Chapter 6. Motion of the Earth
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1. Earth’s Rotation
⑴ Diurnal Motion
① Diurnal motion of stars : Apparent motion of stars due to the Earth’s rotation. Period is 24 hours.
○ Direction of Earth’s rotation : West → East. Counterclockwise when viewed from the celestial North Pole.
○ Direction of diurnal motion : East → West. Clockwise when viewed from the celestial North Pole ( ∵ Considering Earth’s rotation in reverse)
○ Diurnal circle : Path of stars’ apparent motion. Parallel to the celestial equator.
○ The apparent motion of stars can be analyzed with the angular velocity vector in the direction of the celestial South Pole.
○ Reason why the Sun and stars rise in the east and set in the west.
○ Stars in the Southern Hemisphere rise in the west and set in the east.
② Polar Stars, Rising and Setting Stars, Non-Rising Stars
○ For stellar declination δ and observer’s latitude Φ,
○ Polar stars : Stars that never set below the horizon. (90° - Φ) ≤ δ ≤ 90°
○ Rising stars : Stars that rise and set above the horizon. -(90° - Φ) ≤ δ < (90° - Φ)
○ Non-rising stars : Stars that never rise above the horizon. -90° ≤ δ < -(90° - Φ)
○ Phenomenon of midnight sun and polar night occurs when the Sun becomes a polar star and non-rising star respectively.
○ These conditions are not derived from formulas but are understood intuitively.
⑵ Evidence of Earth’s Rotation
① Foucault Pendulum Rotation
○ North Pole
○ Pendulum oscillation plane rotates clockwise : Due to Earth’s counterclockwise rotation and the observer’s motion.
○ Rotation period : Time for A and B to coincide, 24 hours.
Figure 1. Rotation of Foucault Pendulum at the North Pole
○ South Pole : Counterclockwise, 24-hour period.
○ Equator : No rotation of oscillation plane.
○ Arbitrary Latitude : Conclusion combining polar region and equator observations.
Figure 2. Decomposition of Angular Velocity Vector at Arbitrary Latitude
○ Only the component of angular velocity vector perpendicular to the ground affects the rotation of the oscillation plane. Related to sinθ (θ : latitude).
○ Rotation direction of the oscillation plane : Opposite to Earth’s rotation, clockwise in the Northern Hemisphere, counterclockwise in the Southern Hemisphere.
○ Period of oscillation plane rotation
② Coriolis Force
Figure 3. Coriolis Force
○ Fictitious force that deflects the motion of an object.
○ Direction : Right-angle deflection to the direction of motion in the Northern Hemisphere, left-angle in the Southern Hemisphere.
○ Magnitude : Maximum at the poles, 0 at the equator.
○ Formula : For mass m, object’s velocity v, Earth’s angular velocity ω, and latitude Φ, the following holds
③ Satellite Westward Drift Phenomenon : The observed westward drift of a satellite in a fixed orbit as seen from the Earth’s observer, caused by Earth’s rotation.
Figure 4. Satellite Westward Drift Phenomenon
④ Eastern Deflection of Free Fall Phenomenon : The phenomenon where a freely falling object deviates to the east as it falls.
○ Assumption : Earth’s radius R, distance between object A and surface h, latitude θ, angular acceleration Ω.
○ Surface observer’s linear velocity : RΩ cosθ
○ Linear velocity of object A : (R+h)Ω cosθ
○ Object A’s linear velocity is greater than the surface observer’s, causing object A to move further eastward relative to the surface observer.
⑶ Phenomena Resulting from Earth’s Rotation
① Day and night occur.
② Celestial diurnal motion happens, i.e., celestial objects move from west to east by 15 degrees per hour.
③ The precession phenomenon occurs.
2. Earth’s Revolution
⑴ Evidence of Earth’s Revolution
① Aberration
○ Phenomenon where starlight is observed to be inclined due to Earth’s revolution.
○ Aberration decreases as the revolution speed increases.
② Solar Annual Motion
③ Annual Parallax : Phenomenon where the position of stars appears to change periodically with a one-year cycle, represented by the angle formed by the Earth-Star-Sun system.
