(FAQ) Astronomical Calculations for the Amateur
Revised August 12, 1997.
[This FAQ is limited to questions about calculating planetary
positions and related problems of spherical astronomy. Other
areas of interest, such a 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?
9a. How do I convert right ascension and declination to altitude
and azimuth?
9b. What's the hour angle?
9c. What's the sidereal time?
10. How do I predict the ocean tides?
11. How do I calculate the date of Easter?
12. How fast does that comet (or asteroid) move?
======================================================================
1. What is a good source of books and software?
The Willmann-Bell 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
<http://www.willbell.com/>
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 1991,
$24.95. Software supplements in Basic, Pascal and C are
available to purchasers of the book for $24.95 each.
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.
Willmann-Bell once supplied an errata sheet for this book, but
I am told that all known errors have been corrected in the
current edition.
IMPORTANT NOTE: I *strongly* recommend that you get one of
the software supplements. Not only is the source code very
illuminating, but Meeus relies on some modern table-driven
models which would be unfeasible to type in yourself. Pay the
$24.95 and get a diskette. The software is for DOS machines, but
I had no trouble translating the C-language version to
Macintosh. (I did have to write a small DOS program to unpack
the large data tables).
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 an electronic journal called _The Digital
Lens_. It is available via email subscription in either plain text
or Adobe PDF format. To subscribe, send a request to Mike McClure
at:
<mwmccl1@pop.uky.edu>.
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@geocities.com> maintains an "Approximate
astronomical positions" web page containing algorithms and
many links:
<http://www.geocities.com/CapeCanaveral/Lab/4749/>
Paul Schlyter <pausch@saaf.se> has a "Calculating Planetary
Positions" web page at:
<http://spitfire.ausys.se/psr/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:
Astronomical Data Center home page
<http://adc.gsfc.nasa.gov/>
The WWW Virtual Library: Astronomy and Astrophysics
<http://www.w3.org/hypertext/DataSources/bySubject/astro/astro.html>
The Space FAQ
<http://www.cis.ohio-state.edu/hypertext/faq/usenet/space/top.html>
12 Year Planetary Ephemeris: 1995 - 2006
<http://planets.gsfc.nasa.gov/TYPE/TYPE.html>
8. What commercial and shareware programs are available?
[Readers: I have not been paying attention to announcements of
these programs in s.a.a. 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...].
* * * * *
The freeware ephemeris program "ephem" (version 4.27, sic!) for PC
by Elwood Charles Downey (and VGA Watch plots by J.D. McDonald) is
available by anonymous ftp at ftp.funet.fi directory
/pub/astro/pc/solar, filename ephem423.exe (self extracting
archive.) The same site carries many other ephemeris programs also
for other platforms. [Harald Lang <lang@math.kth.se> Dec 3 1995]
[There is a Web page for the Motif version at:
<http://iraf.noao.edu:80/~ecdowney/xephem.html>
Jan 2 1996]
[Eric Muller, <mullere@in2p3.fr>, reports a Macintosh port of
the PC version at:
<ftp://ftp.u-tokyo.ac.jp/mac/umich/misc/astronomy/macephem4.21c.cpt.hqx>
Jun 9, 1997]
* * * * *
I have recently completed a freeware program which might interest
you. It's called the "Windows Ephemeris Tool" and it calculates
tables of positions (and other data) for comets and asteroids.
It's available at <http://fox.nstn.ca/~ecu/ecu.html> or
<ftp://hercules.stmarys.ca/pub/ecu/wetinst.exe>
Regards, Dave Lane
Nova Astronomics [<dlane@ap.stmarys.ca>, Dec 7 1995]
* * * * *
I am very impressed with a program called ASTROWIN, sometimes
referred to as ASTROMEUSS (It uses Meuss' algorithms). I got my
copy from Starbase One BBS, but I am sure it must be on the WEB
somewhere. It is simple, fast and accurate. Text-only output. I
use it a lot.
Stephen Tonkin <sft@aegis1.demon.co.uk>
[This is for DOS and Windows, and is on the web at:
Hawaiian Astronomical Society
<http://www.aloha.net/~prh/master.htm>
Caution: there is another program called ASTROWIN for astrology.
Jun 1 1996]
* * * * *
Willmann-Bell sells several software supplements which have
ephemeris capabilities. See their catalog ([1] above) for details.
* * * * *
Bill Arnet <billa@znet.com> maintains links to planetarium programs
that can be found on the net at:
<http://www.seds.org/billa/astrosoftware.html>
[Jan 31 1997]
* * * * *
9a. 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) )
9b. 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.
9c. What's the sidereal time?
Everything seems to depend on something else, doesn't it? Better
get the Meeus book described in [2] above.
10. How do I predict the ocean tides?
I have never heard of an amateur doing this. The _Explantory
Supplement_ has a small section on the subject and the method
seems quite complex.
11. 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 very informative Royal Greenwich Observatory leaflet
on Easter at:
<http://www.ast.cam.ac.uk:80/pubinfo/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.pip.dknet.dk/~pip10160/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.
12. 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.