Unit 2: Celestial Mechanics The position of the Earth Ptolemy (90 168 AD) Made tables that allowed a user to locate the position of a planet at any past, present, or future date. In order to maintain circular orbits and yet explain retrograde motions, changes in angular size, etc. Ptolemy used a system of deferents, epicycles, an equant, and an eccentric to predict planetary motions and positions. His system was geocentric (placed Earth in the center of the solar system). Copernicus (1473 1543 AD) In his book, Dē revolutionibus orbium coelestium, (On the Revolutions of the Celestial Spheres), Copernicus summarized a heliocentric theory of the solar system. In this work he states: The sun is the center of the universe The Earth has several motions which account for the apparent motions of the sun, moon, planets and stars The firmament is so far away that the distance from the Earth to the sun is insignificant by comparison Tycho Brahe (1546 1601 AD) Made catalogs of planetary and star positions based on his naked eye observations that were unparalleled He was the last astronomer to record naked eye data He was very jealous of his compilations Kepler (1571 1630 AD) Was an excellent mathematician Assisted Brahe in hopes of inheriting Tycho vast compilations of data Wrote three laws of planetary motion from equations based on Tycho s data 1. The planetary orbits are elliptical 2. A line from the focus to a planet sweeps equal areas in equal times 3. P 2 α a 3 or P 2 = k a 3 where P is the orbital period and a is the semi-major axis of the ellipse Example: PMars = 1.88 yr, find a: a = P 2/3 = (1.88) 2/3 = 1.52 AU Galileo (1564 1642 AD) Known for his telescopic work in astronomy (he refined the telescope but did not invent it) His contributions to astronomy are: Discovery of the four largest moons of Jupiter (named the Galilean moons in his honor) Discovery of sunspots Observed light and shadows caused by mountains on the lunar surface Confirmation of the link between the phases and the angular size of Venus The regular changes in phase and brightness argue a heliocentric solar system as the only means of explanation
Newton (1642 1727 AD) Newton s universal gravitation allows a complete formulation of Kepler s third law: P 2 4π 2 a 3 = G(M S + M P ) Ole Rømer (1644 1710 AD) While collecting data on the times of Io disappearing behind Jupiter, Rømer, discovered that at certain times of the year Io appeared to be several minutes behind schedule. He postulated that the difference was not in Io but rather in light traveling at a finite speed required more time to reach Earth in position 2 below than it did to travel to Earth while occupying position 1. Rømer calculated that the time required by light to travel one diameter of Earth s orbit around the sun would require 22 minutes. The actual time is 16⅔ minutes. Rømer s time was approximately 32% too high meaning his value for the speed of light was 32% too low. Rømer did get a value that was the right magnitude and definitely showed that light is not instantaneous; it has a finite speed. Lunar motions The phases of the moon Sidereal and synodic months
Ascending and descending lunar nodes The orbit of the moon is tilted 5 relative to the plane of the ecliptic (defined by Earth s revolution about the sun). Eclipses cannot happen unless the moon is near one of its nodes. Eclipses of the sun and moon Lunar eclipses only occur if a full moon is 11 ½ before reaching or after passing a node There are three types of lunar eclipses: Total: at maximum phase the moon is totally within the umbral shadow Partial: at maximum phase the moon is only part way in the umbral shadow Penumbral: at maximum phase the moon only enters the penumbral shadow Solar eclipses only occur if a new moon is 17 ¼ before reaching or after passing a node There are four types of solar eclipses: Total: the observer is within the umbral shadow Partial: the observer is in the penumbral shadow Annular: the moon is at apogee and the observer is below the cone of the umbral shadow Hybrid: the eclipse changes between annular and total (not shown below)
Earth s gyrations Axial tilt and the seasons Seasons are more a result of Earth s 23.5 axial tilt than from being at closest approach (perihelion) or furthest approach (aphelion) from the sun. In fact, winter in the northern hemisphere occurs when Earth is at perihelion. Note that a line marking a latitude in the northern hemisphere shows more time in the night side in winter (the longest part is in the darker night side) and more time in the daylight side during the summer. The short days and long nights of winter cause the season to be cold in the northern latitudes, not the fact that we are near perihelion. Precession: The gravity of the sun and moon pull on the Earth s equatorial bulge in such a way as to cause the direction in which the poles point to move in a circular motion through the celestial sphere (a complete circle requires about 26 000 years) Nutation: Due to the elliptical and tilted orbit of the moon, its gravitational and tidal forces cause the path of precession to nod or be rough and bumpy. From NASA Goddard Space Flight Center Jet Propulsion Laboratory http://space-geodesy.gsfc.nasa.gov/multimedia/earthorientationanimations/nutationandprecession.html Go to this link to see an animation of precession and nutation.
