Orbital MechanicsCTLA Earth & Environmental Science

The Earth❖ Spherical body that is flattened near the poles due to centrifugal

force (rotation of the Earth)

‣ 40,074 KM across at the Equator

‣ 40,0007 KM across at the Poles

❖ Rotation: spinning of Earth around its axis (~24 hours to complete)

❖ Revolution: movement of Earth around the Sun (~365.25 days to complete)

Barycenter❖ The center of mass of two or more celestial

bodies that are orbiting one another

❖ This is the point around which the bodies orbit

❖ Point is based on the masses of the objects and distance between them

r1 = distance between body 1 & the barycenter m1 = mass of body 1 m2 = mass of body 2

a = distance between the centers of the two bodies

Barycenter

The Earth❖ Axis: imaginary line that runs through Earth from the North Pole to the South Pole, on which the

planet rotates around

❖ Earth’s axis is tilted an average of 23.5°

❖ Nutation: wobbling of the tilt of Earth’s axis, which can strengthen or weaken the effects of our seasons

‣ Tilt can increase to 24° or decrease to 22.5°

‣ Maximum cycle of nutation occurs over an 18 year period, due to the Moon’s effects

The Earth❖ Precession: change in the pointed direction of Earth’s axis

‣ Cycle is completed once every 26,000 years

‣ Does not affect intensity of our seasons, but only the time of year they occur

‣ Changes the North Pole stars

๏ Polaris currently (reaches max on March 24, 2100)

๏ Thuban was the N. Pole star around 3000 BC (one-fifth the brightness of Polaris)

๏ Alpha Cephei will be the next star, around 5000 AD

The Seasons❖ Equinox: 12 hours of daylight & 12 hours of darkness (Sun is overhead at the

Equator)

‣ Vernal equinox is currently March 21-22

‣ Autumnal equinox is currently September 21-22

‣ Will be reversed in about 12,000 years due to precession

The Seasons❖ Summer Solstice

‣ Occurs June 21-22

‣ Longest day of the year in the N. Hemisphere

‣ Sun is directly overhead the Tropic of Cancer (~23.5°N)

‣ 24 hours of sunlight for six months at the North Pole

❖ Winter Solstice

‣ Occurs December 21-22

‣ Shortest day of the year in the N. Hemisphere

‣ Sun is directly overhead the Tropic of Capricorn (~23.5°S)

‣ Zero hours of sunlight for six months at the North Pole

Will be reversed in about 12,000 years due to precession!

What about the Moon?❖ A satellite that orbits the Earth (our only natural satellite)

❖ Its gravity is 1/6th of Earth (too weak to hold an atmosphere for itself)

❖ Temperatures reach 134°C in sunlight and -170°C on the dark side of the Moon

❖ Its rate of rotation = its rate of revolution…also called “tidal lock”

Phases of the Moon❖ Full

❖ Waning Gibbous

❖ Last Quarter Half

❖ Waning Crescent

❖ New

❖ Waxing Crescent

❖ First Quarter Half

❖ Waxing Gibbous

Tidal Effects❖ Both the Moon & the Sun affect

Earth’s oceans

❖ Spring Tide: all three bodies (Earth, Moon, & Sun) are lined up, causing high tides to be higher and low tides to be lower

❖ Neap Tide: the bodies are at 90° angles, causing high tides to be lower and low tides to be higher

Eclipses

❖ Lunar eclipse occurs when the Moon travels through Earth’s shadow

❖ Solar eclipse occurs when the Earth travels through the Moon’s shadow

What is an ellipse?❖ A geometric shape with 2 focal points (a circle

has just 1 focal point)

❖ Has two axes (major & minor)

❖ Perihelion = when a planet is closest to the Sun

❖ Aphelion = when a planet is farthest from the Sun

Kepler’s Laws History❖ Tyco Brahe (1546-1601) was a Danish astronomer who found his

measurements showed that Mars did not orbit in a perfect circle

❖ Johannes Kepler (1571-1630) discovered that the orbit of Mars instead fit an elliptical shape

Kepler’s Laws

❖ 1st Law: orbital paths are ellipses and the Sun is at one focal point (nothing is at the 2nd point)

Kepler’s Laws❖ 2nd Law: the line joining the planets to the Sun sweeps out “equal

areas in equal times” as the planet travels around the ellipse

❖ This is due to a planet’s changing revolutionary speed during orbit (faster when close to the Sun & slower when farther from the Sun)

Kepler’s Laws❖ 3rd Law: the ratio of the square of the revolution time for two

planets is equal to the ratio of the cubes of their semi-major axes

❖ T2 = R3 (where T=time of orbit and R=average distance from the Sun)

❖ OR…see the equation below

Planet T R T2 R3