Astronomical FAQ

Path: senator-bedfellow.mit.edu!bloom-beacon.mit.edu!newsxfer3.itd.umich.edu!how land.erols.net!cpk-news-hub1.bbnplanet.com!su-news-feed4.bbnplanet.com!news.bbnp lanet.com!enews.sgi.com!news.sgi.com!news.msfc.nasa.gov!bcm.tmc.edu!rice!uw-beav er!cornellcs!newsstand.cit.cornell.edu!ism From: lazio@spacenet.tn.cornell.edu Newsgroups: sci.astro,sci.answers,news.answers Subject: [sci.astro] Time (Astronomy Frequently Asked Questions) (3/8) Followup-To: poster Date: Wed, 06 Aug 97 0:49:59 GMT Organization: Cornell University Lines: 1185 Sender: jl26@cornell.edu (Verified) Approved: news-answers-request@MIT.EDU Expires: Fri, 22 Aug 97 0:49:59 GMT Message-ID: <59490007Aug97_3@ism.tn.cornell.edu> References: <59490007Aug97_0@ism.tn.cornell.edu> NNTP-Posting-Host: ism.tn.cornell.edu Summary: This posting addresses frequently asked questions about time, calendars, and and related terrestrial phenomena. Xref: senator-bedfellow.mit.edu sci.astro:193082 sci.answers:6859 news.answers:1 09292 Last-modified: $Date: 1997/07/07 12:14:45 $ Version: $Revision: 2.10 $ URL: http://astrosun.tn.cornell.edu/students/lazio/sci.astro.html Posting-frequency: semi-monthly (Wednesday) Archive-name: astronomy/faq/part3 ------------------------------ Subject: Introduction sci.astro is a newsgroup devoted to the discussion of the science of astronomy. As such its content ranges from the Earth to the farthest reaches of the Universe. However, certain questions tend to appear fairly regularly. This document attempts to summarize answers to these questions. This document is posted on the first and third Wednesdays of each month to the newsgroup sci.astro. It is also available via anonymous ftp in the directory <URL:ftp://seti.tn.cornell.edu/pub/lazio/> and it is on the World Wide Web at <URL:http://astrosun.tn.cornell.edu/students/lazio/sci.astro.html>. Like many other FAQs it is also available from <URL:ftp://rtfm.mit.edu/pub/usenet/news.answers>, <URL:http://www.cis.ohio-state.edu/hypertext/faq/usenet/top.html>, and their worldwide mirrors. Questions/comments/flames should be directed to the FAQ maintainer, Joseph Lazio (lazio@spacenet.tn.cornell.edu). ------------------------------ Subject: Copyright This document, as a collection, is Copyright 1995, 1996, 1997 by T. Joseph W. Lazio (lazio@spacenet.tn.cornell.edu). The individual articles are copyright by the individual authors listed. All rights are reserved. Permission to use, copy and distribute this unmodified document by any means and for any purpose EXCEPT PROFIT PURPOSES is hereby granted, provided that both the above Copyright notice and this permission notice appear in all copies of the FAQ itself. Reproducing this FAQ by any means, included, but not limited to, printing, copying existing prints, publishing by electronic or other means, implies full agreement to the above non-profit-use clause, unless upon prior written permission of the authors. This FAQ is provided by the authors "as is," with all its faults. Any express or implied warranties, including, but not limited to, any implied warranties of merchantability, accuracy, or fitness for any particular purpose, are disclaimed. If you use the information in this document, in any way, you do so at your own risk. ------------------------------ Subject: C.00 Time, Calendars, and Terrestrial Phenomena [Dates in brackets are last edit.] C.01 When is 02/01/04? or is there a standard way of writing dates? [95-10-04] C.02 What are all those different kinds of time? [96-06-13] C.03 How do I compute astronomical phenomena for my location? [96-01-05] C.04 What's a Julian date? [97-06-04] C.05 Will 2000 be a leap year? [95-07-05] C.06 When will the new millennium start? [95-07-05] C.07 Easter: 07.1 When is Easter? [96-05-01] 07.2 Can I calculate the date of Easter? [96-12-11] C.08 What is a "blue moon?" [97-02-04] C.09 What is the Green Flash (or Green Ray)? [95-07-03] C.10 Why isn't the earliest Sunrise (and latest Sunset) on the longest day of the year? [97-06-11] C.11 How do I calculate the phase of the moon? [96-10-08] C.12 What is the time delivered by a GPS receiver? [95-10-04] C.13 Why are there two tides a day and not just one? [96-02-12] There is also a calendar FAQ maintained by Claus Tondering <c-t@pip.dknet.dk>, <URL:http://www.pip.dknet.dk/~pip10160/calendar.html>. ------------------------------ Subject: C.01 When is 02/01/04? or is there a standard way of writing dates? Author: Markus Kuhn <mskuhn@cip.informatik.uni-erlangen.de> The international standard date notation is: YYYY-MM-DD For example, February 4, 1995 is written as 1995-02-04. This notation is standardized in International Standard ISO 8601. For a detailed description of this standard, please download from <URL:ftp://ftp.uni-erlangen.de/pub/doc/ISO/ISO8601.ps.Z> or visit <URL:http://www.ft.uni-erlangen.de/~mskuhn/iso-time.html>. Other commonly used notations are e.g., 2/4/95, 4/2/95, 4.2.1995, 04-FEB-1995, 4-February-1995, and many more. Especially the first two examples are dangerous, because as both are used quite often and can not be distinguished, it is unclear whether 2/4/95 means 1995-04-02 or 1995-02-04. Advantages of the ISO standard date notation are: - easily parsed by software (no 'JAN', 'FEB', ... table necessary) - easily sortable with a trivial string compare - language independent - can not be confused with other popular date notations - consistent with 24h time notation hh:mm:ss which comes also with the most significant component first and is consequently also easily sortable (e.g., write 1999-12-31 23:59:59). - short and has constant length (makes keyboard data entry easier) - identical to the Chinese date notation, so the largest cultural group (>25%) on this planet is already familiar with it. - 4-digit year representation avoids overflow problems after 1999-12-31. In shell scripts, use date "+%Y-%m-%d %H:%M:%S" in order to print the date and time in ISO format. In C, use the string "%Y-%m-%d %H:%M:%S" as the format specifier for strftime(). On some old systems were the designers did not expect that it will be still operational in the year 2000, you will have to use the format string "19%y-%m-%d %H:%M:%S" instead. The year 2000 might be a good opportunity to change to a common worldwide date notation on 2000-01-01. ------------------------------ Subject: C.02 What are all those different kinds of time? Author: Paul Schlyter <pausch@saaf.se>, Markus Kuhn <mskuhn@cip.informatik.uni-erlangen.de>, Paul Eggert <eggert@twinsun.com> In the beginning there was only solar days: sunset was considered to be the end of the day and the beginning of the next day. The Jewish and Moslem calendars, which nowadays are used only for religious purposes, still start a new date at sunset instead of midnight. Later, the solar days were divided into hours: 12 hours for the day and 12 hours for the night. The different lengths of day/night were ignored, therefore the daylight hours were longer in summer than in winter. APPARENT (or TRUE) SOLAR TIME: Still later, the hours were made equally long: the day+night was 24 hours. The "day" now started at midnight, not at sunset, which was marked as 00:00 (or 12:00 midnight in English time format). Noon was at 12:00 (or 12:00 noon in English time format). This is what we now refer to as "true solar time"---it is the time shown by a properly setup sundial. This time is local, it is different for different longitudes. (In strict English construction, 12:00 cannot be given either an A.M. = ante meridiem or P.M. = post meridiem designation, but it has become common to use 12 A.M. to mean midnight and 12 P.M. to mean noon. In traditional English, 12 M. = meridies means _noon_; nowadays one is just as likely to see 12 M. = midnight and 12 N. = noon.) (In general, the old English A.M./P.M. notation is extremely problematic. A shorter and more obvious time notation is the modern 24h notation in which the hours in the day range from 00:00 to 23:59. This notation even allows one to distinguish midnight at the start of the day [00:00] from midnight at the end of the day [24:00], while the old English notation requires kludges like starting a contract at 12:01 A.M. in order to make clear which of the two midnights associated with a date had been intended. The 24h notation is the official international standard time notation (ISO 8601) and displayed by almost all digital clocks outside the U.S.A. The 24h notation is also recommended by the U.S. Naval Observatory in Washington, which defines official time in the U.S.) MEAN SOLAR TIME: True Solar Time isn't a uniform time. The time difference between one noon and the next noon varies through the year, due to two causes: 1. The earth's orbit is elliptical, not perfectly circular, and the Earth's speed in its orbit is greater when closer to the sun. This makes the solar days shorter in July and longer in January. 2. The Earth's axis of rotation does not point in the same direction as the axis of the Earth's orbit round the Sun. (The angle between these two is called the "obliquity of the ecliptic" and is about 23.45 degrees.) This makes the solar days shorter in March and September and longer in June and December. To account for these effects, a fictitious sun, "The Mean Sun," was invented: it moves with uniform velocity in the plane of the Earth's equator, with the same average speed as the true Sun. This Mean Sun defines Mean Solar Time: When the Mean Sun is due south (for northern hemisphere observers), it is noon Mean Solar Time. Now the time difference between two consecutive local noons is always the same (ignoring small irregularities in the Earth's rotation---more about that later). SIDEREAL TIME: Closely connected with the Mean Solar Time is the Sidereal Time, which is defined as the RA (Right Ascension) of the Local Meridian: when the Vernal Point passes the meridian it is 00:00 Sidereal Time. When Orion is at its maximum altitude, it is between 5h and 6h Sidereal Time; when the Big Dipper can be seen close to the zenith it is about 12h Sidereal Time; and when Sagittarius, with all its glories close to the center of our Galaxy, reaches maximum altitude it is around 18h Sidereal Time. The Sidereal Time at a particular place and location is the same as the local Mean Solar Time, plus 12 hours, plus the Right Ascension of the Mean Sun (which is the same as the Mean Longitude of the true sun). It can be computed from this formula: LST(hours) = 6.6974 + 2400.051336 * T + 24 * FRAC(JD+0.5) + long/15 where: LST = Local Sidereal Time in hours JD = the Julian Day Number for the moment, including fractions of a day Note that a new Julian Day starts at Greenwich Noon T = ( JD - 2451545.0 ) / 36525.0 long = your local longitude: east positive, west negative FRAC = a function discarding the integral part and returning only the fractional part of a real number. STANDARD TIME ZONES: Some 100+ years ago the railway made fast transportation possible for the first time. Quite soon it became awkward for the travellers to continually have to adjust their clocks when travelling between different places, and the railway companies had the problem to select which city's time to use for their own schedules. An interim solution was to use a specific "railway time," but soon standard time zones were created. At first the time to be used within a country was the local time of the capital of the country. A few very large countries employed several time zones. It took a few decades to arrive at a worldwide agreement here, and in particular there was a "battle" between England and France whether the world's prime meridian was to be the meridian of the Greenwich or the Paris observatory. England won this battle, and "Greenwich Mean Time" (GMT) was