APERTURE

M = Alpha/Theta

where M is the magnification
Alpha is the apparent field
Theta is the true field

Apparent Field: the closest separation eye can see is 4', more practically
8-25', 1-2' for good eyes. The Zeta Ursae Majoris double (Mizar/Alcor) is 11.75'; Epsilon Lyrae is 3'.

True Field (in o) = 0.25 * time * cos of the declination
(in ') = 15 * time * cos of the declination
where time is the time to cross the ocular field in minutes
A star therefore moves westward at the following rates:
15o /h (1.25o/5 min) at 0o declination
13o /h (1.08o/5 min) at 30o declination
7.5o/h (0.63o/5 min) at 60o declination.

MAGNIFICATION: BY FOCAL LENGTHS

M = F/f

where M is the magnification
F is the focal length of the objective
f is the focal length of the ocular

At prime focus (ground glass), magnification is 1x for each 25 mm of F

MAGNIFICATION: BY DIAMETER AND EXIT PUPIL

M = D/d

where M is the magnification
D is the diameter of the objective
d is the exit pupil (5-6 mm is best; 7 mm will not produce a sharp outer image)

The scotopic (dark-adapted) aperture of the human pupil is typically 6
(theoretically 7, 5 if over age 50) mm. Since the human pupil has a focal
length of 17 mm, it is f/2.4 and yields 0.17 per mm of aperture. 2.5 mm is
the photopic (light-adapted) diameter of the eye.

EXIT PUPIL

d = f/f-number

(by substituting F/f for M)
where d is the exit pupil
f is the focal length of the ocular
f-number is the f-number (f/) of the objective

By substituting d=7 (the scotopic aperture of the human pupil) and multiplying it by the f-number, the longest useful focal length of the ocular is given.

LOW-POWER LAW FOR LIMITING MAGNIFICATION

M = D/6 = 17*D

(by substituting 6 mm for d and taking the reciprocal)
where M is the minimum magnification without wasting light for a dark-adapted eye (17x per mm of aperture)
D is the diameter of the objective in mm

HIGH-POWER LAW FOR LIMITING MAGNIFICATION

M = D/0.63 = 158*D

(by substituting 0.63 mm, the minimum diameter to which the average human pupil can contract, for d and taking the reciprocal)

where M is the maximum theoretical magnification (158x per mm of aperture);
the maximum practical magnification is +50%).

LIMITING VISUAL MAGNITUDE (LIGHT-GATHERING POWER)

m = 6.5-5 log Delta+5 log D = 2.7+5 log D (assuming transparent dark-sky conditions and magnification >= 1D in mm)

where m is the approximate limiting visual magnitude
Delta is the pupillary diameter in mm (accepted value 7.5)
D is the diameter of the objective in mm

RELATIVE LIGHT EFFICIENCY (TWILIGHT FACTOR)

Relative Brightness Value = d^2 = (D/M)^2

where the larger the relative brightness value, the better the instrument (e.g., binoculars) is for viewing in twilight or for astronomical use after dusk (low light conditions only)
d is the diameter of the exit pupil
D is the diameter of the objective
M is the magnification

ANGULAR RADIUS OF AIRY (DIFFRACTION) DISC

r = (1.12*Lambda*206265)/D    (for any wavelength, Lambda)

r = 127.1/D    (for yellow light in which Lambda = 0.00055mm)

where r is the angular radius (one-half the angular diameter) of the Airy disc (irreducible minimum size of a star disc) in seconds.
Lambda is the wavelength of the light in mm
206265 is the number of seconds in a radian
D is the diameter of the objective in mm

The Airy disc in visual appearance is brighter at the center, dimmer at the edges.

LINEAR RADIUS OF AIRY (DIFFRACTION) DISC

r = 0.043*Lambda*f

where r is the linear radius (one-half the linear diameter) of the Airy disc in mm
Lambda is the wavelength of light in mm (yellow light 0.00055)
f is the f-number (f/) of the objective

DAWES LIMIT (SMALLEST RESOLVABLE ANGLE, RESOLVING POWER)

Theta = 115.8/D

where Theta is the smallest resolvable angle in seconds
D is the diameter of the objective in mm

Atmospheric conditions seldom permit Theta < 0.5". The Dawes Limit is one-half the angular diameter of the Airy (diffraction) disc, so that the edge of one disc does not extend beyond the center of the other). The working value is two times the Dawes Limit (diameter of the Airy disc), so that the edges of the two stars are just touching.

MAGNIFICATION NEEDED TO SPLIT A DOUBLE STAR

M = 480/d

where M is the magnification required
480 is number of seconds of arc for an apparent field of 8 minutes of arc
d is the angular separation of the double star

About the closest star separation that the eye can distinguish is 4 minutes of arc (240 seconds of arc). Twice this distance, or an 8-minute (480-second) apparent field angle, is a more practical value for comfortable viewing. In cases where the secondary is more than five magnitudes fainter than the primary, you will need a wider separation: 20 or 25 minutes of arc, nearly the width of the moon seen with the naked eye.

RESOLUTION OF LUNAR FEATURES

Resolution = (2*Dawes Limit*3476)/1800)
Dawes Limit * 38.8

where Resolution is the smallest resolvable lunar feature in km
2*Dawes Limit is the Airy disc (a more practical working value is twice this)
1800 is the angular size of the moon in seconds
3476 is the diameter of the moon in km

LIGHT GRASP

Light Grasp = (D/d)^2*Pi = 7*D^2

where Light Grasp is times that received by the retina
D is the diameter of the objective in mm
d is the diameter of the eye's pupillary aperture in mm (accepted value 7.5)
pi is the transmission factor (approximately equal to 62.5% for the average telescope, up to approximately 180 mm)

To compare the relative light grasp of two main lenses used at the same magnification, compare the squares of their diameters.