DEMONSTRATIONS.

EXAMPLE 1. During one Earth rotation period, The Moon revolves 0.0365 revolutions.

DEMONSTRATION. Earth rotation period = 0.9972697 Earth days, and Moon orbital period = 27.32166 Earth days.

During one Earth rotation period, The Moon revolves                      

  (0.9972697 ÷ 27.32166) = 0.0365 revolutions.

To verify the above time periods, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 2. Earth’s (polar) diameter is equal to 3.6573 times The Moon’s (equatorial) diameter. 

DEMONSTRATION. Wikipedia gives Earth’s polar radius as 6356.8 kilometers. Here is a scan from the relevant portion of Wikipedia (Internet search term “Wikipedia Earth”) to verify this figure.

Wikipedia gives The Moon’s equatorial radius as 1738.1 kilometers. Here is a scan from Wikipedia (Internet search term “Wikipedia The Moon”) to verify this figure.

(6356.8 x 2) ÷ (1738.1 x2) = 3.6573 (Note:- Diameter = Radius x 2.)

EXAMPLE 3 The SUM of the rotation periods of The Sun, The Moon, and The Planets = 365.304 Earth sidereal days.

DEMONSTRATION. The rotation periods of Sun, Moon, and Planets (expressed in Earth days are as follows:- Sun 24.66225 and Moon 27.32166 (which is equal to its orbital period, due to “tidal locking”) and Mercury 58.6462 and Venus 243.0187 and Earth 0.9972697 and Mars 1.02596 and Jupiter 0.41354 and Saturn 0.44401 and Uranus 0.71833 and Neptune 0.671252 and Pluto 6.387230 The SUM of the periods = 364.3064 Earth SOLAR days. To convert to Earth sidereal days, divide by Earth’s (sidereal) rotation period, which is 0.9972697 Earth (solar) days.

To verify the above time periods, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 4. During the precise time period of HALF an Earth Year, The Four Inner Solar System Planets rotate altogether a total of 365.00 rotations.

DEMONSTRATION. The Earth Year = Earth’s orbital period = 365.25636 Earth days.

The Four Inner Solar System Planets are:- Mercury, Venus, Earth, and Mars. Their rotation periods (expressed in Earth days) are – respectively – 58.6462 and 243.0187 and 0.997269663 and 1.025957 The calculation is carried out in the following manner:-

[(365.25636 ÷ 58.6462) + (365.25636 ÷ 243.0187) + (365.25636 ÷ 0.997269663) + (365.25636 ÷ 1.025957)] ÷ 2 = 365.00 rotations.

To verify the above time periods, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 5. During the precise time period of HALF an Earth Year, The Four Inner Solar System Planets revolve altogether a total of 3.6547 revolutions.

DEMONSTRATION. The Earth Year = 365.25636 Earth days.

The Four Inner Solar System Planets are:- Mercury, Venus, Earth, and Mars. Their orbital periods (expressed in Earth days) are – respectively – 87.9692 and 224.70067 and 365.25636 and 686.9782 The calculation is carried out in the following manner:-

[(365.25636 ÷ 87.9692) + (365.25636 ÷ 224.70067) + (365.25636 ÷ 365.25636) + (365.25636 ÷ 686.9782)] ÷ 2 = 3.6547 revolutions.

To verify the above time periods, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 6. During the precise time period of A QUARTER of an Earth Year, The Four Inner Solar System NON-Planetary Bodies rotate altogether a total of 365.7078 rotations.

DEMONSTRATION. The Earth Year = 365.25636 Earth days.

The Four Inner Solar System NON-Planetary Bodies are:- The Sun, The Moon, and The Two Mars Satellites – Phobos and Deimos. The rotation periods (expressed in Earth days) of these four bodies are – respectively – 24.66225 and 27.321661 and 0.31891023 and 1.2624407 (Bear in mind that the rotation period of the above named satellites are equal to their orbital periods, due to “tidal locking”)

The calculation is carried out in the following manner:-

[(365.25636 ÷ 24.66225) + (365.25636 ÷ 27.321661) + (365.25636 ÷ 0.31891023) + (365.25636 ÷ 1.2624407)] ÷ 4 = 365.7078 rotations.

To verify The Four Inner Solar System Non-Planetary Bodies, and to verify The Sun’s rotation period, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

For the orbital periods of The Moon, Phobos, and Deimos, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 7. During ONE QUARTER of An Earth Day, The LARGE Satellites of The Solar System, together with their Parent Planets rotate altogether a total of 3.6597 rotations.

DEMONSTRATION. The planets that have LARGE Satellites are:- Earth, Jupiter, Saturn, Uranus, Neptune, and Pluto. (Mercury and Venus have no satellites, and Mars has only small satellites.) The satellite rotation periods are equal to their orbital periods, due to “tidal locking”.

Earth has one LARGE Satellite – The Moon. Earth rotation period = 0.997269663 Earth days, and The Moon rotation period = 27.321661 Earth days.

Jupiter’s rotation period = 0.41353831 Earth days. The Four LARGE Satellites of Jupiter (The Four “Galilean” Satellites) (arranged in ascending orbit order) are:- Io, Europa, Ganymede, and Callisto. Their rotation periods (expressed in Earth days) are – respectively – 1.769137786 and 3.551181041 and 7.15455296 and 16.6890184 

Saturn rotation period = 0.44401 Earth days. Saturn’s Eight LARGE Satellites (arranged in ascending orbit order) are:- Mimas, Enceladus, Tethys, Dione, Rhea, Titan, Hyperion, and Iapetus. Their rotation periods (expressed in Earth days) are Mimas 0.942421813 and Enceladus 1.370217855 and Tethys 1.887802160 and Dione 2.736914742 and Rhea 4.517500436 and Titan 15.94542068 and Iapetus 79.3301825 Hyperion has a rotation period that is “chaotic”, ie:- it has no fixed rotation period, so we must exclude Hyperion from our calculations.

Uranus rotation period = 0.71833 Earth days. Uranus has Five LARGE Satellites. These are (arranged in ascending orbit order) Miranda, Ariel, Umbriel, Titania, and Oberon. Their rotation periods (expressed in Earth days) are – respectively – 1.4134840 and 2.52037932 and 4.1441765 and 8.7058703 and 13.4632423 

Neptune rotation period = 0.671252 Earth days. Neptune has just one LARGE Satellite, ie:- Triton, which has a rotation period of 5.8768441 Earth days.

Pluto rotation period = 6.387230 Earth days. Pluto has just one LARGE Satellite – Charon, which has a rotation period of 6.387230 Earth days.

We now have 25 rotation periods. For each rotation period, the same procedure (or calculation) must be carried out in the following manner:-

During one Earth day Earth rotates (1 ÷ 0.9972697) = 1.0027378 rotations.

During one Earth day The Moon rotates (1 ÷ 27.32166) = 0.036601 rotations.

During one Earth day Jupiter rotates (1 ÷ 0.41354) = 2.4181556 rotations.

During one Earth day Io rotates (1 ÷ 1.769137786) = 0.5652471 rotations.

During one Earth day Europa rotates (1 ÷ 3.551181041) = 0.281596 rotations.                                                                                                                           etc etc - - - -

You have to add all the results together, getting the grand total of 14.638747 rotations. ONE QUARTER of this number is 3.6597 rotations.

To verify which are the LARGE Satellites of The Solar System, and to verify Hyperion’s “chaotic” rotation, and to verify planet rotation periods (except for Pluto), and to confirm that Mercury and Venus have no satellites, and that Mars has only small satellites, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

For Pluto satellite, Charon – go to Wikipedia (search term – Moons of Pluto).

For Pluto rotation period – go to Wikipedia (search term – Pluto).

For LARGE Satellite orbital periods, except for (Pluto satellite) Charon,click on the following link:- 

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 8. A PRIMARY Satellite is the LARGEST Satellite for a particular planet. During the precise time period of One Earth Day, The PRIMARY Satellites of The Solar System revolve altogether a total of 3.6598 revolutions.

DEMONSTRATION. Here is a list of The PRIMARY Satellites of The Solar System:- Mercury and Venus have no satellites. Earth’s PRIMARY Satellite is The Moon. Mars’ PRIMARY Satellite is Phobos. Jupiter’s PRIMARY Satellite is Ganymede. Saturn’s PRIMARY Satellite is Titan. Uranus’ PRIMARY Satellite is Titania. Neptune’s PRIMARY Satellite is Triton. In this present example, we will consider Pluto to be a Trans-Neptunian Object, rather than a planet, so it is excluded from the calculation.

The orbital periods (expressed in Earth days) of The Moon, Phobos, Ganymede, Titan, Titania, and Triton are – respectively – 27.321661 and 0.31891023 and 7.15455296 and 15.94542068 and 8.7058703 and 5.8768441

We now have 6 orbital periods. For each orbital period, the same procedure (or calculation) must be carried out in the following manner:- During One Earth day, The Moon revolves (1 ÷ 27.321661) = 0.036601 revolutions.

During One Earth day, Phobos revolves (1 ÷ 0.31891023) = 3.135679 revolutions. etc etc - - - -

You have to add all the results together, getting the grand total of 3.6598 revolutions.

To verify the above time periods, go to the following web pages:-

To verify satellite sizes (and confirm the Primary Satellites), and to confirm that Mercury and Venus have no satellites, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

For Satellite orbital periods,click on the following link:- 

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 9. FOUR times the DIRECTED Number SUM (ie:- subtract for retrogrades) of The Orbital Periods of The LONG Period PRIMARY Satellites of The Solar System = 365.8097 Earth Days.

DEMONSTRATION. A Primary Satellite is the LARGEST Satellite for a particular planet. Here is a list of The PRIMARY Satellites of The Solar System:- Mercury and Venus have no satellites. Earth’s PRIMARY Satellite is The Moon. Mars’ PRIMARY Satellite is Phobos. Jupiter’s PRIMARY Satellite is Ganymede. Saturn’s PRIMARY Satellite is Titan. Uranus’ PRIMARY Satellite is Titania. Neptune’s PRIMARY Satellite is Triton. Binary Planet Pluto/Charon’s PRIMARY Satellite is Hydra.

The orbital periods (expressed in Earth days) of The Moon, Phobos, Ganymede, Titan, Titania, Triton, and Hydra are – respectively – PLUS 27.321661 and IGNORE 0.31891023 (because Phobos has a very SHORT period – less than The Earth Day – and is therefore not one of the LONG Period Primary Satellites) and PLUS 7.15455296 and PLUS 15.94542068 and PLUS 8.7058703 and MINUS 5.8768441 (because Triton has retrograde revolution) and PLUS 38.20177    

The DIRECTED Number SUM of these six revolution periods is 91.45243 Earth days. 4 x this sum = 365.8097 Earth days. 

To verify satellite sizes (and confirm the Primary Satellites), and to confirm that Mercury and Venus have no satellites, and to verify Triton’s retrograde revolution (see the letter R directly preceding Triton’s orbital period), click on the following link:- 

www.solarsystemtimeperiods.com/planet-orbital

To verify satellite orbital periods, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

To verify sizes and orbital periods of Pluto satellites, click on the following link:-

 www.solarsystemtimeperiods.com/pluto-sats

 EXAMPLE 10.  The SUM of the orbital periods of The Moon and the Three Inferior Bodies = 365.65 Earth sidereal days.

DEMONSTRATION. The Three Inferior Bodies are:- The Sun, Mercury, and Venus. (The term “inferior” means more central than Earth in The Solar System.) The orbital periods of The Moon, Mercury, and Venus (expressed in Earth days), are – respectively – 27.32166 and 87.9694 and 224.695 The Sun has an orbital period only in the sense that a point on The Sun’s equator “revolves” around The Sun’s centre. In that sense, The Sun’s “orbital” period is the same as its rotation period, which is 24.66225 Earth days.

The SUM of these four periods = 364.648 Earth solar days. To convert to Earth SIDEREAL days, divide by Earth’s (sidereal) rotation period, which is 0.9972697 Earth (solar) days. The result you will obtain is 365.65 Earth sidereal days.

To verify the above time periods, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 11. TWICE The SUM of The Adjacent Orbit Pass Periods of The Nineteen LARGE Satellites of The Outer Solar System = 365.28 Earth sidereal days.

DEMONSTRATION. A Pass Period is the length of time between successive occasions when two satellites (or two planets) pass one another in their separate orbital motions.

Jupiter, Saturn, and Uranus are the only three planets that have multiple LARGE Close Satellites.

The Four LARGE Satellites of Jupiter (The Four “Galilean” Satellites) (arranged in ascending orbit order) are:- Io, Europa, Ganymede, and Callisto. Their orbital periods (expressed in Earth days) are – respectively – 1.769137786 and 3.551181041 and 7.15455296 and 16.6890184

Saturn’s Eight LARGE Satellites (arranged in ascending orbit order) are:- Mimas, Enceladus, Tethys, Dione, Rhea, Titan, Hyperion, and Iapetus. Their orbital periods (expressed in Earth days) are Mimas 0.942421813 and Enceladus 1.370217855 and Tethys 1.887802160 and Dione 2.736914742 and Rhea 4.517500436 and Titan 15.94542068 and Hyperion 21.2766088 and Iapetus 79.3301825

Uranus has Five LARGE Satellites. These are (arranged in ascending orbit order) Miranda, Ariel, Umbriel, Titania, and Oberon. In fact, Miranda is much smaller than any of the other four satellites, having a mass less than one twentieth of any of the other four individual satellites. In that case, for the purposes of this present example, we will consider only Ariel, Umbriel, Titania, and Oberon to be Uranus’ LARGE Satellites. Their orbital periods (expressed in Earth days) are – respectively – 2.52037932 and 4.1441765 and 8.7058703 and 13.4632423

I will now demonstrate the formula for adjacent satellite Pass Periods, using Jupiter satellites Io and Europa (whose orbital periods are 1.769137786 and 3.551181041 Earth days, respectively).

The Io/Europa Pass Period = 1 ÷ [(1 ÷ 1.769137786) minus (1 ÷ 3.551181041)] = 3.5254636 Earth days.  

(For The Jupiter Satellites) it is necessary to perform this same calculation with Europa and Ganymede, and then with Ganymede and Callisto.

Then (for The Saturn Satellites) it is necessary to perform this same calculation with Mimas and Enceladus, and then with Enceladus and Tethys, and then with Tethys and Dione, and then with Dione and Rhea, and then with Rhea and Titan, and then with Titan and Hyperion, and then with Hyperion and Iapetus. 

Then (for The Uranus Satellites) it is necessary to perform this same calculation with Ariel and Umbriel, and then with Umbriel and Titania, and then with Titania and Oberon.

