.

Using these values of 'K' and 'M' the time integral equation is set up and tested for the age of the system.

$$\int \left[ \frac{1}{\upsilon(a\_{\rm lap})} da, a\_{\rm G1}, a\_{\rm lap\text{reset}} \right] = \text{transit time from } a\_{\rm G1} \text{ to the present value of } a\_{\rm lap} \tag{5}$$

This transit time should be of the order of 4.5Gy in the case of Iapetus because that is the age of Iapetus.[ Castillo- Rogez et al (2007)]. Through several iterations we arrive at the correct value of K.

#### **10.3 Theoretical verification of the experimentally observed 'lengthening of day' curve of our planet Earth by primary-centric analysis3**

Since the birth of Earth-Moon System, Earth's spin has been slowing down and Moon has been receding. Earth's spin has slowed down from 5 hours to 24 hours today and Moon has receded from 15,000Km to the present Lunar Orbit of 384,400Km.

<sup>3 [</sup>Sharma, B. K. and Ishwar, B " Basic Mechanics of Planet-Satellite Interaction with special reference to Earth-Moon System", 2008, http://arXiv.org/abs/0805.0100].

Enigma of the Birth and Evolution of Solar Systems

Fig. 4. Lengthening of Day Curve w.r.t. time by Observation

Fig. 5. Lengthening of day curve w.r.t. time by Theory assuming constant C.

Fig. 6. Superposition of the two curves, one by observation and the other by calculation,

**observation** 

with constant C.

May Be Solved by Invoking Planetary-Satellite Dynamics 93

As seen from the superposition of the two lengthening of day curves, there is remarkable match between Observation and Theory in the recent past after the Pre-Cambrian Explosion

**10.4 Comparative study of lengthening of day curve of our Earth by theory and** 

John West Wells through the study of daily and annual bands of Coral fossils and other marine creaturs in bygone era has obtained ten length of day of bygone eras [Wells 1963, Wells 1966]. These benchmarks are tabulated in Table (8).

Leschiuta & Tavella [Leschitua & Tavella 2001] have given the estimate of the synodic month. From the synodic month we can estimate the length of the Solar Day as given in SOM\_Appendix [C]. The results are tabulated in Table (9). [Leschitua & Tavella 2001 based on the study of marine creature fossils]

Kaula & Harris [1975] have determined the synodic month through the studies of marine creatures. The results are tabulated in Table (10).

One benchmark has been provided by Charles P. Sonnett et al through the study of tidalies in ancient canals and estuaries [Sonett & Chan 1998 ]. He gives an estimate of 4 18.9 *TE* hours mean solar day length at about 900 million years B.P. in Proterozoic Eon, pre-Cambrian Age.


Table 8. The Observed lod based on the study of Coral Fossils.


Table 9. Observed Synodic Month


Table 10. Observed Synodic Month (Kaula & Harris 1975) based on the studies of Marine creatures.

92 Space Science

John West Wells through the study of daily and annual bands of Coral fossils and other marine creaturs in bygone era has obtained ten length of day of bygone eras [Wells 1963,

Leschiuta & Tavella [Leschitua & Tavella 2001] have given the estimate of the synodic month. From the synodic month we can estimate the length of the Solar Day as given in SOM\_Appendix [C]. The results are tabulated in Table (9). [Leschitua & Tavella 2001 based

Kaula & Harris [1975] have determined the synodic month through the studies of marine

One benchmark has been provided by Charles P. Sonnett et al through the study of tidalies in ancient canals and estuaries [Sonett & Chan 1998 ]. He gives an estimate of 4 18.9 *TE* hours mean solar day length at about 900 million years B.P. in Proterozoic Eon, pre-Cambrian Age.

T (yrs B.P.) T\* (yrs after the Giant Impact) Length of obs. Solar Day \* *TE* (hrs)

Observed Synodic Month (modern days)

(modern days) Estimated Solar Day (hrs).

Estimated Solar Day (hrs).

13.67 (with modern C) 16.86 (with C = 9.99\* 37 2 10 *kg m* )

65 Ma 4.46456G 23.627 135 Ma 4.39456G 23.25 180 Ma 4.34956G 23.0074 230 Ma 4.29956G 22.7684 280 Ma 4.24956G 22.4765 345 Ma 4.18456G 22.136 380 Ma 4.14956G 21.9 405 Ma 4.12456G 21.8 500 Ma 4.02956G 21.27 600 Ma 3.92956 G 20.674

Table 8. The Observed lod based on the study of Coral Fossils.

