June 03, 2009

Tinted Visor vs. Trigonometry

When Wolfram Alpha, Google and Wikipedia all came through... to prove me wrong.

I suppose after the initial hype, everybody just relegated "Wolfram Alpha" to a cute toy, and not by any stretch a "replacement for google" as it was touted. Today, I had a chance to put Wolfram Alpha to good use. And it worked beautifully.

On the way to work today, the sun just seemed too bright. I barely open my eyes, even with my mildly tinted visor. It didn't seem as bright yesterday, but today was really harsh.

It got me thinking about the sun's declination. On the 21st of June, summer solstice, the declination of the sun at the tropic of cancer (approximately Calcutta) is zero degrees. So maybe today, May 28th, maybe the declination in Bangalore is zero. I wanted to verify this hypothesis.

First Google yielded very little information. Even Wikipedia was of little help. I got some information on the terminology (the words declination, solstice) and some numbers (tropic of cancer is 23.44° N, Bangalore is 12.97° N)

I turned to Wolfram Alpha. I tried combination of words like sun, solstice, equator, declination. It threw about a few numbers, but for the most part, it said "I dunno how to process your input".

Ok, fine. Since everybody was being so uninformative, I decided to use a little bit of high-school trigonometry and find out the answer myself.

First, let's analyze what I already knew.

  1. I knew that the point at which the sun's declination is zero follows a sinusoidal waveform, with the peaks at the tropics and the centre at the equator.
  2. I knew that the peak was at summer solstice, which is Jun 21st.
  3. And of course, I knew a year is 365.25 days long, and Bangalore & Calcutta latitudes.

Back to Wolfram.

12.97 / 23.44 = 0.5533

Bangalore is 0.5533 of the angular distance from the equator to the tropic.

Arc Cosine of that = 56.40 degress

It was also 0.98 radians, but I decided to use degrees in my calculations.

(56.40 / 360) * 365.25 = 57.23 days

So basically Bangalore is zero declination 57.23 days before and after the summer solstice.

57 days before Jun 21st = April 25th
57 days after Jun 21st = August 17th.

So there we are. The sun is closest to Bangalore on those two days, April 25th and August 17th. Today is neither of those days, and the sun just happened to seem bright to me today.

But it does feel good to have "proved" it.