Astronomical surveying: for a star to be circumpolar as seen by an observer at latitude φ (in either hemisphere), which condition must be satisfied?

Difficulty: Medium

Correct Answer: Its angular distance from the celestial pole must be less than the observer's latitude

Explanation:


Introduction / Context:
In astronomical surveying and navigation, a star is called circumpolar if it never sets below the local horizon and remains perpetually above it throughout a sidereal day. Recognizing the condition for circumpolarity is essential for planning observations and understanding star trails and azimuth determinations.



Given Data / Assumptions:

  • Observer at latitude φ (north positive, south negative).
  • Declination of the star is δ, and its angular distance from the nearest celestial pole is 90° − |δ|.
  • Standard spherical astronomy geometry, neglecting refraction near the horizon for the conceptual condition.



Concept / Approach:
A star never sets if the lower culmination still lies above the horizon. Geometrically, the star’s circular path around the celestial pole remains entirely above the horizon when the angular radius of that circle (90° − |δ|) is less than the altitude of the pole above the horizon (|φ|). This yields the circumpolarity criterion: |δ| > 90° − |φ|, which is equivalent to the star’s angular distance from the pole being less than |φ|.



Step-by-Step Solution:
At the observer’s site, the pole altitude equals |φ|.A star’s polar distance = 90° − |δ|.For the star’s entire diurnal circle to be above the horizon: 90° − |δ| < |φ|.Rearrange: |δ| > 90° − |φ|. This means its distance from the pole is less than |φ|.Hence, the condition is: angular distance from pole < latitude.



Verification / Alternative check:
At the North Pole (φ = 90°), all stars with δ > 0° are circumpolar because polar distance is ≤ 90°. At the equator (φ = 0°), no star is circumpolar, matching the condition.



Why Other Options Are Wrong:

  • Declination = 0°: stars on the celestial equator rise and set.
  • Declination = 90°: only the pole itself; overly restrictive.
  • Hour angle = 180°: defines lower culmination moment, not circumpolarity.
  • RA equals LST: specifies transit time, not circumpolar nature.



Common Pitfalls:
Confusing declination with altitude; overlooking sign conventions for southern hemisphere; ignoring that circumpolarity depends on both δ and φ.



Final Answer:
Its angular distance from the celestial pole must be less than the observer's latitude

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