When you add together all the Pass Periods that you have calculated, the result will be as follows:- Jupiter Satellites 23.0996 Earth days, and Saturn Satellites 120.06 Earth days and Uranus Satellites 38.979 Earth days. The total SUM is 182.139 Earth solar days. To convert to Earth sidereal days, divide by Earth’s (sidereal) rotation period, which is 0.997269663 Earth (solar) days. Now multiply the total result by 2 to obtain the result of 365.28 Earth sidereal days.  

To verify satellite sizes, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify Satellite orbital periods, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 12. HALF The SUM of The Adjacent Orbit PASS PERIODS of Earth and The Superior Planets = 365.3208 Earth Years. DEMONSTRATION. A Pass Period is the length of time between successive occasions when two satellites (or two planets) pass one another in their separate orbital motions.

Here is a list of Earth and The Superior Planets (ie:- planets that are further from The Sun than Earth is), listed in ascending orbit order, together with their orbital periods (expressed in Earth days):- Earth 365.25636 and Mars 686.9782 and Jupiter 4332.5235 and Saturn 10727.160 and Uranus 30717.682 and Neptune 60215.912 and Pluto 90803.66

The Earth/Mars PASS PERIOD is calculated in the following manner:-

1 ÷ [(1 ÷ 365.25636) minus (1 ÷ 686.9782)] = 779.9343 Earth days.

 You need to carry out the same calculation, using the same formula, for the following adjacent orbit pairs:- Earth and Mars (already demonstrated), then Mars and Jupiter, then Jupiter and Saturn, then Saturn and Uranus, then Uranus and Neptune, then Neptune and Pluto. Then you add together all the PASS PERIODS that you have calculated. Then divide this sum by 365.25636 (because there are 365.25636 Earth days in One Earth Year – Earth’s orbital period being 365.256 Earth days) to convert your result from Earth days into Earth Years. Finally divide the result by TWO, and the final result you will get is 365.23 Earth Years.  

To verify the above time periods, and to confirm the order of the planets moving outwards from The Sun, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 13. During the precise time period of TWO Moon orbital periods, The Naked-Eye-Visible Planets rotate and revolve altogether a total of 365.5254 times.

DEMONSTRATION. The Moon orbital period = 27.32166 Earth days.

The Naked-Eye-Visible Planets are:- Mercury, Venus, Earth, Mars, Jupiter, and Saturn. (Uranus, Neptune, and Pluto are “telescopic” planets.). Their orbital periods (expressed in Earth days) are – respectively – 87.9694 and 224.695 and 365.256 and 686.98 and 4330.595 and 10727.160

Their rotation periods (expressed in Earth days) are – respectively – 58.6462 and 243.0187 and 0.997269663 and 1.02596 and 0.41354 and 0.44401 

The calculation is carried out in the following manner:-

2 x [(27.32166 ÷ 87.9694) + (27.32166 ÷ 224.695) + (27.32166 ÷ 365.256) + (27.32166 ÷ 686.98) + (27.32166 ÷ 4330.595) + (27.32166 ÷ 10727.160) + (27.32166 ÷ 58.6462) + (27.32166 ÷ 243.0187) + (27.32166 ÷ 0.9972697) + (27.32166 ÷ 1.02596) + (27.32166 ÷ 0.41354) + (27.32166 ÷ 0.44401)] = 365.52 times.

To verify the above time periods, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 14. TWICE the DIRECTED Number SUM (ie:- subtract for retrogrades) of the rotation periods of The Four Inner Solar System Planets = MINUS 365.697 Earth sidereal days.

DEMONSTRATION. The Four Inner Solar System Planets are – Mercury, Venus, Earth, and Mars. Mercury rotation period = 58.6462 and Venus rotation period = MINUS 243.0187 (because Venus has retrograde rotation, ie:- it rotates in the “wrong” direction, In that case, you have to SUBTRACT for Venus. (This Directed Number Addition is a commonplace feature in the Solar System!) and Earth rotation period = 0.9972697, and Mars rotation period = 1.02596 Add these periods together, and multiply by 2, and divide by Earth’s (sidereal) rotation period (which is the Earth sidereal day), which is 0.9972697 Earth (solar) days, to convert to Earth sidereal days. You will get the result of 365.697 Earth sidereal days. 

To verify the above time periods, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 15. TWICE the DIRECTED Number SUM of the rotation periods of the Four Giant Planets and the orbital periods of their 18 LARGE satellites = 365.70196 Earth sidereal days.

DEMONSTRATION. (All periods are expressed in Earth days.)

The Four Giant Planets are:- Jupiter, Saturn, Uranus, and Neptune, (rotation periods – respectively – 0.41353831 and 0.44401 and MINUS 0.71833 – because Uranus has retrograde rotation – and 0.671252).

Jupiter’s Four LARGE Satellites are Io, Europa, Ganymede, and Callisto (Orbital periods – respectively – 1.769137786 and 3.551181041 and 7.15455296 and 16.6890184).

Saturn’s Eight LARGE Satellites are Mimas, Enceladus, Tethys, Dione, Rhea, Titan, Hyperion, and Iapetus. (Orbital periods – respectively – 0.942421813 and 1.370217855 and 1.887802160 and 2.736914742 and 4.517500436 and 15.94542068 and 21.2766088 and 79.3301825

The Five LARGE Uranus satellites are:- Miranda, Ariel, Umbriel, Titania, and Oberon. (Orbital periods respectively – 1.4134840 and 2.52037932 and 4.1441765 and 8.7058703 and 13.4632423

The single LARGE Neptune Satellite Triton has retrograde revolution, so its orbital period = MINUS 5.8768441

If you add together all these above listed periods, and multiply the result by 2, and convert to Earth sidereal days by dividing by Earth’s (sidereal) rotation period, which is 0.9972697 Earth (solar) days, you will get the result of 365.70196 Earth sidereal days.

To verify planet and satellite sizes, and to verify the rotation periods of The Four Giant Planets, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of the LARGE satellites mentioned above, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 16. TWICE The Directed Number SUM of the rotation periods (ie:- subtracting for retrogrades) of The Planets excluding Earth (and excluding Pluto, treating it as a Trans-Neptunian Object, rather than as a planet) = MINUS 365.072 Earth days.

DEMONSTRATION. Here are the rotation periods of the planets (excluding Earth, and excluding Pluto) expressed in Earth days:- Mercury + 58.6462 and Venus MINUS 243.0187 (because Venus has retrograde rotation, and Mars +1.025957  and Jupiter + 0.41353831 and Saturn + 0.44401 and Uranus MINUS 0.71833 (because Uranus has retrograde rotation,  and Neptune + 0.671252 The Directed Number SUM of these seven periods = MINUS 182.5360 Earth days. TWICE this time period = MINUS 365.072 Earth days.

To verify the above time periods, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

(Note:- In the tables scanned into the above web page, the letter R directly preceding a planet’s rotation period indicates that that planet has retrograde rotation.)

EXAMPLE 17. Mars and Jupiter are neighbor planets. Mars and Earth are neighbor planets The Mars/Jupiter Pass Period exceeds The Mars/Earth Pass Period by 36.5 Earth days

DEMONSTRATION. A Pass Period is the length of time between successive occasions when two planets (or satellites) pass one another.

The Mars/Jupiter Pass Period = 816.506 Earth days, and the Mars/Earth Pass Period = 779.93 Earth days.

816.506 minus 779.93 = 36.58 Earth days.

The Pass Periods are calculated in the following manner:-Mars orbital period = 686.98 Earth days, and Jupiter orbital period = 4330.595 Earth days, and Earth orbital period = 365.2566 Earth days.

The Mars/Jupiter Pass Period is calculated thus:-

1 ÷ [(1 ÷ 686.98) minus (1 ÷ 4330.595)] = 816.506 Earth days.

The Mars/Earth Pass Period is calculated thus:-

1 ÷ [(1 ÷ 365.256) minus (1 ÷ 686.98)] = 779.93 Earth days.

To verify the above time periods, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 18. Mercury and Venus are neighbor planets, and Venus and Earth are neighbor planets.

HALF The Mercury/Venus Pass Period PLUS HALF The Venus/Earth Pass Period = 365.22 Earth sidereal days.

DEMONSTRATION. A Pass Period is the length of time between successive occasions when two planets (or satellites) pass one another. The Mercury/Venus Pass Period = 144.57 Earth days. The Venus/Earth Pass Period is 583.88 Earth days.

(144.57 + 583.88) ÷ 2 = 364.22 Earth solar days. To convert to Earth sidereal days, divide by Earth’s (sidereal) rotation period, which is 0.9972697 Earth (solar) days, to obtain the result of 365.22 Earth sidereal days.

The Pass Periods are calculated in the following manner:- Mercury orbital period = 87.9694 Earth days, and Venus orbital period = 224.695 Earth days, and Earth orbital period = 365.256 Earth days. The Mercury/Venus Pass Period is

1 ÷ [(1 ÷ 87.9694) minus (1 ÷224.695)] = 144.57 Earth days.

The Venus/Earth Pass Period is

1 ÷ [(1 ÷ 224.695) minus (1 ÷ 365.256)] = 583.88 Earth days.

To verify the above time periods, click on the following link:- 

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 19. Half the SUM of Venus synodic revolution period plus Venus synodic rotation period = 365.92 Earth sidereal days.

DEMONSTRATION. (Periods expressed in Earth (solar) days. Venus synodic revolution period = 583.9205 and Venus synodic rotation period = 145.9276 Half the SUM of these two periods = 364.92405 Earth (solar) days. To convert from Earth solar days to Earth sidereal days, divide by Earth’s (sidereal) rotation period, which is 0.9972697 Earth (solar) days.                                                                                                      

364.92405 ÷ 0.9972697 = 365.92 Earth sidereal days.

To verify Venus’ synodic revolution period, click on the following link:- 

www.solarsystemtimeperiods.com/planet-synodic

Venus’ synodic rotation period is calculated in the following manner:-

Venus, which has retrograde rotation, has a (sidereal) rotation period which is 243.0187 Earth days.

Earth’s orbital period = 365.256 Earth days.

1 ÷ [(1 ÷ 243.0187) + (1 ÷ 365.256)] = 145.9276 Earth days.

(Note:- The PLUS sign is because Venus has retrograde rotation.)

To verify Venus’ (sidereal) rotation period, and that Venus has retrograde rotation, and to verify Earth’s orbital period, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 20. ONE QUARTER of Venus synodic rotation period = 36.5818 Earth sidereal days.

DEMONSTRATION. Venus synodic rotation period = 145.9276 Earth solar days. One quarter of this time period = 36.4819 Earth solar days. To convert to Earth sidereal days, divide by Earth’s (sidereal) rotation period, which is 0.997269663 Earth (solar) days, and obtain the result of 36.5818 Earth sidereal days.

Venus’ synodic rotation period is calculated in the following manner:-

Venus, which has retrograde rotation, has a (sidereal) rotation period which is 243.0187 Earth days.

Earth’s orbital period = 365.256 Earth days.

1 ÷ [(1 ÷ 243.0187) + (1 ÷ 365.256)] = 145.9276 Earth days.

(Note:- The PLUS sign is because Venus has retrograde rotation.)

To verify Venus’ (sidereal) rotation period, and that Venus has retrograde rotation, and to verify Earth’s orbital period, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 21. Mars and Jupiter are neighbor planets. Jupiter’s Orbital Period exceeds Mars’ Orbital Period by 3653.6 Earth sidereal days.

DEMONSTRATION. The Earth Sidereal Day = Earth’s (sidereal) rotation period, which is 0.9972697 Earth (solar) days.

Jupiter’s Orbital Period = 4330.595 Earth days, and Mars’ Orbital Period = 686.98 Earth days.

(4330.595 minus 686.98) ÷ 0.9972697 = 3653.6 Earth sidereal days.

To verify the above time periods, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 22. There are just Two Inferior Planets, ie:- Mercury and Venus. TWICE The SUM of The Five Periods of each of these two planets = 3655.19 Earth sidereal days.

DEMONSTRATION. The Five periods of Mercury and Venus (expressed in Earth days) are as follows:-

 Solar Day – Mercury 175.938 and Venus 116.749

Orbital period – Mercury 87.9694 and Venus 224.695

Sidereal Rotation Period – Mercury 58.6462 and Venus 243.0187

Synodic Revolution Period – Mercury 115.8774 and Venus 583.9205

Synodic Rotation Period – Mercury 69.8636 and Venus 145.9276

The SUM of these ten periods = 1822.605 Earth solar days, which is (1822.605 ÷ 0.9972697) = 1827.595 Earth sidereal days. (One Earth sidereal day = Earth’s sidereal rotation period, which is 0.9972697 Earth solar days.)  If you multiply this result by 2, the result is 3655.19 Earth sidereal days.

The solar days of Mercury and Venus are calculated in the following manner;-

Mercury 1 ÷ [(1 ÷ 58.6462) minus (1 ÷ 87.9694)] = 175.938 Earth days.

Venus 1 ÷ [(1 ÷ 224.695) plus (1 ÷ 243.0187)] = 116.749 Earth days. (Note:- it is plus because Venus has retrograde rotation.)

Venus’ synodic rotation period is calculated in the following manner:-

Venus, which has retrograde rotation, has a (sidereal) rotation period which is 243.0187 Earth days.

Earth’s orbital period = 365.256 Earth days.

Venus’ synodic rotation period = 1 ÷ [(1 ÷ 243.0187) + (1 ÷ 365.256)] = 145.9276 Earth days.

(Note:- The PLUS sign is because Venus has retrograde rotation.)

Mercury’s synodic rotation period is calculated in the following manner:-

Mercury’s rotation period = 58.6462 Earth days.

Earth’s orbital period = 365.256 Earth days.

Mercury’s synodic rotation period = 1 ÷ [(1 ÷ 58.6462) minus (1 ÷ 365.256)] = 69.8636 Earth days.

To verify the synodic revolution periods of Mercury and Venus, click on the following link:-

www.solarsystemtimeperiods.com/planet-synodic

To verify Venus’ (sidereal) rotation period, and that Venus has retrograde rotation, and to verify Earth’s orbital period, and to verify all the other time periods for this example, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 23. During the precise time period of One Earth Year, The Eight Inner Solar System  Bodies revolve and rotate altogether a total of 3650 times (expressed to three significant figures). 

DEMONSTRATION. The Earth Year = Earth’s orbital period = 365.25636 Earth days.

The Eight Inner Solar System Bodies consist of Four Planets and Four Non-Planetary Bodies.

