SUNDIAL EQUATIONS

 “All this information is secured by means of instruments suitable for these purposes, and by tables and by canons…. For everything works through innate forces shown by lines, angles and figures” Opus Majus, Roger Bacon (1220-1292)

 Sundial equations are published in many of the standard sources. Those reproduced here use the preferred symbols and definitions of the various parameters as described in the glossary. They are also self-consistent, and follow the sign conventions of the glossary, i.e. if the correct signs of the angles are input, and proper note is taken of the signs of the trigonometrical functions, the outputs will also have the correct signs.Notes. In the equations for the hour line angle X, the equations are for -90º < h < 90º, i.e. between 6 a.m. and 6 p.m. L.A.T. For other times, the true hour line angle is given by:X’ = X ± 180ºThe hour angle, h, in degrees, is given byh = (T24 – 12) x 15ºwhere T24 is the time in 24-hour clock notation (hours after midnight) in decimal hours.

 1. Horizontal dial (i = 0º )Style height: Hour line angles: 2. Vertical direct S dial(i = 90º )Style height: Hour line angles: 3. Declining dial(i = 90º ) Style height: Sub-style angle: Hour line angles: 4. Declining-reclining dial The case considered here is for i > and d < dcrit where i.e. the common case of a roughly south-facing dial reclining slightly from the vertical. Then: Noon line angle (with respect to the line of greatest slope): Style height: Sub-style angle: Hour line angles (with respect to the noon line): 5. Sun’s azimuth 6. Sun’s altitude 7. Sunrise/sunset The time (hour angle) of sunrise/sunset is given by: The azimuth of the rising/setting sun is given by: Note that these times and azimuths are for astronomical sunrise/sunset, i.e. when the centre of the sun is on the true horizon, neglecting atmospheric refraction. For other definitions of sunrise/sunset, the corresponding altitudes should be used in the equations of (5) and (6) 8. EoT (best fit equations) A full calculation of the EoT for any time in any epoch is complex and the reader is referred to Meeus (see Sources), an Astronomical Almanac, or the NASS Dialist’s Companion computer program , or use the on-line solar calculator at www.gcstudio.com/suncalc.html . Mean daily values of the EoT (over the period 2000 – 2099) are available from: http://www.chabot.demon.nl/sundials/sunmeangmt.htm. For many practical purposes, the fourier transform approximation given below, which has a worst-case error of 0.0025 radians (35 seconds of time), will be sufficient. where Ea is in radians at 12:00 UT and w is calculated from day number nd (ranging from 1 on 1 January to 365 on 31 December) by: To convert to the EoT in seconds (of time), multiply Ea by 43200/ . For detailed values of the EoT (and many other solar parameters) on any particular day, use the NASS Diallers’ Companion program. 9. Declination (best fit equations) The comments made above for the EoT also apply to the Sun’s declination. The fourier transform approximation below yields a maximum error of 0.0006 radians (less than 3 arcminutes) or, if the final two terms are omitted, 0.0035 radians (12 arcminutes: where is in radians and w is as defined for the EoT above. 10. Sun’s refraction Ro is the refraction in arcmins for a temperature of 10ºC and an atmospheric pressure of 1010 mb. For other conditions, a multiplying factor of 0.28P/T is required, where P is the pressure in mb and T is the temperature in kelvins ( = temp in ºC + 273). 11. Babylonian and Italian Hours The Babylonian hour tB> and Italian hour tI (in hours) are given by: and where h is in degrees and 12. Seasonal or Temporal Hours The temporal hour tT (in hours) is given by: If h< – : If  : Otherwise: where all angles are in degrees and is as defined for Babylonian hours above.