universally agreed upon as the world's standard time zones. Almost all other parts of the world were assigned time zones, which usually differ from GMT by an integral number of hours. Some countries (e.g., India) use use differences that are not an integral number of hours. GMT (Greenwich Mean Time): This term is a historic term which is in a strict sense obsolete, though often used (although not in astronomy, e.g., BBC still uses this abbreviation for patriotic reasons ;-) as a synonym for UTC. In 1972, an international atomic time scale has been introduced and since then, the time on the zero meridian, which goes through the old observatory in Greenwich, London, UK, has been called Universal Time (UT). Prior to 1925, it was reckoned for astronomical purposes from Greenwich mean noon (12h UT). Sometimes GMT is referred to as Z ("Zulu"). (This arises from the military custom of writing times as hours and minutes run together and suffixed with a single letter designating the time zone: 2100Z = 21:00 UTC. The word "zulu" was the phonetic word associated with the letter "z.") UT (Universal time): Defined by the Earth's rotation and determined by astronomical observations. This time scale is slightly irregular. There are several different definitions of UT, but the difference between them is always less than about 0.03 s. Usually one means UT2 when saying UT. UT2 is UT corrected for pole wandering and seasonal variations in the Earth's rotational speed. If you are interested in time more precisely than 1 s, then you'll have to differentiate between the following versions of Universal Time: UT0 is the precise solar local time on the zero meridian. It is today measured by radio telescopes which observe quasars. UT1 is UT0 corrected by a periodic effect known as Chandler wobble or "polar wandering", i.e., small changes in the longitude/latitude of all places on the Earth due to the fact that the geographical poles of the Earth "wander" in semi-regular patterns: the poles follow (very approximately) small circles, about 10--20 meters in diameter, with a period of approximately 400--500 days. The changes in the longitude/latitude of all places of Earth due to this amounts to fractions of an arc second (1 arc second = 1/3600 degree). UT2 is an even better corrected version of UT0 which accounts for seasonal variations in the Earth's rotation rate and is sometimes used in astronomy. UTC is a time defined not by the movement of the earth, but by a large collection of atomic clocks located all over the world, the atomic time scale TAI. When UTC and UT1 are about to drift apart more than 0.9 s, a leap second will be inserted (or deleted, but this never has happened) into UTC to correct this. When necessary, leap seconds are inserted as the 61th second of the last UTC minute of June or December. During a leap second, a UTC clock (e.g., a GPS receiver) shows: 1995-12-31 23:59:59 1995-12-31 23:59:60 1996-01-01 00:00:00 Today, practically all national civil times are defined relative to UTC and differ from UTC by an integral number of hours (sometimes also half- or quarter-hours). UTC is defined in ITU-R Recommendation TF.460-4 and was introduced in 1972. If you are interested in UTC more precisely than a microsecond, then you also have to consider the following differences: The abbreviation UTC can be followed by an abbreviation of the organization who publishes this time reference signal. For example, UTC(USNO) is the US reference time published by the US Naval Observatory, UTC(PTB) is the official German reference time signal published (via a 77.5 kHz long-wave broadcast) by the Physikalisch Technische Bundesanstalt in Braunschweig and UTC(BIPM) is the most official time published by the Bureau International des Poids et Mesures in Paris, however UTC(BIPM) is only a filtered paper clock published each year that is used by the other time maintainers to resynchronize their clocks against each other. All these UTC versions do not differ by more than a few nanoseconds. The acronym UTC stands for Coordinated Universal Time. In 1970 when this system was being developed by the International Telecommunication Union, it felt it was best to designate a single abbreviation for use in all languages in order to minimize confusion. Unanimous agreement could not be achieved on using either the English word order, CUT, or the French word order, TUC, so a compromise using neither, UTC, was adopted. DUT1 is the difference between UTC and UT1 as published by the US Naval Observatory rounded to 0.1 s each week. This results in the UT1 which is used e.g., for space navigation. ET (Ephemeris Time): Somewhere around 1930--1940, astronomers noticed that errors in celestial positions of planets could be explained by assuming that they were due to slow variations on the Earth's rotation. Starting in 1960, the time scale Ephemeris Time (ET) was introduced for astronomical purposes. ET closely matches UT in the 19th century, but in the 20th century ET and UT have been diverging more and more. Currently ET is running almost precisely one minute ahead of UT. In 1984, ET was replaced by Dynamical Time and TT. For most purposes, ET up to 1983-12-31 and TDT from 1984-01-01 can be regarded as a continuous time-scale. TT and Dynamical Time: Introduced in 1984 as a replacement for ET, it defines a uniform astronomical time scale more accurately, taking relativistic effects into account. There are two kinds of Dynamical Time: TDT (Terrestrial Dynamical Time), which is a time scale tied to the Earth, and TDB (Barycentric Dynamical Time), used as a time reference for the barycenter of the solar system. The difference between TDT and TDB is always smaller than a few milliseconds. When the difference TDT-TDB is not important, TDT is referred to as TT. For most purposes, TDT can be considered equal to TAI + 32.184 seconds. TAI (Temps Atomique International = International Atomic Time): Defined by the same worldwide network of atomic clocks that defines UTC. In contrast to UTC, TAI has no leap seconds. TAI and UTC were identical in the late 1950s. The difference between TAI and UTC is always an integral number of seconds. TAI is the most uniform time scale we currently have available. RELATION BETWEEN THE TIME SCALES -------------------------------- TDT = TAI+32.184s ==> UT-UTC = TAI-UTC - (TDT-UT) + 32.184s Starting at TAI-UTC ET/TDT-UT UT-UTC 1972-01-01 +10.00 +42.23 -0.05 1972-07-01 +11.00 +42.80 +0.38 1973-01-01 +12.00 +43.37 +0.81 1973-07-01 -"- +43.93 +0.25 1974-01-01 +13.00 +44.49 +0.69 1974-07-01 -"- +44.99 +0.19 1975-01-01 +14.00 +45.48 +0.70 1975-07-01 -"- +45.97 +0.21 1976-01-01 +15.00 +46.46 +0.72 1976-07-01 -"- +46.99 +0.19 1977-01-01 +16.00 +47.52 +0.66 1977-07-01 -"- +48.03 +0.15 1978-01-01 +17.00 +48.53 +0.65 1978-07-01 -"- +49.06 +0.12 1979-01-01 +18.00 +49.59 +0.59 1979-07-01 -"- +50.07 +0.11 1980-01-01 +19.00 +50.54 +0.64 1980-07-01 -"- +50.96 +0.22 1981-01-01 -"- +51.38 -0.20 1981-07-01 +20.00 +51.78 +0.40 1982-01-01 -"- +52.17 +0.01 1982-07-01 +21.00 +52.57 +0.61 1983-01-01 -"- +52.96 +0.22 1983-07-01 +22.00 +53.38 +0.80 1984-01-01 -"- +53.79 +0.39 1984-07-01 -"- +54.07 +0.11 1985-01-01 -"- +54.34 -0.16 1985-07-01 +23.00 +54.61 +0.57 1986-01-01 -"- +54.87 +0.31 1986-07-01 -"- +55.10 +0.08 1987-01-01 -"- +55.32 -0.14 1987-07-01 -"- +55.57 -0.39 1988-01-01 +24.00 +55.82 +0.36 1988-07-01 -"- +56.06 +0.12 1989-01-01 -"- +56.30 -0.12 1989-07-01 -"- +56.58 -0.40 1990-01-01 +25.00 +56.86 +0.32 1990-07-01 -"- +57.22 -0.04 1991-01-01 +26.00 +57.57 +0.61 1991-07-01 -"- +57.94 +0.24 1992-01-01 -"- +58.31 -0.13 1992-07-01 +27.00 +58.72 +0.46 1993-01-01 -"- +59.12 +0.06 1993-07-01 +28.00 +59.5 +0.7 1994-01-01 -"- +59.9 +0.3 1994-07-01 +29.00 +60.3 +0.9 1995-01-01 -"- +60.7 +0.5 1995-07-01 -"- +61.1 +0.1 For the latest status concerning leap seconds in UTC, send email to <adsmail@tycho.usno.navy.mil> with a Subject: line of 'leap' and no text. You will receive in reply a list of past and provisional future leap seconds. Additional information about the world time standard UTC (e.g., when will the next leap second be inserted in time, etc.) is available from the US Naval Observatory on the Web <URL:http://tycho.usno.navy.mil/time.html> and from the International Earth Rotation Service (IERS) with anonymous ftp from <URL:ftp://mesiom.obspm.fr/iers/>. Also <URL:http://www.eecis.udel.edu/~ntp/> is a good start if you want to learn more about time standards. ------------------------------ Subject: C.03 How do I compute astronomical phenomena for my location? Author: Paul Schlyter <pausch@saaf.se> 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 (1--12) and Date (1--31), 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 unit---degrees---throughout. 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() function---then 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 lacks 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 value---one 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 library---that 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 above---name 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. 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 - 3.563E-7 * d w = 282.9494 + 4.70935E-5 * d M = 356.0470 + 0.9856002585 * d e = 0.016709 - 1.151E-9 * 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. Semi-major 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 it---no 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 Duffet-Smith, 3rd edition. Cambridge University Press 1988. ISBN 0-521-35699-7. 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 Duffet-Smith's "Practical Astronomy with your Computer", which has degenerated into being filled with Basic program listings. "Astronomical Formulae for Calculators", Jean Meeus, 4th ed, Willmann-Bell 1988, ISBN 0-943396-22-0 "Astronomical Algorithms", Jean Meeus, 1st ed, Willmann-Bell 1991, ISBN 0-943396-35-2 Two standard references for many kinds of astronomical computations. Meeus' is an undisputed authority here---many 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. ------------------------------ Subject: C.04 What's a Julian date? Author: Edward Wright <wright@eggneb.astro.ucla.edu>, William Hamblen <william.hamblen@nashville.com> It's the number of days since noon 4713 BC January 1. What's so special about this date? Joseph Justus Scaliger (1540--1609) was a noted Italian-French philologist and historian who was interested in chronology and reconciling the dates in historical documents. Before the western civil calendar was adopted by most countries, each little city or principality reckoned dates in its own fashion, using descriptions like "the 5th year of the Great Poo-bah Magnaminus." Scaliger wanted to make sense out of these disparate references so he invented his own era and reckoned dates by counting days. He started with 4713 BC January 1 because that was when solar cycle of 28 years (when the days of the week and the days of the month in the Julian calendar coincide again), the Metonic cycle of 19 years (because 19 solar years are roughly equal to 235 lunar months) and the Roman indiction of 15 years (decreed by the Emperor Constantine) all coincide. There was no recorded history as old as 4713 BC known in Scaliger's day, so it had the advantage of avoiding negative dates. Joseph Justus's father was Julius Caesar Scaliger, which might be why he called it the Julian Cycle. Astronomers adopted the Julian cycle to avoid having to remember "30 days hath September ...." For reference, 1995 October 9 was Julian day 2450000. ------------------------------ Subject: C.05 Will 2000 be a leap year? Author: Steve Willner <swillner@cfa.harvard.edu> Yes. Oh, you wanted to know more? The reason for