# 4π 2 G(M1+M2 ) p2 = Newton s 3 Laws of Motion AST 105. Introduction to Astronomy: The Solar System. Newton's Version of Kepler's Third Law

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1 REVIEW Newton s 3 Laws of Motion AST 105 Introduction to Astronomy: The Solar System Announcement: First Midterm this Thursday 02/25 Newton's Version of Kepler's Third Law Newton's Version of Kepler's Third Law Orbital Time = Total Distance / Speed r = a for circular P = 2πa / V Orbit 1. An object moves at constant velocity if there is no net force acting on it 2. When a force, F, acts on a body of mass M, it produces in it an acceleration, A, equal to the force divided by the mass. Or A = F/M 3. For any force, there is always an equal and opposite reaction force. circ P2= (2πa)2 / (Vcirc)2 P2 = 4π2a2 / (GM/a) circular orbits P2 = {4π2/GM} a3 What is M again? M = Mass of the OrbitEE p2 = General expression when M1 and M2 are comparable 4π 2 a3 G(M1+M2 ) p = orbital period a = average orbital distance (between centers) = semi-major axis (M1 + M2) = sum of object masses! In most cases, M1+M2 M1 (Mass of the bigger object)

2 Clicker Question Kepler's 3rd Law Newton's Version of Kepler's 3rd Law P2 = a3 P2 = {4π2/GM} a3 Years A.U. sec kg meters G = 6.67 x m3 / (kg s2) ONLY works for orbiting the SUN works for ANYTHING orbiting ANYTHING Clicker Question In Newton s version of Kepler s 3rd Law (p2=4π2a3/g(m1+m2)), why is M1+M2 usually written as just M? A. Because Newton s version only works for one mass. B. Newton s version was designed for planets. C. You only need the mass of the object you want the period of. D. Because usually one of the masses is much smaller than the other. E. The gravitational constant, G, only contains kg, not kg2. In Newton s version of Kepler s 3rd Law (p2=4π2a3/g(m1+m2)), why is M1+M2 usually written as just M? A. Because Newton s version only works for one mass. B. Newton s version was designed for planets. C. You only need the mass of the object you want the period of. D. Because usually one of the masses is much smaller than the other. E. The gravitational constant, G, only contains kg, not kg2. Clicker Question The mass of Jupiter can be calculated by A. Knowing the Sun s mass and measuring the average distance of Jupiter from the Sun B. Measuring the orbital speed of one of Jupiter s moons. C. Measuring the orbital period and distance of Jupiter s orbit around the Sun D. Knowing the Sun s mass and measuring how Jupiter s speed changes during its elliptical orbit around the Sun E. Measuring the orbital period and distance (from Jupiter) of one of Jupiter s moons.

3 Clicker Question The mass of Jupiter can be calculated by A. Knowing the Sun s mass and measuring the average distance of Jupiter from the Sun B. Measuring the orbital speed of one of Jupiter s moons. C. Measuring the orbital period and distance of Jupiter s orbit around the Sun D. Knowing the Sun s mass and measuring how Jupiter s speed changes during its elliptical orbit around the Sun E. Measuring the orbital period and distance (from Jupiter) of one of Jupiter s moons. NVK3L can be used on anything that orbits! Ida (Asteroid) Using Newton s Version of Kepler s 3rd Law (NVK3L) M= {4π2/G} Measure a Direct Observation, Parallax Calculate M = mass of orbitee NVK3L can be used on anything that orbits! 3 M = 4Gπ 2 a 2 P 3 & 1000m ) 100km ( + 4π 2 ' 1km * M= 3 2 m & 86400s ) day + kg s2 (' 1day * = {4π2/G}a3/P2 a P Dactyl (asteroid moon) a ~ 100 km Dactyl (asteroid moon) kg Timing: Count seconds to make 1 orbit Ida (Asteroid) P ~ 1.5 days Three variables If we know two, we can solve for the third! Measure P Mass of Ida a a3/p2 M = kg

4 NVK3L can be used on anything that orbits! How to weigh a black hole Watch stars orbit Even if you can t see the object in the center! an invisible center Mass of supermassive black hole = {4π2/G}a3/P2~ 6 millions of Suns!! PS0-2~ 16 years as0-2 ~ 1000 AU Discovery of of dark matter in the Milky Way Dark matter halo for galaxies Dark matter extends beyond visible part of the galaxy -- mass is ~10x stars and gas! DARK MATTER Probably not normal mass that we know of (protons, neutrons, electrons). Most likely subatomic particles, as yet unidentified!!! BIG MISTERY!!!

