Top Document: [sci.astro] Time (Astronomy Frequently Asked Questions) (3/9) Previous Document: C.02 What are all those different kinds of time? Next Document: C.04 What's a Julian date? modified Julian date? See reader questions & answers on this topic!  Help others by sharing your knowledge COMPUTING AZIMUTH AND ELEVATION  To compute the azimuth and elevation of an object, you first must compute the Local Sidereal Time of the place and time in question. First convert your local time to UT (Universal Time), with the date adjusted if needed. Now suppose that the time is Y,M,D,UT where Y,M,D is the calendar Year, Month (112) and Date (131), and UT is the Universal Time in hours+fractions. Also suppose your position is lat,long, where lat is counted as + if north and  if south, and long is counted as + if east and  if west. Now, first compute a "day number", d: 7*(Y + INT((M+9)/12)) d = 367*Y  INT() + INT(275*M/9) + D  730530 + UT/24 4 where INT is a function that discards the fractional part and returns the integer part of a function. d is zero at 2000 Jan 0.0 Now compute the Local Sidereal Time, LST: LST = 98.9818 + 0.985647352 * d + UT*15 + long (east long. positive). Note that LST is here expressed in degrees, where 15 degrees corresponds to one hour. Since LST really is an angle, it's convenient to use one unitdegreesthroughout. Now, suppose your object resides at a known RA (Right Ascension) and Dec (Declination). Convert both RA and Dec to degrees + decimals, remembering that 1 hour of RA corresponds to 15 degrees of RA. Next, compute the Hour Angle: HA = LST  RA Now you can compute the Altitude, h, and the Azimuth, az: sin(h) = sin(lat) * sin(Dec) + cos(lat) * cos(Dec) * cos(HA) sin(HA) tan(az) =  cos(HA) * sin(lat)  tan(Dec) * cos(Lat) Here az is 0 deg in the south, 90 deg in the west etc. If you prefer 0 deg in the north and 90 deg in the east, add 180 degrees to az. A NOTE ON TRIGONOMETRIC FUNCTIONS ON YOUR COMPUTER  If you have an atan2() function (or equivalent) available on your computer, compute the numerator and denominator separately and feed them both to your atan2() function, instead of dividing and feeding them to your atan() functionthen you'll get the correct quadrant immediately. In the "C" language you would thus write: az = atan2( sin(HA), cos(HA)*sin(lat)tan(Dec)*cos(Lat) ); instead of: az = atan( sin(HA) / (cos(HA)*sin(lat)tan(Dec)*cos(Lat)) ); On a scientific calculator, there is often a "rectangular to polar" coordinate conversion function that does the same thing. Users of Pascal and other programming languages that lack an atan2() function are strongly encouraged to write such a function of their own. In Pascal it would be (pi is assumed to have been assigned an appropriate valueone way is to compute: pi := 4.0*arctan(1) ): function atan2( y : real, x : real ) real; (* Compute arctan(y/x), selecting the correct quadrant *) begin if x > 0 atan2 := arctan(y/x) else if x < 0 atan2 := arctan(y/x) + pi (* Below x is zero *) else if y > 0 atan2 := pi/2 else if y < 0 atan2 := pi/2 /* Below both x and y are zero *) else atan2 := 0.0 (* atan2( 0.0, 0.0 ) is really an error though.. *) end Another trick I also use is to add a set of trig functions that work in degrees instead of radians to my function librarythat will make life a lot easier when you're working in degrees as the basic unit. I name them sind, cosd, atan2d, etc. If you don't do that, you'll have to convert between degrees and radians when calling the standard trig functions. COMPUTING RISE AND SET TIMES  To compute when an object rises or sets, you must compute when it passes the meridian and the HA of rise/set. Then the rise time is the meridian time minus HA for rise/set, and the set time is the meridian time plus the HA for rise/set. To find the meridian time, compute the Local Sidereal Time at 0h local time (or 0h UT if you prefer to work in UT) as outlined abovename that quantity LST0. The Meridian Time, MT, will now be: