Nutation In Ecliptical Longitude and RA (IAU 1980 Theory)

Algorithm to compute nutation in ecliptical longitude (dPsiDeg) for a given JD argument.


 T  = (JD - 2451545) / 36525
 T2 = T*T
 T3 = T*T2

 DegToRad = 3.1415926535897932 / 180

 w1 = 297.85036 + 445267.11148*T - 0.0019142*T2 + (T3 / 189474)
 w1 = DegToRad*(w1)

 w2 = 357.52772 + 35999.05034*T - 0.0001603*T2 - (T3 / 300000)
 w2 = DegToRad*(w2)

 w3 = 134.96298 + 477198.867398*T + 0.0086972*T2 + (T3 / 56250)
 w3 = DegToRad*(w3)

 w4 = 93.27191 + 483202.017538*T - 0.0036825*T2 + (T3 / 327270)
 w4 = DegToRad*(w4)

 w5 = 125.04452 - 1934.136261*T + 0.0020708*T2 + (T3 / 450000)
 w5 = DegToRad*(w5)

  w = sin(w5)*(-174.2*T - 171996)
  w = w + sin(2*(w4 + w5 - w1))*(-1.6*T - 13187)
  w = w + sin(2*(w4 + w5))*(-2274 - 0.2*T)
  w = w + sin(2 * w5)*(0.2*T + 2062)
  w = w + sin(w2)*(1426 - 3.4*T)
  w = w + sin(w3)*(0.1*T + 712)
  w = w + sin(2*(w4 + w5 - w1) + w2)*(1.2*T - 517)
  w = w + sin(2 * w4 + w5)*(-0.4*T - 386)
  w = w + sin(2*(w4 + w5 - w1) - w2)*(217 - 0.5*T)
  w = w + sin(2*(w4 - w1) + w5)*(129 + 0.1*T)
  w = w + sin(w3 + w5)*(0.1*T + 63)
  w = w + sin(w5 - w3)*(-0.1*T - 58)
  w = w + sin(2*w2)*(17 - 0.1*T)
  w = w + sin(2*(w2 + w4 + w5 - w1))*(0.1*T - 16)
  w = w - 301*sin(2*(w4 + w5) + w3)
  w = w - 158*sin(w3 - 2*w1)
  w = w + 123*sin(2*(w4 + w5) - w3)
  w = w +  63*sin(2*w1)
  w = w -  59*sin(2*(w1 + w4 + w5) - w3)
  w = w -  51*sin(2 * w4 + w3 + w5)
  w = w +  48*sin(2*(w3 - w1))
  w = w +  46*sin(2*(w4 - w3) + w5)
  w = w -  38*sin(2*(w1 + w4 + w5))
  w = w -  31*sin(2*(w3 + w4 + w5))
  w = w +  29*sin(2*w3)
  w = w +  29*sin(2*(w4 + w5 - w1) + w3)
  w = w +  26*sin(2*w4)
  w = w -  22*sin(2*(w4 - w1))
  w = w +  21*sin(2*w4 + w5 - w3)
  w = w +  16*sin(2*w1 - w3 + w5)
  w = w -  15*sin(w2 + w5)
  w = w -  13*sin(w3 + w5 - 2*w1)
  w = w -  12*sin(w5 - w2)
  w = w +  11*sin(2*(w3 - w4))
  w = w -  10*sin(2*(w4 + w1) + w5 - w3)
  w = w -   8*sin(2*(w4 + w1 + w5) + w3)
  w = w +   7*sin(2*(w4 + w5) + w2)
  w = w -   7*sin(w3 - 2*w1 + w2)
  w = w -   7*sin(2*(w4 + w5) - w2)
  w = w -   7*sin(2*w1 + 2*w4 + w5)
  w = w +   6*sin(2*w1 + w3)
  w = w +   6*sin(2*(w3 + w4 + w5 - w1))
  w = w +   6*sin(2*(w4 - w1) + w3 + w5)
  w = w -   6*sin(2*(w1 - w3) + w5)
  w = w -   6*sin(2*w1 + w5)
  w = w +   5*sin(w3 - w2)
  w = w -   5*sin(2*(w4 - w1) + w5 - w2)
  w = w -   5*sin(w5 - 2*w1)
  w = w -   5*sin(2*(w3 + w4) + w5)
  w = w +   4*sin(2*(w3 - w1) + w5)
  w = w +   4*sin(2*(w4 - w1) + w2 + w5)
  w = w +   4*sin(w3 - 2*w4)
  w = w -   4*sin(w3 - w1)
  w = w -   4*sin(w2 - 2*w1)
  w = w -   4*sin(w1)
  w = w +   3*sin(2*w4 + w3)
  w = w -   3*sin(2*(w4 + w5 - w3))
  w = w -   3*sin(w3 - w1 - w2)
  w = w -   3*sin(w2 + w3)
  w = w -   3*sin(2*(w4 + w5) + w3 - w2)
  w = w -   3*sin(2*(w1 + w4 + w5) - w2 - w3)
  w = w -   3*sin(2*(w4 + w5) + 3*w3)
  w = w -   3*sin(2*(w1 + w4 + w5) - w2)

  dPsiDeg = w / 36000000.0

PHP function to compute nutation in ecliptical longitude (dPsi) for a given JD argument.


    function delta_psi ($JD)
 {

 // -----------------------
 // READ JD ARGUMENT VALUE.
 // ERROR IF ARGUMENT IS NOT NUMERIC.

    if (!Is_Numeric(trim($JD))) {return FALSE;}

 // -----------------------------------------------------
 // COMPUTE POWERS OF TIME IN JULIAN CENTURIES FROM J2000

    $T1 = ($JD - 2451545) / 36525;
    $T2 = $T1*$T1;
    $T3 = $T2*$T1;

 // -----------------------------------------
 // COMPUTE MEAN LUNAR ELONGATION IN RADIANS.

