Sidereal, Synodic Time and Lunar Cycle

Sidereal and Synodic Time

• Sidereal Day: The time it takes for Earth to make one complete rotation relative to background stars is 23 hours 56 minutes.
• Solar Day: The time it takes for Earth to make one complete rotation relative to the Sun is 24 hours, also known as synodic days.
• Synodic Time: The time it takes for a celestial body to return to the same angular position from the Sun as seen from the Earth.
• Sidereal Time: The time required for a celestial body within the solar system to complete one revolution with respect to distant stars.

The Lunar Cycle

• During the sidereal month, the Earth-Moon system has moved an angle of 27º around the sun.
• It takes the Moon an extra 2.2 days to return to its position between the Earth and Sun.
• The full lunar cycle takes 29.5 days, which is a synodic month.
• The Moon has eight phases.
• The Moon rotates anticlockwise around the Earth.
• The way the Moon looks during the lunar cycle is influenced by the relative positional changes between the Earth, the Moon, and the Sun.
• The phases show the Moon being illuminated from the right (east) side first, then the west.
• Observers in the Northern Hemisphere see this as true.
• Observers in the Southern Hemisphere effectively see the moon 'upside down.'

Apparent Solar Time (AST)

• Sun dials measure the apparent motion of the Sun.
• The fundamental unit is a (solar) day, which is the interval between two successive returns of the Sun to the local meridian.
• The length of a solar day (24 hours) and apparent solar time are not consistent throughout the year.
  • This is due to Earth's elliptical orbit and Earth's axial tilt of 23.5°.
• The length of clocks do not vary because of the defined mean sun/mean solar time (MST).
• Mean Sun: An imaginary Sun moves along the celestial equator at a constant speed throughout the year.
• Mean Solar Time: The average time it takes for the Sun to complete its apparent daily motion across the sky, ensuring consistent 24-hour days.

The Effect of Earth's Elliptical Orbit on Apparent Solar Time

• The Earth travels faster when closer to the Sun.
• Between Earth's rotation (23 hours 56 minutes), it travels a larger distance.
• Earth has to rotate through a larger angle for the Sun to return to the same position.
• This increases the length of a solar day.

The Effect of Earth's Axial Tilt and Obliquity of the Ecliptic on Apparent Solar Time

• Throughout one year, the Sun varies in declination between ± 23.5° from + 23.5 (Tropic of Cancer) on June 21st and - 23.5 (Tropic of Capricorn) on December 21st, passing through 0 (equator) at the two equinoxes.
• Throughout one year, the Sun will also cycle through all 24 hours of Right Ascension.
• The resulting path of the sun over one year in Dec and RA is shown.
• Slope is smallest - larger East to West motion between sidereal days so the Sun appears to move faster (Solar days are longer).
• Slope is greatest - smaller East to West motion between sidereal days so the Sun appears to move slower (Solar days are shorter).
• Analemma: The asymmetrical figure-of-eight that can be traced in the sky at a given place, showing the position of the sun at the same standard time (usually twelve o'clock) on successive days of a year.

The Equation of Time

• The correction between the mean sun and the real sun is given by the equation of time (EOT), represented as an equation but is a value.
  • Equation of time(EOT) = Apparent Solar Time(AST) - Mean Solar Time (MST)
• Y axis: the correction, value of the EOT.
• X axis: Time of year
• If EOT is negative you have to add minutes to AST to catch up with the mean sun.
• If EOT is positive subtract minutes from AST to get back to the mean sun.
• Equation of time is a combination of the effects of:
  • Earth's elliptical orbit.
  • Obliquity of the ecliptic (axial tilt).
• On 4 dates the equation of time is zero.
  • The two causes (obliquity and elliptical orbit) have the same magnitude.
  • One is positive, and one is negative (one is making the solar day longer, and one is making it shorter).
  • The effects exactly cancel out, and the real sun is traveling at the same speed as the mean sun.

Local Mean Time

• Local mean time: The Mean Solar Time at a particular longitude.
• Greenwich mean time (GMT): Local mean time at Greenwich.
• It takes Earth 24 hours to rotate with respect to the Mean Sun.
  • 360° = 24 hours
  • 360° = 24 x 60 x 60 s
  • 1° = (24 x 60 x 60) / 360 s
  • 1° = 4 minutes
• Observers separated by 1º longitude will have a difference of ± 4 minutes in their local mean time.
• Local mean time (LMT) is based off when the Sun reaches local noon crossing the meridian.
• Locations that are East of the prime meridian reach local noon first as the sun rises in the East and sets in the West.
• Locations West of the Prime Meridian reach local noon later.
• There are 24 time zones established based on political and social ease--not exactly aligned with the LMT changes in Longitude.