[sci.astro] Time (Astronomy Frequently Asked Questions) (3/9)@import 'http://faqs.org/abstracts/css/default.css';@import 'http://faqs.org/search.css';[ Usenet FAQs | Search | Web FAQs | Documents | RFC Index ] Search the FAQ Archives Part0 - Part1 - Part2 - Part3 - Part4 - Part5 - Part6 - Part7 - Part8 - MultiPage[sci.astro] Time (Astronomy Frequently Asked Questions) (3/9)There are reader questions on this topic!Helpothers by sharing your knowledgeFrom: jlazio@patriot.netNewsgroups: sci.astroSubject: [sci.astro] Time (Astronomy Frequently Asked Questions) (3/9)Date: 15 Aug 2007 16:37:00 -0400Message-ID: <ypzwsvwr0ur.fsf@adams.patriot.net>Sender: jlazio@adams.patriot.netSummary: This posting addresses frequently asked questions about time, calendars, and and related terrestrial phenomena.User-Agent: Gnus/5.09 (Gnus v5.9.0) Emacs/21.4Last-modified: $Date: 2003/07/16 00:00:01 $Version: $Revision: 4.5 $URL: http://sciastro.astronomy.net/Posting-frequency: semi-monthly (Wednesday)Archive-name: astronomy/faq/part3Subject: Introduction sci.astro is a newsgroup devoted to the discussion of the science ofastronomy. As such its content ranges from the Earth to the farthestreaches of the Universe. However, certain questions tend to appear fairly regularly. Thisdocument attempts to summarize answers to these questions. This document is posted on the first and third Wednesdays of eachmonth to the newsgroup sci.astro. It is available via anonymous ftpfrom <URL:ftp://rtfm.mit.edu/pub/usenet/news.answers/astronomy/faq/>,and it is on the World Wide Web at<URL:http://sciastro.astronomy.net/> and<URL:http://www.faqs.org/faqs/astronomy/faq/>. A partial list ofworldwide mirrors (both ftp and Web) is maintained at<URL:http://sciastro.astronomy.net/mirrors.html>. (As a general note,many other FAQs are also available from<URL:ftp://rtfm.mit.edu/pub/usenet/news.answers/>.)Questions/comments/flames should be directed to the FAQ maintainer,Joseph Lazio (jlazio@patriot.net).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? [2001-12-14] C.02 What are all those different kinds of time? [2002-05-07] C.03 How do I compute astronomical phenomena for my location? [2002-05-04] C.04 What's a Julian date? modified Julian date? [1998-05-06] C.05 Was 2000 a leap year? [2000-03-17] C.06 When will the new millennium start? [2001-01-01] C.07 Easter: 07.1 When is Easter? [1996-05-01] 07.2 Can I calculate the date of Easter? [1996-12-11] C.08 What is a "blue moon?" [2001-10-02] C.09 What is the Green Flash (or Green Ray)? [1999-01-01] C.10 Why isn't the earliest Sunrise (and latest Sunset) on the longest day of the year? [2002-01-30] C.11 How do I calculate the phase of the moon? [1996-10-08] C.12 What is the time delivered by a GPS receiver? [2002-05-07] C.13 Why are there two tides a day and not just one? [1999-12-15]There is also a calendar FAQ maintained by Claus Tondering<c-t@pip.dknet.dk>,<URL:http://www.tondering.dk/claus/calendar.html>.Subject: C.01 When is 02/01/04? or is there a standard way of writing dates?Author: Markus Kuhn <Markus.Kuhn@cl.cam.ac.uk>The international standard date notation is: YYYY-MM-DDFor example, February 4, 1995 is written as 1995-02-04. This notationis standardized in International Standard ISO 8601. For more detailsregarding this standard, please<URL:http://www.cl.cam.ac.uk/~mgk25/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 twoexamples are dangerous, because as both are used quite often and cannot be distinguished, it is unclear whether 2/4/95 means 1995-04-02 or1995-02-04.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 thestring "%Y-%m-%d %H:%M:%S" as the format specifier for strftime().Other useful information on the ISO standard is at <URL:http://dmoz.org/Science/Reference/Standards/Individual_Standards/ISO_8601/>.Subject: C.02 What are all those different kinds of time?Author: Paul Schlyter <pausch@saaf.se>, Markus Kuhn <Markus.Kuhn@cl.cam.ac.uk>, Paul Eggert <eggert@twinsun.com>In the beginning there were only solar days: sunset was considered tobe the end of the day and the beginning of the next day. The Jewishand Moslem calendars, which nowadays are used only for religiouspurposes, still start a new date at sunset instead of midnight.Later, the solar days were divided into hours: 12 hours for the day and12 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 madeequally long: the day+night was 24 hours. The "day" now started atmidnight, not at sunset, which was marked as 00:00 (or 12:00 midnightin English time format). Noon was at 12:00 (or 12:00 noon in Englishtime format). This is what we now refer to as "true solar time"---itis the time shown by a properly setup sundial. This time is local, itis different for different longitudes. (In strict Englishconstruction, 12:00 cannot be given either an A.M. = ante meridiem orP.