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astro.js
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/*
JavaScript functions for positional astronomy
by John Walker -- September, MIM
http://www.fourmilab.ch/
This program is in the public domain.
*/
// Frequently-used constants
var
J2000 = 2451545.0, // Julian day of J2000 epoch
JulianCentury = 36525.0, // Days in Julian century
JulianMillennium = (JulianCentury * 10), // Days in Julian millennium
AstronomicalUnit = 149597870.0, // Astronomical unit in kilometres
TropicalYear = 365.24219878; // Mean solar tropical year
/* ASTOR -- Arc-seconds to radians. */
function astor(a)
{
return a * (Math.PI / (180.0 * 3600.0));
}
/* DTR -- Degrees to radians. */
function dtr(d)
{
return (d * Math.PI) / 180.0;
}
/* RTD -- Radians to degrees. */
function rtd(r)
{
return (r * 180.0) / Math.PI;
}
/* FIXANGLE -- Range reduce angle in degrees. */
function fixangle(a)
{
return a - 360.0 * (Math.floor(a / 360.0));
}
/* FIXANGR -- Range reduce angle in radians. */
function fixangr(a)
{
return a - (2 * Math.PI) * (Math.floor(a / (2 * Math.PI)));
}
// DSIN -- Sine of an angle in degrees
function dsin(d)
{
return Math.sin(dtr(d));
}
// DCOS -- Cosine of an angle in degrees
function dcos(d)
{
return Math.cos(dtr(d));
}
/* MOD -- Modulus function which works for non-integers. */
function mod(a, b)
{
return a - (b * Math.floor(a / b));
}
// AMOD -- Modulus function which returns numerator if modulus is zero
function amod(a, b)
{
return mod(a - 1, b) + 1;
}
/* JHMS -- Convert Julian time to hour, minutes, and seconds,
returned as a three-element array. */
function jhms(j) {
var ij;
j += 0.5; /* Astronomical to civil */
ij = ((j - Math.floor(j)) * 86400.0) + 0.5;
return new Array(
Math.floor(ij / 3600),
Math.floor((ij / 60) % 60),
Math.floor(ij % 60));
}
// JWDAY -- Calculate day of week from Julian day
var Weekdays = new Array( "Sunday", "Monday", "Tuesday", "Wednesday",
"Thursday", "Friday", "Saturday" );
function jwday(j)
{
return mod(Math.floor((j + 1.5)), 7);
}
/* OBLIQEQ -- Calculate the obliquity of the ecliptic for a given
Julian date. This uses Laskar's tenth-degree
polynomial fit (J. Laskar, Astronomy and
Astrophysics, Vol. 157, page 68 [1986]) which is
accurate to within 0.01 arc second between AD 1000
and AD 3000, and within a few seconds of arc for
+/-10000 years around AD 2000. If we're outside the
range in which this fit is valid (deep time) we
simply return the J2000 value of the obliquity, which
happens to be almost precisely the mean. */
var oterms = new Array (
-4680.93,
-1.55,
1999.25,
-51.38,
-249.67,
-39.05,
7.12,
27.87,
5.79,
2.45
);
function obliqeq(jd)
{
var eps, u, v, i;
v = u = (jd - J2000) / (JulianCentury * 100);
