Astro 113-02: Ptolemaic System to Kepler's Laws PDF

Astro 113-02 Winter 2025 Slide 4 Topics Early Astronomers Ptolemy Copernicus Tycho Brahe Kepler’s Laws Galileo’s observations Kepler’s Laws Tides The Solar System Let us travel back approximately 2,000 years. The Ancient Greeks observed that th...

Astro 113-02 Winter 2025 Slide 4 Topics Early Astronomers Ptolemy Copernicus Tycho Brahe Kepler’s Laws Galileo’s observations Kepler’s Laws Tides The Solar System Let us travel back approximately 2,000 years. The Ancient Greeks observed that the celestial sphere, along with the Sun and Moon, appeared to revolve around a motionless Earth. The Sun and Moon followed straightforward paths at fairly consistent speeds relative to the background stars. The Sun completed one journey around the celestial sphere in a year, while the Moon did so in about 28 days. Note: However, the movements of the known planets were far more complex. Direct and Retrograde Motion Direct motion: refers to the usual eastward movement of a celestial object across the sky relative to the stars, as seen from Earth. Retrograde Motion: is the apparent westward (backward) motion of a celestial object relative to the stars. Note: This is an optical illusion that happens when Earth passes an outer planet (such as Mars or Jupiter) in its orbit, or when an inner planet (like Venus or Mercury) moves ahead of Earth, caused by their relative positions and orbital speeds. The Ptolemaic System The Ancient Greeks developed a system to explain the motion of the planets, placing the Earth at the center. This was known as the Geocentric model. An alternative model with the Sun at the center was proposed by Aristarchos of Samos but was eventually dismissed. Ptolemy improved the system to accurately predict the positions of the planets. In his model, each planet moved along a smaller circle called an epicycle, while the center of the epicycle followed a larger circular path known as a deferent, which was offset from the Earth's position. The Ptolemaic System The Ptolemaic System The Earth is a stationary center of the universe. The issue with the Ptolemaic system was its lack of simplicity, as it did not provide a universal rule applicable to all planets. https://science4fun.info/ptolemy/ Heliocentric model In the early 1500s, Nicolaus Copernicus introduced a heliocentric model of the Solar System, where the Sun was at the center. Earth was treated as just another planet, moving around the Sun in a circular orbit. In this model, the Moon continued to orbit Earth. Earth rotated on its axis once every 24 hours, explaining the apparent motion of the stars and the Sun’s daily rise and set. Heliocentric model This concept was opposed by the Catholic Church at the time, as it shifted the center of the Universe away from Earth. However, the Copernican system was no better than the Ptolemaic system at predicting planetary positions since it still relied on perfect circular motions. Despite this, Copernicus believed it was a more natural representation of the cosmos. Copernicus The Copernican System The Sun is the center of the universe. http://www.inquiriesjournal.com/articles Tycho Brahe Around the 1580s, the Danish astronomer Tycho Brahe began recording highly precise measurements of planetary positions and other celestial bodies. Tycho Brahe proposed a hybrid model of the Solar System, often ,Tychonic system, as a compromise between the geocentric Ptolemaic model and the heliocentric Copernican model. Earth remained stationary at the center of the universe. The Sun orbited around the Earth. All other planets (Mercury, Venus, Mars, Jupiter, and Saturn) revolved around the Sun, which, in turn, revolved around Earth. Tycho Brahe Tycho Brahe constructed the first dedicated astronomical observatory, "Uraniborg," on the island of Hven near the Danish coast. Despite the lack of telescopes, which had yet to be invented, he designed astronomical instruments capable of achieving far greater accuracy than any prior observations. Like the Ptolemaic model, Tycho’s system placed the stars on a fixed, distant celestial sphere Tycho Brahe Tycho Brahe was uneasy with the Copernican system, which placed the Sun at the center of the universe. Instead, he devised the Brahe System, which kept Earth at the center while allowing the planets to orbit the Sun. Tycho dedicated much of his life to meticulously measuring the positions of the planets, aiming to demonstrate that his model was superior to both the Ptolemaic and Copernican systems. However, lacking strong mathematical skills, he was unable to fully analyze the vast data he had gathered. After relocating to Prague, Tycho gained the assistance of a talented German mathematician, Johannes Kepler, who helped him further his work. Tycho Brahe The Man with the Golden Nose. He lost part of his real nose in a duel over a mathematical dispute. Book Recommendation: The Watershed: A Biography of Johannes Kepler