science: winter solstice 2007

this year's (2007) winter solstice was on 22 dec... so happy winter solstice!!!

on winter solstice...

• the earth is close to the sun: 147,098,074 km⁵
• for locations on the tropic of capricorn, the sun appears to be directly overhead at noon
• for locations on the antarctic circle, the sun will set at midnight
• for locations bellow the antarctic circle, the sun will not set

according to nasa⁵ the distance between earth and sun on winter solstice - dec 22 is 147,152,000 km

according to nasa⁵ the distance between earth and sun on summer solstice - jun 23 is 152,006,000 km

i was having a huge argument with my cus about the two solstices...
he claimed that the earth's orbit around the sun was near circular and hence there is no significant change in the amount of incident solar radiation

for a local area (like a city)... the significant factor that causes seasonal temperature changes is the earth's tilt in relation to the orbital plane... but if you take the northern hemisphere and the southern hemisphere... the southern gets more solar radiation during its summer than what the northern hemisphere would get during its summer... by a significant 6.5% more! if we were in the equator, december would be warmer than july...

readings made from satellites have shown that the solar constant⁴ varies between 1399 W/m² to 1310 W/m²¹... wikipedia claims these values to be 1412 W/m² to 1321 W/m²²... solar constant varies with time because the sun follows a cycle... there are years when the sun is hotter than normal and there are times when its cooler than normal...

so these measured values have a difference of about 6.5%
1- (1310/1399) = 6.36%
and 1- (1321/1412) = 6.44%

we know that the power intensity density (radiation flux or power per unit area) of the sun follows a inverse square law
solar constant G_S is given by^1,7...
(4πL²) . G_S = (4πr²)σT_S⁴
where T_S is the black body radiation temperature of the sun, T_S = 5762 K or 6035°C and L is the distance between the sun and earth

earth's orbits around the sun is elliptical... with a aphelion of 152,097,701 km and perihelion of 147,098,074 km³

so the difference in solar constants is proportional to L²

147,098,074/152,097,701 = 0.967

the difference of L² is the square of the above value which is 0.935

so there is a difference of 1-0.935 which is 6.5% between the solar constant during summer solstice and winter solstice!

if you're still confused here is the math

(4πL²) . G_S = (4πr²)σT_S⁴

G_S = (r²/L²)σT_S⁴

during summer solstice G_{S, summer} = (r²/L_summer²)σT_S⁴
during winter solstice G_{S, winter} = (r²/L _winter ²)σT_S⁴

G_{S, summer}/G_{S, winter} = L_winter² / L_summer²
G_{S, summer}/G_{S, winter} = (147,098,074/152,097,701)² = 0.967² = 0.935

so the percentage change is (1-0.935) x 100% = 6.5%

and this corresponds to the measured values of G_{S, winter} / G_{S, summer} which is 6.36% (or 6.44%)

notes

1. Yunus A Cengel: Hear Transfer - A Practical Approach - 2002 edition
2. Wikipedia: Solar Radiation
3. Wikipedia: Earth
4. read my prev blog post on summer solstice
5. Wikipedia: Solar Constant
6. Nasa: Moon Phases
7. Nasa: JPL Solar System Simulator
8. Nasa: Solar Radiation and the Earth System

may be i should convince marissaj to write a musical about winter solstice!!!

oh btw... although it was poya yesterday (23rd)... full moon falls today (24th) (according to nasa⁵ that is)