June 20 (or 21 or 22) a.k.a the
Summer Solstice was the longest day of the year in some countries.
For various reasons (please specify in the solution) the date varies from year to year, but (good news!) it is fully predictable.
Please describe how to find the dates of
Solstices and Equinoxes in this century and for that matter for any year, but list them for this century, to allow comparison to published lists, specifying the name and location
of your choice, Northern or Southern Hemisphere and the Time Zone.
Default: Paris, France.
A more accurate version, based on Meeus's Astronomical Algorithms is
clearvars , clc
global year A B C
A = [485,203,199,182,156,136,77,74,70,58,52,50,45,44,29,18,17,16,14,12,12,12,9,8];
B = [324.96,337.23,342.08,27.85,73.14,171.52,222.54,296.72,243.58,119.81,297.17,21.02, ...
247.54,325.15,60.93,155.12,288.79,198.04,199.76,95.39,287.11,320.81,227.73,15.45];
C = [1934.136,32964.467,20.186,445267.112,45036.886,22518.443, ...
65928.934,3034.906,9037.513,33718.147,150.678,2281.226, ...
29929.562,31555.956,4443.417,67555.328,4562.452,62894.029, ...
31436.921,14577.848,31931.756,34777.259,1222.114,16859.074];
for year=2000:2100
Y=(year-2000)/1000;
JDE0 = 2451623.80984 + 365242.37404*Y + 0.05169*Y^2 - 0.00411*Y^3 - 0.00057*Y^4;
JD1=adjust(JDE0);
JDE0 = 2451716.56767 + 365241.62603*Y + 0.00325*Y^2 + 0.00888*Y^3 - 0.00030*Y^4;
JD2=adjust(JDE0);
JDE0 = 2451810.21715 + 365242.01767*Y - 0.11575*Y^2 + 0.00337*Y^3 + 0.00078*Y^4;
JD3=adjust(JDE0);
JDE0 = 2451900.05952 + 365242.74049*Y - 0.06223*Y^2 - 0.00823*Y^3 + 0.00032*Y^4;
JD4=adjust(JDE0);
dt1=datetime(JD1,'ConvertFrom','juliandate');
dt1.Format=('yyyy-MM-dd HH:mm');
dt2=datetime(JD2,'ConvertFrom','juliandate');
dt3=datetime(JD3,'ConvertFrom','juliandate');
dt4=datetime(JD4,'ConvertFrom','juliandate');
disp([dt1 dt2 dt3 dt4])
end
function JDE=adjust(JDE0)
global year A B C
T = ( JDE0 - 2451545.0) / 36525;
W = 35999.373*T - 2.47;
dL = 1 + 0.0334*cosd(W) + 0.0007*cosd(2*W);
S=0;
for i=1:24
S=S+A(i)*cosd(B(i)+C(i)*T);
end
t=(year-2000)/100;
if( year < 948)
deltaT = 2177 + 497*t + 44.1*t^2;
else
deltaT = 102 + 102*t + 25.3*t^2;
end
JDE=JDE0+.00001*S/dL - (deltaT-30)/(1440*60);
% -30 is really double negative, thus positive, to compensate for date format
% truncating seconds instead of rounding to nearest minute.
end
% Formulae from Astronomical Algorithms
% by Jean Meeus, Willmann-Bell, Richmond, VA, USA 2015
% which says the times will generally be no more than 51 seconds off.
Array A contains the amplitudes of the perturbations in hundred-thousanths of a day--on the order of magnitude of a second, and array C has the number of degrees per century that the perturbation moves along its sinusoidal cycle. Some of the perturbations are easy to identify. The fourth, whose cycle moves at 445267.112°/century, if you divide it by 100 to get degrees/year, then by 360 to get cycles/yr you get 12.368530 cyc/yr, recognizable as the effects of the moon in its orbit. Why this perturbation? It's not the earth that is in orbit around the sun, but rather the earth-moon system, 1/81 of whose mass is in the moon and since the moon is about 60 earth radii from the earth, the actual point that's in orbit around the sun is about 3/4 of the way from the center of the earth to its surface. At last quarter phase the moon is leading the earth in earth's orbit and to balance this see-saw, the earth is lagging by about 3000 miles, or about 1/ 194779 of its total orbit or about 162 seconds worth. The fourth amplitude in the A array shows 182/100000 day as the amplitude which comes out to 157 seconds, a good match considering we used rounded values for the mass of the moon and the size of the earth.
