Rotation of Earth , Sun and Planets

Spinning and Rotation of Earth, Sun and planets

By Samy Esmaeil Email : [email protected]

WWW.kandil.com

If a small body of mass (m) orbits a large body of mass (M) then, may be we can calculate the sidereal rotation period of the large body (M)

by the rotational equation. The rotational equation can calculate the rotation period for a Star of mass (M) which circled by another mass (m).

The nine Planets orbit our sun so, we can calculate the rotation period of our sun. Also, from the rotation Equation we can calculate the rotation

period of Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto because all these planets are circled by another moons.

The Earth rotation period is going to be faster as a result of increasing Earth- Moon distance.

The Earth rotation period increases when the moon orbit inclination plane is perpendicular to the Earth equator

The Earth rotation period decreases when the moon orbit inclination plane reaches to it's max. inclination

1-The rotation Equation:

Equation 1

For spherical planets and moons (the small body orbits the large body in a circular pass)

From equation 1 and a planet density=mass/volume

Equation 2

The small body orbits the large body in a circular pass

From equation 2

Equation 3

The small body orbits the large body in a circular pass

Where:

The angular velocity of The large body

The density of the large body

The density of the small body

T=Sidereal rotation period of the large body (sec)

K=Moment of inertia factor of the large body.

G=gravitational constant = 6.67300 � 10-11 m3 kg-1 s-2.

M=Large body mass (kg)

Re= Equatorial radius for the large body (meter)

a=Semi major axis of the mass center of all satellites that orbit large body (meter)

m=Total mass of all satellites that orbit the large body (kg)

re =Equatorial radius of all satellites that orbit large body (meter)

rp =Polar radius of all satellites that orbit large body (meter)

1-2-The equation results:

The rotation period (T) results from equation (2):

Table (1)

Large body

Rotation period

Observed

Calculated

Earth

23.9

24.0

Mars

24.6

24.4

Jupiter

9.93

9.95

Saturn

10.7

9.4

Uranus

17.2

17.1

Neptune

16.1

16.9

Figure (1). The observed and calculated rotation periods (T) from equation (2)

Sun

609.1

Table (2)

Large body

Rotation period

Observed

Calculated

Sun

609.1

Figure (2). The observed and calculated rotation periods (T) from equation (2)

1-1The moment of inertia factor (K) results from equation (3):

Table (3)

Large body

Mom. Of inertia factor

Observed

Calculated

Earth

0.331

Mars

0.364

0.362

Jupiter

0.260

Saturn

0.200

0.192

Uranus

0.260

0.259

Neptune

0.270

0.275

Pluto

0.310

0.283

Figure (3). The observed and calculated moment of inertia factor (K) from equation (3)

Table (4)

Large body

Mom. Of inertia factor

Observed

Calculated

Sun

0.060

Figure (4). The observed and calculated moment of inertia factor (K) from equation (3)

1-2-The errors of equation (2):

Table (5) The rotation period (T) results from equation (2)

Large body

Rotation period

Observed

Calculated

Error

Earth

23.9345

24.007

-0.3%

Mars

24.623

24.360

1.1%

Jupiter

9.925

9.95

-0.3%

Saturn

10.656

9.442

11.4%

Uranus

17.24

17.097

0.8%

Neptune

16.11

16.944

-5.2%

Pluto

153.3

116.200

24.2%

Sun

609.12

609.086

0.0%

Figure (5). The errors in (T) form equation (2).

1-3-The errors of equation (3):

Table (6) The moment of inertia factor (K) results from equation (3)

Large body

Moment of inertia factor (K)

Measured

Calculated

Error

Earth

0.3308

0.3312

-0.1%

Mars

0.364

0.362

0.5%

Jupiter

0.260

-0.1%

Saturn

0.200

0.192

3.9%

Uranus

0.260

0.259

0.5%

Neptune

0.270

0.275

-1.7%

Sun

0.060

0.0%

Pluto

0.310

0.283

8.7%

Haumea

0.37

Figure (6). The errors in (K) form equation (3).

2-Determination of Earth rotation period:

M=5.9736E+24 (kg)

Re= 6.3781E+6 (meter)

The Earth has one satellite (the Moon)

a=3.844E+8 (meter)

m=7.35E+22 (kg)

re =1738100 (meter)

rp =1736000 (meter)

From equation 2 (T) of Earth is 24.007

From equation 3 (K) of Earth is 0.3312

2-1-The Earth rotation

The Earth rotation period is going to be faster as a result of increasing Earth- Moon distance.

