PHP Science Labs - Obliquity of the Ecliptic

OBLIQUITY OF THE ECLIPTIC AND NUTATION IN OBLIQUITY NUTATION COMPUTED USING BOTH IAU 1980 SERIES AND 2000B SERIES DATE ISO YMD = 20250113 = 2025 January 13 = Monday (Gregorian) TIME UT HMS = 00 00 00.000 = 0 day ±DELTA T HMS = +00 00 00.000 = 0 day JD NUMBER FOR DATE, TIME AND DELTA T JD = 2460688.5 TIME VARIABLE CORRESPONDING TO JD T = +0.250335386721 = Julian centuries reckoned from J2000.0 ------------------------------------------------------------- OBLIQUITY OF THE ECLIPTIC Eps Mean = 23.4360361104 = 23° 26' 09.730" (Laskar) Eps True = 23.4384252647 = 23° 26' 18.331" (Using IAU 1980 nutation series) Eps True = 23.4384269752 = 23° 26' 18.337" (Using IAU 2000B nutation series) NUTATION IN OBLIQUITY dEps = +0.0023891543° = +8.601" (IAU 1980 series) = +0.0023908648° = +8.607" (IAU 2000B series)

The obliquity of the ecliptic is essentially the angle of inclination of the Earth's equator with respect to the plane of its orbit, which to us, produces the apparent relative tilt of the Earth's polar axis which causes the annual seasons.

This angle is approximately ±23.45 degrees and is the apparent maximum/minimum angle of the sun above or below the equator during the year as viewed from Earth. This value varies very slowly as the orbit of the Earth evolves over time.

This is one of the values we need to know when computing the apparent position of a planet as viewed from the Earth as it defines the instantaneous inclination of the Earth's orbit.

http://en.wikipedia.org/wiki/Ecliptic

The algorithm used here is based on data published by J. Laskar in
Astronomy and Astrophysics, Vol 157, p68 (1986),
New Formulas for the Precession, Valid Over 10000 years,
Table 8.

Excerpt From Paper

Excerpt From Paper

Provided by the NASA Astrophysics Data System

J. Laskar's Formula For The Mean Obliquity

In this case, the time variable (t) of Table 8 is reckoned in terms
of 10000 Julian years from J2000.0 corresponding to the JD and
may be found from

Time In Units of 100000 Julian Years

Time In Units of 100000 Julian Years

The value 84381.448 seconds of arc equates to 23° 26' 21.448"
which is the mean obliquity for the ecliptic of J2000.0

Based on the Table 8 data above (bottom NGT column), the
mean obliquity (in arc seconds) of the ecliptic at time (t) is:

J. Laskar's Mean Obliquity Formula

J. Laskar's Mean Obliquity Formula

This value of the obliquity is the mean value. For the true, apparent obliquity of the date, a small correction for nutation in obliquity must be applied.

The following functions return the mean obliquity in decimal degrees for any given JD argument, based on J. Laskar's long-term mean obliquity equation.

See also:
Computing Nutations

PHP function to compute the mean obliquity in decimal degrees for any given JD argument


/*

   This PHP function computes the mean obliquity of the ecliptic
   given a JD argument corresponding to any given date and time.

   Author: Jay Tanner - 2010

   The algorithm used here is based on work published by J. Laskar
   Astronomy and Astrophysics, Vol 157, p68 (1986),
   New Formulas for the Precession, Valid Over 10000 years,
   Table 8.

   Source code provided under the provisions of the
   GNU Affero General Public License (AGPL), version 3.
   http://www.gnu.org/licenses/agpl.html

*/


   function epsilon_mean ($JD)
{

// -----------------------------------------------------------
// Compute the (t) value in Julian decamillennia corresponding
// to the JD argument and reckoned from J2000.
   $t = ($JD - 2451545.0) / 3652500.0;