leap days is that the year---the time it takes the Earth to go round the Sun---is not an integral multiple of the day---the time it takes the Earth to rotate once on its axis. In this case, the year of interest is the "tropical year," which controls the seasons. The tropical year is defined as the interval from one spring equinox to the next: very close to 365.2422 days. The Julian calendar, instituted by Julius Caesar (who else?), has a 365-day ordinary year with a 366-day leap year every fourth year. This gives a mean year length of 365.25 years, not a very large error. However, the error builds up, and by the sixteenth century, reform was considered desirable. A new calendar was established in most Roman Catholic countries in 1582 under the authority of Pope Gregory XIII; in that year, the date October 4 was followed by October 15---a correction of 10 days. Most non-Catholic countries adopted this "Gregorian" calendar somewhat later (Great Britain and the American colonies in 1752), and by then the difference between Julian and Gregorian dates was even greater than 10 days. (Russia didn't adopt the Gregorian calendar until after the "October Revolution"---which took place in November under the new calendar!) Many of the calendar changeovers elicited strong emotional reactions from the populations involved; people objected to "losing ten (or more) days of our lives." The rule for leap years under the Gregorian calendar is that all years divisible by four are leap years EXCEPT century years NOT divisible by 400. Thus 1700, 1800, and 1900 were not leap years, while 2000 will be one. This rule gives 97 leap years in 400 years or a mean year length of exactly 365.2425 days. The error in the Gregorian calendar will build up to a full day in roughly 3000 years, by which time another reform will be necessary. Various schemes have been proposed, some taking account of the changing lengths of the day and/or the tropical year, but none has been internationally recognized. Leaving a reform to our descendants seems reasonable, since there is no obvious need to make a correction now. ------------------------------ Subject: C.06 When will the new millennium start? Author: Steve Willner <swillner@cfa.harvard.edu>, Paul Schlyter <pausch@saaf.se> Steve Willner <swillner@cfa.harvard.edu> writes: The big "end of millennium" party will undoubtedly be held on 1999-12-31. The psychological significance of changing the first digit in the year must not be discounted. (The big headache will occur---appropriately enough---the next day, as the date code in everybody's computer breaks.) However, the third millennium A.D. in fact begins on 2001-01-01; there was no year zero, and thus an interval of 2000 years from the arbitrary beginning of "A.D." dates will not have elapsed until then. More details may be found in an article by Ruth Freitag in the 1995 March newsletter of the American Astronomical Society. I am seeking permission to include the article in the FAQ. A view to the contrary is expressed by Paul Schlyter <pausch@saaf.se>: On Jan 1, 2000 of course! Some people argue that it should be Jan 1, 2001 just because Roman Numerals lacks a symbol for zero, but I find that irrelevant, because: 1. Our year count wasn't introduced until A.D. 525---thus the people who lived at A.D. 1 were completely unaware that we label that year "A.D. 1." 2. No real known event occurred at either 1 B.C. or A.D. 1---Christ was born some 6--7 years earlier. Thus the new millennium should _really_ have been celebrated already, at least of we want to celebrate 2000 years since the event that supposedly started our way of counting years.... (Yes, the Julian calendar _was_ around at 1 B.C. and 1 A.D., but at that time the years was counted since the "foundation of Rome") ------------------------------ Subject: C.07 Easter: ------------------------------ Subject: C.07.1 When is Easter? Author: Jim Van Nuland <Jim.Van.Nuland@pctie.microbbs.us.com>, John Harper <John.Harper@vuw.ac.nz> The "popular" rule (for Roman Catholics and most Protestant denominations) is that Easter is on the first Sunday after the first full moon after the March equinox. The actual rule is similar, except that the astronomical equinox is not used; the date is fixed at March 21. And the astronomical full moon is not used; an "ecclesiastical" new moon is determined by adopted tables based on the Metonic cycle, and "full" is taken as the 14th day of that lunation. There are auxiliary rules that make March 22 the earliest possible date for Easter and April 25 the latest. The intent of these rules is that the date will be incontrovertibly fixed and determinable indefinitely in advance. In addition it is independent of longitude or time zones. The popular rule works surprisingly well. When the two rules give different dates, that occurs in only part of the world because two dates separated by the international date line are simultaneously in progress. The Eastern Churches (most Orthodox and some others, e.g., Uniate Churches in Palestine) use the same system, but based on the old (Julian) calendar. In that calendar, Easter Day is also between March 22 and April 25, but in the western (Gregorian) calendar those days are at present April 3 and May 8. Whenever the Gregorian calendar skips a leap year, those dates advance one day. Some Eastern Churches find both movable feasts like Easter and fixed ones like Christmas with the Julian calendar; some use the Julian for movable and the Gregorian for fixed feasts; and the Finnish Orthodox use the Gregorian for all purposes. To explain the Eastern system one must begin with the Jews in Alexandria at the time of the Christian Council of Nicaea in 325, who appear to have been celebrating Passover on the first "full moon" after March 21, as specified by the 19-year Metonic cycle and the Julian calendar (with its leap year