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
hr hr
T
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 609.1 609.1
Table (2)
Large body Rotation period
hr hr
T
Observed  Calculated
Sun 609.1 609.1
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
K
Observed Calculated
Earth 0.331 0.331
Mars 0.364 0.362
Jupiter 0.260 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
K
Observed  Calculated
Sun 0.060 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
T
Observed  Calculated Error
hr hr %
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)
K
Measured Calculated Error
hr hr %
Earth 0.3308 0.3312 -0.1%
Mars 0.364 0.362 0.5%
Jupiter 0.260 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 0.060 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"
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

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
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
Observed  Calculated Error G M Re K m a re rp
hr hr %   Kg meter   Kg meter meter 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 1.74E+06
Moon 7.35E+22 3.84E+08 1.74E+06 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)
Io 8.93E+22 4.22E+08 1.82E+06 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 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
Tethys  6.22E+20 2.95E+08 5.36E+05 5.26E+04
Dione  1.10E+21 3.77E+08 5.64E+05 5.64E+05
Rhea 2.31E+21 5.27E+08 7.68E+05 7.68E+05
Titan 1.35E+23 1.22E+09 2.58E+06 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 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 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 5.85E+05
Titania 3.52E+21 4.36E+08 7.89E+05 7.89E+05
Oberon 3.01E+21 5.84E+08 7.61E+05 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 1.35E+06 K= 0.27 for Neptune (Ref.5)
Triton 2.14E+22 3.55E+08 1.35E+06 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 6.41E+05  (Ref.9,12)
Charon 1.50E+21 1.96E+07 6.42E+05 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 6.42E+05  (Ref.16)
Hi'iaka 1.79E+19 4.99E+07 1.55E+05 1.55E+05  (Ref.16)
Namaka 1.79E+19 2.57E+07 8.50E+04 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 2.44E+06
Venus 4.87E+24 1.08E+11 6.05E+06 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 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
T G M Re Value Value .% m a re rp
hr Sec.   Kg meter       Kg meter meter 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 1.74E+06
Moon     7.35E+22 3.84E+08 1.74E+06 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
Io     8.93E+22 4.22E+08 1.82E+06 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 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
Tethys    6.22E+20 2.95E+08 5.36E+05 5.26E+04
Dione    1.10E+21 3.77E+08 5.64E+05 5.64E+05
Rhea   2.31E+21 5.27E+08 7.68E+05 7.68E+05
Titan   1.35E+23 1.22E+09 2.58E+06 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 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 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 5.85E+05
Titania     3.52E+21 4.36E+08 7.89E+05 7.89E+05
Oberon     3.01E+21 5.84E+08 7.61E+05 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 1.35E+06
Triton     2.14E+22 3.55E+08 1.35E+06 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 6.41E+05
Charon     1.50E+21 1.96E+07 6.42E+05 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 1.63E+05
Hi'iaka     1.79E+19 4.99E+07 1.55E+05 1.55E+05
Namaka     1.79E+19 2.57E+07 8.50E+04 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 2.44E+06
Venus     4.87E+24 1.08E+11 6.05E+06 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 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
Observed  Calculated Error G M Re K m a re rp
hr hr %   Kg meter   Kg meter meter 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 1.74E+06
Moon 7.35E+22 3.84E+08 1.74E+06 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
Io 4.625   6.7E-11 1.9E+27 71492000 0.260 8.93E+22 4.22E+08 1.82E+06 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 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 5.64E+05
Rhea 17.651   6.7E-11 5.68E+26 60268000 0.200 2.31E+21 5.27E+08 7.68E+05 7.68E+05
Titan 8.934   6.7E-11 5.68E+26 60268000 0.200 1.35E+23 1.22E+09 2.58E+06 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 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 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 5.85E+05
Titania 12.838   6.7E-11 8.68E+25 25559000 0.26 3.52E+21 4.36E+08 7.89E+05 7.89E+05
Oberon 12.050   6.7E-11 8.68E+25 25559000 0.26 3.01E+21 5.84E+08 7.61E+05 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 1.35E+06
Triton               2.14E+22 3.55E+08 1.35E+06 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 6.41E+05
Charon               1.50E+21 1.96E+07 6.42E+05 6.42E+05
Haumea 3.915 - - 6.7E-11 4.01E+21 1531000 ???? 3.58E+19 3.78E+07 1.63E+05 1.63E+05
Hi'iaka               1.79E+19 4.99E+07 1.55E+05 1.55E+05
Namaka               1.79E+19 2.57E+07 8.50E+04 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 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 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 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
Observed  Calculated Error G M Re K m a re rp
hr hr %   Kg meter   Kg meter meter 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
Observed  Calculated Error G M Re K m a re rp
hr hr %   Kg meter   Kg meter meter 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 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 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
Observed  Calculated Error G M Re K m a re rp
hr hr %   Kg meter   Kg meter meter 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 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 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