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 traditionalEnglish, 12 M. = meridies means _noon_; nowadays one is just as likelyto see 12 M. = midnight and 12 N. = noon.)(In general, the old English A.M./P.M. notation is extremelyproblematic. A shorter and more obvious time notation is the modern24h 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 ofthe day [00:00] from midnight at the end of the day [24:00], while theold English notation requires kludges like starting a contract at12:01 A.M. in order to make clear which of the two midnightsassociated with a date had been intended. The 24h notation is theofficial international standard time notation (ISO 8601) and displayedby almost all digital clocks outside the U.S.A. The 24h notation isalso recommended by the U.S. Naval Observatory in Washington, whichdefines official time in the U.S.)MEAN SOLAR TIME: True Solar Time isn't a uniform time. The timedifference between one noon and the next noon varies through the year,due to two causes: 1. The earth's orbit is elliptical, not perfectlycircular, and the Earth's speed in its orbit is greater when closer tothe sun. This makes the solar days shorter in July and longer inJanuary. 2. The Earth's axis of rotation does not point in the samedirection as the axis of the Earth's orbit round the Sun. (The anglebetween these two is called the "obliquity of the ecliptic" and isabout 23.45 degrees.) This makes the solar days shorter in March andSeptember and longer in June and December. To account for theseeffects, a fictitious sun, "The Mean Sun," was invented: it moves withuniform velocity in the plane of the Earth's equator, with the sameaverage speed as the true Sun. This Mean Sun defines Mean Solar Time:When the Mean Sun is due south (for northern hemisphere observers), itis noon Mean Solar Time. Now the time difference between twoconsecutive local noons is always the same (ignoring smallirregularities in the Earth's rotation---more about that later).SIDEREAL TIME: Closely connected with the Mean Solar Time is theSidereal Time, which is defined as the RA (Right Ascension) of theLocal Meridian: when the Vernal Point passes the meridian it is 00:00Sidereal Time. When Orion is at its maximum altitude, it is between5h and 6h Sidereal Time; when the Big Dipper can be seen close to thezenith it is about 12h Sidereal Time; and when Sagittarius, with allits glories close to the center of our Galaxy, reaches maximumaltitude it is around 18h Sidereal Time. The Sidereal Time at aparticular place and location is the same as the local Mean SolarTime, plus 12 hours, plus the Right Ascension of the Mean Sun (whichis the same as the Mean Longitude of the true sun). It can becomputed from this formula: LST(hours) = 6.6974 + 2400.051336 * T + 24 * FRAC(JD+0.5) + long/15where: 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 fasttransportation possible for the first time. Quite soon it becameawkward for the travellers to continually have to adjust their clockswhen travelling between different places, and the railway companieshad the problem to select which city's time to use for their ownschedules. An interim solution was to use a specific "railway time,"but soon standard time zones were created. At first the time to beused within a country was the local time of the capital of thecountry. A few very large countries employed several time zones. Ittook a few decades to arrive at a worldwide agreement here, and inparticular there was a "battle" between England and France whether theworld's prime meridian was to be the meridian of the Greenwich or theParis 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, whichusually differ from GMT by an integral number of hours. Somecountries (e.g., India) use differences that are not an integralnumber of hours.GMT (Greenwich Mean Time): This term is a historic term which is in astrict sense obsolete, though often used (although not in astronomy,e.g., BBC still uses this abbreviation for patriotic reasons ;-) as asynonym for UTC. In 1972, an international atomic time scale has beenintroduced and since then, the time on the zero meridian, which goesthrough the old observatory in Greenwich, London, UK, has been calledUniversal Time (UT). Prior to 1925, it was reckoned for astronomicalpurposes from Greenwich mean noon (12h UT). Sometimes GMT is referredto as Z ("Zulu"). (This arises from the military custom of writingtimes as hours and minutes run together and suffixed with a singleletter designating the time zone: 2100Z = 21:00 UTC. The word "zulu"is the phonetic word associated with the letter "z.")UT (Universal time): Defined by the Earth's rotation and determined byastronomical observations. This time scale is slightly irregular.There are several different definitions of UT, but the