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ASTRONOMICAL FORMULAE

First, the important telescope formulas - i.e. The ones you should try to remember :-)

 

Magnification = Objective Focal Length / Eyepiece Focal Length

Example: A 2000 mm focal length telescope using a 20 mm eyepiece yields a magnification of 100 times:

Example: Magnification = 2000 mm / 20 mm = 100

 

True Field of View = Eyepiece Apparent Field of View / Magnification of view

Example:  If we are still using the 20 mm eyepiece from the previous example on the same telescope, and we know from looking up in the manufacture's specifications that the eyepiece has a 50 degree 'apparent field of view':

Example: True Field of View = 50 / 100 = 1/2 degree

In other words, the entire full Moon would just fit in the view since it is 1/2 degree across, but you would see only the central portion of the Andromeda Galaxy which is about 4 1/2 degrees across.

 

Time for a star to cross field:

The time in minutes that it would take a star at the celestial equator (DEC=0) to go completely through the field of view when the telescope is not tracking, could be called "T"  The true field of view can then be calculated as

    True Field of View = 4 * T

 

f/number = Objective Focal Length / Objective Diameter.  Example: A 2000 mm focal length telescope that has 200 mm (8 inch) diameter yields a value of f/10.

 

Exit pupil = Objective Diameter / Magnification = Eyepiece Focal Length / Objective f/number

Or by switching the terms around:

Magnification = Objective Diameter / Exit Pupil Diameter.

 

Dawes limit = 4.56 Arc Seconds / Objective Diameter (inches)

 

Aperture gain = (Objective Diameter / Eye Pupil Diameter) 2

 

ASTRONOMICAL FORMULAE FOR REFERENCE PURPOSES

These are here when and if you ever need them...

 

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 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 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 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.

 

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

bulletwhere M is the magnification required
bullet480 is # of seconds of arc for an apparent field of 8 minutes of arc
bulletd 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

bullet2 * Dawes Limit is the Airy disc (more practical working value: 2x this)
bullet1800 is the angular size of the moon in "
bullet3476 is the diameter of the moon in km
bullet 

APPARENT ANGULAR SIZE OF AN OBJECT

 Apparent Angular Size = (Linear Width / Distance) * 57.3

 where Apparent Angular Size of the object is expressed in degrees

bulletLinear Width is the linear width of the object in m
bulletDistance is the distance of the object in m

A degree is the apparent size of an object whose distance is 57.3 x its diameter.

 

SIZE OF IMAGE (CELESTIAL)

    h = (Theta*F)/K

    Theta = K*(h/F)

    F = (K*h)/Theta

where

bulleth is the linear height in mm of the image at prime focus of an objective or a telephoto lens
bulletTheta is the object's angular height (angle of view) in units corresponding to K
bulletF is the effective focal length (focal length times Barlow  magnification) in mm
bulletK 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)

 h = (Linear Width / Distance) * F

 Linear Width = (Distance * h) / F

 Distance = (Linear Width * F) / h

 F = (Distance * h) / Linear Width

bulletwhere 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
bulletDistance is the distance of the object in m
bulletF is the effective focal length (focal length times Barlow magnification) in mm

 

(STAR TRAILS ON FILM)

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)

 

SURFACE BRIGHTNESS OF AN EXTENDED OBJECT ("B" VALUE)

B = 100.4(9.5-M)/D2

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 = (100.4(M+13))/S*a2

bulletwhere e is the exposure duration in seconds for an image size of >= 0.1 mm
bulletM is the magnitude of the object
bulletS if the film's ISO speed
bulleta is the aperture of the objective

 

MISCELLANEOUS FORMULAE

HOUR ANGLE

H = Theta - Delta

bulletwhere H is the hour angle
bulletTheta is sidereal time
bulletDelta 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(2n))/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)

 

ANGULAR SIZE

Theta = (55*h)/d

bulletwhere Theta is the angular size of the object in degrees
bulleth is the linear size of the object in m
bulletd 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

 

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 approx 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  "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 "pointer stars".)
 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

 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
 

GEOGRAPHIC DISTANCE

Geographic distance of one second of arc = 30 m * COS of the latitude,

where COS(Latitude)=1 on lines of constant longitude.

