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How does a planetary orbit subtend an arc?
Let's ramble around a bit and see if we can get to a place where the answer appears. Ready? Let's jump..
Planets rotate about the sun in their orbits. Some of the orbits are …pretty circular, and others are eliptical. You can picture these different orbits with the sun in the center and the planet moving around the sun in its circular or eliptical orbit. Got a picture? Good. Let's jump again..
If we take a planet with a circular orbit and draw a line from the sun to where it is, that's a radius. If we look at the planet some time later and draw another line from the sun to the planet, we have another radius. The path the planet has traveled during the time we waited before drawing that second line is an arc. It's a portion of the planet's orbital path. In this case, that path is a circle (though no planet has a perfectly circular orbit). If we pick the right amount of time, the planet will have moved through 1/360th of its orbit around the sun. That's one degree of of its orbit or one degree of arc. The (amount of) arc subtends a one degree angle as regards the planet's orbit..
That which subtends an angle is a line or path (in this case, it's an arc), that lies inside two lines drawn out from a central point that form that given angle. We spoke of the arc subtended by the angle, and we can apply this idea to an eliptical path of orbit as well as a circular one. It's a bit "weird" because the eliptical orbit makes for some odd "bits" of arc at different points along the curve of its circumference, but no biggie. We can visualize it..
Oh, and we can talk about the area between the two lines that form that angle and the arc that subtends the angle. It is the area of an orbit subtended by the angle we created (regardless of its size). And check this out! Any planet moving in any kind of stable orbit (regardless of eccentricity) sweeps out equal subtended areas for the same amount of time through any part of its orbit. Let's have a look at this just a bit more closely..
In a circular orbit, the area swept out in a given subtended angle will be the same because the identical "slices of the pie" (refering to the circular shape of the orbit) will have the same area for the same angle. Also, in a circular orbit, a planet has a fairly constant speed along its path of travel. It will sweep out equal areas from anywhere in its orbit in the same amount of time. But in an eliptical orbit, the planet is moving more slowly when it is farther away from the sun. As it is farther away, the lines we draw to set up an angle (and a subtended arc) are longer. Yes, the lines are longer, but the planet is moving more slowly, and it will still sweep out an identical area in the same amount of time as it would closer it. For comets, the orbits are even more eccentric. But the same idea applies. Simple and easy..
How about a nice drawing to illustrate the point? Wikipedia has a good article on Kepler's laws of planetary motion, and the drawing is found there. You'll need a link, and one is posted below. Surf on over and check it out. (MORE)