every 4 years, end of century or not). The Bishop of Alexandria was made responsible for the Christian calendar; he specified that Easter be the Sunday after that Passover. Eastern Christians still say that Easter must follow Passover, but that Passover is the one that is meant, not the Passover defined by the present Jewish calendar. Subsequently the Jews reformed their calendar (in 358 or in the early 6th century according to different sources; possibly at different times in different places), in order to improve the fit between astronomy and their arithmetic, but the Christians did not follow suit. In 1996, for example, Passover was on April 4 but the Orthodox Easter was on Sunday April 14, not April 7 (which as it happens was the Western Easter.) The Eastern Easter is 0, 1, 4, or 5 weeks after the Western Easter. The Western Easter can precede the (modern) Jewish Passover, as in 1967, 1970, 1978, 1986, 1989 and 1997, and can even coincide with it, as in 1981. Much of this information was taken from the Explanatory Supplement to the Astronomical Ephemeris, page 420, 1974 reprint of the 1961 edition. There is more in the Explanatory Supplement, specifically a series of tables that can be used to determine the Easter date for both the Julian (Eastern and pre-1582 Western) and Gregorian calendars. However, the Explanatory Supplement is misleading on the subject of the Eastern Easters, though its tables are correct. Jean Meeus has published a program to compute Easter in "Astronomical Algorithms," also see below. Simon Kershaw has written one in C, available at <URL:http://www.ely.anglican.org/cgi-bin/easter>. The most easily available published source for what the Jews and Christians were doing in ancient Alexandria appears to be Otto Neugebauer's "Ethiopic Easter Computus" in his _Astronomy and History Selected Essays_, Springer, New York, 1983, pp. 523--538. John Harper acknowledges the help of Archimandrite Kyril Jenner, Simon Kershaw, and Dr. Brian Stewart concerning Eastern Easters. ------------------------------ Subject: C.07.2 Can I calculate the date of Easter? Author: Bill Jefferys <bill@clyde.as.utexas.edu> John Horton Conway (the Princeton mathematician who is responsible for "the Game of Life") wrote a book with Guy and Berlekamp, _Winning Ways_, that describes in Volume 2 a number of useful calendrical rules, including How to Calculate the Day of the Week, Given The Date, and Easter. Here's a brief precis of how to calculate Easter: G(the Golden Number) = Year_{mod 19} + 1 (never forget to add the 1!) C(the Century term) = +3 for all Julian years (i.e., if using the Julian Calendar) -4 for 15xx, 16xx } -5 for 17xx, 18xx } Gregorian -6 for 19xx, 20xx, 21xx } The general formula for C in a Gregorian year Hxx is C = -H + [H/4] + [8*(H+11)/25] (brackets [] mean integer part) 1) The Paschal Full Moon is given by the formula (Apr 19 = Mar 50) - (11*G+C)_{mod 30} Except when the formula gives Apr 19 you should take Apr 18, and when it gives Apr 18 and G>=12 you should take Apr 17. Easter is then the following Sunday, since Easter always falls on the next Sunday that is _strictly later_ than the Paschal Full Moon. Example: 1945 = 7 mod 19, so G = 8 and we find for the Paschal Full Moon Mar 50 - (88-6)_{mod 30} = Mar 50 - 22 = Mar 28. This happens to be a Wednesday (by Horton's "Doomsday" rule for Day of the Week, see below). Therefore, Easter 1945 took place on Sunday, April 1. Conway's "Doomsday" method for finding the day of the week, given the date, is needed for his Easter method. To every year there is a distinguished day of the week, which Conway calls the "Doomsday", D. In any year, if March 0 (the last day of February) falls on a particular DOW, then the following dates also fall on the same DOW: 4/4, 6/6, 8/8, 10/10, 12/12. Also 5/9, 9/5, 7/11, 11/7 (for which he has devised the mnemonic "I went to my nine-to-five job at the Seven-Eleven. Note to non-US readers: "Seven-Eleven" is the name of a ubiquitous chain of convenience stores.) In non-leap years, Jan 3 and Feb 0 (Jan 31) also fall on that DOW; in leap years, Jan 4 and Feb 1. Conway calls this DOW the "doomsday" for that year. For example, in 1995 Doomsday is Tuesday. Columbus Day (10/12) is two days after 10/10, a Tuesday, so 10/12 is a Thursday. All that remains is a rule for calculating the Doomsday for any year. In any century, this is done by taking the last two digits of the year, call them xx, dividing by 12 to get a quotient Q and remainder R. Divide R by 4 to get a second quotient Q2. Then this century, the Doomsday for that year is given by Wednesday + Q + R + Q2. In 1995, for example, we have 95/12 = 7 with remainder 11; 11/4 gives quotient 2; Wednesday + 7 + 11 + 2 = Tuesday (cf. above). In other years on the Gregorian calendar, one uses instead of Wednesday, the century day as follows: 16xx and 20xx: Tuesday; 17xx and 21xx: Sunday; 18xx and 22xx: Friday; 15xx, 19xx and 23xx: Wednesday. The cycle repeats over a 4 century period. If you need the DOW on the Julian calendar, the rules are the same except that the century rule is different: for a date in the year ccxx, use -cc for the century day of week, where Sunday = 0. For example, October 4, 1582 (the last day of the Julian calendar in countries that followed Pope Gregory's institution of the Gregorian calendar) took place as follows: 82/12 = 6 remainder 10; 10/4 gives remainder 2; 6+10+2-15= 3, which is Wednesday. 10/10 was Wednesday, 10/3 was Wednesday, so 10/4/1582 (Julian) was a Thursday. The following day was October 15, 1582 (Gregorian). Again we can check: 6+10+2+Wed = Sunday. 