Physics 2113 Physics 2113 Lecture 03: FRI 29 AUGLecture 03: FRI 29 AUG

CH13: Gravitation IIICH13: Gravitation III

Physics 2113

Jonathan Dowling

Michael Faraday (1791–1867)Version: 04/18/23

Isaac Newton (1642–1727)

13.7: Planets and Satellites: Kepler’s 1st Law

1. THE LAW OF ORBITS: All planets move in elliptical orbits, with the Sun at one focus.

Johannes Kepler1571–1630Tycho Brahe

1546–1601

Laws were based on data fits!

13.7: Planets and Satellites: Kepler’s 2nd Law

2. THE LAW OF AREAS:A line that connects a planet to the Sun sweeps out equal areas in the plane of the planet’s orbit in equal time intervals; that is, the rate dA/dt at which it sweeps out area A is constant.

Angular momentum, L:

13.7: Planets and Satellites: Kepler’s 3rd Law

3. THE LAW OF PERIODS: The square of the period of any planet is proportional to the cube of the semi-major axis of its orbit.

Consider a circular orbit with radius r (the radius of a circle is equivalent to the semimajor axis of an ellipse). Applying Newton’s second law to the orbiting planet yields

Using the relation of the angular velocity, ω, and the period, T, one gets:

13.7: Planets and Satellites: Kepler’s 3rd Law

ICPP:

(a) The larger the orbit the longer the period: SAT-2.

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(b) The smaller the orbit the greater the speed: SAT-1.

13.7: Newton Derived Kepler’s Laws from Inverse Square Law!http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/KeplersLaws.htm

Kepler’s Second Law First:Equal Areas Proportional to Equal Time!

Rate of sweeping out of area,

is proportional to the angular momentum L, and equal to L/2m = Constant = C.

Angular Momentum

13.7: Newton Derived Kepler’s Laws from Inverse Square Law!http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/KeplersLaws.htm

Kepler’s First Law: Ellipse with Sun at Focus

This is equivalent to the standard (r, q ) equation of an ellipse of semi-major axis a and eccentricity e, with the origin — the Sun — at one focus. Note 1/L2 is from inverse Square Law.

13.7: Newton Derived Kepler’s Laws from Inverse Square Law!http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/KeplersLaws.htm

Kepler’s 3rd Law:For Ellipse

Example, Halley’s Comet

ICPP: Estimate comet’s speed at farthest distance?

13.8: Satellites: Orbits and Energy

As a satellite orbits Earth in an elliptical path, the mechanical energy E of the satellite remains constant. Assume that the satellite’s mass is so much smaller than Earth’s mass.The potential energy of the system is given by

For a satellite in a circular orbit,

Thus, one gets:

For an elliptical orbit (semimajor axis a),

(a)path 1: As E decreases (dE < 0); r decreases (dr < 0)

(b) Less: As r decreases (dr < 0); T decreases (dT < 0)

ICPP

NASA Gravity Recovery and Climate Experiment What Do the Two Satellites Measure? Changing g field!

http://www.jpl.nasa.gov/missions/gravity-recovery-and-climate-experiment-grace/

Earth is NOT a Uniform Sphere —> Gravitational Field Changes in Orbit.

• Changing field Δg give rise to changing velocity Δv.• Changing Δv gives changing satellite-to-sattellite distance. • Microwave link measures changing distance between satellites. • Measuring Δg allows computation of ΔM — Earth’s Mass Distribution.

ΔM

Rocky Mtn. High

mid-Atl. Low

Example, Mechanical Energy of a Bowling Ball

13.9: Einstein and Gravitation: Curvature of Space

13.9: Einstein and Gravitation: Gravity Waves

LIVINGSTON LASER INTERFEROMETER GRAVITATIONAL-WAVE OBSERVATORY

Disturbances in the Gravitational Field Move Outward As Waves

Two Orbiting Black Holes

HW01 DUE TONIGHT: 11:59PM FRI 29 AUG!WEB ASSIGN CLASS KEY FOR SECTION 2: lsu 8181 3713

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