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The angular diameter of an object refers to the size that it appears to be. Usually defined as an angle, it is usually in proportion to an object’s actual size divided by its distance. The angular distance from one side of something to another is often used to measure the size of objects in the night sky. A circle is 360° around, while the distance across the sky is equal to half a circle, or 180°. Objects in space are typically measured using an arc second, which is equal to 1/3,600th of 1°.
Another measurement often used with angular diameter is a radian, which is 180° divided by Pi, or 3.14 times the diameter of a circle. Angular diameter, therefore, can be determined by taking the answer to that calculation, which is 206,265, and dividing this by the actual distance. An object’s apparent size can be similar to another that may be much larger, but it is often the comparative distances from the Earth which make them look equal.
The Sun and the Moon are usually equal in angular diameter when seen from Earth, even though one is about 400 times wider than the other. Each appears to be about ½° across from ground-based observers. While 1/3,600th of 1° is an arc second, an arc minute is 1/60th of 1°. People with normal visual acuity can see something that is an arc minute in diameter; this is generally similar to viewing a 0.01 US Dollar (USD) coin from 226 feet (about 70 meters) away.
Astronomers often use telescopes that can clarify objects that are 1 arc second across. The most powerful telescopes can be used to view objects that are down to 0.1 arc seconds wide. Such an apparent diameter can be used in a calculation, along with its known distance, to calculate how wide it really is. Similar calculations have often been used to compare the size of the Sun to other stars, and to determine the size of other galaxies, nebula, and other objects which can be seen in space.
Angular diameter can be estimated with the human hand. By holding one’s arm straight out, the pinky finger typically covers 1° of the night sky. Generally, the fist is about 10° across in relation to objects such as stars and galaxies. Astronomers have calculated the angular diameter for the planets in the solar system as well as many other well-known objects in the sky.
Please provide the relationship between angular diameter and apparent magnitudes of celestial objects.