    $w1 = 297.85036 + 445267.11148*$T1
        - 0.0019142*$T2 + ($T3 / 189474);
    $w1 = Deg2Rad($w1);

 // --------------------------------------
 // COMPUTE MEAN SOLAR ANOMALY IN RADIANS.

    $w2 = 357.52772 + 35999.05034*$T1
        - 0.0001603*$T2 - ($T3 / 300000);
    $w2 = Deg2Rad($w2);

 // --------------------------------------
 // COMPUTE MEAN LUNAR ANOMALY IN RADIANS.

    $w3 = 134.96298 + 477198.867398*$T1
        + 0.0086972*$T2 + ($T3 / 56250);
    $w3 = Deg2Rad($w3);

 // ----------------------------------------------
 // COMPUTE LUNAR ARGUMENT OF LATITUDE IN RADIANS.

    $w4 = 93.27191 + 483202.017538*$T1
        - 0.0036825*$T2 + ($T3 / 327270);
    $w4 = Deg2Rad($w4);

 // ----------------------------------------
 // COMPUTE LUNAR ASCENDING NODE IN RADIANS.

    $w5 = 125.04452 - 1934.136261*$T1 + 0.0020708*$T2
        + ($T3 / 450000);
    $w5 = Deg2Rad($w5);

 // ---------------------------------------------
 // COMPUTE THE NUTATION IN ECLIPTICAL LONGITUDE.

    $w  = sin($w5)*(-174.2*$T1 - 171996);
    $w += sin(2*($w4 + $w5 - $w1))*(-1.6*$T1 - 13187);
    $w += sin(2*($w4 + $w5))*(-2274 - 0.2*$T1);
    $w += sin(2 * $w5)*(0.2*$T1 + 2062);
    $w += sin($w2)*(1426 - 3.4*$T1);
    $w += sin($w3)*(0.1*$T1 + 712);
    $w += sin(2*($w4 + $w5 - $w1) + $w2)*(1.2*$T1 - 517);
    $w += sin(2 * $w4 + $w5)*(-0.4*$T1 - 386);
    $w += sin(2*($w4 + $w5 - $w1) - $w2)*(217 - 0.5*$T1);
    $w += sin(2*($w4 - $w1) + $w5)*(129 + 0.1*$T1);
    $w += sin($w3 + $w5)*(0.1*$T1 + 63);
    $w += sin($w5 - $w3)*(-0.1*$T1 - 58);
    $w += sin(2*$w2)*(17 - 0.1*$T1);
    $w += sin(2*($w2 + $w4 + $w5 - $w1))*(0.1*$T1 - 16);
    $w -= 301*sin(2*($w4 + $w5) + $w3);
    $w -= 158*sin($w3 - 2*$w1);
    $w += 123*sin(2*($w4 + $w5) - $w3);
    $w +=  63*sin(2*$w1);
    $w -=  59*sin(2*($w1 + $w4 + $w5) - $w3);
    $w -=  51*sin(2 * $w4 + $w3 + $w5);
    $w +=  48*sin(2*($w3 - $w1));
    $w +=  46*sin(2*($w4 - $w3) + $w5);
    $w -=  38*sin(2*($w1 + $w4 + $w5));
    $w -=  31*sin(2*($w3 + $w4 + $w5));
    $w +=  29*sin(2*$w3);
    $w +=  29*sin(2*($w4 + $w5 - $w1) + $w3);
    $w +=  26*sin(2*$w4);
    $w -=  22*sin(2*($w4 - $w1));
    $w +=  21*sin(2*$w4 + $w5 - $w3);
    $w +=  16*sin(2*$w1 - $w3 + $w5);
    $w -=  15*sin($w2 + $w5);
    $w -=  13*sin($w3 + $w5 - 2*$w1);
    $w -=  12*sin($w5 - $w2);
    $w +=  11*sin(2*($w3 - $w4));
    $w -=  10*sin(2*($w4 + $w1) + $w5 - $w3);
    $w -=   8*sin(2*($w4 + $w1 + $w5) + $w3);
    $w +=   7*sin(2*($w4 + $w5) + $w2);
    $w -=   7*sin($w3 - 2*$w1 + $w2);
    $w -=   7*sin(2*($w4 + $w5) - $w2);
    $w -=   7*sin(2*$w1 + 2*$w4 + $w5);
    $w +=   6*sin(2*$w1 + $w3);
    $w +=   6*sin(2*($w3 + $w4 + $w5 - $w1));
    $w +=   6*sin(2*($w4 - $w1) + $w3 + $w5);
    $w -=   6*sin(2*($w1 - $w3) + $w5);
    $w -=   6*sin(2*$w1 + $w5);
    $w +=   5*sin($w3 - $w2);
    $w -=   5*sin(2*($w4 - $w1) + $w5 - $w2);
    $w -=   5*sin($w5 - 2*$w1);
    $w -=   5*sin(2*($w3 + $w4) + $w5);
    $w +=   4*sin(2*($w3 - $w1) + $w5);
    $w +=   4*sin(2*($w4 - $w1) + $w2 + $w5);
    $w +=   4*sin($w3 - 2*$w4);
    $w -=   4*sin($w3 - $w1);
    $w -=   4*sin($w2 - 2*$w1);
    $w -=   4*sin($w1);
    $w +=   3*sin(2 * $w4 + $w3);
    $w -=   3*sin(2*($w4 + $w5 - $w3));
    $w -=   3*sin($w3 - $w1 - $w2);
    $w -=   3*sin($w2 + $w3);
    $w -=   3*sin(2*($w4 + $w5) + $w3 - $w2);
    $w -=   3*sin(2*($w1 + $w4 + $w5) - $w2 - $w3);
    $w -=   3*sin(2*($w4 + $w5) + 3*$w3);
    $w -=   3*sin(2*($w1 + $w4 + $w5) - $w2);

 // ------------------------------------
 // DONE.  RETURN NUTATION IN ECLIPTICAL
 // LONGITUDE IN DECIMAL DEGREES.

    return $w / 36000000.0;

  }

CPP function to compute nutation in ecliptical longitude (dPsi) for a given JD argument.


   double delta_psi (double JD)
{

// -------------------------
// DEFINE WORKING VARIABLES.

   double T, T2, T3;

   double w, w1, w2, w3, w4, w5;

   double DegToRad, dPsiDeg;

// ---------------------------------
// COMPUTE (T) IN JULIAN CENTURIES
// FROM J2000.0 AND POWERS OF (T).
   T = (JD - 2451545.0) / 36525.0;
   T2 = T*T;
   T3 = T*T*T;

// ------------------------------------------------
// DEFINE CONVERSION FACTOR FOR DEGREES TO RADIANS.
// MULTIPLY DEGREES BY THIS FACTOR TO GET RADIANS.

