My Math Forum Effective and actual angular diameter of the Sun

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March 7th, 2012, 09:50 AM   #1
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Effective and actual angular diameter of the Sun

Hi,

We are using a radio interferometer to measure the angular diameter of the Sun.

An in depth description of how this is done is not relevant to solve the immediate problem I have. But briefly, we rotate the interferometer in azimuth so that the direction it's looking at traverses across the center of the Sun. We take readings of an interference pattern from two mirrors off to the sides during this traverse. Then we use this pattern to find out the angular diameter of the Sun that we saw.

My problem is that the interferometer rotates in azimuth around an axis that is perpendicular to the surface of the Earth, but the Sun is up at some latitude. So the interferometer doesn't trace a straight line diameter across the Sun. It traces an arc. The length of this arc is going to be the effective angular diameter that we finally obtain.

This is described in the attached diagram. The real angular diameter of the Sun is the length of the straight line. But the value for the angular diameter that we actually obtain is the length of the curved line. That curved line is the path the interferometer actually traverses.

I want to find a relationship between the length of the curved line, the length of the straight line, and the latitude of the Sun, so I can find the real angular diameter of the Sun from the value that I obtained.

Could someone please suggest a way I can do this? I think it has to do with spherical trigonometry, but I don't know where to start. I don't want to go through a whole textbook of Spherical Trigonometry just to solve this. What I need is a relevant equation.

Thanks.
[attachment=0:1lz52qv1]sun.jpg[/attachment:1lz52qv1]
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 sun.jpg (19.1 KB, 95 views)

 March 7th, 2012, 01:56 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,931 Thanks: 2205 Why does the interferometer's axis of rotation have to be perpendicular to the surface of the Earth?
 March 7th, 2012, 02:15 PM #3 Newbie   Joined: Jun 2011 Posts: 9 Thanks: 0 Re: Effective and actual angular diameter of the Sun ( I should have explained earlier that this is an experiment in an Astronomy lab we are doing ) Well, the interferometer is located on a level platform. Think of it as a telescope. There are two motors that move it in relation to the platform. One rotates it so it can it can increase and decrease its azimuth (longitude). The other rotates it so it can increase and decrease its elevation ( latitude ). We don't have a sophisticated control mechanism to control these motors. So what we do is first point it directly towards the Sun by adjusting azimuth and elevation, and then use only the azimuth motor to first go back 5 degrees, and then keep gong through 5 degrees past the center of the Sun, all the while keeping the elevation at the same level. So we are only using the azimuth motor to rotate it when we take data. So the rotation is around the axis of a line that is perpendicular to the surface of the Earth. For an example, it is like the rotation of the turret of an armored tank, which is keeping its gun at a constant elevation, and has its lower body level with the ground. So when we traverse the interferometer through the center of the Sun using only the azimuth motor, it traces a curved line as shown in the diagram, not a straight line.

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