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# Astronomical Formulae

 TELESCOPES ASTROPHOTOGRAPHY MISCELLANEOUS

### TELESCOPES

#### APERTURE

D = F/f

where D 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

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 °) = 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:

15° /h (1.25°/5 min) at 0° declination
13° /h (1.08°/5 min) at 30° declination
7.5°/h (0.63°/5 min) at 60° 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 may 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 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 as 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 at 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
= 127.1/D

(the second formula is based on Lambda = 0.00055 for yellow) where r is the angular radius (one-half the angular diameter) of the Airy disc (irreducible min. size of a star disc in “)
Lambda is the wavelength of the light in mm
206265 is the number of ” in a radian
D is the diameter of the objective in mm

#### 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 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 ”
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 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

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 ”
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.

### FORMULAE FOR ASTROPHOTOGRAPHY

#### F-NUMBER: PRIME FOCUS (ERECT IMAGE)

f/ = F/D

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)

f/ = F’/D = (M*Fc)/D = ((F/Fe)*Fc)/D = (F/D)*(Fc/Fe) = (M/D)*Fc

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 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)

f/ = F’/D = (F/D)*(B/A) = (F/D)*(((M+1)*Fe)/A)
= (F/D)*((B/Fe)-1)

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)

f/ = F’/D = (F/D) * (B/A)

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 mag.)

#### EXPOSURE COMPARISON FOR EXTENDED OBJECTS

Exposure Compensation = (f/S)^2/(f/E)^2 = ((f/S)/(f/E))^2

(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

Exposure Compensation = De^2/Ds^2 = (De/Ds)^2

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

Power = r^2/f^2

(the light-recording power is directly proportional to the square of the radius of the objective and inversely proportional 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

#### EFFICIENCY OF LENS FOR PHOTOGRAPHING AN AVERAGE METEOR

Efficiency = F/f^2

where Efficiency is the efficiency of the lens for photographing an average meteor (in a meteor shower)
F is the focal length of the lens
f is the f-number (f/) of the lens

#### PRINT’S EFFECTIVE FOCAL LENGTH

Print EFL = Camera FL * Print Enlargement

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

Guidescope M ~ f/12.5

where Guidescope M is the magnification needed
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 camera 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

Guiding Tolerance = 0.076 * Guidescope M

where Guiding Tolerance is in mm
0.076 is one ” at a 254-mm reading distance from the print (a crosshair is usually 0.05 mm)

#### MAXIMUM ALLOWABLE TRACKING (SLOP) ERROR

S ~ 8250/(F*E)

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)
h = 0.04 mm of image-drift tolerance (an empirical value from astrophotographs).

#### CONVERSION OF PLATE SCALE TO EFFECTIVE FOCAL LENGTH

EFL = mm per degree * 57.3 = 206265/” per mm

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

Resolving Power = 4191″/F

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

Maximum Resolution = 1600/f

where Maximum Resolution is the 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 min 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

Minimum Resolution = Maximum Resolution * Print Enlargement

where Minimum Resolution is the min resolution necessary
Maximum Resolution is the max resolution for a perfect lens
Print Enlargement is the amount of enlargement of the print (3x is the standard for 35-mm film)

#### SIZE OF IMAGE (ANGULAR)

h = (Theta*F)/k
Theta = k*(h/F)
F = (k*h)/Theta

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 27 deg, 41 deg, and 49 deg
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).

#### LENGTH OF A STAR TRAIL ON FILM

Length = F*T*0.0044

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

T = 5455/F

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

T = (1397/F)

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 0°. For other declinations, multiply lengths and divide exposure times by the following cosines of the respective declination angles:

0.98 (10°), 0.93 (20°), 0.86 (30°), 0.75 (40°), 0.64 (50°), 0.50 (60°), 0.34 (70°), 0.18 (80°), 0.10 (85°).