Giant Impact)

900 Ma (Proterozoic) 3.62956G 25.0 19.2 600Ma (Proterozoic) 3.92956G 26.2 20.7 300Ma (Carboniferous) 4.22956G 28.7 22.3 0 (Neozoic) 4.52956G 29.5 24

45 Ma 4.48456G 29.1 23.566

Table 10. Observed Synodic Month (Kaula & Harris 1975) based on the studies of Marine

Observed Synodic Month

T (yrs. B.P.) T\* (yrs. After the

Giant Impact)

2.8 Ga 1.72956G 17

Table 9. Observed Synodic Month

T (yrs. B.P.) T\* (yrs. After the

creatures.

Wells 1966]. These benchmarks are tabulated in Table (8).

on the study of marine creature fossils]

creatures. The results are tabulated in Table (10).

#### **10.4 Comparative study of lengthening of day curve of our Earth by theory and observation**

As seen from the superposition of the two lengthening of day curves, there is remarkable match between Observation and Theory in the recent past after the Pre-Cambrian Explosion

Fig. 4. Lengthening of Day Curve w.r.t. time by Observation

Fig. 5. Lengthening of day curve w.r.t. time by Theory assuming constant C.

Fig. 6. Superposition of the two curves, one by observation and the other by calculation, with constant C.

Enigma of the Birth and Evolution of Solar Systems

Fig. 7. The profile of the assumed evolving C.

Fig. 8. Theoretical lengthening of day curve with evolving C.

evolving C.

Fig. 9. Superposition of the observed curve and theoretical lengthening of day curve with

May Be Solved by Invoking Planetary-Satellite Dynamics 95

of plant and animal life but in the remote past, particularly in early Archean Eon, Earth seems to be spinning much slower than predicted by theory. This implies that rotational inertia was much higher than what has been assumed in this analysis. In fact there are evidence to show that early Earth was much less stratified as compared to modern Earth. It was more like Venus [Allegre, Calnde 1994, Taylor, Rose & Mclennan 1996].

Through out the analysis C, the Principal Moment of Inertia, has been assumed to be constant whereas infact it was evolving since the Giant Impact [Runcorn 1966].

In the first phase of planet formation, Earth was an undifferentiated mass of gas, rocks and metals much like Venus. At the point of Giant Impact, the impactor caused a massive heating which led to melting and magmatic formation of total Earth. The heavier metals, Iron and Nickel, settled down to the metallic core and lighter rocky materials formed the mantle. The mantle consisted of Basalt and Sodium rich Granite.

Due to Giant Impact, Earth gained extra angular momentum. This led to a very short spin period of 5 hours. It has been calculated that the oblateness at the inception must have been 1% [SOM\_Appendix D, Kamble 1966] whereas the modern oblateness is 0.3%. Taking these two factors into account C of Earth must have been much higher than the modern value of 37 2 8.02 10 *kg m* . In this paper the early C has been taken as 37 2 9.9 10 *kg m* .

After Achaean Eon the general cooling of Earth over a period of 2 billion years led to slower plate-tectonic movement. The 100 continental-oceanic plates coalesced into 12 plates initially and into 13 plates subsequently. The slower plate tectonic engine led to deep recycling of the continental crust and hence to complete magmatic distillation and differentiation of the internal structure into multi-layered onion like structure. Thus at the boundary of Archean Eon and Proterozoic Eon a definite transition occurred in the internal structure.

Before this boundary, the mantle and the outer crust was less differentiated. It was composed of a mixture of Basalt and Sodium-rich granite. After this boundary a slower plate-tectonic dynamo helped create the onion-like internal structure with sharply differentiated basaltic mantle and potassium-rich granitic crust. This highly heterogenous internal structure and less oblate geometry leads to the modern value of C equal to 37 2 8.02 10 *kg m* .