Sidereal and solar days Note that during one day the Earth revolves about 1 in its orbit about the sun. Any observer at high noon rotating 360 (one sidereal day) during that time will not reach high noon on the next day but will have to rotate almost 1 more to go from noon to noon (a solar day). Sidereal day 23 hr 56 04 Mean solar day 24 hours Revolution: Earth revolves or orbits about the sun in an elliptical path that is a nearly circular orbit Eccentricity: defines how much the elliptical path varies from a true circle Let: e eccentricity rp the position of the focus at perihelion Then the sun will be at 1 e The current eccentricity for Earth is about 0.0167 but due to gravitational effects of the other planets has varied in the past from about 0.0034 to 0.058. Revolution of Earth (continued) Because of precession of Earth s axis of rotation (see animation above), the seasons also precess. But the axial precession occurs in a clockwise fashion, the only celestial motion for the Earth and the moon that moves in that direction. This causes the sidereal year to be longer than the mean tropical year (sometimes called the solar year). Sidereal year: time required for Earth to revolve 360 relative to the fixed stars. 365.25636 SI days for the J2000.0 epoch. Mean tropical year 365.242189 SI days for the J2000.0 epoch. This is 20 min 24.5 s shorter for the J2000.0 epoch but only 19 min 57.8 s shorter for the average year of the Gregorian calendar.
Planetary motions Conjunction: occurs when another planet as seen from earth lies on a straight line with the sun. This term is also used when any two objects as seen from Earth are at closest approach. Inferior planets have two points of conjunction Inferior conjunction when the order is Earth, the planet, then the sun. Superior conjunction when the order is Earth, the sun, then the planet. Oppositions: occurs when a superior planet is directly opposite the Earth relative to the sun. It is at opposition that superior planets are closest to Earth. Elongation: occurs when an inferior planet is farthest from the sun as viewed from Earth Quadrature: occurs when a superior planet, Earth, and the sun form a right angle. Face the sun to determine if a planet is at eastern or western quadrature. Or, a planet rising at noon is eastern. Synodic period: the time required for a planet to return to its same position relative to the Earth and sun.
Tides Tides result from the differential gravitational forces of the moon and sun. The moon has a greater effect on tides than the sun because it is much closer to the Earth than is the sun. First, consider the forces of the sun (FS) and the moon (FM) on the Earth F S = Gm Sm E R 2 S and F M = Gm Mm E R 2 M so, the ratio F S F M = Gm S m E R 2 S Gm M m E = ( m S ) ( R M m M R 2 M ) 2 = ( 1.98855 1030 kg 384399 km ) ( R S 7.3477 10 22 kg 1.496 10 8 km )2 = 178 Next, consider the difference in the ratios of re /RM and re/rs (To do this properly requires calculus, but a first approximation can be obtained this way with simple ratios) F S = Gm Sm E 2 R S and F M = Gm Mm E 2 R M so, the ratio r E R M re = R S = 390 (as seen in Unit I) R M R S So, we see that the sun applies 178 times more force but the moon is 390 times more effective at generating tides due to its smaller distance. This first approximation gives the effect of the sun as about 45% that of the moon, the same answer a more careful approach would have given. Spring tides occur when the moon is at conjunction or opposition and will produce higher high tides and lower low tides than average. Neap tides occur near first and third quarter moons when the sun-earthmoon angle is 90. Spring and neap tides are usually about two days behind the lunar positions shown at the right because it takes time to move that much water.
The solar system Orbiting inside the Milky Way: our Sol system (the solar system) actually orbits the center of our Milky Way galaxy with an orbital period of 225 to 250 million years. There are theories that our solar system was captured by the Milky Way when the Sagitarius galaxy and the Milky Way collided billions of years ago. That might explain why we do not orbit smoothly within the disk of the Milky Way, but rather we ride above then below the disk like a horse on a merry-go-round ride.