Celestial Coordinates

The celestial sphere is an imaginary sphere where all stars are treated as if at the same distance, and Earth's rotation makes it seem to spin once a day. The altitude of the celestial north pole equals the observer's latitude, and hour angle zero is culmination when a star is highest on the meridian. Culmination altitude is h = 90° - |φ - δ|, and latitude also separates circumpolar from never-rising stars. Here you can move latitude, hour-angle, and declination sliders to track diurnal paths and compute culmination altitude.

What is the celestial sphere?

💡 Imagine the sky as a giant sphere

①An imaginary sphere on which all stars are assumed to lie at the same distance = celestial sphere

②As Earth rotates → the sphere appears to spin once a day

③Center of the sphere = observer / celestial equator = extension of Earth's equator

④Celestial north pole = extension of the polar axis (near Polaris)

⑤To express stellar positions, we need coordinate systems

Comparison

ChartHorizontal vs Equatorial coordinates

Item	Horizontal	Equatorial
Reference	observer's horizon	celestial equator
Coord. 1	azimuth (A)	right ascension α (h)
Coord. 2	altitude (h)	declination δ (°)
Time variation	changes with time	unchanged
Use	find positions when observing	star catalogs, observatories

Diurnal motion and hour angle

Observer latitude φ (°)37

Hour angle H (°)0

🌍 Key relations

①Altitude of celestial north pole = observer's latitude φ (Seoul: φ=37.5° → Polaris altitude = 37.5°)

②Hour angle H = angle measured westward from the meridian

③H = 0 → star on meridian = culmination (highest point)

④Stars rise in the east, transit, then set in the west

⑤Sidereal time (ST) = right ascension (α) + hour angle (H)

Calculating culmination altitude

Declination δ (°)23

Culmination altitude formula

h_culm = 90° - |φ - δ|

culmination altitude = 90° − |latitude − declination|

📐 Example: culmination altitude in Seoul

①Seoul latitude φ = 37.5°

②Summer solstice Sun (δ = +23.5°): h = 90° − |37.5° − 23.5°| = 76°

③Winter solstice Sun (δ = −23.5°): h = 90° − |37.5° − (−23.5°)| = 29°

④This is why the Sun is high in summer and low in winter!

Circumpolar and never-rising stars

Circumpolar condition

δ > 90° − φ

star never sets when its declination exceeds (90° − latitude)

Never-rising condition

δ < φ − 90°

star never rises when its declination is below (latitude − 90°)

🌟 Examples in Seoul

①Seoul (φ=37.5°): circumpolar if δ > 52.5° → never sets

②Never-rising if δ < −52.5° → never visible

③Equator (φ=0°): no circumpolar/never-rising — all stars rise and set

④North pole (φ=90°): all stars with δ>0° are circumpolar — observable 24h

Worked Examples and Exam Practice

Example 1

In Seoul at latitude 37.5°N, what is the meridian (culmination) altitude of a star with declination δ = +23.5° (or the summer-solstice Sun)? (altitude = 90° − |φ − δ|)

Meridian altitude: h = 90° − |latitude − declination| = 90° − |φ − δ|.

h = 90° − |37.5° − 23.5°| = 90° − 14° = 76°.

▸ 76°

In summer (high-δ Sun) the culmination is high; in winter (negative δ) it is low. The same formula applies to any object.

Example 2

At latitude 37.5°N, what declination δ must a star have to be circumpolar (never setting)?

A star is circumpolar when its declination δ > 90° − latitude φ.

With φ = 37.5°, δ > 90° − 37.5° = 52.5°. So a star with declination above 52.5° never sets.

▸ δ > 52.5° (= 90° − 37.5°)

The celestial pole altitude equals the latitude, so stars near the pole (high δ) never set. At the equator there are no circumpolar stars.

CSAT-style

Which statement about celestial coordinates and meridian altitude is correct?

①The altitude of the celestial north pole equals the observer's longitude

②Meridian altitude is found from 90° − |latitude − declination|

③In the equatorial system, right ascension and declination change continuously with time

④A smaller declination makes a star more likely to be circumpolar

⑤In the horizontal system, a star's coordinates do not change over time

▸ ② Meridian altitude is found from 90° − |latitude − declination|

The meridian-altitude formula h = 90° − |φ − δ| is the key equation for any culmination problem.

The celestial pole altitude equals the latitude (①), right ascension and declination are fixed (③), higher declination makes a star circumpolar (④), and horizontal coordinates change with time (⑤).

Summary

Culmination altitude

h = 90° - |φ - δ|

core formula for all culmination problems

🎯 Exam Points

①Culmination altitude = 90° − |φ − δ| (memorize!)

②Altitude of celestial north pole = φ

③Circumpolar: δ > 90° − φ (never sets)

④Never-rising: δ < φ − 90° (never rises)

⑤Sidereal time (ST) = α (RA) + H (hour angle)

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