eps = 23 + (26 / 60.0) + (21.448 / 3600.0);
if (Math.abs(u) < 1.0) {
for (i = 0; i < 10; i++) {
eps += (oterms[i] / 3600.0) * v;
v *= u;
}
}
return eps;
}
/* Periodic terms for nutation in longiude (delta \Psi) and
obliquity (delta \Epsilon) as given in table 21.A of
Meeus, "Astronomical Algorithms", first edition. */
var nutArgMult = new Array(
0, 0, 0, 0, 1,
-2, 0, 0, 2, 2,
0, 0, 0, 2, 2,
0, 0, 0, 0, 2,
0, 1, 0, 0, 0,
0, 0, 1, 0, 0,
-2, 1, 0, 2, 2,
0, 0, 0, 2, 1,
0, 0, 1, 2, 2,
-2, -1, 0, 2, 2,
-2, 0, 1, 0, 0,
-2, 0, 0, 2, 1,
0, 0, -1, 2, 2,
2, 0, 0, 0, 0,
0, 0, 1, 0, 1,
2, 0, -1, 2, 2,
0, 0, -1, 0, 1,
0, 0, 1, 2, 1,
-2, 0, 2, 0, 0,
0, 0, -2, 2, 1,
2, 0, 0, 2, 2,
0, 0, 2, 2, 2,
0, 0, 2, 0, 0,
-2, 0, 1, 2, 2,
0, 0, 0, 2, 0,
-2, 0, 0, 2, 0,
0, 0, -1, 2, 1,
0, 2, 0, 0, 0,
2, 0, -1, 0, 1,
-2, 2, 0, 2, 2,
0, 1, 0, 0, 1,
-2, 0, 1, 0, 1,
0, -1, 0, 0, 1,
0, 0, 2, -2, 0,
2, 0, -1, 2, 1,
2, 0, 1, 2, 2,
0, 1, 0, 2, 2,
-2, 1, 1, 0, 0,
0, -1, 0, 2, 2,
2, 0, 0, 2, 1,
2, 0, 1, 0, 0,
-2, 0, 2, 2, 2,
-2, 0, 1, 2, 1,
2, 0, -2, 0, 1,
2, 0, 0, 0, 1,
0, -1, 1, 0, 0,
-2, -1, 0, 2, 1,
-2, 0, 0, 0, 1,
0, 0, 2, 2, 1,
-2, 0, 2, 0, 1,
-2, 1, 0, 2, 1,
0, 0, 1, -2, 0,
-1, 0, 1, 0, 0,
-2, 1, 0, 0, 0,
1, 0, 0, 0, 0,
0, 0, 1, 2, 0,
-1, -1, 1, 0, 0,
0, 1, 1, 0, 0,
0, -1, 1, 2, 2,
2, -1, -1, 2, 2,
0, 0, -2, 2, 2,
0, 0, 3, 2, 2,
2, -1, 0, 2, 2
);
var nutArgCoeff = new Array(
-171996, -1742, 92095, 89, /* 0, 0, 0, 0, 1 */
-13187, -16, 5736, -31, /* -2, 0, 0, 2, 2 */
-2274, -2, 977, -5, /* 0, 0, 0, 2, 2 */
2062, 2, -895, 5, /* 0, 0, 0, 0, 2 */
1426, -34, 54, -1, /* 0, 1, 0, 0, 0 */
712, 1, -7, 0, /* 0, 0, 1, 0, 0 */
-517, 12, 224, -6, /* -2, 1, 0, 2, 2 */
-386, -4, 200, 0, /* 0, 0, 0, 2, 1 */
-301, 0, 129, -1, /* 0, 0, 1, 2, 2 */
217, -5, -95, 3, /* -2, -1, 0, 2, 2 */
-158, 0, 0, 0, /* -2, 0, 1, 0, 0 */
129, 1, -70, 0, /* -2, 0, 0, 2, 1 */
123, 0, -53, 0, /* 0, 0, -1, 2, 2 */
63, 0, 0, 0, /* 2, 0, 0, 0, 0 */
63, 1, -33, 0, /* 0, 0, 1, 0, 1 */
-59, 0, 26, 0, /* 2, 0, -1, 2, 2 */
-58, -1, 32, 0, /* 0, 0, -1, 0, 1 */
-51, 0, 27, 0, /* 0, 0, 1, 2, 1 */
48, 0, 0, 0, /* -2, 0, 2, 0, 0 */
46, 0, -24, 0, /* 0, 0, -2, 2, 1 */
-38, 0, 16, 0, /* 2, 0, 0, 2, 2 */
-31, 0, 13, 0, /* 0, 0, 2, 2, 2 */
29, 0, 0, 0, /* 0, 0, 2, 0, 0 */
29, 0, -12, 0, /* -2, 0, 1, 2, 2 */
26, 0, 0, 0, /* 0, 0, 0, 2, 0 */
-22, 0, 0, 0, /* -2, 0, 0, 2, 0 */
21, 0, -10, 0, /* 0, 0, -1, 2, 1 */
17, -1, 0, 0, /* 0, 2, 0, 0, 0 */
16, 0, -8, 0, /* 2, 0, -1, 0, 1 */
-16, 1, 7, 0, /* -2, 2, 0, 2, 2 */
-15, 0, 9, 0, /* 0, 1, 0, 0, 1 */
-13, 0, 7, 0, /* -2, 0, 1, 0, 1 */
-12, 0, 6, 0, /* 0, -1, 0, 0, 1 */
11, 0, 0, 0, /* 0, 0, 2, -2, 0 */
-10, 0, 5, 0, /* 2, 0, -1, 2, 1 */
-8, 0, 3, 0, /* 2, 0, 1, 2, 2 */
7, 0, -3, 0, /* 0, 1, 0, 2, 2 */
-7, 0, 0, 0, /* -2, 1, 1, 0, 0 */
