Why is the altitude of the pole star equal to my latitude?

The rotation axis of Earth points at the celestial pole. From the North Pole the pole is overhead (altitude 90°), from the equator it lies on the horizon (0°), and in between the angle between the horizon and the axis equals the geographic latitude. Polaris sits 0.74° from the pole, so its altitude oscillates by that much around |φ| during the day.

Which stars never set, and which never rise?

A star never sets if its declination is closer to the visible pole than the horizon is: |δ| > 90° − |φ| on the observer’s side of the celestial equator. Stars equally far on the other side never rise. At the equator both ranges vanish — every star rises and sets; at the poles half the sky is permanently up and half permanently hidden.

Why are star trails vertical at the equator but tilted at 45° N?

Stars move on circles parallel to the celestial equator. At the equator that plane is perpendicular to the horizon, so stars rise straight up in the east; at latitude φ the diurnal circles are tilted by 90° − φ to the horizon, and at the pole they run parallel to it — stars neither rise nor set.

What is sidereal time and why not use clock time?

Local sidereal time is the right ascension on the meridian right now; a star culminates when LST equals its α. One sidereal day is 23 h 56 min, so a given star rises about 4 minutes earlier each solar day. The simulator sets LST directly so the geometry does not depend on the date.

Celestial Sphere: Daily Motion of the Stars

Observer at latitude φ: pole altitude = |φ|, equator, meridian, circumpolar and never-rising caps; stars turn with sidereal time.

Observer

Latitude φ (+N / −S)45°

Sidereal time (LST)6 h

Clock speed0.5 h/s

View

Show celestial equatorShow eclipticShow labelsStar trails (last 2 h)

Selected star

You can also click a named star on the canvas. Altitude and azimuth update live as sidereal time advances.

Shortcuts

Space / Enter — run / stop the sidereal clock
R — reset latitude, time and view
Drag the sphere to rotate it · click a named star to select it

Measured values

Altitude of celestial pole = |φ|45.0°

Circumpolar starsδ > +45°

Never-rising starsδ < −45°

Limit 90° − |φ|45.0°

Sidereal time06:00

StarPolaris

RA / Dec02:31 / +89.3°

Altitude+45.4°

Azimuth (N=0°, E=90°)359.2°

Hour angle H = LST − α+52.0°

Visibilitycircumpolar (never sets)

Hours above horizon per day24.0h

About this model

This simulator draws the celestial sphere for an observer at a chosen latitude φ and turns it with local sidereal time. Each star keeps fixed equatorial coordinates (right ascension α, declination δ); its hour angle is H = LST − α, and its altitude and azimuth follow from the spherical-triangle relation sin a = sin φ sin δ + cos φ cos δ cos H. The visible celestial pole stands at altitude |φ| above the north point (southern hemisphere: the south pole above the south point), the celestial equator crosses the meridian at 90° − |φ|, and the horizon, meridian and zenith are drawn as reference circles. Two shaded caps mark the declination ranges that never set (circumpolar, |δ| > 90° − |φ| on the observer’s side of the equator) and that never rise (the same range on the opposite side). The field contains ~200 pseudo-random stars plus ten real bright stars with J2000 coordinates (Polaris, Vega, Sirius, Betelgeuse, Canopus, Arcturus, Capella, Rigel, Deneb, Altair); click one to read its altitude, azimuth and hours above the horizon from cos H₀ = −tan φ tan δ. The “sky dome from outside” view is an orthographic 3D sphere you can rotate by dragging; the “looking at horizon” view is a flat altitude–azimuth panel 180° wide toward N, E, S or W, where the slope of star trails shows how the diurnal motion depends on latitude. Omitted: atmospheric refraction, precession, proper motion, the Sun, Moon and planets, and the conversion from clock time to sidereal time (sidereal time is set directly).

Who it's for: Grades 7–11 astronomy and geography: horizon coordinates, the pole star and latitude, circumpolar stars, and why the sky looks different at the equator, mid-latitudes and the poles.

Key terms

Celestial sphere
Celestial pole
Celestial equator
Hour angle
Sidereal time
Altitude and azimuth
Circumpolar star
Meridian

How it works

The celestial sphere is an imaginary sphere centred on the observer; stars sit on it at fixed right ascension α and declination δ and the whole sphere turns once per sidereal day about the celestial poles. For an observer at latitude φ the visible pole stands at altitude |φ| above the north (or south) horizon, the celestial equator crosses the meridian at altitude 90° − |φ|, and a star’s altitude follows sin a = sin φ sin δ + cos φ cos δ cos H with hour angle H = LST − α. Stars with δ beyond 90° − |φ| on the observer’s side of the equator never set (circumpolar); those equally far on the other side never rise. The model ignores refraction, precession and proper motion.

Key equations

sin a = sin φ · sin δ + cos φ · cos δ · cos H

H = LST − α (hour angle, 1 h = 15°)

altitude of celestial pole = |φ|

circumpolar: |δ| > 90° − |φ| (same hemisphere) · never rises: |δ| > 90° − |φ| (opposite)

rise / set hour angle: cos H₀ = −tan φ · tan δ

Frequently asked questions

Why is the altitude of the pole star equal to my latitude?: The rotation axis of Earth points at the celestial pole. From the North Pole the pole is overhead (altitude 90°), from the equator it lies on the horizon (0°), and in between the angle between the horizon and the axis equals the geographic latitude. Polaris sits 0.74° from the pole, so its altitude oscillates by that much around |φ| during the day.
Which stars never set, and which never rise?: A star never sets if its declination is closer to the visible pole than the horizon is: |δ| > 90° − |φ| on the observer’s side of the celestial equator. Stars equally far on the other side never rise. At the equator both ranges vanish — every star rises and sets; at the poles half the sky is permanently up and half permanently hidden.
Why are star trails vertical at the equator but tilted at 45° N?: Stars move on circles parallel to the celestial equator. At the equator that plane is perpendicular to the horizon, so stars rise straight up in the east; at latitude φ the diurnal circles are tilted by 90° − φ to the horizon, and at the pole they run parallel to it — stars neither rise nor set.
What is sidereal time and why not use clock time?: Local sidereal time is the right ascension on the meridian right now; a star culminates when LST equals its α. One sidereal day is 23 h 56 min, so a given star rises about 4 minutes earlier each solar day. The simulator sets LST directly so the geometry does not depend on the date.

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