Astronomy 217

 
 

Prof. Andrew W. Steiner

 
 
Aug. 25, 2021
 
 

TA Bryce Fennig

Last Time

  • Constellations
  • Clusters
  • Ecliptic
  • Alt-az coordinates
  • Equatorial coordinates
  • Circumpolar stars
  • Tropics and the polar tilt

Today

  • Solar time
  • Universal time and time zones
  • Sidereal and synodic periods
  • Seasons and leap days
  • Precession of the equinoxes
  • Epochs and stellar motion

Solar Day

  • The rising and setting of the sun are natural markers for the day
  • However, the variation in the length of the day make sunrise and sunset make very irregular time keepers
  • The upper transit, the passage of the sun across the local zenith meridian, AKA noon, is a more reliable marker
  • The solar day is defined to be the length of time for the Sun to return to your celestial meridian

Mean Solar Time

  • Two effects cause the actual solar day to deviate over the course of the year:
  • 1) The Earth’s slightly elliptical orbit causes the Earth to move faster in winter and slower in summer
  • 2) The projection of the solar motion onto the Celestial Equator causes the Sun to move slower near the equinox by \( \cos(23.5^{\circ}) \approx 0.9 \) .
  • To work around these variation, we define the mean solar day to be average length of solar day.

Analemma

  • If you were to take a picture of the sun each day at the same time over the course of a year, you’d get a figure 8 on the sky.
  • This is called the analemma.
  • This image shows the ruins at Ancient Nemea in Greece, but the shape of the figure would be similar in Knoxville, because of the similar latitude.

Universal Time

  • Historically, every locality had it’s own local time, based on the solar upper transit at local noon.
  • In the late nineteenth century, with railroads and telegraphs connecting many local times at higher and higher speeds, it became necessary to establish a common standard of time.
  • Since the Prime Meridian was already established to run through Greenwich, England, Greenwich Mean Time (GMT) became the standard, or Universal time. It is now generally denoted UT (or UTC).
  • UT is inconvenient as a local time away from the Prime Meridian. For example, today 11:30 AM at UTK is 3:30 PM UTC.

Time Zones

  • As a compromise between a multitude of local times and a single universal times, 24 one hour wide time zones were devised as a local correction to the universal time.
  • This allows the solar upper transit to occur near 12 pm. Ideally, each time zone is \( 360^{\circ}/24 = 15^{\circ} \) wide
  • For the Eastern Time Zone, local time is UT − 5 hr, except that it is UT − 4 when daylight savings time is in effect.
  • The center of this timezone is then at \(5 \times 15^{\circ} = 75^{\circ} \) W Longitude (Philadelphia, PA) and it ideally covers from \(67.5^{\circ}\) W (Bar Harbor, ME) to \(82.5^{\circ}\) W (Asheville, NC).

Eastern Time

  • Knoxville is situated near the Western edge of the Eastern Time Zone at \( 84^{\circ} \) W.
  • What is the Mean Solar Time of the solar upper transit? Center of timezone is \(75^{\circ}\) W, so Knoxville is \(9^{\circ}\) west of center. \(60 \mathrm{min} \times 9^{\circ}/15^{\circ} = 36\) min. So the answer is 12:36 pm.
  • Can the Actual Solar Time of the upper transit be 12 noon?
  • Maximum deviations are −14 minutes in February and +14 minutes in October, so our astronomical noon varies from 12:22 pm to 12:50 pm EST.

Sidereal Time

  • As the earth rotates on its axis, it is also revolving around the sun, thus it must rotate further to align with the sun than it would to align with a distant star.
  • We call the period between pointing at a fixed star the sidereal period. For the earth’s rotation, this is the sidereal day. This causes there to be one more sidereal day than solar day per year, thus the sidereal day is 1/365th shorter (~4 min.)

The Month

  • Traditionally, the month is defined by the period of the Moon’s orbit. The lunar month is the time to complete the cycle of lunar phases (29.53 days).
  • This is the average period of the Moon's revolution with respect to the sun, the synodic period. This is not the moon’s true orbital period.