5 Clicker Question Even on complex problems like NVK3L, you can use the logical method to compare with known quantities How long would one orbit (for Earth) take if the Sun was 4 times as massive as it currently is? A. B. C. D. 1/16th of a year (~3 weeks) 1/4th of a year (3 months) 1/2 of a year (6 months) 1 year (12 months, 52 weeks, 365 days, π 107 seconds) E. 2 years Clicker Question How long would one orbit (for Earth) take if the Sun was 4 times as massive as it currently is? A. B. C. D. 1/16th of a year (~3 weeks) 1/4th of a year (3 months) 1/2 of a year (6 months) 1 year (12 months, 52 weeks, 365 days, π 107 seconds) E. 2 years How long would one orbit (for Earth) take if the Sun was 4 times as massive as it currently is? p2 = 4π 2 a3 GM M 4 1/M 4 p2 4 p 4 = 2

6 What if the orbiter and orbitee are of similar masses? What if the orbiter and orbitee are of similar masses? Pluto & Charon Center of Mass P 2 = 4π 2 a GM P 2 = 4π 2 a G(M 1 +M 2 ) What if the orbiter and orbitee are of similar masses? Pluto & Charon P 2 = 4π 2 a GM Midterm I Covering Lectures Notes 2 through 9 Strictly individual effort P 2 = 4π 2 a G(M 1 +M 2 ) Next: Midterm I review

7 Earth Scales in the Universe: our Cosmic Address Lookback Time The image of a galaxy spreads across 100,000 years of time Sun/Solar System Milky Way Galaxy Try to think of what we SEE NOW as different from what may EXIST now Local Group Local Supercluster Daily Motions - Apparent Daily Motions - Actual Apparent motion = Stars appear to move counterclockwise (when looking north) Rise in the east Set in the west Actual motion = Earth spins with axis pointed in fixed direction - towards Polaris West to East: NY leads LA

8 Apparent Motion of the Sun Do the planets reverse course? Planets usually move slightly eastward from night to night (not in the course of one night!!) relative to the stars. But sometimes they go westward relative to the stars for a few weeks: apparent retrograde motion Equinox We see apparent retrograde motion when we pass by a planet in its orbit. The REAL Reason for the Seasons

9 When the Sun is high in When the Sun is low in the sky, the amount of the sky, the amount of direct sunlight received is direct sunlight received is greater. This results in less. This results in SUMMER WINTER NP Although the Moon is always ½ lit by the Sun, we see different amounts of the lit portion from Earth depending on where the Moon is located in its orbit. Moon is illuminated (always ½) by Sun Phases of Moon Lunar Eclipse We see a changing combination of the bright and dark faces as Moon orbits the Earth See this demonstration: The Earth lies directly between the Sun and the Moon

10 Solar Eclipse Next: The Science of Astronomy The Moon lies directly between the Sun and the Earth Geocentric vs Heliocentric How did the Greeks explain planetary motion? Plato Earth-Centered (Geocentric) Sun-Centered (Heliocentric) Aristotle Greek geocentric model (c. 400 B.C.)

11 How did Copernicus challenge the Earth-centered idea? Copernicus (early 1500s): Kepler first tried to match Tycho's observations with circular orbits Proposed Sun-centered (heliocentric) model with circular orbits to determine layout of solar system But... Model was no more accurate than Ptolemy's model in predicting planetary positions Tycho Brahe Johannes Kepler If I had believed that we could ignore those eight minutes of arc, I would have patched up my hypothesis accordingly. But, since this was not permissible to ignore, those eight minutes pointed the road to a complete reformation in astronomy Eventually he decided to try ellipses These orbits allowed him to match the data (and predict future observations) nearly perfectly 41 Kepler's 3 Laws of Planetary Motion Kepler's First Law: The orbit of each planet around the Sun is an ellipse with the Sun at one focus. Kepler's 3 Laws of Planetary Motion Kepler's Second Law: As a planet moves around its orbit, it sweeps out equal areas in equal times 30 days 30 days 30 days } a The areas swept out during all equal time periods are equal

12 Kepler's 3 Laws of Planetary Motion Kepler's Third Law: Planetary orbits follow the mathematical relationship: p2 = a 3 p = orbital period in years a = avg. distance from Sun in AU = semi-major axis of elliptical orbit Newton s 3 Laws of Motion 1. An object moves at constant velocity if there is no net force acting on it 2. When a force, F, acts on a body of mass, M, it produces in it an acceleration, A, equal to the force divided by the mass. Or A = F/M 3. For any force, there is always an equal and opposite reaction force. Planet's mass doesn't matter! Acceleration Due to Earth's Gravity Universal Law of Gravity F = MA A = F/M G G = 6.67 x m3/(kg s2) "BIG G" Ac = M Earth M cabbage 2 REarth M cabbage GM Ac = 2 E RE M2 = M c d M1= MEarth d = radius of Earth=RE

13 How is mass different from weight? Mass the amount of matter in an object Weight the force that acts upon an object (based on acceleration and mass) Your weight can change a lot depending on where you are but your mass doesn't. Orbital Speed Planets (orbiters) are constantly trying to get away Gravity (from the orbitee) is constantly pulling them back We can use Newton's law of gravity to calculate how fast they move 49 Escape Velocity The velocity needed to escape the gravity of the orbitee Vescape = 2GM orbitee r Earth Vcirc= 8 km/s Vesc= 11 km/s Vcircular = GM orbitee r Orbiter Orbitee Vcircular = GM orbitee r

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