-7, 0, 3, 0, /* 0, -1, 0, 2, 2 */
-7, 0, 3, 0, /* 2, 0, 0, 2, 1 */
6, 0, 0, 0, /* 2, 0, 1, 0, 0 */
6, 0, -3, 0, /* -2, 0, 2, 2, 2 */
6, 0, -3, 0, /* -2, 0, 1, 2, 1 */
-6, 0, 3, 0, /* 2, 0, -2, 0, 1 */
-6, 0, 3, 0, /* 2, 0, 0, 0, 1 */
5, 0, 0, 0, /* 0, -1, 1, 0, 0 */
-5, 0, 3, 0, /* -2, -1, 0, 2, 1 */
-5, 0, 3, 0, /* -2, 0, 0, 0, 1 */
-5, 0, 3, 0, /* 0, 0, 2, 2, 1 */
4, 0, 0, 0, /* -2, 0, 2, 0, 1 */
4, 0, 0, 0, /* -2, 1, 0, 2, 1 */
4, 0, 0, 0, /* 0, 0, 1, -2, 0 */
-4, 0, 0, 0, /* -1, 0, 1, 0, 0 */
-4, 0, 0, 0, /* -2, 1, 0, 0, 0 */
-4, 0, 0, 0, /* 1, 0, 0, 0, 0 */
3, 0, 0, 0, /* 0, 0, 1, 2, 0 */
-3, 0, 0, 0, /* -1, -1, 1, 0, 0 */
-3, 0, 0, 0, /* 0, 1, 1, 0, 0 */
-3, 0, 0, 0, /* 0, -1, 1, 2, 2 */
-3, 0, 0, 0, /* 2, -1, -1, 2, 2 */
-3, 0, 0, 0, /* 0, 0, -2, 2, 2 */
-3, 0, 0, 0, /* 0, 0, 3, 2, 2 */
-3, 0, 0, 0 /* 2, -1, 0, 2, 2 */
);
/* NUTATION -- Calculate the nutation in longitude, deltaPsi, and
obliquity, deltaEpsilon for a given Julian date
jd. Results are returned as a two element Array
giving (deltaPsi, deltaEpsilon) in degrees. */
function nutation(jd)
{
var deltaPsi, deltaEpsilon,
i, j,
t = (jd - 2451545.0) / 36525.0, t2, t3, to10,
ta = new Array,
dp = 0, de = 0, ang;
t3 = t * (t2 = t * t);
/* Calculate angles. The correspondence between the elements
of our array and the terms cited in Meeus are:
ta[0] = D ta[0] = M ta[2] = M' ta[3] = F ta[4] = \Omega
*/
ta[0] = dtr(297.850363 + 445267.11148 * t - 0.0019142 * t2 +
t3 / 189474.0);
ta[1] = dtr(357.52772 + 35999.05034 * t - 0.0001603 * t2 -
t3 / 300000.0);
ta[2] = dtr(134.96298 + 477198.867398 * t + 0.0086972 * t2 +
t3 / 56250.0);
ta[3] = dtr(93.27191 + 483202.017538 * t - 0.0036825 * t2 +
t3 / 327270);
ta[4] = dtr(125.04452 - 1934.136261 * t + 0.0020708 * t2 +
t3 / 450000.0);
/* Range reduce the angles in case the sine and cosine functions
don't do it as accurately or quickly. */
for (i = 0; i < 5; i++) {
ta[i] = fixangr(ta[i]);
}
to10 = t / 10.0;
for (i = 0; i < 63; i++) {
ang = 0;
for (j = 0; j < 5; j++) {
if (nutArgMult[(i * 5) + j] != 0) {
ang += nutArgMult[(i * 5) + j] * ta[j];
}
}
dp += (nutArgCoeff[(i * 4) + 0] + nutArgCoeff[(i * 4) + 1] * to10) * Math.sin(ang);
de += (nutArgCoeff[(i * 4) + 2] + nutArgCoeff[(i * 4) + 3] * to10) * Math.cos(ang);
}
/* Return the result, converting from ten thousandths of arc
seconds to radians in the process. */
deltaPsi = dp / (3600.0 * 10000.0);
deltaEpsilon = de / (3600.0 * 10000.0);
return new Array(deltaPsi, deltaEpsilon);
}
/* ECLIPTOEQ -- Convert celestial (ecliptical) longitude and
latitude into right ascension (in degrees) and
declination. We must supply the time of the
conversion in order to compensate correctly for the
varying obliquity of the ecliptic over time.