The Four Inner Solar System Planets are:- Mercury, Venus, Earth, and Mars. Their orbital periods (expressed in Earth days) are – respectively – 87.9694 and 224.695 and 365.256 and 686.98 and their rotation periods (expressed in Earth days) are – respectively – 58.6462 and 243.0187 and 0.9972697 and 1.02596

The Four Inner Solar System NON-Planetary Bodies are:- The Sun, The Moon, and The Two Mars Satellites Phobos and Deimos. Their rotation periods (expressed in Earth days) are – respectively – 24.66225 and 27.321661 and 0.31891023 and 1.2624407 Their orbital periods (expressed in Earth days) are as follows:- The Sun (for the purposes of this demonstration) does not have an orbital period, only a rotation period. The Moon orbital period = 27.321661 and Phobos and Deimos orbital periods are – respectively – 0.31891023 and 1.2624407 (Note:- the rotation periods of The Moon, Phobos, and Deimos are equal to their orbital periods, due to “tidal locking”.)

We now have fifteen (revolution and rotation) periods. The calculation is carried out in the following manner:-

In One Earth Year, Mercury revolves (365.256 ÷ 87.9694) = 4.1521 revolutions.

In One Earth Year, Venus revolves (365.256 ÷ 224.695) = 1.6256 revolutions.

You must perform this same calculation for all fifteen periods (including the rotation periods), and then add all the fifteen results together. The SUM that you will arrive at will be 3648.16 times, which is 3650 when expressed to three significant figures.

To verify orbital and rotation periods of The Moon, Phobos, and Deimos, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

For all the other periods mentioned above, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 24. The Sun, The Moon, and Mercury are the only prograde bodies that can ever be inferior to The Earth (ie:- closer to the Sun than Earth is) (Venus has retrograde rotation, and is therefore not a prograde body.)

During the precise time period of TWO Earth days, The Sun, The Moon, and Mercury rotate and revolve altogether a total of 0.3654 times in all.

DEMONSTRATION. The rotation periods (expressed in Earth days) of the Sun, The Moon, and Mercury are – respectively:- 24.66225 and 27.32166 and 58.6462 and their orbital periods (expressed in Earth days) are:- Sun orbital period (on the basis that a point on The Sun’s equator “revolves” round The Sun’s centre) = 24.66225 and The Moon orbital period = 27.32166 and Mercury orbital period = 87.9694  

The calculation is carried out in the following manner:-

(2 ÷ 24.66225) + (2 ÷ 27.32166) + (2 ÷ 58.6462) + (2 ÷ 24.66225) + (2 ÷ 27.32166) + (2 ÷ 87.9694) = 0.3654 rotations.

To verify all the periods mentioned above, and to verify that Venus has retrograde rotation (see the letter R directly preceding Venus’ rotation period in the table), click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 25. There are only three planets that (each) have only just one single LARGE satellite. These are Earth (whose one single LARGE satellite is The Moon), and Neptune (whose one single LARGE satellite is Triton), and Pluto (whose one single LARGE satellite is Charon). (All the other planets have either no satellites, or no LARGE satellites, or MULTIPLE LARGE satellites.).

During the precise time period of one Earth SIDEREAL Day, These three planets and their (single) LARGE satellites revolve altogether a total of 0.36509 revolutions.

DEMONSTRATION. The Earth sidereal day = Earth’s (sidereal) rotation period, which is 0.9972697 Earth (solar) days. The orbital periods (expressed in Earth solar days) of Earth, Moon, Neptune, Triton, Pluto, and Charon are – respectively – 365.256 and 27.32166 and 60215.912 and 5.8768441 and 90803.66 and 6.387230

Mercury and Venus have no satellites. Mars has no LARGE satellites. Jupiter has four LARGE satellites. Saturn has eight LARGE satellites.  Uranus has five LARGE satellites.  

 We have six orbital periods in all. The calculation is carried out in the following manner:-

During One Earth Sidereal Day, Earth revolves (0.997269663 ÷ 365.25636) = 0.0027303 revolutions.

During One Earth Sidereal Day, The Moon revolves (0.997269663 ÷ 27.321661) = 0.036501 revolutions.

During One Earth Sidereal Day, Neptune revolves (0.997269663 ÷ 60215.912) = 0.00001656 revolutions.   etc. etc. - - - -

You must carry out this same operation for all six orbital periods. Then you must add together all your six separate results. The SUM will be 0.36509

To verify Earth’s rotation period, and the orbital periods of Earth, The Moon, Neptune, and Pluto, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify Charon’s orbital period, click on the following link:-

www.solarsystemtimeperiods.com/pluto-sats

To verify Triton’s orbital period,click on the following link:- 

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 26. THE NAKED-EYE-VISIBLE PLANETS.     The Naked-Eye-Visible Planets (arranged in ascending orbit order) are:- Mercury, Venus, Earth, Mars, Jupiter, and Saturn. Here are their orbital and rotation periods (expressed in Earth days):-

Mercury orbital period = 87.9694 and rotation period = 58.6462 and The SUM of its orbital and rotation periods = 146.6156

Venus orbital period = 224.695 and rotation period = 243.0187 and The SUM of its orbital and rotation periods = 467.714

Earth orbital period = 365.256 and rotation period = 0.9972697 and The SUM of its orbital and rotation periods = 366.253

Mars orbital period = 686.98 and rotation period = 1.02596 and The SUM of its orbital and rotation periods = 688.006

Jupiter orbital period = 4330.595 and rotation period = 0.41354 and The SUM of its orbital and rotation periods = 4331.009

Saturn orbital period = 10727.160 and rotation period = 0.44401 and The SUM of its orbital and rotation periods = 10727.60

With these SUMS of revolution + rotation periods in mind, look at the following graphic. (Note:- all ratios are Superior Body: Inferior Body)

 ONE QUARTER of the SUM of The Ratios = 3.656

To verify the naked-eye-visible planets, click on the following link:-

www.solarsystemtimeperiods.com/naked-eye

To verify the orbital and rotation periods of the planets, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 27. The Earth Neighbors (arranged in ascending orbit order) are:- Venus, The Moon, and Mars.

The mass of each of these bodies and Earth (expressed in units of 1024 kilograms) are as follows:- Venus 4.8685 and Earth 5.9736 and The Moon 0.07349 and Mars 0.64185

With these figures in mind, look at the following graphic.

FOUR times The SUM of these three ratios = 364.98, which is 365 when expressed to three significant figures.

To verify the masses of the planets, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 28. This example involves the masses of The Planets (excluding Pluto, treating it as a Trans-Neptunian Object). The masses are expressed in units of 1024 kilograms. The Inner Solar System (arranged in ascending orbit order) consists of Mercury, Venus, Earth, and Mars. Mercury mass = 0.3302 and Venus mass = 4.8685 and Earth mass = 5.9736 and Mars mass = 0.64185 The Outer Solar System (arranged in ascending orbit order) consists of Jupiter, Saturn, Uranus, and Neptune. Jupiter mass = 1898.6 and Saturn mass = 568.46 and Uranus mass = 86.625 and Neptune mass = 102.43

Now look at the following graphic.

SUM of Ratios = 36.49217, ie:- 36.5 expressed to three significant figures. 

To verify the masses of the planets, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 29. Jupiter has Four Large Satellites (The Four “Galilean” Satellites). Inferior to them (ie:- closer to Jupiter) are Jupiter’s Four Small “Inner” Satellites. The names of these Four Small “Inner” Satellites, arranged in ascending orbit order are:- Metis, Adrastea, Amalthea, and Thebe. A Pass Period is the length of time between successive occasions when two planets (or satellites) pass one another. The Metis/Adrastea Pass Period = 25.26468 Earth days, and The Adrastea/Amalthea Pass Period = 0.743235 Earth days, and The Amalthea/Thebe Pass Period = 1.90545 Earth days.

The following graphic shows the Pass Periods (expressed in Earth days) between adjacent orbit satellites.

The SUM of The Ratios = 36.5566

 DEMONSTRATION. The orbital periods of Metis, Adrastea, Amalthea, and Thebe (expressed in Earth days) are – respectively – 0.294780 and 0.298260 and 0.498179 and 0.674536  The Adjacent Orbit Pass Periods are calculated in the following manner:-

The Metis/Adrastea Pass Period = 1 ÷ [(1 ÷ 0.294780) minus (1 ÷ 0.298260)] = 25.26468

The Adrastea/Amalthea Pass Period = 1 ÷ [(1 ÷ 0.298260) minus (1 ÷ 0.498179)] = 0.743235

The Amalthea/Thebe Pass Period = 1 ÷ [(1 ÷ 0.498179) minus (1 ÷ 0.674536)] = 1.90545

To verify the sizes of The Jupiter Satellites, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of Jupiter’s Four Small “Inner” Satellites, click on the following link:-

www.solarsystemtimeperiods.com/jup-inner

EXAMPLE 30. Uranus has Five Large Satellites. Arranged in ascending orbit order, these are:- Miranda, Ariel, Umbriel, Titania, and Oberon. Their orbital periods (expressed in Earth days) are – respectively – 1.4134840 and 2.52037932 and 4.1441765 and 8.7058703 and 13.4632423  In the graphic below the abbreviation DIFF stands for DIFFERENCE.

The SUM of The DIFFERENCES between The DIFFERENCES = 3.650477

To verify the sizes of The Uranus Satellites, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of The Five Large Uranus satellites, click on the following link:- 

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 31. Jupiter has Four Large Satellites (The Four “Galilean” Satellites). Arranged in ascending orbit order, their names are:- Io, Europa, Ganymede, and Callisto. Their orbital periods (expressed in Earth days) are – respectively – 1.769137786 and 3.551181041 and 7.15455296 and 16.6890184 With these periods in mind, look at the following graphic. DIFF stands for DIFFERENCE. 

The DIFFERENCE between The DIFFERENCES = 1.821329 Earth days. TWICE this time period is 3.65263 Earth days.

To verify the sizes of The Jupiter Satellites, click on the following link:- 

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of Jupiter’s Four Large Satellites, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 32. During the precise time period of One Eclipse Year, Earth, Moon, and The Two Inferior Planets (Mercury and Venus) rotate synodically altogether a total of 365.6944 rotations.

DEMONSTRATION. The Eclipse Year is 346.620080 Earth days.

The Two Inferior Planets (ie:- closer to The Sun than Earth is) are Mercury and Venus. Their synodic rotation periods (expressed in Earth days) are – respectively – 69.8636 and 145.9276  The Moon’s synodic rotation period (which is equal to its synodic revolution period, due to “tidal locking”) = 29.5305882 Earth days, and Earth’s synodic rotation period is 1 Earth (solar) day, ie:- 24 hours.

The Eclipse Year is defined as “The period of time between two successive passages of The Sun through the same node of The Moon’s orbit - - - being less than The Sidereal Year because of regression of The Moon’s nodes”.

The calculation is carried out in the following manner:-

During One Eclipse Year Mercury rotates synodically (346.620080 ÷ 69.8636) = 4.9614 rotations.

During One Eclipse Year Venus rotates synodically (346.620080 ÷ 145.9276) = 2.3753 rotations.

During One Eclipse Year The Moon rotates synodically (346.620080 ÷ 29.5305882) = 11.7377 rotations.

During One Eclipse Year Earth rotates synodically (346.620080 ÷ 1) = 346.620080 rotations.

4.9614 + 2.3753 + 11.7377 + 346.620080 = 365.6944

Venus’ synodic rotation period is calculated in the following manner:-

Venus, which has retrograde rotation, has a (sidereal) rotation period which is 243.0187 Earth days.

Earth’s orbital period = 365.256 Earth days.

Venus’ synodic rotation period = 1 ÷ [(1 ÷ 243.0187) + (1 ÷ 365.256)] = 145.9276 Earth days.

(Note:- The PLUS sign is because Venus has retrograde rotation.)

Mercury’s synodic rotation period is calculated in the following manner:-

Mercury’s rotation period = 58.6462 Earth days.

Earth’s orbital period = 365.256 Earth days.

Mercury’s synodic rotation period = 1 ÷ [(1 ÷ 58.6462) minus (1 ÷ 365.256)] = 69.8636 Earth days.

To verify Venus’ (sidereal) rotation period, and that Venus has retrograde rotation, and to verify Mercury’s (sidereal) rotation period, and to verify Earth’s orbital period, click on the following link:- 

www.solarsystemtimeperiods.com/periods-inner

To verify the definition of the term “Eclipse Year”, and its precise value, click on the following link:-

www.solarsystemtimeperiods.com/eclipse-year

To verify The Moon’s synodic revolution period, click on the following link:- 

www.solarsystemtimeperiods.com/planet-synodic

EXAMPLE 33. The DIRECTED NUMBER SUM (ie:- subtract for retrogrades) of The LONG sidereal Periods of The Inner Solar System Planets, excluding Earth, is 365.88 Earth days. 

DEMONSTRATION. The Inner Solar System Planets, excluding Earth, are:- Mercury, Venus, and Mars. The LONG sidereal periods of these planets (expressed in Earth days) are :- Mercury orbital period +87.9694 and Mercury rotation period +58.6462 and Venus orbital period MINUS 224.695 and Venus rotation period MINUS 243.0187 (because Venus has retrograde rotation) and Mars orbital period 686.98 and Mars rotation period is a SHORT period, and is therefore excluded (1.02596 Earth days)

87.9694 + 58.6462 minus 224.695 minus 243.0187 + 686.98 = 365.88 Earth days.

To verify the above time periods, and to verify that Venus has retrograde rotation (See the letter R directly preceding Venus’ rotation period in the table), click on the following link:- 

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 34. Saturn and Uranus are neighbor planets.

FOUR times the SUM of the orbital periods of The INNERMOST Satellites of Saturn, and Uranus (ie:- Saturn’s Pan, and Uranus’ Cordelia) = 3.6503 Earth sidereal days.

DEMONSTRATION. Saturn’s INNERMOST Satellite is Pan, whose orbital period = 0.57505 Earth days, and Uranus’ INNERMOST Satellite is Cordelia, whose orbital period = 0.335034 Earth days. 

The calculations are carried out in the following manner:-

4 x (0.57505 + 0.335034) = 3.64033 Earth solar days, which is equal to (3.64033 ÷ 0.997269663) = 3.6503 Earth sidereal days. (One Earth sidereal day = 0.9972697 Earth solar days.) 

