What is angular diameter of Earth as seen from moon whose distance from Earth is 60 times the radius of Earth?

Astronomers use angular measure to describe the apparent size of an object in the night sky. An angle is the opening between two lines that meet at a point and angular measure describes the size of an angle in degrees, designated by the symbol °. A full circle is divided into 360° and a right angle measures 90°. One degree can be divided into 60 arcminutes (abbreviated 60 arcmin or 60'). An arcminute can also be divided into 60 arcseconds (abbreviated 60 arcsec or 60"). 

The angle covered by the diameter of the full moon is about 31 arcmin or 1/2°, so astronomers would say the Moon's angular diameter is 31 arcmin, or the Moon subtends an angle of 31 arcmin. 

If you extend your hand to arm's length, you can use your fingers to estimate angular distances and sizes in the sky. Your index finger is about 1° and the distance across your palm is about 10°.

The Small-Angle Formula

The angular sizes of objects show how much of the sky an object appears to cover. Angular size does not, however, say anything about the actual size of an object. If you extend your arm while looking at the full moon, you can completely cover the moon with your thumb, but of course, the moon is much larger than your thumb, it only appears smaller because of its distance. How large an object appears depends not only on its size, but also on its distance. The apparent size, the actual size of an object, and the distance to the object can be related by the small angle formula:

D = θ d / 206,265

D = linear size of an objectθ = angular size of the object, in arcsec

d = distance to the object

Example:

A certain telescope on Earth can see details as small as 2 arcsec. What is the greatest distance you could see details as small the the height of a typical person (1.6 m)?

d = 206,265 D / θ = 206,265 × 1.6 m / 2 = 165,012 m = 165.012 km

This is much less than the distance to the Moon (approximately 384,000 km) so this telescope would not be able to see an astronaut walking on the moon. (In fact, no Earth based telescope could.)

Practice Questions

1. The average distance to the Moon is approximately 384,000 km. The Moon subtends and angle of 31 arcminutes, or about 1/2°. Use this information and the small-angle formula to find the diameter of the moon in kilometers.
2. At what distance would you have to hold a quarter (which has a diameter of about 2.5 cm) for it to subtend and angle of 1°?

Answers:

1. The diameter of the Moon is about 3,463 km
2. You would have to hold it at a distance of 1.43 meters.

The average angular diameter of the Moon, as seen from the Earth, is about 31 arcminutes.

The angular diameter depends on the distance between the two objects and the diameter of the object being viewed. Specifically, for small angles, it is the diameter divided by the distance. When the distance is the same, the angular size is proportional to the diameter.

The distance remains the same when viewing the Earth from the Moon, but the Earth is larger. According to NASA, the diameter of the Moon is 3,476 km, and the diameter of the Earth is 12,756 km.

So, because it's proportional, the angular diameter can be calculated as follows:

$a_{Earth} = a_{Moon} \times {d_{Earth}\over d_{Moon}}$

$ = 31 arcminutes \times {12,756 km \over 3,476 km}$

$ \approx 114 arcminutes$, or just under 2 degrees.

This is approximate, because not only is this valid only for small degrees, where the tangent of an angle can be approximated by the angle itself (in radians), the Earth-Moon distance varies because the Moon's orbit around the Earth is an ellipse.

How large a sphere or circle appears

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The angular diameter, angular size, apparent diameter, or apparent size is an angular distance describing how large a sphere or circle appears from a given point of view. In the vision sciences, it is called the visual angle, and in optics, it is the angular aperture (of a lens). The angular diameter can alternatively be thought of as the angular displacement through which an eye or camera must rotate to look from one side of an apparent circle to the opposite side. Humans can resolve with their naked eyes diameters of up to about 1 arcminute (approximately 0.017° or 0.0003 radians).[1] This corresponds to 0.3 m at a 1 km distance, or to perceiving Venus as a disk under optimal conditions.

Formula

The angular diameter of a circle whose plane is perpendicular to the displacement vector between the point of view and the center of said circle can be calculated using the formula[2]

δ = 2 arctan ⁡ ( d 2 D ) , {\displaystyle \delta =2\arctan \left({\frac {d}{2D}}\right),}

in which δ {\displaystyle \delta }

d {\displaystyle d}

D {\displaystyle D}

D ≫ d {\displaystyle D\gg d}

δ ≈ d / D {\displaystyle \delta \approx d/D}

For a spherical object whose actual diameter equals d a c t , {\displaystyle d_{\mathrm {act} },}

δ = 2 arcsin ⁡ ( d a c t 2 D ) {\displaystyle \delta =2\arcsin \left({\frac {d_{\mathrm {act} }}{2D}}\right)}

The difference is due to the fact that the apparent edges of a sphere are its tangent points, which are closer to the observer than the center of the sphere. The difference is significant only for spherical objects of large angular diameter, since the following small-angle approximations hold for small values of x {\displaystyle x}

arcsin ⁡ x ≈ arctan ⁡ x ≈ x . {\displaystyle \arcsin x\approx \arctan x\approx x.}

