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Gravity • Quiz: based on your observations of yesterday’s lab: • 1. As a planet falls towards the sun what happens to its apparent velocity? • 2. Where do you think the earth travels faster in its elliptical orbit, as it approaches the sun or as it leaves the sun? • 3. What do you think would happen if a large meteorite hit the moon and slowed it down significantly? Astronomy: Part I Ptolemy was an astronomer, mathematician and geographer. He codified the Greek geocentric view of the universe, and rationalized the apparent motions of the planets as they were known in his time. Ptolemy synthesized and extended Hipparchus's system of epicycles and eccentric circles to explain his geocentric theory of the solar system. Ptolemy's system involved at least 80 epicycles to explain the motions of the Sun, the Moon, and the five planets known in his time. The circle was considered as the ideal orbit even if Hipparchus proposed an eccentric motion. It was only Kepler who finally showed that the planet orbits are elliptic and not spherical. Egypt, from approx. 87 to probably 170 AD. • Copernicus was a Polish astronomer and mathematician who was a proponent of the view of an Earth in daily motion about its axis and in yearly motion around a stationary sun. This theory profoundly altered later workers' view of the universe, but was rejected by the Catholic church History of Astronomy - Part II • After the Copernican Revolution, astronomers strived for more observations to help better explain the universe around them • During this time (1600-1750) many major advances in science and astronomy occurred – Kepler's Laws of Planetary Motion – Newton's Laws of Motion and Gravity • Warning! - Math and Equations Ahead! Tycho Brahe - An Observer • Tycho Brahe was a prominent scholar and aristocrat in Denmark in the mid-late 1500's • He made a huge number of observations of the stars and planets, all with the naked eye – Even without a telescope, he was very accurate in his measurements • Also recorded the appearance of comets and supernovae – The Tycho supernova remnant is still visible today Tycho (1546-1601) Johannes Kepler - A Theorist • Shortly before his death, Tycho began working with another scientist named Kepler • Kepler was put to the task of creating a model to fit all of Tycho's planetary data • Kepler spent the remainder of his life formulating a set of laws that explained the motion of the planets Kepler (1571 - 1630) Kepler's First Law • Kepler first noted that the orbital path of a planet around the Sun is an ellipse, not a perfect circle • The Sun lies at one of the foci of the ellipse • The eccentricity of an ellipse is a measure of how 'squished' from a circle the shape is • Most planets in the Solar System are very close to a perfect circle – Eccentricity, e ~ 0 for a circle Focus Focus Kepler's 1st Law: The orbital paths of the planets are elliptical with the Sun at one focus. Kepler's First Law =closest to the Sun =farthest from the Sun Kepler's Second Law • Kepler also noticed that the planets sweep out equal areas in their orbit over equal times • Notice that this means the planet must speed up and slow down at different points • If it takes the same amount of time to go through A as it does C, at what point is it moving faster? – C, when it is closest to the Sun Kepler's 2nd Law: An imaginary line connecting the Sun to any planet sweeps out equal areas of the ellipse over equal intervals of time. Kepler's Third Law • Finally, Kepler noticed that the period of planet's orbit squared is proportional to the cube of its semi major axis Kepler's 3rd Law Simplified P a 2 • This law allowed the orbits of all the planets to be calculated • It also allowed for the prediction of the location of other possible planets 3 NOTE: In order to use the equation as shown, you must be talking about a planet in the Solar System, P must be in years, and a must be in A.U. !!! A.U. = astronaumical unit: How many A.U.s is the earth from the sun? Kepler's Third Law - Examples • Suppose you found a new planet in the Solar System with a semi major axis of 3.8 A.U. P 2 a3 P 2 3.83 54.872 P 54.872 1 2 54.872 7.41 years • A planet with a semi major axis of 3.8 A.U. would have an orbital period of 7.41 years Kepler's Third Law - Examples • Suppose you want to know the semi major axis of a comet with a period of 25 years a3 P2 a 3 252 625 a 625 1 3 3 625 8.55 A.U. • A planet with an orbital period of 25 years would have a semi major axis of 8.55 A.U. Your turn: 2B turned in • 1. Describe Kepler’s first Law • 2. Describe Kepler’s second law • 3. Describe Kepler’s third law 4. Jupiter is 5.1 A.U. from the sun. How many earth years does it take for Jupiter to orbit the sun? 5. Name the scientist who first believed that all planets rotate around our sun Quiz: Kepler’s Law 1.Pluto is 39.1 amu from the sun. How many earth years does it take for Pluto to make one revolution around the sun? • P2 = a3 2. A meteorite has just been spotted orbiting between the Earth and Jupiter. • If it takes 7 