To verify that Saturn and Uranus are neighbor planets, click on the following link:- 

www.solarsystemtimeperiods.com/planet-orbital

To verify that Pan is Saturn’s Innermost Satellite, and to verify its orbital period, click on the following link:-

www.solarsystemtimeperiods.com/saturn-sats

To verify that Cordelia is Uranus’ Innermost Satellite, and to verify its orbital period, click on the following link:-

www.solarsystemtimeperiods.com/uranus-innermost

To verify Earth’s (sidereal) rotation period (ie:- The Earth sidereal day), click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 35. The OUTERMOST LARGE PROGRADE Satellites of The Four Giant Planets. The ARITHMETIC MEAN of their orbital periods is 36.59406 Earth sidereal days.

DEMONSTRATION. The Four Giant Planets are:- Jupiter, Saturn, Uranus, and Neptune. The OUTERMOST LARGE Satellites of Jupiter, Saturn, Uranus and Neptune are – respectively – Callisto, Iapetus, Oberon, and Triton. The orbital periods (expressed in Earth days) of these four satellites are – respectively – 16.6890184  and 79.3301825 and 13.4632423  and we ignore Triton in this calculation because Triton has RETROGRADE Revolution, and we are only including Progrades in this example. The SUM of the periods 16.6890184 and 79.3301825 and 13.4632423 is 109.48244 Earth solar days, which is equal to (109.48244 ÷ 0.997269663) = 109.78219 Earth sidereal days. (One Earth sidereal day = 0.9972697 Earth solar days.) Divide this number by three to calculate the arithmetic mean, and the result is 36.59406 Earth sidereal days.

To verify planet sizes and satellite sizes, and the OUTERMOST LARGE Satellites of the Four Giant Planets, and to verify Triton’s retrograde revolution (See the letter R immediately preceding Triton’s orbital period), and to verify Earth’s (sidereal) rotation period – ie:- The Earth sidereal day), click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of the large satellites, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 36. The SUM of the orbital periods of The Inner Solar System Planets, excluding Earth is 2.7368 Earth Years.                      

The reciprocal of the number 2.7368 is 0.365

DEMONSTRATION. The Earth Year = 365.25636 Earth days (which is Earth’s orbital period).

The Four Inner Solar System Planets are:- Mercury, Venus, Earth, and Mars. Their orbital periods (expressed in Earth days) are – respectively – 87.9694 and 224.695  and 365.256 and 686.98  The SUM of 87.9694 + 224.695 + 686.98 = 999.644 Earth days, which is (999.644 ÷ 365.256) = 2.7368 Earth Years. The reciprocal of the number 2.7368 is 0.365

To verify the above mentioned planetary orbital periods, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 37. HALF the SUM of The Moon’s Orbital Period PLUS The Moon’s Orbital Oscillation Period = 0.27422 Earth Years.

The reciprocal of the number 0.27422 is 3.65 (when expressed to three significant figures.

DEMONSTRATION. The Moon’s Orbital Oscillation Period = 173 Earth days.

The Moon’s Orbital Period = 27.321661 Earth days.

The Earth (sidereal) Year = 365.25636 Earth days.

[(173 + 27.321661) ÷ 2] ÷ 365.25636 = 0.27422 Earth years.

1 ÷ 0.27422 = 3.6467 = 3.65 expressed to three significant figures.

To verify The Moon’s orbital period, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

To verify The Moon’s oscillation period, click on the following link:-

www.solarsystemtimeperiods.com/moon-oscillation

EXAMPLE 38. HALF The SUM of The Orbital Periods of The LARGE Outer Solar System Satellites = 0.273345 Earth years. The reciprocal of the number 0.273345 is 3.6584

DEMONSTRATION. The Five Outer Solar System Planets are:- Jupiter, Saturn, Uranus, Neptune, and Pluto.

The Four LARGE Satellites of Jupiter (The Four “Galilean” Satellites) (arranged in ascending orbit order) are:- Io, Europa, Ganymede, and Callisto. Their orbital periods (expressed in Earth days) are – respectively – 1.769137786 and 3.551181041 and 7.15455296 and 16.6890184

Saturn’s Eight LARGE Satellites (arranged in ascending orbit order) are:- Mimas, Enceladus, Tethys, Dione, Rhea, Titan, Hyperion, and Iapetus. Their orbital periods (expressed in Earth days) are Mimas 0.942421813 and Enceladus 1.370217855 and Tethys 1.887802160 and Dione 2.736914742 and Rhea 4.517500436 and Titan 15.94542068 and Hyperion 21.2766088 and Iapetus 79.3301825 

  Uranus has Five LARGE Satellites, ie:- Miranda, Ariel, Umbriel, Titania, and Oberon. Their orbital periods (expressed in Earth days) are – respectively – 1.4134840 and 2.52037932 and 4.1441765 and 8.7058703 and 13.4632423

Neptune has just one LARGE Satellite, ie:- Triton, whose orbital period = 5.8768441 Earth days.

Pluto has just one LARGE Satellite, ie:- Charon, with an orbital period of 6.387230 Earth days. 

The SUM of these nineteen periods = 199.68218 Earth days. HALF this time period = 99.84109 Earth days, which is equal to (99.84109 ÷ 365.25636) = 0.273345 Earth years. (There are 365.25636 Earth days in One Earth Year.) The reciprocal of the number 0.273345 is 3.6584

To verify satellite sizes, and to verify the order of the planets, and therefore which planets are “Outer Solar System” Planets (ie:- everything superior to Mars), and to verify The Earth year, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of these satellites, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 39. During HALF a Moon Orbital Period, The Four Inner Solar System Planets REVOLVE altogether a total of 0.27337 revolutions. The reciprocal of the number 0.27337 is 3.658

DEMONSTRATION. The Moon orbital period = 27.32166 Earth days. The Four Inner Solar System Planets (arranged in ascending orbit order) are:- Mercury, Venus, Earth, and Mars. Their orbital periods (expressed in Earth days) are – respectively – 87.9694 and 224.695 and 365.256 and 686.98

The calculations are carried out in the following manner:- 

[(27.32166 ÷ 87.9694) + (27.32166 ÷ 224.695) + (27.32166 ÷ 365.256) + (27.321661 ÷ 686.98)] ÷ 2 = 0.27337 revolutions. The reciprocal of the number 0.27337 is 3.658

To verify the above orbital periods, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 40. During A QUARTER of The Moon’s Orbital Period, The Four Inner Solar System NON-Planetary Bodies rotate altogether a total of 27.3554 rotations.

The reciprocal of the number 27.3554 is 0.036556

DEMONSTRATION. The Four Inner Solar System NON-Planetary Bodies are:- The Sun, The Moon, and The Two Mars Satellites, Phobos and Deimos. Their rotation periods (expressed in Earth days) are – respectively – 24.66225 and 27.321661 and 0.31891023 and 1.2624407

(27.321661 ÷ 24.66225) + (27.321661 ÷ 27.321661) + (27.321661 ÷ 0.31891023) + (27.321661 ÷ 1.2624407) = 109.4217 rotations. ONE QUARTER of this number is 27.3554 and the reciprocal of the number 27.3554 is 0.036556

To verify the orbital periods of The Moon, Phobos, and Deimos, (bearing in mind that these satellites have their rotation periods equal to their orbital periods, due to “tidal locking”), click on the following link:-

www.solarsystemtimeperiods.com/large-sats

To verify which are The Four Inner Solar System NON-Planetary Bodies, and to verify The Sun’s rotation period, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 41. The Moon’s orbital period is 27.321661 Earth days.

During ONE THIRD of this time period, The Four Inner Solar System NON-Planetary Bodies rotate altogether a total of 36.5 rotations.) (expressed to three significant figures).

DEMONSTRATION. The Four Inner Solar System Non-Planetary Bodies are:- The Sun, The Moon, and The Two Mars Satellites – Phobos and Deimos. Their rotation periods (expressed in Earth days) are as follows:- The Sun 24.66225 and The Moon 27.321661 and Phobos 0.31891023 and Deimos 1.2624407 (Note:- For the three satellites, the orbital and rotation periods are the same, due to “tidal locking”.)

During One Moon orbital period, The Four Inner Solar System Non-Planetary Bodies rotate altogether a total of

(27.321661 ÷ 24.66225) + (27.321661 ÷ 27.3216610) + (27.321661 ÷ 0.31891023) + (27.321661 ÷ 1.2624407) = 109.4217 rotations.

109.4217 ÷ 3 = 36.4739 rotations, which is 36.5 rotations, expressed to three significant figures.

To verify the orbital periods of The Moon, Phobos, and Deimos, (bearing in mind that these satellites have their rotation periods equal to their orbital periods, due to “tidal locking”), click on the following link:-

www.solarsystemtimeperiods.com/large-sats

To verify which are The Four Inner Solar System NON-Planetary Bodies, and to verify The Sun’s rotation period, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 42. THE EARTH NEIGHBORS.

The Earth Neighbors are:- Venus, The Moon, Mars, and The Two Mars Satellites – Phobos and Deimos.

The SUM of the rotation periods of Earth and its neighbors  = 273.94494 Earth days.

 The reciprocal of the number 273.94494 is 3.6504

DEMONSTRATION. The rotation periods of Venus, Earth, Moon, Mars, Phobos, and Deimos (expressed in Earth days) are – respectively – 243.0187 and 0.9972697 and 27.321661 and 1.02596 and 0.31891023 and 1.2624407 The SUM of these five periods is 273.94494 Earth days.

1 ÷ 273.94494 = 0.0036504

To verify the rotation periods of Venus, Earth, and Mars, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

 To verify the orbital periods of The Moon, Phobos, and Deimos, (bearing in mind that these satellites have their rotation periods equal to their orbital periods, due to “tidal locking”), click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 43. During the precise time period of TWO Earth Days, Jupiter and its Eight Close “Regular” Satellites rotate altogether a total of 27.39929 rotations.

 The reciprocal of the number 27.39929 is 0.0365 (when expressed to three significant figures).

DEMONSTRATION. Jupiter has Four Large Satellites (The “Galilean” Satellites), and inferior to them (ie:- closer to Jupiter) are Jupiter’s Four Small “Inner” Satellites (sometimes called The Four “Amalthean” Satellites). These two satellite groups represent Jupiter’s Eight Close “Regular” Satellites. 

Jupiter rotation period = 0.41354 Earth days.

Jupiter’s Four “Inner” Satellites are Metis, Adrastea, Amalthea, and Thebe. Their rotation periods (expressed in Earth days) are – respectively – 0.294780 and 0.298260 and 0.498179 and 0.674536

Jupiter’s Four LARGE (“Galilean”) Satellites are:- Io, Europa, Ganymede, and Callisto. Their rotation periods (expressed in Earth days) are – respectively – 1.769137786 and 3.551181041 and 7.15455296 and 16.6890184 

The calculations are carried out in the following manner:-

(2 ÷ 0.41354) + (2 ÷ 0.294780) + (2 ÷ 0.298260) + (2 ÷ 0.498179) + (2 ÷ 0.674536) + (2 ÷ 1.769137786) + (2 ÷ 3.551181041) + (2 ÷ 7.15455296) + (2 ÷ 16.6890184) = 27.39929 rotations.

1 ÷ 27.39929 = 0.036497 which is 0.0365 when expressed to three significant figures.

To verify the sizes of Jupiter’s satellites, and to verify Jupiter’s rotation period, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of Jupiter’s Four Large Satellites, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

To verify the orbital periods of Jupiter’s Four Small “Inner” Satellites, click on the following link:-

www.solarsystemtimeperiods.com/jup-inner

(Note:- the orbital periods of the above mentioned satellites are the same as their rotation periods, due to “tidal locking”.)

EXAMPLE 44. FOUR times The SUM of the synodic rotation periods of The Moon and The Planets (excluding Pluto, which is sometimes regarded as being a “Trans-Neptunian Object”, rather than a “proper” planet) = 2.73338 Earth Years.

The reciprocal of the number 27.3338 is 0.36585

DEMONSTRATION. The Moon and The Planets are as follows:- The Moon, Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune (excluding Pluto). Their synodic rotation periods (expressed in Earth days) are – respectively – 29.530588 and 69.8636 and 145.9276 and 1 and 1.02749 and 0.41358 and 0.44403 and 0.71835 and 0.671259 The SUM of these nine periods = 249.5965 Earth days. If you multiply this by four, the result is 998.386 Earth days, which is (998.386 ÷ 365.256) = 2.73338 Earth Years. (The Earth Year, which is Earth’s orbital period = 365.256 Earth days.) The reciprocal of the number 27.3339 is 0.36585

The synodic rotation periods of The Planets are calculated in the following manner:-

Here are the (sidereal) rotation periods (expressed in Earth days) of The Planets:- Mercury 58.6462 and Venus 243.0187 and Earth 0.9972697 and Mars 1.02596 and Jupiter 0.41354 and Saturn 0.44401 and Uranus 0.71833 and Neptune 0.671252

Here are the orbital periods (expressed in Earth days) of Earth and The Superior Planets:- Earth 365.256 and Mars 686.98 and Jupiter 4330 595 and Saturn 10727.160 and Uranus 30717.682 and Neptune 60215.912

Mercury’s synodic rotation period =

1 ÷ [(1 ÷ 58.6462) minus (1 ÷ 365.256)] = 69.8636 Earth days.

Venus’ synodic rotation period =

1 ÷ [(1 ÷ 243.0187) PLUS (1 ÷ 365.256)] = 145.9276 Earth days. (Note:- The PLUS is because Venus has retrograde rotation.)

Earth’s synodic rotation period =

1 ÷ [(1 ÷ 0.9972697) minus (1 ÷ 365.256)] = 1.000000 Earth days.

Mars’ synodic rotation period =

1 ÷ [(1 ÷ 1.02596) minus (1 ÷ 686.98)] = 1.02749 Earth days.

Jupiter’s synodic rotation period =

1 ÷ [(1 ÷ 0.41354) minus (1 ÷ 4330.595)] = 0.41358 Earth days.

Saturn’s synodic rotation period =

1 ÷ [(1 ÷ 0.44401) minus (1 ÷ 10727.160)] = 0.44403 Earth days.

Uranus’ synodic rotation period =

1 ÷ [(1 ÷ 0.71833) minus (1 ÷ 30717.682)] = 0.71835 Earth days.

Neptune’s synodic rotation period =

1 ÷ [(1 ÷ 0.671252) minus (1 ÷ 60215.912)] = 0.671259 Earth days.

The Moon’s synodic rotation period is equal to its synodic revolution period (due to “tidal locking”), which is 29.530588 Earth days.