by Arthur Koestler (1960). https://www.britannica.com/bio graphy/Tycho-Brahe-Danish- astronomer Galileo Galileo was the first to point a telescope toward the sky in 1609. When Galileo utilized his telescope, he made several discoveries that supported the heliocentric model of the Solar System. He observed the phases of Venus, similar to the phases of the Moon. These observations demonstrated that Venus must revolve around the Sun. He observed moons orbiting Jupiter, which clearly showed that the Earth was not the center of all motion in the Universe. Galileo In addition to his telescope discoveries, Galileo made numerous other significant scientific observations and measurements. One of these was his observation that, when friction is minimal, all objects fall to the Earth with the same acceleration. https://www.astronomy.com/science/how-galileo Johannes Kepler Johannes Kepler, a German mathematician and astronomer, revolutionized celestial mechanics by refining the Copernican model with elliptical orbits, forming the Keplerian Model of the Solar System, based on Tycho Brahe's precise data. What he found are summarized in what we now call Kepler’s Laws. https://study.com/learn/lesson/keplers Kepler’s Laws 1) Elliptical Orbits: Planets move in elliptical orbits with the Sun at one focus, rejecting the long-held belief in perfect circular orbits 2) Equal Areas in Equal Times: A line drawn from a planet to the Sun sweeps out equal areas in equal intervals of time. This means planets move faster when closer to the Sun and slower when farther away. 3) Harmonic Law: The square of a planet’s orbital period is proportional to the cube of the semi-major axis of its orbit. This law established a precise relationship between the time a planet takes to orbit the Sun and its distance from the Sun. Kepler’s First Laws The orbit of each planet around the Sun is an ellipse, with the Sun at one of the two foci. An ellipse is a closed curve around two points called Focal points. Note: Planets do not move in perfect circles but in slightly elongated paths called ellipses. Note: The Sun is not at the center of the ellipse but at a focus point, which explains variations in orbital speed.  Ellipse An ellipse has two foci. The longest diameter of an ellipse, called the major axis, passes through both foci. Half of that distance is called the semi-major axis. Note: A circle is a special case of an ellipse in which the two foci are at the same point. https://byjus.com/maths/ellipse/ Ellipse The shape of an ellipse is described by its eccentricity, shown by e. The value of e can range from 0 (a circle) to just under 1 (nearly a straight line). The semimajor axis is also the average distance between the Sun and the planet. If x = Distance between a focal point and the center of the ellipse, then 𝑥 eccentricity is defined as 𝑒 = ,where b is the semimajor. 𝑏 Note: Most of the planets have eccentricities close to zero. (Their orbits are almost circular) e.g. Venus: e is around 0.007 Kepler’s Second Law (Law of Equal Areas) A line joining a planet and the Sun sweeps out equal areas in equal intervals of time Kepler found a way to describe how a planet’s speed varies as it moves along its orbit. A planet moves most rapidly when it is nearest the Sun, at a point on its orbit called perihelion. Conversely, a planet moves most slowly when it is farthest from the Sun, at a point called aphelion. Kepler’s Second Law Question: What happens to the speed of a planet moving in a circular orbit? Kepler’s Third Law The square of the sidereal period of a planet is directly proportional to the cube of the semimajor axis of the orbit. Which means that the larger a planet’s orbit (larger the semimajor axis, or average distance from the planet to the Sun), the longer the sidereal period, which is the time it takes the planet to complete an orbit. Kepler’s Third Law as an equation: 𝑃2 = 𝑎3 Where “𝑃” is the sidereal period measured in years, and “𝑎” is the length of its semimajor axis measured in astronomical unit (AU).  Kepler’s Laws Planet Semimajor axis Sidereal Period P2 a3 a (AU) P (Earth years) Mercury 0.387 0.24 0.99 Venus 0.723 0.61 0.98 Earth 1.000 1.00 1.00 Mars 1.524 1.88 1.00 Jupiter 5.204 11.86 0.998 Saturn 9.554 29.46 0.995 NOTE: In fact the ratio P2/a3 = constant no matter what units you use. Kepler’s Laws helped to justify the Heliocentric model of the Solar System. His laws allowed the positions of the Planets to be calculated more accurately than ever before. Example The average distance from Venus to the Sun (that is, the semimajor axis “a” of Venus’s orbit) is 0.72 AU. Determine the sidereal period of Venus. Solutions: 𝑃2 = 𝑎3 𝑎3 = 0.72 3 = 0.373 𝑃 = √𝑎3 = 0.373 = 0.61 𝑦𝑒𝑎𝑟𝑠 Question Two planets are in circular orbits around the same star, both at an identical distance from it. One planet is the size of Jupiter, while the other is the size of Earth. Which of the following statements is accurate? a. The