Longitude–time conversion in astro-surveying: the difference in longitude between two places, when expressed in time units, equals the difference in which time scales at those places?

Introduction / Context:
Longitude is fundamentally linked to time because Earth completes one rotation (360 degrees) in 24 hours. This principle underpins astronomical navigation and geodetic surveying, where longitude differences are routinely converted into time differences.

Given Data / Assumptions:

• Earth's rotation rate is assumed uniform for practical purposes.
• Time can be referenced to different celestial markers: mean sun, apparent (true) sun, or distant stars (sidereal).

Concept / Approach:
The angular–temporal relationship is 360 degrees ↔ 24 hours, or 15 degrees ↔ 1 hour. Therefore, 1 degree of longitude corresponds to 4 minutes of time. This conversion is independent of the chosen time reference (sidereal, apparent solar, or mean solar) because each is ultimately tied to Earth's rotation relative to a celestial reference.

Step-by-Step Solution:
Relate angle to time: 1 degree = 4 minutes.Apply to any longitude difference Δλ to obtain Δt = Δλ * (4 minutes/degree).Recognize that the same Δt manifests in sidereal, apparent solar, and mean solar time comparisons between the two places.

Verification / Alternative check:
Standard astro-navigation texts define local time = Greenwich time ± (longitude in time). This formulation is valid across the three time scales with the appropriate celestial reference.

Why Other Options Are Wrong:

• No single option is exclusive: the relationship holds for all listed time systems.

Common Pitfalls:

• Using longitude in degrees without converting to time (4 minutes per degree).
• Confusing east/west longitude sign conventions when adding/subtracting time.

Final Answer:
all the above