#### SIZE OF IMAGE (LINEAR)

i = (h/D)*F
h = (D*i)/F
D = (h*F)/i
F = (D*i)/h

where i is the linear image size in mm of the image at prime focus of an objective or telephoto lens (for terrestrial objects, 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])
h is the linear height of the object in units corresponding to D
D is the distance of the object in units corresponding to h
F is the effective focal length (focal length times Barlow magnification) in mm

The last formula gives the focal length necessary to photograph a recognizable celestial (Linear Width in km) or terrestrial (Linear
Width in m).

#### EXPOSURE DURATION FOR EXTENDED OBJECTS

E = f^2/(S*B)

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)

B = 10^0.4(9.5-M)/D^2

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

e = (10^0.4(M+13))/S*a^2

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

### MISCELLANEOUS FORMULAE

#### HOUR ANGLE

H = Theta – Delta

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

(4 + 3(2^n))/10 in AU at aphelion

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)

#### APPARENT ANGULAR SIZE OF AN OBJECT

Theta = (h/D)*k

where Theta is the object’s apparent angular size in units corresponding to k
h is the linear height of the object in units corresponding to D
D is the distance of the object in units corresponding to h
Theta is the object’s angular height (angle of view) in units corresponding to k
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)

A degree is the apparent size of an object whose distance is 57.3 times its diameter. The formula holds for celestial or terrestrial objects. E.g., for the width of a quarter at arm’s length: (57.3*25 mm)/700 mm = 2°.

Under ideal conditions, the human eye can resolve anything subtending more than a 1′ angle, i.e., see an object as an extended object or see a double star as two stars rather than a single point of light, provided that the two components are of nearly equal brightness. A more practical value would be 4′; 8′ is an even more practical value for comfortable viewing. The best earthbound telescopes are usually limited by atmospheric effects to objects 1″ or larger (0.25″ with excellent seeing) in apparent size (before magnification). In theory, a telescope could see everything with a magnification of 60x (1″ magnified to 1′).

#### LENGTH OF A METEOR TRAIL

h = (Theta*D)/57.3

where h is the linear height of the meteor in km
Theta is the object’s apparent angular size in degrees
D is the distance of the object in km

#### GEOGRAPHIC DISTANCE

Geographic distance of one sec of arc = 30 m * cos of the latitude

where cos(Latitude)=1 on lines of constant longitude

#### 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 ……………….. 1°
Dime at arm’s length ……………………………….. 1°
Quarter at arm’s length …………………………….. 2.5°
Width of Orion’s belt ………………………………. 3°
Alpha Ursae Majoris (Dubhe) to Beta Ursae Majoris (Merak) . 5°
(Height of Big Dipper’s cup. These are the “pointer stars” to Polaris.)
Alpha Geminorum (Castor) to Beta Geminorum (Pollux) ……. 5°
Width of fist at arm’s length ……………………….. 10°
Alpha Ursae Majoris (Dubhe) to Delta Ursae Majoris (Megrez) 10°
(Width of Big Dipper’s cup.)
Height of Orion ……………………………………. 16°
Length of palm at arm’s length ………………………. 18°
Width of thumb to little finger at arm’s length ……….. 20°
Alpha Ursae Majoris (Dubhe) to Eta Ursae Majoris (Alkaid) . 25°
(Length of Big Dipper.)
Alpha Ursae Majoris (Dubhe) to Alpha Ursae Minoris (Polaris) ……………………………………… 27°

#### ESTIMATING MAGNITUDES

Big Dipper, from cup to handle

Alpha (Dubhe) ………………… 1.8
Beta (Merak) ………………….. 2.4
Gamma (Phecda) …………… 2.5
Delta (Megrez) ……………….. 3.4
Epsilon (Alioth) ………………. 1.8
Zeta (Mizar) …………………… 2.2
Eta (Alkaid) …………………… 1.9

Little Dipper, from cup to handle

Beta (Kochab) ……………………… 2.0
Gamma (Pherkad) ……………….. 3.1
Eta …………………………………….. 5.0
Zeta …………………………………… 4.3
Epsilon ………………………………. 4.4
Delta (Pherkard) …………………. 4.4
Alpha (Polaris) ……………………. 2.0

#### RANGE OF USEFUL MAGNIFICATION OF A TELESCOPE

D = diameter of aperture in mm

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 turbulence ……… 500