Keplers Laws What are the shapes and important

Kepler’s Laws • What are the shapes and important properties of the planetary orbits? • How does the speed of a planet vary as it orbits the sun? • How does the period of a planet's orbit depend on its distance from the Sun?

Kepler’s First Law • The orbits of the planets are elliptical (not circular) with the Sun at one focus of the ellipse. • 'a' = semi-major axis: Avg. distance between sun and planet

Kepler’s Second Law ¬Kepler determined that a planet moves faster when near the Sun, and slower when far from the Sun. Planet Faster Sun Slower

Kepler's Second Law A line connecting the Sun and a planet sweeps out equal areas in equal times. slower faster Translation: planets move faster when closer to the Sun.

Kepler's Third Law The square of a planet's orbital period is proportional to the cube of its semi-major axis. P 2 is proportional to a 3 or P 2 (in Earth years) = a 3 (in A. U. ) 1 A. U. = 1. 5 x 108 km Translation: The further the planet is from the sun, the longer the period.

Correction to Kepler’s Third Law ¬ Earth and sun actually rotate about their common center of mass ¬ Corresponds to a point inside sun ¬ Used to detect extrasolar planets

Why? ¬Kepler’s Laws provided a complete kinematical description of planetary motion (including the motion of planetary satellites, like the Moon) - but why did the planets move like that?

The Apple & the Moon ¬Isaac Newton realized that the motion of a falling apple and the motion of the Moon were both actually the same motion, caused by the same force - the gravitational force.

Universal Gravitation ¬Newton’s idea was that gravity was a universal force acting between any two objects.

Gravitational Field Strength ¬To measure the strength of the gravitational field at any point, measure the gravitational force, F, exerted on any “test mass”, m. ¬Gravitational Field Strength, g = F/m

Gravitational Field Strength ¬Near the surface of the Earth, g = F/m = 9. 8 N/kg = 9. 8 m/s 2. ¬In general, g = GM/r 2, where M is the mass of the object creating the field, r is the distance from the object’s center, and G = 6. 67 x 10 -11 Nm 2/kg 2.

Gravitational Force ¬If g is the strength of the gravitational field at some point, then the gravitational force on an object of mass m at that point is Fgrav = mg. ¬If g is the gravitational field strength at some point (in N/kg), then the free fall acceleration at that point is also g (in m/s 2).

Gravitational Field Inside a Planet ¬If you are located a distance r from the center of a planet: – all of the planet’s mass inside a sphere of radius r pulls you toward the center of the planet. – All of the planet’s mass outside a sphere of radius r exerts no net gravitational force on you.

Gravitational Field Inside a Planet ¬The blue-shaded part of the planet pulls you toward point C. ¬The grey-shaded part of the planet does not pull you at all.

Gravitational Field Inside a Planet ¬Half way to the center of the planet, g has one-half of its surface value. ¬At the center of the planet, g = 0 N/kg.

Kepler’s Laws are just a special case of Newton’s Laws! ¬Newton explained Kepler’s Laws by solving the law of Universal Gravitation and the law of Motion ¬Ellipses are one possible solution, but there are others (parabolas and hyperbolas)

Kepler’s Laws are just a special case of Newton’s Laws! ¬Newton explained Kepler’s Laws by solving the law of Universal Gravitation and the law of Motion ¬Ellipses are one possible solution, but there are others (parabolas and hyperbolas)

Bound and Unbound Orbits Unbound (comet) Unbound (galaxy-galaxy) Bound (planets, binary stars)

Understanding Kepler’s Laws: conservation of angular momentum L = mv x r = constant r larger distance smaller v planet moves slower smaller distance smaller r bigger v planet moves faster

Understanding Kepler’s Third Law Newton’s generalization of Kepler’s Third Law is given by: p 2 = 4 p 2 a 3 G(M 1 + M 2) Mplanet << Msun, so p 2 = 4 p 2 a 3 GMsun

This has two amazing implications: ¬The orbital period of a planet depends only on its distance from the sun, and this is true whenever M 1 << M 2

An Astronaut and the Space Shuttle have the same orbit!

Second Amazing Implication: ¬If we know the period p and the average distance of the orbit a, we can calculate the mass of the sun!