Derivation of planetary system parameters from Kepler’s Laws, Cassini’s determination of the AU (astronomical unit), Newton’s Laws and Cavendish’s gravitational constant determination.

Kepler first two laws were published in 1609 and the third law in 1619 after toiling 10 years to find this “harmony of the spheres”. Keplers 3rd Law: The semi major axis (a) of the elliptical orbit cubed is equal to the orbital period (P) squared.

a^3=P^2

With this formula the ratios of the planetary distances from the sun could be determined with simple mathematics. However actual distances to the sun had to wait another 53 years until the earth’s distance to the sun had been measured with reasonable accuracy by Giovanni Cassini and Jean Richer.

Planet | Period in Earth Days | Period Squared | a = cube root of period squared | Ratio |

Mercury | 88 | 7744 | 19.78 | 0.39 |

Venus | 225 | 50,625 | 36.99 | 0.72 |

Earth | 365 | 133,225 | 51.07 | 1 |

Mars | 687 | 471,969 | 77.85 | 1.52 |

Jupiter | 4333 | 18,774,889 | 275.78 | 5.2 |

Saturn | 10756 | 115,756,081 | 487.35 | 9.54 |

In 1672 Giovanni Cassini and Jean Richer made the first somewhat accurate and well documented determination of the AU (astronomical unit) the average distance of the earth to the sun. This was done by the parallax method when mars was closest to the earth. Richer was sent to Cayenne – French Guyana to observe mars in its closest approach to the earth, while Giovanni Cassini did the same observation from Paris. The result of the parallax calculation was about 140,000,000 km which was 92.5 % of the actual modern AU of 150,000,000 km

This was enough to establish the actual distances of the planets to the sun. The vast distances astounded the science community.

Planet | Ratio | Distance km |
---|---|---|

Mercury | 0.39 | 58,000,000 |

Venus | 0.72 | 108,000,000 |

Earth | 1 | 150,000,000 |

Mars | 1.52 | 228,000,000 |

Jupiter | 5.2 | 780,000,000 |

Saturn | 9.54 | 1,431,000,000 |

In 1687 Newton published his blockbuster scientific work Principia where he laid the foundation of gravitation and mechanics. In this publication he stated that the attraction force between two masses is proportional to the product of the masses divided by the square of the distance between them. Newton never worked out the universal gravitational constant that would have made the formula below a true mathematical equality rather than a proportionality.

\begin {aligned}F\propto\frac{M1*M2}{R^2}\end{aligned}

Only after Henry Cavendish did his landmark experiments on the gravitational constant in 1798, more than a century after the publication of Principia, was Newton’s law on gravitation complete . The modern value of the constant is

G=6.67430*10^{-11} m^3 kg^{-1}s^{-2}

With Newton’s completed formula for gravitation, it was now simple to calculate the mass of our planet.

The mass of the Earth can be established by using a standard 1 kg weight in the gravitational field of the Earth at sea level. Where M1 is the mass of the Earth to be solved, M2 is the standard weight of 1 kg, R is the Earth radius and G is the gravitational constant as determined by Henry Cavendish. The Gravitational force in Newtons of a 1 kg weight at sea level is m*a=1*9,807 (the gravitational acceleration for the earth as established by Galileo and Newton). The distance to the center of the Earth “R” at sea level is 6,378,000 m

\begin{aligned}F=1*9.807=\frac{G*M1*M2}{R^2}=\frac{6.67430*10^{-11}*x*1}{6378000^2} \frac{kg m}{sec^2}\end{aligned}

\begin{aligned}x=\frac{9.807*6378000^2}{6.67430*10^{-11}}=5.9772*10^{24}kg\end{aligned}

Now that we have the mass of the Earth we can derive the mass of the sun through the force equality of the gravitational attraction and the centrifugal force. Where M1 is the mass of the sun and M2 is the mass of the much smaller planet.

\begin{aligned}F=M2*a=M2*\frac{v^2}{R}=\frac{G*M1*M2}{R^2}\end{aligned}

The equality now depends only on the mass M1, the major astronomical body, the sun. The mass of the earth M2 cancels out. Just the earth’s orbit velocity and the distance to the sun matters.

\begin{aligned}v^2=\frac{G*M1}{R} \text{ or solved for the unknown solar mass: }M1=\frac{v^2*R}{G}\end{aligned}

To solve this we need the orbital velocity, which is 30,000 meters per second, The distance to the sun is 150,000,000,000 meters and the gravitational constant.

\begin{aligned}\text{The mass of our sun} = \frac{30,000^2*150,000,000,000}{6.67430*10^{-11}}=2*10^{30}\text{ kg}\end{aligned}

The weight of the other planets cannot be derived from the gravitational – centrifugal equality, as the planet mass drops out of the equation and leaving only the planets orbital velocity as the solution.

To solve for the mass of the planet we need to be able to measure the orbit size and period of a satellite or a moon. This is fairly easily available through an average telescope for Jupiter and Saturn. For Mars you will need at least a 12″ Dobson to resolve the tiny satellites Phobos and Deimos. The original discovery of the Mars satellites in 1877 were with the US Naval observatory 26″ telescope. However for Venus and Mercury there are no satellites and the only estimate can be made by measuring the planet size and assuming the same density as the Earth