Tilt, Ecliptic, Equinoxes, and Seasons — Study Notes

Overview: Tilt, the ecliptic, and seasons

• The tilt of the Earth's axis remains fixed relative to the stars as the Earth orbits the Sun, which keeps the orientation of the axis consistent (tilt is “intact”).
• This fixed tilt causes the Sun’s apparent path across the sky (the ecliptic) to move up and down relative to the celestial equator as the Earth travels around the Sun.
• Because the ecliptic sits at an angle to the celestial equator, the Sun appears higher in the sky at some times of year and lower at others, generating seasonal changes in sunlight and temperature.
• Direct vs. indirect sunlight changes with the tilt: the tilt controls how directly sunlight hits different hemispheres, not proximity alone.
• Even though the Earth’s orbit is slightly elliptical, the eccentricity is tiny enough that, for practical purposes, the orbit can be treated as nearly circular; distance variation does not drive the seasons.
• Despite being closer to the Sun in the Northern Hemisphere winter than in summer (due to the elliptical orbit), this proximity does not cause the seasons; tilt and Sun angle do.
• The concept of declination (latitude on the celestial sphere) and right ascension (RA) (the eastward angle from the vernal equinox) describe where the Sun appears on the sky over the year.

The Ecliptic, Equator, and Equinoxes/Solstices

• The ecliptic goes up and down because the axis tilt remains fixed while the Earth moves around the Sun.
• The Sun’s apparent path (the ecliptic) crosses the celestial equator at the equinoxes.
• Equinoxes are the moments when the ecliptic intersects the celestial equator:
  • Vernal (春分) equinox around March 21 (the start of spring in the Northern Hemisphere).
  • Autumnal (秋分) equinox around September 23.
• Solstices describe the points of maximum and minimum Sun height in the sky:
  • Summer solstice: Sun is high in the sky.
  • Winter solstice: Sun is low in the sky.
• The Sun’s declination (positive = north of the celestial equator; negative = south) varies over the year as the tilt interacts with the orbital position.
• Right ascension (RA) is measured in hours from the vernal equinox, increasing eastward: 0h at the vernal equinox, up to 24h around the year.
• The vernal equinox occurs around March 21 and marks the reference point for RA = 0.
• The autumnal equinox occurs around September 23 and corresponds to RA ≈ 12h.
• For each hour of RA, the sky is rotated by $1 ext{ h} = 15^ ext{0}$ of angle: $1 ext{ h} = 15^ ext{0}$ (i.e., $ext{angle} = ext{RA}_{ ext{h}} imes 15^ ext{0}$).
• Therefore, $24 ext{ h} imes 15^ ext{0}/ ext{h} = 360^ ext{0}$, i.e., a full circle in RA corresponds to 360 degrees.
• Why use hours for RA? Because as the stars rise and set, the sky appears to rotate relative to the Sun and Earth; using hours aligns with the actual observed diurnal motion relative to distant stars (see below).
• The line of sight to the Sun cycles through all declinations between the extreme northern and southern limits set by the tilt over a year.

The Diurnal Motion, Right Ascension, and Declination

• Positive declination means north of the celestial equator; negative declination means south.
• Right ascension increases as time passes, moving the coordinate reference point eastward along the celestial sphere.
• The Sun’s path is described by its declination δ and its RA α, both changing predictably over the year due to the tilt and orbital position.
• The equinoxes mark the times when δ = 0° (on average), and RA transitions through 0h and 12h at those times.
• The relationship between time of day and sky position is tied to the Earth’s rotation, which makes the stars rise about 4 minutes earlier per day (see next section).

The Diurnal Shift: Why Stars Rise Earlier Each Night

• Because the Earth rotates once per day, stars complete a sky circle relative to an observer every 24 hours.
• The night sky drifts relative to the Sun by about 4 minutes earlier each day: $ext{daily shift} \approx 4 ext{ minutes}$.
• Over weeks and months, this causes seasonal changes in which constellations are visible at night.
• Example: Orion is typically seen in the evening during winter; by summer, Orion is not visible at night due to the Earth’s position in its orbit.
• The extremes of the Earth’s orbit help explain which constellations are visible at midnight at different times of year.

Practical question: Can the Sun be directly overhead at a latitude of 40° (e.g., Muncie, IN)?

• Let φ denote the observer’s latitude; for Muncie, φ = 40°.
• The Sun is directly overhead when the Sun’s declination δ equals the observer’s latitude: δ = φ.
• The Sun’s declination δ varies over the year between a northern extreme δmax and a southern extreme δmin, centered around 0°, due to the tilt of the axis.
• From the geometry described in this lecture, δ is bounded: oxed{|\,oldsymbol{
  delta}\,| \, ext{max} \,= \varepsilon} where ε is the tilt angle of the Earth's axis relative to the orbital plane. In common terms, the Sun’s maximum declination is about oxed{|\,oldsymbol{
  delta}\,| \, ext{max} \approx 23.5^ ext{0}}.
• Therefore, for φ = 40°, there is no solution to δ = φ, since δ cannot reach 40°.
• Conclusion: The Sun is never directly overhead at Muncie (40° latitude).
• Note (context): The Sun can be directly overhead only for latitudes between the Tropic of Cancer and the Tropic of Capricorn (roughly within ±23.5° of the equator).

Quick reference: Dates, angles, and conversions

• Vernal equinox: around March 21; RA ≈ 0h; δ ≈ 0°.
• Autumnal equinox: around September 23; RA ≈ 12h; δ ≈ 0°.
• RA to angle relation: $1 ext{ h} = 15^ ext{0}$, so $ext{Angle} = ext{RA in hours} imes 15^ ext{0}.$
• Full circle in RA: 24 ext{ h} imes 15^ ext{0}/ ext{h} = 360^ ext{0}.
• Daily star rise shift: ext{rise time shift} \approx 4 ext{ minutes/day}.$$