// --------------------------------------
// Compute mean obliquity in arc seconds.
   $w  = 84381.448;   $p  = $t;
   $w -=  4680.93*$p; $p *= $t;
   $w -=     1.55*$p; $p *= $t;
   $w +=  1999.25*$p; $p *= $t;
   $w -=    51.38*$p; $p *= $t;
   $w -=   249.67*$p; $p *= $t;
   $w -=    39.05*$p; $p *= $t;
   $w +=     7.12*$p; $p *= $t;
   $w +=    27.87*$p; $p *= $t;
   $w +=     5.79*$p; $p *= $t;
   $w +=     2.45*$p;

// ----------------------------------
// Compute mean ecliptic obliquity in
// degrees from arc seconds value.
   $EpsMeanDeg = $w / 3600.0;

   return $EpsMeanDeg;

} // End of  epsilon_mean()

CPP function to compute the mean obliquity in decimal degrees for any given JD argument


/*

   This CPP/C++ function computes the mean obliquity of the ecliptic
   given a JD argument corresponding to any given date and time.

   Author: Jay Tanner - 2010

   The algorithm used here is based on work published by J. Laskar
   Astronomy and Astrophysics, Vol 157, p68 (1986),
   New Formulas for the Precession, Valid Over 10000 years,
   Table 8.

   Source code provided under the provisions of the
   GNU General Public License (GPL), version 3.
   http://www.gnu.org/licenses/gpl.html

*/


   double epsilon_mean (double JD)
{

// Define working variables.
   double t, p, w, EpsMeanDeg;

// -----------------------------------------------------------
// Compute the (t) value in Julian decamillennia corresponding
// to the JD argument and reckoned from J2000.
   t = (JD - 2451545.0) / 3652500.0;

// --------------------------------------
// Compute mean obliquity in arc seconds.
   w  = 84381.448;  p  = t;
   w -=  4680.93*p; p *= t;
   w -=     1.55*p; p *= t;
   w +=  1999.25*p; p *= t;
   w -=    51.38*p; p *= t;
   w -=   249.67*p; p *= t;
   w -=    39.05*p; p *= t;
   w +=     7.12*p; p *= t;
   w +=    27.87*p; p *= t;
   w +=     5.79*p; p *= t;
   w +=     2.45*p;

// ----------------------------------
// Compute mean ecliptic obliquity in
// degrees from arc seconds value.
   EpsMeanDeg = w / 3600;

   return EpsMeanDeg;

} // End of  epsilon_mean()

VB.NET function to compute the mean obliquity in decimal degrees for any given JD argument


' THIS VB.NET FUNCTION COMPUTES THE MEAN OBLIQUITY OF THE ECLIPTIC
' GIVEN A JD ARGUMENT CORRESPONDING TO ANY GIVEN DATE AND TIME.
'
' AUTHOR: Jay Tanner - Revised 2011 Oct 23
'
' The algorithm used here is based on work published by J. Laskar
' Astronomy and Astrophysics, Vol 157, p68 (1986),
' New Formulas for the Precession, Valid Over 10000 years,
' Table 8.
'
' SOURCE CODE PROVIDED UNDER THE PROVISIONS OF THE
' GNU GENERAL PUBLIC LICENSE (GPL), VERSION 3.
' http://www.gnu.org/licenses/gpl.html


Function Epsilon_Mean (ByVal JD As Double) As Double

  Dim p, t, s As Double


' -----------------------------------------------
' COMPUTE TIME IN JULIAN DECAMILLENNIA FROM J2000
' CORRESPONDING TO THE JD ARGUMENT.

  t = (JD - 2451545.0) / 3652500.0

' --------------------------------------
' COMPUTE MEAN OBLIQUITY IN ARC SECONDS.

  s  = 84381.448  : p  = t
  s -=  4680.93*p : p *= t
  s -=     1.55*p : p *= t
  s +=  1999.25*p : p *= t
  s -=    51.38*p : p *= t
  s -=   249.67*p : p *= t
  s -=    39.05*p : p *= t
  s +=     7.12*p : p *= t
  s +=    27.87*p : p *= t
  s +=     5.79*p : p *= t
  s +=     2.45*p

' ------------------------------
' RETURN MEAN ECLIPTIC OBLIQUITY
' EXPRESSED IN DECIMAL DEGREES.

  Return s / 3600.0

End Function

© Jay Tanner - PHP Science Labs - 2025