# Planetary Spin Calculator

This web page and the maths associated with this calculator are copyright protected by

Keith Dixon-Roche {© 14/03/17}

## Terminology

1) Gravitational energy is the same as potential energy. As this web page deals with Newtonian mechanics only the term gravitational energy will be deployed.

2) Suffixes apply as follows:

'ₒ' = satellite orbit; '₁' = force-centre; '₂' = satellite; '₃' = secondary satellite

3) Refer to our Planetary Spin web page on for a detailed breakdown of the maths used in this calculator

## Planetary System (Fig 1)

Fig 1. Planetary System

The body being investigated is a satellite

(e.g. as sun, planet or moon)

Its force-centre is a galactic star, solar sun or planet

Its secondary satellite(s) is a planet or moon

## What Is Planetary Spin?

It is the spin rate of a force-centre or satellite. It has both [velocity] and [rotational] direction.

Whilst it is not part of Newtonian mechanics, the energy that generates spin is found in Isaac Newton's laws of motion.

## What Are Spin Energies?

Spin energy is the rotational energy inducing spin in a satellite.

### Orbital Energy (Eₒ)

Assuming no other [energy] influences, gravitational energy will ensure that a satellite orbiting a force-centre will always present the same face to its force-centre causing it to spin at angular velocity; ωₒ

### Force-Centre Energy (E₁)

This is the rotational energy induced in a satellite due to its own orbital kinetic energy, which varies with its distance from its force-centre according to Isaac Newton's inverse-square relationship. This energy will cause a satellite to rotate in the opposite direction to ωₒ

This input value for Spin comes from the Newton output data for the satellite

### Force-Centre Energy (E₃)

This is the rotational energy induced in a satellite by its own secondary satellites. This energy will cause a satellite to rotate in the same direction as ωₒ. Its rotational direction is defined by the relative angle between the satellite's orbital plane and that of its secondary satellites.

This input value for Spin comes from the [sum of] Newton output data for the satellite's own secondary satellite(s)

## Moons or No Moons!

Apart from Mars, all the planets in our solar system possess spin energy from secondary satellite(s) at least a thousand times greater than that induced by the satellite's own kinetic energy. This is why Venus rotates in the opposite direction to the other planets; it has no secondary satellites to overcome the influence of E₁

Mars is a very special case because it appears to be hollow. The influence of its secondary satellites is only 2.5 times greater than E₁. Phobos actually rotates faster than its force-centre (Mars), which must be due to Mar's unusually low 'Δ' value

## Example Calculation 1

Mercury & Venus have no moons, so we shall calculate the spin in both planets (Fig 2 shows the calculation for Venus)

Property | units | Mercury | Venus |

Newton Calculation | |||

m₁ | kg | 1.9885E+30 | 1.9885E+30 |

m₂ | kg | 3.30110E+23 | 4.86737E+24 |

r₂ | m | 2439700 | 6051800 |

T₂ | s | 7600522 | 19413907 |

R̂ | m | 4.60012E+10 | 1.07477E+11 |

E₁ | J | 5.76563E+23 | 2.51533E+23 |

Spin Calculation | |||

m₂ | kg | 3.30110E+23 | 4.86737E+24 |

r₂ | m | 2439700 | 6051800 |

Δ₂ | 0.812863 | 0.68118 | |

T₂ | s | 7600522 | 19413907 |

θ | ° | 0 (no moons) | 0 (no moons) |

E₁ | J | 5.76563E+23 | 2.51533E+23 |

E₃ | J | 0 (no moons) | 0 (no moons) |

J₂ | kg.m² | 5.19309E+35 | 5.19309E+35 |

E₂ | J | 3.99116E+23 | -1.48128E+24 |

ω₂ | ᶜ/s | 1.2398E-06 | -2.99233E-07 |

Planetary spin in planets with no moons |

Fig 2. Angular Velocity Calculation for the Planet Venus

## Example Calculation 2

Neptune is a little more complicated as it has many moons, three of which orbit in the opposite direction (retrograde) to Neptune's own orbit (prograde)

Neptune's Moon | E₃ (J) |

Naiad | -1.41526E+25 |

Thalassa | -2.72711E+25 |

Despina | -1.30039E+26 |

Galatea | -2.20708E+26 |

Larissa | -2.30921E+26 |

S/2004 N 1 | -1.63551E+23 |

Proteus | -1.44926E+27 |

Triton | -2.05897E+29 x -1 |

Nereid | 1.12539E+26 |

Halimede | 2.56883E+21 |

Laomedeia | 1.15675E+22 |

Sao | 4.34646E+21 |

Neso | 1.84075E+22 x -1 |

Psamathe | 1.80844E+21 x -1 |

ΣE₃ | 2.0393706E+29 # |

Planetary spin energies calculated using the Newton calculator |

Property | units | Neptune | |

Newton Calculation | |||

m₁ | kg | 1.9885E+30 | |

m₂ | kg | 1.024134E+26 | |

r₂ | m | 24622000 | |

T₂ | s | 5200329600 | |

R̂ | m | 4444450000000 | |

E₁ | J | 2.09361E+21 | |

Spin Calculation | |||

m₂ | kg | 1.0241340E+26 | |

r₂ | m | 24622000 | |

Δ₂ | 0.0374096 | ||

T₂ | s | 5200329600 | |

θ | ° | 23.8 | |

E₁ | J | 2.09361E+21 | |

E₃ | J | 2.039371E+29 # | |

J₂ | kg.m² | 3.4756E+37 | |

E₂ | J | 2.03937E+29 | |

ω₂ | ᶜ/s | 1.0833E-04 | |

Planetary spin in the planet Neptune |

Fig 3. Angular Velocity Calculation for the Planet Neptune

## Planetary Spin Calculator - Technical Help

### Units

You may use whatever units you like (except: time must be in seconds and angles must be in degrees), but you will get out whatever you enter, so be consistent.

### Input Data

m₂: mass of the satellite

r₂: volumetric radius of the satellite

Δ₂: delta value of the satellite. If you not know this value but you do know ω₂ for the satellite, you must iterate this value until ω₂ is correct. Then you will have the satellite's Δ value for analysis using Core Pressure theory.

T₂: orbital period of the satellite [seconds]

θ: angle between the orbital plane of the satellite and that of its secondary satellites [degrees]

E₁: the energy induced into the satellite by its force-centre. This value may be obtained from CalQlata's Newton calculator from a calculation of the satellite itself

E₃: the energy induced into the satellite by its secondary satellites. This value may be obtained from CalQlata's Newton calculator from a calculation of its secondary satellites (the sum of all satellites ΣE₃)

### Output Data

J₂: polar moment of inertia of the satellite

E₂: total spin energy generating ω₂

ω₂: angular velocity of the satellite

### Applicability

This calculator can be used for any satellite orbiting a force-centre

### Accuracy

This calculator is as accurate as Newton's own laws of motion

### Further Reading

You will find further reading on this subject in reference publications^{(55, 61, 62 & 64)}

**Go to the calculator**