Angular diameter

From Wikipedia, the free encyclopedia
  (Redirected from Apparent diameter)
Jump to: navigation, search

The angular diameter or apparent size is an angular measurement describing how large a sphere or circle appears from a given point of view. In the vision sciences it is called the visual angle. The angular diameter can alternately be described as the angle an eye or camera must rotate to look from one side of an apparent circle to the opposite side.

Formula[edit]

Diagram for the formula of the angular diameter

The angular diameter of a circle whose plane is normal the displacement vector between the point of view and the centre of said circle can be calculated using the formula:

\delta = 2 \arctan \left( \tfrac{d}{2D}\, \right),

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

For a spherical object whose actual diameter equals d_\mbox{act}, and where D is the distance to the centre of the sphere, the angular diameter can be found by the formula:

\delta = 2 \arcsin \left( \tfrac{d_\mbox{act}}{2D}\, \right)

The reason for the difference is that when you look at a sphere, the edges are the tangent points, which are closer to the observer than the centre of the sphere. arctan refers to the ratio opposite/adjacent, whereas arcsin refers to the ratio opposite/hypotenuse. For practical use, the distinction is only significant for spherical objects that are relatively close.

For very distant or stellar objects, the Small-angle approximation can also be used:

\begin{align}
  \sin \theta &\approx \theta \approx \arcsin \theta \\
  \tan \theta &\approx \theta \approx \arctan \theta
\end{align}

which simplifies the above equations to:

\delta \approx d / D (for small \delta)

Use in astronomy[edit]

Angular diameter: the angle subtended by an object

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. Since these angular diameters are typically small, it is common to present them in arcseconds. An arcsecond is 1/3600th of one degree, and a radian is 180/\pi degrees, so one radian equals 3600*180/\pi arcseconds, which is about 206265 arcseconds. Therefore, the angular diameter of an object with visual diameter d at a distance D, expressed in arcseconds, is given by:[1]

\delta = 206265 d / D arcseconds.

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:

Celestial body Angular diameter Relative size (10 pixels per arcsecond)
Sun 31.6′ – 32.7′ 28.7–29.7 times the maximum value for Venus (orange bar below) / 1896–1962″
Moon 29.3′ – 34.1′ 26.6–31.0 times the maximum value for Venus (orange bar below) / 1758–2046″
Venus 9.565″ – 66.012″

Jupiter 29.800″ – 50.115″

Saturn 14.991″ – 20.790″

Mars 3.492″ – 25.113″

Mercury 4.535″ – 13.019″

Uranus 3.340″ – 4.084″

Neptune 2.179″ – 2.373″

Ceres 0.330″ – 0.840″

Vesta 0.20" – 0.64"

Pluto 0.063″ – 0.115″

R Doradus 0.052″ – 0.062″

Betelgeuse 0.049″ – 0.060″

Eris 0.034" – 0.089″

Alphard 0.00909″
Alpha Centauri A 0.007″
Canopus 0.006″
Sirius 0.005936″
Altair 0.003″
Deneb 0.002″
Proxima Centauri 0.001″
Comparison of angular diameter of the Sun, Moon and planets. To get a true representation of the sizes, view the image at a distance of 103 times the width of the "Moon: max." circle. For example, if this circle is 10 cm wide on your monitor, view it from 10.3 m away.

The table shows that the angular diameter of Sun, when seen from Earth is approximately 32 arcminutes (1920 arcseconds or 0.53 degrees), as illustrated above.

Thus the angular diameter of the Sun is about 250,000 times that of Sirius (Sirius has twice the diameter and its distance is 500,000 times as much; the Sun is 10^10 times as bright, corresponding to an angular diameter ratio of 10^5, so Sirius is roughly 6 times as bright per unit solid angle).

The angular diameter of the Sun is also about 250,000 times that of Alpha Centauri A (it has about the same diameter and the distance is 250,000 times as much; the Sun is 4×10^10 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 Sun's diameter is 400 times as large and its distance also; the Sun is 200,000 to 500,000 times as bright as the full Moon (figures vary), corresponding to an angular diameter ratio of 450 to 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 4.5 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:

To put this in perspective, the full moon viewed from earth is about 12 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.

In astronomy, it is typically difficult to directly measure the distance to an object. But the object may have a known physical size (perhaps it is similar to a closer object with known distance) and a measurable angular diameter. In that case, the angular diameter formula can be inverted to yield the Angular diameter distance to distant objects as

d \equiv 2 D \tan \left( \frac{\delta}{2} \right).

In non-Euclidean space, such as our expanding universe, the angular diameter distance is only one of several definitions of distance, so that there can be different "distances" to the same object. See Distance measures (cosmology).

Non-circular objects[edit]

Many deep sky objects such as galaxies and nebulas appear as non-circular, and are thus typically given two measures of diameter: Major Diameter and Minor Diameter. For example, the Small Magellanic Cloud has a visual apparent diameter of 5° 20′ × 3° 5′.

Defect of illumination[edit]

Defect of illumination is the maximum angular width of the unilluminated part of a celestial body seen by a given observer. For example, if an object is 40 seconds of arc across and is 75 percent illuminated, the defect of illumination is 10 seconds of arc.

See also[edit]

References[edit]

  1. ^ Michael A. Seeds; Dana E. Backman (2010). Stars and Galaxies (7 ed.). Brooks Cole. p. 39. ISBN 978-0-538-73317-5. 

External links[edit]