These are my complete notes for the Origin of Modern Astronomy.
I color-coded my notes according to their meaning - All numbered notes (which I call rules) are red, and include examples and the basis for understanding a topic. Definitions are written in green, and other important information (such as large-scale drawings that are better visualized than explained) was written in blue. All of this information is preserved on this page, with logical flow and breaks. I use ascii line drawings sparingly - If I can convey information or a graph using an image online, I will do so.
All of the knowledge present in these notes are filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes. In the unlikely event there are any egregious errors, contact me at jdlacabe@gmail.com.
Origin of Modern Astronomy
IV. Origin of Modern Astronomy.
# Modern Astronomy:
While Europe was going througn an anti-intellectual phase during the Middle Ages, there was a flowering of Astronomical discovery and progress during the Islamic Golden Age, which preserved and elaborated on the ideas of the Greeks. It was the end of the Dark Ages in Europe and the ushering in of the Renaissance that saw a revival in intellectual interest in sceintific, and importantly Astronomical, ideas in Europe. Copernicus embodied the spirit of European revival in interest in Astronomy (see below).
A. Rule 21. Nicolaus Copernicus was a Polish astronomer who flourished during the early 1500s. He led the critical reappraisal of Ptolemy's ancient model of the Universe, which had stood as the dominant explanation of the cosmos for over a thousand years. He led the formation of the Heliocentric model of the solar system, which radically considered Earth itself to be a planet that orbited the sun, alongside all other planets. Additionally, he theorized that only the moon orbits the Earth.
As evident, he still assumed that the orbits of the planets would be uniformally circular. However, his ideas were revolutionary enough to provoke discussion across the scientific world, and the tenets of heliocentrism would eventually be popularly accepted over a century after his death. Controversiallly (as with everything he theorized), he explained the precession of the celestial sphere aas being the rotation of the Earth along its axis, while the sphere itself is stationary.
A. Rule 22. Copernicus's masterpiece, De Revolutionibus, elaborates on Earth as one of six planets, and correctly lists the Planets in order of their proximity to the sun, and was able to deduce that the closer to the sun the planet, the greater the orbital speed. Furthermore, complexities like retrograde motion were explained through this new theory with much simpler explanations than Ptolemy. This theory conflicted with the thousands of years of established common sense: the greek/classical schools of thought, and the dogma pushed by the Catholic Church.
A. Rule 23. """"common sense"""" is the propaganda of human nature - half of the time it is fake news and ignorance, it is much more commmon than sensical. Aristotle believed that heavier objects will fall faster than lighter objects, and it was just so deducible from common sense that he did not even bother to test his theories before writing them down, and it was a long philosophical tradition to place high value on basic fundamental truths, 'axioms', which could be deduced by 'Basic' human thought, combined with """"divine revelation"""", DO NOT FALL FOR IT. The advancement of science is very easy to see as the progressive challenging of previously accepted axioms and theories - Classical Mechanics is useful for building the skills and understanding of general topics of Physics, but what happens to gravity on the Quantum Level? Relativity was born from the need to reconcile the Principle of Relativity with Electrodynamics, which had failed to be accomplished through the standard Newtonian model (see the failure of Emission Theory ([[[[)). Theory will only take you so far - dogmatic rationalism can be an impediment to scientific progress, think of how the Catholic Church persecuted Galileo. As soon as you theorize, you should theorize a complementary experiment to test its validity.
A. Rule 24. The scientific method, nonexistant during the time of Copernicus, was developed by Galileo Galileo during the 17th century. he contributed greatly to mexhanics, classical Physics of motion and forces on bodies. Galileo theorized the basics of Inertia, that rest is no more natural a state for a body than motion. He argued that a force is required not only to start an object moving from rest, but also for slowing down, stopping, speeding up, and changing direction in any form. Additionally, in studying acceleration (see P. Rule 12), he found that objects will accelerate uniformally as they free fall or roll down a ramp. These laws would be formulated in exact mathematical expressions. Most importantly, he adopted the heliocentric model of the universe in the 1590s and began lecturing on the topic. The church pushed back, decreeing that heliocentrism was "false and absurd" in 1616 and disavowing its defense.
Galileo would drastically improve upon the telescope and its powers of magnification. He used this invention to observe the cosmos, beginning in 1609, and he discovered many distant stars too faint to be seen with the naked eye, and that the Mily Way across the night sky was made out of many stars as well. He found that Jupiter had its own moons, circling Jupiter at different orbit speeds, proving that centers of motion could themseves be in motion (a theory argued against by geocentrics). Venus proved to go through phases like the moon, showing it revolved around the sun. For these discoveries, Galileo was rewarded with a house arrest on the orders of the Catholic Church, who found his work heretical, and this would last until his death. In this confinement he would write on of his greatest masterpieces, the "Two New Sciences" which would form the foundations for modern Physics.
