452 lines
19 KiB
Plaintext
452 lines
19 KiB
Plaintext
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ASTRONOMICAL FORMULAE - [These are GREAT - no author! - S.H.]
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---------------------
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MAGNIFICATION: BY FIELDS
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M = Alpha/Theta
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where M is the magnification
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Alpha is the apparent field
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Theta is the true field
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Apparent Field: the closest separation eye can see is 4', more practically
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8-25', 1-2' for good eyes. The Zeta Ursae Majoris double (Mizar/Alcor) is
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11.75'; Epsilon Lyrae is 3'.
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True Field (in o) = 0.25 * time * cos of the declination
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(in ') = 15 * time * cos of the declination
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where time is the time to cross the ocular field in minutes
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A star therefore moves westward at the following rates:
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15o /h (1.25o/5 min) at 0o declination
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13o /h (1.08o/5 min) at 30o declination
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7.5o/h (0.63o/5 min) at 60o declination.
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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MAGNIFICATION: BY DIAMETER AND EXIT PUPIL
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M = D/d
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where M is the magnification
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D is the diameter of the objective
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d is the exit pupil (5-6 mm is best; 7 mm not produce a sharp outer
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image)
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The scotopic (dark-adapted) aperture of the human pupil is typically 6
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(theoretically 7, 5 if over age 50) mm. Since the human pupil has a focal
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length of 17 mm, it is f/2.4 and yields 0.17 per mm of aperture. 2.5 mm is
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the photopic (light-adapted) diameter of the eye.
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DAWES LIMIT (SMALLEST RESOLVABLE ANGLE, RESOLVING POWER)
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Theta = 115.8/D
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where Theta is the smallest resolvable angle in "
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D is the diameter of the objective in mm
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Atmospheric conditions seldom permit Theta < 0.5". The Dawes Limit is one-
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half the angular diameter of the Airy (diffraction) disc, so that the edge
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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of one disc does not extend beyond the center of the other). The working
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value is two times the Dawes Limit (diameter of the Airy disc), so that the
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edges of the two stars are just touching.
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MAGNIFICATION NEEDED TO SPLIT A DOUBLE STAR
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M = 480/d
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where M is the magnification required
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480 is # of seconds of arc for an apparent field of 8 minutes of arc
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d is the angular separation of the double star
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About the closest star separation that the eye can distinguish is 4 minutes
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of arc (240 seconds of arc). Twice this distance, or an 8-minute (480-
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second) apparent field angle, is a more practical value for comfortable
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viewing. In cases where the comes is more than five magnitudes fainter
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than the primary, you will need a wider separation: 20 or 25 minutes of
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arc, nearly the width of the moon seen with the naked eye.
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RESOLUTION OF LUNAR FEATURES
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Resolution = (2*Dawes Limit*3476)/1800)
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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Dawes Limit * 38.8
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where Resolution is the smallest resolvable lunar feature in km
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2*Dawes Limit is the Airy disc (more practical working value: 2x this)
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1800 is the angular size of the moon in "
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3476 is the diameter of the moon in km
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APPARENT ANGULAR SIZE OF AN OBJECT
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Apparent Angular Size = (Linear Width / Distance) * 57.3
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where Apparent Angular Size of the object is expressed in degrees
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Linear Width is the linear width of the object in m
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Distance is the distance of the object in m
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A degree is the apparent size of an object whose distance is 57.3 x its
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diameter.
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SIZE OF IMAGE (CELESTIAL)
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h = (Theta*F)/K
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Theta = K*(h/F)
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F = (K*h)/Theta
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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where h is the linear height in mm of the image at prime focus of an
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objective or a telephoto lens
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Theta is the object's angular height (angle of view) in units
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corresponding to K
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F is the effective focal length (focal length times Barlow
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magnification) in mm
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K is a constant with a value of 57.3 for Theta in degrees, 3438 in
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minutes of arc, 206265 for seconds of arc (the number of the
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respective units in a radian)
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The first formula yields image size of the sun and moon as approximately 1%
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of the effective focal length (Theta/K = 0.5/57.3 = 0.009).
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The second formula can be used to find the angle of view (Theta) for a
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given film frame size (h) and lens focal length (F). Example: the 24 mm
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height, 36 mm width, and 43 mm diagonal of 35-mm film yields an angle of
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view of 27o, 41o, and 49o for a 50-mm lens.
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The third formula can be used to find the effective focal length (F)
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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required for a given film frame size (h) and angle of view (Theta).