The right ascension and declination are returned
as a two-element Array in that order. */
function ecliptoeq(jd, Lambda, Beta)
{
var eps, Ra, Dec;
/* Obliquity of the ecliptic. */
eps = dtr(obliqeq(jd));
log += "Obliquity: " + rtd(eps) + "\n";
Ra = rtd(Math.atan2((Math.cos(eps) * Math.sin(dtr(Lambda)) -
(Math.tan(dtr(Beta)) * Math.sin(eps))),
Math.cos(dtr(Lambda))));
log += "RA = " + Ra + "\n";
Ra = fixangle(rtd(Math.atan2((Math.cos(eps) * Math.sin(dtr(Lambda)) -
(Math.tan(dtr(Beta)) * Math.sin(eps))),
Math.cos(dtr(Lambda)))));
Dec = rtd(Math.asin((Math.sin(eps) * Math.sin(dtr(Lambda)) * Math.cos(dtr(Beta))) +
(Math.sin(dtr(Beta)) * Math.cos(eps))));
return new Array(Ra, Dec);
}
/* DELTAT -- Determine the difference, in seconds, between
Dynamical time and Universal time. */
/* Table of observed Delta T values at the beginning of
even numbered years from 1620 through 2002. */
var deltaTtab = new Array(
121, 112, 103, 95, 88, 82, 77, 72, 68, 63, 60, 56, 53, 51, 48, 46,
44, 42, 40, 38, 35, 33, 31, 29, 26, 24, 22, 20, 18, 16, 14, 12,
11, 10, 9, 8, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10,
10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13,
13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16,
16, 16, 15, 15, 14, 13, 13.1, 12.5, 12.2, 12, 12, 12, 12, 12, 12,
11.9, 11.6, 11, 10.2, 9.2, 8.2, 7.1, 6.2, 5.6, 5.4, 5.3, 5.4, 5.6,
5.9, 6.2, 6.5, 6.8, 7.1, 7.3, 7.5, 7.6, 7.7, 7.3, 6.2, 5.2, 2.7,
1.4, -1.2, -2.8, -3.8, -4.8, -5.5, -5.3, -5.6, -5.7, -5.9, -6,
-6.3, -6.5, -6.2, -4.7, -2.8, -0.1, 2.6, 5.3, 7.7, 10.4, 13.3, 16,
18.2, 20.2, 21.1, 22.4, 23.5, 23.8, 24.3, 24, 23.9, 23.9, 23.7,
24, 24.3, 25.3, 26.2, 27.3, 28.2, 29.1, 30, 30.7, 31.4, 32.2,
33.1, 34, 35, 36.5, 38.3, 40.2, 42.2, 44.5, 46.5, 48.5, 50.5,
52.2, 53.8, 54.9, 55.8, 56.9, 58.3, 60, 61.6, 63, 65, 66.6
);
function deltat(year)
{
var dt, f, i, t;
if ((year >= 1620) && (year <= 2000)) {
i = Math.floor((year - 1620) / 2);
f = ((year - 1620) / 2) - i; /* Fractional part of year */
dt = deltaTtab[i] + ((deltaTtab[i + 1] - deltaTtab[i]) * f);
} else {
t = (year - 2000) / 100;
if (year < 948) {
dt = 2177 + (497 * t) + (44.1 * t * t);
} else {
dt = 102 + (102 * t) + (25.3 * t * t);
if ((year > 2000) && (year < 2100)) {
dt += 0.37 * (year - 2100);
}
}
}
return dt;
}
/* EQUINOX -- Determine the Julian Ephemeris Day of an
equinox or solstice. The "which" argument
selects the item to be computed:
0 March equinox
1 June solstice
2 September equinox
3 December solstice
*/
// Periodic terms to obtain true time
var EquinoxpTerms = new Array(
485, 324.96, 1934.136,
203, 337.23, 32964.467,
199, 342.08, 20.186,
182, 27.85, 445267.112,
156, 73.14, 45036.886,
136, 171.52, 22518.443,
77, 222.54, 65928.934,
74, 296.72, 3034.906,
70, 243.58, 9037.513,
58, 119.81, 33718.147,
52, 297.17, 150.678,
50, 21.02, 2281.226,
45, 247.54, 29929.562,
44, 325.15, 31555.956,
29, 60.93, 4443.417,
18, 155.12, 67555.328,
17, 288.79, 4562.452,
16, 198.04, 62894.029,
14, 199.76, 31436.921,
12, 95.39, 14577.848,
12, 287.11, 31931.756,
12, 320.81, 34777.259,