The Sidereal Month

  • As with the day, the period of the moon’s rotation with respect to the fixed stars is less. $$ \frac{1}{P_{\mathrm{sid}}} = \frac{1}{P_{\mathrm{lun}}} + \frac{1}{P_{\mathrm{year}}} $$
  • Plun = 29.53 days and Pyear = 365.24 days so Psid = 27.32 days

The Year

  • Historically, the definition of a year is the interval between two successive returns of the Sun to the vernal equinox. This is called the tropical year, because the Sun is at the zenith on the tropic of Cancer on this anniversary. The tropical year has a length of 365.2422 mean solar days.
  • The period of revolution of the Earth around the Sun as referenced to the distant stars is called the sidereal year. It has a length of 365.2564 mean solar days. The difference (1 part in 25,721) is due to the precession of the equinoxes.
  • The sidereal year is the "true" year, but our calendar is based on the tropical year because the seasons, which are important to agrarian societies, are correlated with the tropical year.

Seasons

  • Ultimately, the seasonal variations we experience are the result of the misalignment of the Earth’s rotational and orbital axes.
  • More direct sunlight and additional hours of sunlight result in warmer summers.
  • In fact the Earth is slightly closer to the Sun during January (147,098,290 km) than in June (152,098,232 km).

Leap Days

  • To keep their calendars aligned with the seasons, many cultures adopted leap days (or months) to periodically bring their calendar into alignment with the Heavens.
  • In 46 BC, Julius Caesar introduced a calendar with a leap day added to February every fourth year, for a 365 + 1/4 day average year (the Julian Calendar).
  • Though accurate to better than 11 minutes per year, the errors in the Julian calendar added up over the centuries, amounting to 10 days by 1582 AD when Pope Gregory decreed a new calendar (the Gregorian Calendar).
  • By dropping 3 leap days per 400 years, the Gregorian calendar achieves an accuracy of better than 30 sec./ year.

Precession of the Equinoxes

  • Another discovery by Hipparchus of Nicea near 130 BC was a precession of the equinoxes. Hipparchus found a rate of precession > 1° per century, or a period < 36,000 years).
  • The equinoxes move about 1° in 72 years or a period of roughly 26,000 years.
  • As a result, the tropical year is about 20 minutes shorter than the sidereal year.

Precession of the Equinoxes

  • The cause of the precession of the equinoxes is a precession in the Earth’s rotational axis.
  • As a result, the Earth’s Poles trace circles on the Celestial Sphere and the “Pole” star becomes time-dependent.

Oblate Earth

  • Because it is rotating, the Earth is slightly flattened at the poles, forming an Oblate Spheroid.
  • Because the Pole is inclined to the Ecliptic, there is an unbalanced component of the Sun’s gravity on the near bulge and a smaller component on the far bulge.
  • This unbalanced force (plus a contribution from the Moon) on the rotating Earth, produces a torque that drives the polar precession.

Epochs

  • To account for slow variations like the precession of the Pole and nutation, every astronomical coordinate has a timestamp.
  • An Epoch is a moment in time used as a reference point for some time-varying astronomical quantity.
  • The current standard Epoch is J2000.0 which marks January 1, 2000, 11:58:55.816 UT or 12:00:00 Terrestrial Time (TT).
  • Terrestrial Time does not include leap seconds.
  • Intervals from J2000.0 are counted in Julian days.
  • Values for constants are define from J2000, e.g., Greenwich Sidereal Time of J2000 (GST J2000 ) = 280.46061837°.
  • Star charts from older epochs like J1950.0 are still occasionally found.

Stellar Motion

  • True motions are limited because very large distances turn large velocities into small angles.
  • The orbital motions of the “wanderers”; planets, moons and other solar system bodies, are comparable (1-1000°/yr) to the apparent motion caused by the Earth.
  • The proper motions of the distant stars, true motions relative to the Solar System, are much smaller.

Proper Motion

  • Stars do move relative to the Solar System, but only in unusual circumstances is this discernible in human time. But the effects can be spectacular.
  • Mira, a red Giant in the constellation Cetus, 350 ly from Earth, is moving across the sky at 130 km/s (291,000 miles/hr).
  • Mira is losing gas, leaving a tail 13 ly long behind it over 30,000 years.
  • Even the fastest proper motion, like Barnard’s star (1°/ 350 yrs), are small compared to the Earth’s motions