From equa.2, the Earth is fastening it's rotation period by -0.43 milliseconds/century (3.82 cm/year) fig.7 (If the moon orbits the Earth in a circular pass)

Lambeck estimated that the Length of the Day decreases by about -0.7 milli-seconds per century. (Ref.20)

http://rsta.royalsocietypublishing.org/content/287/1347/545.abstract

Figure (7). The rotation period of Earth and moon distance relationship.

2-3-The Earth rotation is going to be faster according to international Earth rotation service data. (fig 8)

http://tycho.usno.navy.mil/systime.html

http://tycho.usno.navy.mil/leapsec.html

Figure (8). The rotation period of Earth is decreased by 2.7 milliseconds.

3-Determination of Mars rotation period:

M=6.4185E+23 (kg)

Re= 3397000 (meter)

There are two satellites orbit Mars

m= Total mass of all satellites that orbit the large body Mars (kg)

= m1 (Phobos)+m2 (Deimos) =1.3E+16

a=semi major axis of the center of mass of all satellites that orbit Mars (meter)

=(m1 a1 +m2 a2 )/(m1 +m2 )=1.1977E+7 (meter)

re= Equatorial radius of the total satellites (meter)

re=[(r1)^3+(r2)^3]^0.3333=[(13400)^3+(7500)^3)]^0.3333=14128 (meter)

rp= Polar radius of the total satellites (meter)

rp=[(r1)^3+(r2)^3]^0.3333=[(9200)^3+(5200)^3)]^0.3333=9714 (meter)

From equation 2 (T) of Mars is 24.36

From equation 3 (K) of Mars is 0.362

3-2-Mars rotation is going to be slower:

Mars rotation period is going to be slower as a result of decreasing the distance of it's moons' mass center.

From the rotation equation, Mars is slowing it's rotation period by 0.67 second/century (1.8 meter/year)

A positive 0.67 leap second may be inserted every 100 year

3-3-Determination of Jupiter, Saturn, Uranus, Neptune, Pluto, Haumea and sun rotation periods:

By the same way, we can calculate the rotation period (T) from equation 2 as shown in table (7)

By the same way, we can calculate the moment of inertia factor (K) from equation 3 as shown in table (8)

All moons of small masses are negligible according to the equation results.

4-The relationship between Earth rotation rate and lunar declination

4-1-The period is from 10/Jul/2006 to 23/Nov./2006 (lunar declination is 29 degree)

The Earth diameter is in the moon inclination plane

From UT-UtC, the Earth rotation period increases when the moon orbit inclination plane is perpendicular to the Earth equator

From UT-UTC, the Earth rotation period decreases when the moon orbit inclination plane reaches to it's max. inclination

The Low points in Fig.(9) represents the highest speed of the earth rotation. (close to the earth equator)

The Low points in Fig.(10) represents the Min. Lunar declination.

The Peak points in Fig.(11) represents the Max.earth radius in the lunar orbital plane.(the equator radius)

The Earth rotation rate reaches to it's Max. when lunar declination is zero. (the highest Earth diameter)

There is a perfect correlation in the three curves.

From Fig.(9), (10) and (11), the Earth speed increased when the lunar declination decreased.

The Earth diameter is in the moon inclination plane

4-2-The period is from 9/Oct/1997 to 22/Feb./1998 (moon declination is 18 degree)

The Earth diameter is in the moon inclination plane

From UT-UtC, the Earth rotation period increases when the moon orbit inclination plane is perpendicular to the Earth equator

From UT-UTC, the Earth rotation period decreases when the moon orbit inclination plane reaches to it's max. inclination

The Low points in Fig.(12) represents the highest speed of the earth rotation.(close to the earth equator)

The Low points in Fig.(13) represents the Min. Lunar declination.

The Peak points in Fig.(14) represents the Max.earth radius in the lunar orbital plane.

The Earth rotation rate reaches to it's Max. when lunar declination is zero. (the highest Earth diameter)

There is a perfect correlation in the three curves.

From Fig.(12), (13) and (14), the Earth rotation speed increased when the lunar declination decreased.

The Earth diameter is in the moon inclination plane

5-Acknowledgements

Thanks to Eng. Franz J. Heeke; (Germany) for his useful internet web.