A. Rule 25. Tycho Brahe was a Danish astronomer who worked concurrent to Galileo, and produced a sound mathematical basis (along with Kepler) of Copernicus's Theory of Heliocentrism. He established an observatory on a Northe Sea Island, and he is said to be the last of the pre-telescopic observers in Europe. He took extremely detailed records of the positions of the Sun, Moon, and Planets for almost 20 years, the greatest collection of such data since Hipparchus. When he fled Denmark because of all the enemies he'd accumulated, he became the court Astronomer in Prague, where he enlisted the help of the young Astronomer Johannes Kepler in analyzing the data. Johannes Kepler was a German astronomer who served as Brahe' assistant in Prague. It was only after Brahe's death did Kepler gain access to the entirety of the records, which occupied him for the next 20 years. In this analysis, he developed three laws that dictate the motion of planets through Space, known as Kepler's Laws of Planetary Motion.
A. Rule 26.
Law 1 - The Orbit of a planet is an ellipse with the Sun at one of the two foci.
Explanation: The path of an object through space is called its orbit. Kepler assumed, at first, that the orbits of planets were cirlces, but doing so did not allow him to find orbits that were consistent with Brahe's observations. He discovered that the orbit of Mars was elliptical rather than circular. All of these shpaes are closed curves, belonging to a family of curves known as conic sections.
See [[[[[ for more information about focus points (foci) and ellipses/hyperbola. The center of a circle, of course, is a special point: the distance from the center of a cicle to anywhere on the circle is always the same - that is the radius. in an ellipse there are TWO special points, they are called the foci, or the focus points of the ellipse. The sum of the distance from the focus points to any position on the ellipse is always the same:
The foci will, of course, change their postion depending on the size/nature of the ellipse. The widest diameter of the ellipse is called its Major Axis, while half that distance, the distance from the center of the ellipse from end to end, is called the Semimajor Axis. The smallest diameter of the ellipse is the Minor Axis (of Symmetry), perpendicular to the Major Axis, and it has two semimajor axes at either side of the center as well:
The Minor Axis – the shortest diameter of an ellipse, each end point is called a co-vertex.
The Semi-major Axis (a) – Half of the major axis.
The Semi-minor Axis (b) – Half of the minor axis.
Eccentricity (e) – the distance between the two focal points, F1 and F2, divided by the length of the major axis.
ae – the distance between one of the focal points and the centre of the ellipse (the length of the semi-major axis multiplied by the eccentricity). Courtesy of the Science & Math Zone.
The semimajor axis of the orbit of Mars (also the planet's average distance from the Sun) is 228 million kilometers.
The shape/roundness of an ellipse depends on how close together the two foci are, compared with the Major Axis. The ratio of the distance between the foci to the elngth of the major axis is called the eccentricity of the ellipse. If the eccentricity is zero, then the foci will be in the same spot and the ellipse will be a circle. Thus, in elliptical terms, a cricle is an ellipse of zero eccentricity with the semimajor-axis as the radius.
The greater the eccentricity, the more elongated the ellipse, up to a maximum eccentricity of 1.0, which is just a flat line. The size and shape of an ellipse are completely specificed by its semimajor axis and its eccentricity. Mars has an elliptical orbit of 0.1, wth the sun at one of the foci, the other being empty. This discovery was generalized by Kepler to apply to all orbits, with different eccentricities.
A. Rule 27.
Law 2 - A line segment joining a planet and the sun sweeps out equal areas during equal intervals of time.
The second law deals with the orbital speed, or the speed with which each planet moves along its ellipse. Kepler determined that Mars moves faster as it comes closer to the Sun, and slows down as it pulls away from the Sun. Visualize an elastic band connecting a celestial body with the Sun. As the body gets farther from the sun, the band gets stretched, and thus moves slower until the band will pull it back to the sun. When it is closer to the sun, the band is not stretched as much and thus movs rapidly. Additionally, if you were to imagine the area sweeped in the ellipse (centered from the sun) by the orbit, then, given equal intervals of time, any two seweeped areas of the orbit will be equal.
While a circular orbit would cause a planet to move at the same speed throughout its orbit, the orbits and differing speeds of the planets make it evident that their obits are elliptical.
# Astronomical Units: The average distance between the Sun & the Earth, generalized as 1.5 × 10⁸ km.
# Kinematics: The science of movement and motion, such as Newton's & Kepler's Laws.
A. Rule 28. Law 3 - The square of a planet's orbital period is proportional to the cube of the length of the semni-major axis of its orbit.
While the first two laws deal largely with the shape and speeds of a planet's orbit, there is yet to be a mathematical model for the spacing of the planets, how they ended up like that, known as the "Harmony of the Spheres". It was Kepler's 1619 discovery of a basic relationship relating the orbits of the planets to their relative distance from the sun that a mathematical model governing planetary spacing was formed. The Orbital Period P is the time it takes a planet to travel once around the Sun, and the semimajor axis of a planet's orbit is equivalent to its average distance from the sun. In mathematical terms, this law can be defined as the following:
P² ∝ a³
With regard to the Earth (and all other planets where years and AU are the given units), where P is measure in years and a is expressed in AU, the sides of the formula are not only proportional, but equal. The Third Law of Planetary Motion applies for all objects orbiting the sun, and enables the determination of an objects relative distance from the Sun based on how long the Orbital Period is. Since Mars has an orbital period of 1.88 years, the third law tells us that (1.88)² = a³, using years and AU as our units. Therefore, the semimajor axis of Mars has a length of ∛3.53 AU, or 1.52 AU, thus orbiting farther from the Sun than Earth.
