Date: Sat, 13 Dec 1997
From: Simon Cassidy <scassidy@EARTHLINK.NET>
Subject: Calendar question for Simon Cassidy
Comments: To: Miguel Barcenas Cardenas <miguelin@usa.net>

A few weeks ago Miguel Barcenas Cardenas <miguelin@usa.net> wrote:
> I'd like to ask: Do you know of a web site with a formula or table to
> calculate the interval between equinoxes for the last 2000 years?

Simon responds:
Do you mean the interval between sucessive Spring Equinoxes? Or do you mean the half year intervals between Spring Equinoxes and Fall Equinoxes (or vice versa)? I have given Meeus' formula for the instants of Spring Equinox (expressed in Ephemeris or Dynamical Julian Day Numbers) from 1000 to 3000 AD (in a posting to CALNDR-L) and give it again below, since I know of no other Web site that gives the tools to calculate this:

***************************************************************************

Meeus ("Astronomical Algorithms", 1991) gives the following formulae for the instant of Vernal Equinox (in Dynamical time) as a function of the integral A.D. year-number Y (good from about 1000 to about 3000 AD):

M = (Y-2000) / 1000                 :convert AD year to millenia, from 2000 AD.

JDME = 2,451,623.80984 + 365,242.37404*M + 0.05169*M^2 - 0.00411*M^3 - 0.00057*M^4

T = (JDME - 2451,545.0) / 36525                :Julian centuries from 2000 (of equ/sol).

S = sum, for all A,B and C below, of A*COSINE( B + C*T )                :perturbations.

```   A    B in degrees    C in degrees
485     324.96          1,934.136
203     337.23         32,964.467
199     342.08             20.186
182      27.85        445,267.112
156      73.14         45,036.886
136     171.52         22,518.443
77     222.54         65,928.934
74     296.72          3,034.906
70     243.58          9,037.513
58     119.81         33,718.147
52     297.17            150.678
50      21.02          2,281.226
45     247.54         29,929.562
44     325.15         31,555.956
29      60.93          4,443.417
18     155.12         67,555.328
17     288.79          4,562.452
16     198.04         62,894.029
14     199.76         31,436.921
12      95.39         14,577.848
12     287.11         31,931.756
12     320.81         34,777.259
9     227.73          1,222.114
8      15.45         16,859.074```

W = ( 35,999.373*T - 2.47 ) degrees

L = 1 + 0.0334*COSINE(W) + 0.0007*COSINE(2*W)

JD = JDME + (0.00001*S/L)                :The final result in Julian Dynamical Days.

*****************************************************************************

Simon continues:

For the years 1000 AD back to -1000 AD (1001 BC) the above formulae can be used if the equation to calculate JDME is changed appropriately:

Replace this line above:
JDME = 2,451,623.80984 + 365,242.37404*M + 0.05169*M^2 - 0.00411*M^3 - 0.00057*M^4

with this equation (from Meeus):
JDME = 1,721,139.29189 + 365,242.13740*M + 0.06134*M^2 + 0.00111*M^3 - 0.00071*M^4

and use the following formula for finding M (the millenium):
M = Y/1000                :convert AD year to millenia, from 0 AD (1 BC).
(instead of: M = (Y-2000) / 1000 shown above)

Remember that these formulae give results in Dynamical time (Ephemeris days). To know these instants in true Universal time (i.e. Calendar time based on the rotation of the Earth) one must use a formula for Delta T and correct by subtracting Delta T from the instant expressed in Dynamical Time. Or interpolate from the following table of approximate values for Delta T (from Meeus, 1995?).

```Year  Minutes     Year  Minutes     Year  Minutes     Year  Minutes
0     177         100   158         200   140         300   123
400   107         500    93         600    79         700    66
800    55         900    45         1000   35         1100   27
1200   20         1300   14         1400    9         1500    5
1600    2         1700    0         1800    0         1980    1
2075    4         2200    8         2300   13         2400   19
2500   26         2600   34         2700   43         2800   53
2900   64         3000   76```

> What are the exact intervals between:
>
> *march equinox and june solstice?
> *june solstice and september equinox?
> *september equinox and december solstice?
> *december solstice and march equinox?
>
> Of course, it would be much better to have the exact Julian Dates for those
> events in the past 2000 years, but I don't know any web site which has that
> 2000 year astronomical almanac. Do you?

Simon responds:
Meeus (1991) gives a table of the "Duration of the astronomical seasons in days" which appears to answer your question about the intervals between equinoxes and solstices. However I have not checked it myself yet and there appears to me to be a problem with it (It has the March equinox to June solstice interval equal to the June solstice to September equinox interval at circa 1250 AD). I will therefore give you the formulae (to use with the above formulae) for calculating the instants of June solstice, September equinox and December solstice:

For June solstice from 3000 AD to 1000 AD use the following for JDME:
JDME = 2,451,716.56767 + 365,241.62603*M + 0.00325*M^2 + 0.00888*M^3 - 0.00030*M^4

For June solstice from 1000 AD to -1000 AD use the following for JDME:
JDME = 1,721,233.25401 + 365,241.72562*M - 0.05323*M^2 + 0.00907*M^3 + 0.00025*M^4
and use the following formula for finding M (the millenium): M = Y/1000                :convert AD year to millenia, from 0 AD (1 BC). (instead of: M = (Y-2000) / 1000 shown above)

For September equinox from 3000 AD to 1000 AD use the following for JDME:
JDME = 2,451,810.21715 + 365,242.01767*M - 0.11575*M^2 + 0.00337*M^3 + 0.00078*M^4

For September equinox from 1000 AD to -1000 AD use the following for JDME:
JDME = 1,721,325.70455 + 365,242.49558*M - 0.11677*M^2 - 0.00297*M^3 + 0.00074*M^4
and use the following formula for finding M (the millenium):
M = Y/1000                :convert AD year to millenia, from 0 AD (1 BC).
(instead of: M = (Y-2000) / 1000 shown above)

For December solstice from 3000 AD to 1000 AD use the following for JDME:
JDME = 2,451,900.05952 + 365,242.74049*M - 0.06223*M^2 - 0.00823*M^3 + 0.00032*M^4

For December solstice from 1000 AD to -1000 AD use the following for JDME:
JDME = 1,721,414.39987 + 365,242.88257*M - 0.00769*M^2 - 0.00933*M^3 - 0.00006*M^4
and use the following formula for finding M (the millenium):
M = Y/1000                 :convert AD year to millenia, from 0 AD (1 BC).
(instead of: M = (Y-2000) / 1000 shown above)

Let me know what results you come up with and I will double check them also as I wish to find the problem with Meeus' table of "Durations..." (or my misunderstanding of the qualitative physics involved).

--Dee's Y'rs, Simon Cassidy, 1053 47th. St. Emeryville Ca. 94608.