○ Closest star’s annual parallax is 0.76”.
○ Parallax decreases for more distant stars.
○ Parallax 1” = Distance 1 parsec, r (pc) = 1 / P”
○ Discovered by German astronomer Bessel in 1838.
④ Doppler Effect of Starlight : Redshift, blueshift.
⑵ Solar Annual Motion
① Solar Annual Motion : The Sun’s daily motion of about 1 degree from west to east among the constellations.
Figure 5. Solar Annual Motion
Figure 6. Constellations around the Ecliptic
○ Distant constellations show less effect due to the revolution.
○ As the Earth revolves from west to east, the Sun also moves from west to east.
○ If we fix the Sun, constellations appear to move from east to west by about 1 degree per day.
○ Example : At noon on June, the constellation where the Sun is located is Taurus.
○ Example : At noon on June, the constellation rising in the eastern sky is Leo.
○ Example : The right ascension of Aquarius in March is approximately 0h, and the right ascension of Taurus in June is approximately 6h.
② Ecliptic : The path of the Sun’s annual motion.
Figure 7. Ecliptic Plane in the Celestial Sphere
○ Horoscope of the Ecliptic (Horoscope): 12 constellations located on the ecliptic
○ The plane of the ecliptic and the equatorial plane of the celestial sphere form an inclination of 23.5°
○ The angular velocity vector due to the annual motion and the angular velocity vector due to the solar apparent motion form an angle of 180° - 23.5°
○ Equinox point at 06:00, Solstice point at 12:00, Autumnal equinox point at 18:00, Winter solstice point at 24:00, with the rest proportionally calculated
○ The above descriptions may contain errors. Subject to future revisions
③ Points of Division and Position
Figure 8. Points of Division and Position
○ Equinox: Intersection of the ecliptic and the celestial equator (e.g., Vernal Equinox, Autumnal Equinox)
○ Solstice: The point where the Sun is farthest from the celestial equator (e.g., Summer Solstice, Winter Solstice)
○ Vernal Equinox: Declination = 0°, Right Ascension = 0h
○ Summer Solstice: Declination = 23.5°, Right Ascension = 6h
○ Autumnal Equinox: Declination = 0°, Right Ascension = 12h
○ Winter Solstice: Declination = -23.5°, Right Ascension = 18h
④ Changes in Solar Altitude
○ Solar noon altitude (Northern Hemisphere)
○ Formula: h = 90° - Φ + δ (Φ: Latitude, δ: Solar Declination)
Figure 9. Formula for Solar Noon Altitude
○ The solar noon altitude forms a 90° angle with the horizon. Highest on the day of the Winter Solstice
○ Solar noon altitude (Southern Hemisphere)
○ Formula: h* = 90° - Φ* + δ*
○ h*: Solar noon altitude in the Southern Hemisphere reference
○ Φ*: Latitude in the Southern Hemisphere reference
○ δ*: Solar declination in the Southern Hemisphere reference
○ Solar energy received per unit area
○ Solar energy received on the surface per unit area when the Sun’s altitude is h
○ Proportional to the product of solar energy at solar noon (E0) and sin h
○ Maximum solar energy when the Sun’s altitude is 90° (noon)
○ Changes in daylight hours by season
○ Vernal Equinox to Autumnal Equinox: Daytime > Nighttime
○ Vernal Equinox day and Autumnal Equinox day: Daytime = Nighttime
○ Autumnal Equinox to Vernal Equinox: Daytime < Nighttime
⑤ Changes in Seasons
○ Cause: Changes in solar altitude and daylight hours
○ Summer: Northern Hemisphere tilts toward the Sun. Sunlight is nearly vertical at around 20°N.
○ Winter: Southern Hemisphere tilts toward the Sun. Sunlight is nearly vertical at around 20°S.
○ Equatorial regions receive the most concentrated sunlight, leading to higher temperatures.