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SIZE OF IMAGE (TERRESTRIAL)
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h = (Linear Width / Distance) * F
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Linear Width = (Distance * h) / F
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Distance = (Linear Width * F) / h
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F = (Distance * h) / Linear Width
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where h is the linear height in mm of the image at prime focus of an
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objective or telephoto lens
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Linear Width is the linear width of the object in m
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Distance is the distance of the object in im
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F is the effective focal length (focal length times Barlow
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magnification) in mm
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(Star trails on film)
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The earth rotates 5' in 20 s, which yields a barely detectable star trail
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with an unguided 50-mm lens. 2-3' (8-12 s) is necessary for an
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undetectable trail, 1' (4 s) for an expert exposure. Divide these values
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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by the proportional increase in focal length over a 50-mm lens. For
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example, for 3' (12 s), a 150-mm lens would be 1/3 (1' and 4 s) and a 1000-
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mm lens would be 1/20 (0.15' and 0.6 s). Note that to compensate for these
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values, the constant in the formula would be 1000 for a barely-detectable
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trail, 600 for an undetectable trail, and 200 for an expert exposure.
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N.B. The above formulae assume a declination of 0o. For other declina-
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tions, multiply lengths and divide exposure times by the following cosines
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of the respective declination angles: 0.98 (10o), 0.93 (20o), 0.86 (30o),
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0.75 (40o), 0.64 (50o), 0.50 (60o), 0.34 (70o), 0.18 (80o), 0.10 (85o).
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SURFACE BRIGHTNESS OF AN EXTENDED OBJECT ("B" VALUE)
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B = 10^0.4(9.5-M)/D^2
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where B is the surface brightness of the (round) extended object
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M is the magnitude of the object (total brightness of the object),
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linearized in the formula
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D is the angular diameter of the object in seconds of arc (D^2 is
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the surface area of the object)
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EXPOSURE DURATION FOR POINT SOURCES
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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e = (10^0.4(M+13))/S*a^2
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where e is the exposure duration in seconds for an image size of >= 0.1 mm
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M is the magnitude of the object
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S if the film's ISO speed
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a is the aperture of the objective
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MISCELLANEOUS FORMULAE
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----------------------
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HOUR ANGLE
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H = Theta - Delta
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where H is the hour angle
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Theta is sidereal time
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Delta is right ascension
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The Hour Angle is negative east of and positive west of the meridian (as
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right ascension increases eastward).
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BODE'S LAW
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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(4 + 3(2^n))/10 in AU at aphelion
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where n is the serial order of the planets from the sun (Mercury's 2n =1,
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Venus's n = 0, Earth's n = 1, asteroid belt = 3)
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ANGULAR SIZE
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Theta = (55*h)/d
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where Theta is the angular size of the object in degrees
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h is the linear size of the object in m
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d is the distance from the eye in m
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e.g., for the width of a quarter at arm's length:
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(55*0.254)/0.711 = 2o
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ESTIMATING ANGULAR DISTANCE
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Penny, 4 km distant ....................................... 1"
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Sun, Moon ................................................. 30'
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(The Moon is approximately 400 times smaller in angular
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diameter than the Sun, but is approx 400 times closer)
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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Width of little finger at arm's length .................... 1o
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Dime at arm's length ...................................... 1o
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Quarter at arm's length ................................... 2.5o
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Width of Orion's belt ..................................... 3o
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Alpha Ursae Majoris (Dubhe) to Beta Ursae Majoris (Merak) . 5o
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(Height of Big Dipper's "pointer stars" to Polaris.)
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Alpha Geminorum (Castor) to Beta Geminorum (Pollux) ....... 5o
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Width of fist at arm's length ............................. 10o
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Alpha Ursae Majoris (Dubhe) to Delta Ursae Majoris (Megrez) 10o
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(Width of Big Dipper's "pointer stars".)
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Height of Orion ........................................... 16o
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Length of palm at arm's length ............................ 18o
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Width of thumb to little finger at arm's length ........... 20o
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Alpha Ursae Majoris (Dubhe) to Eta Ursae Majoris (Alkaid) . 25o
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(Length of Big Dipper.)
