Culmination geometry: for latitude θ and declination δ, the upper culmination of a body is south of the zenith if its zenith distance z equals which expression?

Introduction / Context:
At upper culmination (meridian transit), a celestial body attains its greatest altitude for that day. Whether it passes south or north of the zenith depends on the observer’s latitude θ and the body’s declination δ. This relation is fundamental in astro-navigation and field astronomy.

Given Data / Assumptions:

• Observer is in the same hemisphere as the body (typical surveying cases).
• Upper culmination occurs on the local meridian.
• Refraction is negligible for the conceptual discussion.

Concept / Approach:
For an object culminating on the meridian, the zenith distance is z = |θ − δ|. If δ < θ (typical mid-latitudes with small declinations), the body lies south of the zenith at culmination and z = θ − δ (a positive angle). If δ > θ, the body passes north of the zenith and z = δ − θ.

Step-by-Step Solution:
Use meridian altitude formula: altitude at upper culmination a = 90° − |θ − δ|.Hence z = 90° − a = |θ − δ|.For “south of zenith” in the common case δ < θ → z = θ − δ.Therefore the required expression is θ − δ.

Verification / Alternative check:
Example: θ = 30°, δ = 10° → z = 20° (body is 20° south of zenith). If δ = 40° > θ, the body passes north of zenith with z = 10° but oriented to the north side.

Why Other Options Are Wrong:
θ + δ is not a zenith-distance expression; δ − θ would be negative for δ < θ (the south-of-zenith case); parentheses do not change the value.

Common Pitfalls:
Forgetting the absolute value in z = |θ − δ| and mixing “north of zenith” with “south of zenith.”

Final Answer:
θ − δ