1- Rotation of Sun and planets

Sami Esmaeil *

2-Abstract.

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 this planets are circled by another moons.

3-Keywords.

Rotation, Sun, Planets, Moment, Inertia, Leap, Second, Mass, Radius

4-Main text.

4-1-Introduction.:

The moons are "driving" the rotation of their parent planet in the same way, as sun's rotation is being controlled by the planets. ( Franz J. Heeke; Germany) .

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

There was a paper published by the Astronomical Journal in April, 1965 (vol. 70, page 193) by Paul D. Jose (Ref.2 ).

He concluded that "Certain forces exerted upon the Sun by the planets are the cause of the sunspot cycle" .

4-2-The rotation Equation:

Equation (1)

From equation 1 and Kepler's laws:

Equation (2)

Where:

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

K=Moment of inertia factor of the large body.

G=Gravitational constant.

M=Large body mass (kg)

Re= Equatorial radius for the large body (meter)

a=Semimajor 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)

P=Rotation period of the moons's mass center that orbit the large body (sec.)

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*Bsc, 1981 Mechanical engineering, Egypt, Email: saeles@yahoo.com

4-3-Solar system data sources.

All parameters according to NASA fact sheets (Ref.1)

http://nssdc.gsfc.nasa.gov/planetary/factsheet/index.html

4-4-The equation results:

4-4-1-Determination of Earth rotation period:

The Earth has one satellite (the Moon)

K=0.329865

current value =0.3308

M=5.9736E+24 (kg)

Re= 6.3781E+6 (meter)

a=3.844E+8 (meter)

m=7.35E+22 (kg)

re =1738100 (meter)

rp =1736000 (meter)

From the rotation Equation, the rotation period of Earth=23.93 hr

4-4-2-The Earth rotation is going to be faster:

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 -0.428 milliseconds/century (3.82 cm/year)

A negative leap second may be inserted every 233 000 year

The lower the rotation period of the large body, the higher the orbital one of it's moon.

The rotation period was equal the orbital one= 54.7 hr at a distance of 73 500 000 meter, it is now 384 400 000

4-4-3-Determination of Mars rotation period:

K=0.3581

current value =0.366

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=semimajor 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 the rotation Equation, the rotation period of Mars=24.62 hr

4-4-4-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

The lower the rotation period of the large body, the higher the orbital one of it's moon.

The rotation period was equal to the orbital one= 20.2 hr at a distance of 17 880 000 meter, it is now 12 000 000

4-4-5-Determination of Jupiter, Saturn, Uranus, Neptune, Pluto and sun rotation period:

By the same way, we can calculate their rotation periods from the rotation equation.(table-1)

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

4-5-Conclusion:

1-The slower the rotation period of the large body, the faster the orbital one of it's moon.

2-The slower the rotation period of the large body, the smaller the distance of it's moon.

3-The slower the rotation period of the large body, the higher it's density (D)

4-The slower the rotation period of the large body, the smaller of it's moon density (d)

5-The slower the rotation period of the large body, the smaller of it's moment of inertia factor value (k)

6-Jupiter, Uranus and Neptune may have the same moment of inertia factor approximately.

7-The moment of inertia of Pluto is very close to that of Saturn.

5-Acknowledgements

I express my deep-felt gratitude to Meta Ottevanger, for helping me to compose this page

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

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

6-Appendix

Title Rotation of sun and planets

AUTHOR Sami Esmaeil

Keywords Rotation, Sun, Planets, Moment, Inertia, Leap, Second, Mass, Radius

7-References

1-David R. Williams, 2006. NASA Planetary fact sheet .

http://nssdc.gsfc.nasa.gov/planetary/factsheet/index.html

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

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

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

9-Tables and figures

Large body properties

Small body properties

Rotation

Grav.

Equatorial

Moment

Semi.Maj.

Equatorial

Polar

Observed

Calculated

Const.

Mass

Radius

of inertia

Mass

Axis

Radius

meter

Earth

23.9345

23.9344

7E-11

6.0E+24

6.38E+06

0.329864

7.349E+22

3.84E+08

1.74E+06

Moon

7.35E+22

3.84E+08

1.74E+06

Mars

24.6230

7E-11

6.4E+23

3.40E+06

0.358103

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

9.9354

7E-11

1.9E+27

71492000

0.257

3.93E+23

1.10E+09

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

10.6586

7E-11

5.7E+26

60268000

0.189

1.40E+23

1.23E+09

2.63E+06

2.62E+06

Enceladus

1.08E+20

2.38E+08

2.56E+05

2.45E+05

Tethys

6.18E+20

2.95E+08

5.36E+05

5.26E+04

Dione

1.10E+21

3.77E+08

5.60E+05

Rhea

2.31E+21

5.27E+08

7.64E+05

Titan

1.35E+23

1.22E+09

2.58E+06

Iapetus

1.81E+21

3.56E+09

7.18E+05

Uranus

17.24

17.2380

7E-11

8.7E+25

25559000

0.2562

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

16.1088

7E-11

1.0E+26

24764000

0.2719

2.14E+22

3.55E+08

1.35E+06

Triton

2.14E+22

3.55E+08

1.35E+06

Pluto

153.292

153.2903

7E-11

1.25E+22

1195000

0.21528

1.62E+21

1.96E+07

5.93E+05

Charon

1.62E+21

1.96E+07

5.93E+05

Sun

609.12

609.1094

7E-11

2.0E+30

6.96E+08

0.059447

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

1.30E+22

5.91E+12

1.16E+06

1.15E+06

Table (1) The rotation equation results and solar system parameters.

Download the full calculations from the Excel sheet

http://www.geocities.com/saeles/Rotation.xls