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Alpha Ursae Majoris (Dubhe) to Alpha Ursae Minoris
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(Polaris) ............................................. 27o
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ESTIMATING MAGNITUDES
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Big Dipper, from cup to handle
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Alpha (Dubhe) 1.9
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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Beta (Merak) 2.4
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Gamma (Phecda) 2.5
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Delta (Megrez) 3.4
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Epsilon (Alioth) 1.7 (4.9)
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Zeta (Mizar) 2.4 (4.0)
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Eta (Alkaid) 1.9
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Little Dipper, from cup to handle
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Beta (Kochab) 2.2
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Gamma (Pherkad) 3.1
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Eta 5.0
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Zeta 5.1 (4.3)
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Epsilon 4.4
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Delta 4.4
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Alpha (Polaris) 2.1
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RANGE OF USEFUL MAGNIFICATION OF A TELESCOPE
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D = diameter of aperture in mm
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Minimum useful magnification .................... 0.13*D 0.2*D for better contrast
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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Best visual acuity .............................. 0.25*D
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Wide views ...................................... 0.4*D
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Lowest power to see all detail (resolution of eye
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matches resolution of telescope) ............. 0.5*D
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Planets, Messier objects, general viewing ....... 0.8*D
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Normal high power, double stars ................. 1.2*D to 1.6*D
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Maximum useful magnification .................... 2.0*D
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Close doubles ................................... 2.35*D
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Sometimes useful for double stars ............... 4.0*D
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Limit imposed by atmospheric turbulance ......... 500
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GEOGRAPHIC DISTANCE
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Geographic distance of one second of arc = 30 m * cos of the latitude
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where cos(Latitude)=1 on lines of constant longitude
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ANGULAR SIZE UNITS
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1 degree = 60 arc minutes denoted 60'
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1 ' = 60 arc seconds denoted "
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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1 Radian = 57.2957795 deg
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= 3437.74677'
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= 206264.806"
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# of square degrees in a sphere = 41252.96124
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Ex
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Moon
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1800" = .5 deg = 30' = 3500 km = 2170 miles
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180 " = 350 km
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1.8 " = 35 km = 2.1 miles
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.
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. .
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A radian is defined such that the angle,T,produced
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. c . by setting the length of arc a = to the radius c
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.------ will subtend 1 radian or 57.3 degrees approximately.
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\ T /
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. \ /a
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\ /.
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. \
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. .
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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ANNUAL PARALLAX
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Tan(pi) approx= pi = a/D (by small angle equation)
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Where a = 1 AU or Astronomical Unit = 9.3E7 miles
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D = distance in parsecs
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The distance is therefore related to the parallax definition by:
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D = 1/pi
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The parallax is a measure of distance based on angular displacement of a
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star against much distant background stars over the course of a year's time
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as the earth circles the sun. (A similar affect is obtained by closing one
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eye, holding out a pencil vertically, and alternately closing and opening
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the opposing eyes. The pencil shifts relative to the background which in
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this case is the wall,window,woman, what have you. That is a parallactic
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effect, except the eyes take the place of a camera taking pictures when the
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earth is at opposite ends of its orbit.
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The parsec or PARallax-SECond is defined in terms of the parallax: The
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parsec is the distance a star has to be such that the Earth's motion around
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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the sun would cause the star to shift in the sky by one arc second through
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the course of one year. The parsec is 3.26 light years in measure and is
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obtained by conversion of light years or by taking 1/parallax value.
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STELLAR DISTANCES
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D(pc) = 10^(1+.2(m-M)) or rewritten as
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m = M + 5*Log(D) - 5
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Where as usual:
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D = distance in parsecs. Obtained by taking 1/parallax.
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m = apparent magnitude
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M = absolute magnitude
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m-M = distance modulus
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SPECTRAL CLASS FEATURES
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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Spectral
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Class Special features
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---------------------------------------------------------------------
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O HeII lines visible; lines from highly ionized species, for
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example, CIII, NIII, OIII, SiIV ; H lines relatively weak;
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strong ultraviolet continuum.
