(FAQ) Astronomical Calculations for the Amateur

Revised March 22, 2000.

[This FAQ is limited to questions about calculating planetary positions and related problems of spherical astronomy. Other areas of interest, such as calculations of telescope optics, are beyond the bounds of this document].

Calculation of astronomical events is a vast field with literature stretching back centuries, even to ancient times. This "frequently asked questions" list is directed toward the amateur astronomer who is looking for starting points. If you become familiar with the first two books recommended below, you will be well on your way. You will, in fact, have surpassed the author of the FAQ!

Of course, I invite your suggestions.

  1. What is a good source of books and software?
  2. What is the best beginner's book?
  3. How much computer power does it take to perform these calculations?
  4. What is a more advanced reference work?
  5. Are there any relevant periodicals for amateurs?
  6. Where are online sources of algorithms?
  7. Where are online sources of data?
  8. What commercial and shareware programs are available?
  9. How do I convert right ascension and declination to altitude and azimuth?
  10. What's the hour angle?
  11. What's the sidereal time?
  12. How do I predict the ocean tides?
  13. How do I calculate the date of Easter?
  14. How fast does that comet (or asteroid) move?
  15. How do I find my longitude and latitude?
  1. What is a good source of books and software?
    The Willmann-Bell (http://www.willbell.com/) printed catalog has a large section on "Computational "Astronomy", as well as many other astronomy books, atlases and telescope-making supplies:
    Willmann-Bell Inc
    PO Box 35025
    Richmond VA 23235

    Monday-Friday, 9AM-5PM Eastern time
    800-825-STAR (order only)
    804-320-7016
    24 hour fax: 804-272-5920

    If you have access to a good library, books under the subject headings "Spherical Astronomy" and "Celestial Mechanics" would be the places to start.
  2. What is the best beginner's book?
    Astronomical Algorithms by Jean Meeus, Willmann-Bell, Second Edition 1998, $24.95.

    Although it requires some study, this is the closest thing to a "cookbook" approach I have seen. Better than that, it explains and makes comprehensible many difficult concepts, and has many worked examples and illustrations. It is not restricted to elementary problems, but treats many advanced topics. No calculus is required.

    Beginners face two obstacles before they can calculate anything useful: (1) they must learn to convert between civil and astronomical dates and times (a task made more difficult by the fact that the Earth's rate of rotation is variable), and (2) they must learn a number of translations between coordinate systems (Sun-centered to Earth-centered to location-centered, as well as ecliptic to equatorial to horizon) and the application of corrections for precession and nutation and parallax. This is why questions such as "How do I predict the location of the moon?" do not have simple answers. You must know how to do (1) and (2) before you can start on the moon.

    The proper order of corrections and coordinate conversions had previously been very confusing for me, but Meeus gave me everything I needed to overcome these obstacles.

    He covers the basics of time and coordinate transformations, corrections for precession and nutation, and for the observer's true "topocentric" location as offset from the center of the Earth. For any given time, you can predict the positions of the Sun, Moon and planets and derive all the normal phenomena of the almanac. You can derive physical ephemerides (that is, the orientation of the objects as seen through a telescope) for the Sun, Moon, Jupiter, Mars and Saturn's rings. He provides both low-precision and high-precision techniques for charting Jupter's four largest moons. The Keplerian techniques of dealing with the orbits of new bodies such as comets and asteroids are also given.

    A software supplement was available for the first edition, but this is no longer the case.

  3. How much computer power does it take to perform these calculations?
    Modern personal computers, especially those with floating point hardware, are very capable machines. Calculating the position of all the planets several different ways, using Meeus' techniques, takes my 68040 a small fraction of a second. Performance on a PowerPC or Pentium would be stunning.
  4. What is a more advanced reference work?
    Explanatory Supplement to the Astronomical Almanac, edited by P.K. Seidelmann, University Science Books 1992, 752 pages, $65 (available from Willmann-Bell).

    "Completely Revised and Rewritten", so make you sure you get the 1992 edition.

    This explains how the data in the annual "Astronomical Almanac" is produced. It is also a high-quality spherical astronomy text with many references to the current research literature. If you've read Meeus and want "more", this is the logical next step.

    Note that it contains very few worked examples and the math is much more advanced than in Meeus. Some of the chapters deal with issues of the professional astronomer that will not usually concern the amateur. Examples: plate tectonic motion can cause an observing site to shift its position several centimeters per year. Ocean tidal pressure on the continental shelves, and atmospheric pressure above the continents, can cause elevation to vary by similar amounts.

    Note also that they use a different method of calculating planetary positions than does Meeus.

  5. Are there any relevant periodicals for amateurs?
    Sky & Telescope magazine has an astronomical computing column.

    Astronomy publishes programs from time to time.

    Willmann-Bell sells back issues of Celestial Computing, "A Journal for Personal Computers and Celestial Mechanics", dated from 1988 through 1992, edited by David Eagle. This is no longer published.