Estimating angular diameter using the hand

Estimates of angular diameter may be obtained by holding the hand at right angles to a fully extended arm, as shown in the figure.[4][5][6]

Use in astronomy

In astronomy, the sizes of celestial objects 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 (1°) and a radian is 180/π degrees. So one radian equals 3,600 × 180/ π {\displaystyle \pi }

δ = 206 , 265   ( d / D )   a r c s e c o n d s {\displaystyle \delta =206,265~(d/D)~\mathrm {arcseconds} }

These objects have an angular diameter of 1″:

• an object of diameter 1 cm at a distance of 2.06 km
• an object of diameter 725.27 km at a distance of 1 astronomical unit (AU)
• an object of diameter 45 866 916 km at 1 light-year
• an object of diameter 1 AU (149 597 871 km) at a distance of 1 parsec (pc)

Thus, the angular diameter of Earth's orbit around the Sun as viewed from a distance of 1 pc is 2″, as 1 AU is the mean radius of Earth's orbit.

The angular diameter of the Sun, from a distance of one light-year, is 0.03″, and that of Earth 0.0003″. The angular diameter 0.03″ of the Sun given above is approximately the same as that of a human body at a distance of the diameter of Earth.

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

The table shows that the angular diameter of Sun, when seen from Earth is approximately 32′ (1920″ or 0.53°), 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 1010 times as bright, corresponding to an angular diameter ratio of 105, 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 4×1010 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.

Angular sizes measured in degrees are useful for larger patches of sky. (For example, the three stars of the Belt cover about 4.5° of angular size.) However, much finer units are needed to measure the angular sizes of galaxies, nebulae, or other objects of the night sky.

Degrees, therefore, are subdivided as follows:

• 360 degrees (°) 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 as viewed from Earth is about 1⁄2°, or 30′ (or 1800″). The Moon's motion across the sky can be measured in angular size: approximately 15° every hour, or 15″ per second. A one-mile-long line painted on the face of the Moon would appear from Earth to be about 1″ in length.

In astronomy, it is typically difficult to directly measure the distance to an object, yet 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 ≡ 2 D tan ⁡ ( δ 2 ) . {\displaystyle d\equiv 2D\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

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

Defect of illumination

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″ of arc across and is 75% illuminated, the defect of illumination is 10″.

See also

• Angular diameter distance
• Angular resolution
• Solid angle
• Visual acuity
• Visual angle
• Visual Angle Illusion
• List of stars with resolved images
• Apparent magnitude

References

1. ^ Yanoff, Myron; Duker, Jay S. (2009). Ophthalmology 3rd Edition. MOSBY Elsevier. p. 54. ISBN 978-0444511416.
2. ^ This can be derived using the formula for the length of a cord found at "Circular Segment". Archived from the original on 2014-12-21. Retrieved 2015-01-23.
3. ^ "A Taylor series for the functionarctan" (PDF). Archived from the original (PDF) on 2015-02-18. Retrieved 2015-01-23.
4. ^ "Coordinate Systems". Archived from the original on 2015-01-21. Retrieved 2015-01-21.
5. ^ "Photographing Satellites". 8 June 2013. Archived from the original on 21 January 2015.
6. ^ Wikiversity: Physics and Astronomy Labs/Angular size
7. ^ Michael A. Seeds; Dana E. Backman (2010). Stars and Galaxies (7 ed.). Brooks Cole. p. 39. ISBN 978-0-538-73317-5.
8. ^ "Problem 346: The International Space Station and a Sunspot: Exploring angular scales" (PDF). Space Math @ NASA !. 2018-08-19. Retrieved 2022-05-20.
9. ^ a b Wong, Yan (2016-01-24). "How small can the naked eye see?". BBC Science Focus Magazine. Retrieved 2022-05-23.
10. ^ a b "Sharp eyes: how well can we really see?". Science in School – scienceinschool.org. 2016-09-07. Retrieved 2022-05-23.
11. ^ Graney, Christopher M. (Dec 10, 2006). "The Accuracy of Galileo's Observations and the Early Search for Stellar Parallax". arXiv.org. doi:10.1007/3-540-50906-2_2. Retrieved May 21, 2022.
12. ^ "Galileo's telescope - How it works". Esposizioni on-line - Istituto e Museo di Storia della Scienza (in Italian). Retrieved May 21, 2022.
13. ^ 800 000 times smaller angular diameter than that of Alnitak as seen from Earth. Alnitak is a blue star so it gives off a lot of light for its size. If it were 800 000 times further away then it would be magnitude 31.5, at the limit of what Hubble can see.

External links

• Small-Angle Formula
• Visual Aid to the Apparent Size of the Planets

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