   DegToRad = 3.1415926535897932 / 180;

// ---------------------------------------------------
// COMPUTE THE MEAN ELONGATION OF THE MOON IN RADIANS.

   w1 = DegToRad*(297.85036 + 445267.11148*T
      - 0.0019142*T2 + (T3 / 189474));

// -----------------------------------------------
// COMPUTE THE MEAN ANOMALY OF THE SUN IN RADIANS.

   w2 = DegToRad*(357.52772 + 35999.05034*T
      - 0.0001603*T2 - (T3 / 300000));

// ------------------------------------------------
// COMPUTE THE MEAN ANOMALY OF THE MOON IN RADIANS.

   w3 = DegToRad*(134.96298 + 477198.867398*T
      + 0.0086972*T2 + (T3 / 56250));

// ---------------------------------------------------
// COMPUTE THE MOON'S ARGUMENT OF LATITUDE IN RADIANS.

   w4 = DegToRad*(93.27191 + 483202.017538*T
      - 0.0036825*T2 + (T3 / 327270));

// ----------------------------------------------------------
// COMPUTE THE LONGITUDE OF MOON'S ASCENDING NODE IN RADIANS.

   w5 = DegToRad*(125.04452 - 1934.136261*T + 0.0020708*T2
      + (T3 / 450000));

// ---------------------------------------------
// COMPUTE THE NUTATION IN ECLIPTICAL LONGITUDE.

   w = sin(w5)*(-174.2*T - 171996);

   w += sin(2*(w4 + w5 - w1))*(-1.6*T - 13187);
   w += sin(2*(w4 + w5))*(-2274 - 0.2*T);
   w += sin(2 * w5)*(0.2*T + 2062);
   w += sin(w2)*(1426 - 3.4*T);
   w += sin(w3)*(0.1*T + 712);
   w += sin(2*(w4 + w5 - w1) + w2)*(1.2*T - 517);
   w += sin(2*w4 + w5)*(-0.4*T - 386);
   w -= 301*sin(2*(w4 + w5) + w3);
   w += sin(2*(w4 + w5 - w1) - w2)*(217 - 0.5*T);
   w -= 158*sin(w3 - 2*w1);
   w += sin(2*(w4 - w1) + w5)*(129 + 0.1*T);
   w += 123*sin(2*(w4 + w5) - w3);
   w +=  63*sin(2*w1);
   w += sin(w3 + w5)*(0.1*T + 63);
   w -=  59*sin(2*(w1 + w4 + w5) - w3);
   w += sin(w5 - w3)*(-0.1*T - 58);
   w -=  51*sin(2*w4 + w3 + w5);
   w +=  48*sin(2*(w3 - w1));
   w +=  46*sin(2*(w4 - w3) + w5);
   w -=  38*sin(2*(w1 + w4 + w5));
   w -=  31*sin(2*(w3 + w4 + w5));
   w +=  29*sin(2*w3);
   w +=  29*sin(2*(w4 + w5 - w1) + w3);
   w +=  26*sin(2*w4);
   w -=  22*sin(2*(w4 - w1));
   w +=  21*sin(2*w4 + w5 - w3);
   w += sin(2*w2)*(17 - 0.1*T);
   w +=  16*sin(2*w1 - w3 + w5);
   w += sin(2*(w2 + w4 + w5 - w1))*(0.1*T - 16);
   w -=  15*sin(w2 + w5);
   w -=  13*sin(w3 + w5 - 2*w1);
   w -=  12*sin(w5 - w2);
   w +=  11*sin(2*(w3 - w4));
   w -=  10*sin(2*(w4 + w1) + w5 - w3);
   w -=   8*sin(2*(w4 + w1 + w5) + w3);
   w +=   7*sin(2*(w4 + w5) + w2);
   w -=   7*sin(w3 - 2*w1 + w2);
   w -=   7*sin(2*(w4 + w5) - w2);
   w -=   7*sin(2*w1 + 2*w4 + w5);
   w +=   6*sin(2*w1 + w3);
   w +=   6*sin(2*(w3 + w4 + w5 - w1));
   w +=   6*sin(2*(w4 - w1) + w3 + w5);
   w -=   6*sin(2*(w1 - w3) + w5);
   w -=   6*sin(2*w1 + w5);
   w +=   5*sin(w3 - w2);
   w -=   5*sin(2*(w4 - w1) + w5 - w2);
   w -=   5*sin(w5 - 2*w1);
   w -=   5*sin(2*(w3 + w4) + w5);
   w +=   4*sin(2*(w3 - w1) + w5);
   w +=   4*sin(2*(w4 - w1) + w2 + w5);
   w +=   4*sin(w3 - 2*w4);
   w -=   4*sin(w3 - w1);
   w -=   4*sin(w2 - 2*w1);
   w -=   4*sin(w1);
   w +=   3*sin(2*w4 + w3);
   w -=   3*sin(2*(w4 + w5 - w3));
   w -=   3*sin(w3 - w1 - w2);
   w -=   3*sin(w2 + w3);
   w -=   3*sin(2*(w4 + w5) + w3 - w2);
   w -=   3*sin(2*(w1 + w4 + w5) - w2 - w3);
   w -=   3*sin(2*(w4 + w5) + 3*w3);
   w -=   3*sin(2*(w1 + w4 + w5) - w2);

// ------------------------------------
// COVERT DELTA PSI TO DECIMAL DEGREES.

   dPsiDeg = w / 36000000.0;

// -----
// Done.

   return dPsiDeg;

  }

VB.NET function to compute nutation in ecliptical longitude (dPsi) for a given
JD argument.