The form the evolving C is as follows:

$$\begin{array}{c} \text{f}[(\text{t-2E9})\_{-}] \text{=} \text{If} [(\text{t-2E9}) \ge 0, 1, 0] \\ \text{(9.9E37-(9.9E37-8.02E37))} \text{(1-Exp[-t/16E9])} \text{-f}[(\text{t-2E9})\_{-}] \text{(1.4E37)} \text{(1-Exp[-t/(0.5E9)])} \end{array} \tag{6}$$

Here f[(t-2E9)\_] is defined as a step function which is 0 before 2 billion years and is Unity at 2 billion years and at greater times.

The profile of evolution of C with time is obtained in Fig. (7) :

As can be seen in Fig. (9), there is a much closer fit except for a large deviation at 2.5Gyrs after the Giant impact. This is due to step change in Moment of Inertia, C, at 2Gyrs after the Giant Impact. It would have been more realistic to assume a gradual change in C at the boundary of Archean and Proterozoic Eon. This correction will be made in a sequel paper.

94 Space Science

of plant and animal life but in the remote past, particularly in early Archean Eon, Earth seems to be spinning much slower than predicted by theory. This implies that rotational inertia was much higher than what has been assumed in this analysis. In fact there are evidence to show that early Earth was much less stratified as compared to modern Earth. It

Through out the analysis C, the Principal Moment of Inertia, has been assumed to be

In the first phase of planet formation, Earth was an undifferentiated mass of gas, rocks and metals much like Venus. At the point of Giant Impact, the impactor caused a massive heating which led to melting and magmatic formation of total Earth. The heavier metals, Iron and Nickel, settled down to the metallic core and lighter rocky materials formed the

Due to Giant Impact, Earth gained extra angular momentum. This led to a very short spin period of 5 hours. It has been calculated that the oblateness at the inception must have been 1% [SOM\_Appendix D, Kamble 1966] whereas the modern oblateness is 0.3%. Taking these two factors into account C of Earth must have been much higher than the modern value of

After Achaean Eon the general cooling of Earth over a period of 2 billion years led to slower plate-tectonic movement. The 100 continental-oceanic plates coalesced into 12 plates initially and into 13 plates subsequently. The slower plate tectonic engine led to deep recycling of the continental crust and hence to complete magmatic distillation and differentiation of the internal structure into multi-layered onion like structure. Thus at the boundary of Archean Eon and Proterozoic Eon a definite transition occurred in the

Before this boundary, the mantle and the outer crust was less differentiated. It was composed of a mixture of Basalt and Sodium-rich granite. After this boundary a slower plate-tectonic dynamo helped create the onion-like internal structure with sharply differentiated basaltic mantle and potassium-rich granitic crust. This highly heterogenous internal structure and less

f[(t-2E9)\_]:=If[(t-2E9)>0,1,0] (6) {9.9E37-(9.9E37-8.02E37)}{1-Exp[-t/16E9]}-f[(t-2E9)\_](1.4E37){1-Exp[-t/(0.5E9)]}}

Here f[(t-2E9)\_] is defined as a step function which is 0 before 2 billion years and is Unity at

As can be seen in Fig. (9), there is a much closer fit except for a large deviation at 2.5Gyrs after the Giant impact. This is due to step change in Moment of Inertia, C, at 2Gyrs after the Giant Impact. It would have been more realistic to assume a gradual change in C at the boundary of Archean and Proterozoic Eon. This correction will be made in a sequel paper.

was more like Venus [Allegre, Calnde 1994, Taylor, Rose & Mclennan 1996].

constant whereas infact it was evolving since the Giant Impact [Runcorn 1966].

37 2 8.02 10 *kg m* . In this paper the early C has been taken as 37 2 9.9 10 *kg m* .

oblate geometry leads to the modern value of C equal to 37 2 8.02 10 *kg m* .

The profile of evolution of C with time is obtained in Fig. (7) :

mantle. The mantle consisted of Basalt and Sodium rich Granite.

internal structure.

The form the evolving C is as follows:

2 billion years and at greater times.

Fig. 7. The profile of the assumed evolving C.

Fig. 8. Theoretical lengthening of day curve with evolving C.

Fig. 9. Superposition of the observed curve and theoretical lengthening of day curve with evolving C.

Enigma of the Birth and Evolution of Solar Systems

accretion resulting into terrestrial rocky planets.

belt asteroids.

orbital motion.

