Latitude–pole altitude relationship: how are the geographic latitude (λ) of an observation site and the observed altitude (α) of the celestial pole related?

Introduction / Context:
One of the most elegant results in positional astronomy is that the elevation of the celestial pole above the horizon equals the observer’s geographic latitude. This provides a direct observational route to determining latitude using stellar observations.

Given Data / Assumptions:

• Celestial sphere model with Earth’s rotation axis extended to the celestial poles.
• Observer located at latitude λ on Earth.
• Altitude α is measured from the local horizon to the pole (typically the north celestial pole in the Northern Hemisphere).

Concept / Approach:
Consider the astronomical triangle formed by the celestial pole, zenith, and the observed body. For the pole itself, the geometry simplifies: the zenith distance of the pole equals the co-latitude (90° − λ). Consequently, the pole’s altitude α equals 90° − (90° − λ) = λ.

Step-by-Step Solution:
Define co-latitude = 90° − λ.Relate pole altitude to zenith distance: α = 90° − (zenith distance of pole).Zenith distance of pole = co-latitude → α = 90° − (90° − λ) = λ.

Verification / Alternative check:
At the equator (λ = 0°), the pole lies on the horizon (α = 0°). At the geographic pole (λ = 90°), the celestial pole is at the zenith (α = 90°). These limiting cases confirm the formula.

Why Other Options Are Wrong:

• Expressions involving 90° − α or α − 90° contradict the standard geometry.
• λ = 180° − α is not meaningful in this context.

Common Pitfalls:

• Mixing up declination of Polaris (≈ current NCP vicinity) with exact pole location; Polaris is near, not exactly at, the pole.
• Failing to correct for atmospheric refraction when observing low altitudes.

Final Answer:
λ = α