To verify the orbital and rotation periods of the planets, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify The Moon’s synodic revolution period, click on the following link:-

www.solarsystemtimeperiods.com/planet-synodic

EXAMPLE 45. There are Three Retrograde (rotation) Planets, ie:- Venus, Uranus, and Pluto. FOUR times The SUM of their rotation periods = 2.739164 Earth Years.

The reciprocal of the number 2.739164 is 0.36507

DEMONSTRATION. There are Three Retrograde Planets (ie:- that have retrograde rotation – that is – rotation in the “wrong” direction). These are:- Venus, Uranus, and Pluto. The rotation periods of these three planets (expressed in Earth days) are – respectively – 243.0187 and 0.71833 and 6.387230

The SUM of these three periods = 250.1243 Earth days. If you multiply this time period by FOUR, the result is 1000.4972 Earth days, which is equal to (1000.4972 ÷ 365.25636) = 2.739164 Earth years. (There are 365.25636 Earth days in One Earth Year.) The reciprocal of the number 2.739164 is 0.36507

To verify the above periods, and that these above mentioned planets have retrograde rotation (See the letter R directly preceding the rotation periods of thee planets), click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 46. There are a number of sunspot cycles. One of these is the 250 day cycle. FOUR times this time period = 2.7378 Earth Years.

The reciprocal of this number = 0.365256

DEMONSTRATION. Apart from the 11.13 year Sunspot Cycle, there are various short term Sunspot Cycles, one of which is the 250 day Sunspot Cycle. “A very definite (Sun Spot) cycle of length a little less than 250 days that far out shadows everything else. Its amplitude is so large, and the regularities of the peaks obtained is so excellent - - - - the 250 day cycle - - - has been repeated more than 50 times. The chances of getting such a cycle by pure random fluctuation is only one in billions.”

The above quote is from the article “A Periodogram Investigation of Short Period Sun Spot Cycles”, by Dinsmore Alter (Griffith Observatory, Los Angeles, California, December 1937), published in Monthly Weather Review, issue for July 1938, page 208 to page 212.

This article can be viewed as a pdf file on the internet. Go to the following web page:-

https://docs.lib.noaa.gov/rescue/mwr/066/mwr-066-07-0208.pdf

Here is a scan of the relevant portion of this article:-

4 x 250 = 1000 Earth days, which is (1000 ÷ 365.25636) = 2.7378 Earth Years. (There are 365.25636 Earth days in One Earth Year.) The reciprocal of the number 2.7378 = 0.365256

EXAMPLE 47. THREE times The SUM of the rotation periods of The Moon and The Planets = 2.737175 Earth Years.

The reciprocal of the number 2.737175 is 0.36534

DEMONSTRATION. The rotation periods of the Moon and The Planets (excluding Pluto, which we will treat here as a Trans-Neptunian Object rather than a planet) (expressed in Earth days) are as follows:- The Moon 27.32166 and Mercury 58.6462 and Venus 243.0187 and Earth 0.9972697 and Mars 1.02596 and Jupiter 0.41354 and Saturn 0.44401 and Uranus 0.71833 and Neptune 0.671252 The SUM of these nine periods = 333.2569 Earth days, which is (333.2569 ÷ 365.256) = 0.912392 Earth years (There are 365.25636 Earth days in one Earth year, which is Earth’s orbital period). Multiply by three and the result is 2.737175 Earth years. The reciprocal of the number 2.737175 is 0.36534

To verify the above time periods, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 48. Deimos is one of Mars’ Two Satellites. The Deimosic Day (length of time between successive risings of Deimos on a fixed horizon on Mars) is 5.476952 Earth days.  Twice the reciprocal of the number 5.476952 is 0.36516

DEMONSTRATION. The Deimosic Day is the length of time between successive risings of Deimos on a fixed horizon on Mars.

Mars rotation period = 1.02596 Earth days, and Deimos orbital period = 1.2624407 Earth days.

The Deimosic Day is calculated in the following manner:-

1 ÷ [(1 ÷ 1.02596) minus (1 ÷ 1.2624407)] = 5.47704 Earth days. The reciprocal of the number 5.47704 is 0.18258 and TWICE this number is 0.36516

To verify Mars’ rotation period, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

To verify Deimos’ orbital period, click on the following link:- 

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 49. There are Four Giant Planets. Arranged in ascending orbit order (ie:- in ascending order of distance from The Sun), they are:- Jupiter, Saturn, Uranus, and Neptune. Jupiter and Saturn are “The Two Gas Giants”. Uranus and Neptune are “The Two Ice Giants”.

The “day” of a satellite is the length of time between successive risings of the satellite on a fixed horizon of the parent planet.    

This example involves just “The Two Ice Giants”. These two planets have a total of Six LARGE Satellites. The SUM of The DAYS of these six satellites = 5.4781445 Earth days.

TWICE The RECIPROCAL OF THIS NUMBER = 0.365087  

DEMONSTRATION.

Uranus rotation period = 0.71833 Earth days.

Uranus has Five LARGE Satellites. Here are their names and orbital periods (expressed in Earth days):- Miranda 1.4134840 and Ariel 2.52037932 and Umbriel 4.1441765 and Titania 8.7058703 and Oberon 13.4632423

The Miranda Day =

1 ÷ [(1 ÷ 0.71833) minus (1 ÷ 1.4134840)] = 1.4606087

The Ariel Day =

1 ÷ [(1 ÷ 0.71833) minus (1 ÷ 2.52037932)] = 1.0046695

The Umbriel Day =

1 ÷ [(1 ÷ 0.71833) minus (1 ÷ 4.1441765)] = 0.868949

The Titania Day =

1 ÷ [(1 ÷ 0.71833) minus (1 ÷ 8.7058703)] = 0.7829304

The Oberon Day =

1 ÷ [(1 ÷ 0.71833) minus (1 ÷ 13.4632423)] = 0.75882

Neptune has One LARGE Satellite – Triton, whose orbital period = 5.8768441 Earth days. Triton has RETROGRADE REVOLUTION!

Neptune’s rotation period = 0.671252 Earth days.

The Triton Day =

1 ÷ [(1 ÷ 0.671252) PLUS (1 ÷ 5.8768441)] = 0.6024413 Earth days.

(Note:- The PLUS is because Triton has retrograde revolution!)

The SUM of these six “days” = 5.4784145 Earth days.

(1 ÷ 5.4784145) x 2 = 0.365087

To verify the positions of The Four Giant Planets, and to verify which are their LARGE Satellites, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of The LARGE Satellites of Uranus and Neptune, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 50. There are just FOUR SLOW-ROTATING Bodies in The Inner Solar System, ie:- Mercury, Venus, The Sun, and The Moon. The SUM of their Synodic Revolution Periods and Synodic Rotation Periods = 2.7408 Earth Years.                                                                                               

The reciprocal of this number is 0.365 (when expressed to three significant figures). 

DEMONSTRATION. Here are the names of The Eight Inner Solar System Bodies, together with their sidereal rotation periods (expressed in Earth Days):- Mercury 58.6462 and Venus 243.0187 and Earth 0.9972697 and Mars 1.02596 and The Sun 24.66225 and The Moon 27.321661 and (The Two Mars Satellites) Phobos 0.3189 and Deimos 1.2624 From the above you can see that there are only Four Slow-Rotating Inner Solar System Bodies, ie:- Mercury, Venus, The Sun, and The Moon.

The SYNODIC Rotation periods of Mercury, Venus, Sun, and Moon (expressed in Earth days) are – respectively – 69.8636 and 145.9276 and 26.44803 and 29.53059                                                                                                 The SYNODIC Revolution periods of Mercury, Venus, Sun, and Moon (expressed in Earth days) are – respectively – 115.8774 and 583.9205 and NONE (since The Sun rotates, but does not revolve) and 29.53059

The SUM of these seven periods is 1001.0983 Earth days, which is (1001.0983 ÷ 365.256) = 2.7408 Earth Years. (There are 365.256 Earth days in one Earth Year, which is Earth’s orbital period.) The reciprocal of the number 2.7408 = 0.36486, which is 0.365 when expressed to three significant figures. `

Mercury’s synodic rotation period =

1 ÷ [(1 ÷ 58.6462) minus (1 ÷ 365.256)] = 69.8636 Earth days.

Venus’ synodic rotation period =

1 ÷ [(1 ÷ 243.0187) PLUS (1 ÷ 365.256)] = 145.9276 Earth days. (Note:- The PLUS is because Venus has retrograde rotation.)

The Sun’s synodic rotation period =

1 ÷ [(1 ÷ 24.66225) minus (1 ÷ 365.256)] = 26.44803 Earth days.

The Moon’s synodic rotation period =

1 ÷ [(1 ÷ 27.32166) minus (1 ÷ 365.256)] = 29.53059 Earth days.

To verify the (sidereal) rotation periods of the Eight Inner Solar System Bodies, and to verify Earth’s orbital period, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

To verify the synodic revolution periods of Mercury, Venus, and The Moon, click on the following link:-

www.solarsystemtimeperiods.com/planet-synodic

EXAMPLE 51. There are just Four LARGE SLOW-ROTATING Bodies in The Planetary System, ie:- The Sun, Mercury, Venus, and Pluto. (All the other planets are fast-rotating.)                                                            

THREE times The SUM of the rotation periods of these four bodies = 2.732719 Earth years. The reciprocal of this number = 0.36594

DEMONSTRATION. The Sun rotation period = 24.66225 Earth days.

Mercury rotation period = 58.6462 Earth days.

Venus rotation period = 243.0187 Earth days.

Pluto rotation period = 6.3872 Earth days.

The Earth (sidereal) year (ie:- Earth’s orbital period) = 365.25636 Earth days.

[(24.66225 + 58.6462 + 243.0187 + 6.3872) x 3] ÷ 365.25636 = 2.73272 Earth years.

1 ÷ 2.732719 = 0.36594

To verify sizes of Solar System Bodies, and to verify Earth’s orbital period, and to verify the rotation periods of the above mentioned bodies, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 52. Earth’s Two Neighbor Planets are Venus and Mars.

THREE times The Venus/Mars Pass Period = 2.742635 Earth Years. The reciprocal of the number 2.742635 = 0.365 (when expressed to three significant figures.

DEMONSTRATION. The Pass Period (ie:- the length of time between successive occasions of two planets or satellites directly passing one another – ie:- of one “overtaking” the other) of two planets or satellites is calculated in the following manner:-

Venus orbital period = 224.695 Earth days, and Mars orbital period = 686.98 Earth days.

1 ÷ [(1 ÷ 224.695) minus (1 ÷ 686.98)] = 333.9087 Earth days.

By this calculation, you have determined that The Venus/Mars Pass Period = 333.9087 Earth days, which is (333.9087 ÷ 365.256 = 0.9142 Earth Years. (The Earth Year, which is Earth’s orbital period, = 365.256 Earth days.)

0.9142 x 3 = 2.7426 Earth years. The reciprocal of the number 2.7426 = 0.3646 ie:- 0.365 when expressed to three significant figures.

To verify that Venus and Mars are Earth’s neighbors, and to verify the orbital periods of Venus, Earth, and Mars, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 53. The INNERMOST Satellites out as far as Jupiter are as follows:- The Sun’s INNERMOST Satellite is Mercury. Mercury and Venus have no satellites. Earth’s INNERMOST Satellite is The Moon. Mars’ INNERMOST Satellite is Phobos. Jupiter’s INNERMOST Satellite is Metis. The SUM of the synodic rotation periods of these four bodies = 0.273802 Earth Years. The reciprocal of the number 0.273802 is 3.6523

DEMONSTRATION. The synodic rotation periods (expressed in Earth days) of Mercury, The Moon, Phobos, and Metis are – respectively – 69.8636 and 29.53059 and 0.319058 and 0.29480 The SUM of these four periods = 100.008 Earth days. The Earth Year (which is Earth’s orbital period) = 365.256 Earth days, and (100.008 ÷ 365.256) = 0.273803 Earth Years. The reciprocal of the number 0.273803 is 3.6523

Mercury’s (sidereal) rotation period = 58.6462 Earth days.

Mercury’s synodic rotation period =

1 ÷ [(1 ÷ 58.6462) minus (1 ÷ 365.256)] = 69.8636 Earth days.

The Moon’s (sidereal) rotation period (which is equal to its orbital period, due to “tidal locking) = 27.32166 Earth days.

The Moon’s synodic rotation period =

1 ÷ [(1 ÷ 27.32166) minus (1 ÷ 365.256)] = 29.53059 Earth days.

Phobos’ (sidereal) rotation period (which is equal to its orbital period, due to “tidal locking) = 0.31891023 Earth days; and Mars’ orbital period = 686.98 Earth days.

Phobos’ synodic rotation period =

1 ÷ [(1 ÷ 0.31891023) minus (1 ÷ 686.98)] = 0.319058 Earth days.

 Metis’ (sidereal) rotation period (which is equal to its orbital period, due to “tidal locking) = 0.294780 Earth days; and Jupiter’s orbital period = 4330.595 Earth days.

Metis’ synodic rotation period =

1 ÷ [(1 ÷ 0.294780) minus (1 ÷ 4330.595)] = 0.29480 Earth days.

To verify that Metis is Jupiter’s INNERMOST satellite, and to confirm Metis’ orbital period, click on the following link:-

www.solarsystemtimeperiods.com/jup-inner

To verify that Mercury and Venus have no satellites, and that Mars’ INNERMOST Satellite is Phobos, and that Mercury is the INNERMOST Planet, and to verify the orbital periods of Earth, Mars, The Moon and Jupiter, and  to verify the (sidereal) rotation period of Mercury, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

To verify the orbital period of Phobos, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 54. FOUR times the SUM of the Synodic Revolution Periods and Synodic Rotation Periods of The Sun and The Naked-Eye Visible Planets, excluding Earth = 27.387206 Earth Years. The reciprocal of the number 27.387206 is 0.036513

DEMONSTRATION. Here are the names of The Sun and the Naked-Eye Visible Planets (excluding Earth), together with their synodic revolution periods (expressed in Earth days). The Sun NONE (The Sun rotates, but does not, properly speaking, revolve) and Mercury 115.8774 and Venus 583.9205 and Mars 779.9382 and Jupiter 398.8846 and Saturn 378.0928 (The other planets are “telescopic” planets, ie:- not visible to the naked eye.)

Here are the names of The Sun and the Naked-Eye Visible Planets (excluding Earth), together with their synodic rotation periods (expressed in Earth days). The Sun 26.44803 and Mercury 69.8636 and Venus 145.9276 and Mars 1.027491204 and Jupiter 0.41357779 and Saturn 0.44402  (The other planets are “telescopic” planets, ie:- not visible to the naked eye.)

