Celestial Mechanics AST12 Notes PDF

AST12 - Celestial Mechanics Sir Lhevy Lessons Terms to Remember The Julian Period - This was proposed by Joseph Scaliger in 1583. - The Julian Period was based on the consecutive numbering of days from January 1, 4713 BC (This was chosen in which is the result of three cycles: Solar Cyc...

AST12 - Celestial Mechanics Sir Lhevy Lessons Terms to Remember The Julian Period - This was proposed by Joseph Scaliger in 1583. - The Julian Period was based on the consecutive numbering of days from January 1, 4713 BC (This was chosen in which is the result of three cycles: Solar Cycle, Metonic Cycle, and Indiction Cycle began together). - This idea was adopted by John Herschel (He originated the use of the Julian day system in Astronomy). Solar Cycle - The Solar Cycle of the Julian Calendar lasts for 25 years in which dates recur on the same days of the week. Metonic Cycle - The Metonic Cycle of the Julian Calendar, also known as “Lunar Cycle” lasts for 19 years in which the phases of the moon recur on a particular day in the solar year, or year of the seasons. Indication Cycle - The Indication Cycle of the Julian Calendar lasts for 15 years. Originally, this is the schedule of periodic taxes or government requisitions in Ancient Rome.  CELESTIAL TIMEKEEPING AND NAVIGATION LOCAL SKY Zenith - The point directly overhead, which has an altitude of 90°. Horizon - A boundary that divides what we can see from what we cannot see. North Celestial Pole - The point on the celestial sphere directly above Earth’s North Pole. Meridian - A half-circle extending from your horizon (altitude 0°) due to south, through your zenith, to your horizon due north. Celestial Equator - The extension of Earth’s equator onto the celestial sphere.  LENGTH OF A DAY Sidereal Day - A sidereal day is the time it takes any star to make a circuit of the local sky. It is about 23 hours and 56 minutes. - The period of the rotation of the Earth is approximately 23 hours, 56 minutes, and 4.09 seconds. Solar Day - A solar day is the time it takes the Sun to make a circuit of the local sky. Its precise length varies slightly over the course of the year, but the average is 24 hours. - Solar day is longer than a sidereal day by about 1/360 because Earth moves about 1° in orbit each day.  LENGTH OF A MONTH Sidereal Month - The Moon orbits Earth in 27.3 days. - Earth & Moon travel 30° around the Sun during that time (30°/60° = 1/12). Synodic Month - A cycle of lunar phases; Therefore takes about 29.5 days, 1/12 longer than a sidereal month. LENGTH OF A YEAR Sidereal Year - Time for Earth to complete one orbit of Sun Tropical Year - Time for Earth to complete one cycle of seasons - Tropical year is about 20 minutes (1/26,000) shorter than a sidereal year because of Earth’s precession. Planetary Periods - Planetary periods can be measured with respect to stars (sidereal) or to apparent position of the Sun (synodic). Conjunction - An event in which a planet and the Sun line up in our sky. Opposition - The point at which a planet appears opposite from the Sun in our sky. Greatest Elongation - For Mercury or Venus, the point at which it appears farthest from the Sun in our sky.  HOW DO WE TELL THE TIME OF DAY? Apparent Solar Time - This depends on the position of the Sun in the local sky - An example of a device that gives the apparent solar time is a Sundial. - Length of an apparent solar day changes during the year because Earth’s orbit is slightly elliptical. - It is based on the actual position of the Sun in the sky, accounting for the Earth’s orbital and axial characteristics. Mean Solar Time - Defined according to the Sun’s position relative to the meridian. - It is based on the average length of a day. - Noon is the average time at which the Sun crosses the meridian. - It is a local definition of time. - It is a uniform and average timekeeping system. Universal Time (UT) - Also known as “Greenwich Civil Time”, is based on the rotation of the Earth. - Universal Time is incorrectly called “Greenwich Mean Time” in Great Britain. - UT is necessary for civil life and for the astronomical calculations where local hour angles are involved. - UT begins at midnight while GMT begins at noon (GMT and UT differ by 12 hours). Ephemeris Time (ET) - This was made because the astronomers needed a uniform time scale for their accurate calculations for Celestial Mechanics, Orbits, and Ephemerides. - It was used from 1960 to 1983 in the great astronomical almanacs such as the Astronomical Ephemeris. - It is defined by the laws of dynamics: it was based on the planetary motions. Dynamical Time - This was the replacement of ET in 1984. - It is defined by atomic clocks; In fact, a prolongation of the Ephemeris Time.  TIME BY THE STARS Hour Angle (HA) - Defined as the time since an object in the celestial sphere last crossed the meridian (or the higher of its two meridian crossing points for a circumpolar star). - For example: If a star is crossing the meridian now, its hour angle is 0h. If a star crossed the meridian 3 hours ago, its hour angle is 3h. If a star will cross the meridian 1 hour from now, its hour angle is -1h or, equivalently, 23h. WHEN AND WHY DO WE HAVE LEAP YEARS? The length of a tropical year is about 365.25 days. In order to keep the calendar year synchronized with the seasons, we must add one day every four years (February 29). For precise synchronization, years divisible by 100 (e.g., 1900) are not leap years unless they are divisible by 400 (e.g., 2000)  HOW DO WE LOCATE OBJECT ON THE CELESTIAL SPHERE? Celestial Coordinates Right Ascension - Like Longitude on celestial sphere (measured in hours with respect to spring equinox). Declination - Like latitude on celestial sphere (measured in degrees above celestial equator).  