Ch.4 Newtonian Gravity

Kepler's first law

Planets orbit in ellipses with Sun at one focus.

Kepler's second law

Equal areas are swept in equal times; planets move faster near Sun.

Kepler's third law

For solar orbits, period squared equals semimajor axis cubed (years, AU).

Semimajor axis

Half the long diameter of an ellipse; describes orbit size.

Perihelion

Point in an orbit closest to the Sun.

Aphelion

Point in an orbit farthest from the Sun.

Where is orbital speed fastest?

At perihelion, closest to the Sun.

Where is orbital speed slowest?

At aphelion, farthest from the Sun.

Newton's first law

An object keeps constant velocity unless acted on by net force.

Newton's second law

Net force equals mass times acceleration (F = ma).

Newton's third law

Forces between two objects are equal and opposite.

Mass

Amount of matter; does not depend on location.

Weight

Force of gravity acting on an object.

Why does constant-speed circular motion require force?

Velocity direction changes, so there is acceleration.

Universal gravitation

All masses attract; force scales with mass product and inverse distance squared.

Double distance between two masses

Gravitational force becomes one-quarter as large.

Double one object's mass

Gravitational force doubles, other factors fixed.

Free fall

Motion under gravity alone; apparent weightlessness is possible.

Orbit

Continuous free fall with sideways velocity.

Bound orbit

Object remains gravitationally bound; often an ellipse.

Unbound trajectory

Object has enough energy to escape; parabola or hyperbola.

Angular momentum conservation

An orbiting object speeds up as its distance to center decreases.

Why do outer planets take longer to orbit?

Larger orbits and lower orbital speeds give longer periods.

Planet at 4 AU: orbital period

8 years, from P² = a³.

Planet with 4-year orbital period: orbital size

About 2.5 AU.

If orbital radius doubles, period changes by

A factor of 2^(3/2), about 2.83.

If orbital period doubles, radius changes by

A factor of 2^(2/3), about 1.59.

Newton's version of Kepler's third law

Orbital period and size also depend on total system mass.

Why is solar-system Kepler constant nearly the same?

The Sun dominates the mass of each Sun–planet pair.

Tides

Differential gravity stretches Earth toward and away from Moon.

Spring tides

Largest tidal range near new and full Moon.

Neap tides

Smallest tidal range near quarter Moons.

Tidal friction

Slows Earth's rotation and helps push Moon farther away.

Synchronous rotation

Rotation period equals orbital period, showing same face.

Discovery of Neptune

Predicted from unexplained deviations in Uranus's orbit; discovered in 1846.

Mercury's orbit anomaly

A limitation of Newtonian gravity explained by Einstein's relativity.

Dark matter

Matter inferred from gravity that does not emit detectable light.