Angular diameter

After we talk about parallax, now we will discuss about angular diameter.

I. Definition

The angle that the actual diameter of an object makes in the sky; also known as angular size or apparent diameter. The angular diameter of an object as seen from a given position is the "visual diameter" of the object measured as an angle. The visual diameter is the diameter of the perspective projection of the object on a plane through its center that is perpendicular to the viewing direction. Because of foreshortening, it may be quite different from the actual physical diameter for an object that is seen under an angle. For a disk-shaped object at a large distance, the visual and actual diameters are the same.The Moon, with an actual diameter of 3,476 kilometers, has an angular diameter of 29' 21" to 33' 30", depending on its distance from Earth. If both angular diameter and distance are known, linear diameter can be easily calculated.

The Sun and the Moon have angular diameters of about half a degree, as would a 10-centimeter (4-inch) diameter orange at a distance of 11.6 meters (38 feet). People with keen eyesight can distinguish objects that are about an arc minute in diameter, equivalent to distinguishing between two objects the size of a penny at a distance of 70 meters (226 feet). Modern telescopes allow astronomers to routinely distinguish objects one arc second in diameter, and less. The Hubble Space Telescope, for example, can distinguish objects as small as 0.1 arc seconds. For comparison, 1 arc second is the apparent size of a penny seen at a distance of 4 kilometers (2.5 miles).

The angular diameter is proportional to the actual diameter divided by its distance. If any two of these quantities are known, the third can be determined.

For example if an object is observed to have an apparent diameter of 1 arc second and is known to be at a distance of 5,000 light years, it can be determined that the actual diameter is 0.02 light years.

II. Formulas

The angular diameter of an object can be calculated using the formula:

in which δ is the angular diameter, and d and D are the visual diameter of and the distance to the object, expressed in the same units. When D is much larger than d, δ may be approximated by the formula δ = d / D, in which the result is in radians.

For a spherical object whose actual diameter equals d_act, the angular diameter can be found with the formula:

for practical use, the distinction between d and d_act only makes a difference for spherical objects that are relatively close.

III. Estimating Angular Diameter

This illustration shows how you can use your hand to make rough estimates of angular sizes. At arm's length, your little finger is about 1 degree across, your fist is about 10 degrees across, etc. Credit: NASA/CXC/M.Weiss

IV. Use in Astronomy

In astronomy the sizes of objects in the sky are often given in terms of their angular diameter as seen from Earth, rather than their actual sizes.

The angular diameter of Earth's orbit around the Sun, from a distance of one parsec, is 2" (two arcseconds).

The angular diameter of the Sun, from a distance of one light-year, is 0.03", and that of the Earth 0.0003". The angular diameter 0.03" of the Sun given above is approximately the same as that of a person at a distance of the diameter of the Earth.[1]

This table shows the angular sizes of noteworthy celestial bodies as seen from the Earth:

Sun	31.6' – 32.7'
Moon	29.3′ – 34.1'
Venus	10″ – 66″
Jupiter	30″ – 49″
Saturn	15″ – 20″
Mars	4″ – 25″
Mercury	5″ – 13″
Uranus	3″ – 4″
Neptune	2″
Ceres	0.8″
Pluto	0.1″

* Betelgeuse: 0.049″ – 0.060″
* Alpha Centauri A: ca. 0.007″
* Sirius: ca. 0.007″

This meaning the angular diameter of the Sun is ca. 250,000 that of Sirius (it has twice the diameter and the distance is 500,000 times as much; the Sun is 10,000,000,000 times as bright, corresponding to an angular diameter ratio of 100,000, so Sirius is roughly 6 times as bright per unit solid angle).

The angular diameter of the Sun is also ca. 250,000 that of Alpha Centauri A (it has the same diameter and the distance is 250,000 times as much; the Sun is 40,000,000,000 times as bright, corresponding to an angular diameter ratio of 200,000, so Alpha Centauri A is a little brighter per unit solid angle).

The angular diameter of the Sun is about the same as that of the Moon (the diameter is 400 times as large and the distance also; the Sun is 200,000-500,000 times as bright as the full Moon (figures vary), corresponding to an angular diameter ratio of 450-700, so a celestial body with a diameter of 2.5-4" and the same brightness per unit solid angle would have the same brightness as the full Moon).

Even though Pluto is physically larger than Ceres, when viewed from Earth, e.g. through the Hubble Space Telescope, Ceres has a much larger apparent size.

While angular sizes measured in degrees are useful for larger patches of sky (in the constellation of Orion, for example, the three stars of the belt cover about 3 degrees of angular size), we need much finer units when talking about the angular size of galaxies, nebulae or other objects of the night sky.

Degrees, therefore, are subdivided as follows:

* 360 degrees (º) in a full circle
* 60 arc-minutes (′) in one degree
* 60 arc-seconds (′′) in one arc-minute

To put this in perspective, the full moon viewed from earth is about ½ degree, or 30 arc minutes (or 1800 arc-seconds). The moon's motion across the sky can be measured in angular size: approximately 15 degrees every hour, or 15 arc-seconds per second. A one-mile-long line painted on the face of the moon would appear to us to be about one arc-second in length.

Source : Wikipedia and encyclopedia of science.

Cited from : All About Astronomy