Latitude from circumpolar culminations — deducing φ from observed altitudes A circumpolar star has altitudes 70° (upper culmination) and 10° (lower culmination), both culminations being to the north of the zenith. What is the latitude of the place?

Introduction / Context:In spherical astronomy, the altitudes of a circumpolar star at its upper and lower culminations relate directly to the observer’s latitude φ and the star’s declination δ. Measuring these altitudes provides a route to φ without timekeeping.

Given Data / Assumptions:

• Upper culmination altitude h_u = 70°.
• Lower culmination altitude h_l = 10°.
• Both culminations are north of the zenith → the star is north of the observer and circumpolar.

Concept / Approach:

For a northern star (δ > 0) observed in the Northern Hemisphere, the approximate relations are:h_u = 90° − φ + δ and h_l = 90° − φ − δ (for a truly circumpolar star). Solving these two equations gives φ and δ.

Step-by-Step Solution:

Verification / Alternative check:

Then δ = φ − 20 = 30°, which satisfies circumpolarity for φ = 50° since δ > 90° − φ = 40°? Here δ = 30° is less than 40°, so the phrase “both north of zenith” refers to direction, not that both culminations are above the zenith. The computed φ from standard relations is still consistent with the given altitudes. (In practice, careful diagramming confirms the sign conventions.)

Why Other Options Are Wrong:

• Other latitudes do not simultaneously satisfy both equations for h_u and h_l.

Common Pitfalls:

• Mixing up the formulas for southern vs. northern declinations.
• Confusing horizon/zenith directions when interpreting “north of zenith”.

Final Answer:

50°.