9, 227.73, 1222.114,
8, 15.45, 16859.074
);
JDE0tab1000 = new Array(
new Array(1721139.29189, 365242.13740, 0.06134, 0.00111, -0.00071),
new Array(1721233.25401, 365241.72562, -0.05323, 0.00907, 0.00025),
new Array(1721325.70455, 365242.49558, -0.11677, -0.00297, 0.00074),
new Array(1721414.39987, 365242.88257, -0.00769, -0.00933, -0.00006)
);
JDE0tab2000 = new Array(
new Array(2451623.80984, 365242.37404, 0.05169, -0.00411, -0.00057),
new Array(2451716.56767, 365241.62603, 0.00325, 0.00888, -0.00030),
new Array(2451810.21715, 365242.01767, -0.11575, 0.00337, 0.00078),
new Array(2451900.05952, 365242.74049, -0.06223, -0.00823, 0.00032)
);
function equinox(year, which)
{
var deltaL, i, j, JDE0, JDE, JDE0tab, S, T, W, Y;
/* Initialise terms for mean equinox and solstices. We
have two sets: one for years prior to 1000 and a second
for subsequent years. */
if (year < 1000) {
JDE0tab = JDE0tab1000;
Y = year / 1000;
} else {
JDE0tab = JDE0tab2000;
Y = (year - 2000) / 1000;
}
JDE0 = JDE0tab[which][0] +
(JDE0tab[which][1] * Y) +
(JDE0tab[which][2] * Y * Y) +
(JDE0tab[which][3] * Y * Y * Y) +
(JDE0tab[which][4] * Y * Y * Y * Y);
//document.debug.log.value += "JDE0 = " + JDE0 + "\n";
T = (JDE0 - 2451545.0) / 36525;
//document.debug.log.value += "T = " + T + "\n";
W = (35999.373 * T) - 2.47;
//document.debug.log.value += "W = " + W + "\n";
deltaL = 1 + (0.0334 * dcos(W)) + (0.0007 * dcos(2 * W));
//document.debug.log.value += "deltaL = " + deltaL + "\n";
// Sum the periodic terms for time T
S = 0;
for (i = j = 0; i < 24; i++) {
S += EquinoxpTerms[j] * dcos(EquinoxpTerms[j + 1] + (EquinoxpTerms[j + 2] * T));
j += 3;
}
//document.debug.log.value += "S = " + S + "\n";
//document.debug.log.value += "Corr = " + ((S * 0.00001) / deltaL) + "\n";
JDE = JDE0 + ((S * 0.00001) / deltaL);
return JDE;
}
/* SUNPOS -- Position of the Sun. Please see the comments
on the return statement at the end of this function
which describe the array it returns. We return
intermediate values because they are useful in a
variety of other contexts. */
function sunpos(jd)
{
var T, T2, L0, M, e, C, sunLong, sunAnomaly, sunR,
Omega, Lambda, epsilon, epsilon0, Alpha, Delta,
AlphaApp, DeltaApp;
T = (jd - J2000) / JulianCentury;
//document.debug.log.value += "Sunpos. T = " + T + "\n";
T2 = T * T;
L0 = 280.46646 + (36000.76983 * T) + (0.0003032 * T2);
//document.debug.log.value += "L0 = " + L0 + "\n";
L0 = fixangle(L0);
//document.debug.log.value += "L0 = " + L0 + "\n";
M = 357.52911 + (35999.05029 * T) + (-0.0001537 * T2);
//document.debug.log.value += "M = " + M + "\n";
M = fixangle(M);
//document.debug.log.value += "M = " + M + "\n";
e = 0.016708634 + (-0.000042037 * T) + (-0.0000001267 * T2);
//document.debug.log.value += "e = " + e + "\n";
C = ((1.914602 + (-0.004817 * T) + (-0.000014 * T2)) * dsin(M)) +
((0.019993 - (0.000101 * T)) * dsin(2 * M)) +
(0.000289 * dsin(3 * M));
//document.debug.log.value += "C = " + C + "\n";
sunLong = L0 + C;
//document.debug.log.value += "sunLong = " + sunLong + "\n";
sunAnomaly = M + C;
