Motions in the Sky and Measuring Distances

Motions in the Sky

Earth's Fundamental Motions

• Daily Rotation of Earth: Responsible for the day/night cycle.
• Annual Orbit of Earth: Earth's revolution around the Sun, defining the year and influencing seasons.
• Monthly Orbit of the Moon: The Moon's revolution around Earth, responsible for lunar phases and tides.

Precession of Earth's Axis

• Cause: The Sun's gravitational pull on the non-spherical Earth, similar to gravity wobbling a slanted spinning top.
• Effect: A slow "wobbling" of Earth's rotational axis around the vertical axis relative to the Ecliptic plane.
• Period: This complete wobble takes approximately $26,000$ years.
• Consequence: The celestial North Pole traces a large circular path on the sky over this $26,000$-year period.
  • Currently, the celestial North Pole is closest to Polaris (around A.D. $2100$). Polaris is not inherently special (e.g., neither particularly bright nor nearby) but is significant due to its current alignment with Earth's axis.
  • In about $12,000$ years, the celestial North Pole will be near Vega in the constellation Lyra.

Earth's Rotation

• Direction: When viewed from above the North Pole, Earth rotates counterclockwise on its axis.
• Duration: One complete rotation relative to the Sun (a Solar Day) takes $24$ hours.

Earth's Orbit and Time Scales

• Orbit Shape: Earth's orbit around the Sun is nearly circular.
• Astronomical Unit (AU): The average distance from Earth to the Sun.
  • $1 ext{ AU} = 150 ext{ million km}$. For reference, this is $1.5 imes 10^8 ext{ km}$.
• Day/Night Cycle: Caused directly by Earth's rotation.
• Sidereal Time vs. Solar Time:
  • One Solar Day: The time it takes for the Sun to return to the same position in the sky on successive days. This is exactly $24$ hours.
  • One Sidereal Day: The time it takes for the background stars to return to the same position in the sky. This is approximately $23$ hours and $56$ minutes.
  • Difference: The $4$-minute difference arises because Earth is not only rotating but also simultaneously orbiting the Sun. After one complete $360^{ ext{o}}$ rotation relative to distant stars (a sidereal day), Earth has moved slightly in its orbit, so it needs to rotate a little further (approximately $1^{ ext{o}}$) to face the Sun again.

Length of Daylight Hours

• Dependence on Latitude: The number of daylight hours experienced at a location varies significantly with its latitude on Earth.
• Polar Regions: Areas closer to the poles (e.g., Northern Pole):
  • Receive more daylight hours during the summer (>12 hours).
  • Receive fewer daylight hours during the winter (<12 hours).
  • Within the Arctic Circle (>66.5^{ ext{o}} latitude), some summer days experience the "Midnight Sun" or "White Night", where the Sun never sets below the horizon (it traces a path that dips low but remains visible).
• Equatorial Regions: Regions close to the equator consistently receive approximately $12$ hours of sunlight throughout the entire year.

The Seasons

• Insignificant Factor (Distance): Earth's distance from the Sun has only a very minor influence on seasonal temperature variations.
  • Counterintuitively, Earth is slightly closer to the Sun in January (Northern Hemisphere winter) than in July (Northern Hemisphere summer). The eccentricity of Earth's orbit is greatly exaggerated in diagrams to highlight this.
• Primary Factor (Axial Tilt): The seasons are primarily caused by the tilt of Earth's rotational axis.
  • Earth's axis is not perpendicular to its orbital plane (the ecliptic plane).
  • Instead, Earth's axis is tilted at an angle of roughly $23.5^{ ext{o}}$ relative to the perpendicular of the ecliptic.
• Mechanism of Seasons Due to Tilt:
  • Summer: When a hemisphere is tilted towards the Sun, the angle of incoming sunlight is steeper (closer to perpendicular). This results in solar energy being more concentrated over a smaller area, leading to higher temperatures and longer daylight hours.
  • Winter: When a hemisphere is tilted away from the Sun, the angle of incoming sunlight is shallower. This spreads the solar energy over a larger area, leading to lower temperatures and shorter daylight hours.
  • Hemispheric Opposition: The Southern Hemisphere experiences opposite seasons to the Northern Hemisphere because while one hemisphere is tilted towards the Sun, the other is simultaneously tilted away.

