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Planetary Spin {© 14/03/17}
(a theory for the spin of stars and planets)

This paper, which was released by Keith Dixon-Roche (one of CalQlata's Contributors) on the 7th of April 2017 may well become one of the most important astronomical theories since Isaac Newton's Principia.
Whilst Newton's famous work told us how planets and moons orbited suns and planets, this paper tells us how and why they spin.

It not only accurately predicts all aspects of the spin of every planet and the sun in our solar system, it does so using Newton's own theories and is the only study we (at CalQlata) know on this subject that manifestly works.

Note: All the input data in these calculations has been provided by CalQlata's Solar System Orbits
All the calculations are the sole copyright priority of Keith Dixon-Roche © 2017
Keith Dixon-Roche is also responsible for all the other web pages on this site related to planetary motions
A 'pdf' version of this paper can be found at: Planetary Spin - The Paper

A calculator is now available for Keith Dixon-Roche's planetary spin

Introduction

The purpose of this paper is to answer the following questions:

1) What causes a star or a planet to rotate on its axis?

2) What defines the magnitude and direction of this rotation?

(Refer to Mathematical Symbols & Units for an explanation of the terminology, mathematical symbols and units in this web page)

Conclusions

The calculations in this paper identify the cause of spin in stars, planets and moons in terms of, and according to, Newtonian mechanics

It accurately predicts the angular velocity in all the planets of our solar system along with the earth’s and Mars’ moons and our sun

The reason why Venus spins in the opposite direction to Mercury, for example, is because the sun's influence is greater in Venus than it is in Mercury (neither of which have satellites) and the sun's rotational energy causes its planets to rotate in the opposite direction to their orbital direction.

Planets with satellites (moons) are induced with spin energies so much higher than the induced energy from the sun that this reversal would not materialise.

Planetary spin can also be used to calculate the internal properties of a planet or star, including its magnetic field

Further Work

Is Mars hollow?

The basic solar system of planetary spin

Fig 1. The Basic System

The Basic System

The basic system (Fig 1) comprises;
a force-centre (e.g. a star); and
an orbiting satellite (e.g. a planet); and
a secondary orbiting satellite (e.g. a moon)

Methodology

The following procedure was used to establish the controlling formulas for planetary spin using our solar system for verification:

Isolate and identify the relative angular direction(s) imposed on a planet by its force-centre and its satellite(s) and determine the energy sources responsible for their generation.

It will be assumed that only orbiting bodies and their force-centres can induce spin in each other, which is actually correct as all spin energies can be found from Newtonian mechanics.

Definitions

The bodies are defined in Fig 1 according general understanding. In this paper, however:

1) A force-centre can be galactic, solar or planetary, which may have their own satellites

2) A satellite can be solar, planetary or lunar, which may have their own secondary satellites

Calculations

The polar moment of inertia of any body may be calculated thus:
J = ⅖.m.(Δ.r)²

Spin energy may be calculated thus:
E = ½.J.ω²

The relative angular velocities induced in a planet (or star) are defined below.

The Orbit

ωₒ: the natural angular velocity of a lone planet (with no moons) orbiting a star rotating at the same angular velocity as its planet.

ωₒ = 2π / T
Eₒ = ½.J.ωₒ²

The gravitational energy between the core of a sun and that of its planet will induce spin (ωₒ) in the planet with the same direction and period as that of its sun (e.g. Fig 1; +ve or prograde).
If a sun has only one planet with no moons, they would both have the same angular velocity (ω₁ = ωₒ). Otherwise, the sun and planets would spin at different rates.

This is the starting point for the calculation procedure.

The Force-Centre

ω₁ is angular velocity in a satellite generated by its own orbital kinetic energy and varies with the distance between it and its force-centre according to Isaac Newton's inverse-square relationship

E₁ = δKE . (r/R )²

This energy will causes a planet to rotate in the opposite direction (e.g. Fig 1; -ve or retrograde)

The Satellites

ω₃ is the angular velocity induced in a force-centre by its orbiting satellite(s)

E₃ = Σ(KEᴾ + PEᴬ) . Sign[Cos(θ)]
Σ(KEᴾ + PEᴬ) must be negative before θ is applied

Satellites induce spin throughout the mass of their force-centre in the same direction as their orbit.