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B HeI lines strong; attain maxmimum at B2; HeII lines absent;
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H lines stronger; lower ions, for example, CII, OII, SiIII
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A H lines attain maxmimum strength at A0 and decrease toward later
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types; MgII, SiII strong; CaII weak and increasing in strength
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F H weaker, CaII stronger; lines of neutral atoms and first ions
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of metals appear prominently
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G Solar-type spectra; CaII lines extremely stron; neutral metals
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prominent, ions weaker; G band (CH) strong; H lines weakening
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K Neutral metallic lines dominate; H quite weak; molecular bands
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(CH,CN) developing; continuum weak in blue
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M Strong molecular bands, particularly TiO; some neutral lines for
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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example, CaI quite strong; red continua
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C(R,N) Carbon stars; strong bands of carbon compounds C ,CN,CO;
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TiO absent; temperatures in range of 2 classes K and M
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S Heavy-element stars; bands of ZrO, YO, LaO; neutral atoms strong
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as in classes K and M; overlaps these classes in temperature range
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Ia-0 Most extreme supergiants
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Ia Luminous supergiants
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Iab Moderate supergiants
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Ib Less luminous supergiants
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II Bright giants
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III Normal giants
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IV Subgiants
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V Dwarfs (main sequence)
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VI Subdwarf (below main sequence, extreme metal poor. )
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VII White dwarfs
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COMPLETE DATA FOR THE BRIGHTEST STARS
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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Sp
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Star Name RA Dec m M Cl Lum Rad M Ly Tms
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h m d m *Lo *Ro *Mo E6yr
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a And Alpheratz 00 07 +28 58 2.06 -0.1 B9p 93 3.1 5.0 90 500
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a Ari Hamal 02 06 +23 22 2.00 +0.2 K2III 103 17 5.1 76 500
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a UMi Polaris 02 12 +89 11 1.99 -4.6 F8Ib 1600 80 10 680 62
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b Per Algol 03 07 +40 52 2.06 -0.5 B8V 132 3.2 4.5 105 340
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a Per Mirfak 03 23 +49 47 1.8 -4.4 F5Ib 4800 55 14 570 29
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n Tau Alcyone 03 46 +24 03 2.9 -3.2 B7III 1800 8.5 10.5 410 58
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a Tau Aldeberan 04 35 +16 28 0.86 -1.2 K5III 150 4.5 4.5 68 300
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b Ori Rigel 05 14 -08 13 0.14 -7.1 B8Ia 150000 80 42 900 3
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a Aur Capella 05 15 +45 59 0.05 -0.6 G8III 75 1.2 3.8 45 500
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y Ori Bellatrix 05 24 +06 20 1.64 -4.2 B2III 4000 6.5 14 470 3.5
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a Ori Betelgeuse 05 54 +07 24 0.41 -5.6 M2Ia 13000 800 8.1 520 6.2
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a Car Canopus 06 24 -52 41 -0.72 -3.1 F0Ib 800 40 3.2 98 40
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a CMa Sirius 06 44 -16 42 -1.47 1.45 A1V 23 2.3 2.7 8.6 1174
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a Gem Castor 07 33 +31 56 1.97 1.3 A1V 28 2.3 2.8 45 1000
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a CMi Procyon 07 38 +05 17 0.37 2.7 F5IV 7.6 2 1.8 11.3 2370
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b Gem Pollux 07 44 +28 05 1.16 1.0 K0III 30 16 2.9 35 950
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a Hyd Alphard 09 26 -08 35 1.98 -0.3 K4III 114 162 4.4 94 385
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a Leo Regulus 10 07 +12 04 1.36 -0.7 B7V 140 3 4.7 84 335
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a UMa Dubhe 11 03 +61 52 1.81 -0.7 K0III 140 * 4.7 105 335
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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b Leo Denebola 11 48 +14 41 2.14 1.5 A3V 21 * 2.6 42 1238
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a CVn CorCaroli 12 55 +38 26 2.90 0.1 B9p 77 3.6 3.9 118 500
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a Vir Spica 13 24 -11 03 0.91 -3.3 B1V 1700 3 10.3 220 60
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a Boo Arcturus 14 15 +19 17 -0.06 -0.3 K2III 100 20 4.2 36 420
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a Cen Rigil Kent 14 38 -60 46 0.01 4.4 G2V 1.3 1 1.1 4.3 8500
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a CrB Alphecca 15 34 +26 47 2.23 0.4 A0V 120 3.6 4.5 76 375
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a Sco Antares 16 28 -26 23 0.92 -5.1 M1Ib 9000 800 17.2 520 19
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a Her RasAlgethi 17 14 +14 24 3.10 -2.3 M5II 700 800 7.9 410 112
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a Oph Rasalhague 17 34 +12 35 2.09 0.8 A5III 29 6.4 2.8 60 965
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a Lyr Vega 18 36 +38 46 0.04 0.5 A0V 50 2.5 3.4 27 680
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b Cyg Albireo 19 30 +27 55 3.07 -2.4 K3II 800 59 8.1 410 100
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a Aql Altair 19 50 +08 49 0.77 2.2 A7IV 9.8 1.5 2 16.5 2000
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a Cyg Deneb 20 41 +45 12 1.26 -7.1 A2Ia 100000 40 37 1600 3.7
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a Cep Alderamin 21 18 +62 31 2.44 1.4 A7IV 330 9.5 6.1 52 184
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e Peg Emif 21 43 +09 48 2.38 -4.6 K2Ib 5900 140 15.1 780 25
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a PsA Fomalhaut 22 57 -29 44 1.15 2.0 A3V 12 2 2.2 22.6 1830
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NOTE: A '*' means no data available at this time
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R
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<Press Home PgDn PgUp Down Arrow End Q=Print>
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