differencebetween them is always less than about 0.03 s. Usually one means UT2when saying UT. UT2 is UT corrected for pole wandering and seasonalvariations in the Earth's rotational speed.If you are interested in time more precisely than 1 s, then you'llhave to differentiate between the following versions of UniversalTime: 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 radio clock such as MSF, HBG, or DCF77) 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 noticedthat errors in celestial positions of planets could be explained byassuming that they were due to slow variations on the Earth'srotation. Starting in 1960, the time scale Ephemeris Time (ET) wasintroduced for astronomical purposes. ET closely matches UT in the19th century, but in the 20th century ET and UT have been divergingmore and more. Currently ET is running almost precisely one minuteahead of UT. In 1984, ET was replaced by Dynamical Time and TT. Formost purposes, ET up to 1983-12-31 and TDT from 1984-01-01 can beregarded as a continuous time-scale.TT and Dynamical Time: Introduced in 1984 as a replacement for ET, itdefines a uniform astronomical time scale more accurately, takingrelativistic effects into account. There are two kinds of DynamicalTime: TDT (Terrestrial Dynamical Time), which is a time scale tied to theEarth, and TDB (Barycentric Dynamical Time), used as a time referencefor the barycenter of the solar system. The difference between TDT andTDB is always smaller than a few milliseconds. When the differenceTDT-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 definesUTC. In contrast to UTC, TAI has no leap seconds. TAI and UTC wereidentical in the late 1950s. The difference between TAI and UTC isalways an integral number of seconds. TAI is the most uniform timescale 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 1996-01-01 +30.00 +61.63 +0.55 1996-07-01 -"- +62.0 +0.2 1997-01-01 -"- +62.4 -0.2 1997-07-01 +31.00 +62.8 +0.4 1998-01-01 -"- +63.3 -0.1 1998-07-01 -"- +63.7 -0.5 1999-01-01 +32.00 +64.1 +0.1Additional information about the world time standard UTC (e.g., whenwill the next leap second be inserted in time) is available from theUS Naval Observatory and the International Earth Rotation Service(IERS):<URL:http://tycho.usno.navy.mil/time.html><URL:http://tycho.usno.navy.mil/gps_datafiles.html><URL:http://maia.usno.navy.mil/><URL:ftp://maia.usno.navy.mil/ser7/tai-utc.dat><URL:ftp://tycho.usno.navy.mil/pub/series/ser14.txt><URL:ftp://maia.usno.navy.mil/ser7/deltat.preds> <URL:ftp://mesiom.obspm.fr/iers/>. <URL:ftp://hpiers.obspm.fr/iers/bul/bulc/BULLETINC.GUIDE> Also <URL:http://www.eecis.udel.edu/~ntp/> is a good start if you wantto 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 mustcompute the Local Sidereal Time of the place and time in question.First convert your local time to UT (Universal Time), with the dateadjusted if needed. Now suppose that the time is Y,M,D,UT where Y,M,Dis the calendar Year, Month (1--12) and Date (1--31), and UT is theUniversal Time in hours+fractions. Also suppose your position islat,long, where lat is counted as + if north and - if south, and longis counted as + if east and - if west. Now, first compute a "daynumber", d: 7*(Y + INT((M+9)/12))d = 367*Y - INT(---------------------) + INT(275*M/9) + D - 730530 + UT/24 4where INT is a function that discards the fractional part and returns theinteger part of a function. d is zero at 2000 Jan 0.0Now 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) andDec (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 - RANow 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 prefer0 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 yourcomputer, compute the numerator and denominator separately and feedthem both to your atan2() function, instead of dividing and feedingthem to your atan() function---then you'll get the correct quadrantimmediately. 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 theirown. In Pascal it would be (pi is assumed to have been assigned anappropriate 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.. *) endAnother trick I also use is to add a set of trig functions that workin degrees instead of radians to my function library---that will makelife a lot easier when you're working in degrees as the basic unit. Iname them sind, cosd, atan2d, etc. If you don't do that, you'll haveto convert between degrees and radians when calling the standard trigfunctions. COMPUTING RISE AND SET TIMES ----------------------------To compute when an object rises or sets, you must compute when itpasses the meridian and the HA of rise/set. Then the rise time isthe meridian time minus HA for rise/set, and the set time is themeridian time plus the HA for rise/set.To find the meridian time, compute the Local Sidereal Time at 0h localtime (or 0h UT if you prefer to work in UT) as outlined above---namethat quantity LST0. The Meridian Time, MT, will now be: MT = RA - LST0where "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 fromsidereal to solar time. Now, compute HA for rise/set, name thatquantity 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 purelymathematical 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 = -35/60degrees (-35 arc minutes), and if you want to compute the rise/set timesfor 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 multiplyby 365.2422/366.2422 for stellar objects, to convert from sidereal tosolar time. Finally compute: Rise time = MT - HA0 Set time = MT + HA0convert the times from degrees to hours by dividing by 15.If you'd like to check that your calculations are accurate or justneed 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 tocompute the Sun's position at any time. First compute the "daynumber" d as outlined above, for the desired moment. Next compute: oblecl = 23.4393 - 3.563E-7 * d w = 282.9404 + 4.70935E-5 * d M = 356.0470 + 0.9856002585 * d e = 0.016709 - 1.151E-9 * dThis is the obliquity of the ecliptic, plus some of the elements ofthe 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 strictlytrue, this is still an accurate approximation). Next compute E, theeccentric 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 areneeded because we know e has a sufficiently small value. Next computethe 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 + wSince the Sun is always at the ecliptic (or at least very very close toit), we can use simplified formulae to convert slon (the Sun's eclipticlongitude) 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 anatan2() function to compute sRA. REFERENCES ----------"Practical Astronomy with your Calculator", Peter Duffet-Smith, 3rdedition. 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. Thisbook is also preferable to Duffet-Smith's "Practical Astronomy withyour Computer", which has degenerated into being filled with Basicprogram 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 0-943396-35-2Two standard references for many kinds of astronomical computations.Meeus' is an undisputed authority here---many other authors quote hisbooks. "Astronomical Algorithms" is the more accurate and more modernof the two, and one can also buy a floppy disk containing softwareimplementations (in Basic or C) to that book.Subject: C.04 What's a Julian date? modified Julian date?Author: Edward Wright <wright@eggneb.astro.ucla.edu>, William Hamblen <william.hamblen@nashville.com>It's the number of days since noon GMT 4713 BC January 1. What's sospecial about this date?Joseph Justus Scaliger (1540--1609) was a noted Italian-Frenchphilologist and historian who was interested in chronology andreconciling the dates in historical documents. Before the westerncivil calendar was adopted by most countries, each little city orprincipality reckoned dates in its own fashion, using descriptionslike "the 5th year of the Great Poo-bah Magnaminus." Scaliger wantedto make sense out of these disparate references so he invented his ownera and reckoned dates by counting days. He started with 4713 BCJanuary 1 because that was when solar cycle of 28 years (when the daysof the week and the days of the month in the Julian calendar coincideagain), the Metonic cycle of 19 years (because 19 solar years areroughly equal to 235 lunar months) and the Roman indiction of 15 years(decreed by the Emperor Constantine) all coincide. There was norecorded history as old as 4713 BC known in Scaliger's day, so it hadthe advantage of avoiding negative dates. Joseph Justus's father wasJulius Caesar Scaliger, which might be why he called it the JulianCycle. Astronomers adopted the Julian cycle to avoid having toremember "30 days hath September ...."For reference, Julian day 2450000 began at noon on 1995 October 9.Because Julian dates are so large, astronomers often make use of a"modified Julian date"; MJD = JD - 2400000.5. (Though, sometimes they're sloppy and subtract 2400000 instead.)Subject: C.05 Was 2000 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 theEarth to go round the Sun---is not an integral multiple of theday---the time it takes the Earth to rotate once on its axis. In thiscase, the year of interest is the "tropical year," which controls theseasons. The tropical year is defined as the interval from one springequinox to the next: very close to 365.2422 days.The Julian calendar, instituted by the Roman Emperor Julius Caesar(who else? :), has a 365-day ordinary year with a 366-day leap yearevery fourth year. This gives a mean year length of 365.25 years, nota very large error. However, the error builds up, and by thesixteenth century, reform was considered desirable. A new calendarwas established in most Roman Catholic countries in 1582 under theauthority of Pope Gregory XIII; in that year, the date October 4 wasfollowed by October 15---a correction of 10 days. Most non-Catholiccountries adopted this "Gregorian" calendar somewhat later (GreatBritain and the American colonies in 1752), and by then the differencebetween Julian and Gregorian dates was even greater than 10 days.