Planetary Motion

Kepler’s Laws apply to any celestial body orbiting any other celestial body. Any planet around a sun The moon around the Earth Any satellite around the Earth The international space station Any rings around any planet

Kepler’s Breakthrough Kepler used Brahe’s data to develop three laws that could be used to describe planetary motion. All of the laws are based upon an understanding of the ellipse.

After Tycho Brahe’s death, Johannes Kepler (pictured here with Tycho in the background) used Tycho’s observations to deduce the three laws of planetary motion.

KEPLER’S THREE LAWS OF PLANETARY MOTION LAW #1. The orbit of a planet around the Sun is an ellipse with the Sun at one focus.

The amount of elongation in a planet’s orbit is defined as its orbital eccentricity. An orbital eccentricity of 0 is a perfect circle while an eccentricity close to 1.0 is nearly a straight line. In an elliptical orbit, the distance from a planet to the Sun varies. The point in a planet’s orbit closest to the Sun is called perihelion, and the point farthest from the Sun is called aphelion.

Perihelion – location of planet at its closest position to the star it is orbiting Aphelion– location of planet at its furthest position to the star it is orbiting

KEPLER’S THREE LAWS OF PLANETARY MOTION LAW #2: A line joining the planet and the Sun sweeps out equal areas in equal intervals of time. Planet moves slower in its orbit when farther away from the Sun. Planet moves faster in its orbit when closer to the Sun.

KEPLER’S THREE LAWS OF PLANETARY MOTION LAW #3: The square of a planet’s sidereal period around the Sun is directly proportional to the cube of its semi-major axis. This law relates the amount of time for the planet to complete one orbit around the Sun to the planet’s average distance from the Sun. If we measure the orbital periods (P) in years and distances (a) in astronomical units, then the law mathematically can be written as P2 = a3.

Orbital Periods for Planets Sidereal Period The true orbital period of a planet with respect to the background stars. Synodic Period The period of time that elapses between two successive identical configurations as seen from Earth (example: for Venus, greatest eastern elongation to greatest eastern elongation)

Newton’s Physics—Motion and Gravity Newton’s Three Laws of Motion A body remains at rest or moves in a straight line at a constant speed unless acted upon by an net outside force. (Inertia) The acceleration of an object is proportional to the force acting on it and dependent upon its mass. (F=ma) Whenever one body exerts a force on a second body, the second body exerts and equal and opposite force on the first body. (for every action there is a opposite and equal reaction)

Newton’s Universal Law of Gravitation: Fgravity = G x m1m2 r2 F = gravitational force between two objects m1 = mass of first object m2 = mass of second object r = distance between objects or d G = universal constant of gravitation If the masses are measured in kilograms and the distance between them in meters, then the force is measured in Newtons Laboratory experiments have yielded a value for G of G = 6.67 × 10–11 Newton • m2/kg2

Lets Review

JOHANNES KEPLER THE SOLAR SYSTEM LAWS OF PLANETARY MOTION

Austrian mathematician Johannes Kepler (1571-1630), interested in how the planets move around the sun, went to Tyco’s island to get these accurate measurements.

At that time, many astronomers believed that planets orbited around the sun in perfect circles, but Tyco’s accurate measurements for Mars didn’t fit a circle. Instead, the mathematician Johannes Kepler found that the orbit of Mars fit an ellipse the best…

What is an ellipse? FIRST LAW OF PLANETARY MOTION An ellipse is a geometric shape with 2 foci instead of 1 central focus, as in a circle. The sun is at one focus with nothing at the other focus. 2 foci FIRST LAW OF PLANETARY MOTION (Law of Ellipses)

An ellipse also has… …a major axis …and a minor axis Perihelion Aphelion Semi-major axis Perihelion: When Mars or any another planet is closest to the sun. Aphelion: When Mars or any other planet is farthest from the sun.

SECOND LAW OF PLANETARY MOTION Kepler also found that Mars changed speed as it orbited around the sun: faster when closer to the sun, slower when farther from the sun… But, areas A and B, swept out by a line from the sun to Mars, were equal over the same amount of time. A B SECOND LAW OF PLANETARY MOTION

THIRD LAW OF PLANETARY MOTION Kepler found a relationship between the time it took a planet to go completely around the sun (T, sidereal year), and the average distance from the sun (R, semi-major axis)… T1 R1 T2 R2 T1 2 R1 3 T 2 = T x T ( ) = R3 = R x R x R T2 2 R2 3 THIRD LAW OF PLANETARY MOTION

Earth’s sidereal year (T) and distance (R) both equal 1 Earth’s sidereal year (T) and distance (R) both equal 1. The average distance from the Earth to the sun (R) is called 1 astronomical unit (AU). T2 R2 Kepler’s Third Law, then, changes to T1 2 R1 3 T1 2 R1 3 or T1 2 = R1 3 = or = T2 2 R2 3 1 1

When we compare the orbits of the planets… Planet T(yrs) R(au) T2 R3 Venus 0.62 0.72 0.38 0.37 Earth 1.00 1.00 1.00 1.00 Mars 1.88 1.52 3.53 3.51 Jupiter 11.86 5.20 141 141 We find that T2 and R3 are essentially equal.

Determining the Distances to Astronomical Objects parallax

Parallax Parallax view: the variation in angle that occurs when viewing a nearby object from different places. Importance of parallax: Danish astronomer Tycho Brahe reasoned that the distance of the object may be determined by measuring the amount of parallax. A smaller parallax angle meant the object was further away.

The apparent change in the location of an object due to the difference in location of the observer is called parallax. Their views differ because of a change in position relative to the mountain

Parallax – apparent difference in position of object viewed from two different locations

Because the parallax of the “star” was too small to measure, Tycho knew that it had to be among the other stars, thus disproving the ancient belief that the “heavens” were fixed and unchangeable. http://www.astronomy.ohio-state.edu/~pogge/Ast162/Movies/parallax.gif http://instruct1.cit.cornell.edu/courses/astro101/java/parallax/parallax.html

Limitation to using parallax Eventually, the parallax shift will no longer be measurable. This is because the distance is too great for the effect to be observed.