10/10 was a Sunday (Gregorian) so 10/15/1582 (Gregorian) was a Friday. The nice thing about these algorithms is that they can easily be done in one's head with a little practice (OK, mod 19 for the Golden Number is a bit hairy for me, but I can still do it!). The DOW calculation is very useful if you are caught without a calendar, and it makes a good party trick. Additional information is available at <URL:http://quasar.as.utexas.edu/BillInfo/doomsday.html> and <URL:http://quasar.as.utexas.edu/BillInfo/ReligiousCalendars.html>. ------------------------------ Subject: C.08 What is a "blue moon?" Author: Steve Willner <swillner@cfa.harvard.edu> The alt.english.usage FAQ discusses six different meanings of the term "blue moon." The two definitions most relevant to astronomy are the following: 1. Under certain conditions of atmospheric haze, the moon may actually look blue. A notable example occurred after the explosion of the volcano Krakatoa. The appropriate conditions seem to be extremely rare. 2. The second full moon in a calendar month. Since the synodical month is 29.53 days, this kind of blue moon occurs roughly once out of 60 30-day months and once out of 21 31-day months or about once in 2.5 years on average. It can occur in January and the following March if there is no full moon at all in February. There are some indications that calendars used to put the first moon in the month in red, the second in blue, hence the origin of the term. By extension---probably from the second of these definitions---"a blue moon" is used colloquially to mean "a very long time." The alt.english.usage FAQ is available from <URL: ftp://rtfm.mit.edu/ pub/usenet-by-group/alt.usage.english/alt.usage.english_FAQ> or <URL: http://www.cis.ohio-state.edu/ hypertext/faq/usenet/alt-usage-english-faq/faq.html>. ------------------------------ Subject: C.09 What is the Green Flash (or Green Ray)? Author: Steve Willner <swillner@cfa.harvard.edu>, Geoffrey A. Landis <geoffrey.landis@lerc.nasa.gov> When the sun sets, sometimes the last bit of light from the disk itself is an emerald green. The same is true of the first bit of light from the rising sun. This phenomenon is known as the "green flash" or "green ray." It is not an optical illusion. The green flash is common and will be visible any time the sun is rises or sets on a *clear*, *unobstructed*, and *low* horizon. From our observatory at Mt. Hopkins, I (SW) see the sunset green flash probably 90% of the evenings that have no visible clouds on the western horizon. It typically lasts one or two seconds (by estimate, not stopwatch) but on rare occasions much longer (5 seconds??). I've seen the dawn green flash only once, but a) I'm seldom outside looking, b) the topography is much less favorable, and c) it takes luck to be looking in exactly the right place. If you'd like to see the green flash, the higher you can go, the better (see below). The explanation for the green flash involves refraction, scattering, and absorption. First, the most important of these processes, refraction: light is bent in the atmosphere with the net effect that the visible image of the sun at the horizon appears roughly a solar diameter *above* the geometric position of the sun. This refraction is mildly wavelength dependent with blue light being refracted the most. Thus if refraction were the only effect, the red image of the sun would be lowest in the sky, followed by yellow, green, and blue highest. If I've understood the refraction table properly, the difference between red and blue (at the horizon) is about 1/40 of a solar diameter. Now scattering: the blue light is Rayleigh scattered away (not Compton or Thomson scattering). Now absorption: air has a very weak absorption band in the yellow. When the sun is overhead, this absorption hardly matters, but near the horizon, the light travels through something like 38 "air masses," so even a weak absorption becomes significant. The explanation for the green flash is thus, 1) refraction separates the solar images by color; 2) at just the right instant, the red image has set, 3) the yellow image is absorbed; and 4) the blue image is scattered away. We are left with the upper limb of the green image. Because the green flash is primarily a refraction effect, it lasts longer and is easier to see from a mountain top than from sea level. The amount of refraction is proportional to the path length through the atmosphere times the density gradient (in a linear approximation for the atmosphere's index of refraction). This product will scale like 1+(h/a)^(0.5), where h is your height and a the scale height of the atmosphere. The density scale height averaged over the bottom 10 km of the atmosphere is about 9.2 km, so for a 2 km mountain the increase in refraction is about a factor 1.5; a 3 km mountain gives 1.6 and a 4.2 km mountain (e.g., Mauna Kea) gives 1.7. More details can be found in _The Green Flash and Other Low Sun Phenomena_, by D. J. K. O'Connell and the classic _Light and Color in the Open Air._ A refraction table appears in _Astrophysical Quantities_, by C. W. Allen. ------------------------------ Subject: C.10 Why isn't the earliest Sunrise (and latest Sunset) on the longest day of the year? Author: Steve Willner <willner@cfa183.harvard.edu> This phenomenon is called the "equation of time." This is just a fancy name for the fact that the Sun's speed along the Earth's equator is not constant. In other words, if you were to measure the Sun's position at exactly noon every day, you would see not only the familiar north-south change that goes with the seasons but also an east-west change in the Sun's position. A graphical representation of both positional changes is the analemma, that funny figure 8 that most globes stick in the middle of the Pacific ocean. The short explanation of the equation of time is that it has two causes. The slightly larger effect comes from the obliquity of the ecliptic---the Earth's equator is tilted with respect to the orbital plane. Constant speed along the ecliptic---which is how the "mean sun" moves---translates to varying speed in right ascension (along the