A. Rule 29. :) Kepler's laws are highly qualitative, they only reflect rules of thumb that Kepler discerned about the planetary motion in the solar systems. The laws do not actually explain what forces of nature are causing the celestial body to act in this way. It would be Isaac Newton who would form the sound mathematical framework that explained the oservations and rules formulated by Galileo, Brahe, Kepler, and the rest. As a professor of Mathematics at Cambridge, Newton dedicated much of his time to pondering the cosmos and reality, and from this he developed special mathematical models/equations and laws to govern motion itself, the forces behind Kinematics. He wrote down his theories in a book called the Principia, in 1687, where he unleashed unto an unsuspecting public his three laws of motion:
LAW 1: Every object will continue to be in a state of rest, or move at a constant speed in a straight line, until it is compelled to change by an outside force. The first law is just a restatement of one of Galileo's discoveries, called the Conservation of Momentum (see P. Section VIII.II). This law states that in the abscence of any outside influence, there is a measure of a body's motion, called its momentum, that remains unchanged. The first law is also oftentimes called the Law of Inertia, where Inertia is the tendency of an object to resist a change in the state of motion (see the physics reference), akin to re-electing an incumbent forever. In other words, a stationary object will stay at rest while a moving object will keep moving until some force intervenes.
A. Rule 30. Momentum relies on three factors:
- Speed, how fast an object moves (zero if stationary).
- The direction of the motion, whether using relative, cartesian, or cardinal directions.
- The mass of the body, a measure of the amount of matter in a body.
Momentum, referred to as variable p, can be expressed through the following equation: p = m × v, where m is the object's mass and v is the velocity at any point. It is difficult to see Momentum and the First Law of motion in action in the real world, because of how many forces are acting on a body at any one time, from Air Resistance to Electromagnetism. For example, while a ball rolling on the sidewalk will eventually be slowed down due to Rubbing Friction (see P. Rule 58 and below), but if that ball was moving in space (the vacuum to which the idealized equations of kinematics are most applicable), the friction is so minute and insignificant that it would continue coasting through space forever. Thus, momentum can change only under the action of an outside force, such as gravity.
A. Rule 31. LAW 2: The change of motion of a body is proportional to, and in the direction of, the force acting on it. This law expresses force in terms of its ability to change momentum with time. A force, push or pull, has both direction and magnitude. When a force is applied to a body, the momentum changes in the direction of the applied force. This means that a force is required to change either the speed or the direction of a body, or both - to start it moving, to speed it up, to slow it down, to stop it, or to change its direction.
The rate of change in an object's velocity is called acceleration, and the second law states that the acceleration of a body is proportional to the force being applied to it.
For example: Imagine a table as a smooth, frictionless surface. If you were to push a book across it, it will speed up as long as you keep pushing it with a positive acceleration (or negative acceleration if you are pushing it backwards). The harder you push the book, the more it will speed up. How much an object will accelerated given a force is reliant on the mass of the object; if you pushed a pen on the table with the same force you pushed the book with, it would accelerate to a higher speed.
A. Rule 31. LAW 3: For every action, there is an equal and opposite reaction (or: the mutual actions of two bodies upon each other are always equal and act in opposite directions). This law generalizes the first law in a way that also defines mass. Thus, if there were to be a car system of multiple oblects, isolated from external forces, the first law would stipulate that the total momentum of the objects would remain constant. Any change of momentum in a system must be balanced by another change, equal in force and opposite in direction, so that the momentum of the entire system is not changed. In every situation, there is always a force pair governing the reactions between any two objects. When you fall from a tree, the force pair is you and the Earth, because the Earth is exerting the force of gravity attracting you to the Earth, while the Earth is accelerated by the student's pull. You do not notice the change in momentum of the Earth because of its huge mass, which measures the inertia of the object, the tendency of the object to resist acceleration. The recoil from hitting a ball with a baseball is further proof of the third law.
# Volume: The measurement of how much Physical space an object occupies. Measured in cubic units, like cm³ or liters.
# Density: Mass divided by volume. Common units include g/cm³.
A. Rule 32. Angular Momentum is the measurement of the rotation of a body as it revolves around some fixed point, like a planet rotating around the Sun. Mathematically, the angualr momentum of an object is the product of its mass, velocity, and distance from the fixed point (which isn't always the radius, as known from elliptical orbits). If these three values were to be constant, meaning that an object's motion takes places at a constant velocity at a fixed distance from the spin center, then the angular momentum is also constant.
Kepler's 2nd law is a result of the conservation of angular momentum. As a planet approaches the Sun on an elliptical orbit and the distance to the spin center decreases; the planet speeds up to conserve angular momentum. Similary, when the planet is farther from the sun, it moves slower. When you spin on a swivel chair at the park, when you have your arms outstretched, you move slower, but when you bring your arms inward, you speed up.