If the plane of a satellite's orbit is tilted (θ) greater than 90º relative to the plane of the planet's orbit, or if it is orbiting in the opposite direction to the planet's orbit about its force centre, the energy it induces (E₃) must be multiplied by -1 {i.e. Sign[Cos(θ)] }

The Planet's Angular Velocity

The energy inducing the angular velocity of a planet (E₂) may be calculated thus:

E₂ = E₁ - E₃ - Eₒ
Note: E₃ in the above formula is minus because it is a negative value in Newtonian mechanics

The angular velocity of a planet (ω₂) may be calculated thus:

ω₂ = √[2.E₂ / J₂]

This calculation method predicts all reversed spins; e.g. Venus, Uranus and Pluto

Related Mathematical Relationships

1 + ½e² = π.R ² / Aₒ
In which Aₒ is the area of satellite's orbit as calculated in Newton's Laws of Motion
and R is the average distance of a satellite from its force-centre as calculated in Newton's laws of motion

The orbital energy that defines the angular velocity of a force-centre may be calculated thus:
E = m.Aₒ.ω₂ₒ² / 2.π.(1-½e²)
In which E is the total energy calculated in Newton's Laws of Motion

Calculation Results

The following Table shows the relevant spin energies in some of the bodies in our solar system:

JEₒE₁E₃E₂ω₂
Kg.m²JJJJᶜ/s
Sun3.91229E+461.46587E+165.01045E+32-1.60100E+351.60602E+352.86533E-06
Mercury5.19308E+351.77447E+235.76563E+2303.99116E+231.23980E-06
Venus3.30863E+371.73281E+242.51495E+230-1.48132E+24-2.99237E-07
Earth1.08212E+372.14478E+233.20800E+23-2.87708E+282.87709E+287.29212E-05
Mars1.58326E+318.87109E+161.55612E+22-2.42128E+223.97739E+227.08824E-05
Jupiter1.92586E+392.71288E+232.52842E+26-2.97774E+312.97777E+311.75853E-04
Saturn1.52389E+383.48093E+219.19174E+24-2.04404E+302.04405E+301.63788E-04
Uranus1.38902E+373.90074E+192.80792E+227.11807E+28-7.11807E+28-1.01238E-04
Neptune3.47506E+372.53648E+192.09365E+21-2.03937E+292.03937E+291.08338E-04
Pluto5.48500E+351.76850E+176.27106E+153.55515E+25-3.55515E+25-1.13856E-05
Moon2.73159E+349.67616E+221.93523E+2309.67616E+222.66170E-06
Phobos4.04662E+221.05210E+152.10421E+1501.05210E+152.28033E-04
Deimos4.68802E+187.77805E+091.55566E+1007.77854E+095.76062E-05
Calculated energy values for planetary spin

Claims

Claim 1: The spin in any force-centre or satellite can be calculated using Newtonian mechanics

Claim 2: The spin in a force-centre is induced by its orbiting bodies

Claim 3: A satellite's spin rate will be altered by a force centre rotating at a different angular velocity

Claim 4: Only force-centres and their satellites can influence each other's angular velocity

Mathematical Symbols & Units

A mass orbiting a force-centre will generate a positive kinetic energy (KE) and a negative potential energy (PE) between the force-centre and the orbiting body. The sum of the two is Newton's combined energy (E). Refer to Laws of Motion.

The potential energy (PE) between two or more bodies is also gravitational energy.

'δKE' is the difference between the kinetic energies of a satellite at its perigee and its apogee {J}
i.e. δKE = KEᴾ - KEᴬ

'KEᴾ' is the kinetic energy of a satellite at its perigee {J}
'KEᴬ' is the kinetic energy of a satellite at its apogee {J}

'PEᴬ' is the potential energy between a force centre and its satellite at its apogee {J}

'θ' is the angle of inclination of a satellite's orbital plane relative to its own plane orbital plane {radians}

'E₁' is the spin energy induced in a satellite by its force-centre {J}

'E₂' is the total spin energy in a satellite {kg.m²}

'E₃' is the spin energy induced in a satellite by its secondary satellite(s) {kg.m²}

'Eₒ' is the natural spin energy in a satellite induced by its own orbit {kg.m²}

'ω₁' is the angular velocity induced in a satellite by E₁ {J}

'ω₂' is the total angular energy in a satellite induced by E₂ {kg.m²}

'ω₃' is the total angular energy in a satellite induced by E₃ {kg.m²}

'ωₒ' is the total angular energy in a satellite induced by Eₒ {kg.m²}

'J' is the polar moment of inertia of a body {kg.m²}

'T' is the satellite's orbital period

'r' is the satellite's radius

'R ' is the average orbital distance between the centre's of a satellite and its force-centre

'Δ' radial modifier (factor) for the polar moment of inertia of a rotating body

'₁' refers to the primary force-centre (star)

'₂' refers to the secondary force-centre (planet)

For the purposes of this document, the terms 'rotational' and 'angular' are interchangeable; all such velocities shall be interpreted has having magnitude and direction.

Refer to Laws of Motion for a detailed explanation of Newton's laws of planetary motion and Solar System Orbits for planetary orbit details

Chicken & Egg?

