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An Introduction to Celestial Mechanics


Introduction

Celestial mechanics is the field of astronomy that deals with the motion of celestial objects. Celestial mechanics has its roots in ancient times and in particular ancient Greece, with the beginning of our modern understanding of it only goes back about 500 years. In an case Celestial mechanics forms the bases of space flight opening up our understanding of the Universe.


Ptolemy

Ptolemy was a Greco-Roman Philosopher from the city of Alexandria. He was Born in 90 A.D. and died in 168 A.D. in Alexandria, Egypt. While he did many things Ptolemy is most known for his Geocentric Cosmology.


Here is a video animation of Ptolemy's Geocentric Cosmology
1 year = 10 seconds

Note that there are only five planets, that is because Mercury, Venus, Mars, Jupiter and Saturn the only planets visible to the naked eye.


Ptolemaic epicycle

Here is a simple illustration of the Ptolemaic astronomy basic elements. Planet rotates on an epicycle which rotates around the inside of a crystalline sphere with the Earth is slightly off center of the crystalline sphere. The out sphere was where Ptolemy thought the stars were. This system is an excellent example a philosophically driven model the needed substantial additional elements to make it fit reality.

Nicolaus Copernicus

Copernicus was the Renaissance mathematician and astronomer who developed the heliocentric model of the universe which puts the Sun at the center rather than the Earth. He was born on February 19, 1473 and died: May 24, 1543.


Here is a video animation of Copernicus' Heliocentric Cosmology
1 year = 10 seconds

Note that there are only five planets, that is because Mercury, Venus, Mars, Jupiter and Saturn the only planets visible to the naked eye.


Galileo Galilei

Galileo, was an Italian physicist, mathematician, astronomer, and philosopher. He was born on February 15, 1564 and died on January 8, 1642. He was an early champion of Heliocentrism. Galileo was first know astronomer to use telescope resulting in his discovery of the four largest Moons of Jupiter; Callisto, Europa, Ganymede, Io; resulting in the being  referred to as the Galilean moons.

The biggest myth about Galileo is a bout his trouble with the Catholic Church. The myth is that he was prosecuted for teaching Heliocentrism in opposition to the Bible. This myth is false and it originated with the 19th century anti-Christian bigots William Draper and Dickson White. They tried to portray it as science vs the Bible when in reality it was science vs science and specifically Ptolemy vs Copernicus. At the time Ptolemy was seen as an authority by the scientific establishment of his day. Also in one of his books Galileo had made the pope look like a buffoon thus belittling the pope’s office and authority over Heliocentrism vs Geocentrism. This was happening in Italy at the height of the protestant reformation. His first trial was in1616, and the Pope’s authority over Europe was falling apart and Galileo was yet another challenge to his authority. Proponents of Heliocentrism in protestant counties such as Kepler had no problems from protestant churches on this issue.

Galileo's work helped pave the way for further development. Kepler’s discovery that planets orbit the sun in an ellipse. Newton’s development of the concept of gravity.


Tycho Brahe

Tycho Brahe was well known for his accurate astronomical and planetary observations. Tycho was born in December 14, 1546 and died in October 24, 1601. Most well known for his hybrid geocentric / heliocentric model of the universe.


Tycho’s hybrid geocentric / heliocentric model

Tycho’s cosmology was essentially geocentric, in that every thing ultimately went around the Earth. However it was part heliocentric in that the other planets were seen as going around the sun, while the sun went around the Earth. A modified version of Tycho’s model is held to by modern geocentrics. However despite the fact that heliocentricity is taught as fact in schools it is not absolutely true.

The real answer is that according to General Relativity both geocentricity and heliocentricity can be considered correct. This because according to General Relativity you can choose any frame of reference you want because they are all valid. However this does not included only geocentricity and heliocentricity but Mars-centricity, Venus-centricity and even Kepler-62f-centricity. It is simply a matter of choosing the most convenient frame of reference.


Johannes Kepler

A German astronomer, and mathematician. born on December 27, 1571 and died on November 15, 1630. He is best known for his laws of planetary motion. Kepler's laws were derived by use of the data collected by Tycho Brahe. These laws were for planetary orbits around the sun, but they apply to dwarf planets, asteroids, moons and satellite orbits as well.

First Law


All planets move in elliptical orbits, with the sun at one focus.

Second Law


A line that connects a planet to the sun sweeps out equal areas in equal times.

Third Law

Planet

Semimajor axis
(AU)

Period
(Earth Years)

T2/a3

Mercury 0.386 0.241 1.01
Venus 0.72 0.615 1.01
Earth 1 1 1.00
Mars 1.52 1.88 1.01
Jupiter 5.187 11.9 1.01
Saturn 9.533 29.5 1.01
Uranus 19.1333 84 1.01
Neptune 30 165 1.01

The square of the period of any planet is proportional to the cube of the semi-major axis of its orbit


Isaac Newton

Newton was an English physicist and mathematician. He was born on January 4, 1643 and died on March 31, 1727. Newton is considered one of the most influential scientists of all time. He developed the mathematics of Calculus anf the basic laws of motion. Newton is most famous for developing the Law of Gravity. These developments have paved the way for a full understanding of celestial mechanics. They not only describe how moons and planets move but why they move that way.


Orbital Mechanics

Orbital Mechanics is the physics that describes the orbits of  satellites and planetary bodies. It is base on Kepler’s and Newton’s laws is used to determine future positions of planets, moon, man made objects, and any other object in space.


Here are the main elements of orbital motion.
r = Radius of the motion from the center of the gravitating body.
g = Gravitational acceleration.
v = Velocity of the satellite.
q = Angle of velocity to radius.


Planetary Motion

Planetary motion is a special case of orbital motion. It obeys Kepler’s and Newton’s laws of motion. However planets are large enough to have significant gradational influence on each other as a result only some obits in a planetary system are stable.


Stellar Motion


By Brews ohare (Own work)
[CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0)
or GFDL (http://www.gnu.org/copyleft/fdl.html)],
via Wikimedia Commons

The main motion of a star as seen from Earth is called its proper motion. A star’s proper motion is the angle the star moves across the sky over time do to its motion relative to the sun. It is a function of the stars distance and transverse velocity and radial velocity is found by way of the star’s doppler shift. Transverse and radial velocity combine to give the star’s space velocity which is its actual velocity relative to sun The motion of stars relative to each other are essentially random. While gravity between stars dose affect this motion. it is small but acts over a long period of time. The same main elements apply to stellar motion as does to orbital motion.


Galactic Rotation


Expected (A) and observed (B) star velocities as a function of
distance from the galactic center. (C) Normal orbital curve.
Created by User:PhilHibbs in Inkscape 0.42
http://commons.wikimedia.org/wiki/File:GalacticRotation.png
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License
http://en.wikipedia.org/wiki/en:GNU_Free_Documentation_License

The same basic laws of physics apply to galactic rotation as does any orbital motion, however galaxies are more complex than planetary systems and so exhibit different velocity vs distance curves. The lack of sufficient mass to explain galactic rotation curves has lead to the theoretical idea called dark matter. While often considered an exotic form of matter it could still be some type of undetected normal mater such as rouge planets.


Conclusions

Celestial Mechanics is the study of the motion of heavenly bodies it ultimately it deals with orbits of one sort or another. However understanding orbital mechanics is what allows space craft to orbit the Earth as well as travel between planets.


Ptolemy

Nicolaus Copernicus

Galileo Galilei

Tycho Brahe

Johannes Kepler

Isaac Newton

Planetary Motion

Stellar Motion

Galactic Rotation

Orbital Mechanics


 

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