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urface of the earth. umber of days past December 21. Let h noon, and let 9 be the latitude of the 50 sin (360), 150 cos (360), 0). -

urface of the earth. umber of days past December 21. Let h noon, and let 9 be the latitude of the 50 sin (360), 150 cos (360), 0). -

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Like so many astronomers before us and throughout history, we shall determine, through calculations, the angle of elevation of the sun for any given time of any given day at any given place on the surface of the earth. Let d be the number of days past December 21. Let h be the number of hours (positive or negative) from noon, and let 0 be the latitude of the observer. The vector from the earth to the sun is: 3= (150 sin(), – 150 cos (* (360d), 0). 365 365 We want to find the angle between s and the observer's plane of tangency to the earth. To do this, we will need the normal, n, of this plane of tangency. A. Pick specific values of d, h, and 0 (perhaps the current date and time of your current latitude 0. Use negative values of e for southern latitudes. Calculate s. B. Now to find n, we start by assuming the earth's axis is not tilted. i. Given that 3= (s1, s2, 0), let ni = scos ( ) + (- s2, s1, 0) sin(). 360h Why is this the correct normal for a person on the equator? ii. Let 2 = n cos 0 + (0, 0, | 171] sine). What does represent? C. The earth's axis is tilted p = 23.45° away from the z-axis in the direction of the y-axis. If m= (a, b, c) , then n= (abcos p + csinq, ccos p - bsinp) . Justify this and then calculate n.
Transcribed Image Text:Like so many astronomers before us and throughout history, we shall determine, through calculations, the angle of elevation of the sun for any given time of any given day at any given place on the surface of the earth. Let d be the number of days past December 21. Let h be the number of hours (positive or negative) from noon, and let 0 be the latitude of the observer. The vector from the earth to the sun is: 3= (150 sin(), – 150 cos (* (360d), 0). 365 365 We want to find the angle between s and the observer's plane of tangency to the earth. To do this, we will need the normal, n, of this plane of tangency. A. Pick specific values of d, h, and 0 (perhaps the current date and time of your current latitude 0. Use negative values of e for southern latitudes. Calculate s. B. Now to find n, we start by assuming the earth's axis is not tilted. i. Given that 3= (s1, s2, 0), let ni = scos ( ) + (- s2, s1, 0) sin(). 360h Why is this the correct normal for a person on the equator? ii. Let 2 = n cos 0 + (0, 0, | 171] sine). What does represent? C. The earth's axis is tilted p = 23.45° away from the z-axis in the direction of the y-axis. If m= (a, b, c) , then n= (abcos p + csinq, ccos p - bsinp) . Justify this and then calculate n.
D. Let B be the angle between 3 and n, calculate a = 90° - B. This is the angle of elevation of the sun. Why is a the angle of elevation of the sun, and not B ? E. What does it mean if the angle of elevation is negative? (In practice, the angle between a line and a plane will always be between 0° and 90° . Why?) F. Develop a general formula for a in terms of d, h, and e. How can we find the positions of the stars and other planets?
Transcribed Image Text:D. Let B be the angle between 3 and n, calculate a = 90° - B. This is the angle of elevation of the sun. Why is a the angle of elevation of the sun, and not B ? E. What does it mean if the angle of elevation is negative? (In practice, the angle between a line and a plane will always be between 0° and 90° . Why?) F. Develop a general formula for a in terms of d, h, and e. How can we find the positions of the stars and other planets?
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