Assorted Formulae for Telescopes
APERTURE
where a is the aperture of the objective
F is the focal length of the objective
f is the f-number (f/) of the objective
MAGNIFICATION: BY FIELDS
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
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
where M is the magnification
D is the diameter of the objective
d is the exit pupil
(5-6 mm is best; 7 mm does 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
(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
(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
(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)
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
ANGULAR RADIUS OF AIRY (DIFFRACTION) DISC
(the second formula is based on Lambda = 0.00055 for yellow light)
where r is the angular radius (one-half the angular diameter) of the Airy
disc (irreducible minimum size of a star disc) in arcsecs
Lambda is the wavelength of the light in mm
206265 is the number of arcsecs 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
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)
where Theta is the smallest resolvable angle in "
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
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 comes 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
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 arcsecs
3476 is the diameter of the moon in km
LIGHT GRASP
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.
Formulae for Astrophotography
F-NUMBER: PRIME FOCUS (ERECT IMAGE)
where f/ is the f-number of the system (objective)
F is the focal length of the objective
D is the diameter of the objective
F-NUMBER: AFOCAL, EYEPIECE-CAMERA LENS (REVERSED IMAGE)
where f/ is the f-number of the system
F' is the effective focal length of the system
Fe is the focal length of the ocular (divided by any Barlow
magnification)
D is the diameter of the objective
M is the magnification
Fc is the focal length of the camera
F is the focal length of the objective
Fc/Fe is the projection magnification
M/D is the power per mm
The diameter of the first image equals the film diagonal (44 mm for 35 mm
film) divided by the magnification.
F-NUMBER: EYEPIECE PROJECTION, POSITIVE LENS (REVERSED IMAGE)
where f/ is the f-number of the system
F' is the effective focal length of the system
D is the diameter of the objective
F is the focal length of the objective (times any Barlow
magnification)
B is the secondary image ("throw"), the distance of the ocular
center from the focal plane of the film, equal to ((M+1)*Fe)/A
A is the primary image, the distance of the ocular center from the
focal point of the telescope objective
M is the projection magnification, equal to (B/Fe)-1
Fe is the focal length of the ocular
F-NUMBER: NEGATIVE LENS PROJECTION (ERECT IMAGE)
where f/ is the f-number of the system
F' is the effective focal length of the system
D is the diameter of the objective
B is the distance of the Barlow center from the focal plane of the
film
A is the distance of the Barlow center from the focal point of the
telescope objective
B/A is the projection magnification (Barlow magnification)
EXPOSURE COMPARISON FOR EXTENDED OBJECTS
(the ratio of intensities of illumination is squared according to the
inverse square law)
where Exposure Compensation is the exposure compensation to be made to the
example system
f/S is the f-number (f/) of the subject system
f/E is the f-number (f/) of the example system
EXPOSURE COMPARISON FOR POINT SOURCES
where Exposure Compensation is the exposure compensation to be made to the
example system
De is the objective diameter of the example system
Ds is the objective diameter of the subject system
LIGHT-RECORDING POWER OF A SYSTEM
(the light-recording power is directly proportional to the square of the
radius of the objective and inversely propertional to the square of the
f-number)
where Power is the light-recording power of the system
r is the radius of the objective
f is the f-number (f/) of the system
Example: a 200-mm f/8 system compared with a 100-mm f/5 system
(100^2)/8^2 compared with (50^2)/5^2
156.25 compared with 100, or 1.56 times more light-recording power
PRINT'S EFFECTIVE FOCAL LENGTH
where Print EFL is the print's effective focal length
Camera F. L. is the camera's focal length
Print Enlargement is the amount of enlargement of the print (3x is
the standard for 35-mm film)
GUIDESCOPE MAGNIFICATION
where Guidescope M is the magnification needed for guiding astrophotographs
f is the photographic focal length in mm
Experience indicates that the minimum guiding magnification needed is about
f divided by 12.5, precisely what a 12.5 mm guiding ocular used in an off-
axis guider for prime-focus photography yields. (Since visual magnification
is the ratio of the objective to ocular focal length, the combination of
prime-focus camer and off-axis guider with a 12.5-mm ocular gives a guiding
magnification of f/12.5. f/7.5 (as with a typical focal reducer that
reduces the effective focal length by a factor of 0.6) is a significant
improvement. f/5 or higher magnification is for top-quality guiding.
Guidescope M = Guidescope EFL / Print EFL
where Guidescope M is the guidescope's magnification (should be >= 1,
preferably 5-8)
Guidescope EFL is the guidescope's effective focal length, the
guidescope's focal length times any Barlow
magnification (should be >= to the focal length
of the primary and the guidescope's magnification,
0.2x per mm of focal length of the objective, 0.1x
per mm of the camera lens
Print EFL is the print's effective focal length
GUIDING TOLERANCE
where Guiding Tolerance is in mm
0.076 is one " at a 254-mm reading distance from the print
(a crosshair web is usually 0.05 mm)
MAXIMUM ALLOWABLE TRACKING (SLOP) ERROR
where S is the error ("slop") in "
F is the focal length in mm
E is the amount of enlargement of the print (3x is the standard for
35-mm film)
The slop is derived from the formula Theta = K*(h/F), with K = 206256 (the
number of seconds in a radian) and h = 0.04 mm of image-drift tolerance (an
empirical value from astrophotographs).
