Moon and Sun rise and set for any latitude

Contents

• Overview
  • Definitions and the problem
  • Different algorithms
  • Quadratic interpolation method
• QBASIC listing
• References

Overview

This page describes an algorithm that will enable you to calculate the time of Moon rise and set and Sun rise and set, as well as the twilight times, for any latitude on the Earth's surface. The algorithm presented here will warn you if the Moon or Sun remains above or below the horizon all day (a definite possibility above latitudes of 66.5 degrees North), or if the object does not rise or set within the day chosen (the Moon will exhibit this behaviour on one day in each lunar cycle).

If you just want the results for your observing location, then I recommend the Web page based programs provided by the US Naval Observatory. One of these programs will give you a compact yearly calendar style printout of rise and set times at a given location on a single sheet of paper. I use a set of printouts in my own observation planning - its easier than switching on the computer!

In the implementation of the algorithm on this page, I have used approximate routines to calculate the coordinates of the Sun and Moon. You can choose latitudes where the program will predict a very short day when the Sun really remains below the horizon all day - this inaccuracy affects latitudes close to 67.43 degrees North, and you can imagine a 'fuzzy zone' around the Earth at this latitude (and a mirror image one in the South) where the program may be in error. I suspect that more accurate position routines and better approximations for topocentric parallax, refraction and the diameter of the Sun and Moon would shrink this 'fuzzy zone', at the cost of increased complexity.

The algorithm is implemented in QBASIC, a well structured interpreted basic available free for Windows computers, and easily adapted to Visual Basic syntax. The implementation is based very closely on PASCAL routines taken from Montenbruck and Pfleger.

Definitions and the problem

The rising and setting of the Sun and Moon is defined here to be the instant when the upper limb of the object is seen to cross a mathematically flat horizon by an observer at sea level. Real observing situations on land rarely approach this idealised model, and for my uses calculating rise and set times to the nearest minute is fine.

The following factors need to be addressed

• The Sun and Moon vary in position against the stars over a period of one day - the Moon moves about 12 degrees per day on average.
• There are latitudes on Earth where the Sun and Moon can be above or below the horizon for days at a time. The algorithm has to detect and report these circumstances.
• The time of Moonset will be about 50 minutes later each night after New Moon. There will be at least one day in each lunation during which the Moon does not set, and one day where the Moon does not rise. The algorithm must report these days.
• You must calculate the altitude of the Sun and the Moon as seen by the observer on the surface of the Earth - allowing for parallax from the geocentric coordinates.
• You must correct for the effects of atmospheric refraction (not relevant for the twilights).
• You must allow for the size of the Sun or Moon's disc (except for the twilights).

In Montenbruck and Pfleger's approximate implementation, the last three points above boil down to subtracting a single correction term from the geocentric altitude of the Moon or Sun when looking for rise or setting events.

• Sunrise occurs when the centre of the Sun's disc is at a geocentric altitude of -50 arc minutes.
• Moon rise occurs when the centre of the Moon's disc is at a geocentric altitude of +8 arc minutes.
• Nautical twilight is defined to be when the centre of the Sun is at a geocentric altitude of -12 degrees.

The large difference between the Moon and the Sun correction reflects the much larger parallax effect on Moon positions - roughly 1 degree for the Moon compared to 9 arc seconds for the Sun.

Different algorithms

Possible algorithms include:-

• Use a single position and calculate as for a star
• Use a 'trial and improvement' or 'iterative' method
• Use a table based interpolation method

I used the 'single position and calculate as for star' method on a programmable calculator - the results for the Sun at moderate latitudes were 20 minutes or so wrong. Given the state of the local horizon, this was not much of a problem. This 'method' was very poor for the Moon, almost worse than guessing from the number of days since New Moon.

The 'trial and improvement' method with allowance for refraction, parallax and disc size is much better, and can give good results for the Sun for latitudes below 65N. The method will mostly converge after a few iterations - see the spreadsheet implementation on this Web site, or Paul Schlyter's page.

Tral and improvement methods get fiddly in situations where the object remains above or below the horizon for a number of days, or where the object does not set one day or the next.

A table based method could be very simple - just calculate the altitude of the Moon for each minute of the day and find when the altitude changes sign. There is your rising or setting event.

This simple table based method could detect the 'always above the horizon' or 'always below the horizon' situations as the (corrected) altitude would never change sign over the day. The method could also cope with the Moon missing a rise or set each lunar cycle, but 'grazing' situations, where the Sun never quite dips below the horizon, would be more difficult as there would be no sign change in the altitudes. A real program based on this system would be really slow, requiring at worst case 1440 calculations of altitude for each object for each day.

Quadratic interpolation method

Interpolation based methods use a shorter table of altitudes - in my implementation we use a table with a maximum of 12 rows - and fit a quadratic function to each set of three altitudes in turn. We then use some GCSE algebra to find the times when the quadratic crosses the X axis - has zero altitude - and these times are candidates for a rising or setting event. This system can also deal with the situation when the rising and setting occur within the hour - the quadratic then has two zeros. Finally, if the two zeros are the same, then there must be a 'grazing' event in that interval, and the algorithm can detect the event.