 Public Function Delta_Psi(ByVal JD As Double) As Double

 Dim T, T2, T3, w, w1, w2, w3, w4, w5, DegToRad As Double

 DegToRad = Math.PI / 180

 ' -------------------------------------------
 ' COMPUTE TIME FACTOR (T) IN JULIAN CENTURIES
 ' AS RECKONED FROM J2000 AND POWERS OF T.

 T = (JD - 2451545.0) / 36525.0
 T2 = T*T
 T3 = T2 * T

 ' -----------------------------------------
 ' COMPUTE MEAN LUNAR ELONGATION IN RADIANS.

 w1 = DegToRad * (297.85036 + 445267.11148*T _
               - 0.0019142*T2 + (T3 / 189474))

 ' --------------------------------------
 ' COMPUTE MEAN SOLAR ANOMALY IN RADIANS.

 w2 = DegToRad * (357.52772 + 35999.05034*T
               - 0.0001603*T2 - (T3 / 300000))

 ' --------------------------------------
 ' COMPUTE MEAN LUNAR ANOMALY IN RADIANS.

 w3 = DegToRad * (134.96298 + 477198.867398*T _
               + 0.0086972*T2 + (T3 / 56250))

 ' ----------------------------------------------
 ' COMPUTE LUNAR ARGUMENT OF LATITUDE IN RADIANS.

 w4 = DegToRad * (93.27191 + 483202.017538*T _
               - 0.0036825*T2 + (T3 / 327270))

 ' -----------------------------------------------------
 ' COMPUTE LONGITUDE OF LUNAR ASCENDING NODE IN RADIANS.

 w5 = DegToRad * (125.04452 - 1934.136261*T _
               + 0.0020708*T2 + (T3 / 450000))

 ' --------------------------------------------
 ' COMPUTE THE NUTATION IN ECLIPTICAL LONGITUDE
 ' IN UNITS OF (ARCSEC * 10000).

 w  = Math.Sin(w5)*(-174.2*T - 171996)
 w += Math.Sin(2*(w4 + w5 - w1))*(-1.6*T - 13187)
 w += Math.Sin(2*(w4 + w5))*(-2274 - 0.2*T)
 w += Math.Sin(2*w5)*(0.2*T + 2062)
 w += Math.Sin(w2)*(1426 - 3.4*T)
 w += Math.Sin(w3)*(0.1*T + 712)
 w += Math.Sin(2*(w4 + w5 - w1) + w2)*(1.2*T - 517)
 w += Math.Sin(2*w4 + w5)*(-0.4*T - 386)
 w += Math.Sin(2*(w4 + w5 - w1) - w2)*(217 - 0.5*T)
 w += Math.Sin(2*(w4 - w1) + w5)*(129 + 0.1*T)
 w += Math.Sin(w3 + w5)*(0.1*T + 63)
 w += Math.Sin(w5 - w3)*(-0.1*T - 58)
 w += Math.Sin(2*w2)*(17 - 0.1*T)
 w += Math.Sin(2*(w2 + w4 + w5 - w1))*(0.1*T - 16)
 w -= 301*Math.Sin(2*(w4 + w5) + w3)
 w -= 158*Math.Sin(w3 - 2*w1)
 w += 123*Math.Sin(2*(w4 + w5) - w3)
 w += 63*Math.Sin(2*w1)
 w -= 59*Math.Sin(2*(w1 + w4 + w5) - w3)
 w -= 51*Math.Sin(2*w4 + w3 + w5)
 w += 48*Math.Sin(2*(w3 - w1))
 w += 46*Math.Sin(2*(w4 - w3) + w5)
 w -= 38*Math.Sin(2*(w1 + w4 + w5))
 w -= 31*Math.Sin(2*(w3 + w4 + w5))
 w += 29*Math.Sin(2*w3)
 w += 29*Math.Sin(2*(w4 + w5 - w1) + w3)
 w += 26*Math.Sin(2*w4)
 w -= 22*Math.Sin(2*(w4 - w1))
 w += 21*Math.Sin(2*w4 + w5 - w3)
 w += 16*Math.Sin(2*w1 - w3 + w5)
 w -= 15*Math.Sin(w2 + w5)
 w -= 13*Math.Sin(w3 + w5 - 2*w1)
 w -= 12*Math.Sin(w5 - w2)
 w += 11*Math.Sin(2*(w3 - w4))
 w -= 10*Math.Sin(2*(w4 + w1) + w5 - w3)
 w -= 8*Math.Sin(2*(w4 + w1 + w5) + w3)
 w += 7*Math.Sin(2*(w4 + w5) + w2)
 w -= 7*Math.Sin(w3 - 2*w1 + w2)
 w -= 7*Math.Sin(2*(w4 + w5) - w2)
 w -= 7*Math.Sin(2*w1 + 2*w4 + w5)
 w += 6*Math.Sin(2*w1 + w3)
 w += 6*Math.Sin(2*(w3 + w4 + w5 - w1))
 w += 6*Math.Sin(2*(w4 - w1) + w3 + w5)
 w -= 6*Math.Sin(2*(w1 - w3) + w5)
 w -= 6*Math.Sin(2*w1 + w5)
 w += 5*Math.Sin(w3 - w2)
 w -= 5*Math.Sin(2*(w4 - w1) + w5 - w2)
 w -= 5*Math.Sin(w5 - 2*w1)
 w -= 5*Math.Sin(2*(w3 + w4) + w5)
 w += 4*Math.Sin(2*(w3 - w1) + w5)
 w += 4*Math.Sin(2*(w4 - w1) + w2 + w5)
 w += 4*Math.Sin(w3 - 2*w4)
 w -= 4*Math.Sin(w3 - w1)
 w -= 4*Math.Sin(w2 - 2*w1)
 w -= 4*Math.Sin(w1)
 w += 3*Math.Sin(2*w4 + w3)
 w -= 3*Math.Sin(2*(w4 + w5 - w3))
 w -= 3*Math.Sin(w3 - w1 - w2)
 w -= 3*Math.Sin(w2 + w3)
 w -= 3*Math.Sin(2*(w4 + w5) + w3 - w2)
 w -= 3*Math.Sin(2*(w1 + w4 + w5) - w2 - w3)
 w -= 3*Math.Sin(2*(w4 + w5) + 3 * w3)
 w -= 3*Math.Sin(2*(w1 + w4 + w5) - w2)