HOW DO STARS MOVE THROUGH THE LOCAL SKY? - A star’s path depends on your latitude and the star’s declination. Star Paths HOW DOES THE SUN MOVE THROUGH THE LOCAL SKY? Special Latitudes Sun’s Path  PRINCIPLE OF CELESTIAL NAVIGATION How can you determine your latitude? 1. At night: - Look for the North Star (Polaris) in the Northern Hemisphere. - Measure its altitude. Your latitude is approximately equal to Polaris’s altitude. - For more accuracy, measure the altitude of any stat as it crosses your meridian. 2. During the day: - Observe the Sun’s altitude when it crosses your meridian. - Know the date and the Sun’s declination on that date. - Calculate your latitude using the Sun’s altitude and declination. For example: If Polaris is 17° high, your altitude is between 16°N and 18°N. If Vega is at 78° 44’ altitude and its declination is +38° 44’, your latitude is 50°N. On March 21, if the Sun is at 70° altitude, and its delineation is 0°, your latitude is 20°S. Visualization: How can you determine your longitude? 1. During the day: - Use a sundial to determine your local time. - Compare it with the time in a known longitude (like Greenwich). - The time difference indicates your longitude. - Each hour difference corresponds to 15° of longitude. - If your local time is earlier than the known time, you’re west; if it’s later, you’re east. 2. At night: - Compare the positions of stars in your sky with their positions in a known longitude. - Determine the time difference based on when a specific star crosses your meridian compared to its crossing in the known longitude. - Convert the time difference into longitude, where each hour corresponds to 15°. For example: If your local time is 2 hours earlier than Greenwich, you’re at 30°W longitude. If Vega crosses your meridian 6 hours before Greenwich time, you’re at 90°E longitude. Visualization: CELESTIAL NAVIGATION IN PRACTICE  CALCULATING POSITION OF THE MOON The Moon's Orbit - The Moon orbits Earth in 27.32 days (sidereal month) relative to stars. - Earth’s movement around the Sun adds extra distance for the Moon to reach the same position relative to the Sun. - This takes 29.53 days (synodic month). - The Moon's motion around Earth is prograde (same direction as Earth around Sun). - From a distant viewpoint, the Moon orbits the Sun with Earth, with a slight wobble due to Earth’s influence. - The Sun's gravity is stronger than Earth's, even though Earth is closer to the moon. - Perturbations case two effects: The Moon’s perigee precesses (moves forward) in its orbit, completing a full cycle in about 8.85 years (unlike planetary perihelia which are nearly stationary). The line of nodes (N’1N’2) regresses (moves backward) around the ecliptic, completing a full cycle in about 18.61 years. TERMS Annual equation - A correction for the Moon’s position due to the Earth’s distance from the Sun. Ascending node (Ω) - The point where the Moon’s orbit crosses the ecliptic (Earth’s orbital plane) moving northward. Draconic month (nodal month) - The time it takes for the Moon to return to its ascending node (27.2122 days). Evection - A variation in the apparent shape of the Moon’s orbit due to the Sun’s influence. Perigee (ω) - The point in the Moon’s orbit closest to Earth. Prograde motion - Motion in the same direction, in this case, referring to the Moon’s orbit around Earth and Earth’s orbit around the Sun (both moving counter-clockwise). Retrograde motion - Motion in the opposite direction, referring to the movement of the line of nodes (N’1N’2) which regresses around the ecliptic. Sidereal Month - The time it takes for the Moon to orbit Earth relative to the background stars (27.3217 days). Synodic Month - The time it takes for the moon to return to the same position relative to the Sun (29.5306 days). True Anomaly (v) - The Moon’s angular position in its orbit relative to the perigee. Variation - A correction for the Moon’s position due to its changing distance from the Sun relative to Earth. Formulas Dynamical Time Dynamical Time  DAY OF THE WEEK AND DAY OF THE YEAR Day of the Week Day of the Year DATE OF EASTER Gregorian Easter Calculation  COORDINATE SYSTEMS AND TRANSFORMATIONS General Coordinate Transformations Table  TYPES OF COORDINATE SYSTEMS Horizon Coordinate System Equatorial Coordinate System Ecliptic Coordinate System Galactic Coordinate System PROCESS OF FINDING THE TRANSFORMED COORDINATES PROCESS OF FINDING THE TRANSFORMED COORDINATES  TIME BY THE STARS Hour Angle (HA) CALCULATING THE POSITION OF THE MOON CALCULATING THE POSITION OF THE SUN CELESTIAL MECHANICS OF PLANETS Calculate the right ascension and declination on 22 November 2003 for Jupiter: THE ANGULAR DISTANCE BETWEEN TWO CELESTIAL OBJECTS Sir Sablan Lessons (Handwritten Notes) Deriving Newton’s Laws Three Body Problem Lagrangian Mechanics AST13 - Astrophysics (Handwritten Notes) Basic Astrophysical Measurements Radiation Processes Doppler-Fizeau Effect Tidal Forces and Roche Limit Comets Energy Balance Jean’s Criterion HR Diagram Stars