Heliocentric vs. Geocentric Models

• Astro 113-02 is a course being offered in Winter 2025

Topics

• The topics covered include:
• Early Astronomers Ptolemy, Copernicus, Tycho Brahe, Kepler's Laws, Galileo's observations
• Kepler's Laws
• Tides

The Solar System

• About 2,000 years ago, the Ancient Greeks believed that the celestial sphere, along with the Sun and Moon, revolved around a motionless Earth
• The Sun and Moon followed fairly straightforward paths at consistent speeds relative to background stars
• The path that the Sun makes around the celestial sphere is completed in one year
• The path that the Moon makes around the celestial sphere is completed in about 28 days
• Planetary movements were known to be far more complex

Direct and Retrograde Motion

• Direct motion describes the eastward movement of an object across the sky relative to stars from Earth's point of view
• Retrograde motion describes the westward or backward movement of an object relative to the stars
• Retrograde motion is an optical illusion that occurs when Earth passes an outer planet or an inner planet moves ahead of Earth

The Ptolemaic System

• Ancient Greeks developed a geocentric model that placed Earth at the center of the universe to explain planetary motion
• Aristarchos of Samos proposed an alternative heliocentric model with the Sun at the center, it was dismissed
• Ptolemy improved the geocentric model; in his model, each planet moved along a small circle called an epicycle
• The center of the epicycle, at the same time, moved along a larger circle called the deferent, which was offset from the Earth

Ptolemaic System Details

• In the Ptolemaic System, Earth is a stationary center of the universe
• The Ptolemaic system lacked simplicity and a universal explanation for all planets

Heliocentric Model

• In the early 1500s, Nicolaus Copernicus introduced the heliocentric model, with the Sun at the center
• This model treats Earth as just another planet orbiting in a circle around the Sun
• In the heliocentric model, the Moon continues to orbit Earth
• The Earth rotated on its axis every 24 hours, explaining the motion of the stars and the Sun's rise and set
• The Catholic Church opposed this concept since it shifted the center of the Universe away from Earth
• The Copernican system was not better than the Ptolemaic system for predicting planetary positions, since it relied on perfect circular motions
• However, Copernicus believed it was a more natural representation of the cosmos

Copernicus

• In the Copernican System, the Sun is at the center of the universe

Tycho Brahe

• In the 1580s, Danish astronomer Tycho Brahe began making precise measurements of planetary positions
• Brahe proposed a hybrid model, called the Tychonic system, as a compromise between the geocentric Ptolemaic and heliocentric Copernican models
• The Tychonic system had Earth stationary and at the center of the universe
• The Sun orbited around the Earth; all other planets orbited the Sun, which in turn, revolved around earth.
• Tycho Brahe's observatory, "Uraniborg," was on the island of Hven, near the Danish coast
• Without telescopes, Brahe designed astronomical instruments achieving far greater accuracy
• Much like the Ptolemaic model, Brahe's system placed the stars distant celestial sphere
• Brahe devised the Brahe System, which kept Earth at the center, allowing the planets to orbit the Sun
• Brahe was uneasy with the Copernican system, which placed the Sun at the center of the universe
• He dedicated his life to demonstrating that his Tychonic model was superior to both Ptolemaic and Copernican systems.
• Lacking strong mathematical skills, he enlisted German mathematician Johannes Kepler to help further his work
• Brahe lost part of his real nose in a duel over a mathematical dispute, earning him the nickname "The Man with the Golden Nose"

Galileo

• Galileo first pointed a telescope toward the sky in 1609
• Galileo made discoveries that supported the heliocentric model of the Solar System
• Galileo saw that Venus has phases, demonstrating that Venus must revolve around the Sun
• He observed moons orbiting Jupiter, showing that Earth was not the center of all motion
• Galileo observed that, when friction is minimal, all objects fall to Earth with the same acceleration

Johannes Kepler

• Johannes Kepler, a German mathematician and astronomer, revolutionized mechanics refining the Copernican model with elliptical orbits
• Johannes Kepler based his Keplerian Model on Tycho Brahe's precise data

Kepler's Laws

• Elliptical Orbits: Planets move in elliptical orbits, with the Sun at one focus
• Equal Areas in Equal Times: A line from a planet to the Sun sweeps equal areas in equal time intervals. Planets move faster when closer to the Sun and slower when farther away
• Harmonic Law: States the square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit; this relates orbital time to distance from the Sun