    The Computing Section of the Association of Lunar and Planetary Observers (A.L.P.O.) has a Computing Section and an electronic journal called The Digital Lens:

    http://www.m2c3.com/alpocs/
  6. Where are online sources of algorithms?
    Sky & Telescope maintains an archive of program sources which have appeared in the magazine:
    http://www.skypub.com/software/software.html

    Unfortunately, these consist of uncommented BASIC listings. Pseudo-code articles would be of greater use to those trying to understand the calculations. Astronomy magazine provides a small set of BASIC programs:

    http://www.kalmbach.com/astro/Bytes/Bytes.html
    Keith Burnett (keith@xylem.demon.co.uk) maintains an "Approximate astronomical positions" web page containing algorithms and many links:
    http://www.xylem.demon.co.uk/kepler/
    http://www.stargazing.net/kepler/
    Paul Schlyter (pausch@saaf.se) has a "Calculating Planetary Positions" web page at:
    http://hotel04.ausys.se/pausch/comp/ppcomp.html
    Sites listed in the next topic also have software.
  7. Where are online sources of data?
    There are astronomical amounts of data online. Try these web sites as starting points:
  8. What commercial and shareware programs are available?
    [Readers: I have not been paying attention to announcements of these programs in sci.astro.amateur. Anyone who has such or knows of same, please e-mail me the info and I will include descriptions here. The emphasis is not on "planetarium" or charting programs, but on ephemeris-generating software. Obviously, these categories overlap...].
  9. How do I convert right ascension and declination to altitude and azimuth?
    Given the hour angle H of the object with right ascension RA and declination DEC, and the observer's latitude LAT:
    azimuth = atan2(sin(H), cos(H) * sin(LAT) - tan(DEC) * cos(LAT))

    altitude = asin(sin(LAT) * sin(DEC) + cos(LAT)* cos(DEC) * cos(H))

    where "atan2(x,y)" is C-library function equivalent to "atan(x/y)".

    Bill Owen (wmo@wansor.jpl.nasa.gov) offers the following comments:

    For the azimuth, it might be better to multiply both numerator and denominator by cos(DEC). Granted that the answer should turn out the same either way, since 0/something = something else/infinity, but you'll avoid the overflow that would otherwise result when you compute tan(DEC) near the poles.

    Also, the formula you have here is zero when you're looking south. Although there are different conventions, the most common one reckons azimuth eastward from *north*.

    Combine these nits, and the formula I use is:

    azimuth = atan2 (-sin(H)*cos(DEC), cos(LAT)*sin(DEC) - sin(LAT)*cos(DEC)*cos(H) )
  10. What's the hour angle?
    Given an object with right ascension RA and the observer's longitude LONG, and the sidereal time at Greenwich ST:
    H = ST - LONG - RA
    where LONG is positive to the west and ST is represented as an angle. If you measure longitude to the east:
    H = ST + LONG - RA.
  11. What's the sidereal time?
    Everything seems to depend on something else, doesn't it? Better get the Meeus book described in [2] above.
  12. How do I predict the ocean tides?
    This is not commonly done by amateurs. The Explantory Supplement has a small section on the subject and the method seems quite complex.
  13. How do I calculate the date of Easter?
    Many people know the formula:
    Easter is the first Sunday after the first full Moon following the vernal equinox.
    Caution! This is "astronomical Easter", and it is usually but not always the same day as "ecclesiastical Easter", which is the date used by the churches and printed on calendars. "Ecclesiastical Easter" is determined by a formula codified many years ago.

    Here is the method published in the Explanatory Supplement. Perform integer math and drop all remainders. It is valid for any Gregorian year "Y":

    C = Y / 100
    N = Y - 19 * (Y / 19)
    K = (C - 17) / 25
    I = C - C / 4 - (C - K) / 3 + 19 * N + 15
    I = I - 30 * (I / 30)
    I = I - (I / 28) * (1 - (I / 28) * (29 / (I + 1)) * ((21 - N) / 11))
    J = Y + Y / 4 + I + 2 - C + C / 4
    J = J - 7 * (J / 7)
    L = I - J
    M = 3 + (L + 40) / 44
    D = L + 28 - 31 * (M / 4)
    "M" is the month number (3 -> March, 4 -> April) and "D" is the day of the month.

    There is a short BASIC program at

    http://www.skypub.com/software/software.html
    See also the informative Royal Observatory leaflet on Easter at:
    http://www.rog.nmm.ac.uk/leaflets/easter/easter.html
    There is an HTML Ecclesiatical Calendar generator at:
    http://cssa.stanford.edu/~marcos/ec-cal.html
    See also the Calendar FAQ at:
    http://www.tondering.dk/claus/calendar.html
    Tidbits: the pattern of Gregorian Easter days, one year to the next, repeats in a cycle 5,700,000 years long. March 22 is the earliest date of Easter, April 25 is the latest, and April 19 is the most frequent.
  14. How fast does that comet (or asteroid) move?
    From Harald Lang (lang@math.kth.se).

    The current speed of a body like a comet orbiting the sun, or in a hyperbolic or parabolic orbit, is:

    2 * pi * sqrt(2/r - (1-e)/q) AU/year

    where r is the current distance in AU to the sun, q is the perihelion distance in AU, and e is the eccentricity of the orbit.

  15. How do I find my longitude and latitude?
    Here are some sites that give longitude and latitude information.

    It has been suggested to me that the following precisions are appropriate for the applications shown: 100 miles for most skyviewing work, 2 miles for accurately predicting Iridium flares, 50 feet for occultation work.

This document is archived at:
http://www.salamander.com/~wmcclain/astro_calc.html
Comments?
Bill McClain (wmcclain@salamander.com)