- #1

- 10,295

- 41

A recent thread (https://www.physicsforums.com/showthread.php?t=69970) by DaveC426913 got me thinking about differential geometry. The compass angle at which the sun rises each morning varies with latitude, but not linearly; it actually seems to be a rather complicated relationship. Let's look at a diagram:

http://www.virtualcivilization.org/sun.png

This diagram is drawn above the north pole. It is summer in the northern hemisphere, and the north pole experiences continual sunlight.

Consider two points at dawn, along the terminator. Point A is on the equator. The due-east and normal (altitude) vectors are drawn in black, and the due-north vector points directly out of the page.

Point B is at about 60 degrees north latitude. The due-east, due-north, and normal vectors are also drawn in black. At each point, a vector pointing directly to the sun (assumed to be at infinite distance) is drawn in red.

It can be easily seen that the sun rises more northerly at higher latitudes. That fact that can be understood easily by considering parallel transport. If the vector triple at point A were slid (parallel-transported) from point A to the north pole

On the other hand, if the vector triple at point A is parallel-transported along the terminator, its rotation is complicated. This is because the terminator is not a geodesic (great circle), while the line of longitude is.

I'd like to figure out how to understand this more deeply. I'd like to go from drawing a pretty picture to actually understanding the mathematics -- how to calculate the sunrises's compass angle on the horizon for any latitude. The summer solstice can be assumed.

Can anyone help?

- Warren

http://www.virtualcivilization.org/sun.png

This diagram is drawn above the north pole. It is summer in the northern hemisphere, and the north pole experiences continual sunlight.

Consider two points at dawn, along the terminator. Point A is on the equator. The due-east and normal (altitude) vectors are drawn in black, and the due-north vector points directly out of the page.

Point B is at about 60 degrees north latitude. The due-east, due-north, and normal vectors are also drawn in black. At each point, a vector pointing directly to the sun (assumed to be at infinite distance) is drawn in red.

It can be easily seen that the sun rises more northerly at higher latitudes. That fact that can be understood easily by considering parallel transport. If the vector triple at point A were slid (parallel-transported) from point A to the north pole

*via a line of longitude*, the sun's compass position would vary directly with the latitude -- it'd be an easy problem.On the other hand, if the vector triple at point A is parallel-transported along the terminator, its rotation is complicated. This is because the terminator is not a geodesic (great circle), while the line of longitude is.

I'd like to figure out how to understand this more deeply. I'd like to go from drawing a pretty picture to actually understanding the mathematics -- how to calculate the sunrises's compass angle on the horizon for any latitude. The summer solstice can be assumed.

Can anyone help?

- Warren

Last edited by a moderator: