# Effective angular diameter of the sun for this experiment

• omoplata
In summary, the conversation is discussing the use of a radio interferometer to measure the angular diameter of the Sun. The interferometer rotates in azimuth, causing it to trace an arc instead of a straight line across the Sun due to the Sun's latitude. The length of this arc is the effective angular diameter, which can be calculated using spherical trigonometry. The person is looking for an equation to find the actual angular diameter from the effective one.
omoplata
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.

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• sun_line.jpg
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Sorry, that diagram is wrong. Attached is a better diagram.

#### Attachments

• sun_line.jpg
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OK, I found an expression for the effective angular diameter of the first diagram (but not for the second one yet) using the spherical law of cosines.

If the latitude of the sun is $\theta$, and the actual angular diameter of the sun ( straight line ) is $d$, the effective angular diameter ( curved line ) as shown in the first diagram ( when the direction the interferometer is pointing towards doesn't go through the center of the sun ) is,
$$d \phi = \cos (\theta) \arccos \left( \frac{\cos( d ) - \sin^2 ( \theta )}{\cos^2 ( \theta )} \right)$$

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## 1. What is the effective angular diameter of the sun?

The effective angular diameter of the sun is a measurement that represents the apparent size of the sun as seen from Earth. It takes into account the distance between the Earth and the sun, as well as the sun's actual physical diameter.

## 2. Why is the effective angular diameter of the sun important for this experiment?

The effective angular diameter of the sun is important for this experiment because it allows us to accurately measure the size of the sun and its position in the sky. This information is crucial for understanding the effects of the sun on our planet and conducting experiments related to solar energy and other solar phenomena.

## 3. How is the effective angular diameter of the sun calculated?

The effective angular diameter of the sun is calculated by dividing the physical diameter of the sun by its distance from Earth. This results in a measurement in degrees, minutes, and seconds that represents the perceived size of the sun from our perspective on Earth.

## 4. Does the effective angular diameter of the sun change?

Yes, the effective angular diameter of the sun can vary slightly due to the elliptical orbit of the Earth around the sun. This means that the distance between the two bodies can change, resulting in a slightly different measurement of the sun's apparent size.

## 5. How is the effective angular diameter of the sun used in other scientific fields?

The effective angular diameter of the sun is used in a variety of scientific fields, including astronomy, physics, and meteorology. It is crucial for understanding and predicting solar eclipses, studying the sun's impact on climate and weather patterns, and measuring the size and distance of other celestial bodies.

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