How are Solar Values calculated for SDY Time Values?

Solar Day of Year (SDY) Time values indicate a point in time during the course of a 'standard' 365 day year.

The Sun API functions that accept SDY time values are designed to return a "least error"/ "average value" optimised for the years 2000 to 2100.

Challenges

There are a few challenges in choosing values to return for a 'standard' year.

There is a significant variance between the solar values for a given day over the years of a 4 year leap year cycle.

Taking Solar Declination as an example: The following table shows Solar Declination in degrees for 12 Noon UT on 5 days in the year over two leap year cycles 2001-2004 and 2005-2008. If you compare years that are 4 years apart (they have the same colour in the table), you can see that they are much closer in value than the year immediately before or after.

Day	January 1st	March 1st	June 1st	September 1st	December 1st
2001	-22:58:15.7334	-7:27:11.7200	22:05:59.2182	8:09:07.1846	-21:51:13.2850
(mid cycle year) 2002	-22:59:31.3352	-7:32:47.9003	22:04:02.2775	8:14:27.2356	-21:49:00.4150
2003	-23:00:45.9922	-7:38:14.0202	22:02:10.9598	8:19:40.7343	-21:46:48.9941
(Leap year) 2004	-23:01:58.1718	-7:20:54.2859	22:08:18.1012	8:03:13.6535	-21:53:49.9977
2005	-22:58:13.4104	-7:26:29.2738	22:06:22.4906	8:08:30.7978	-21:51:39.8560
(mid cycle year) 2006	-22:59:27.6680	-7:31:57.4962	22:04:31.0230	8:13:35.0021	-21:49:24.8322
2007	-23:00:38.1843	-7:37:25.0390	22:02:31.2946	8:18:55.9538	-21:47:06.9830
(Leap year) 2008	-23:01:48.2240	-7:20:03.3338	22:08:36.2102	8:02:21.1553	-21:54:04.8307

Mid cycle years give the best approximating values for that leap cycle.

Over the next 100 years, 4 year average solar values will shift. The shift is different for different days of the year. Some days of the year show almost no shift over the period of 100 years, some days show a value increase whereas some days show a value decrease. However for a given day the shift is approximately linear over 100 years. This means that mid century years can be used to give approximate values for the next century.

Taking Solar Declination again as an example: The following table compares solar declination values for 12 Noon UT on 5 days during 5 years over the next century. The long term shift in declination value varies from +16 minutes of arc per 100 years in March to -17 minutes of arc in September.

Day	January 1st	March 1st	June 1st	September 1st	December 1st
2002	-22:59:31.3352	-7:32:47.9003	22:04:02.2775	8:14:27.2356	-21:49:00.4150
2026	-22:58:33.1090	-7:28:31.2579	22:05:37.6571	8:09:58.4513	-21:50:36.5346
2050	-22:57:15.0193	-7:24:12.8886	22:06:52.1907	8:05:30.2980	-21:51:50.3629
2074	-22:56:11.0624	-7:20:15.7856	22:08:13.2382	8:01:17.6825	-21:53:17.4511
2098	-22:55:19.8609	-7:16:10.5533	22:09:50.9967	7:56:55.3747	-21:54:57.0678
Trend/96 years	+0:04:11.4743	+0:16:37.3470	+0:05:48.7192	-0:17:31.8609	-0:05:56.6528

For Eastern timezones (e.g. Almaty -6 gives UT) the UT based SDY value corresponding to 2am January 1st is actually in December! For Western Timezones the UT based SDY value corresponding to 11pm December 31st is actually in January!

For example: 2am January 1st [-6 gives UT]. sdSDYx(1,1,2,0,0,-6) equals 365.8333 which is equivalent to 10pm December 31st UT!

Therefore the values for December 31st, just before midnight and January 1 just after midnight should match and not be discontinuous.

Solution

After investigating many different ways of calculating 21st Century standard values and evaluating their average, median and maximum errors and absolute errors the following method was chosen.

For SDY time values - The Sun API's functions convert the time to a year 2050 MJD time value and return the value for that year. A minimal linear blend is applied to December and January values to achieve a smooth transition from December 31st back to January 1st.

The year 2050 was chosen because it is mid-leap cycle and mid-century and experimentally gives close to minimum error values.