Thanks to Dr. Jerald Lee ; (USA) for his useful internet discussion and advice.

6-References

1-Charles F. Yoder,"Astrometric and Geodetic Properties of Earth and the Solar System"

www.agu.org/reference/gephys/4_yoder.pdf

2-Richard Schmude, Jr., "Uranus, Neptune, Pluto and How to Observe Them"

http://books.google.com/books?id=kqNr7rjw028C&pg=PA20&dq=uranus+moment+of+inertia

3-Mulholland,J.D:1980, "on the polar moment of inertia of a Compressible Body", Celest. Mech. 22/57.

4-Stevenson, D. J.,1982, "Interiors of the giant planets", Annual Reviews, 257/295

http://articles.adsabs.harvard.edu/full/seri/CeMec/0031//0000095,001.html

5-The Planets

http://geophysics.ou.edu/solid_earth/notes/planets.html

6-FRANK SOHL1, GERALD SCHUBERT2,3, TILMAN SPOHN1, 2006, "The Moment-of-Inertia of Mars and Implications for the Planet�s Interior Structure"

, Asia Oceania Geosciences Society, Singapore

7-Franz J. Heeke, 2007 ,Shaker Effects in Celestial Mechanics

http://www.surf2000.de/user/f-heeke/article1.html

8-Paul D. Jose, 1965."Sun's Motion and Sunspots". The Astronomical Journal, Vol. 70, Number 3, April 1965; P. 193-200

9-Hauke Hussmann, Frank Sohl , Tilman Spohn, 2006, "Subsurface oceans and deep interiors of medium-sized outer planet satellites

and large trans-Neptunian objects" Icarus 185 (2006) 258�273

10-C.M. Bertka and Y. Fei, "Implications for Geochemical Models of the Martian Interior", Lunar and Planetary Science XXVIII

www.lpi.usra.edu/meetings/lpsc97/pdf/1147.PDF

11-G. H. A. Cole, M. M. Woolfson, "Planetary science", Page 340

12-Paul R. Weissman, 1991."The angular momentum of the Oort cloud ". ICARUS, Volume 89, Issue 1, Pages 190-193

The angular momentum of the Oort cloud

13-Fundamentals of Geophysics: Ill. : Bib - Google Books Result

http://books.google.com/books?id=7vR2RJSIGVoC&pg=PA7&lpg=PA7&dq=moment+of+inertia+planets

14-The Earth as a Planet.

http://principles.ou.edu/earth_planet/index.html

15-Eliot F. Young and Richard P. Binzel, 1994."A New Determination of Radii and Limb Parameters for Pluto and Charon from Mutual Event Light curves "

ICARUS, Volume 108, Issue 2, Pages 219-224

A New Determination of Radii and Limb Parameters for Pluto and ...

16-D. Ragozzine and M.E.Brown, 2009, "Orbits and Masses of the Satellites of Dwarf Planet Haumea (2003 EL61)", Astronomical Journal, Volume 137, pp. 4766-4776

17-Landerer, F. W., J. H. Jungclaus, and J. Marotzke 2007 ,"Ocean bottom pressure changes lead to a decreasing length-of-day in a warming climate",

Geophys. Res. Lett., 34, L06307,

http://www.agu.org/pubs/crossref/2007/2006GL029106.shtml

18-Dennis D. McCarthy.2004,"Precision time and the rotation of the Earth",U. S. Naval Observatory, Washington, DC 20392, USA

19-S.Segan, I.Damjanov and B.Surlan,2003,"EARTH'S ROTATION IRREGULARITIES DERIVED FROM UTIBLI BY METHOD OF MULTI-COMPOSING OF ORDINATES"

Ser.A.journal

20- Lambeck, K. (1977). "Tidal dissipation in the oceans: Astronomical, geophysical and oceanographic consequences".

Phil. Trans. Royal Society of London A287: 545�594. doi:10.1098/rsta.1977.0159

http://rsta.royalsocietypublishing.org/content/287/1347/545.abstract

7-Another tables

7-1-Table (7) total planets and sun calculation.

Table (7) Total rotation equation results and solar system parameters to calculate (T), (Equation 2).

Large body properties

Small body properties

Rotation period

Grav.

Equatorial

M. of Inertia

Semi.Maj.

