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 16,859.074W = ( 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^4with this equation (from Meeus):

JDME = 1,721,139.29189 + 365,242.13740*M + 0.06134*M^2 + 0.00111*M^3 - 0.00071*M^4and 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 76you also asked:

> 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^4For 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^4For 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^4For 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.

ph.510-547-0684. email: scassidy@earthlink.net

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