Start, Maximum and Stop Influenced Rotation Distance



Mercury's Stop Rotation Boundary



Venus' Stop Rotation Boundary



Earth's Stop Rotation Boundary



Moon's Stop Rotation Boundary 

Rotation Equations of the Sun, Planets and Moons
Peyman Parsa  November 15, 2004  Updated July, 2012
Published by: Journal of Theoretics Volume 6.5 / Extensive Papers,
Nov. 23, 2004
Clear Dark Sky / AstroDocuments , March. 13, 2005 

Abstract
The rotation rates of spherical celestial bodies such as stars, planets and moons and also the state of synchronization of moons can be calculated with a set of equations.
Computing the rotation rate is possible by applying the five following facts; Mass, Density, Orbit Inclination, Axis Tilt Degree and Distance, plus applying five constants.
Based on the author’s theory of celestial rotation any spherical body has its own original rotation “Free Rotation” plus gained or lost rotations “Influenced Rotation” by the influence of its satellites, parent planet or the Sun. The Influenced Rotation, relative to the mass of the parent body and the orbiting body, starts at a relative far away distance and gradually maximizes until certain nearer distance after which in a relatively short distance it comes to an end jointly with the Free Rotation. In Rotation Equations the Free Rotation and any Influenced Rotation is computed primarily and then by totalling all the rotations the actual rotation rate of the body is determined. The number of equations depends on the number of bodies involved in calculations. 

Introduction
Laws of the rotation rates of the Sun, planets and moons have long been one of the long lasting puzzles in astronomy. Most planets with bigger masses have faster rotations, but Venus with more mass than Mars, and Mercury with more mass than Pluto have extremely slower rotations in comparison. Also, the Sun with its tremendous mass compared to planets, has about 1/66th of the Jupiter’s rotation. Moreover, nearly half of the moons of the entire solar system don’t have tropical rotations and their sidereal rotations are synchronized by their evolutions around their parent planet, and consequently they always reveal nearly the same hemisphere to their parent planet. In addition, Pluto and Charon stop each other’s local rotation, and they always face each other with the same hemisphere. The rotation variation of bodies and their irregularities indicate a very complex structure. Therefore, this complexity demands a set of complex rules to describe rotation’s variations.
Rotation Equations is a series of equations for calculating the rotation of spherical celestial bodies and synchronous bodies based on the assumption that primary rotation "Free Rotation" is originated at the center of the mass at the time of its formation, and it can be increased or decreased by the influence of central or orbiting bodies.

