Circumpolar star at elongation: If δ is the star’s declination and φ is the observer’s latitude, which relation gives the azimuth (Z) of the star at its greatest elongation?

Introduction / Context:
In astronomical surveying, stars near the pole (circumpolar) reach points of greatest azimuth from the local meridian, called elongations. The azimuth at elongation is a standard result used to derive observer’s latitude and to check instrument orientation.

Given Data / Assumptions:

• Star is circumpolar for the observer’s latitude φ.
• Declination of the star is δ (north positive).
• Elongation refers to the greatest angular departure from the meridian.

Concept / Approach:
At elongation the diurnal circle of the star is tangent to the vertical circle through the star. The condition dZ/dH = 0 (Z = azimuth, H = hour angle) leads to a pair of canonical relations for circumpolar stars: one for the hour angle and one for the azimuth. The azimuth relation reduces to sin Z = sec φ * cos δ, provided the star is indeed circumpolar (so the right-hand side does not exceed 1 in magnitude).

Step-by-Step Solution:
Use the elongation condition to eliminate H and link Z to φ and δ.The resulting standard formula is sin Z = sec φ * cos δ.Check feasibility: |sec φ * cos δ| ≤ 1 must hold for a valid circumpolar elongation.

Verification / Alternative check:
Treat φ = 45°, δ = 60°: sec φ = 1.414, cos δ = 0.5 → RHS ≈ 0.707 → Z ≈ 45°, a reasonable elongation azimuth.

Why Other Options Are Wrong:
cos Z = sec φ * cos δ / tan Z = sec φ * cos δ: not the standard elongation relation and dimensionally inconsistent with typical values.None of these / ad hoc expression: unnecessary because a standard closed-form relation exists.

Common Pitfalls:
Using formulas valid at meridian transit (culmination) instead of elongation; ignoring the circumpolar condition that constrains δ for a given φ.

Final Answer:
sin Z = sec φ * cos δ.