equator). This gives the overall figure 8 shape of the analemma. Almost as large is the fact that the Earth's orbit is not circular, and the Sun's angular speed along the ecliptic is therefore not constant. This gives the inequality between the two lobes of the figure 8. The long answer is provided by Nick Strobel at <URL: http://bizweb.lightspeed.net/~astronomy/naked-eye/naked-eye.html#A1.2.6.4>, but you'll probably want to start at <URL: http://bizweb.lightspeed.net/~astronomy/naked-eye/naked-eye.html#A1.2.6>. The Earth's analemma will change with time as the Earth's orbital parameters change. This is described by Bernard Oliver (1972 July, _Sky and Telescope_, pp. 20--22) An article by David Harvey (1982 March, _Sky and Telescope_, pp. 237--239) shows the analemmas of all nine planets. ------------------------------ Subject: C.11 How do I calculate the phase of the moon? Author: Bill Jefferys <bill@clyde.as.utexas.edu> John Horton Conway (the Princeton mathematician who is responsible for "the Game of Life") wrote a book with Guy and Berlekamp, _Winning Ways_, that describes in Volume 2 a number of useful calendrical rules. One of these is an easy "in your head" algorithm for calculating the phase of the Moon, good to a day or better depending on whether you use his refinements or not. In the 20th century, calculate the remainder upon dividing the last two digits of the year by 19; if greater than 9, subtract 19 from this to get a number between -9 and 9. Multiply the result by 11 and reduce modulo 30 to obtain a number between -29 and +29. Add the day of the month and the number of the month (except for Jan and Feb use 3 and 4 for the month number instead of 1 and 2). Subtract 4. Reduce modulo 30 to get a number between 0 and 29. This is the age of the Moon. Example: What was the phase of the Moon on D-Day (June 6, 1944)? Answer: 44/19=2 remainder 6. 6*11=66, reduce modulo 30 to get 6. Add 6+6 to this and subtract 4: 6+6+6-4=14; the Moon was (nearly) full. I understand that the planners of D-day did care about the phase of the Moon, either because of illumination or because of tides. I think that Don Olsen recently discussed this in _Sky and Telescope_ (within the past several years). In the 21st century use -8.3 days instead of -4 for the last number. Conway also gives refinements for the leap year cycle and also for the slight variations in the lengths of months; what I have given should be good to +/- a day or so. ------------------------------ Subject: C.12 What is the time delivered by a GPS receiver? Author: Markus Kuhn <mskuhn@unrza3.dialin.rrze.uni-erlangen.de> Navstar GPS (global positioning system) is a satellite based navigation system operated by the US Air Force. The signals broadcasted by GPS satellites, contain all information required by a GPS receiver in order to determine both UTC and TIA highly accurately. Commercial GPS receivers can provide a time reference that is closer than 340 ns to UTC(USNO) in 90% of all measurements, classified military versions are even better. ------------------------------ Subject: C.13 Why are there two tides a day and not just one? Author: Joseph Lazio <lazio@spacenet.tn.cornell.edu>, Paul Zander <paulz@sc.hp.com> An easy way to think of the Moon's effect on the Earth is the following. The Moon exerts a gravitational force on the Earth. The strength of the gravitational force decreases with increasing distance. So, because the surface of the ocean is closer to the Moon than the sea floor, the surface water is attracted more strongly to the Moon. That's the tide that occurs (nearly) under the Moon. What's happening on the other side of the Earth? On the other side of the Earth from the Moon, the sea floor is being pulled more strongly toward the Moon than the surface water. In essence, the surface water is being left behind. Voila, another bulge in the surface water and another tide. In principle, there should be two tides of equal height in a day. In practice, many parts of the earth do not experience two tides of equal height in a day. First, because the Moon's orbit is at an angle to the Earth's equator, one tidal bulge may be in the northern hemisphere, while the other is in the southern hemisphere. Except around Antarctica, the shape of the Earth's continents prevent the tidal bulges from simply following the moon. Each ocean basin has its own individual pattern for the tidal flow. In the South Atlantic Ocean, the tides travel from south to north, taking about 12 hours to go from the tip of Africa to the equator. In the North Atlantic, the tides travel in a counter-clockwise direction going around once in about 12 hours. The effect is similar to water sloshing around in a bowl. Because the two tides are roughly equal, they are called semidaily or semidiurnal. In some parts of the Gulf of Mexico, there is only one high tide and one low tide a day. These are called daily or diurnal tides. In much of the Pacific Ocean, there are two high tides and two low tides each day, but they are of unequal height. These are called mixed tides. The traditional way to predict tides has been to collect data for several years to have enough combinations of positions of the moon and sun to allow accurate extrapolation. More recently, computer models have been made taking into account detailed shapes of the ocean bottoms and coastlines. Even the best predictions can have difficulties. The extremely heavy snow fall during the winter of 1994--95 in California and the associated run-off as it melted were not part of the model for San Francisco Bay. Sail boat races scheduled to take advantage of tidal currents coming into the Golden Gate found the current was still going out! Ref: Oceanography, A View of the Earth, M. Grant Gross, Prentice Hall, Englewood Cliffs, New Jersey, 1972. For even more details, see <URL:ftp://d11t.geo.tudelft.nl/pub/ejo/tides>.