CONVERSION OF PLATE SCALE TO EFFECTIVE FOCAL LENGTH
where EFL is the effective focal length in mm
57.3 is the number of degrees in a radian
206256 is the number of " in a radian
RESOLVING POWER OF A PHOTOGRAPHIC SYSTEM
where Resolving Power is the resolving power of a photographic system with
Kodak 103a or color film
F is the focal length of the system in mm
MAXIMUM RESOLUTION FOR A PERFECT LENS
where Maximum Resolution is the maximum resolution for a perfect lens
f is the f-number (f/) of the lens
Most films, even fast ones, resolve only 60 lines/mm; the human eye resolves
6 lines/mm (less gives a "wooly" appearance). 80 lines/mm for a 50-mm lens
is rated excellent (equal to 1 minute of arc); a 200-mm lens is rated
excellent with 40 lines/mm. 2415 films yields 320 line pairs (160 lines)/mm
(equal to 1 second of arc); Tri-X yields 80 lines/mm.
MINIMUM RESOLUTION NECESSARY FOR FILM
where Minimum Resolution is the minimum resolution necessary for film
Maximum Resolution is the maximum resolution for a perfect lens
Print Enlargement is the amount of enlargement of the print (3x is the
standard for 35-mm film)
APPARENT ANGULAR SIZE OF AN OBJECT
where Apparent Angular Size is the apparent angular size of the object in
degrees
Linear Width is the linear width of the object in m
Distance is the distance of the object in m
A degree is the apparent size of an object whose distance is 57.3 times its
diameter.
SIZE OF IMAGE (CELESTIAL)
where h is the linear height in mm of the image at prime focus of an
objective or a telephoto lens
Theta is the object's angular height (angle of view) in units
corresponding to K
F is the effective focal length (focal length times Barlow
magnification) in mm
K is a constant with a value of 57.3 for Theta in degrees, 3438 in
minutes of arc, 206265 for seconds of arc (the number of the
respective units in a radian)
The first formula yields image size of the sun and moon as approximately 1%
of the effective focal length (Theta/K = 0.5/57.3 = 0.009).
The second formula can be used to find the angle of view (Theta) for a given
film frame size (h) and lens focal length (F). Example: the 24 mm height,
36 mm width, and 43 mm diagonal of 35-mm film yields an angle of view of
27o, 41o, and 49o for a 50-mm lens.
The third formula can be used to find the effective focal length (F)
required for a given film frame size (h) and angle of view (Theta).
SIZE OF IMAGE (TERRESTRIAL)
where h is the linear height in mm of the image at prime focus of an
objective or telephoto lens
Linear Width is the linear width of the object in m
Distance is the distance of the object in im
F is the effective focal length (focal length times Barlow
magnification) in mm
LENGTH OF A STAR TRAIL ON FILM
where Length is the length in mm of the star trail on film
F is the focal length of the lens in mm
T is the exposure time in minutes
0.0044 derives from (2*Pi)/N for minutes (N = 1440 minutes per day)
EXPOSURE TIME FOR STAR TRAIL ON 35-MM FILM
where T is the exposure time in minutes for a length of 24 mm (the smallest
dimension of 35-mm film)
F is the focal length of the lens in mm
MAXIMUM EXPOSURE TIME WITHOUT STAR TRAIL
where T is the maximum exposure time in seconds without a star trail
1397 derives from 1' at reading distance (254 mm), the smallest
angular quantity that can be perceived by the human eye without
optical aid ("limiting resolution") and is equal to < 0.1 mm. This
quantity also applies to the moon. 2-3x yields only a slight
elongation. Use 20x for a clock drive.
F is the focal length of the lens in mm
The earth rotates 5' in 20 s, which yields a barely detectable star trail
with an unguided 50-mm lens. 2-3' (8-12 s) is necessary for an undetectable
trail, 1' (4 s) for an expert exposure. Divide these values by the
proportional increase in focal length over a 50-mm lens. For example, for
3' (12 s), a 150-mm lens would be 1/3 (1' and 4 s) and a 1000-mm lens would
be 1/20 (0.15' and 0.6 s). Note that to compensate for these values, the
constant in the formula would be 1000 for a barely-detectable trail, 600 for
an undetectable trail, and 200 for an expert exposure.
N.B. The above formulae assume a declination of 0o. For other declinations,
multiply lengths and divide exposure times by the following cosines of the
respective declination angles: 0.98 (10o), 0.93 (20o), 0.86 (30o), 0.75
(40o), 0.64 (50o), 0.50 (60o), 0.34 (70o), 0.18 (80o), 0.10 (85o).