Equatorial

Polar

Const.

Mass

Radius

Factor

Mass

Axis

Radius

Observed

Calculated

Error

meter

Earth

23.9345

24.007

-0.30%

6.67E-11

5.97E+24

6.38E+06

0.3308

7.35E+22

3.84E+08

1.74E+06

Moon

7.35E+22

3.84E+08

1.74E+06

Mars

24.623

24.360

1.1%

6.67E-11

6.42E+23

3.40E+06

0.364

1.30E+16

1.20E+07

1.41E+04

9.72E+03

K= 0.3635 for Mars (Ref.6)

Phobos

1.06E+16

9.38E+06

1.34E+04

9.20E+03

Deimos

2.40E+15

2.35E+07

7.50E+03

5.20E+03

Jupiter

9.925

9.950

-0.3%

6.67E-11

1.90E+27

7.15E+07

0.260

3.93E+23

1.10E+09

3.48E+06

3.47E+06

K= 0.26 for Jupiter (Ref.12)

8.93E+22

4.22E+08

1.82E+06

Europa

4.82E+22

6.71E+08

1.56E+06

1.57E+06

Ganymede

1.48E+23

1.07E+09

2.63E+06

Callisto

1.08E+23

1.88E+09

2.41E+06

2.40E+06

Saturn

10.656

9.442

11.4%

6.67E-11

5.68E+26

6.03E+07

0.200

1.40E+23

1.23E+09

2.64E+06

2.63E+06

K= 0.2 for Saturn (Ref.12), (Ref.5)

Mimas

3.97E+19

1.86E+08

2.09E+05

1.91E+05

Enceladus

7.30E+19

2.38E+08

2.56E+05

2.45E+05

Tethys

6.22E+20

2.95E+08

5.36E+05

5.26E+04

Dione

1.10E+21

3.77E+08

5.64E+05

Rhea

2.31E+21

5.27E+08

7.68E+05

Titan

1.35E+23

1.22E+09

2.58E+06

Hyperion

5.50E+18

1.48E+09

1.85E+05

1.13E+05

Iapetus

1.59E+21

3.56E+09

7.26E+05

Uranus

17.24

17.097

0.8%

6.67E-11

8.68E+25

2.56E+07

0.260

9.12E+21

4.24E+08

1.10E+06

K= 0.26 for Uranus (Ref.13)

Miranda

6.60E+19

1.29E+08

2.40E+05

2.33E+05

Ariel

1.35E+21

1.91E+08

5.81E+05

5.78E+05

Umbriel

1.17E+21

2.66E+08

5.85E+05

Titania

3.52E+21

4.36E+08

7.89E+05

Oberon

3.01E+21

5.84E+08

7.61E+05

Neptune

16.11

16.944

-5.2%

6.67E-11

1.02E+26

2.48E+07

0.270

2.14E+22

3.55E+08

1.35E+06

K= 0.27 for Neptune (Ref.5)

Triton

2.14E+22

3.55E+08

1.35E+06

Pluto

153.292

116.212

24.2%

6.67E-11

1.31E+22

1.13E+06

0.310

1.50E+21

1.96E+07

6.41E+05

(Ref.9,12)

Charon

1.50E+21

1.96E+07

6.42E+05

(Ref.9,12)

Haumea

3.915

6.67E-11

4.01E+21

1.53E+06

????

3.58E+19

3.78E+07

6.42E+05

(Ref.16)

Hi'iaka

1.79E+19

4.99E+07

1.55E+05

(Ref.16)

Namaka

1.79E+19

2.57E+07

8.50E+04

Sun

609.12

609.085

0.0%

6.67E-11

1.99E+30

6.96E+08

0.060

2.67E+27

1.13E+12

8.51E+07

7.88E+07

K= 0.06 for Sun (Ref.12,5)

Mercury

3.30E+23

5.79E+10

2.44E+06

Venus

4.87E+24

1.08E+11

6.05E+06

Earth

5.97E+24

1.50E+11

6.38E+06

6.36E+06

Mars

6.42E+23

2.28E+11

3.40E+06

3.38E+06

Jupiter

1.90E+27

7.79E+11

7.15E+07

6.69E+07

Saturn

5.68E+26

1.43E+12

6.03E+07

5.44E+07

Uranus

8.68E+25

2.87E+12

2.56E+07

2.49E+07

Neptune

1.02E+26

4.50E+12

2.48E+07

2.43E+07

Pluto

1.31E+22

5.91E+12

1.13E+06

7-2-Table (8) total planets and sun calculation.