Rotation Properties
Rotating bodies in Rotation Equations are divided in two categories.
Spherical Bodies:
Consists of bodies with adequate mass, roughly above 1.0+8 Kg, in order to become spherical in shape under the influence of gravity. These bodies have their own inner spin “Free Rotation” and they rotate continuously without the influence of any nearby spherical body, if alone in space
Nonspherical Bodies:
Consists of nonspherical and irregular body shapes formed mainly out of the fragments of exploded planets or moons. These bodies depending on their shape might have slower Free Rotations or lack any Free Rotations, and their natural rotations are mainly caused by external forces. Therefore due to the complexity of rotation sources of nonspherical bodies their rotation rates are calculated only if their rotation is synchronized due to tidal forces. 
Free Rotation
The Free Rotation is originated at the center of spherical bodies and partially at the center of nonspherical bodies at the time of formation and is transferred to the outer layers. The Free Rotation of a body is depended on its Mass and Density. A bigger mass results in a slower Free Rotation in contrast with a denser mass that results in a faster Free Rotation. 
Influenced Rotation
All rotating bodies, if close enough to each other can significantly influence each other’s rotation and this gain or loss of rotation is called “Influenced Rotation”. All bodies can either decrease or increase each other’s rotations, and bodies with more mass in certain conditions can totally stop the tropical rotations of smaller bodies. The Sun owes less about 25% of its average surface rotations to four of its planets Jupiter, Saturn, Venus and Earth. Planets relevant to their masses owe most portions of their rotations to the Sun. Moons due to their relatively small masses do not affect the Sun’s rotation but they mostly minimally affect the rotation rate of their parent planets, either reducing or increasing their rotations and if their mass quantity is relatively substantial and they are close enough to their parent planets, as it is in the case of Charon/Pluto, they can stop the local tropical rotation of their parent planet by their tidal forces. Planets on the other hand have large effects on their moons’ rotations either preventing them from tropical rotations or decreasing/increasing their rotations. For example, our moon has no tropical rotations due to the influence of the Earth’s tidal forces and consequently it has a synchronized sidereal rotation and always reveals nearly one side toward the earth. Planets’ effects on each others and on foreign moons and also moons’ effects on each others and on foreign planets are so minimal and variable that can be neglected in Rotation Equations. 
Rotation Factors
The Free Rotation rate or the Influenced Rotation rate of a body depends on several factors. The Free Rotation rate is influenced by mass and density whereas the influenced rotation rate is influenced also by additional factors such as orbit inclination, axis tilt degree and distance to the influencing body. It is important to note that a body’s Free Rotation or Influenced Rotation in Rotation Equations corresponds only to the body’s surface rotation.
1 Mass:
While mass has an increasing effect on the central inner Free Rotation it inversely affects the surface Free Rotation. In fact when comparing bodies with the same density, the more massive bodies have slower surface Free Rotation.
In case of two bodies interacting with each other, the body with a bigger mass has more influence on the rotation of the body with a smaller mass. A bigger mass has the capability of stopping the tropical rotation of the smaller mass by its tidal forces whereas the smaller mass can force the bigger mass to stop its rotation only if it has a minimum of certain amount of mass and right distance. Both bigger and smaller masses can decrease or increase each others’ rotation
2 Density:
Density has deferent effects on Free Rotation and Influenced Rotation. With respect to bodies with similar masses the denser mass results in a faster Free Rotation. In fact, as a body is denser its radius is shrunk and its center is more under pressure to spin faster and consequently the surface Free Rotation rate is increased too. Conversely in a less dense mass the outer layers have less friction in order to be moved easier under the influence of other bodies and consequently it results in boosting the Influenced Rotation.
3 Orbit Inclination:
Every rotating body throws its influencing energy to the space along the extension of its equatorial plane and this energy declines gradually toward both poles. Therefore as an orbiting body is inclined from the direction and vicinity of the central equatorial plane its rotation is less influenced by the central body.
4 Axis Tilt Degree:
As an orbiting body tilts its axis, it begins to change the direction of its rotation, and in proportion to its degree of til its Free Rotation and also its Influenced Rotation decreases. In fact as the axis tilt degree is increased and the body’s direction of rotation is changed, the influenced rotation proportionally cancels the orbiting body’s opposite Free Rotation.
5 Distance:
One of the most important factors of rotation variation is the distance between interacting bodies. There are three critical spots in a distance between two bodies. The distance of these three spots to the central body varies by both bodies’ Mass and the orbit Inclination Degree of the orbiting body.
A: Start Influenced Rotation Distance “”:
This is the maximum distance to the central body at which the Influenced Rotation begins. In fact it is at this point that the Influenced Rotation begins to be added to the Free Rotation.
B: Maximum Influenced Rotation Distance “”:
This is the distance at which the Influenced Rotation reaches its maximum rate. The rotation rate increases linear from the direction of the “Start Influenced Rotation Distance” toward the “Maximum Influenced Rotation Distance”. This is also the start point of the tidal forces which increases sharply toward "Stop Influenced Rotation Distance".
C: Stop Influenced Rotation Distance “”:
This is the point at which tidal force causes all rotations whether Free ones or Influenced ones come to an end. The rotation rate decreases non linear from the “Maximum Influenced Rotation Distance” toward the “Stop Influenced Rotation Distance”. In fact, the total rotation falls sharply as the body is closer to its Stop Rotation Distance.
Start, Maximum and Stop Influenced Rotation Distance

Constants
In order to calculate the rotation rate of a body we need to apply five constants;
Free Rotation Constant “fR”
Start Influenced Rotation Distance Constant “iSt”
Maximum Influenced Rotation Distance Constant “iMa"
Stop Influenced Rotation Distance Constant “iSp”
Maximum Influenced Rotation Constant “iM” “iMs”
Free Rotation Constant is the first constant to be used in calculations as a prerequisite for calculating other facts and it is calculated based on the Sun’s Free Rotation. 
Categories for Computing Rotation Rates
With a set of equations we can calculate the rotation rate of the Sun, planets and moons. The Rotation Equations consist of three categories;
 Planet’s Rotation
 Consisting of equations to compute the Sun’s influence and moon’s influence.
 Moon’s Rotation
Consisting of equations to compute parent planet’s influence and the Sun’s influence.
 Sun’s Rotation
Consisting of equations to compute each planet’s influence plus Earth's moon.

