The Organization of the Solar System

and Planetary Motion

At first… The Egocentric Earth

• Ancient Greece: planets, Sun, Moon, and stars orbit around the Earth– Pythagoras and followers used math to

describe natural phenomena– Aristotle: the universe is governed by

regular laws

Ptolomy’s Geocentric Model• Complex interactions of circles • Each planet orbits around a circle -

the epicycle• The epicycle orbits on a bigger

circle - the deferent - around the Earth

• The center of the deferent – a point near the midpoint of the

distance between Earth and the Equant

• The equant – the point at which an epicycle’s center

always seems to move at the same speed.

– each planet has a different equant.

http://www.astro.utoronto.ca/~zhu/ast210/geocentric.html

Problems with the Geocentric Model

• Retrograde motion of planets– Planets move eastward (left) relative to background of

stars in the northern hemisphere– Occasionally, the planets seem to stop and then back

up ofor several weeks or months (westward movement)

• The mechanical description of a geocentric model was complex

• As observations improved, the model increasingly failed to fit the data

• Aristarchus: all planets orbit the Sun (ignored for 1800 years)

Copernicus: The Heliocentric Model

• Assumed that everything orbits the Sun (circular orbits)

• Mercury and Venus are always observed fairly near the Sun, then they must be closer to the Sun

• Mars, Jupiter, and Saturn are high in the sky in the middle of the night, when the Sun is far below the horizon - they are farther way

Kepler’s First Law

• Based on planets’ positions against the background of distant stars, orbits must be elliptical

• First Law:– The orbit of a planet about the

Sun is an ellipse with the Sun at one focus

– Orbital Eccentricity (e):• Roundest ellipse is a circle

(e = 0)

• Straight line (e = 1)

1/2 major axis = semimajor axis (a) - the avg distance between a planet andthe Sun

The longest diameter (passing through the 2 foci) is called the major axis.

Kepler’s Second Law

• A line joining a planet and the Sun sweeps out equal areas in equal intervals of time (Law of Equal Areas)

Kepler’s Third Law• The square of a planet’s sidereal period around the

Sun is directly proportional to the cube of the length of its orbit’s semimajor axis– Sidereal period: orbital period of one object about

another measured with respect to the stars– Rotation versus revolution

• A solar day (defined as noon-to-noon) is different from a sideral day (defined as one Earth rotation).

• Mean Solar day: 24hrs• Sidereal day: 23hrs, 56min• This means that a fixed star rises 4 mins earlier each successive

night, or two hours earlier each month.

– P2 = a3 – A planet closer to the Sun has a shorter year than does a

planet further from the sun

Galileo

• Venus appears smallest at gibbous phase and largest at crescent phase

• This observation supported the fact that Venus orbits the Sun

Galilean Moons

• Orbit Jupiter because they move across from one side of the planet to the other

• Jupiter’s four moons obey Kepler’s third law

Newton’s First Law• Law of Inertia:

– A body remains at rest or moves in a straight line at constant speed unless acted upon by a net outside force

– There must be an outside force acting on the planets, otherwise they would move away from the Sun along straight-line paths at constant speeds

– Some force confines the planets to their elliptical orbits

Due to momentum (how much the object tends to stay in motion):

Momentum = Mass x Velocity

Newton’s Second Law• The acceleration of an object

is proportional to the force acting on it

• The harder you push on an object, the greater the resulting acceleration

• F = m x a• If an object revolves around

the Sun in circular orbit at constant speed, then it is constantly accelerating in order to change the direction of its motion

Net force causes acceleration

Newton’s Third Law

• Whenever one body exerts a force on a second body, the second body exerts an equal and opposite force on the first body

• The Sun exerts a force on each planet to keep it in orbit, and each planet exerts an equal and opposite force on the Sun

Conservation of Angular Momentum

• Sun is pulling the planets• Planets don’t fall onto the Sun due to

conservation of angular momentum• Measure of how much energy is stored in an

object due to its rotation and revolution– Depends on mass, rotation, revolution, and how

spread out the mass is

• As the orbiting planets fall Sunward, their angular momentum provides them with motion perpendicular to that infall thus continuously missing the Sun

Newton’s Law of Universal Gravitation

• Two bodies attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

• Gravitational force decreases with distance• F = G (m1m2/r2)

– G = 6.668 x 10-11 N m2 kg-2 (universal constant of gravitation)

– M = masses– R = radius

Example• mEarth = 6.0 x 1024 kg

• mSun = 2.0 x 1030 kg

• rS-E = 1.5 x 1011 m

• Then,– F = 3.6 x 1022 N– Since, F = m x a,

• aEarth = 6.0 x 10-3 m/s2

• aSun = 1.8 x 10-8m/s2

• Earth pulls on the Sun, causing the Sun to move toward it, but due to the Sun’s greater mass, the Sum accelerates the Earth 300,000x more than the Earth accelerates the Sun