Sky Coordinate Systems: Local Horizon vs Universal (RA/Dec), Ecliptic, and Precession

Horizon as reference; two coordinates: altitude and azimuth.
Altitude: angular distance from the horizon up to the object. 0° at horizon, 90° at zenith. Expressed as an angle: $Alt = \angle(\text{horizon}, \text{object})$ .
Azimuth: direction along the horizon from due north, measured clockwise. 0° = north, 90° = east, 180° = south, 270° = west. Expressed as $Az = \phi$ with 0° ≤ $\phi$ < 360°.
Meridian: the north–south line that passes overhead; helps relate altitude on the meridian to latitude.
Relationship to latitude: altitude of the celestial equator on the meridian is $Alt_{CE} = 90^{\circ} - \lambda$ , where $\lambda$ is the observer’s latitude.
Example: at latitude $\lambda = 40^{\circ}$ , $Alt_{CE} = 50^{\circ}$ .
Altitude of Polaris (North Star) ≈ observer’s latitude in the Northern Hemisphere; Polaris has azimuth ≈ 0° (due north).

Local horizon coordinates are location-specific; universal description uses Right Ascension (RA) and Declination (Dec).
The vernal equinox marks RA = 0h; the Sun’s position along the ecliptic defines RA/Dec over the year.
The ecliptic is the Sun’s apparent path across the sky; it is tilted by $\varepsilon = 23.5^{\circ}$ relative to the celestial equator.
The Sun appears to move north–south along the ecliptic, causing seasons as the Sun’s declination changes.

The celestial equator is the projection of Earth’s equator onto the sky.
The altitude of the celestial equator on the meridian is $Alt_{CE} = 90^{\circ} - \lambda$ (reiterated for clarity).
The latitude of the observer is given by the altitude of Polaris: for Northern Hemisphere, $Alt_{Polaris} \approx \lambda\;\text{(degrees above northern horizon)}$ .
The ecliptic path and the zodiac constellations the Sun passes through: Leo, Gemini, Taurus, Aries, Pisces, etc.
There are 12 zodiac constellations; technically there are 88 official constellations total. There is occasionally reference to a 13th constellation (Ophiuchus) in popular lore, but it is not part of the traditional zodiac.

Equinoxes: Sun crosses the celestial equator twice a year.
- Vernal (spring) equinox: around $March\;21$ , RA = 0h, Sun on the celestial equator.
- Autumnal (fall) equinox: around $Sept\;23$ .
Solstices: extremes of Sun’s declination.
- Summer solstice: around $June\;21$ , Sun at maximum + $23.5^{\circ}$ declination.
- Winter solstice: around $Dec\;21$ , Sun at maximum - $23.5^{\circ}$ declination.
The apparent path of the Sun (the ecliptic) is inclined to the celestial equator by $\varepsilon = 23.5^{\circ}$ .
The Sun’s position on the ecliptic changes through the year, producing seasonal changes in day length.

Maximum solar declination is $\pm 23.5^{\circ}$ ; never reaches 90° declination from the horizon.
Thus, the Sun is never directly overhead at latitude $\lambda = 40^{\circ}$ ; you’d need a latitude of 90° to have overhead noon sun at some time of year.

The ecliptic line on star charts shows the Sun’s path through the sky and the constellations it passes through.
Example: on a given date, the Sun may be in Leo (around August/September).
Star charts show the ecliptic and the equal-day/night line (the celestial equator) as flat representations; the tilt along the ecliptic is due to Earth’s axial tilt.

Polaris location: near the North Celestial Pole; used to find due north.
Finding Polaris: use the two stars at the end of the Big Dipper’s cup, draw a straight line away from the cup, and the first bright star you encounter is Polaris.
Polaris is part of Ursa Minor (Little Dipper) and is the brightest star in that faint constellation; Polaris is the tip of the Little Dipper’s tail.
The Big Dipper is an asterism (a recognizable pattern within a larger constellation, Ursa Major).
Concept: as the Earth rotates, the sky appears to rotate around the North Celestial Pole; star trails illustrate this rotation.
Altitude of Polaris equals the observer’s latitude: $Alt_{Polaris} \approx \lambda\; (\text{in the Northern Hemisphere})$ .

Axial precession: the Earth's rotation axis slowly traces a circle with a period of about $2.6\times 10^{4}$ years (roughly 26,000 years).
Due to precession, the North Celestial Pole points toward different stars over millennia; Polaris is not always the North Star.
Precession also shifts the orientation of the celestial equator relative to the ecliptic; this causes the observed zodiac signs to drift relative to the constellations over long timescales.
The motion also explains why “astrological signs” do not align with where the Sun is today on star charts, and why people’s sun signs would shift if you go back/forward thousands of years.

Astrology is a pseudoscience; it uses stars and the Sun’s position to assign horoscopes but does not reflect modern astronomical understanding.
In astronomy, we use accurate celestial coordinates (RA/Dec) and account for precession, proper motion, and the tilt of the ecliptic.

Altitude vs azimuth: Alt is vertical angle from horizon; Az is horizontal angle from due north clockwise.
Latitude ➜ celestial equator altitude on the meridian: $Alt_{CE} = 90^{\circ} - \lambda$ .
Polaris altitude ≈ your latitude; Polaris azimuth ≈ 0° (north).
The Sun’s path: the ecliptic, tilted by $\varepsilon = 23.5^{\circ}$ ; equinoxes at RA = 0h (vernal) and RA = 12h (autumnal, roughly; dates given on charts).
Sidereal day: $23\,\text{h}\,56\,\text{m}$ ; solar day is 24h; stars rise about $4\text{ minutes}$ earlier each day.
Precession: one full cycle ≈ $2.6\times 10^{4}$ years; North Star changes over millennia.
Always cross-check your birthday position on the ecliptic when reading star charts to see which constellation the Sun would be in on that date.