ASTRONOMICAL FORMULAE - [These are GREAT - no author! - S.H.] --------------------- 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 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 where M is the magnification required 480 is # 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 (more practical working value: 2x this) 1800 is the angular size of the moon in " 3476 is the diameter of the moon in km 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 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 x its diameter. SIZE OF IMAGE (CELESTIAL) 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 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 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 (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 declina- tions, 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 = 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) ANGULAR SIZE Theta = (55*h)/d 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 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 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 ANGULAR SIZE UNITS 1 degree = 60 arc minutes denoted 60' 1 ' = 60 arc seconds denoted " 1 Radian = 57.2957795 deg = 3437.74677' = 206264.806" # of square degrees in a sphere = 41252.96124 Ex Moon 1800" = .5 deg = 30' = 3500 km = 2170 miles 180 " = 350 km 1.8 " = 35 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: D = distance in parsecs. Obtained by taking 1/parallax. m = apparent magnitude M = absolute magnitude m-M = distance modulus 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 R