The steps in the procedure are

1. Calculate the sine of the altitude (corrected for refraction, parallax and the limb radius) for the first three hours in the day. Using the sine of the altitude saves some calculation and means that the values will always be within the range +1 to -1.
2. Save sign of the altitude at 0h
3. Take the values of the sine of the altitude for each set of three consecutive hours, correct for the refraction, parallax and fit a quadratic curve through these three points. See Montenbruck and Pfleger for algebra.
4. Calculate the 'discriminant' for the quadratic, and classify as
   • No roots within the 2 hour interval - move onto the next 2 hour range and start from step 1
   • Two roots, one root within the interval - classify as rising or setting event and appropriate flag
     • If two flags set, i.e. rising and setting events both found, go to step 5
     • If one flag set, then return to step 1 for next two hour interval, to search for remaining event.
   • Two roots, both within the interval - set the rising and setting flags and go to step 5.
   • One root (i.e. both roots same) within interval - set both rising and setting flags and go to step 5.
5. If neither rise nor set flag is set, then warn user that object is 'always above' or 'always below' horizon depending on the sign of the altitude in step 2, and move onto next object or quit.
6. Print rise and set times, or message if one event missing in current day.
7. Move onto next object or quit.

QBASIC listing and test data

You can download the tested QBASIC code below. The QBASIC editor insists on adding functions and subroutines to the end of the listing in alphabetical order - which does not make reading the code any easier. I have added comments in the 'main' program that should help you link the code to the description of the algorithm above.

• Link to QBASIC code (12 K) (some long lines)
• Code in ZIP file format (4 K)

Test data

Below are some samples of program input and output, together with accurate results from Chris Marriott's SkyMap Pro 6.0 and with a year 2000 calendar for the Sun, Moon and Nautical Twilight generated from the USNO web based program.

Case 1

Rise and set times for today in Birmingham UK - simple case with no ambiguities. Below is the dialogue with the QBASIC program. Times of events are given in form hhmm, and are UT times.

   Rise and set for Sun and Moon
   =============================

   Year (yyyy) - - - - - - - - :2000
   Month  (mm) - - - - - - - - :1
   Day    (dd) - - - - - - - - :3
   Time zone (East +) - - - -  :0
   Longitude (w neg, decimals) :-1.91667
   Latitude  (n pos, decimals) :52.5

   Moon
     501
    1409

   Sun
     818
    1606

   Nautical twilight
     653
    1731

Skymap Pro gives the following times for the Sun and Moon on 2000 Jan 3 - agreement is fine to nearest minute, except for Moonrise that rounds the wrong way to give a result one minute different from the USNO tables.

Sunrise and twilight
Date           Rise        Set         Naut Twi E    Naut Twi B  
03 Jan 2000    08:18:11    16:05:50    17:30:56      06:53:06 

Moonrise
Rise: 5h 0m 23s
Set: 14h 9m 18s

Case 2

   Rise and set for Sun and Moon
   =============================

   Year (yyyy) - - - - - - - - :1999
   Month  (mm) - - - - - - - - :12
   Day    (dd) - - - - - - - - :25
   Time zone (East +) - - - -  :0
   Longitude (w neg, decimals) :0
   Latitude  (n pos, decimals) :67.43

   Moon
    1747
    1206

   Sun
    1156
    1204

   Nautical twilight
     747
    1613

Skymap Pro shows that the Sun remains below the horizon for the whole of that day at the latitude given. SkyMap Pro gives the altitude of the (centre of) Sun at transit for this day as roughly 53 minutes below the horizon. This implies an error of roughly 30 minutes in Sun position at that time for the QBASIC program. This seems rather large and I will be puggling about with this a bit more.

The Moon times and times for twilight are within a minute of those produced by SkyMap Pro. You might treat output from the QBASIC program presented here with caution for locations with latitudes between 67 and 68 North.

Case 3

For Narvik in Norway (further North), the program gives correct results give or take a minute:-

   Rise and set for Sun and Moon
   =============================

   Year (yyyy) - - - - - - - - :2000
   Month  (mm) - - - - - - - - :1
   Day    (dd) - - - - - - - - :3
   Time zone (East +) - - - -  :1
   Longitude (w neg, decimals) :17.42
   Latitude  (n pos, decimals) :68.43

   Moon
     629
    1157

   Sun
    always below horizon

   Nautical twilight
     743
    1607

Moon
Rise: 6h 27m 59s
Set: 11h 57m 47s

Date           Rise        Set         Naut Twi E    Naut Twi B  
03 Jan 2000    --:--:--    --:--:--    16:07:06      07:42:21    

References

Oliver Montenbruck and Thomas Pfleger,
Astronomy on the Personal Computer
Springer
1994, 3rd edition
ISBN 3-540-63521-1

Section 3.8 of this book is the basis of the QBASIC listing here. I have translated the PASCAL routines into QBASIC one by one. This book does not contain an exhaustive explanation of the quadratic fit method, and the test data the authors provide will not test all branches of the code. This book comes with a software disc with all the PASCAL routines - but does not cover all the ground in, say, Meeus' Astronomical Algorithms.

Sky and Telescope's software pages contain listings in 'spaghetti basic' that implement a similar method for the Sun and the Moon - there is a clever fix used to avoid re-calculating positions for the Sun and Moon up to 12 times each day at the cost of some increase in the 'fuzzy zone' when the programs may predict a Sunrise and set when there is none. The links below are to the BASICA listings.

http://www.skypub.com/resources/software/basic/programs/sunup.bas

http://www.skypub.com/resources/software/basic/programs/moonup.bas

Seidelmann, P. Kenneth (ed)
Explanatory Supplement to the Astronomical Almanac
University Science Books
1992, completely revised
ISBN 0-935702-68-7

Definitive reference on all aspects of the ephemeris and associated calculations. No hostages taken, no example calculations and modern vectoral notation used throughout. Sections 9.311 (p484) and 9.33 (p486) present the iterative algorithm for Sunrise for moderate latitudes.

Duffett-Smith, Peter
Practical Astronomy with your calculator
Cambridge University Press
3rd edition 1988
ISBN 0-521-35699-7

Good treatment of the basics of coordinate systems. Presents the formulas for rise and set time of a star, and presents an interesting linear interpolation scheme for Moonrise.

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