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A circumpolar star is a star that, as viewed from a given latitude on Earth, never sets (that is, never disappears below the horizon), due to its proximity to one of the celestial poles. Circumpolar stars are therefore visible from said location toward nearest pole for the entire night on every night of the year (and would be continuously visible throughout the day too, were they not overwhelmed by the Sun's glare).
All circumpolar stars are within the circumpolar circle. This was in fact the original meaning of "Arctic Circle", before the current geographical meaning, meaning "Circle of the Bears" (Ursa Major, the Great Bear; and Ursa Minor, the Little Bear), from Greek αρκτικός (arktikos), "near the Bear", from the word άρκτος (arktos) bear.
As Earth spins daily on its axis, the stars appear to rotate in circular paths around one of the celestial poles (the north celestial pole for observers in the northern hemisphere, or the south celestial pole for observers in the southern hemisphere). Stars far from a celestial pole appear to rotate in large circles; stars located very close to a celestial pole rotate in small circles and hence hardly seem to engage in any diurnal motion at all. Depending on the observer's latitude on Earth, some stars — the circumpolar ones — are close enough to the celestial pole to remain continuously above the horizon, while other stars dip below the horizon for some portion of their daily circular path (and others remain permanently below the horizon).
The circumpolar stars appear to lie within a circle that is centered at the celestial pole and tangential to the horizon. At the Earth's North Pole, the north celestial pole is directly overhead, and all stars that are visible at all (that is, all stars in the northern celestial hemisphere) are circumpolar. As one travels south, the north celestial pole moves towards the northern horizon. More and more stars that are at a distance from it begin to disappear below the horizon for some portion of their daily "orbit", and the circle containing the remaining circumpolar stars becomes increasingly small. At the Earth's equator this circle vanishes to a single point – the celestial pole itself – which lies on the horizon, and there are therefore effectively no circumpolar stars at all.
As one travels south of the equator the opposite happens. The south celestial pole appears increasingly high in the sky, and all the stars lying within an increasingly large circle centered on that pole become circumpolar about it. This continues until one reaches the Earth's South Pole where, once again, all visible stars are circumpolar.
The north celestial pole is located very close to the Pole star (Polaris or North Star), so, from the northern hemisphere all circumpolar stars appear to rotate around Polaris. Polaris itself remains almost stationary, always at the north (i.e., the azimuth is 0°), and always at the same altitude (angle from the horizon), equal to the latitude of the point of observation on Earth.
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Whether a star is circumpolar depends upon the observer's latitude. Since the altitude of the north celestial pole (NCP) or south celestial pole (SCP) (whichever is visible) is the same as the absolute value of the observer's latitude, any star whose angular distance from the visible celestial pole is less than the absolute latitude will be circumpolar. For example, if the observer has latitude +50°, any star will be circumpolar if it is less than 50° from the north celestial pole. If the observer's latitude is −35°, then all stars within 35° of the south celestial pole will be circumpolar. Stars on the celestial equator are not circumpolar when observed from any latitude in either hemisphere of the Earth.
Whether a given star is circumpolar at the observer's latitude (θ) may be calculated in terms of the star's declination (δ). The star is circumpolar if θ+δ is greater than +90° (observer in northern hemisphere), or θ+δ is less than −90° (observer in southern hemisphere).
Similarly, the star will never rise above the local horizon if δ−θ is less than −90° (observer in northern hemisphere), or δ−θ is greater than +90° (observer in southern hemisphere).
Some stars within the far northern constellations, such as Cassiopeia, Cepheus, Ursa Major, and Ursa Minor, roughly north of the Tropic of Cancer (+23½°), will be circumpolar stars that never rise or set.
For British observers, for example, the first magnitude stars Capella (declination +45° 59) and Deneb (+45° 16') do not set from anywhere in the country. Vega (+38° 47') is technically circumpolar north of latitude +51° 13' (just south of London); taking atmospheric refraction into account it will probably only be seen to set at sea level from Cornwall and the Scilly Isles.
Stars (and constellations) that are circumpolar in one hemisphere are always invisible in the high latitudes of the opposite hemisphere, and these never rise above the horizon. For example, the southern circumpolar star Acrux is invisible from most of the Continental United States, likewise, the seven stars of the northern circumpolar Big Dipper asterism are invisible from most of the Patagonia region of South America.