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Great Directions (Posted on 2005-03-16) Difficulty: 3 of 5

I live in Boston, which is at latitude 42 degrees North, longitude 70 degrees west.

a) I happen to know that on the spring equinox (roughly March 22), the sun is directly over the equator. So why it do I see it rising directly east (true east, not magnetic), even though the equator is far south of me?

b) Similarly, on the summer solstice (roughly June 22), the sun is directly over the Tropic of Cancer (latitude 23.5 degrees North). In fact, this is the furthest North the sun ever gets. So why do I see the sun rising roughly East-North-East (halfway between true East and true NorthEast), even though the Tropic of Cancer is also south of me?

See The Solution Submitted by Steve Herman    
Rating: 3.5000 (2 votes)

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Solution Explanation | Comment 1 of 8

Part a:

The sun is 93,000,000 miles, or 150,000,000, km away from the earth, so that its rays are essentially parallel when they strike anywhere on earth, as the earth is only 4000 miles in radius, and arctan(4,000/93,000,000) is less than 9 arc seconds.  The major lack of parallelism in the sun's rays that strike the earth arises from the sun's large diameter, about 100 times that of earth, so we see a solar disk rather than a point source of light.   But if we consider the rays parallel, they will still trace back to a point quite near the center of the apparent solar disk (only at most 9 arcseconds away, out of a 900 arcsecond apparent radius).

So now consider your point of view from directly above the equator, on the meridian where the sun is rising when the sun is directly above the equator also.  The sun is way out to your right. You are far enough back from the plane containing the earth's axis and the sun so that the earth appears in orthographic projection.  The essentially parallel rays of the sun are coming from the right.  The parallels of latitude on the earth also all appear horizontal to you as they are seen edge-on, including the 42nd north parallel, where Steve is watching the sun rise (he's on the meridian facing us). A ray from the sun is coming in directly from the right, and therefore parallel to the parallels of latitude, and striking Steve.  As it is tangent to the surface and coming directly parallel to his parallel of latitude, he sees it as coming directly from the east.

Going back to our scale consideration, while such a ray hitting the equator came from the central part of the solar disk as seen from the sun, the one hitting Steve came from 4000*sin(42-deg) or about 2700 miles farther north on the sun's surface, but that's only a small fraction of the total size of the sun.  So essentially the parallelism is legitimate. Of course any observer will see rays from all over the sun, but that just fills in the full disk compared to just a point nighttime-starlike source in the sky.

Part b:

Consider the parallel rays coming down to earth, perpendicular to a point on the surface on the tropic of Cancer (actually closer to 23.4 degrees north than to 23.5 degrees north). The sunrise/set line is a great circle centered on this point. As with all great circles, it has a radius of 90 degrees.  Consider also a plane containing the sun, the center of the earth and Steve in Boston.  As the rays are parallel, Steve is 90 degrees away from the point where the sun is overhead.  At latitude 42 north, Boston is 48 degrees from the north pole. The point directly beneath the sun is 66.6 degrees from the north pole.  From there it's a matter of spherical trigonometry, to find the angle between the direction of the north pole from Steve's point of view and the direction of the sun from Steve's point of view.  It is similar to the situation of finding the great circle route from Boston to whatever point the sun is overhead. It starts out going somewhat north of east and then becomes east-west, as it's turning southward to get to that tropical point.

Doing the spherical trig, the angle opposite the 66.6-deg side is about 58 degrees, indicating the angle, from Steve's point of view, between the direction to the north pole and the direction to the subsolar point, is about 58 degrees, so that the sun is rising 32 degrees north of east, the same direction you'd have to get to Saudi Arabia via great circle, which further spherical trig calculations show is the place where the sun is overhead while it is rising in Boston on the day of the Summer solstice. (So by coincidence, the Boston sunrise watcher is almost facing Mecca.)

An aside:

When the sun is rising, atmospheric refraction makes it appear about 1/2 degree higher in the sky than it would be strictly geometrically with straight rays. Also, as the sun's apparent radius is about 1/4 degree, the sun starts to rise when its center is still geometrically 3/4 degrees below the horizon, and has been seen to have completely risen when the center is still 1/4 degree below the horizon, so in all the spherical trig, 90 degrees should be replaced by between 90.75 and 90.25 degrees, during the course of sunrise.  So on the day of the equinox, the sun will rise a fraction of a degree north of east, and on the day of the solstice the figure given above will be off a fraction of a degree.

  Posted by Charlie on 2005-03-16 14:42:34
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