○ In the Northern Hemisphere, proximity to the Sun in winter doesn’t dictate the season
⑶ Motion of Stars
① Stars near the ecliptic plane appear as shapes closer to a straight line
② Stars near the ecliptic poles appear as shapes closer to a circle
③ If a star is not in the same position after one year, it has proper motion
3. Earth’s Motion
⑴ Precession
① Generally refers to the precession of the equinoxes
○ Precession of the Equinoxes: Precession caused by the Sun and the Moon
○ Planetary Precession: Precession caused by other planets
② Earth’s Precession: Earth’s axial rotation around the ecliptic pole
○ Inclined at an angle of 23.5°
○ Rotates 50 arc seconds per year from east to west: Vernal Equinox moves along the ecliptic by about 50 arc seconds per year
○ Approximately 26,000-year cycle
③ Cause: Gravitational effects of the Moon and the Sun near the equator of the Earth
④ Result: Tries to upright the Earth’s tilted axis of 23.5°
⑵ Results of Precession
① North Polar Motion of the Celestial Sphere
○ The north pole of the celestial sphere rotates counterclockwise around the ecliptic north pole every 26,000 years
○ After about 12,000 years, the north pole of the celestial sphere will be near the North Celestial Pole
② Movement of the Vernal Equinox: Moves 50 arc seconds per year (East to West) → This affects the celestial coordinates
③ Difference between Sidereal Year and Tropical Year (Solar Year)
○ Sidereal Year: Time for the Sun to return to the same position relative to the constellations = 365.2564 days
○ Tropical Year (Solar Year): Time for the Sun to return to the same position relative to the vernal equinox = 365.2422 days
○ Sidereal Year is longer than the Tropical Year due to the precession of the equinoxes
⑶ Nutation
⑷ Wobbling
4. Earth’s Spacetime
⑴ Time
① Sidereal Time and Solar Time
○ Sidereal Day: The interval between successive transits of the vernal equinox (actual Earth’s rotation period), referred to as sidereal day
○ Sidereal Time: The hour angle of a celestial object at the time of its transit
○ Solar Day: The interval between successive solar transits, not exactly constant, but its average value over a year, the mean solar day (24 hours), is used
○ Solar Time = Solar Hour Angle + 12 hours
○ Equation of Time: Solar Time - Mean Solar Time, due to the non-constant Earth’s axial tilt and orbital speed
② Standard Time
○ Local Time: Time set to 12 when the mean Sun transits the local meridian
○ Standard Time: Time set to 12 when the Sun transits the standard meridian at 15° intervals
○ Universal Time: Local Time at the Royal Greenwich Observatory
○ International Date Line
○ Approximately 180° meridian of East/West longitude
○ Crossing from East to West (West to East) results in a day gained (or lost)
⑵ Location
① Determination of Longitude: Defined with the Prime Meridian as 0°, East as 180°, West as -180°
○ Method using two stars transiting at the same time: If stars A and B transit local times L and G simultaneously
○ Longitude of L = (Right Ascension of star A - Right Ascension of star B) × 15° per hour = (Sidereal Time of L - Sidereal Time of G) × 15° per hour
○ L > 0: East Longitude, L < 0: West Longitude
○ Method using the time difference of a star’s transit: If a star transits L and G at times L and G respectively
○ Longitude of L = (Time of G - Time of L) × 15° per hour
○ L > 0: East Longitude, L < 0: West Longitude
② Determination of Latitude
○ Altitude of the North Star = Latitude
○ Solar Altitude: h = 90° - Φ + δ, where Φ is the celestial latitude
⑶ Strategy
① Lunar Calendar: Based on the lunar phase cycle (synodic month) of 29.5 days, with 1 year as 354 days
○ Metonic Cycle: 19 years with 7 intercalary months to synchronize with seasons
○ 24 Solar Terms: Dividing the year into 24 parts based on the Sun’s ecliptic position
② Solar Calendar: Based on the Sun, accurate with seasons
○ Julian Calendar: A year is 365.25 days, with a leap year every 4 years
○ Gregorian Calendar
○ Introduced by Pope Gregory XIII to rectify the inaccuracies of the Julian Calendar
○ Leap years divisible by 4 but not divisible by 100, except if divisible by 400
○ The Gregorian Calendar is accurate to 365.2422 days per year
Input : 2019.03.23 00:55