The largest amplitude, 485/100000 day or 6.984 minutes, has a cycle time of 18.61 years, the period that the nodes of the moon's orbit take to do one rotation about the earth. The other cycles are probably due to other planets pulling on the earth in its orbit.
Another factor considered is the slowing of the earth's rotation. What is considered the time of day depends on this, while the earth goes on in its orbit without slowing down. That's the deltaT in the formula; it's actually irregular but a smoothed version is acceptably accurate.
The improved table (GMT):
2000-03-20 07:35 2000-06-21 01:47 2000-09-22 17:27 2000-12-21 13:37
2001-03-20 13:30 2001-06-21 07:37 2001-09-22 23:04 2001-12-21 19:21
2002-03-20 19:16 2002-06-21 13:24 2002-09-23 04:55 2002-12-22 01:14
2003-03-21 01:00 2003-06-21 19:10 2003-09-23 10:46 2003-12-22 07:03
2004-03-20 06:48 2004-06-21 00:56 2004-09-22 16:29 2004-12-21 12:41
2005-03-20 12:33 2005-06-21 06:45 2005-09-22 22:22 2005-12-21 18:35
2006-03-20 18:25 2006-06-21 12:25 2006-09-23 04:03 2006-12-22 00:21
2007-03-21 00:07 2007-06-21 18:05 2007-09-23 09:50 2007-12-22 06:07
2008-03-20 05:48 2008-06-20 23:59 2008-09-22 15:44 2008-12-21 12:03
2009-03-20 11:43 2009-06-21 05:45 2009-09-22 21:18 2009-12-21 17:46
2010-03-20 17:31 2010-06-21 11:28 2010-09-23 03:09 2010-12-21 23:38
2011-03-20 23:20 2011-06-21 17:16 2011-09-23 09:04 2011-12-22 05:29
2012-03-20 05:14 2012-06-20 23:08 2012-09-22 14:48 2012-12-21 11:11
2013-03-20 11:01 2013-06-21 05:03 2013-09-22 20:43 2013-12-21 17:10
2014-03-20 16:56 2014-06-21 10:51 2014-09-23 02:29 2014-12-21 23:02
2015-03-20 22:44 2015-06-21 16:37 2015-09-23 08:20 2015-12-22 04:47
2016-03-20 04:29 2016-06-20 22:33 2016-09-22 14:20 2016-12-21 10:43
2017-03-20 10:28 2017-06-21 04:23 2017-09-22 20:01 2017-12-21 16:27
2018-03-20 16:14 2018-06-21 10:06 2018-09-23 01:53 2018-12-21 22:21
2019-03-20 21:58 2019-06-21 15:53 2019-09-23 07:49 2019-12-22 04:18
2020-03-20 03:49 2020-06-20 21:43 2020-09-22 13:30 2020-12-21 10:02
2021-03-20 09:37 2021-06-21 03:31 2021-09-22 19:20 2021-12-21 15:58
2022-03-20 15:32 2022-06-21 09:13 2022-09-23 01:03 2022-12-21 21:47
2023-03-20 21:24 2023-06-21 14:57 2023-09-23 06:49 2023-12-22 03:27
2024-03-20 03:06 2024-06-20 20:50 2024-09-22 12:43 2024-12-21 09:19
2025-03-20 09:01 2025-06-21 02:41 2025-09-22 18:19 2025-12-21 15:02
2026-03-20 14:45 2026-06-21 08:24 2026-09-23 00:04 2026-12-21 20:49
2027-03-20 20:24 2027-06-21 14:10 2027-09-23 06:00 2027-12-22 02:41
2028-03-20 02:16 2028-06-20 20:01 2028-09-22 11:44 2028-12-21 08:19