Jupiter-size planet has a much longer period than the Earth-size planet. b. The Earth-size planet has a much longer period than the Jupiter-size planet. c. Both planets have the same period. d. Not enough information Question Two planets are in circular orbits around the same star, both at an identical distance from it. One planet is the size of Jupiter, while the other is the size of Earth. Which of the following statements is accurate? a. The Jupiter-size planet has a much longer period than the Earth-size planet. b. The Earth-size planet has a much longer period than the Jupiter-size planet. c. Both planets have the same period. d. Not enough information Everything Started with an Apple! Isaac Newton Building on the observations of Galileo, Newton was able to formulate 3 laws that govern all motion in the Universe. A force is a push or a pull. First Law: An object moves with a velocity that is constant in magnitude and direction unless a non-zero NET force acts on it. Second Law: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Third Law: For every action, there is an equal and opposite reaction. Newton’s Laws   a F Remember: m    F  ma F x  max ,  Fy  ma y ,  Fz  maz Note Weight and Mass are different things. Mass is an intrinsic property of an object. Weight is the force of gravity on the object. The mass of an object always remains the same! Newton and Kepler’s Laws Using his three laws of motion and his law of gravity, Newton realized that he could prove Kepler's three laws mathematically. Kepler’s first law concerning the elliptical shape of planetary orbits, proved to be a direct consequence of the 1/𝑟 2 factor in the gravitational law. Kepler’s second law (the law of equal areas), is a consequence of the Sun’s gravitational force on a planet being directed straight toward the Sun. Newton and Kepler’s Laws Newton’s form of Kepler’s third law It is valid whenever two objects orbit each other because of their mutual gravitational attraction. 2 4𝜋2 𝑃 = 𝐺 𝑚1 +𝑚2 𝑎3 Where: P= sidereal period of orbit, in seconds. a= semimajor axis of orbit, in meters. 𝑚1 = mass of first object, in kilograms. 𝑚2 = mass of second object, in kilograms. G= gravitational constant Note: 𝑃2 = 𝑎3 is only valid for objects that orbit the Sun! Newton and Kepler’s Laws For the case when one mass is very much larger than the other (this is the case for the planets orbiting the Sun), we can write: m1  m2  mSun  mPlanet  mSun Therefore, the ratio: P2 4 2 3  a GmSun This is a constant that does not depend on the mass of each planet. This is exactly what Kepler had discovered—that this ratio is the same for all planets orbiting the Sun. However, later when we study binary star systems, we will not be able to ignore the mass of the smaller object. Important note on the vector nature of force Since velocity is a vector quantity, changing its direction also results in acceleration. Example: circular motion. With this knowledge, answer the next question. Question: Which one is correct? a. A planet moving at constant speed in an unchanging direction has no force exerted on it. b. A planet traveling at constant speed but changing direction of travel has a force exerted on it. c. A planet traveling with changing speed and changing direction of travel has a force exerted on it. d. B and C are correct, but not A. e. A, B, and C are all correct. Question: Which one is correct? a. A planet moving at constant speed in an unchanging direction has no force exerted on it. b. A planet traveling at constant speed but changing direction of travel has a force exerted on it. c. A planet traveling with changing speed and changing direction of travel has a force exerted on it. d. B and C are correct, but not A. e. A, B, and C are all correct. Tides Gravity also explains the occurrence of tides on Earth. Tides are primarily caused by the Moon's gravitational pull. Since the gravitational force increases as the distance to the Moon decreases, the force on the side of Earth closest to the Moon is stronger than at the Earth's center or on the opposite side. Length of arrow represents size of force (exaggerated). Earth moon Tides From the Earth's P.O.V. forces look like: The Earth is fairly solid so it does not deform very much, but the oceans can change shape easily. Note that this a simplification, as we have ignored the effect of the Earth’s rotation, continents and wind patterns on the Earth, and the effect of the Sun. Tides The Sun also exerts tidal forces on Earth's oceans, though its effect is about half as strong as the Moon's. As a result, when the Sun and Moon align, we experience tides that are larger than usual, which are called Spring Tides. Note: This has nothing to do with springtime – it comes from the notion that the water “springs up”. When the Sun and Moon form a right angle relative to each other, their tidal forces partially cancel each other, resulting in smaller-than- normal tides, which are called Neap Tides.