Table (8) Moment of inertia results to calculate (k), (Equation 3).

Large body properties

Small body properties

Rotation

Grav.

Equatorial

Moment of inertia factor K

Semi.Maj.

Equatorial

Polar

Observed

Const.

Mass

Radius

Calculated

Current

Error

Mass

Axis

Radius

Value

Sec.

meter

Earth

23.9345

86164

6.67E-11

6.0E+24

6.38E+06

0.3312

0.3308

0.1%

7.35E+22

3.84E+08

1.74E+06

Moon

7.35E+22

3.84E+08

1.74E+06

Mars

24.6230

88643

6.67E-11

6.4E+23

3.40E+06

0.3623

0.3635

-0.3%

1.30E+16

1.20E+07

1.41E+04

9.72E+03

Phobos

1.06E+16

9.38E+06

1.34E+04

9.20E+03

Deimos

2.40E+15

2.35E+07

7.50E+03

5.20E+03

Jupiter

9.9250

35730

6.67E-11

1.9E+27

71492000

0.2603

0.260

0.1%

3.93E+23

1.10E+09

3.48E+06

3.47E+06

8.93E+22

4.22E+08

1.82E+06

Europa

4.82E+22

6.71E+08

1.56E+06

1.57E+06

Ganymede

1.48E+23

1.07E+09

2.63E+06

Callisto

1.08E+23

1.88E+09

2.41E+06

2.40E+06

Saturn

10.656

38362

6.67E-11

5.7E+26

60268000

0.1921

0.20

-3.9%

1.40E+23

1.23E+09

2.64E+06

2.63E+06

Mimas

3.97E+19

1.86E+08

2.09E+05

1.91E+05

Enceladus

7.30E+19

2.38E+08

2.56E+05

2.45E+05

Tethys

6.22E+20

2.95E+08

5.36E+05

5.26E+04

Dione

1.10E+21

3.77E+08

5.64E+05

Rhea

2.31E+21

5.27E+08

7.68E+05

Titan

1.35E+23

1.22E+09

2.58E+06

Hyperion

5.50E+18

1.48E+09

1.85E+05

1.13E+05

Iapetus

1.59E+21

3.56E+09

7.26E+05

Uranus

17.24

62064

6.67E-11

8.7E+25

25559000

0.2593

0.260

-0.3%

9.12E+21

4.24E+08

1.10E+06

Miranda

6.60E+19

1.29E+08

2.40E+05

2.33E+05

Ariel

1.35E+21

1.91E+08

5.81E+05

5.78E+05

Umbriel

1.17E+21

2.66E+08

5.85E+05

Titania

3.52E+21

4.36E+08

7.89E+05

Oberon

3.01E+21

5.84E+08

7.61E+05

Neptune

16.11

57996

6.67E-11

1.0E+26

24764000

0.2746

0.2700

1.7%

2.14E+22

3.55E+08

1.35E+06

Triton

2.14E+22

3.55E+08

1.35E+06

Pluto

153.292

551851

6.67E-11

1.3E+22

1133000

0.2827

0.31

-8.8%

1.50E+21

1.96E+07

6.41E+05

Charon

1.50E+21

1.96E+07

6.42E+05

Haumea

3.915

14096

6.67E-11

4.0E+21

980000

0.3696

3.58E+19

3.78E+07

1.63E+05

Hi'iaka

1.79E+19

4.99E+07

1.55E+05

Namaka

1.79E+19

2.57E+07

8.50E+04

Sun

609.12

2192832

6.67E-11

2.0E+30

6.96E+08

0.060

0.06

0.02%

2.67E+27

1.13E+12

8.51E+07

7.88E+07

Mercury

3.30E+23

5.79E+10

2.44E+06

Venus

4.87E+24

1.08E+11

6.05E+06

Earth

5.97E+24

1.50E+11

6.38E+06

6.36E+06

Mars

6.42E+23

2.28E+11

3.40E+06

3.38E+06

Jupiter

1.90E+27

7.79E+11

7.15E+07

6.69E+07

Saturn

5.68E+26

1.43E+12

6.03E+07

5.44E+07

Uranus

8.68E+25

2.87E+12

2.56E+07

2.49E+07

Neptune

1.02E+26

4.50E+12

2.48E+07

2.43E+07

Pluto

1.31E+22

5.91E+12

1.13E+06

7-2-Table (9) the effect of one moon on the rotation period when the other moons omitted.