2029-03-20 08:01 2029-06-21 01:47 2029-09-22 17:37 2029-12-21 14:13
2030-03-20 13:51 2030-06-21 07:30 2030-09-22 23:26 2030-12-21 20:08
2031-03-20 19:40 2031-06-21 13:16 2031-09-23 05:14 2031-12-22 01:55
2032-03-20 01:21 2032-06-20 19:08 2032-09-22 11:10 2032-12-21 07:55
2033-03-20 07:22 2033-06-21 01:00 2033-09-22 16:51 2033-12-21 13:45
2034-03-20 13:16 2034-06-21 06:43 2034-09-22 22:39 2034-12-21 19:33
2035-03-20 19:02 2035-06-21 12:32 2035-09-23 04:38 2035-12-22 01:30
2036-03-20 01:02 2036-06-20 18:30 2036-09-22 10:22 2036-12-21 07:12
2037-03-20 06:49 2037-06-21 00:21 2037-09-22 16:12 2037-12-21 13:07
2038-03-20 12:39 2038-06-21 06:08 2038-09-22 22:01 2038-12-21 19:01
2039-03-20 18:31 2039-06-21 11:56 2039-09-23 03:48 2039-12-22 00:40
2040-03-20 00:10 2040-06-20 17:45 2040-09-22 09:44 2040-12-21 06:32
2041-03-20 06:06 2041-06-20 23:35 2041-09-22 15:26 2041-12-21 12:17
2042-03-20 11:52 2042-06-21 05:15 2042-09-22 21:10 2042-12-21 18:03
2043-03-20 17:27 2043-06-21 10:57 2043-09-23 03:05 2043-12-22 00:00
2044-03-19 23:19 2044-06-20 16:49 2044-09-22 08:46 2044-12-21 05:43
2045-03-20 05:06 2045-06-20 22:32 2045-09-22 14:31 2045-12-21 11:34
2046-03-20 10:57 2046-06-21 04:14 2046-09-22 20:21 2046-12-21 17:27
2047-03-20 16:52 2047-06-21 10:02 2047-09-23 02:07 2047-12-21 23:06
2048-03-19 22:32 2048-06-20 15:52 2048-09-22 07:59 2048-12-21 05:01
2049-03-20 04:28 2049-06-20 21:46 2049-09-22 13:41 2049-12-21 10:51
2050-03-20 10:18 2050-06-21 03:32 2050-09-22 19:27 2050-12-21 16:37
2051-03-20 15:58 2051-06-21 09:17 2051-09-23 01:26 2051-12-21 22:33
2052-03-19 21:55 2052-06-20 15:15 2052-09-22 07:14 2052-12-21 04:16
2053-03-20 03:46 2053-06-20 21:03 2053-09-22 13:05 2053-12-21 10:08
2054-03-20 09:33 2054-06-21 02:45 2054-09-22 18:58 2054-12-21 16:08
2055-03-20 15:27 2055-06-21 08:38 2055-09-23 00:47 2055-12-21 21:55
2056-03-19 21:09 2056-06-20 14:27 2056-09-22 06:38 2056-12-21 03:50
2057-03-20 03:07 2057-06-20 20:18 2057-09-22 12:22 2057-12-21 09:41
2058-03-20 09:04 2058-06-21 02:03 2058-09-22 18:07 2058-12-21 15:24
2059-03-20 14:43 2059-06-21 07:45 2059-09-23 00:02 2059-12-21 21:17
2060-03-19 20:37 2060-06-20 13:44 2060-09-22 05:47 2060-12-21 03:00
2061-03-20 02:25 2061-06-20 19:31 2061-09-22 11:30 2061-12-21 08:48
2062-03-20 08:06 2062-06-21 01:10 2062-09-22 17:19 2062-12-21 14:41
2063-03-20 13:58 2063-06-21 07:01 2063-09-22 23:07 2063-12-21 20:20
2064-03-19 19:37 2064-06-20 12:44 2064-09-22 04:56 2064-12-21 02:07
2065-03-20 01:27 2065-06-20 18:31 2065-09-22 10:41 2065-12-21 07:59