(Russia didn't adopt the Gregorian calendar until after the "OctoberRevolution"---which took place in November under the new calendar!)Many of the calendar changeovers elicited strong emotional reactionsfrom the populations involved; people objected to "losing ten (ormore) days of our lives."The rule for leap years under the Gregorian calendar is that all yearsdivisible by four are leap years EXCEPT century years NOT divisible by400. Thus 1700, 1800, and 1900 were not leap years, while 2000 will beone. This rule gives 97 leap years in 400 years or a mean year lengthof exactly 365.2425 days.The error in the Gregorian calendar will build up to a full day inroughly 3000 years, by which time another reform will be necessary.Various schemes have been proposed, some taking account of the changinglengths of the day and/or the tropical year, but none has beeninternationally recognized. Leaving a reform to our descendants seemsreasonable, 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>There is a difference of opinion. Steve Willner writes:Big "end of millennium" parties were held on 1999-12-31. Thepsychological significance of changing the first digit in the yearmust not be discounted. (Preceeding these parties were the bigheadaches that occurred as everybody rushed to ensure---appropriatelyenough---that the date code in everybody's computer did not break onthe next day.) However, the third millennium A.D. in fact begins on2001-01-01; there was no year zero, and thus an interval of 2000 yearsfrom the arbitrary beginning of "A.D." dates will not have elapseduntil then.More details may be found in an article by Ruth Freitag in the 1995March newsletter of the American Astronomical Society. I am seekingpermission to include the article in the FAQ.A view to the contrary is expressed by Paul Schlyter <pausch@saaf.se>:On 2000 January 1 of course! Some people argue that it should be 2001January 1 just because Roman Numerals lacks a symbol for zero, but Ifind 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---Jesus 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 thattime the years was counted since the "foundation of Rome.")Interested readers may also want to check the Web sites of The RoyalObservatory Greenwich <URL:http://www.rog.nmm.ac.uk/> and the US NavalObservatory <URL:http://www.usno.navy.mil/>.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 Protestantdenominations) is that Easter is on the first Sunday after the firstfull moon after the March equinox.The actual rule is similar, except that the astronomical equinox isnot used; the date is fixed at March 21. And the astronomical fullmoon is not used; an "ecclesiastical" new moon is determined byadopted tables based on the Metonic cycle, and "full" is taken as the14th day of that lunation. There are auxiliary rules that make March22 the earliest possible date for Easter and April 25 the latest. Theintent of these rules is that the date will be incontrovertibly fixedand determinable indefinitely in advance. In addition it isindependent of longitude or time zones.The popular rule works surprisingly well. When the two rules givedifferent dates, that occurs in only part of the world because two datesseparated by the international date line are simultaneously in progress.The Eastern Churches (most Orthodox and some others, e.g., UniateChurches in Palestine) use the same system, but based on the old(Julian) calendar. In that calendar, Easter Day is also between March22 and April 25, but in the western (Gregorian) calendar those daysare at present April 3 and May 8. Whenever the Gregorian calendarskips a leap year, those dates advance one day.Some Eastern Churches find both movable feasts like Easter and fixedones like Christmas with the Julian calendar; some use the Julian formovable and the Gregorian for fixed feasts; and the Finnish Orthodoxuse the Gregorian for all purposes.To explain the Eastern system one must begin with the Jews inAlexandria at the time of the Christian Council of Nicaea in 325, whoappear to have been celebrating Passover on the first "full moon"after March 21, as specified by the 19-year Metonic cycle and theJulian calendar (with its leap year every 4 years, end of century ornot). The Bishop of Alexandria was made responsible for the Christiancalendar; he specified that Easter be the Sunday after that Passover.Eastern Christians still say that Easter must