Angular Diameter Calculator

How Angular Diameter Is Calculated

Angular diameter is how large an object appears in the sky, measured in degrees, arcminutes, or arcseconds. It depends on two things and only two things: how big the object actually is, and how far away it sits. That is why the Sun and the Moon, wildly different in size, happen to look almost identical from here.

Angular Diameter Formula: δ = 2 × arctan(d / 2D)

For small angles (where d « D), this simplifies to: δ (radians) ≈ d / D

Converting to arcseconds: δ (arcseconds) = (d / D) × 206,265

Where:

• δ = angular diameter
• d = actual physical diameter of the object
• D = distance from observer to object

Worked Example, the Moon:

• Diameter: 3,474 km
• Average distance: 384,400 km
• δ = (3,474 / 384,400) × 206,265 = 0.0090375 × 206,265 = 1,864 arcseconds = 31.07 arcminutes ≈ 0.518°

The Sun works out at ~0.53°, almost the same, and that near-coincidence is the reason total solar eclipses look the way they do. The Sun is about 400 times wider than the Moon and sits about 400 times farther away, so the two discs land on nearly the same angle from here.

Reference Angular Sizes:

• Moon: ~31 arcmin
• Sun: ~32 arcmin
• Jupiter (opposition): 44 to 50 arcsec, depending on whether the opposition falls near Jupiter’s perihelion
• Venus (max): ~64 arcsec
• Alpha Centauri A: ~0.0085 arcsec. Well below what Hubble can resolve, since its diffraction limit at visible wavelengths is around 0.06 arcsec. Stellar diameters at this scale are measured by interferometry, not by imaging
• Human eye resolution limit: ~1 arcminute

Resolving Power: Telescope resolving power (Rayleigh criterion): θ = 1.22 × λ / D where λ is wavelength and D is aperture diameter. In green light that comes to about 138.4 / D(mm) arcseconds, so a 200 mm scope resolves 0.69" and cannot show the disc of any star. The Rayleigh criterion calculator works it out in full, and the telescope angular resolution calculator does the same in the units amateurs actually use.