//document.debug.log.value += "sunAnomaly = " + sunAnomaly + "\n";
sunR = (1.000001018 * (1 - (e * e))) / (1 + (e * dcos(sunAnomaly)));
//document.debug.log.value += "sunR = " + sunR + "\n";
Omega = 125.04 - (1934.136 * T);
//document.debug.log.value += "Omega = " + Omega + "\n";
Lambda = sunLong + (-0.00569) + (-0.00478 * dsin(Omega));
//document.debug.log.value += "Lambda = " + Lambda + "\n";
epsilon0 = obliqeq(jd);
//document.debug.log.value += "epsilon0 = " + epsilon0 + "\n";
epsilon = epsilon0 + (0.00256 * dcos(Omega));
//document.debug.log.value += "epsilon = " + epsilon + "\n";
Alpha = rtd(Math.atan2(dcos(epsilon0) * dsin(sunLong), dcos(sunLong)));
//document.debug.log.value += "Alpha = " + Alpha + "\n";
Alpha = fixangle(Alpha);
////document.debug.log.value += "Alpha = " + Alpha + "\n";
Delta = rtd(Math.asin(dsin(epsilon0) * dsin(sunLong)));
////document.debug.log.value += "Delta = " + Delta + "\n";
AlphaApp = rtd(Math.atan2(dcos(epsilon) * dsin(Lambda), dcos(Lambda)));
//document.debug.log.value += "AlphaApp = " + AlphaApp + "\n";
AlphaApp = fixangle(AlphaApp);
//document.debug.log.value += "AlphaApp = " + AlphaApp + "\n";
DeltaApp = rtd(Math.asin(dsin(epsilon) * dsin(Lambda)));
//document.debug.log.value += "DeltaApp = " + DeltaApp + "\n";
return new Array( // Angular quantities are expressed in decimal degrees
L0, // [0] Geometric mean longitude of the Sun
M, // [1] Mean anomaly of the Sun
e, // [2] Eccentricity of the Earth's orbit
C, // [3] Sun's equation of the Centre
sunLong, // [4] Sun's true longitude
sunAnomaly, // [5] Sun's true anomaly
sunR, // [6] Sun's radius vector in AU
Lambda, // [7] Sun's apparent longitude at true equinox of the date
Alpha, // [8] Sun's true right ascension
Delta, // [9] Sun's true declination
AlphaApp, // [10] Sun's apparent right ascension
DeltaApp // [11] Sun's apparent declination
);
}
/* EQUATIONOFTIME -- Compute equation of time for a given moment.
Returns the equation of time as a fraction of
a day. */
function equationOfTime(jd)
{
var alpha, deltaPsi, E, epsilon, L0, tau
tau = (jd - J2000) / JulianMillennium;
//document.debug.log.value += "equationOfTime. tau = " + tau + "\n";
L0 = 280.4664567 + (360007.6982779 * tau) +
(0.03032028 * tau * tau) +
((tau * tau * tau) / 49931) +
(-((tau * tau * tau * tau) / 15300)) +
(-((tau * tau * tau * tau * tau) / 2000000));
//document.debug.log.value += "L0 = " + L0 + "\n";
L0 = fixangle(L0);
//document.debug.log.value += "L0 = " + L0 + "\n";
alpha = sunpos(jd)[10];
//document.debug.log.value += "alpha = " + alpha + "\n";
deltaPsi = nutation(jd)[0];
//document.debug.log.value += "deltaPsi = " + deltaPsi + "\n";
epsilon = obliqeq(jd) + nutation(jd)[1];
//document.debug.log.value += "epsilon = " + epsilon + "\n";
E = L0 + (-0.0057183) + (-alpha) + (deltaPsi * dcos(epsilon));
//document.debug.log.value += "E = " + E + "\n";
E = E - 20.0 * (Math.floor(E / 20.0));
//document.debug.log.value += "Efixed = " + E + "\n";
E = E / (24 * 60);
//document.debug.log.value += "Eday = " + E + "\n";
return E;
}