follow Passover, butthat Passover is the one that is meant, not the Passover defined bythe present Jewish calendar.Subsequently the Jews reformed their calendar (in 358 or in the early6th century according to different sources; possibly at differenttimes in different places), in order to improve the fit betweenastronomy and their arithmetic, but the Christians did not followsuit. In 1996, for example, Passover was on April 4 but the OrthodoxEaster was on Sunday April 14, not April 7 (which as it happens wasthe Western Easter.)The Eastern Easter is 0, 1, 4, or 5 weeks after the WesternEaster. The Western Easter can precede the (modern) Jewish Passover,as in 1967, 1970, 1978, 1986, 1989 and 1997, and can even coincidewith it, as in 1981.Much of this information was taken from the Explanatory Supplement tothe Astronomical Ephemeris, page 420, 1974 reprint of the 1961edition. There is more in the Explanatory Supplement, specifically aseries of tables that can be used to determine the Easter date forboth the Julian (Eastern and pre-1582 Western) and Gregoriancalendars. However, the Explanatory Supplement is misleading on thesubject of the Eastern Easters, though its tables are correct.Jean Meeus has published a program to compute Easter in "AstronomicalAlgorithms," 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, SimonKershaw, 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, _WinningWays_, that describes in Volume 2 a number of useful calendricalrules, 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 itgives Apr 18 and G>=12 you should take Apr 17. Easter is then thefollowing 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 ofthe Week, see below). Therefore, Easter 1945 took place on Sunday,April 1.Conway's "Doomsday" method for finding the day of the week, given thedate, is needed for his Easter method.To every year there is a distinguished day of the week, which Conwaycalls the "Doomsday", D. In any year, if March 0 (the last day ofFebruary) falls on a particular DOW, then the following dates alsofall 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 mynine-to-five job at the Seven-Eleven. Note to non-US readers:"Seven-Eleven" is the name of a ubiquitous chain of conveniencestores.) In non-leap years, Jan 3 and Feb 0 (Jan 31) also fall onthat 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 twodays 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 theyear, call them xx, dividing by 12 to get a quotient Q and remainderR. 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. In1995, for example, we have 95/12 = 7 with remainder 11; 11/4 givesquotient 2; Wednesday + 7 + 11 + 2 = Tuesday (cf. above).In other years on the Gregorian calendar, one uses instead ofWednesday, the century day as follows: 16xx and 20xx: Tuesday; 17xxand 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 sameexcept 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 thatfollowed Pope Gregory's institution of the Gregorian calendar) tookplace 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 inone's head with a little practice (OK, mod 19 for the Golden Number is abit hairy for me, but I can still do it!). The DOW calculation is veryuseful if you are caught without a calendar, and it makes a good partytrick.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>, Jay Respler <jrespler@superlink.net>Colloquially the term "blue moon" is used to mean "a very long time."In fact, there have been at least seven different uses of the term"blue moon" in the past several hundred years.The alt.usage.english FAQ discusses these different meanings of theterm "blue moon." The two definitions most relevant to astronomy arethe following:1. Under certain conditions of atmospheric haze, the moon may actuallylook blue. A notable example occurred after the explosion of thevolcano Krakatoa. The appropriate conditions are extremely rare.2. The second full moon in a calendar month. Since the synodic monthis 29.53 days, this kind of blue moon occurs roughly once out of 6030-day months and once out of 21 31-day months or about once in 2.5years on average. It can occur in January and the following March ifthere is no full moon at all in February. There are some indicationsthat some calendars used to put the first moon in the month in red,the second in blue, hence the origin of the term. Philip Hiscock, writing in the 1999 March issue of Sky & Telescope,expands upon the history of this definition. This definition of "bluemoon" is of fairly recent vintage and came into widespread use in thelate 1980s as a result of the board game Trivial Pursuit. He was ableto trace its origin to an (incorrect) entry in the 1937 edition of the_Maine Farmer's Almanac_.The alt.usage.english 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 itselfis an emerald