laws of planetary motion

24 days of Books (1)

There are 24 days to Christmas and I thought I would do something a little bit different. So, everyday till Christmas I will post a book recommendation. It will be a math or math related book of course ^_^ Enjoy

First is Mathematical Principles of Natural Philosophy

Philosophiæ Naturalis Principia Mathematica (Latin for “Mathematical Principles of Natural Philosophy”), often referred to as simply the Principia, is a work in three books by Sir Isaac Newton, in Latin, first published 5 July 1687. The Principia states Newton’s laws of motion, forming the foundation of classical mechanics, also Newton’s law of universal gravitation, and a derivation of Kepler’s laws of planetary motion (which Kepler first obtained empirically). The Principia is “justly regarded as one of the most important works in the history of science”.

The French mathematical physicist Alexis Clairaut assessed it in 1747: “The famous book of mathematical Principles of natural Philosophy marked the epoch of a great revolution in physics. The method followed by its illustrious author Sir Newton … spread the light of mathematics on a science which up to then had remained in the darkness of conjectures and hypotheses.”

Level of mathematics: I would say a high level, this is basically calculus. 

Kepler is an iconic figure in the history of astronomy. He formulated three major laws of planetary motion which enabled Isaac Newton to devise the law of gravitation.

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“Johannes Kepler, one of the forefathers of modern astronomy (he determined the laws of planetary motion, which allowed Newton to determine his law of universal gravitation; Kant later called Kepler “the most acute thinker ever born”), wrote in 1603 that “philosophy, and therefore genuine astrology is a testimony of God’s works, and is therefore holy. It is by no means a frivolous thing.” Three years later, in 1606, he declared: “Somehow the images of celestial things are stamped upon the interior of the human being, by some hidden method of absorption … The character of the sky flowed into us at birth.”

Mars-Sized Exoplanet Discovered

An interesting method has revealed the mass of a small planet orbiting a faraway red dwarf star: Kepler-138.

The planet in question, Kepler-138b, had a known diameter and volume but getting mass is no simple task, especially for something that causes so little a tug on its host star.

Astronomers therefore decided to look at how the planet tugged on other planets.

Planetary orbits are extremely predictable. It’s how NASA can send a spacecraft all the way to Pluto, launching it on the correct trajectory almost a decade before it will even arrive.

It was discovered that two of the other planets around Kepler-138 had orbits that were somehow off by an hour.

Bingo. By observing how much their orbits shifted SETI Institute astronomers were able to decipher the force of gravity the small planet was exerting upon its neighbors.

The force of gravity can then be put in these terms: 

Fg=G(M1M2)/r2

With the important parts being r, G and the two masses. G is just a constant. r is known thanks to Kepler’s third law of planetary motion. The mass of each of the two larger planets was known already (larger exoplanets are easier to observe and work with).

With this equation, force of gravity becomes dependent on two things: distance and mass.

As any grade school student can tell you, if you have an equation and you need to solve for x, you just move x to one side (or in this case one of the “M”s). This is all they did here and it resulted in a groundbreaking, beautiful discovery.

What we now know is that this planet is rocky and likely shares some fundamental similarities to Mars.

(Image credit: Danielle Futselaar)