 ' ------------------------------------
 ' DONE.  RETURN NUTATION IN ECLIPTICAL
 ' LONGITUDE EXPRESSED IN DEGREES.

 Return w / 36000000.0

 End Function

Nutation In Ecliptical Longitude (Δ_Ψ) and Right Ascension (Δ_RA) (1980 IAU Theory) PHP Program by Jay Tanner
Date: ±Y m d	Time: H M S UT	±ΔT: H M S

Calendar Mode: Julian Gregorian

Elements
JD	General JD number for given date, time and ΔT
T	Time factor in Julian centuries reckoned from J2000.0, corresponding to JD
DegToRad	This is a conversion factor used to convert degrees into equivalent radians. We multiply degrees by this factor to obtain the equivalent angle expressed in radians. Not every programming language has functions to convert between radians and degrees.
w	Nutation series accumulator (arc seconds x 10000).
w1	Mean elongation of the moon from the sun.
w2	Mean solar anomaly.
w3	Mean lunar anomaly.
w4	Lunar argument of latitude.
w5	Longitude of ascending node of lunar orbit on the ecliptic as measured from the mean equinox of date.
dPsiDeg	Nutation in ecliptical longitude expressed in degrees.
Algorithm to compute nutation in ecliptical longitude (dPsiDeg) for a given JD argument. T = (JD - 2451545) / 36525 T2 = TT T3 = TT2 DegToRad = 3.1415926535897932 / 180 w1 = 297.85036 + 445267.11148T - 0.0019142T2 + (T3 / 189474) w1 = DegToRad(w1) w2 = 357.52772 + 35999.05034T - 0.0001603T2 - (T3 / 300000) w2 = DegToRad(w2) w3 = 134.96298 + 477198.867398T + 0.0086972T2 + (T3 / 56250) w3 = DegToRad(w3) w4 = 93.27191 + 483202.017538T - 0.0036825T2 + (T3 / 327270) w4 = DegToRad(w4) w5 = 125.04452 - 1934.136261T + 0.0020708T2 + (T3 / 450000) w5 = DegToRad(w5) w = sin(w5)(-174.2T - 171996) w = w + sin(2(w4 + w5 - w1))(-1.6T - 13187) w = w + sin(2(w4 + w5))(-2274 - 0.2T) w = w + sin(2 w5)(0.2T + 2062) w = w + sin(w2)(1426 - 3.4T) w = w + sin(w3)(0.1T + 712) w = w + sin(2(w4 + w5 - w1) + w2)(1.2T - 517) w = w + sin(2 w4 + w5)(-0.4T - 386) w = w + sin(2(w4 + w5 - w1) - w2)(217 - 0.5T) w = w + sin(2(w4 - w1) + w5)(129 + 0.1T) w = w + sin(w3 + w5)(0.1T + 63) w = w + sin(w5 - w3)(-0.1T - 58) w = w + sin(2w2)(17 - 0.1T) w = w + sin(2(w2 + w4 + w5 - w1))(0.1T - 16) w = w - 301sin(2(w4 + w5) + w3) w = w - 158sin(w3 - 2w1) w = w + 123sin(2(w4 + w5) - w3) w = w + 63sin(2w1) w = w - 59sin(2(w1 + w4 + w5) - w3) w = w - 51sin(2 w4 + w3 + w5) w = w + 48sin(2(w3 - w1)) w = w + 46sin(2(w4 - w3) + w5) w = w - 38sin(2(w1 + w4 + w5)) w = w - 31sin(2(w3 + w4 + w5)) w = w + 29sin(2w3) w = w + 29sin(2(w4 + w5 - w1) + w3) w = w + 26sin(2w4) w = w - 22sin(2(w4 - w1)) w = w + 21sin(2w4 + w5 - w3) w = w + 16sin(2w1 - w3 + w5) w = w - 15sin(w2 + w5) w = w - 13sin(w3 + w5 - 2w1) w = w - 12sin(w5 - w2) w = w + 11sin(2(w3 - w4)) w = w - 10sin(2(w4 + w1) + w5 - w3) w = w - 8sin(2(w4 + w1 + w5) + w3) w = w + 7sin(2(w4 + w5) + w2) w = w - 7sin(w3 - 2w1 + w2) w = w - 7sin(2(w4 + w5) - w2) w = w - 7sin(2w1 + 2w4 + w5) w = w + 6sin(2w1 + w3) w = w + 6sin(2(w3 + w4 + w5 - w1)) w = w + 6sin(2(w4 - w1) + w3 + w5) w = w - 6sin(2(w1 - w3) + w5) w = w - 6sin(2w1 + w5) w = w + 5sin(w3 - w2) w = w - 5sin(2(w4 - w1) + w5 - w2) w = w - 5sin(w5 - 2w1) w = w - 5sin(2(w3 + w4) + w5) w = w + 4sin(2(w3 - w1) + w5) w = w + 4sin(2(w4 - w1) + w2 + w5) w = w + 4sin(w3 - 2w4) w = w - 4sin(w3 - w1) w = w - 4sin(w2 - 2w1) w = w - 4sin(w1) w = w + 3sin(2w4 + w3) w = w - 3sin(2(w4 + w5 - w3)) w = w - 3sin(w3 - w1 - w2) w = w - 3sin(w2 + w3) w = w - 3sin(2(w4 + w5) + w3 - w2) w = w - 3sin(2(w1 + w4 + w5) - w2 - w3) w = w - 3sin(2(w4 + w5) + 3w3) w = w - 3sin(2(w1 + w4 + w5) - w2) dPsiDeg = w / 36000000.0 