EXPOSURE DURATION FOR EXTENDED OBJECTS
where e is the exposure duration in seconds for an image size of >= 0.1 mm
f is the f-number (f/) of the lens
S is the film's ISO speed
B is the brightness factor of the object (Venus 1000, Moon 125, Mars
30, Jupiter 5.7)
Thus, a 2-minute exposure at f/1.4 is equivalent to a 32-minute exposure at
f/5.6 (4 stops squared times 2 minutes), ignoring the effects of reciprocity
failure in the film, which would mean that the 32-minute exposure would have
to be even longer.
SURFACE BRIGHTNESS OF AN EXTENDED OBJECT ("B" VALUE)
where B is the surface brightness of the (round) extended object
M is the magnitude of the object (total brightness of the object),
linearized in the formula
D is the angular diameter of the object in seconds of arc (D^2 is
the surface area of the object)
EXPOSURE DURATION FOR POINT SOURCES
where e is the exposure duration in seconds for an image size of >= 0.1 mm
M is the magnitude of the object
S if the film's ISO speed
a is the aperture of the objective
FOCAL LENGTH NECESSARY TO PHOTOGRAPH A RECOGNIZABLE OBJECT
where F is the focal length in mm necessary to photograph a recognizable
object
Distance is the distance of the object in m
Linear Field is the linear field of the object in m
Image Size is the image size in mm (equal to 24 mm divided by the
amount of enlargement of the print [3x is the standard for
35-mm film] for the smallest dimension of 35-mm film)
where H is the hour angle
Theta is sidereal time
Delta is right ascension
The Hour Angle is negative east of and positive west of the meridian (as
right ascension increases eastward).
BODE'S LAW
where n is the serial order of the planets from the sun (Mercury's 2n =1,
Venus's n = 0, Earth's n = 1, asteroid belt = 3)
ANGULAR SIZE
where Theta is the angular size of the object in degrees
h is the linear size of the object in m
d is the distance from the eye in m
e.g., for the width of a quarter at arm's length:
(55*0.254)/0.711 = 2o
RELATIVE LIGHT EFFICIENCY (TWILIGHT FACTOR)
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
LENGTH OF A METEOR TRAIL
where L is the linear size, or actual length in space, in km
A is the maximum angular length as observed in degrees
D is the known altitude of the meteor as it enters the
atmosphere in km
EFFICIENCY OF LENS FOR PHOTOGRAPHING AN AVERAGE METEOR
where Efficiency is the efficiency of the lens for photographing
an average (in a meteor shower)
F is the focal length of the lens
f is the f-number (f/) of the lens
ESTIMATING ANGULAR DISTANCE
Penny, 4 km distant ....................................... 1"
Sun, Moon ................................................. 30'
(The Moon is approximately 400 times smaller in angular
diameter than the Sun, but is approximately 400 times
closer.)
Width of little finger at arm's length .................... 1o
Dime at arm's length ...................................... 1o
Quarter at arm's length ................................. 2.5o
Width of Orion's belt ..................................... 3o
Alpha Ursae Majoris (Dubhe) to Beta Ursae Majoris (Merak) . 5o
(Height of Big Dipper's cup. These are the "pointer
stars" to Polaris.)
Alpha Geminorum (Castor) to Beta Geminorum (Pollux) ....... 5o
Width of fist at arm's length ............................. 10o
Alpha Ursae Majoris (Dubhe) to Delta Ursae Majoris (Megrez) 10o
(Width of Big Dipper's cup.)
Height of Orion ........................................... 16o
Length of palm at arm's length ............................ 18o
Width of thumb to little finger at arm's length ........... 20o
Alpha Ursae Majoris (Dubhe) to Eta Ursae Majoris (Alkaid) . 25o
(Length of Big Dipper.)
Alpha Ursae Majoris (Dubhe) to Alpha Ursae Minoris
(Polaris) .............................................. 27o
ESTIMATING MAGNITUDES
Big Dipper, from cup to handle
Alpha (Dubhe) 1.9
Beta (Merak) 2.4
Gamma (Phecda) 2.5
Delta (Megrez) 3.4
Epsilon (Alioth) 1.7 (4.9)
Zeta (Mizar) 2.4 (4.0)
Eta (Alkaid) 1.9
Little Dipper, from cup to handle
Beta (Kochab) 2.2
Gamma (Pherkad) 3.1
Eta 5.0
Zeta 5.1 (4.3)
Epsilon 4.4
Delta 4.4
Alpha (Polaris) 2.1
RANGE OF USEFUL MAGNIFICATION OF A TELESCOPE
Minimum useful magnification .................... 0.13*D
(0.2*D for better contrast)
Best visual acuity .............................. 0.25*D
Wide views ...................................... 0.4*D
Lowest power to see all detail (resolution of eye
matches resolution of telescope) ............. 0.5*D
Planets, Messier objects, general viewing ....... 0.8*D
Normal high power, double stars ................. 1.2*D to 1.6*D
Maximum useful magnification .................... 2.0*D
Close doubles ................................... 2.35*D
Sometimes useful for double stars ............... 4.0*D
Limit imposed by atmospheric turbulance ......... 500
GEOGRAPHIC DISTANCE
Geographic distance of one second of arc = 30 m * cos of the latitude
where cos(Latitude)=1 on lines of constant longitude