The results as shown in table (9)

1-The rotation period of Mars may be 8.4 when Deimos only orbits Mars.

2-The rotation period of Jupiter may be 4.63 when IO only orbits Jupiter

3-The rotation period of Saturn may be 0.3 when Tethys only orbits Saturn.

4-The rotation period of Sun may be 19.2 when Mercury only orbits sun.

5-The rotation period of Sun may be 8.3 when Pluto only orbits sun.

Why this small planet (Pluto) can rotates alone the sun with 8.3 hr rotational period?

The distance between Sun and Pluto is too large (the torque arm is too large).

Table (9) Rotation equation results when one small body rotates around the large body, (equation 2).

Large body properties

Small body properties

Rotation period

Grav.

Equatorial

M. of Inertia

Semi.Maj.

Equatorial

Polar

Const.

Mass

Radius

Factor

Mass

Axis

Radius

Observed

Calculated

Error

meter

Earth

23.9345

24.007

-0.30%

6.67E-11

5.97E+24

6.38E+06

0.3308

7.35E+22

3.84E+08

1.74E+06

Moon

7.35E+22

3.84E+08

1.74E+06

Mars

24.623

24.360

1.1%

6.67E-11

6.42E+23

3.40E+06

0.364

1.30E+16

1.20E+07

1.41E+04

9.72E+03

Phobos

29.093

6.67E-11

6.42E+23

3.40E+06

0.364

1.06E+16

9.38E+06

1.34E+04

9.20E+03

Deimos

8.416

6.67E-11

6.42E+23

3.40E+06

0.364

2.40E+15

2.35E+07

7.50E+03

5.20E+03

Jupiter

9.925

9.950

-0.26%

6.7E-11

1.9E+27

71492000

0.260

3.93E+23

1.10E+09

3.48E+06

3.47E+06

4.625

6.7E-11

1.9E+27

71492000

0.260

8.93E+22

4.22E+08

1.82E+06

Europa

4.676

6.7E-11

1.9E+27

71492000

0.260

4.82E+22

6.71E+08

1.56E+06

1.57E+06

Ganymede

11.601

6.7E-11

1.9E+27

71492000

0.260

1.48E+23

1.07E+09

2.63E+06

Callisto

9.325

6.7E-11

1.9E+27

71492000

0.260

1.08E+23

1.88E+09

2.41E+06

2.40E+06

Saturn

10.656

9.442

11.4%

6.7E-11

5.68E+26

60268000

0.200

1.40E+23

1.23E+09

2.64E+06

2.63E+06

Mimas

17.812

6.7E-11

5.68E+26

60268000

0.200

3.97E+19

1.86E+08

2.09E+05

1.91E+05

Enceladus

19.078

6.7E-11

5.68E+26

60268000

0.200

7.30E+19

2.38E+08

2.56E+05

2.45E+05

Tethys

0.303

6.7E-11

5.68E+26

60268000

0.200

6.22E+20

2.95E+08

5.36E+05

5.26E+04

Dione

12.365

6.7E-11

5.68E+26

60268000

0.200

1.10E+21

3.77E+08

5.64E+05

Rhea

17.651

6.7E-11

5.68E+26

60268000

0.200

2.31E+21

5.27E+08

7.68E+05

Titan

8.934

6.7E-11

5.68E+26

60268000

0.200

1.35E+23

1.22E+09

2.58E+06

Hyperion

66.404

6.7E-11

5.68E+26

60268000

0.200

5.50E+18

1.48E+09

1.85E+05

1.13E+05

Iapetus

9.944

6.7E-11

5.68E+26

60268000

0.200

1.59E+21

3.56E+09

7.26E+05

Uranus

17.24

17.097

0.829%

6.7E-11

8.68E+25

25559000

0.26

9.12E+21

4.24E+08

1.10E+06

Miranda

27.807

6.7E-11

8.68E+25

25559000

0.26

6.60E+19

1.29E+08

2.40E+05

2.33E+05

Ariel

17.864

6.7E-11

8.68E+25

25559000

0.26

1.35E+21

1.91E+08

5.81E+05

5.78E+05

Umbriel

21.216

6.7E-11