PHP* function to compute nutation in ecliptical longitude (dPsi) for a given JD argument. function delta_psi ($JD) { // ----------------------- // READ JD ARGUMENT VALUE. // ERROR IF ARGUMENT IS NOT NUMERIC. if (!Is_Numeric(trim($JD))) {return FALSE;} // ----------------------------------------------------- // COMPUTE POWERS OF TIME IN JULIAN CENTURIES FROM J2000 $T1 = ($JD - 2451545) / 36525; $T2 = $T1$T1; $T3 = $T2$T1; // ----------------------------------------- // COMPUTE MEAN LUNAR ELONGATION IN RADIANS. $w1 = 297.85036 + 445267.11148$T1 - 0.0019142$T2 + ($T3 / 189474); $w1 = Deg2Rad($w1); // -------------------------------------- // COMPUTE MEAN SOLAR ANOMALY IN RADIANS. $w2 = 357.52772 + 35999.05034$T1 - 0.0001603$T2 - ($T3 / 300000); $w2 = Deg2Rad($w2); // -------------------------------------- // COMPUTE MEAN LUNAR ANOMALY IN RADIANS. $w3 = 134.96298 + 477198.867398$T1 + 0.0086972$T2 + ($T3 / 56250); $w3 = Deg2Rad($w3); // ---------------------------------------------- // COMPUTE LUNAR ARGUMENT OF LATITUDE IN RADIANS. $w4 = 93.27191 + 483202.017538$T1 - 0.0036825$T2 + ($T3 / 327270); $w4 = Deg2Rad($w4); // ---------------------------------------- // COMPUTE LUNAR ASCENDING NODE IN RADIANS. $w5 = 125.04452 - 1934.136261$T1 + 0.0020708$T2 + ($T3 / 450000); $w5 = Deg2Rad($w5); // --------------------------------------------- // COMPUTE THE NUTATION IN ECLIPTICAL LONGITUDE. $w = sin($w5)(-174.2$T1 - 171996); $w += sin(2($w4 + $w5 - $w1))(-1.6$T1 - 13187); $w += sin(2($w4 + $w5))(-2274 - 0.2$T1); $w += sin(2 * $w5)(0.2$T1 + 2062); $w += sin($w2)(1426 - 3.4$T1); $w += sin($w3)(0.1$T1 + 712); $w += sin(2($w4 + $w5 - $w1) + $w2)(1.2$T1 - 517); $w += sin(2 $w4 + $w5)(-0.4$T1 - 386); $w += sin(2($w4 + $w5 - $w1) - $w2)(217 - 0.5$T1); $w += sin(2($w4 - $w1) + $w5)(129 + 0.1$T1); $w += sin($w3 + $w5)(0.1$T1 + 63); $w += sin($w5 - $w3)(-0.1$T1 - 58); $w += sin(2$w2)(17 - 0.1$T1); $w += sin(2($w2 + $w4 + $w5 - $w1))(0.1$T1 - 16); $w -= 301sin(2($w4 + $w5) + $w3); $w -= 158sin($w3 - 2$w1); $w += 123sin(2($w4 + $w5) - $w3); $w += 63sin(2$w1); $w -= 59sin(2($w1 + $w4 + $w5) - $w3); $w -= 51sin(2 $w4 + $w3 + $w5); $w += 48sin(2($w3 - $w1)); $w += 46sin(2($w4 - $w3) + $w5); $w -= 38sin(2($w1 + $w4 + $w5)); $w -= 31sin(2($w3 + $w4 + $w5)); $w += 29sin(2$w3); $w += 29sin(2($w4 + $w5 - $w1) + $w3); $w += 26sin(2$w4); $w -= 22sin(2($w4 - $w1)); $w += 21sin(2$w4 + $w5 - $w3); $w += 16sin(2$w1 - $w3 + $w5); $w -= 15sin($w2 + $w5); $w -= 13sin($w3 + $w5 - 2$w1); $w -= 12sin($w5 - $w2); $w += 11sin(2($w3 - $w4)); $w -= 10sin(2($w4 + $w1) + $w5 - $w3); $w -= 8sin(2($w4 + $w1 + $w5) + $w3); $w += 7sin(2($w4 + $w5) + $w2); $w -= 7sin($w3 - 2$w1 + $w2); $w -= 7sin(2($w4 + $w5) - $w2); $w -= 7sin(2$w1 + 2$w4 + $w5); $w += 6sin(2$w1 + $w3); $w += 6sin(2($w3 + $w4 + $w5 - $w1)); $w += 6sin(2($w4 - $w1) + $w3 + $w5); $w -= 6sin(2($w1 - $w3) + $w5); $w -= 6sin(2$w1 + $w5); $w += 5sin($w3 - $w2); $w -= 5sin(2($w4 - $w1) + $w5 - $w2); $w -= 5sin($w5 - 2$w1); $w -= 5sin(2($w3 + $w4) + $w5); $w += 4sin(2($w3 - $w1) + $w5); $w += 4sin(2($w4 - $w1) + $w2 + $w5); $w += 4sin($w3 - 2$w4); $w -= 4sin($w3 - $w1); $w -= 4sin($w2 - 2$w1); $w -= 4sin($w1); $w += 3sin(2 $w4 + $w3); $w -= 3sin(2($w4 + $w5 - $w3)); $w -= 3sin($w3 - $w1 - $w2); $w -= 3sin($w2 + $w3); $w -= 3sin(2($w4 + $w5) + $w3 - $w2); $w -= 3sin(2($w1 + $w4 + $w5) - $w2 - $w3); $w -= 3sin(2($w4 + $w5) + 3$w3); $w -= 3sin(2($w1 + $w4 + $w5) - $w2); // ------------------------------------ // DONE. RETURN NUTATION IN ECLIPTICAL // LONGITUDE IN DECIMAL DEGREES. return $w / 