Season Markers and Sun's Path

• Summer Solstice (June 21):
  • Occurs when the Northern Hemisphere is maximally tilted towards the Sun.
  • The Sun's path across the sky is highest for Northern Hemisphere observers.
  • The Sun rises and sets at its most extreme north of due east and west, respectively.
• Winter Solstice (December 21):
  • Occurs when the Northern Hemisphere is maximally tilted away from the Sun.
  • The Sun's path across the sky is lowest for Northern Hemisphere observers.
  • The Sun rises and sets at its most extreme south of due east and west, respectively.
• Equinoxes (Vernal/March 21 & Autumnal/September 21):
  • Occur when Earth's axis is neither tilted towards nor away from the Sun (it is perpendicular to the Earth-Sun line).
  • Day and night hours are approximately equal all over Earth.
  • The Sun rises precisely due east and sets precisely due west.
  • The Sun's path through the sky is midway between the extremes of the solstices.
• Annual Cycle: The time it takes for the Sun's highest noon position to shift from its highest point (Summer Solstice) to its lowest point (Winter Solstice) and back again defines one year.

Measuring Astronomical Distances

The Challenge of Distance Measurement

• Problem: How to measure the distance to objects that are immensely far beyond the reach of physical measuring instruments?
• Solution: Employ geometry, specifically triangulation.
  • Triangulation: Involves measuring a known baseline (the distance between two observation points) and the angles from each end of the baseline to the distant object. With these three pieces of information, the distance to the object can be calculated.

Angular Measurements in Astronomy

• Purpose: Astronomers use angles to quantify the apparent sizes of celestial objects and the apparent distances between them in the sky.
• Basic Unit: The fundamental unit for angular measure is the degree ($^{ ext{o}}$).
  • The angular size of the Moon, for example, is approximately $0.5^{ ext{o}}$.
• Angular Distance: If lines are drawn from an observer's eyes to two different stars, the angle formed between these lines represents the angular distance between those two stars.
• Estimation: The human hand held at arm's length can be used to roughly estimate angles in the sky.
• Subdivisions of a Degree:
  • $1^{ ext{o}} = 60'$ (arcminutes)
  • $1' = 60''$ (arcseconds)
  • Therefore, $1'' = \frac{1}{60}' = \frac{1}{60 imes 60}^{ ext{o}} = \frac{1}{3600}^{ ext{o}}$.

Trigonometric Parallax (Stellar Parallax)

• Concept: This is a geometric method used to measure the distance to relatively nearby stars.
  • It capitalizes on the apparent shift in a star's position against a more distant background as the Earth orbits the Sun.
  • Think of it like holding a finger close to your face and observing its apparent shift against a distant lamppost as you move your head from side to side.
  • This principle is also used in $3 ext{D}$ movies, where two slightly offset images are shown to each eye, simulating depth perception.
• Definition: The stellar parallax ($p$) of a star is defined as half the angular shift in the star's apparent position over a six-month period, as observed from two opposite points in Earth's orbit (a baseline of $2 ext{ AU}$ relative to the Sun).
  • $p$ is typically measured in arcseconds ($''$).
• Parallax Decreases with Distance: The closer a star is to Earth, the larger its trigonometric parallax (apparent shift) will be. Conversely, more distant stars exhibit smaller parallax angles.

Parsec: The Unit of Parallax Distance

• Definition: A parsec (pc) is the distance to a star that has a stellar parallax of exactly one arcsecond ($1''$).
• Conversion: $1 ext{ parsec} ext{ (pc)} ext{ is approximately equivalent to } 3 imes 10^{13} ext{ km} ext{ or about } 3.26 ext{ light-years}$.
• Formula for Distance: The distance $d$ to a star in parsecs can be calculated using its parallax angle $p$ in arcseconds:
  $d( ext{parsec}) = \frac{1}{p('')}$
• Examples:
  • If a star has a parallax $p = 0.01''$
    $d = \frac{1}{0.01} = 100 ext{ pc}$
  • If a star has a parallax $p = 0.05''$
    $d = \frac{1}{0.05} = 20 ext{ pc}$