 

ANGULAR SIZE UNITS

 1 degree = 60 arc minutes denoted 60'

 1 '          = 60 arc seconds denoted 60"

 1 Radian = 57.2957795 deg

              = 3437.74677'

              = 206264.806"

Number of square degrees in a sphere = 41252.96124

 Ex Moon

         1800" = 0.5 deg = 30' = 3500 km = 2170 miles
         180 " = 350 km
         1.8 " = 3.5  km = 2.1 miles

              .
           .      .
                        A radian is defined such that the angle,T, produced
         .        c .   by setting the length of arc a = to the radius c
              .------   will subtend 1 radian or 57.3 degrees approximately.
              \ T   /
         .     \   /a
                \ /.
           .     \
              .  .

ANNUAL PARALLAX

 Tan(pi) approx= pi = a/D  (by small angle equation)

Where a = 1 AU or Astronomical Unit = 9.3E7 miles

        D = distance in parsecs

The distance is therefore related to the parallax definition by:

        D = 1/pi

The parallax is a measure of distance based on angular displacement of a star against much distant background stars over the course of a year's time as the earth circles the sun. (A similar affect is obtained by closing one eye, holding out a pencil vertically, and alternately closing and opening the opposing eyes. The pencil shifts relative to the background which in this case is the wall, window, woman, what have you. That is a parallactic effect, except the eyes take the place of a camera taking pictures when the earth is at opposite ends of its orbit.

The parsec or PARallax-SECond is defined in terms of the parallax: The parsec is the distance a star has to be such that the Earth's motion around
the sun would cause the star to shift in the sky by one arc second through the course of one year. The parsec is 3.26 light years in measure and is
obtained by conversion of light years or by taking 1/parallax value.

 

STELLAR DISTANCES

        D(pc) = 10(1+.2(m-M)) or rewritten as:

        m = M + 5*Log(D) - 5

Where as usual:

bulletD = distance in parsecs. Obtained by taking 1/parallax.
bulletm = apparent magnitude
bulletM = absolute magnitude
bulletm-M = distance modulus
bullet 

SPECTRAL CLASS FEATURES

Spectral
 Class     Special features
 ---------------------------------------------------------------------
 O         HeII lines visible; lines from highly ionized species, for
           example, CIII, NIII, OIII, SiIV ; H lines relatively weak;
           strong ultraviolet continuum.

 B         HeI lines strong; attain maxmimum at B2; HeII lines absent;
           H lines stronger; lower ions, for example, CII, OII, SiIII

 A         H lines attain maxmimum strength at A0 and decrease toward later
           types; MgII, SiII strong; CaII weak and increasing in strength

 F         H weaker, CaII stronger; lines of neutral atoms and first ions
           of metals appear prominently

 G         Solar-type spectra; CaII lines extremely stron; neutral metals
           prominent, ions weaker; G band (CH) strong; H lines weakening

 K         Neutral metallic lines dominate; H quite weak; molecular bands
           (CH,CN) developing; continuum weak in blue

 M         Strong molecular bands, particularly TiO; some neutral lines for
           example, CaI quite strong; red continua

 C(R,N)    Carbon stars; strong bands of carbon compounds C  ,CN,CO;
           TiO absent; temperatures in range of 2 classes K and M

 S         Heavy-element stars; bands of ZrO, YO, LaO; neutral atoms strong
           as in classes K and M; overlaps these classes in temperature range


 Ia-0      Most extreme supergiants
 Ia        Luminous supergiants
 Iab       Moderate supergiants
 Ib        Less luminous supergiants
 II        Bright giants
 III       Normal giants
 IV        Subgiants
 V         Dwarfs (main sequence)
 VI        Subdwarf (below main sequence, extreme metal poor. )
 VII       White dwarfs
 