2066-03-20 07:19 2066-06-21 00:15 2066-09-22 16:26 2066-12-21 13:45
2067-03-20 12:53 2067-06-21 05:54 2067-09-22 22:18 2067-12-21 19:42
2068-03-19 18:48 2068-06-20 11:52 2068-09-22 04:06 2068-12-21 01:32
2069-03-20 00:44 2069-06-20 17:40 2069-09-22 09:51 2069-12-21 07:21
2070-03-20 06:34 2070-06-20 23:21 2070-09-22 15:44 2070-12-21 13:18
2071-03-20 12:34 2071-06-21 05:19 2071-09-22 21:37 2071-12-21 19:03
2072-03-19 18:20 2072-06-20 11:13 2072-09-22 03:27 2072-12-21 00:55
2073-03-20 00:12 2073-06-20 17:06 2073-09-22 09:14 2073-12-21 06:50
2074-03-20 06:08 2074-06-20 22:57 2074-09-22 15:02 2074-12-21 12:35
2075-03-20 11:45 2075-06-21 04:39 2075-09-22 20:57 2075-12-21 18:26
2076-03-19 17:38 2076-06-20 10:36 2076-09-22 02:49 2076-12-21 00:12
2077-03-19 23:30 2077-06-20 16:22 2077-09-22 08:34 2077-12-21 06:00
2078-03-20 05:10 2078-06-20 21:57 2078-09-22 14:24 2078-12-21 11:57
2079-03-20 11:00 2079-06-21 03:48 2079-09-22 20:12 2079-12-21 17:43
2080-03-19 16:43 2080-06-20 09:33 2080-09-22 01:55 2080-12-20 23:31
2081-03-19 22:34 2081-06-20 15:15 2081-09-22 07:37 2081-12-21 05:21
2082-03-20 04:30 2082-06-20 21:02 2082-09-22 13:22 2082-12-21 11:04
2083-03-20 10:09 2083-06-21 02:42 2083-09-22 19:11 2083-12-21 16:52
2084-03-19 15:58 2084-06-20 08:39 2084-09-22 00:58 2084-12-20 22:40
2085-03-19 21:52 2085-06-20 14:32 2085-09-22 06:42 2085-12-21 04:28
2086-03-20 03:34 2086-06-20 20:08 2086-09-22 12:31 2086-12-21 10:22
2087-03-20 09:28 2087-06-21 02:05 2087-09-22 18:28 2087-12-21 16:08
2088-03-19 15:16 2088-06-20 07:56 2088-09-22 00:17 2088-12-20 21:55
2089-03-19 21:06 2089-06-20 13:42 2089-09-22 06:06 2089-12-21 03:51
2090-03-20 03:01 2090-06-20 19:35 2090-09-22 11:58 2090-12-21 09:43
2091-03-20 08:41 2091-06-21 01:17 2091-09-22 17:49 2091-12-21 15:38
2092-03-19 14:33 2092-06-20 07:14 2092-09-21 23:41 2092-12-20 21:31
2093-03-19 20:33 2093-06-20 13:06 2093-09-22 05:28 2093-12-21 03:20
2094-03-20 02:21 2094-06-20 18:41 2094-09-22 11:16 2094-12-21 09:12
2095-03-20 08:14 2095-06-21 00:38 2095-09-22 17:11 2095-12-21 15:00
2096-03-19 14:02 2096-06-20 06:30 2096-09-21 22:54 2096-12-20 20:45
2097-03-19 19:47 2097-06-20 12:12 2097-09-22 04:35 2097-12-21 02:36
2098-03-20 01:39 2098-06-20 18:02 2098-09-22 10:23 2098-12-21 08:20
2099-03-20 07:17 2099-06-20 23:41 2099-09-22 16:10 2099-12-21 14:04
2100-03-20 13:03 2100-06-21 05:31 2100-09-22 21:59 2100-12-21 19:50
Edited on July 6, 2021, 3:08 pm
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Posted by Charlie
on 2021-07-06 15:06:16 |
No Subject
