ASTRONOMICAL FORMULAE

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 = 2o. 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 second 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 .................... 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) 1.8 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 turbulance ..................... 500

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Maintained by Ray Shapp Page last updated 06/23/2001