Space Telescopes
and
Minimum Observable Objects

Astronomy Angle Notes:

1 degree = 60 arc-minutes or 3,600 arc-seconds (written as 3600”)
1 arc-minute = 60 arc-seconds
1 radian = 57.295 degrees

Astronomy Wavelength Notes:

Visible Light: 0.39 to 0.75 micrometres
Near Infrared IR: 0.78 to 3 micrometres
Medium Wavelength IR: 3 to 50 micrometres
Long Wavelength IR: 50 to 1000 micrometres
1 inch: 25,400 micrometers
1 cm: 10,000 micrometres

A Microsoft Excel file with the most important equations already laid out is available HERE.

Converting from Arc-Seconds to Radians

Degrees = ArcSeconds / 3600
Radians = Degrees • ( π / 180)

EXAMPLE: Our Space telescope can resolve down to 0.05 arc-seconds. What is this in radians?

First, we convert arc-seconds to Degrees.

0.05 / 3600 = 1.389E-05 degrees

Then to Radians

1.389E-05 • ( π / 180) = 2.42E-07 radians

Calculating Angular Diameter for distant objects when given Actual Diameters and Distances

δ = 2 • arctan (½ • d / D)

Where:

δ: Angular Diameter of object in radians
d: Diameter of object
D: Distance between observer and object.
NOTE: d/D must be expressed in the same unit.

EXAMPLE: Earth's moon has a diameter of about 3,474 km and is about 384,000~ km from Earth. What is its angular size?

2 • arctan (½ • 3,474 / 384,000) = 0.009 radians

Calculating Distance when given Angular and Actual Diameters for distant objects

D = d / [ 2 • TAN(δ/2) ]

Where:

δ: Angular Diameter of object in radians
d: Diameter of object
D: Distance between observer and object.
NOTE: d/D must be expressed in the same unit.

EXAMPLE: The Hubble Space Telescope has a visual resolution of 0.1 arc-seconds (0.00000048 radians). How far away can it see an object 100 meters in diameter?

100 / [ 2 * TAN(0.00000048/2) ] = 208,333,333.33 meters or 208,333.33 kilometers

Dawes Limit for Resolution of an Visual Light Optical Telescope

This was based off of empirical studies done by W.R. Dawes. It is for light with a wavelength of about 562 nm.

Resolution = 4.56 / Diameter

Where:

Resolution: Resolution of the telescope in arc seconds
Diameter: Diameter of the Telescope optic in inches.

EXAMPLE: What is the resolution in arc seconds of a 8-foot (96-inch) diameter optical telescope?

4.56 / 96 = 0.0475 arc-seconds

Rayleigh Criterion

sin θ = 1.220 * (λ / D)

Where:

θ: Angular resolution of imaging system in radians
λ: Wavelength of observed radiation
D: Diameter of the lens aperture.
NOTE: Both the lens aperture diameter and the wavelength must use the same measurement units.

Quick Angular Resolution Approximation via Wavelength for a Single Telescope

θ = (λ / D)

Where:

θ: Angular resolution of imaging system in radians
λ: Wavelength of observed radiation
D: Diameter of the lens aperture.
NOTE: Both the lens aperture diameter and the wavelength must use the same measurement units.

Quick Angular Resolution Approximation via Wavelength for a Telescope Array (Interferometer)

θ = (λ / B)

Where:

θ: Angular resolution of imaging system in radians
λ: Wavelength of observed radiation
D: Length of maximum physical separation of telescopes in the array (baseline)
NOTE: Both the baseline and wavelength must use the same measurement units.