Sunrise and set and the equation of time

The equation of time calculated for 2008, drawn by Nick Lomb

The equation of time calculated for 2008, drawn by Nick Lomb

In this morning’s Column 8 in the Sydney Morning Herald, David Kerr of Hornsby reports that on the three dates bracketing August 15 this year the times of sunrise and set are symmetric with respect to 12 noon. In other words on those dates the Sun is due north and at its highest point at 12 noon. Mr Kerr is quite correct, so hopefully he was using the 2007 Australian Sky Guide for his sunrise and set times!

On most days during the year the Sun is not due north at 12 noon, but either early or late. This is represented by the equation of time, which is a combination of the effects of the tilt of the Earth’s axis and the oval-shape of its path around the Sun. If you have a look at a sundial that usually has an engraved table of values from the equation of time so that sundial time can be corrected to ordinary standard time.

There is a further complication. To actually use the equation of time we have to allow for our longitude or specifically the difference between our longitude and the meridian on which our time zone is based. In Sydney for example the longitude is 151°12′ which is 1°12′ greater than the 150° longitude on which eastern standard time is based. Hence the Sun is due north always 4 minutes and 48 seconds earlier than at the standard meridian.

As can be seen on the above graphical representation there are four days during the year when the Sun is due north exactly at 12 noon EST in Sydney: 5 January, 30 March, 9 July and 15 August.

David Kerr asked what those days should be called and suggests “middle noon” or “equiday”? I am not sure about either of those. Any suggestions?

12 responses to “Sunrise and set and the equation of time

  • could someone please help explain this calculation . I have taken a section from a book that discusses a calculation relating to the true time of sunrise . The Example is :

    Time of sunrise 30th april 1911 at greenwich
    Suns declination from the ephemeris at 6 a.m = 14 Deg 26min N
    Lat of greenwich 51 Deg 28 min N
    then tan log 14 Deg 26 min = 9.41057 ?
    + tan log 51 Deg 28 min = 0.09888 ?
    = 9.50945
    = sine 18 Deg 51 min
    + 90Deg 0 min
    = 108 Deg 51 min

    I can understand combining the suns declination at time of sunrise with the latitude of greenwich but could someone please explain how the use of the tan and log co ordinates in this example and how he arrives at the two figures marked with question marks thankyou

  • Hello Chris.There is a simple formula for determining the direction of sunrise and set. All you need for the formula is your latitude and the declination of the Sun for the day – declination is like latitude on Earth, it indicates how far north or south the Sun appears from the celestial equator. For the solstices the Sun’s declination is +23°26′ or -23°26′ for winter or summer, respectively.

    cos (azimuth) = sin (declination)/cos (latitude)

    here azimuth is the bearing on the horizon measured from north towards the east.

    For a latitude of -34°23′ I calculate sunrise 61°12′ east of north at the winter solstice and 118°48′ at summer solstice. Sunset is the same with the same values west of north.

    At the equinoxes the Sun’s declination is zero and trivially from the above formula the azimuth is 90°. That is, the Sun rises and sets due east and west.

  • Ah, thank you, Nick.

    I have been wondering how to calculate the direction of both sunrise and sunset at the summer and winter solstices and also the equinoxes. is there a simple calculation based on latitudes or alternatively, would you please let me know what the relevant directions are at 34 deg 23 min South? Or is there no significant difference between this latitude and Sydney?


  • Hello Sam. The direction of sunset on any date depends on where you live, in particular the latitude. In Sydney the Sun sets 28.5° south of true west at the summer solstice on ~21 December, due west at both the spring and the autumn equinoxes (~20 March and 23 September) and sets 28.5° north of true west at the winter solstice on ~21 June. With regard to readings provided by a compass note that currently in Sydney magnetic north is 12.5° east of true north.

    • HI Nick
      I’m planning a garden at a new place, and have come in in autumn.
      1. Re: ” In Sydney the Sun sets 28.5° south of true west at the summer solstice on ~21 December, due west at both the spring and the autumn equinoxes (~20 March and 23 September) and sets 28.5° north of true west at the winter solstice on ~21 June”
      Is it also true that sunRISE in Sydney
      a) summer solstice 28.5′ south of true E?
      b) winter solstice 28.5′ north of true E?

      2. then, Re: “currently in Sydney magnetic north is 12.5° east of true north”
      what would be the compass readings for a) and b) currently?
      thank you!

      • Sam, The answer to both 1a and 1b is Yes. And for Q2 you need to subtract 12.5 from the ‘true’ bearings. That gives a) At summer solstice the Sun rises at a compass bearing of 90+28.5-12.5 = 106 degrees; and b) At winter solstice Sun rises at a compass bearing of 90-28.5-12.5 = 49 degrees.

  • I would like to know where the sun sets in relation to true west and magnetic west during the different times of the year. I have heard that it sets at about 31 degrees south of true west in winter and 31 north of true west during summer (maybe other way around??). Could you please enlighten me (pardon the pun).

  • Hello Lloyd. You are not the only one to be agape over the derivation of formulae by Smart as they take a major effort to follow. I find the formulae given in the yearly The Astronomical Almanac published by the US Naval Observatory and the UK Hydrographic Office generally much easier to apply – “Low precision formulas for the Sun’s coordinates and the equation of time” page C24. You should be able to find the almanac in a local library. For the equation of time it gives

    E (apparent time minus mean time) in minutes = (L – RA), in degrees, multiplied by 4

    Here L is the mean longitude of the Sun and RA is its right ascension. The almanac gives formulae for both quantities, but you will need to look them up.

  • I am very much taken with your “Star Gazing by Night” thing, and will recommend the site to my Grandson.

    I came upon your site while surfing for a formula for the equation of time.

    I have W.M. Smart’s text (5th ed, 1965, 22s. 6d.net $3.95 !!) and his derivation for 1931. I wonder if I could use that with current sun’s longitude of perigee, but his “after some reduction” leaves me agape. So, with several sites’ data vice the ephemeris, I’ll give it a try, lacking any guidance from you, which whould not be unappreciated.


    Lloyd Gilman
    Mercer Island, Washington USA

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