Basic Orbital Ellipse Parameters

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/*
   ###########################################################################
   This function computes the perimeter of an ellipse or elliptical orbit with
   the given parameters (R, r).  It is a high-precision, brute-force numerical
   solution to the complete elliptic integral of the second kind.

   Where:
   r = Semiminor axis
   R = Semimajor axis
   C = Circumference of Ellipse

   (r, R) can be in any convenient units, such as AUs, kilometers, miles, etc.

   Internally, arbitrary-precision arithmetic is used to help preserve
   accuracy, with the final result rounded at 16 decimals.

   NOTE:
   Since this function uses BC arithmetic, the numeric values used should be
   numeric strings,  like '3210.123465978'  to represent EXACT values and to
   avoid confusion with the default fixed double-precision numbers.

   Author : Jay Tanner - 2026
   ###########################################################################
*/

   function Ellipse_Circum ($RParam='1', $rParam='1')
{
// -----------------------------------------------
// Read input (R, r) ellipse parameter arguments.
// they are assumed to be purely numeric strings.

   $R = trim($RParam); // Corresponds to (a)
   $r = trim($rParam); // Corresponds to (b)

// -----------------------------------------------------
// Return FALSE (ERROR) if either (R, r) is non-numeric.

   if (!Is_Numeric($R) or !Is_Numeric($r))   {return FALSE;}

// --------------------------------------------
// Define local constant (2*pi) to 50 decimals.

   $TwoPi = '6.28318530717958647692528676655900576839433879875';

// ------------------------------------------
// Set to 16 decimals final output precision.

   $decimals = 16;

/* ----------------------------------------------
   Use 8 extra decimals for internal computations
   to reduce accumulative rounding errors in the
   final summation.
*/
   $d = $decimals + 8;

// ---------------------------------------------
// If (r > R), then swap values so that (R > r).

   if (bcComp($R, $r, $d) < 0)  {$w=$R;  $R=$r;  $r=$w;}

// ------------------------------------------------------
// Compute the square of the elliptic eccentricity value.

   $e2 = bcSub('1', bcDiv(bcMul($r,$r,$d), bcMul($R,$R,$d),$d),$d);

/* -------------------------------------------------
   Perform power series summation loop while the
   nth term is > the 16-decimal (PrecisionLimit).
   This limit could be changed, but 16 decimals
   should generally be sufficient.
*/
   $n = $ePow2n = $PrecisionLevel = 1;   $sum = 0;

   $PrecisionLimit = '0.' . Str_Repeat('0', $d-1) . '1';

   while (abs($PrecisionLevel) > $PrecisionLimit)
  {
   $N = 2*$n - 1;
   $X = 1;

   for ($n=0;  $n <= ceil($N/2)-1;  $n++)
       {
        $X  = bcMul($X, $N - 2*$n); // (2n-1)!!
       }
        $X2 = bcMul($X,$X); // (2n-1)!!^2

   $N = 2*$n;
   $Y = 1;

   for ($n=0;  $n <= ceil($N/2)-1;  $n++)
       {
        $Y  = bcMul($Y, $N - 2*$n); // (2n)!!
       }
        $Y2 = bcMul($Y,$Y); // (2n)!!^2

   $ePow2n         = bcMul($ePow2n, $e2,  $d); // e^(2n)
   $Numer          = bcMul($X2, $ePow2n,  $d); // (2n-1)!!^2 * e^(2n)
   $Denom          = bcMul($Y2, 2*$n - 1, $d); // (2n)!!^2 * (2n-1)
   $CurrSumTerm    = bcDiv($Numer, $Denom, $d);
   $sum            = bcAdd($sum, $CurrSumTerm, $d);
   $PrecisionLevel = bcSub($CurrSumTerm, $PrecisionLimit, $d);
   $n++;

  } // END OF while (...)

/* -------------------------------------------------
   The summation part is done.  Finish computing the
   final elliptic circumference and round the result
   to the indicated number of decimals.
*/
   $C = bcMul($TwoPi, bcMul($R, bcSub('1', $sum, $d),$d),$d);

   return bcAdd($C, '0.'.str_repeat('0', $decimals).'5', $decimals);

} // END OF  Ellipse_Circum (...)