MT = RA  LST0 where "RA" is the object's Right Ascension (in degrees!). If negative, add 360 deg to MT. If the object is the Sun, leave the time as it is, but if it's stellar, multiply MT by 365.2422/366.2422, to convert from sidereal to solar time. Now, compute HA for rise/set, name that quantity HA0: sin(h0)  sin(lat) * sin(Dec) cos(HA0) =  cos(lat) * cos(Dec) where h0 is the altitude selected to represent rise/set. For a purely mathematical horizon, set h0 = 0 and simplify to: cos(HA0) =  tan(lat) * tan(Dec) If you want to account for refraction on the atmosphere, set h0 = 35/60 degrees (35 arc minutes), and if you want to compute the rise/set times for the Sun's upper limb, set h0 = 50/60 (50 arc minutes). When HA0 has been computed, leave it as it is for the Sun but multiply by 365.2422/366.2422 for stellar objects, to convert from sidereal to solar time. Finally compute: Rise time = MT  HA0 Set time = MT + HA0 convert the times from degrees to hours by dividing by 15. If you'd like to check that your calculations are accurate or just need a quick result, check the USNO's Sun or Moon Rise/Set Table, <URL:http://aa.usno.navy.mil/AA/data/docs/RS_OneYear.html>. COMPUTING THE SUN'S POSITION  To be able to compute the Sun's rise/set times, you need to be able to compute the Sun's position at any time. First compute the "day number" d as outlined above, for the desired moment. Next compute: oblecl = 23.4393  3.563E7 * d w = 282.9404 + 4.70935E5 * d M = 356.0470 + 0.9856002585 * d e = 0.016709  1.151E9 * d This is the obliquity of the ecliptic, plus some of the elements of the Sun's apparent orbit (i.e., really the Earth's orbit): w = argument of perihelion, M = mean anomaly, e = eccentricity. Semimajor axis is here assumed to be exactly 1.0 (while not strictly true, this is still an accurate approximation). Next compute E, the eccentric anomaly: E = M + e*(180/pi) * sin(M) * ( 1.0 + e*cos(M) ) where E and M are in degrees. This is itno further iterations are needed because we know e has a sufficiently small value. Next compute the true anomaly, v, and the distance, r: r * cos(v) = A = cos(E)  e r * sin(v) = B = sqrt(1  e*e) * sin(E) and r = sqrt( A*A + B*B ) v = atan2( B, A ) The Sun's true longitude, slon, can now be computed: slon = v + w Since the Sun is always at the ecliptic (or at least very very close to it), we can use simplified formulae to convert slon (the Sun's ecliptic longitude) to sRA and sDec (the Sun's RA and Dec): sin(slon) * cos(oblecl) tan(sRA) =  cos(slon) sin(sDec) = sin(oblecl) * sin(slon) As was the case when computing az, the Azimuth, if possible use an atan2() function to compute sRA. REFERENCES  "Practical Astronomy with your Calculator", Peter DuffetSmith, 3rd edition. Cambridge University Press 1988. ISBN 0521356997. A good introduction to basic concepts plus many useful algorithms. The third edition is much better than the two previous editions. This book is also preferable to DuffetSmith's "Practical Astronomy with your Computer", which has degenerated into being filled with Basic program listings. "Astronomical Formulae for Calculators", Jean Meeus, 4th ed, WillmannBell 1988, ISBN 0943396220 "Astronomical Algorithms", Jean Meeus, 1st ed, WillmannBell 1991, ISBN 0943396352 Two standard references for many kinds of astronomical computations. Meeus' is an undisputed authority heremany other authors quote his books. "Astronomical Algorithms" is the more accurate and more modern of the two, and one can also buy a floppy disk containing software implementations (in Basic or C) to that book. User Contributions:Comment about this article, ask questions, or add new information about this topic:Top Document: [sci.astro] Time (Astronomy Frequently Asked Questions) (3/9) Previous Document: C.02 What are all those different kinds of time? Next Document: C.04 What's a Julian date? modified Julian date? 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Last Update March 27 2014 @ 02:11 PM