Kepler's First Law

• The orbit of each planet around the Sun is an ellipse, with the Sun at one of the foci
• An ellipse is a closed curve around two points called focal points
• Planets do not move in perfect circles but in slightly elongated paths called ellipses
• Kepler's first law explains variations in orbital speed

Ellipse Details

• An ellipse has two foci
• The longest diameter, the major axis, passes through both foci; half of that distance is called the semi-major axis
• A circle is a special case of an ellipse with both foci at the same point

Ellipse - additional details

• The shape of an ellipse is described by its eccentricity, shown by "e" where e ranges from 0 (a circle) to just under 1 (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 eccentricity is defined as e = x/b, where b is the semimajor
• E.g. Venus: e is around 0.007

Kepler's Second Law

• A line from a planet to the Sun sweeps equal areas in equal time intervals
• Kepler's second law describes how a planet's speed varies along orbit
• A planet moves most rapidly when nearest to the Sun, at a point called perihelion
• A planet moves most slowly when farthest from the Sun, at a point called aphelion

Kepler's Second Law - Question

• No change occurs to the speed of a planet moving in a perfect circular orbit

Kepler's Third Law

• The square of the sidereal period of a planet is proportional to cube of the semimajor axis of the orbit
• The larger a planet's orbit, the longer the period to complete an orbit
• Kepler's Third Law is written as P^2 = a^3 where P is the sidereal period in years and a is the length of its semimajor axis in AU

Kepler's Laws - Additional Info

• NOTE: P^2/a^3 = constant no matter what units you use
• Kepler's Laws helped to justify the Heliocentric model of the Solar System
• Kepler's Laws allowed for the calculations of planets more accurately than ever before

Example Application of Kepler's Third Law

• Venus has a distance from the Sun of 0.72 AU, use P^2=a^3 to determine that the sidereal period, P=0.61 years

Question relating planet size to length of orbital period

• Conclusion is that the two planets have the same period

Isaac Newton

• Newton followed observations of Galileo and formulated 3 laws that govern all motion
• A force is a push or a pull
• Newton's First Law: An object moves with a constant velocity unless a non-zero net force acts on it
• Newton's Second Law: An object's acceleration is directly proportional to the net force acting on it, and inversely proportional to its mass
• Newton's Third Law: For every action, there is an equal and opposite reaction

Newton's Laws expressed as formula

• Newton's Laws can be summarized by: a = ΣF/ m or ΣF = ma
• 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 Relations

• Newton could prove Kepler's three laws mathematically using his laws of motion and gravity
• Kepler's first law, concerning the elliptical shape of planetary orbits, is a direct consequence of 1/r^2 in the gravitational law.
• Kepler's second law, (equal areas), states that the Sun's gravitational force is straight toward the Sun

Newton's form of Kepler's Third law

• Newton's form of Kepler's third law is valid whenever two objects orbit each other because of their mutual gravitational attraction
• p^2 = [4π^2/G(m₁+m₂)] a^3 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: P^2 = a^3 is only valid for objects that orbit the Sun!

Another view to Newton and Kepler's laws

• The previous formula can be expressed assuming one one mass is significantly larger than the other
• This happens in planet sun relations, from the suns point of view.
• For this case the ratio P^2/a^3 = 4π^2/Gmsun
• Kepler had discovered this ratio

Vector Nature of Force

• Velocity is a vector quantity
• Changing its direction results in acceleration
• Circular motion is an example

Tides - Introduction

• The first explanation here is "A, a planet moving at constant speed in an unchanging direction has no force exerted on it."
• The tides occur on Earth and are explained primarily by gravity
• Tides are caused by the Moon's gravitational pull
• As distance to the Moon decreases, gravitational force increases; this is the side of Earth closest to the Moon
• In opposition, the side furthest away from Earth has the weakest gravitational force acting on it.
• Earth is fairly solid so the oceans can change shape easily
• This explanation is a simplification - and leaves out Earth's rotation, continents, wind patterns, and the Sun

Sun and tides

• The Sun also exerts tidal forces, about half as strong as the Moon's
• Larger tides occur when the Sun and Moon align. These are Spring Tides.
• When the Sun and Moon form a right angle relative to each other, tidal forces partially cancel each other. These are Neap Tides
• Spring Tides and Neap Tides have nothing to do with the season of Spring

Explore the shift from geocentric to heliocentric models of the solar system. Understand the contributions of Ptolemy, Copernicus, and Brahe as well as challenges faced by each model. Discover now how our understanding of the cosmos has evolved.