May Be Solved by Invoking Planetary-Satellite Dynamics 97

b. Through computer simulation studies [Tsiganis, Gomes, Morbidelli & Lavison 2005] it has been shown that our planetary system, with initial quasi-circular, coplanar orbits, would have evolved to the current orbital configurations provided Jupiter and Saturn crossed the 1:2 mean motion resonance (MMR). When the ratio of the orbital periods of Jupiter and Saturn is 1:2 it is the strongest resonance point. At all integer ratios resonance is obtained but the maximum is obtained at 1:2. The resonance crossings excite the orbital eccentricities and mutual orbital inclinations to the present values. Jupiter ,Saturn and Uranus have the present eccentricities of 6%, 9% and 8% respectively. The present mutual inclination of the orbital planes of Saturn, Uranus and Neptune take the maximum values of approximately 2º with respect to that of Jupiter. The simulation was started with the initial positions of Jupiter and Saturn at 5.45AU and 8AU respectively. 1:2MMR crossing occurs at 8.65AU. The present orbital semi major axes of Jupiter, Saturn, Uranus and Neptune are 10AU, 15AU, 19.3AU and 30AU respectively. This simulation reproduces all aspects of the orbits of the giant planets: existence of natural satellites, distribution of Jupiter's Trojans and the presence of main

c. The presence of Jupiter's Trojans can be explained only by 1:2MMR crossing by Jupiter and Saturn[Morbidelli, Levison, Tsiganis and Gomes 2005]. These are asteroids which are in he same orbit as that of Jupiter but they are leading or lagging by 60º in their co-

d. The petrology record on our Moon suggests that a cataclysmic spike in the cratering rate occurred approximately 700 million years after the planets formed[Gomes, Levison, Tsiganis and Morbidelli 2005]. With the present evidence we assume the birth of our Solar Nebula at 4.56Gya. The formation of Gas Giants and Ice Giants was completed in first 5 millon years and Earth was completed in first 30 million years. This puts the date of completion of Giant Planets at 4.555Gya and the date of completion of the Terrestrial Planets particularly Earth at 30 million years after the solar nebula was born that is at 4.53Gya. At 4.53Gya, the Giant Impact occurred and from the impact generated circumterrestrial debris, Moon was born beyond Roche's Limit at 16,000Km orbital radius. By gravitational sling shot effect it was launched on an outward spiral path. Presently Moon is at the semi-major axis of 3,84,400Km with a recession velocity of 3.7cm/year. Towards the end of planet formation phase, the residual debris of the solar nebula was being rapidly sucked in or swept out of the system. This resulted in heavy meteoritic bombardment of all the big sub-stellar objects including our Moon. Through Apollo Mission studies it has been determined that there is a sharp increase in the bombardment rate and hence in the cratering rate around the period of 4.5 to 3.855Gya. From this it is concluded that

called asteroids and they lie between Mars and Jupiter orbit between a radii of 3AU to 10AU. Most of the asteroids are in near circular orbits. There are 700 odd asteroids known as Hilda which are in highly elliptical orbit and these eccentricities could have been imparted only by a migrating Jupiter set on an expanding spiral path. The migrating Jupiter first ejected some proto-Hilda asteroids out of the system and next elongated the orbits of the residual asteroids. The migrating Jupiter could have also set the planetary embryos on unruly chaotic paths which led to frequent collisions and

#### **10.5 A new perspective of birth and evolution of our solar system & extra solar systems**

The new perspective holds that:


#### **11. Observational proofs in support of gravitational sling shot model**

In recent days four observations strongly suggest that in remote past Jupiter and the gas giants may have experienced gravitational sling shot and they may have been launched on an outward spiral path just the way Moon has been launched or for that matter all planetary natural satellites have been launched.

