Tycho Brahe (1546-1601) Danish nobility; lost nose in duel (so had metal one). Got King Frederick II to give him a little island and build the world’s best observatory on it. Designed, built and used very accurate instruments for measuring sky positions. Kept voluminous records for years. Hired Kepler to try to understand motion of Mars. Had model with Sun going around Earth, but planets orbit Sun. Found that comets moved between planetary orbits (not Ptolemaic). Motion of Mars still not fully explained. Fell out of favor; moved. Uraniburg

Johannes Kepler (1571-1630) Born sickly and poor. Smart: got scholarships. Became Lutheran minister; learned Copernicus. Went to work with Tycho to escape 30 Years War. Tycho withheld important data until he died in 1601. Kepler proposed geometrical heliocentric model with imbedded polygons (clever and aesthetic, but not better). With full Mars data, Kepler found his laws of planetary motion in 1605 and published in 1609. Had to keep moving around, but kept publishing better predictions of planetary positions, which were confirmed observationally. A recent note: it turns out that the 1609 publication did not contain real data, but data generated using the laws (which constitutes no independent support at all… Bad Science!).

Ellipses An ellipse is an example of a “conic section”. Circles and hyperbolas are others. All are possible forms for orbits. You can make an ellipse with 2 tacks and a string. The tacks are the “foci”, and if you put them further apart, the ellipse is more “eccentric” (one tack makes a circle).

Kepler’s Laws of Planetary Motion The planets move in elliptical orbits, with the Sun at one focus. 2) A line between a planet and the Sun sweeps out equal areas of the ellipse in equal amounts of time. Notes: There is nothing at the other focus or in the center. The Second Law means that planets swing around the Sun faster when they are closer to it. These laws work for anything orbiting around anything due to gravity.

Kepler’s Second Law Animated

Kepler’s Third Law 3) The orbital period of a planet is proportional to its semi-major axis, in the relation P2 ~ a3 The more general form of this law (crucial for determining all masses in Astronomy is For the planets (with the Sun as the central mass), you can take the units to be AU for a (semi-major axis) and years for P (with M in solar masses). Then all the numbers are “1” for the Earth. Example: if Jupiter is at 5 AU, how long is its orbital period? Kepler didn’t understand the physical basis of these laws (though he suspected they arose because the Sun attracted the planets, perhaps through magnetism he speculated.

Isaac Newton (1642-1727) One of world’s greatest scientists. Co-inventor of calculus. Discovered the law of Universal Gravitation. Newton's 3 laws of motion. Corpuscular theory of light. Law of cooling. Professor, Theologian, Alchemist, Warden of the Mint, President of Royal Society, member of Parliment. Personally rather obnoxious, poor relations with women, lots of odd stuff with the great stuff. Did most of it in before he turned 25! Trinity College, Cambridge

Newton’s Three Laws 1) The Law of Inertia: objects will move at a constant velocity unless acted upon by forces. (really Galileo’s law) 2) The Force Law: a force will cause an object to change its velocity (accelerate) in proportion to the force and inversely in proportion to the mass of the object. This can be expressed: F = m * a or a = F / m 3) The Law of Reaction: forces must occur in equal and opposite pairs – for every action there is an equal and opposite reaction.

Newton’s Law of Gravitation

Astro Quiz Why are astronauts in the Space Shuttle weightless? 1) The extra inertial of the Shuttle just compensates for the extra gravitational pull on it, so it falls at the same rate as the astronauts. 2) The Shuttle is sufficiently high in its orbit that the Earth’s gravitational pull is negligible. 3) The Shuttle’s engines keep it on a path that matches the Earth’s curve, and there is no air resistance.

Newton explains Kepler’s Laws Newton was able to show mathematically (using his calculus), that for inverse square forces, the orbits are ellipses and obey Kepler’s laws. He realized this must apply to all celestial bodies. In particular, he could show that the period and size of an orbit are given by: Where P is period, a is semi-major axis, M is central mass, and G is the “gravitational constant” that expresses the strength of gravity (in the right units, of course). Thus, this law (or Kepler’s Third Law) can be used to find the mass of any body in which an orbiting body’s period and distance can be measured (starting with the Earth-Moon system).

Finding the mass of the Earth We know that the Sun is about 400 times further away than the Moon, and takes a month to orbit the Earth. Thus, its semi-major axis is about 1/400 AU, and its period is about 1/12 years. Since we used AU and years, the mass is in solar masses. So the Earth is about a million times less massive than the Sun. To find out how many kilograms (or whatever) it has, we have to use the form of Kepler’s Law given by Newton, and put in all the physical units [like P(sec), a(meters), G (mks units)]. In this class, we will always use ratios and avoid units (so we get relative comparisons).

Orbital Motion Gravity always makes things fall. The question is whether the path of the fall intersects any surface. The shape of the orbit depends on the velocity the body has at a given point. Low velocity will make the point the highest, high velocity will make it the lowest (circular orbits mean it has to be “just right”). If the velocity is too high, the orbit will be a hyperbola instead of an ellipse, and the body will not return.

Orbital and Escape velocity Escape velocity depends on the mass and size of the body. It is about 11 km/s from the Earth. You have a black hole when it is the speed of light (you need a lot of mass in a little size). Note these velocities do not depend on the mass of the escaping or orbiting body.