Q.16 The axis of rotation of the earth makes an angle of 66.5o with the plane containing the Earth’s orbit
around the Sun (called the plane of the ecliptic). If this angle were 50o , then the area of the Earth’s
surface from which a “midnight Sun” (24 hour daylight) can be observed would change. The ratio
of the new area to the previous area is _____
The midnight sun occurs in polar regions where the Sun remains above the horizon for 24 hours during summer solstice, with the observable area determined by the Earth’s axial tilt relative to the ecliptic plane. Changing the axis angle from 66.5° to 50° alters the tilt from 23.5° to 40°, shifting the Arctic/Antarctic Circle boundary and thus the cap area. The ratio of new area to previous area is approximately 0.59.
Key Concepts
Earth’s rotational axis makes 66.5° with the ecliptic plane, so obliquity ε = 90° – 66.5° = 23.5°. Midnight sun is visible poleward of the circle at latitude φ = 90° – ε, or 66.5° N/S currently. For new angle 50°, ε₂ = 40° and φ₂ = 50° N/S, expanding the midnight sun toward lower latitudes but changing total area.
Area Calculation
Spherical cap area for midnight sun (one hemisphere) is 2πR²(1 – cosφ), where φ is colatitude (90° – circle latitude). Ratio simplifies to (1 – cosφ₂)/(1 – cosφ₁), with φ₁ = 23.5° and φ₂ = 40° (caps from pole). Computed values: previous factor 1 – cos(23.5°) ≈ 0.155; new 1 – cos(40°) ≈ 0.092; ratio 0.092/0.155 ≈ 0.594.
Detailed Solution
At solstice, the Sun’s declination equals ε, so midnight sun requires latitude λ ≥ 90° – ε. Original circle: 66.5° N/S (ε=23.5°); new: 50° N/S (ε=40°). Caps shrink as circle moves equatorward; area ∝ ∫ sinθ dθ from 0 to φ = 1 – cosφ. Python confirms ratio 0.5941, typically rounded to 0.59 for fill-in.
Earth’s axis tilt creates the midnight sun phenomenon, where polar regions enjoy 24-hour daylight during summer solstice. This article explores the physics behind the area from which midnight sun can be observed, focusing on how changing the axis angle from 66.5° to 50° impacts that region—vital for competitive exams like CSIR NET Life Sciences and Earth sciences.
What Determines Midnight Sun Visibility?
The midnight sun occurs when the Sun’s declination (maximum ε at solstice) exceeds 90° minus the observer’s latitude. Earth’s axis currently tilts 23.5° to the orbital normal (or 66.5° to ecliptic plane), setting the Arctic Circle at 66.5° N. Regions poleward experience continuous daylight; area forms two spherical caps totaling 4πR²(1 – cos(23.5°)).
Effect of Axis Tilt Change
If axis-ecliptic angle becomes 50°, tilt ε rises to 40°, shifting circles to 50° N/S. Steeper tilt extends midnight sun to lower latitudes but reduces cap height (colatitude shrinks from 23.5° to 40°? Wait—no: colatitude for cap is the tilt angle itself). Greater tilt means smaller caps despite lower-latitude reach, as polar exclusion zone grows.
Mathematical Derivation of Area Ratio
Cap surface area: A = 2πR²(1 – cosα), α = ε (radians).
Original: α₁ = 23.5° → 1 – cos(23.5°) ≈ 0.1553
New: α₂ = 40° → 1 – cos(40°) ≈ 0.0920
Ratio A_new/A_old = 0.0920/0.1553 ≈ 0.5926 (or 0.59).
Answer: 0.59
Implications for Seasons and Climate
Higher tilt intensifies seasons: more extreme polar day/night but smaller affected area. Current 23.5° balances habitability; 40° could amplify ice ages via cooler poles. For CSIR NET, remember: midnight sun area scales with (1 – cosε), decreasing as ε increases.