a. 700 Hilda asteroids in elliptical orbit [Franklin et al 2004].The asteroid belt is populated with hundred thousands of rocky remnants leftover from planet formation. These are 96 Space Science

i. before the thermonuclear furnace turns on that is before full scale fusion reaction begins, in the inner region of the solar system by a process of agglomeration-accretion the icy-rocky core is formed. As soon as it reaches a critical mass of 10ME , it rapidly wraps itself with Hydrogen and Helium gas which is available in abundance in the gasdust debris . As it grows to 300ME , a gaping hole is formed in the disk. This paucity of gas terminates the runaway gas accretion. As we see, the necessity of a snow line does not arise as the inner region is sufficiently cold (100 kelvin) to keep the dust coated with amorphous ice which eliminates impact rebounce and permits agglomeration to take

ii. by the above process sequentially the four jovian planets are born i.e. one after another. As the first Gas Giant is formed, because of initial slingshot effect, caused by our Sun, Jupiter spirals out and makes space for the formation of the next gas giant namely Saturn. As Saturn spirals out, the Ice Giants namely Neptune and Uranus are

iii. Just as Jupiter spirals out to wide orbits, it is equally probable that the gas giant may be perturbed within the inner geo-orbit in other solar systems. Those tumbling short of inner geo-orbit get launched on inward collapsing spiral path doomed to their certain distruction. They become hot jupiters in scorchingly tight orbit. In course of planet

iv. the planet formation sequence follows the descending order of mass. The heaviest

v. the time factor of evolution is inversely proportional to the mass i.e. the massive giants evolve out of their initial orbit very rapidly whereas the lightest one remaining almost stay put. This implies that Jupiter spirals out of the maternity ward very rapidly

vi. in the first phase, Gas Giants and Ice Giants are formed when there is abundance of gas and dust. In course of birth and evolution of these massive planets the disk is dissipated of gas partly due to the accretion by the jovian cores and partly due to photoevaporation. The remnant disk is largely populated by planetismals. In the second phase the rocky planetismals gravitationally collapse to form the terrestrial planets in a sequence according to the descending order of mass. Earth was formed the first and Mercury the last. Pluto is a recently captured body. It has not been formed

In recent days four observations strongly suggest that in remote past Jupiter and the gas giants may have experienced gravitational sling shot and they may have been launched on an outward spiral path just the way Moon has been launched or for that matter all planetary

a. 700 Hilda asteroids in elliptical orbit [Franklin et al 2004].The asteroid belt is populated with hundred thousands of rocky remnants leftover from planet formation. These are

(i.e.Jupiter) being born the first and the lightest ( i.e. Neptune) the last;

whereas the terrestrial planets remain orbiting where they were born;

**11. Observational proofs in support of gravitational sling shot model** 

**10.5 A new perspective of birth and evolution of our solar system & extra solar** 

**systems** 

formed.

insitu.

natural satellites have been launched.

The new perspective holds that:

place unhindered to km size planetismal.

discovery, many examples of hot jupiters have turned up.

called asteroids and they lie between Mars and Jupiter orbit between a radii of 3AU to 10AU. Most of the asteroids are in near circular orbits. There are 700 odd asteroids known as Hilda which are in highly elliptical orbit and these eccentricities could have been imparted only by a migrating Jupiter set on an expanding spiral path. The migrating Jupiter first ejected some proto-Hilda asteroids out of the system and next elongated the orbits of the residual asteroids. The migrating Jupiter could have also set the planetary embryos on unruly chaotic paths which led to frequent collisions and accretion resulting into terrestrial rocky planets.


Enigma of the Birth and Evolution of Solar Systems

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there was a cataclysmic Late Heavy Bombardment of all big sub-stellar bodies, including our Moon, at about 700 My after the completion of formation of Jupiter and Saturn.

As the planet formation was completed , the gaseous circumsolar nebula was dissipated by gravity accretion and finally by photoevaporation. According to Tsiganis et al [2005], Jupiter and Saturn were born at 5.45AU and 8AU respectively where the orbital period ratio that PS/ PJ was less than 2. According to them the resulting interaction with massive disk of residual planetismals Jupiter and Saturn spiraled out on diverging path crossing 1:2MMR(PS/ PJ = 2) point at 8.65AU and today the ratio is little less than 2.5. At the 1:2MMR crossing due to gravitational resonance their orbits became eccentric. This abrupt transition temporarily destabilized the giant planets, leading to a short phase of close encounters among Saturn, Uranus and Neptune. As a result of these encounters, and of the interactions of the ice giants with the disk, Uranus and Neptune reached their current heliocentric distances of 19.3AU and 30AU. And Jupiter and Saturn evolved to the current orbital eccentricities of 6% and 9%. The same planetary evolution can explain LHB provided Jupiter and Saturn crossed 1:2MMR 700My after their formation. That is LHB occurred at 3.855Gya.