Polar moment of inertia of a planet

Fig 2. Polar Moment of Inertia

It is generally believed that a sun rotates under its own steam pulling its planets around with it. If this were the case, it would need a suitable energy source to do so, moreover, the same claim must also apply to rotating planets. Whilst this claim may (or may not) be made for our sun and even Earth itself, it cannot be made for planets such as Pluto, which is a solid lump of rock and ice with no internal energy source. Moreover, Pluto’s local orbit (Appendix 7) could not be explained by such an internal energy source.

It is therefore claimed (by the author) that as an orbiting body induces far greater rotational energy spin in its force centre than vice-versa and the initiation of a solar system must be due to a force-centre that was not initially rotating being caused to orbit by its orbiting bodies.

Subsequent rotational influence by a force-centre on its satellites will occur once in motion, but is insufficient to generate the energies required to maintain their orbits.

Polar Moment of Inertia (Δ)

The basic formula for the polar moment of inertia (J) of a sphere is:
J = ⅖.m.r²
Where 'm' is the mass of the sphere and 'r' its radius (Fig 2)

However, this formula only applies to a sphere that comprises the same homogeneous material throughout its structure. Planets, however, are anything but homogeneous as gravitational energy generates very high densities at their cores (Fig 3)

'Δ' in the above formula can provide us with an equivalent representative radius of an homogeneous sphere with the same mass and the correct polar moment of inertia (J) as follows:

Eₒ = ½.J.ωₒ²

J = ⅖.m.(Δ.r)²

The calculated variable; ’Δ’ for various solar system bodies is provided in the following Table

Variable density in a planet

Fig 3. Variable Density

BodyΔ
Sun0.318782372247959
Mercury0.812862196423113
Venus0.681180492057101
Earth0.33428172721771
Mars0.00231707805666362
Jupiter0.0227806693989634
Saturn0.014059868482105
Uranus0.0249372276830553
Neptune0.0374067226435373
Pluto8.64241935542982
Moon0.554903433736135
Phobos0.275895222790585
Deimos0.014346539805995
Delta values for solar system bodies

This value (Δ) can be used to establish the construction of a planet, star or moon, as described on our web page for Core Pressure

Mars

The author refers to Mars as a rocky (not an iron) planet because of its perceived density. However its exceptionally low 'Δ' value along with its largest moon (Phobos) orbiting faster than the spin in its force-centre and the planet’s red colour appear to show that it is may well be a hollow iron planet, that has at some time blown all of its ferrite-rich material from its core out onto its surface through its volcano; ‘Olympus Mons’.

Stars, Gas & Ice Planets

As can be seen in Fig 3 the pressure/density of gas increases exponentially under gravity and inversely proportional with radius indicating that the vast majority of the mass in a gas planet must be at its centre.

Relative Densities

As we can only guestimate the structures of our sun or the ice and gas planets, we can only guestimate their polar moments of inertia. To do this, we may use the known values for ωₒ and Eₒ to establish a representative radial modifier 'Δ' (see Polar Moment of Inertia (Δ) above). We can then use 'Δ' to estimate the expected surface density for each planet based upon its average density.

For each 'Δ' to be representative, it must reflect the structure of the planet concerned. A reasonable estimate can be made from the average densities of each planet.

By way of illustration, it is possible to estimate for most planets from their relative densities
'ρˢ = Δ.ρᵅ'
Where: ρᵅ is the average density and ρˢ is the surface density

Using this argument for the planets in our solar system with moons, the surface densities of each are estimated as follows:

Given their respective surface temperatures and despite the unknown nature or composition of each planet’s inner material(s), with the exception of Mars (Appendix 5) and Pluto (Appendix 7), each is representative of its expected surface materials.

Table 1 shows the relevant properties of the bodies in our solar system:

ΔSurface Density (kg/m³)
Earth0.3342817271840.672632
Mars0.0023170789.115572455
Jupiter0.02278066930.21210674
Saturn0.0140598689.660859203
Uranus0.02493722831.68063158
Neptune0.03740672361.26999625
Pluto8.64241935516074.55597
Density of surface materials

Pluto

Pluto's local orbit due to Charon

Fig 4. Pluto's Local Orbit

Pluto's principal moon; Charon, is so large (>12% of the Pluto's mass) it is pulling Pluto into a local orbit (Fig 4) and is the reason why its effective radial modifier (Δ) is greater than 1

Pluto is the only planet in these calculations with a 'Δ' value greater than 1 and the only planet being pulled by its moon into a significant localised orbit, thereby vindicating a value of; 'Δ>1' and the use of this variable in these calculations.

Further Reading

You will find further reading on this subject in reference publications(55, 60, 61, 62, 63 & 64)

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