36000000.0; } CPP* function to compute nutation in ecliptical longitude (dPsi) for a given JD argument. double delta_psi (double JD) { // ------------------------- // DEFINE WORKING VARIABLES. double T, T2, T3; double w, w1, w2, w3, w4, w5; double DegToRad, dPsiDeg; // --------------------------------- // COMPUTE (T) IN JULIAN CENTURIES // FROM J2000.0 AND POWERS OF (T). T = (JD - 2451545.0) / 36525.0; T2 = TT; T3 = TTT; // ------------------------------------------------ // DEFINE CONVERSION FACTOR FOR DEGREES TO RADIANS. // MULTIPLY DEGREES BY THIS FACTOR TO GET RADIANS. DegToRad = 3.1415926535897932 / 180; // --------------------------------------------------- // COMPUTE THE MEAN ELONGATION OF THE MOON IN RADIANS. w1 = DegToRad(297.85036 + 445267.11148T - 0.0019142T2 + (T3 / 189474)); // ----------------------------------------------- // COMPUTE THE MEAN ANOMALY OF THE SUN IN RADIANS. w2 = DegToRad(357.52772 + 35999.05034T - 0.0001603T2 - (T3 / 300000)); // ------------------------------------------------ // COMPUTE THE MEAN ANOMALY OF THE MOON IN RADIANS. w3 = DegToRad(134.96298 + 477198.867398T + 0.0086972T2 + (T3 / 56250)); // --------------------------------------------------- // COMPUTE THE MOON'S ARGUMENT OF LATITUDE IN RADIANS. w4 = DegToRad(93.27191 + 483202.017538T - 0.0036825T2 + (T3 / 327270)); // ---------------------------------------------------------- // COMPUTE THE LONGITUDE OF MOON'S ASCENDING NODE IN RADIANS. w5 = DegToRad(125.04452 - 1934.136261T + 0.0020708T2 + (T3 / 450000)); // --------------------------------------------- // COMPUTE THE NUTATION IN ECLIPTICAL LONGITUDE. w = sin(w5)(-174.2T - 171996); w += sin(2(w4 + w5 - w1))(-1.6T - 13187); w += sin(2(w4 + w5))(-2274 - 0.2T); w += sin(2 * w5)(0.2T + 2062); w += sin(w2)(1426 - 3.4T); w += sin(w3)(0.1T + 712); w += sin(2(w4 + w5 - w1) + w2)(1.2T - 517); w += sin(2w4 + w5)(-0.4T - 386); w -= 301sin(2(w4 + w5) + w3); w += sin(2(w4 + w5 - w1) - w2)(217 - 0.5T); w -= 158sin(w3 - 2w1); w += sin(2(w4 - w1) + w5)(129 + 0.1T); w += 123sin(2(w4 + w5) - w3); w += 63sin(2w1); w += sin(w3 + w5)(0.1T + 63); w -= 59sin(2(w1 + w4 + w5) - w3); w += sin(w5 - w3)(-0.1T - 58); w -= 51sin(2w4 + w3 + w5); w += 48sin(2(w3 - w1)); w += 46sin(2(w4 - w3) + w5); w -= 38sin(2(w1 + w4 + w5)); w -= 31sin(2(w3 + w4 + w5)); w += 29sin(2w3); w += 29sin(2(w4 + w5 - w1) + w3); w += 26sin(2w4); w -= 22sin(2(w4 - w1)); w += 21sin(2w4 + w5 - w3); w += sin(2w2)(17 - 0.1T); w += 16sin(2w1 - w3 + w5); w += sin(2(w2 + w4 + w5 - w1))(0.1T - 16); w -= 15sin(w2 + w5); w -= 13sin(w3 + w5 - 2w1); w -= 12sin(w5 - w2); w += 11sin(2(w3 - w4)); w -= 10sin(2(w4 + w1) + w5 - w3); w -= 8sin(2(w4 + w1 + w5) + w3); w += 7sin(2(w4 + w5) + w2); w -= 7sin(w3 - 2w1 + w2); w -= 7sin(2(w4 + w5) - w2); w -= 7sin(2w1 + 2w4 + w5); w += 6sin(2w1 + w3); w += 6sin(2(w3 + w4 + w5 - w1)); w += 6sin(2(w4 - w1) + w3 + w5); w -= 6sin(2(w1 - w3) + w5); w -= 6sin(2w1 + w5); w += 5sin(w3 - w2); w -= 5sin(2(w4 - w1) + w5 - w2); w -= 5sin(w5 - 2w1); w -= 5sin(2(w3 + w4) + w5); w += 4sin(2(w3 - w1) + w5); w += 4sin(2(w4 - w1) + w2 + w5); w += 4sin(w3 - 2w4); w -= 4sin(w3 - w1); w -= 4sin(w2 - 2w1); w -= 4sin(w1); w += 3sin(2w4 + w3); w -= 3sin(2(w4 + w5 - w3)); w -= 3sin(w3 - w1 - w2); w -= 3sin(w2 + w3); w -= 3sin(2(w4 + w5) + w3 - w2); w -= 3sin(2(w1 + w4 + w5) - w2 - w3); w -= 3sin(2(w4 + w5) + 3w3); w -= 3sin(2(w1 + w4 + w5) - w2); // ------------------------------------ // COVERT DELTA PSI TO DECIMAL DEGREES. dPsiDeg = w / 36000000.0; // ----- // Done. return dPsiDeg; } VB.NET* function to compute