green. The same is true of the first bit of light fromthe rising sun. This phenomenon is known as the "green flash" or "greenray." It is not an optical illusion.The green flash is common and will be visible any time the sun isrises or sets on a *clear*, *unobstructed*, and *low* horizon. Fromour observatory at Mt. Hopkins, I (SW) see the sunset green flashprobably 90% of the evenings that have no visible clouds on thewestern horizon. It typically lasts one or two seconds (by estimate,not stopwatch) but on rare occasions much longer (5 seconds??). I'veseen the dawn green flash only once, but a) I'm seldom outsidelooking, b) the topography is much less favorable, and c) it takesluck to be looking in exactly the right place. If you'd like to seethe 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 thatthe visible image of the sun at the horizon appears roughly a solardiameter *above* the geometric position of the sun. This refractionis mildly wavelength dependent with blue light being refracted themost. Thus if refraction were the only effect, the red image of thesun would be lowest in the sky, followed by yellow, green, and bluehighest. If I've understood the refraction table properly, thedifference between red and blue (at the horizon) is about 1/40 of asolar diameter.Now scattering: the blue light is Rayleigh scattered away (not Comptonor 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 thehorizon, the light travels through something like 38 "air masses," soeven a weak absorption becomes significant.The explanation for the green flash is thus, 1) refraction separatesthe solar images by color; 2) at just the right instant, the red imagehas set, 3) the yellow image is absorbed; and 4) the blue image isscattered away. We are left with the upper limb of the green image.Because the green flash is primarily a refraction effect, it lastslonger and is easier to see from a mountain top than from sea level.The amount of refraction is proportional to the path length throughthe atmosphere times the density gradient (in a linear approximationfor the atmosphere's index of refraction). This product will scalelike 1+(h/a)^(0.5), where h is your height and a the scale height ofthe 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 theincrease in refraction is about a factor 1.5; a 3 km mountain gives1.6 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 SunPhenomena_, by D. J. K. O'Connell and the classic _Light and Color inthe Open Air_. A refraction table appears in _AstrophysicalQuantities_, by C. W. Allen. There's also an on-line resource at<URL:http://mintaka.sdsu.edu/GF>.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 afancy name for the fact that the Sun's speed along the Earth's equatoris not constant. In other words, if you were to measure the Sun'sposition at exactly noon every day, you would see not only thefamiliar north-south change that goes with the seasons but also aneast-west change in the Sun's position. A graphical representation ofboth positional changes is the analemma, that funny figure 8 that mostglobes stick in the middle of the Pacific ocean.The short explanation of the equation of time is that it has twocauses. The slightly larger effect comes from the obliquity of theecliptic---the Earth's equator is tilted with respect to the orbitalplane. Constant speed along the ecliptic---which is how the "meansun" moves---translates to varying speed in right ascension (along theequator). 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 notconstant. This gives the inequality between the two lobes of thefigure 8.Some additional discussion, with illustrations, is provided by NickStrobel at <URL:http://www.astronomynotes.com/nakedeye/s9.htm>, thoughyou may want to start with the section on time at<URL:http://www.astronomynotes.com/nakedeye/s7.htm>. MattthiasReinsch provides an analytic expression for determining the number ofdays between the winter solstice and the day of the latest sunrise forNorthern Hemisphere observers,<URL:http://arXiv.org/abs/astro-ph/?0201074>.The Earth's analemma will change with time as the Earth's orbitalparameters 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. A simulationof the Martian analemma is at<URL:http://apod.gsfc.nasa.gov/apod/ap030626.html>, and illustrationsof other planetary analemmas is at <URL:http://www.analemma.com/>.