Latitude by meridian observation — which sky region implies φ = δ − z? If an observer computes latitude by subtracting a star’s zenith distance z from its declination δ (i.e., φ = δ − z), then the observed star must lie between which two reference directions?

Introduction / Context:
At meridian transit, the relationship between latitude φ, declination δ, and zenith distance z depends on where the star lies relative to the observer's zenith and the celestial equator. Choosing the correct sign is crucial for accurate latitude.

Given Data / Assumptions:

• Star observed on the meridian.
• Declination and zenith distance are known in sign and magnitude.

Concept / Approach:

For a northern observer and a star north of the equator, if the star is closer to the pole than to the zenith (i.e., it lies between the zenith and the pole), then z = δ − φ, giving φ = δ − z. If the star is between the equator and the zenith, then z = φ − δ and φ = δ + z. Correctly identifying the region resolves the sign choice.

Step-by-Step Solution:

Verification / Alternative check:

Sky diagrams of diurnal circles at meridian transit confirm the sign switch at the zenith relative to the equator and pole.

Why Other Options Are Wrong:

• Other sky regions lead to φ = δ + z or other sign choices; they are inconsistent with the stated computation.

Common Pitfalls:

• Ignoring the star’s bearing (north/south) when assigning signs to δ and z.

Final Answer:

Zenith and pole.