FOUR times the SUM of these eleven periods = 10003.3513 Earth days. If you divide this number by 365.25636 (ie:- the number of Earth days in The Earth Year), the result is 27.3872 Earth Years. The reciprocal of the number 27.387206 is 0.036513

This is how the synodic rotation periods of the relevant planets are calculated:- Here are the orbital periods (expressed in Earth days) of Earth and The Naked-Eye Visible Superior Planets:- Earth 365.256 and Mars 686.98 and Jupiter 4330 595 and Saturn 10727.160

Here are the (sidereal) rotation periods (expressed in Earth days) of The Sun and The Naked-Eye Visible Planets:- The Sun 24.66225 and Mercury 58.6462 and Venus (retrograde rotation) 243.0187 and Mars 1.02596 and Jupiter 0.41354 and Saturn 0.44401

The Sun’s synodic rotation period =

1 ÷ [(1 ÷ 24.66225) minus (1 ÷ 365.256)] = 26.44803 Earth days.

Mercury’s synodic rotation period =

1 ÷ [(1 ÷ 58.6462) minus (1 ÷ 365.256)] = 69.8636 Earth days.

Venus’ synodic rotation period =

1 ÷ [(1 ÷ 243.0187) PLUS (1 ÷ 365.256)] = 145.9276 Earth days. (Note:- The PLUS is because Venus has retrograde rotation.)

Mars’ synodic rotation period =

1 ÷ [(1 ÷ 1.02596) minus (1 ÷ 686.98)] = 1.02749 Earth days.

Jupiter’s synodic rotation period =

1 ÷ [(1 ÷ 0.41354) minus (1 ÷ 4330.595)] = 0.41358 Earth days.

Saturn’s synodic rotation period =

1 ÷ [(1 ÷ 0.44401) minus (1 ÷ 10727.160)] = 0.44403 Earth days.

To verify The Naked-Eye Visible Planets, click on the following link:-

www.solarsystemtimeperiods.com/naked-eye

To verify the synodic revolution periods of Mercury, Venus, Mars, Jupiter, and Saturn, click on the following link:-

www.solarsystemtimeperiods.com/planet-synodic

To verify the (sidereal) rotation periods of The Sun and the planets, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 55. A PRIMARY Satellite is the LARGEST Satellite for a particular planet.

During FOUR Moon orbital periods, The PRIMARY Satellites of The Superior Naked-Eye Visible Planets revolve altogether a total of 365 revolutions (expressed to three significant figures). 

DEMONSTRATION. The Superior (ie:- with orbits outside Earth’s orbit) Naked-Eye Visible Planets are:- Mars, Jupiter, and Saturn. 

Here is a list of The PRIMARY Satellites of The Superior Naked-Eye Visible Planets:- Mars’ PRIMARY Satellite is Phobos. Jupiter’s PRIMARY Satellite is Ganymede. Saturn’s PRIMARY Satellite is Titan.

The orbital periods (expressed in Earth days) of Phobos, Ganymede, and Titan are – respectively – 0.31891023 and 7.15455296 and 15.94542068

The Moon’s orbital period = 27.321661 Earth days.

4 x [(27.321661 ÷ 0.31891023) + (27.321661 ÷ 7.15455296) + (27.321661 ÷ 15.94542068)] = 364.8167 revolutions, ie:- 365 revolutions expressed to three significant figures.

To verify which are the Naked-eye-visible planets, click on the following link:-

www.solarsystemtimeperiods.com/naked-eye

To verify which are the superior planets, and to verify satellite sizes, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of the above mentioned satellites, click on the following link:- 

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 56. A PRIMARY Satellite is the LARGEST Satellite for a particular planet.

During One LUNAR DAY, The Sun (which is The PRIMARY Body of The Solar System) and The PROGRADE PRIMARY Satellites (excluding Pluto satellites – treating Pluto as a Trans-Neptunian Object, rather than as a planet) rotate altogether a total of 3.6539 rotations.

DEMONSTRATION. The Lunar Day = 1.0350501 Earth days.

The PROGRADE PRIMARY Satellites of The Solar System are as follows:- Mercury and Venus have no satellites. Earth’s PRIMARY Satellite is The Moon. Mars’ PRIMARY Satellite is Phobos. Jupiter’s PRIMARY Satellite is Ganymede. Saturn’s PRIMARY Satellite is Titan. Uranus’ PRIMARY Satellite is Titania. Neptune’s PRIMARY Satellite is Triton, but this satellite has retrograde revolution, and is therefore excluded. In this present example, we will consider Pluto to be a Trans-Neptunian Object, rather than a planet, so it is excluded from the calculation.

The rotation periods of The Sun and The PROGRADE PRIMARY Satellites (expressed in Earth days) are as follows:- The Sun 24.66225 and The Moon 27.321661 and Phobos 0.31891023 and Ganymede 7.15455296 and Titan 15.94542068 and Titania 8.7058703. (Note:- The rotation periods of these satellites are the same as their orbital periods, due to “tidal locking”.)

The lunar day is calculated in the following manner:-

Earth’s (sidereal) rotation period = 0.9972697 Earth (solar) days; and the Moon’s orbital period = 27.321661 Earth days.

[(1 ÷ 0.9972697) minus (1 ÷ 27.321661)] = 1.0350501 Earth days.

The general calculation is carried out in the following manner:-

(1.0350501 ÷ 24.66225) + (1.0350501 ÷ 27.321661) + (1.0350501 ÷ 0.31891023) + (1.0350501 ÷ 7.15455296) + (1.0350501 ÷ 15.94542068) + (1.0350501 ÷ 8.7058703) = 3.6539 rotations.

To verify satellites sizes, and to verify that Triton has retrograde revolution (See the letter R directly preceding Triton’s orbital period in the table), and to verify Earth’s rotation period, and to verify that Mercury and Venus have no satellites, and to verify The Sun’s rotation period, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of the above mentioned satellites, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 57. A PRIMARY Satellite is the LARGEST Satellite for a particular planet.

During One Mars Solar Day, The Sun (which is The PRIMARY Body of The Solar System) and The PROGRADE PRIMARY Satellites of The Solar System rotate altogether a total of 3.654122 rotations.

DEMONSTRATION. Here is a list of The PRIMARY Satellites of The Solar System:- The Sun (which is The PRIMARY Body of The Solar System). Mercury and Venus have no satellites. Earth’s PRIMARY Satellite is The Moon. Mars’ PRIMARY Satellite is Phobos. Jupiter’s PRIMARY Satellite is Ganymede. Saturn’s PRIMARY Satellite is Titan. Uranus’ PRIMARY Satellite is Titania. Neptune’s PRIMARY Satellite is Triton, but this satellite has retrograde revolution, and is therefore excluded. In this example, we will consider Pluto to be a binary planet – Pluto/Charon, whose PRIMARY Satellite is Hydra.  

The rotation periods of The Sun and The PROGRADE PRIMARY Satellites (expressed in Earth days) are as follows:- The Sun 24.66225 and The Moon 27.321661 and Phobos 0.31891023 and Ganymede 7.15455296 and Titan 15.94542068 and Titania 8.7058703 and Hydra 38.20177 (Note:- Hydra’s rotation period is not precisely known, but we will use Hydra’s orbital period to stand proxy for its rotation period.) . (Note:- The rotation periods of these satellites are the same as their orbital periods, due to “tidal locking”.)

The Mars solar day = 1.02749 Earth days.

The Mars Solar Day is calculated in the following manner:-

Mars’ (sidereal) rotation period = 1.02596 Earth days; and Mars’ orbital period = 686.98 Earth days.

[(1 ÷ 1.02596) minus (1 ÷ 686.98)] = 1.02749 Earth days.

The general calculation is carried out in the following manner:-

 (1.02749 ÷ 24.66225) + (1.02749 ÷ 27.321661) + (1.02749 ÷ 0.31891023) + (1.02749 ÷ 7.15455296) + (1.02749 ÷ 15.94542068) + (1.02749 ÷ 8.7058703) + (1.02749 ÷ 38.20177) = 3.654122 rotations.

To verify satellite sizes, and to verify the Sun’s rotation period, and to verify the orbital and rotation periods of Mars, and to verify that Triton has retrograde revolution (See the letter R directly preceding Triton’s orbital period in the table), click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of the above mentioned satellites (except for Charon), click on the following link:-

www.solarsystemtimeperiods.com/large-sats

To verify Charon’s orbital period, click on the following link:-

www.solarsystemtimeperiods.com/pluto-sats

EXAMPLE 58. A PRIMARY Satellite is the LARGEST Satellite for a particular planet.

During FOUR Sun rotation periods, The Sun (which is The PRIMARY Body of The Solar System) and The PRIMARY Satellites of The Solar System (excluding Pluto satellites – treating Pluto as a Trans-Neptunian Object, rather than as a planet) rotate altogether a total of 365.0345 rotations.

DEMONSTRATION. Here is a list of The PRIMARY Satellites of The Solar System:- The Sun (which is The PRIMARY Body of The Solar System). Mercury and Venus have no satellites. Earth’s PRIMARY Satellite is The Moon. Mars’ PRIMARY Satellite is Phobos. Jupiter’s PRIMARY Satellite is Ganymede. Saturn’s PRIMARY Satellite is Titan. Uranus’ PRIMARY Satellite is Titania. Neptune’s PRIMARY Satellite is Triton. 

The rotation periods of The Sun and The PRIMARY Satellites of The Solar System (expressed in Earth days) are as follows:- The Sun 24.66225 and The Moon 27.321661 and Phobos 0.31891023 and Ganymede 7.15455296 and Titan 15.94542068 and Titania 8.7058703 and Triton 5.8768441

The calculation is carried out in the following manner:-

4 x [(24.66225 ÷ 24.66225) + (24.66225 ÷ 27.321661) + (24.66225 ÷ 0.31891023) + (24.66225 ÷ 7.15455296) + (24.66225 ÷ 15.94542068) + (24.66225 ÷ 8.7058703) + (24.66225 ÷ 5.8768441)] = 365.0345

To verify satellite sizes, and to verify the Sun’s rotation period, and to verify the orbital and rotation periods of Mars, and to verify that Mercury and Venus have no satellites, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of the above mentioned satellites, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 59. During the precise time period of One Mars orbital period, The Inner Solar System Revolving Bodies revolve altogether a total of 2737.2011 revolutions.

The reciprocal of the number 2737.2011 = 0.0003653

DEMONSTRATION. One Mars orbital Period = 686.98 Earth days. The Inner Solar System Revolving Bodies are:- Mercury, Venus, Earth, Mars, The Moon, and The Two Mars Satellites – Phobos and Deimos. Their orbital periods (expressed in Earth days) are – respectively – 87.9694 and 224.695 and 365.256 and 686.98 and 27.321661 and 0.31891023 and 1.2624407                                                                                                       The calculations are carried out in the following manner:-

(686.98 ÷ 87.9694) + (686.98 ÷ 224.70067) + (686.98 ÷ 365.256) + (686.9782 ÷ 686.98) + (686.98 ÷ 27.321661) + (686.98 ÷ 0.31891023) + (686.98 ÷ 1.2624407) = 2737.21 revolutions. The reciprocal of the number 2737.21 = 0.0003653

To verify The Inner Solar System revolving Bodies (ie:- everything inferior to Jupiter), and to verify the orbital periods of the planets, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of the above mentioned satellites, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 60. This example involves Earth and The Two Inferior Planets (ie:- closer to The Sun than Earth is), Mercury and Venus, comparing them to The Sun. A Solar Day is the time period between two successive risings of The Sun on a fixed horizon of the planet in question. The Mercury Solar Day is 175.939 Earth days, and The Venus Solar Day is 116.7505 Earth days, and The Earth Solar Day is 1 Earth (solar) day. The Sun’s rotation period = 24.66225 Earth days. Now, with the above figures in mind, look at the following graphic.

 The SUM of the Ratios = 36.53

DEMONSTRATION:- To calculate The Mercury Solar Day:- Mercury’s orbital period = 87.9694 Earth days; and Mercury’s rotation period = 58.6462 Earth days.

The Mercury Solar Day =

1 ÷ [(1 ÷ 58.6462) minus (1 ÷ 87.9694)] = 175.938 Earth days.

To calculate the Venus Solar Day:- Venus’ orbital period = 224.695 Earth days; and Venus’ rotation period = 243.0187 Earth days.

1 ÷ [(1 ÷ 224.695) PLUS (1 ÷ 243.0187)] = 116.749 Earth days. (Note:- The PLUS is because Venus has retrograde rotation.)

To verify the orbital and rotation periods of Mercury and Venus, and The Sun’s rotation period, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 61. Mars has two satellites (Phobos and Deimos). The SUM of the SIDEREAL revolution (ie:- orbital) periods of Mars and its two satellites = 688.5596 Earth days. Mars SYNODIC revolution period = 779.9382 Earth days.

4 x (779.9382 minus 688.5596) = 365.51 Earth days.

DEMONSTRATION. Mars SIDEREAL Revolution (ie:- orbital) Period = 686.98 Earth days. Mars’ two satellites are Phobos and Deimos. Phobos SIDEREAL Revolution (ie:- orbital) Period = 0.31891023 Earth days, and Deimos SIDEREAL Revolution (ie:- orbital) Period = 1.2624407 Earth days. Mars SYNODIC Revolution Period = 779.9382 Earth days.

686.98 + 0.31891023 + 1.2624407 = 688.56 Earth days.

4 x (779.9382 minus 688.5596) = 365.51 Earth days.

To verify Mars’ orbital period, and to verify that Mars has just two satellites, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify Mars’ synodic revolution period, click on the following link:-

www.solarsystemtimeperiods.com/planet-synodic

To verify the orbital periods of Phobos and Deimos, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 62. Saturn and Uranus are neighbor planets.

Uranus’ rotation period exceeds Saturn’s rotation period by 0.27432 Earth days.

The reciprocal of the number 0.27432 is 3.65 when expressed to three significant figures.

DEMONSTRATION. Saturn rotation period = 0.44401 Earth days, and Uranus rotation period = 0.71833 Earth days. 

0.71833 minus 0.44401 = 0.27432 Earth days.

The reciprocal of the number 0.27432 is 3.64537 which is 3.65 when expressed to three significant figures.

To verify the rotation periods of Saturn and Uranus, and to verify that they are neighbor planets, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 63. The Moon’s orbital plane oscillates in relation to Earth’s orbital plane with a period of 173 Earth days.                                                 During the time period of 173 Earth days, (neighbor planets) Mercury and Venus revolve altogether a total of 2.7365 revolutions.

