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.
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
dact, the angular diameter can be found with the formula:
For practical use, the distinction between d and dact only makes a difference for spherical objects that are relatively close.
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.
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.
This stats 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.