Astronomical refraction — first-order correction as a function of altitude If α is the observed altitude of a star, which expression approximates the refraction correction (in arcseconds) for standard conditions?

Introduction / Context:
Astronomical refraction bends light downward, making celestial objects appear higher than they truly are. A simple first-order formula provides a quick correction for moderate altitudes.

Given Data / Assumptions:

• Standard pressure and temperature near sea level.
• Moderate zenith distances (not very near the horizon).
• Small-angle approximation using a first-order term.

Concept / Approach:

The classic first-order refraction formula uses the zenith distance z = 90° − α. Refraction R (arcseconds) ≈ 58″ * tan z. Substituting tan z = cot α gives R ≈ 58″ * cot α, which is easy to apply when altitude is known.

Step-by-Step Solution:

Verification / Alternative check:

More accurate models add higher-order terms (e.g., −0.066″ * tan^3 z). For typical field work at altitudes above about 15°, the first-order formula is sufficiently accurate.

Why Other Options Are Wrong:

• (b) swaps tan z with tan α and is incorrect.
• (c) and (d) have wrong functional form and units.
• (e) has the right function of z but with the wrong coefficient.

Common Pitfalls:

• Applying the first-order formula near the horizon where refraction becomes large and nonlinear.

Final Answer:

58″ * cot α.