The reciprocal of the number 2.7365 is 0.36543 

DEMONSTRATION. Mercury orbital period = 87.9694 Earth days. Venus orbital period = 224.695 Earth days.

(173 ÷ 87.9694) + (173 ÷ 224.695) = 2.7365 revolutions.

The reciprocal of the number 2.7365 is 0.36543 

Note:- The figure 173 Earth days specifies no decimal places. This means that any value between 172.5 and 173.5 Earth days would “count” as being 173 Earth days. If the true value were any greater than 173.4 Earth days, then this “echoing” of 365 would no longer hold good. The probability that this reiteration of the number 365 is invalid (due to “error bars”) is of the order of 0.10 ie:- a 10 percent chance of being invalid, on the basis of the figures currently available.

To verify The Moon’s oscillation period, click on the following link:-

www.solarsystemtimeperiods.com/moon-oscillation

To verify that Mercury and Venus are neighbor planets, and to verify their orbital periods, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 64. Jupiter has Four LARGE Satellites (The “Galilean” Satellites). Inferior to these (ie:- closer to the parent planet) are Jupiter’s Four SMALL “Inner” Satellites (sometimes known as The Four “Amalthean” Satellites).

The SUM of the orbital periods of Jupiter’s Four LARGE “Galilean” Satellites exceeds the SUM of the orbital periods of Jupiter’s Four SMALL “Inner” Satellites by 27.398135 Earth days.

The reciprocal of the number 27.398135 is 0.0365 when expressed to three significant figures.

DEMONSTRATION. Jupiter’s Four LARGE “Galilean” Satellites are:- Io, Europa, Ganymede, and Callisto. Their orbital periods (expressed in Earth days) are – respectively – 1.769137786 and 3.551181041 and 7.15455296 and 16.6890184 The SUM of these four periods = 29.16389 Earth days.

Jupiter’s Four SMALL “Inner” Satellites are:- Metis, Adrastea, Amalthea, and Thebe. Their orbital periods (expressed in Earth days) are – respectively – 0.294780 and 0.298260 and 0.498179 and 0.674536 The SUM of these four periods = 1.765755 Earth days.

29.16389 minus 1.765755 = 27.398135 Earth days.  

The reciprocal of the number 27.398135 is 0.0364988, which is 0.0365 when expressed to three significant figures.

To verify the sizes of Jupiter’s satellites, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of Jupiter’s Four Large Satellites, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

To verify the orbital periods of Jupiter’s Four Small “Inner” Satellites, click on the following link:-

www.solarsystemtimeperiods.com/jup-inner

EXAMPLE 65. The SUM of the orbital periods of The INNERMOST LARGE Satellites of The Four Giant Planets = 0.027383 Earth years.

The reciprocal of the number 0.027383 is 36.5187

DEMONSTRATION. The Four Giant Planets are:- Jupiter, Saturn, Uranus, and Neptune. Jupiter’s INNERMOST LARGE Satellite is Io, and Saturn’s INNERMOST LARGE Satellite is Mimas, and Uranus’ INNERMOST LARGE Satellite is Miranda, and Neptune’s INNERMOST LARGE Satellite is Triton. The orbital periods (expressed in Earth days) of Io, Mimas, Miranda, and Triton are – respectively – 1.769137786 and 0.942421813 and 1.4134840 and 5.8768441 The SUM of these four periods is 10.001887 Earth days.

The Earth year (ie:- Earth’s orbital period) is 365.25636 Earth days. In that case, divide 10.001887 by 365.25636 and the result is 0.027383 Earth years. The reciprocal of the number 0.027383 (ie:- one divided by that number) is 36.5187

To verify planet sizes (ie:- to verify which are the Four Giant Planets), and to verify satellite sizes (ie:- to verify which are the INNERMOST LARGE Satellites of The Four Giant Planets), and to verify Earth’s orbital period, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of the above mentioned satellites, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 66. In this example, we compare The Four Inner Solar System Planets with Jupiter’s Four Large (“Galilean”) Satellites.

The Four Inner Solar System Planets, arranged in ascending orbit order, are:- Mercury, Venus, Earth, and Mars

Jupiter’s Four LARGE “Galilena” Satellites, arranged in ascending orbit order, are:- Io, Europa, Ganymede, and Callisto. 

The masses of The Four Inner Solar System Planets (expressed in units of 1024 kilograms) are as follows:- Mercury 0.3302 and Venus 4.8685 and Earth and Moon together 6.04709 and Mars 0.64185

The masses of Jupiter’s Four Large (“Galilean”) Satellites (expressed in units of 1024 kilograms) are as follows:- Io 0.08918 and Europa 0.04791 and Ganymede 0.14817 and Callisto 0.10766 Now look at the following graphic. 

To clarify:- Sum of ratios plus ratio between the sums = 182.3485 and TWICE this number = 364.697 which is 365 when expressed to 3 significant figures. The Four Inner Solar System Planets and Jupiter’s Four Large Galilean Satellites are the only bodies whose masses are known accurately enough to do a calculation of this kind. 

To verify the masses of planets and satellites, and to verify the order of The Planets and The Large Jupiter Satellites (in ascending order of distance from The Sun, and from Jupiter), click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 67. There are just three Naked-Eye Visible Planets that have LARGE Satellites. These are:- Earth, Jupiter, and Saturn, which have between them altogether Thirteen LARGE Satellites.

TWICE The SUM of the orbital periods of these Thirteen LARGE Satellites exceeds TWICE The SUM of the rotation periods of their parent planets (Earth, Jupiter, and Saturn) by 365.2756 Earth days. 

DEMONSTRATION. The Earth’s LARGE Satellite is The Moon, whose revolution period = 27.321661 Earth days.

Jupiter’s Four LARGE “Galilean” Satellites are:- Io, Europa, Ganymede, and Callisto. Their orbital periods (expressed in Earth days) are – respectively – 1.769137786 and 3.551181041 and 7.15455296 and 16.6890184

Saturn has Eight LARGE Satellites. They are:- (arranged in ascending orbit order) Mimas, Enceladus, Tethys, Dione, Rhea, Titan, Hyperion, and Iapetus. Their orbital periods (expressed in Earth days) are – respectively – 0.942421813 and 1.370217855 and 1.887802160 and 2.736914742 and 4.517500436 and 15.94542068 and 21.2766088 and 79.3301825 

The SUM of these thirteen orbital periods = 184.4926 Earth days. 

 The rotation periods of Earth, Jupiter, and Saturn (expressed in Earth days) are – respectively – 0.997269663 and 0.41353831 and 0.44401

The SUM of these three rotation periods = 1.8548 Earth days.

(184.4926 minus 1.8548) x 2 = 365.2756 Earth days.

(Note:- Planets further out than Saturn are “telescopic” planets, and not visible to the naked eye.)

To verify The Naked-Eye-Visible Planets, click on the following link:-

www.solarsystemtimeperiods.com/naked-eye

To verify satellites sizes, and to verify planetary rotation periods, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of the above mentioned satellites, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 68. Earth and Mars are neighbor planets.

TWICE The SUM of the Synodic Rotation Periods of Earth and Mars is equal to 36.4948 Sun oscillations, which is 36.5 Sun oscillations, when expressed to three significant figures. 

DEMONSTRATION. Earth and Mars are neighbor planets.

The Sun’s Oscillation Period = 0.111111111 Earth days. 

Mars synodic rotation period = 1.027491204 Earth days and Earth synodic rotation period = 1 Earth day. TWICE The SUM of these two periods = 4.0549824 Earth days, which is (4.0549824 ÷ 0.11111111) = 36.4948 Sun oscillations, which is 36.5 Sun oscillations, when expressed to three significant figures.  

Mars’ synodic rotation period is calculated in the following manner:-

Mars’ orbital period = 686.98 Earth days; and Mars’(sidereal) rotation period = 1.02596 Earth days.

Mars’ synodic rotation period =

1 ÷ [(1 ÷ 1.02596) minus (1 ÷ 686.98)] = 1.02749 Earth days.

To verify the orbital and rotation periods of Mars, and to verify that Earth and Mars are neighbor planets, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

To verify The Sun’s Oscillation Period, click on the following link:-

www.solarsystemtimeperiods.com/sun-oscillation

EXAMPLE 69. The SUM of the rotation periods of The Four Inner Solar System Planets is equal to 2733.19 Sun oscillations. The reciprocal of the number 2733.19 is 0.0003659

DEMONSTRATION. The Sun’s Oscillation Period = 0.111111111 Earth days.

The Four Inner Solar System Planets are:- Mercury, Venus, Earth, and Mars. Their rotation periods (expressed in Earth days) are – respectively – 58.6462 and 243.0187 and 0.9972697 and 1.02596

The SUM of these four periods = 303.6881 Earth days, which is equal to (303.6881 ÷ 0.111111111) = 2733.19 Sun oscillations. The reciprocal of the number 2733.19 is 0.0003659

To verify The Sun’s Oscillation Period, click on the following link:-

www.solarsystemtimeperiods.com/sun-oscillation

To verify the rotation periods of The Four Inner Solar System Planets, and to verify which planets are The Four Inner Solar System Planets (ie:- everything inferior to Jupiter), click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 70. ONE THIRD of The SUM of the orbital periods of Pluto’s Five Satellites is equal to 365.3183 Sun oscillations.

DEMONSTRATION. Pluto’s Five Satellites, together with their orbital periods (expressed in Earth days) are:- Charon 6.387230 and Styx 20.16155 and Nix 24.85463 and Kerberos 32.16756 and Hydra 38.20177

The SUM of these five periods = 121.77274 Earth days.

The Sun’s oscillation period = 0.111111 Earth days.

(121.77274 ÷ 0.11111) ÷ 3 = 365.3182

To verify The Sun’s Oscillation Period, click on the following link:-

www.solarsystemtimeperiods.com/sun-oscillation

To verify the orbital periods of The Pluto Satellites, click on the following link:-

www.solarsystemtimeperiods.com/pluto-sats

EXAMPLE 71. The Solar System is full of apparently “meaningful” coincidences, ie:- coincidences that are apparently deliberately configured for OUR attention, so as to act as “deliberate clues” to the fact that The Solar System is artificially created, rather than naturally formed. Here is just one of these numerical coincidences – but the coincidence is even “better” than at first appears.

(A). The Sun’s diameter is 109.237 Earth diameters.

(B). Earth aphelion distance to The Sun is 109.27 Sun diameters.

This, in itself is remarkable, and appears to be a deliberately configured “coincidence”. However, there is a third related fact.

(C). The Mean Earth/Moon distance (ie:- The Moon’s semi-major axis) is 110.64 Moon diameters.

A remarkable series of numerical coincidences. However, it is even better than first appearances would suggest.

The arithmetic mean of these three numbers is 109.71567

ONE THIRD of The Number 109.71567 is 36.572  

DEMONSTRATION.

(A). The Sun’s radius = 695950 kilometers and Earth radius = 6371.01 kilometers. 

695950 ÷ 6371.01 = 109.237

(Note:- Diameter = Radius x 2)

(B). Sun radius = 695950 kilometers, and Sun diameter = 695950 x 2 = 1391900 kilometers.

Earth Aphelion Distance to The Sun is 152100000 kilometers (expressed to four significant figures). 

152100000 ÷ 1391900 = 109.27 Sun diameters.

The formula for Aphelion distance is as follows:-

Aphelion distance = [semi-major axis x (1 – e2)] ÷ (1 – e) 

Where e is eccentricity, and semi-major axis means the mean distance (of Earth) from The Sun, which is 149.598 x 106 kilometers. Earth’s eccentricity e = 0.0167 

Aphelion distance of Earth from the Sun =

[149.598 x 106 x (1 –  0.01672)] ÷ (1 – 0.0167) = 152100000 kilometers (expressed to four significant figures).   

(C). Mean Earth/Moon distance = Moon semi-major axis = 0.38440 x 106 kilometers.

Moon radius = 1737.1 kilometers, and Moon diameter = 1737.1 x 2 = 3474.2 kilometers.

(0.38440 x 106 ) ÷ 3474.2 = 110.64

The arithmetic mean of the three numbers 109.237 and 109.27 and 110.64 = The SUM of these three numbers divided by three = 109.71567

One third of the number 109.71567 is 36.572

To verify The Earth’s semi-major axis, and to verify Earth’s radius, and The Moon’s radius, and the Sun’s radius, and the semi-major axes of Earth and Moon (column a 106 km), and Earth’s orbital eccentricity (column e), click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 72. Earth and Mars are neighbor planets.

There are 1339.1945 Mars sidereal days in Two Mars Years. There are 366.256 Earth sidereal days in One Earth Year.

1339.1949 ÷ 366.256 = 3.6564

DEMONSTRATION. The Mars Year = Mars Orbital Period, which is 686.98 Earth days. The Mars Sidereal Day = Mars sidereal rotation period = 1.02596 Earth days. 

There are (686.98 ÷ 1.02596) x 2 = 1339.1945 Mars sidereal days in Two Mars Years.

The Earth Year is Earth’s orbital period. There are 365.256 Earth solar days in One Earth Year. There are (365.256 ÷ 0.9972697) = 366.256 Earth sidereal days in One Earth Year. (One Earth sidereal day, ie:- Earth’s sidereal rotation period = 0.9972697 Earth solar days.

1339.1949 ÷ 366.25636 = 3.6564

To verify the orbital and rotation periods of Earth and Mars, and to verify that they are neighbor planets, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 73. The SUM of The Moon’s Nodal Revolution Period + The Moon’s Perigeal Revolution Period exceeds The Moon’s revolution period (orbital period) by 27.3862 Earth Years.

The reciprocal of the number 27.3862 is 0.036515 

DEMONSTRATION. The Earth Year (ie:- Earth’s orbital period) is 365.256 Earth days. The Moon’s Node (ie:- the point where The Moon’s orbital plane intersects The Earth’s orbital plane) revolves (round The Earth – retrogradely) with a period of 6798.375113 Earth days. The Moon’s Perigee (ie:- the point of closest approach of The Moon to Earth in The Moon’s elliptical orbit) revolves (round The Earth – progradely) with a period of 3231.492837 Earth days. The Moon’s orbital period is the length of time taken for The Moon to complete one 360 degree revolution round The Earth. The Moon’s orbital period = 27.321661 Earth days.

(6798.375113 + 3231.492837) minus 27.32166 = 10002.5463 Earth days, which is equal to 10002.5463 ÷ 365.256 = 27.385 Earth Years.