8.68E+25

25559000

0.26

1.17E+21

2.66E+08

5.85E+05

Titania

12.838

6.7E-11

8.68E+25

25559000

0.26

3.52E+21

4.36E+08

7.89E+05

Oberon

12.050

6.7E-11

8.68E+25

25559000

0.26

3.01E+21

5.84E+08

7.61E+05

Neptune

16.11

16.944

-5.2%

6.7E-11

1.02E+26

24764000

0.27

2.14E+22

3.55E+08

1.35E+06

Triton

2.14E+22

3.55E+08

1.35E+06

Pluto

153.292

116.212

24.2%

6.67E-11

1.31E+22

1.13E+06

0.310

1.50E+21

1.96E+07

6.41E+05

Charon

1.50E+21

1.96E+07

6.42E+05

Haumea

3.915

6.7E-11

4.01E+21

1531000

????

3.58E+19

3.78E+07

1.63E+05

Hi'iaka

1.79E+19

4.99E+07

1.55E+05

Namaka

1.79E+19

2.57E+07

8.50E+04

Sun

609.12

609.085

0.0%

6.7E-11

1.99E+30

6.96E+08

0.06

2.67E+27

1.13E+12

8.51E+07

7.88E+07

Mercury

19.193

6.7E-11

1.99E+30

6.96E+08

0.06

3.30E+23

5.79E+10

2.44E+06

Venus

23.698

6.7E-11

1.99E+30

6.96E+08

0.06

4.87E+24

1.08E+11

6.05E+06

Earth

18.709

6.7E-11

1.99E+30

6.96E+08

0.06

5.97E+24

1.50E+11

6.38E+06

6.36E+06

Mars

21.733

6.7E-11

1.99E+30

6.96E+08

0.06

6.42E+23

2.28E+11

3.40E+06

3.38E+06

Jupiter

474.557

6.7E-11

1.99E+30

6.96E+08

0.06

1.90E+27

7.79E+11

7.15E+07

6.69E+07

Saturn

1196.192

6.7E-11

1.99E+30

6.96E+08

0.06

5.68E+26

1.43E+12

6.03E+07

5.44E+07

Uranus

159.701

6.7E-11

1.99E+30

6.96E+08

0.06

8.68E+25

2.87E+12

2.56E+07

2.49E+07

Neptune

76.215

6.7E-11

1.99E+30

6.96E+08

0.06

1.02E+26

4.50E+12

2.48E+07

2.43E+07

Pluto

8.254

6.7E-11

1.99E+30

6.96E+08

0.06

1.31E+22

5.91E+12

1.13E+06

7-3-Table (10) the effect of the giant planets on the rotation of Sun.

The results as shown in table (10)

From the rotation equation, it is very clear that the giant planets control the sun rotation,

The rotation period of the sun may be 612 hr when the giant planets only orbit the sun.

Table (10) Rotation equation results when the Giant planets only orbit Sun, (equation 2).

Large body properties

Small body properties

Rotation period

Grav.

Equatorial

M. of Inertia

Semi.Maj.

Equatorial

Polar

Const.

Mass

Radius

Factor

Mass

Axis

Radius

Observed

Calculated

Error

meter

Sun

609.12

612.033

-0.5%

6.7E-11

1.99E+30

6.96E+08

0.06

2.66E+27

1.13E+12

8.51E+07

7.88E+07

Mercury

Venus

Earth

Mars

Jupiter

474.557

6.7E-11

1.99E+30

6.96E+08

0.06

1.90E+27

7.79E+11

7.15E+07

6.69E+07

Saturn

1196.192

6.7E-11

1.99E+30

6.96E+08

0.06

5.68E+26

1.43E+12

6.03E+07

5.44E+07

Uranus

159.701

6.7E-11

1.99E+30

6.96E+08

0.06

8.68E+25

2.87E+12

2.56E+07

2.49E+07

Neptune

76.215

6.7E-11

1.99E+30

6.96E+08

0.06

1.02E+26

4.50E+12

2.48E+07

2.43E+07

Pluto

7-4-Table (11) the effect of the terrestrial planets on the rotation of sun.

The results as shown in table (11)

The rotation period of the sun may be 25 hr when the terrestrial planets only orbit the sun.

