Star elevation at elongation: which relation correctly gives the elevation α of a circumpolar star at elongation in terms of latitude φ and declination δ?

Introduction / Context:
Elongation of a circumpolar star occurs when its azimuth is extremal (east or west), i.e., the star lies on or near the prime vertical. Determining the star's elevation at this instant is a classic problem in spherical astronomy used in surveying reductions.

Given Data / Assumptions:

• Observer's latitude = φ.
• Star's declination = δ (circumpolar).
• Elongation condition corresponds to extreme azimuth (prime vertical crossing).

Concept / Approach:
From spherical trigonometry for the astronomical triangle, the general altitude formula is: sin h = sin φ sin δ + cos φ cos δ cos H. At elongation on the prime vertical, the condition for extremal azimuth yields cos H = (sin φ cos δ) / (cos φ sin δ) = tan φ cot δ. Substituting back gives sin h = sin φ * (sin^2 δ + cos^2 δ) / sin δ = sin φ * cosec δ.

Step-by-Step Solution:
Write sin h = sin φ sin δ + cos φ cos δ cos H.Use elongation condition: cos H = tan φ cot δ.Substitute and simplify → sin h = sin φ cosec δ.

Verification / Alternative check:
Alternative derivations via azimuth differentiation (dA/dH = 0) lead to the same expression.

Why Other Options Are Wrong:

• Other combinations (using secant with sin φ or cos φ) do not satisfy the elongation condition when substituted into the altitude formula.

Common Pitfalls:

• Using H = 90° (cos H = 0) indiscriminately; the true elongation hour angle depends on φ and δ.

Final Answer:
sin α = sin φ cosec δ