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, _WinningWays_, that describes in Volume 2 a number of useful calendricalrules. One of these is an easy "in your head" algorithm forcalculating the phase of the Moon, good to a day or better dependingon whether you use his refinements or not.In the 20th century, calculate the remainder upon dividing thelast two digits of the year by 19; if greater than 9, subtract19 from this to get a number between -9 and 9. Multiply the result by 11 and reduce modulo 30 to obtain anumber between -29 and +29.Add the day of the month and the number of the month (exceptfor Jan and Feb use 3 and 4 for the month number instead of1 and 2).Subtract 4.Reduce modulo 30 to get a number between 0 and 29. This isthe 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 phaseof the Moon, either because of illumination or because of tides. Ithink 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 alsofor the slight variations in the lengths of months; what I havegiven should be good to +/- a day or so.Subject: C.12 What is the time delivered by a GPS receiver?Author: Markus Kuhn <Markus.Kuhn@cl.cam.ac.uk>Navstar GPS (global positioning system) is a satellite basednavigation system operated by the US Air Force. The signals broadcastby GPS satellites, contain all information required by a GPS receiverin order to determine both UTC and TIA highly accurately. CommercialGPS receivers can provide a time reference that is closer than 340 nsto UTC(USNO) in 90% of all measurements, classified military versionsare even better.Subject: C.13 Why are there two tides a day and not just one?Author: Joseph Lazio <jlazio@patriot.net>, Paul Zander <paulz@sc.hp.com>An easy way to think of the Moon's effect on the Earth is thefollowing. The Moon exerts a gravitational force on the Earth. Thestrength of the gravitational force decreases with increasingdistance. So, because the surface of the ocean is closer to the Moonthan the sea floor, the surface water is attracted more strongly tothe 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 ofthe Earth from the Moon, the sea floor is being pulled more stronglytoward the Moon than the surface water. In essence, the surface wateris being left behind. Voila, another bulge in the surface water andanother tide. In principle, there should be two tides of equal height in a day. Inpractice, many parts of the earth do not experience two tides of equalheight 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 isin the southern hemisphere.Except around Antarctica, the shape of the Earth's continents preventthe tidal bulges from simply following the moon. Each ocean basin hasits own individual pattern for the tidal flow. In the South AtlanticOcean, the tides travel from south to north, taking about 12 hours togo from the tip of Africa to the equator.In the North Atlantic, the tides travel in a counter-clockwisedirection going around once in about 12 hours. The effect is similarto water sloshing around in a bowl. Because the two tides are roughlyequal, they are called semidaily or semidiurnal.In some parts of the Gulf of Mexico, there is only one high tide andone low tide a day. These are called daily or diurnal tides. In muchof the Pacific Ocean, there are two high tides and two low tides eachday, but they are of unequal height. These are called mixed tides.The traditional way to predict tides has been to collect data forseveral years to have enough combinations of positions of the moon andsun to allow accurate extrapolation. More recently, computer modelshave been made taking into account detailed shapes of the oceanbottoms and coastlines.Even the best predictions can have difficulties. The extremely heavysnow fall during the winter of 1994--95 in California and theassociated run-off as it melted were not part of the model for SanFrancisco Bay. Sail boat races scheduled to take advantage of tidalcurrents coming into the Golden Gate found the current was still goingout!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> and<URL:http://www.co-ops.nos.noaa.gov/restles1.html>.Subject: CopyrightThis document, as a collection, is Copyright 1995--2005 by T. JosephW. Lazio (jlazio@patriot.net). The individual articles are copyrightby the individual authors listed. All rights are reserved.Permission to use, copy and distribute this unmodified document by anymeans and for any purpose EXCEPT PROFIT PURPOSES is hereby granted,provided that both the above Copyright notice and this permissionnotice appear in all copies of the FAQ itself. Reproducing this FAQby any means, included, but not limited to, printing, copying existingprints, publishing by electronic or other means, implies fullagreement to the above non-profit-use clause, unless upon priorwritten permission of the authors. This FAQ is provided by the authors "as is," with all its faults. Anyexpress or implied warranties, including, but not limited to, anyimplied warranties of merchantability, accuracy, or fitness for anyparticular purpose, are disclaimed. If you use the information inthis document, in any way, you do so at your own risk. Part0 - Part1 - Part2 - Part3 - Part4 - Part5 - Part6 - Part7 - Part8 - MultiPage
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