Latitude–zenith distance–declination relation: When a star is situated between the pole and the horizon (circumpolar side), which relation correctly connects latitude (φ), zenith distance (z), and declination (δ)?

Introduction / Context:
In astronomical surveying, the meridian altitude method leads to different formulas for latitude depending on the body’s position. The “star between the pole and the horizon” case (circumpolar side) is a classic configuration requiring care with signs and supplements.

Given Data / Assumptions:

• Observation at culmination on the celestial meridian.
• Star lies on the side of the elevated pole, between pole and horizon (not crossing the zenith).
• Angles measured in degrees; z is 90° − altitude.

Concept / Approach:
When the star is on the polar side, the arc geometry on the celestial meridian yields a supplementary relation. One can construct the spherical triangle (pole–zenith–star) and, by inspecting the arcs along the meridian, obtain φ + δ + z = 180°. Rearranging gives φ = 180° − (z + δ).

Step-by-Step Solution:
Write the meridian relation for the polar-side configuration: φ + δ + z = 180°.Solve for latitude: φ = 180° − (z + δ).Check plausibility: large z and δ reduce φ, which is consistent with a low-altitude circumpolar star.

Verification / Alternative check:
Draw the meridian with pole at 90°, equator at 0°. For a star below the zenith and toward the pole, the three arcs along the same great circle indeed sum to 180°, confirming the expression.

Why Other Options Are Wrong:
φ = z + δ or φ = z − δ: apply to equatorial-side cases.φ = δ − z or (z + δ) − 180°: incorrect sign/supplement handling for the polar-side configuration.

Common Pitfalls:
Forgetting the 180° supplement when the body is on the polar side; not sketching the geometry leads to sign mistakes.

Final Answer:
φ = 180° − (z + δ).