Table (11) Rotation equation results when the terrestrial planets only orbit sun, (equation 2).

Large body properties

Small body properties

Rotation period

Grav.

Equatorial

M. of Inertia

Semi.Maj.

Equatorial

Polar

Const.

Mass

Radius

Factor

Mass

Axis

Radius

Observed

Calculated

Error

meter

Sun

609.12

25.316

95.8%

6.7E-11

1.99E+30

6.96E+08

0.06

1.15E+25

1.38E+11

8.12E+06

8.10E+06

Mercury

19.193

6.7E-11

1.99E+30

6.96E+08

0.06

3.30E+23

5.79E+10

2.44E+06

Venus

23.698

6.7E-11

1.99E+30

6.96E+08

0.06

4.87E+24

1.08E+11

6.05E+06

Earth

18.709

6.7E-11

1.99E+30

6.96E+08

0.06

5.97E+24

1.50E+11

6.38E+06

6.36E+06

Mars

21.733

6.7E-11

1.99E+30

6.96E+08

0.06

6.42E+23

2.28E+11

3.40E+06

3.38E+06

Jupiter

Saturn

Uranus

Neptune

Pluto

7-5-Table (12) the effect of Pluto on the rotation of sun.

The results as shown in table (12), and (9)

Table (12) Rotation equation results when all planets except Pluto orbit sun, (equation 2).

Large body properties

Small body properties

Rotation period

Grav.

Equatorial

M. of Inertia

Semi.Maj.

Equatorial

Polar

Const.

Mass

Radius

Factor

Mass

Axis

Radius

Observed

Calculated

Error

meter

Sun

609.12

609.094

0.0%

6.7E-11

1.99E+30

6.96E+08

0.06

2.67E+27

1.13E+12

8.51E+07

7.88E+07

Mercury

19.193

6.7E-11

1.99E+30

6.96E+08

0.06

3.30E+23

5.79E+10

2.44E+06

Venus

23.698

6.7E-11

1.99E+30

6.96E+08

0.06

4.87E+24

1.08E+11

6.05E+06

Earth

18.709

6.7E-11

1.99E+30

6.96E+08

0.06

5.97E+24

1.50E+11

6.38E+06

6.36E+06

Mars

21.733

6.7E-11

1.99E+30

6.96E+08

0.06

6.42E+23

2.28E+11

3.40E+06

3.38E+06

Jupiter

474.557

6.7E-11

1.99E+30

6.96E+08

0.06

1.90E+27

7.79E+11

7.15E+07

6.69E+07

Saturn

1196.192

6.7E-11

1.99E+30

6.96E+08

0.06

5.68E+26

1.43E+12

6.03E+07

5.44E+07

Uranus

159.701

6.7E-11

1.99E+30

6.96E+08

0.06

8.68E+25

2.87E+12

2.56E+07

2.49E+07

Neptune

76.215

6.7E-11

1.99E+30

6.96E+08

0.06

1.02E+26

4.50E+12

2.48E+07

2.43E+07

Pluto

When all planets orbit the Sun, the rotation period of the Sun =609.085 hr.

When all planets except Pluto orbit the Sun, the rotation period of the Sun =609.094 hr.

So, Pluto can fasting the rotation period of Sun by only 0.009 hr. if it rotates with all planets around Sun.

This is because the Giant planets can control the rotation of Sun very well.

8-Conclusion:

If a small body of mass (m) orbits a large body of mass (M) then, may be we can calculate the sidereal rotation period of the large body (M).

The moons are "driving" the rotation of their parent planet in the same way, as sun's rotation is being controlled by the planets.

The Earth rotation period is going to be faster as a result of increasing Earth- Moon distance.

From the rotation equation, the Earth is fastening it's rotation period by -0.43 milliseconds/century.(if the moon orbits the Earth in a circular pass)

If a small body of mass (m) orbits a large body of mass (M) then, the rotation period of (M) increased by decreasing the moment of inertia factor (K) for (M).

The Moment of inertia factor of Haumea may be expected to be 0.37 from the rotation equation.

A perfect correlation between lunar declination and the changes of leap seconds.(UT1-UTC)-(UT1-UTC)

From UT-UtC, the Earth rotation period increases when the moon orbit inclination plane is perpendicular to the Earth equator

From UT-UTC, the Earth rotation period decreases when the moon orbit inclination plane reaches to it's max. inclination