Planetary Motion!How We Know Where They’re Going

Early Philosophy

Geocentric Model Earth is the center of the universe Philosophy at the time leads to the idea of

perfection and harmony in the heavens, thus orbits are perfect circles

The Problem? Retrograde Motion:

MarsUranus

Claudius Ptolemy

He still has a geocentric model BUT Earth not exactly at the center

Planets move in circles called epicycles Able to predict future locations Explains retrograde motion

(90-168)

Epicycles and Retrograde Motion

Epicycles and Retrograde Motion

Epicycles and Retrograde Motion

Nicolas Copernicus Proposes heliocentric model to explain

retrograde motion The Sun is at the center of the solar system

now Still assumes perfect circles Able to calculate periods and relative distances Not better than Ptolemaic predictions.

(1473-1543)

Tycho Brahe Recognized the need for new model and

dedicated his life to making more precise measurements Built first modern observatory Recorded planetary positions from 1576 – 1591 2.5x more accurate than any previous records!

(1546-1601)

Johannes Kepler Continued the work of Brahe by trying to

use his data to prove the Copernican model Now recognizes that planetary orbits are

elliptical Developed 3 Laws of Planetary Motion

Answered “what” but not “why”

(1571-1630)

Galileo Galilei Challenges the belief that the heavens

are perfect In 1604, he observes a nova. Why can new

stars appear if everything is perfect as it is? 1609 refines the telescope (didn’t invent

it) 1610, publishes findings:

The surface of the moon isn’t perfect Stars found in Pleiades that are “invisible” to naked eye

(1564-1642)

Galileo Galilei Discovers Galilean moons (1610)

(1564-1642)

Telescope Photograph of Jupiter and the Galilean Moons

Galileo Galilei Venus has phases like the moon

In 1613, publishes a letter documenting sun spots!

(1564-1642)

Shoulders of Giants Isaac Newton (1642-1727)

Formulated 3 Laws of Motion and 1 Law of Universal Gravitation▪ Now we’ve answered the question of “why”▪ Theory matches observation, so we must reexamine

our beliefs Happened again with Einstein in 1911

▪ General Relativity offers new explanation of gravity and explain phenomena that couldn’t be explained by past theories.

▪ Verified experimentally

Kepler’s 1st Law The Law of Orbits

“The orbit of every planet is an ellipse with the Sun at one of the two foci”

The eccentricity of an orbit tells you have elliptical it is▪ An eccentricity of 0 is a circle▪ The further from 0, the more elliptical the

orbit

Better Accuracy

Eccentricity of the Planets

Mercury 0.2056 Jupiter 0.0489 Ceres 0.0789

Venus 0.0067 Saturn 0.0565 Pluto 0.2488

Earth 0.0167 Uranus 0.0457 Haumea 0.1913

Mars 0.0935 Neptune 0.0113 Makemake 0.1559

Eris 0.4407

Copy Into Planet Packets

Kepler’s 2nd Law The Law of Areas

“A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.”

Planets move faster when they are closest to the Sun, and slower when they are farther away

▪ Closest approach is called perihelion▪ For Earth, it occurs on January 3 when we’re

1.46 x 108 miles from the Sun! ▪ Furthest point is called aphelion

▪ For Earth, it occurs on July 4 when we’re 1.50 x 108 miles from the Sun!

Kepler’s 2nd Law

If it sweeps out equal areas in equal times, does it travel faster or slower when it is far from the Sun?

If is sweeps out equal areas in equal times, does it travel faster or slower when far from the Sun?

If is sweeps out equal areas in equal times, does it travel faster or slower when far from the Sun?

Same Areas

Kepler’s 3rd Law The Law of Periods

“The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.”

T in Earth years, D in astronomical units

Kepler’s third law comes from gravitation – Newton used Kepler’s third law to formulate his theory of gravity.

Kepler’s 3rd LawThe farther away a planet is from the Sun, the longer

it takes for that planet to go around the Sun once

Practice

While gazing at the planets that are visible with the naked eye, you tell a friend that the farther a planet is from the Sun, the longer its solar year is. Your friend first asks what a solar year is. After explaining that it’s the time required for a planet to return to its same position relative to the Sun, your friend then asks, “Why does it take longer for the outermost planets to orbit the Sun?” What is your reply?

Newton’s 1st Law of Motion

“An object at rest will remain at rest, and an object in motion will remain in motion with a constant speed and in a straight line unless acted upon by an external force.”

An astronaut floating in space will continue to float forever in a straight line unless some

external force is changing his/her motion.

Newton’s 2nd Law of Motion

The acceleration is inversely proportional to mass and directly proportional and in the same direction to the net force.

Acceleration is a change in velocity or a change in

direction of velocity.

Newton’s laws classify objects as accelerating or non-accelerating, not

as moving or stationary.

Newton’s 3rd Law of Motion

To every action, there is an equal and opposite reaction.

The same force that is accelerating the rocket

forward, is accelerating the exhaust backward.

Gravitation?

So how did Newton revolutionize our understanding of planetary motion using gravitation?

Gravitation?

So how did Newton revolutionize our understanding of planetary motion using gravitation?

Planets/Moons have curved paths↓

Their velocity is changing↓

They’re accelerating↓

There must be a force causing the acceleration!

Weight v. Mass

Starting at Newton’s 2nd Law…

↓…we can look at the force due to gravity

(weight) in Earth’s gravitational field

Newton’s Universal Law of Gravitation

Mass attracts mass The magnitude of the force of attraction is

proportional to the product of their masses and the inverse of the square of the distance between them

F = gravitational force between two objectsm1 = mass of first object

m2 = mass of second objectr = distance between objects

G = universal constant of gravitation

Gravitational Force + Distance

If the bodies are twice as far apart, the gravitational force of each body on the other is ¼ of their previous values.

This is called an “inverse-square law.”

Newton’s description of gravity accounts for Kepler’slaws and explains the motions of the planets and

other orbiting bodies

Orbital Speed

For a satellite:

G = universal constant of gravitationM central = mass of central objectr = distance between objects

Practice

The Earth exerts a gravitational force on an orbiting satellite. Use Newton’s third law to compare the force of the satellite on the Earth. Draw a picture similar to the ones I drew for object A and object B.

According to Newton’s second law, compare the accelerations of the satellite and Earth as a result of their interaction.

Orbital Motion

In order to stay on a closed orbit, an object

has to be within a certain range of

velocities:

Too slow → Object falls back down to Earth

Too fast → Object escapes Earth’s gravity

Geosynchronous Orbit