years for it to orbit the Sun how far away is it in astonaumical units from Earth? Isaac Newton • Kepler's Laws were a revolution in regards to understanding planetary motion, but there was no explanation why they worked • That explanation would have to wait until Isaac Newton formulated his laws of motion and the concept of gravity • Newton's discoveries were important because they applied to actions on Earth and in space • Besides motion and gravity, Newton also developed calculus Newton (1642-1727) Some terms • Force: the push or pull on an object that in some way affects its motion • Weight: the force which pulls you toward the center of the Earth (or any other body) • Inertia: the tendency of an object to keep moving at the same speed and in the same direction • Mass: basically, the amount of matter an object has • The difference between speed and velocity – These two words have become identical in common language, but in physics, they mean two different things – Speed is just magnitude of something moving (25 km/hr) – Velocity is both the magnitude and direction of motion (35 km/hr to the NE) Newton's First Law • Newton's first law states: An object at rest will remain at rest, an object in uniform motion will stay in motion UNLESS acted upon by an outside force Outside Force • This is why you should always wear a seat belt! Newton's Second Law • Acceleration is created whenever there is a change in velocity – Remember, this can mean a change in magnitude AND/OR direction • Newton's Second Law states: When a force acts on a body, the resulting acceleration is equal to the force divided by the object's mass F a m or F ma • Notice how this equation works: – The bigger the force, the larger the acceleration – The smaller the mass, the larger the acceleration Newton's Third Law • Newton's Third Law states: For every action, there is an equal and opposite reaction • Simply put, if body A exerts a force on body B, body B will react with a force that is equal in magnitude but opposite direction • This will be important in astronomy in terms of gravity – The Sun pulls on the Earth and the Earth pulls on the Sun Newton and the Apple - Gravity • After formulating his three laws of motion, Newton realized that there must be some force governing the motion of the planets around the Sun • Amazingly, Newton was able to connect the motion of the planets to motions here on Earth through gravity • Gravity is the attractive force two objects place upon one another The Gravitational Force Gm1m2 Fg r2 • G is the gravitational constant – G = 6.67 x 10-11 N m2/kg2 • m1 and m2 are the masses of the two bodies in question • r is the distance between the two bodies Gravity - Examples • Weight is the force you feel due to the gravitational force between your body and the Earth – We can calculate this force since we know all the variables Let’s say you have a mass (m1) of 72kg, and the earth’s mass is as indicated with radius to the surface r. N m 2 Gm1m2 Fg 2 r (6.67 10 11 )(72kg)(5.97 10 24 kg) kg 2 6 2 (6.378 10 m) Fg 705 N 1 Newton is approximately 0.22 pounds 0.22lbs Fg 705 N 155lbs 1N Gravity - Examples • What if we do the same calculation for a person standing on the Moon? – All we have to do is replace the Earth's mass and radius with the Moon's Gm1m2 Fg 2 r (6.67 10 11 N m 22 )( 72 kg )( 7 . 35 10 kg) 2 kg 6 2 (1.738 10 m) 2 Fg 117 N 1 Newton is approximately 0.22 pounds 0.22lbs Fg 117 N 26lbs 1N Your turn: 2B turned in • 6. How much would you weigh on the moon in pounds? 2.2 lbs = 1kg Gravity - Examples • If gravity works on any two bodies in the universe, why don't we all cling to each other? – Replace the info from previous examples with two people and the distance with 5 meters Gm1m2 Fg 2 r (6.67 10 11 N m )(72kg)(65kg) 2 kg 2 (5m) 2 8 Fg 0.0000000125N 1.25 10 N 1 Newton is approximately 0.22 pounds 0.22lbs Fg 1.25 10 N 2.75 10 9 lbs 1N 8 Last one: 2B turned in 7. Pick the person sitting next to you and calculate Fg between you and that person. Estimate the distance to that person to the nearest meter. 2.2 lbs. = 1kg. Revisions to Kepler's 1st Law • Newton's law of gravity required some slight modifications to Kepler's laws • Instead of a planet rotating around the center of the Sun, it actually rotates around the center of mass of the two bodies • Each body makes a small elliptical orbit, but the Sun's orbit is much much smaller than the Earth's because it is so much more massive Revisions to Kepler's 3rd Law • Gravity also requires a slight modification to Kepler's 3rd Law 3 a P M1 M 2 2 • The sum of the masses of the two bodies is now included in the equation • For this equation to work, the masses must be in units of solar mass (usually written as M ) • Why did this equation work before? Remember - for this equation to work: P must be in years! a must be in A.U. M1 and M2 must be in solar masses Homework • Complete the homework questions provided • Note that on the back of the sheet is information concerning the velocity of satellites in orbit around a body. Read it and be prepared to answer questions next class.