nutation in ecliptical longitude (dPsi) for a given JD argument. Public Function Delta_Psi(ByVal JD As Double) As Double Dim T, T2, T3, w, w1, w2, w3, w4, w5, DegToRad As Double DegToRad = Math.PI / 180 ' ------------------------------------------- ' COMPUTE TIME FACTOR (T) IN JULIAN CENTURIES ' AS RECKONED FROM J2000 AND POWERS OF T. T = (JD - 2451545.0) / 36525.0 T2 = TT T3 = T2 T ' ----------------------------------------- ' COMPUTE MEAN LUNAR ELONGATION IN RADIANS. w1 = DegToRad * (297.85036 + 445267.11148T _ - 0.0019142T2 + (T3 / 189474)) ' -------------------------------------- ' COMPUTE MEAN SOLAR ANOMALY IN RADIANS. w2 = DegToRad * (357.52772 + 35999.05034T - 0.0001603T2 - (T3 / 300000)) ' -------------------------------------- ' COMPUTE MEAN LUNAR ANOMALY IN RADIANS. w3 = DegToRad * (134.96298 + 477198.867398T _ + 0.0086972T2 + (T3 / 56250)) ' ---------------------------------------------- ' COMPUTE LUNAR ARGUMENT OF LATITUDE IN RADIANS. w4 = DegToRad * (93.27191 + 483202.017538T _ - 0.0036825T2 + (T3 / 327270)) ' ----------------------------------------------------- ' COMPUTE LONGITUDE OF LUNAR ASCENDING NODE IN RADIANS. w5 = DegToRad * (125.04452 - 1934.136261T _ + 0.0020708T2 + (T3 / 450000)) ' -------------------------------------------- ' COMPUTE THE NUTATION IN ECLIPTICAL LONGITUDE ' IN UNITS OF (ARCSEC * 10000). w = Math.Sin(w5)(-174.2T - 171996) w += Math.Sin(2(w4 + w5 - w1))(-1.6T - 13187) w += Math.Sin(2(w4 + w5))(-2274 - 0.2T) w += Math.Sin(2w5)(0.2T + 2062) w += Math.Sin(w2)(1426 - 3.4T) w += Math.Sin(w3)(0.1T + 712) w += Math.Sin(2(w4 + w5 - w1) + w2)(1.2T - 517) w += Math.Sin(2w4 + w5)(-0.4T - 386) w += Math.Sin(2(w4 + w5 - w1) - w2)(217 - 0.5T) w += Math.Sin(2(w4 - w1) + w5)(129 + 0.1T) w += Math.Sin(w3 + w5)(0.1T + 63) w += Math.Sin(w5 - w3)(-0.1T - 58) w += Math.Sin(2w2)(17 - 0.1T) w += Math.Sin(2(w2 + w4 + w5 - w1))(0.1T - 16) w -= 301Math.Sin(2(w4 + w5) + w3) w -= 158Math.Sin(w3 - 2w1) w += 123Math.Sin(2(w4 + w5) - w3) w += 63Math.Sin(2w1) w -= 59Math.Sin(2(w1 + w4 + w5) - w3) w -= 51Math.Sin(2w4 + w3 + w5) w += 48Math.Sin(2(w3 - w1)) w += 46Math.Sin(2(w4 - w3) + w5) w -= 38Math.Sin(2(w1 + w4 + w5)) w -= 31Math.Sin(2(w3 + w4 + w5)) w += 29Math.Sin(2w3) w += 29Math.Sin(2(w4 + w5 - w1) + w3) w += 26Math.Sin(2w4) w -= 22Math.Sin(2(w4 - w1)) w += 21Math.Sin(2w4 + w5 - w3) w += 16Math.Sin(2w1 - w3 + w5) w -= 15Math.Sin(w2 + w5) w -= 13Math.Sin(w3 + w5 - 2w1) w -= 12Math.Sin(w5 - w2) w += 11Math.Sin(2(w3 - w4)) w -= 10Math.Sin(2(w4 + w1) + w5 - w3) w -= 8Math.Sin(2(w4 + w1 + w5) + w3) w += 7Math.Sin(2(w4 + w5) + w2) w -= 7Math.Sin(w3 - 2w1 + w2) w -= 7Math.Sin(2(w4 + w5) - w2) w -= 7Math.Sin(2w1 + 2w4 + w5) w += 6Math.Sin(2w1 + w3) w += 6Math.Sin(2(w3 + w4 + w5 - w1)) w += 6Math.Sin(2(w4 - w1) + w3 + w5) w -= 6Math.Sin(2(w1 - w3) + w5) w -= 6Math.Sin(2w1 + w5) w += 5Math.Sin(w3 - w2) w -= 5Math.Sin(2(w4 - w1) + w5 - w2) w -= 5Math.Sin(w5 - 2w1) w -= 5Math.Sin(2(w3 + w4) + w5) w += 4Math.Sin(2(w3 - w1) + w5) w += 4Math.Sin(2(w4 - w1) + w2 + w5) w += 4Math.Sin(w3 - 2w4) w -= 4Math.Sin(w3 - w1) w -= 4Math.Sin(w2 - 2w1) w -= 4Math.Sin(w1) w += 3Math.Sin(2w4 + w3) w -= 3Math.Sin(2(w4 + w5 - w3)) w -= 3Math.Sin(w3 - w1 - w2) w -= 3Math.Sin(w2 + w3) w -= 3Math.Sin(2(w4 + w5) + w3 - w2) w -= 3Math.Sin(2(w1 + w4 + w5) - w2 - w3) w -= 3Math.Sin(2(w4 + w5) + 3 w3) w -= 3Math.Sin(2(w1 + w4 + w5) - w2) ' ------------------------------------ ' DONE. RETURN NUTATION IN ECLIPTICAL ' LONGITUDE EXPRESSED IN DEGREES. Return w / 36000000.0 End Function Jay Tanner - PHP Science Labs - 2024 - v1.7