Tychonic Model  Tycho Brahe - late 1500’s  Combination Copernican & Ptolemaic Models  Introduced as Tycho could not measure parallax  Strong biblical influence  Accepted by many clerics as Copernican system banned

PLANETARY ORBITS Chapter 2

CONIC SECTIONS Circles and Ellipses are Conic Sections - plane cutting a cone produces a circle (plane cuts parallel to the base) or an ellipse (plane cuts at an angle to the base)

PLANETARY GEOMETRY Definition of a Circle A Circle is a figure for which all points on it are the same distance from the centre. Definition of an Ellipse An Ellipse is a figure for which the sum of the distances from any point on the figure to two points inside the figure is always the same

HOW to DRAW an ELLIPSE

Major parts of an ellipse Ellipses How to draw an ellipse ellipticity e = distance between foci / major axis Major parts of an ellipse

HOW to DRAW an ELLIPSE (ellipticity e = distance between foci / major axis – eEarth 0.02, eJupiter 0.05, eMercury 0.21) Major Axis Focus Minor Axis

German mathematician, astronomer His 3 laws led the way to Newtonian KEPLER’S THREE LAWS Johannes Kepler 1571 - 1630 German mathematician, astronomer and astrologer His 3 laws led the way to Newtonian physics

KEPLER’S THREE LAWS First Law: The orbit of each planet is an ellipse with the Sun at one focus Foci Planet Sun

PLANETARY ORBITS

KEPLER’S THREE LAWS First Law: The orbit of each planet is an ellipse with the Sun at one focus Second Law: The line joining the planet and the Sun sweeps out equal areas in equal times.

KEPLER’S SECOND LAW Sun Aphelion Perihelion A B C D Area 1 = Area 2 If Area 1 = Area 2 time to go from A to B is same as time to go from C to D

WHERE DOES A PLANET MOVE FASTEST IN ITS ORBIT? FLASHCARD WHERE DOES A PLANET MOVE FASTEST IN ITS ORBIT? A) At A B) At perihelion C) At aphelion D) At D Sun Aphelion Perihelion A B C D

KEPLER’S THREE LAWS First Law: The orbit of each planet is an ellipse with the Sun at one focus Second Law: The line joining the planet and the Sun sweeps out equal areas in equal times. Third Law: The squares of the planets’ periods of revolution are in proportion to the cubes of the semimajor axes of their orbits

Period must be in years and distance in AU KEPLER’S THIRD LAW (period)2 = (distance)3 Eg. Mars: p = 1.88 year, d = 1.52 AU (1.88)2 = (1.52)3 Sun Planet Semimajor Axis Period must be in years and distance in AU

Flashcard If perihelion of a comet is 2 AU and aphelion is 4 AU, what is the comet’s orbital period around the Sun? (A) 2.1 years (B) 3.2 years (C) 5.2 years (D) 8.0 years

KEPLER’S THIRD LAW

When was comet last at perihelion? Three comments about the orbit? COMET HALLEY’S ORBIT When was comet last at perihelion? Where is the comet now? Three comments about the orbit?

WOULD YOU EXPECT HALLEY’S COMET TO OBEY KEPLER’S LAWS? FLASHCARD WOULD YOU EXPECT HALLEY’S COMET TO OBEY KEPLER’S LAWS? A) Yes B) No

FLASHCARD SPECIFICALLY, DOES HALLEY’S COMET OBEY KEPLER’S THIRD LAW? PERIOD = 76 YEARS, SEMI-MAJOR AXIS = 18 AU A) Yes B) No

ORBITAL DATA Planet Semimaj Axis (AU) Period (years) d3 p2 Mercury 0.39 0.24 0.0593 0.0576 Venus 0.72 0.62 0.3732 0.3844 Erath 1.00 1.000 Mars 1.52 1.88 3.5118 3.5344 Asteroid 2.77 4.60 21.254 21.160 Jupiter 5.20 1.86 140.61 140.66 Saturn 9.54 29.4 868.25 867.89 Uranus 19.19 84.07 7,066 7,068 Neptune 30.06 164.80 27,162 27,159 Pluto 39.60 248.60 62,099 1,802

KEPLER’S THREE LAWS First Law: The orbit of each planet is an ellipse with the Sun at one focus Second Law: The line joining the planet and the Sun sweeps out equal areas in equal times. Third Law: The squares of the planets’ periods of revolution are in proportion to the cubes of the semimajor axes of their orbits

NEWTON’S THREE LAWS 1643 - 1727 perhaps greatest scientist of all time Described theory gravitation which removed last doubts about heliocentric model Showed Kepler’s Laws could be derived from his theory Invented reflecting telescope and theory colour Invented calculus (with Leibniz) In 2005 voted person with greatest influence on science (beat out Einstein).

NEWTON’S THREE LAWS First Law: Every body continues what it is doing (at rest or in motion in a straight line) unless acted upon by an outside force. (Note break here with tradition) Second Law: Change in motion of a body is proportional to the force acting on it and is in the direction that the force is acting. Third Law: For every action there is an equal and opposite reaction.

It is Newton’s third law that allows jets to fly and rockets to be flown to the Moon and beyond.

NEWTON’S UNIVERSAL LAW of GRAVITATION Force = GM1M2/D2 M1 and M2 are the masses of the two objects D is the distance between them G is a constant called the Gravitational Constant Newton found that the orbits derived from this force were exactly those found by Kepler Kepler’s 3rd Law becomes D3 = (M1 + M2) x P2

Flashcard Force = GM1M2/D2 If the distance between two masses increases by 3 times and one mass decreases by 2 times, how will the gravitational force between the masses change? (A) Decreases by 2/3 (B) Decreases by 18 times (C) Increases by 1.5 times (D) Increases by 6 times