COMPLETE DATA FOR THE BRIGHTEST STARS

                                             Sp
 Star  Name        RA     Dec     m     M    Cl    Lum    Rad   M   Ly   Tms
                   h m    d  m                     *Lo    *Ro  *Mo       E6yr

 a And Alpheratz  00 07  +28 58  2.06  -0.1  B9p   93     3.1  5.0  90   500
 a Ari Hamal      02 06  +23 22  2.00  +0.2  K2III 103    17   5.1  76   500
 a UMi Polaris    02 12  +89 11  1.99  -4.6  F8Ib  1600   80   10   680  62
 b Per Algol      03 07  +40 52  2.06  -0.5  B8V   132    3.2  4.5  105  340
 a Per Mirfak     03 23  +49 47  1.8   -4.4  F5Ib  4800   55   14   570  29
 n Tau Alcyone    03 46  +24 03  2.9   -3.2  B7III 1800   8.5  10.5 410  58
 a Tau Aldeberan  04 35  +16 28  0.86  -1.2  K5III 150    4.5  4.5  68   300
 b Ori Rigel      05 14  -08 13  0.14  -7.1  B8Ia  150000 80   42   900  3
 a Aur Capella    05 15  +45 59  0.05  -0.6  G8III 75     1.2  3.8  45   500
 y Ori Bellatrix  05 24  +06 20  1.64  -4.2  B2III 4000   6.5  14   470  3.5
 a Ori Betelgeuse 05 54  +07 24  0.41  -5.6  M2Ia  13000  800  8.1  520  6.2
 a Car Canopus    06 24  -52 41 -0.72  -3.1  F0Ib  800    40   3.2  98   40
 a CMa Sirius     06 44  -16 42 -1.47   1.45 A1V   23     2.3  2.7  8.6  1174
 a Gem Castor     07 33  +31 56  1.97   1.3  A1V   28     2.3  2.8  45   1000
 a CMi Procyon    07 38  +05 17  0.37   2.7  F5IV  7.6    2    1.8  11.3 2370
 b Gem Pollux     07 44  +28 05  1.16   1.0  K0III 30     16   2.9  35   950
 a Hyd Alphard    09 26  -08 35  1.98  -0.3  K4III 114    162  4.4  94   385
 a Leo Regulus    10 07  +12 04  1.36  -0.7  B7V   140    3    4.7  84   335
 a UMa Dubhe      11 03  +61 52  1.81  -0.7  K0III 140    *    4.7  105  335
 b Leo Denebola   11 48  +14 41  2.14   1.5  A3V   21     *    2.6  42   1238
 a CVn CorCaroli  12 55  +38 26  2.90   0.1  B9p   77     3.6  3.9  118  500
 a Vir Spica      13 24  -11 03  0.91  -3.3  B1V   1700   3    10.3 220  60
 a Boo Arcturus   14 15  +19 17 -0.06  -0.3  K2III 100    20   4.2  36   420
 a Cen Rigil Kent 14 38  -60 46  0.01   4.4  G2V   1.3    1    1.1  4.3  8500
 a CrB Alphecca   15 34  +26 47  2.23   0.4  A0V   120    3.6  4.5  76   375
 a Sco Antares    16 28  -26 23  0.92  -5.1  M1Ib  9000   800  17.2 520  19
 a Her RasAlgethi 17 14  +14 24  3.10  -2.3  M5II  700    800  7.9  410  112
 a Oph Rasalhague 17 34  +12 35  2.09   0.8  A5III 29     6.4  2.8  60   965
 a Lyr Vega       18 36  +38 46  0.04   0.5  A0V   50     2.5  3.4  27   680
 b Cyg Albireo    19 30  +27 55  3.07  -2.4  K3II  800    59   8.1  410  100
 a Aql Altair     19 50  +08 49  0.77   2.2  A7IV  9.8    1.5  2    16.5 2000
 a Cyg Deneb      20 41  +45 12  1.26  -7.1  A2Ia  100000 40   37   1600 3.7
 a Cep Alderamin  21 18  +62 31  2.44   1.4  A7IV  330    9.5  6.1  52   184
 e Peg Emif       21 43  +09 48  2.38  -4.6  K2Ib  5900   140  15.1 780  25
 a PsA Fomalhaut  22 57  -29 44  1.15   2.0  A3V   12     2    2.2  22.6 1830

 NOTE: A '*' means no data available at this time

 

Introduction to the Night Sky - Part II

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11/2011