Solar parallax correction in altitude – select the correct expression For the Sun, what geocentric parallax in altitude should be added to the observed altitude (approximate horizontal parallax ≈ 9″)?

Introduction / Context:
Parallax correction converts a topocentric (observer-based) altitude to a geocentric one by accounting for the observer’s displacement from Earth’s center. For the Sun, the horizontal parallax is small (about 8.8–9.0 arcseconds), yet the correct functional form with altitude matters in precise astronomical reductions.

Given Data / Assumptions:

• Small-angle approximation suitable for solar observations.
• Horizontal parallax of the Sun HP ≈ 9″.
• α denotes observed altitude of the Sun.

Concept / Approach:
Parallax in altitude = HP × cos(altitude). This arises because the component of the Earth-center to observer displacement perpendicular to the line of sight scales with cos α. The correction is added (positive) to the observed altitude to arrive at the geocentric altitude.

Step-by-Step Solution:
Write correction: p_alt = HP * cos α.Insert HP ≈ 9″ → p_alt ≈ 9″ cos α.This is added to the observed altitude to reduce to the Earth’s center.

Verification / Alternative check:
At the horizon (α ≈ 0°), cos α ≈ 1, so parallax ≈ HP (maximum). At the zenith (α ≈ 90°), cos α ≈ 0, so parallax ≈ 0; this matches physical intuition.

Why Other Options Are Wrong:

• sin α, tan α, cot α, sec α do not match the geometric projection of the Earth-center offset onto the line of sight.

Common Pitfalls:
Confusing sign conventions or using sine instead of cosine; forgetting that solar parallax is small but non-zero.

Final Answer:
9″ cos α