The reciprocal of the number 27.385 is 0.036516 

To verify The Earth (sidereal) Year (which is Earth’s orbital period) – and The Moon’s orbital period, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

To verify the values of The Moon’s Nodal Revolution Period and The Moon’s Perigeal Revolution Period, go to the following TWO  web pages:-

www.solarsystemtimeperiods.com/lunar-node

www.solarsystemtimeperiods.com/lunar-perigee

EXAMPLE 74.  The SUM of The Moon’s Perigeal revolution period (3231.492 Earth days) and The Moon’s Nodal revolution period (6798.375 Earth days) = 10029.87 Earth days. During this time period, the number of revolutions of The Moon exceeds the number of revolutions of The Moon’s Node by 365.6 revolutions.                                                  (Moon's revolution period = 27.32166 Earth days.) 

DEMONSTRATION. During the time period of 10029.87 Earth days, The Moon revolves (10029.87 ÷ 27.321661) = 367.1032 revolutions.

During the time period of 10029.87 Earth days, The Moon’s Node revolves (10029.87 ÷ 6798.375) = 1.4753 revolutions.

367.1032 minus 1.4753 = 365.6 revolutions.

To verify The Moon’s orbital period, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

To verify the values of The Moon’s Nodal Revolution Period and The Moon’s Perigeal Revolution Period, go to the following TWO  web pages:-

www.solarsystemtimeperiods.com/lunar-node

www.solarsystemtimeperiods.com/lunar-perigee

EXAMPLE 75. Mercury and Mars are, respectively, The INNERMOST and The OUTERMOST of The Inner Solar System Planets.

Mercury synodic revolution period, as viewed from Mars, exceeds Mars synodic rotation period, as viewed from Mercury by 0.2734 Earth Years.

The reciprocal of the number 0.2734 is 3.6577 

DEMONSTRATION. There are 365.25636 Earth days in One Earth Year 

Mercury orbital period = 87.9694 Earth days, and Mars orbital period = 686.98 Earth days, and Mars sidereal rotation period = 1.02596 Earth days.

Mercury synodic revolution period, as viewed from Mars =                

1 ÷ [(1 ÷ 87.9694) minus (1 ÷ 686.98)] = 100.888 Earth days.

Mars synodic rotation period as viewed from Mercury =

1 ÷ [(1 ÷ 1.02596) minus (1 ÷ 87.9694)] = 1.03807 Earth days.

(100.888  minus 1.03807) ÷ 365.256 = 0.2734 Earth years.

The reciprocal of the number 0.2734 is 3.658 

To verify The Earth sidereal year (which is Earth’s orbital period) – the orbital periods of Mercury and Mars – and Mars sidereal rotation period, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 76. The LARGE PROGRADE Revolving Inner Solar System Bodies (apart from Earth itself) are:- Mercury (NOT Venus, because Venus has retrograde rotation), The Moon, and Mars (NOT The Two Mars Satellites, which are very small)

Mercury synodic revolution period = 115.8774 Earth days.  

The Moon synodic revolution period = 29.5305882 Earth days.  

Mars synodic revolution period = 779.9382 Earth days.  

The Earth (sidereal) year = 365.25636 Earth days.

The Earth year exceeds Mercury synodic revolution period by 249.3790 Earth days. 

The Earth year exceeds The Moon’s synodic revolution period by 335.7258 Earth days. 

Mars synodic revolution period exceeds The Earth year by 414.6818 Earth days.

The SUM of these three periods = 999.7866 Earth days, which is equal to (999.7866 ÷ 365.25636) = 2.73722 Earth years. The reciprocal of this number = 1 ÷ 2.73722 = 0.36533

To verify the synodic revolution periods of Mercury, The Moon, and Mars, click on the following link:- 

www.solarsystemtimeperiods.com/planet-synodic

 To verify The Earth (sidereal) year (which is Earth’s orbital period), click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 77. Each planet has four periods, ie:- orbital period, rotation period, synodic revolution period, and synodic rotation period.

The Inner Solar System has three planets, apart from Earth. These are:- Mercury, Venus, and Mars.

The SUM of the four periods of each of these planets = 2998.8937 Earth days, ie:- an average of 999.6312 Earth days for each planet, which is 2.73679 Earth years.

The reciprocal of the number 2.73679 is 0.36539 

DEMONSTRATION. The orbital periods of Mercury, Venus, and Mars (expressed in Earth days) are:- Mercury 87.9694 and Venus 224.695 and Mars 686.98

The rotation periods of Mercury, Venus, and Mars (expressed in Earth days) are:- Mercury 58.6462 and Venus 243.0187 and Mars 1.02596

The synodic revolution periods of Mercury, Venus, and Mars (expressed in Earth days) are:- Mercury 115.8774 and Venus 583.9205 and Mars 779.9382

The synodic rotation periods of Mercury, Venus, and Mars (expressed in Earth days) are:- Mercury 69.8636 and Venus 145.9276 and Mars 1.027491204

The SUM of these sixteen periods is 2998.8937 Earth days.

This is an average of (2998.8937 ÷ 3) = 999.6312 Earth days.

The Earth (sidereal) year (ie:- Earth’s orbital period) = 365.25636 Earth days.

(999.6312 ÷ 365.25636) = 2.73679 Earth years.

1 ÷ 2.73679 = 0.36539

The calculation of the synodic rotation periods of Mercury, Venus, and Mars are carried out in the following manner:-

Earth’s orbital period = 354.256 Earth days.

Mercury’s synodic rotation period =

1 ÷ [(1 ÷ 58.6462) minus (1 ÷ 365.256)] = 69.8636 Earth days.

Venus’ synodic rotation period =

1 ÷ [(1 ÷ 243.0187) PLUS (1 ÷ 365.256)] = 145.9276 Earth days. (Note:- The PLUS is because Venus has retrograde rotation.)

Mars’ synodic rotation period =

1 ÷ [(1 ÷ 1.02596) minus (1 ÷ 686.98)] = 1.02749 Earth days.

To verify the Inner Solar System Planets (ie:- everything inferior to Jupiter), and to verify planet orbital periods and planet (sidereal) rotation periods, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

To verify the synodic revolution periods of the planets, click on the following link:-

www.solarsystemtimeperiods.com/planet-synodic

EXAMPLE 78. Earth and Mars are neighbor planets.

During One Lunar Node Revolution Period, Mars and its two satellites rotate altogether a total of 33328.98 rotations. 

TWICE The SQUARE ROOT of this number = 365.1245

DEMONSTRATION.

The Lunar Node Revolution Period = 6798.375113 Earth days.

The rotation periods of Mars and its two satellites (expressed in Earth days) are:- Mars 1.02596 and Phobos 0.31891023 and Deimos 1.2624407

(6798.375113 ÷ 1.02596) + (6798.375113 ÷ 0.31891023) + (6798.375113 ÷ 1.2624407) = 33328.98 rotations.

TWICE The SQUARE ROOT of this number = 365.1245

To verify the values of The Moon’s Nodal Revolution Period, go to the following TWO web pages:-

www.solarsystemtimeperiods.com/lunar-node

www.solarsystemtimeperiods.com/lunar-perigee

To verify Mars’ rotation period, and to verify that Mars has just two satellites, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

To verify the rotation periods of Phobos and Deimos (which are equal to their orbital periods, due to “tidal locking”), click on the following link:- 

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 79. Venus and Mars are Earth’s Two Neighbor Planets.                                            Mars synodic revolution period, as viewed from Venus is 333.91 Earth days.

TWICE The SQUARE ROOT of this number = 36.5470      

DEMONSTRATION. Mars orbital period = 686.98 Earth days.

Venus orbital period = 224.695 Earth days.                                                                  Mars synodic revolution period, as viewed from Venus =

1 ÷ [(1 ÷ 224.695) minus (1 ÷ 686.98)] = 333.91 Earth days.

TWICE The SQUARE ROOT of this number = 36.546

To verify the orbital periods of Mars and Venus, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 80. There are Four inner Solar System Planets:- Mercury, Venus, Earth, and Mars.

Mars synodic revolution period, as viewed from Mercury is 100.8882 Earth days; and Mars synodic revolution period, as viewed from Venus is 333.9216 Earth days; and Mars synodic revolution period, as viewed from Earth is 779.9382 Earth days.

The SUM of these three periods = 1214.748 Earth days, which is equal to 3.32574 Earth years. TWICE The SQUARE ROOT of this number = 3.65 when expressed to three significant figures.

DEMONSTRATION. The orbital periods of Mercury, Venus, Earth, and Mars (expressed in Earth days) are:- Mercury 87.9694 and Venus 224.695 and Earth 365.256 and Mars 686.98

Mars synodic revolution period –---------------  as viewed from Mercury = 1 ÷ [(1 ÷ 87.9694) minus (1 ÷ 686.98)] = 100.888 Earth days.

Mars synodic revolution period -------------------- as viewed from Venus = 1 ÷ [(1 ÷ 224.695) minus (1 ÷ 686.98)] = 333.909 Earth days.

Mars synodic revolution period --------------------- as viewed from Earth = 1 ÷ [(1 ÷ 365.256) minus (1 ÷ 686.98)] = 779.934 Earth days.

100.888 + 333.909 + 779.934 = 1214.73 Earth days.

1214.73 ÷ 365.256 = 3.3257 Earth years. (The Earth year = 365.256 Earth days.); and TWICE The SQUARE ROOT of the number 3.3257 = 3.6473 which is 3.65 when expressed to three significant figures.

To verify the orbital periods of the above mentioned planets, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

EXAMPLE 81. The SUM of the orbital periods of The Four LARGE RETROGRAXDE Bodies = 333.333 Earth years. TWICE The SQUARE ROOT of this number = 36.5148

DEMONSTRATION. The Four LARGE RETROGRADE Bodies of The Solar System are:- Venus, Uranus, Pluto (all with retrograde rotation), and (Neptune satellite) Triton (which has retrograde revolution round Neptune). Here are the orbital periods of these four bodies (expressed in Earth days):- Venus 243.0187 and Uranus 30717.682 and Pluto 90803.66 and Triton 5.8768441 The SUM of these four periods = 121751.9195 Earth days.

The Earth (sidereal) year (ie:- Earth’s orbital period) = 365.25636 Earth days.

121751.9195 ÷ 365.25636 = 333.333 Earth years. TWICE The SQUARE ROOT of this number = 36.5148

To verify planet and satellite sizes, and to verify planet orbital periods, and to verify that Venus, Uranus, and Pluto have retrograde rotation (See the letter R directly preceding their rotation period), and to verify that Triton has retrograde revolution (See the letter R directly preceding the orbital period), click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify Triton’s orbital period, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 82. The Four Giant planets are divided into two portions, ie:- Jupiter and Saturn, The Two Gas Giants – and Uranus and Neptune, The Two Ice Giants.

Here are the semi-major axes (ie:- the distance to The Sun of each planet) of these four planets (expressed in units of 106 kilometers):-

TWICE The SQUARE ROOT of the number 3.3368 = 3.65338

and

TWICE The SQUARE ROOT of the number 3.3313 = 3.6504

To verify planet sizes, and order of the planets (ie:- ascending orbit order from The Sun), and to verify planet semi-major axes, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 83. Jupiter and Saturn are neighbor planets.

Jupiter’s mass is 3.3399 times Saturn’s mass.

TWICE The SQUARE ROOT of the number 3.3399 = 3.65508

DEMONSTRATION. Here are the masses of Jupiter and Saturn (expressed in units of 1024 kilograms):- Jupiter 1898.6 and Saturn 568.46

1898.6 ÷ 568.46 = 3.3399

TWICE The SQUARE ROOT of the number 3.3399 = 3.65508

To verify the masses of these two planets, and to verify that they are neighbor planets, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

EXAMPLE 84. The SUM of Mercury’s Four Periods = 333.2663 Earth sidereal days.

TWICE The SQUARE ROOT of the number 333.2663 = 36.511

DEMONSTRATION.

Every planet has four periods, ie:- Orbital Period, Rotation Period, Synodic Revolution Period, and Synodic Rotation Period.

Here are Mercury’s four periods, expressed in Earth days:- Orbital period 87.9694 and Rotation period 58.6462 and Synodic revolution period 115.8774 and Synodic rotation period 69.8636

The SUM of these four periods = 332.3566 Earth (solar) days.

The Earth sidereal day = 0.9972697 Earth (solar) days.

332.3566 ÷ 0.9972697 = 333.2665 Earth sidereal days. 

TWICE The SQUARE ROOT of the number 333.2665 = 36.511

Earth’s orbital period = 365.256 Earth days.

Mercury’s synodic rotation period =

1 ÷ [(1 ÷ 58.6462) minus (1 ÷ 365.256)] = 69.8636 Earth days.

To verify the orbital and (sidereal) rotation periods of Mercury, and to verify Earth’s orbital period, click on the following link:-

www.solarsystemtimeperiods.com/periods-inner

To verify Mercury’s synodic revolution period, click on the following link:-

www.solarsystemtimeperiods.com/planet-synodic

EXAMPLE 85.

Uranus has Five LARGE Satellites, but one of these satellites – Miranda – is only (approximately) one twentieth the mass of any of the other four satellites, and it can therefore be considered as only an intermediate sized satellite.

Here are the names and orbital periods of Uranus’ Four (really) LARGE Satellites, expressed in Earth days:- Ariel 2.52037932 and Umbriel 4.1441765 and Titania 8.7058703 and Oberon 13.4632423

With these periods in mind, now look at the following graphic:-

TWICE The SQUARE ROOT of the number 3.32642 = 3.64769

Which is 3.65 when expressed to three significant figures.

and also:-

The SUM of these two ratios = 3.6472, ie:- 3.65 when expressed to three significant figures.

To verify satellite masses, and to verify the order of The Large Uranus satellites, click on the following link:-

www.solarsystemtimeperiods.com/planet-orbital

To verify the orbital periods of the above mentioned satellites, click on the following link:-

www.solarsystemtimeperiods.com/large-sats

EXAMPLE 86. The orbital periods of The Three Inner Solar System Planets (apart from Earth itself), expressed in Earth days are:- Mercury 87.9694 and Venus 224.695 and Mars 686.98

The arithmetic mean of these three periods = 333.2148 Earth days.

TWICE The SQUARE ROOT of the number 333.2148 = 36.508

DEMONSTRATION.

(87.9694 + 224.695 + 686.98) ÷ 3 = 333.2148 Earth days.

TWICE The SQUARE ROOT of the number 333.2148 = 36.508

To verify the orbital periods of The Inner Solar System Planets, and to verify which planets are the Inner Solar System Planets (ie:- everything inferior to Jupiter), click on the following link:-

www.solarsystemtimeperiods.com/periods-inner