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)

The longest diameter (passing through the 2 foci) is called the major axis. 1/2 major axis = semimajor axis (a) - the avg distance between a planet and the Sun

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