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/* keptest.cpp: test code for Kepler-solving functions
Copyright (C) 2010, Project Pluto
This program is free software; you can redistribute it and/or
modify it under the terms of the GNU General Public License
as published by the Free Software Foundation; either version 2
of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
02110-1301, USA. */
#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define THRESH 1.e-8
#define PI 3.141592653589793238462643383279502884197169399375105
#define CUBE_ROOT( X) (exp( log( X) / 3.))
/* 'asinh' = 'arc-hyperbolic sine.' Most compilers now implement this. */
#ifdef _MSC_VER
static double asinh( const double z)
{
return( log( z + sqrt( z * z + 1.)));
}
#endif
/* If the eccentricity is very close to parabolic, and the eccentric
anomaly is quite low, you can get an unfortunate situation where
roundoff error keeps you from converging. Consider the just-barely-
elliptical case, where in Kepler's equation,
M = E - e sin( E)
E and e sin( E) can be almost identical quantities. To
around this, near_parabolic( ) computes E - e sin( E) by expanding
the sine function as a power series:
E - e sin( E) = E - e( E - E^3/3! + E^5/5! - ...)
= (1-e)E + e( -E^3/3! + E^5/5! - ...)
It's a little bit expensive to do this, and you only need do it
quite rarely. (I only encountered the problem because I had orbits
that were supposed to be 'pure parabolic', but due to roundoff,
they had e = 1+/- epsilon, with epsilon _very_ small.) So 'near_parabolic'
is only called if we've gone seven iterations without converging. */
static double near_parabolic( const double ecc_anom, const double e)
{
const double anom2 = (e > 1. ? ecc_anom * ecc_anom : -ecc_anom * ecc_anom);
double term = e * anom2 * ecc_anom / 6.;
double rval = (1. - e) * ecc_anom - term;
int n = 4;
while( fabs( term) > 1e-15)
{
term *= anom2 / (double)(n * (n + 1));
rval -= term;
n += 2;
}
return( rval);
}
/* For a full description of this function, see KEPLER.HTM on the Guide
Web site, http://www.projectpluto.com. There was a long thread about
solutions to Kepler's equation on sci.astro.amateur, and I decided to
go into excruciating detail as to how it's done below. */
#define MAX_ITERATIONS 7
static double kepler( const double ecc, double mean_anom)
{
double curr, err, thresh, offset = 0., delta_curr = 1.;
int is_negative = 0, n_iter = 0;
if( !mean_anom)
return( 0.);
if( ecc < .3) /* low-eccentricity formula from Meeus, p. 195 */
{
curr = atan2( sin( mean_anom), cos( mean_anom) - ecc);
/* two correction steps, and we're done */
for( n_iter = 2; n_iter; n_iter--)
{
err = curr - ecc * sin( curr) - mean_anom;
curr -= err / (1. - ecc * cos( curr));
}
return( curr);
}
if( ecc < 1.)
if( mean_anom < -PI || mean_anom > PI)
{
double tmod = fmod( mean_anom, PI * 2.);
if( tmod > PI) /* bring mean anom within -pi to +pi */
tmod -= 2. * PI;
else if( tmod < -PI)
tmod += 2. * PI;
offset = mean_anom - tmod;
mean_anom = tmod;
}
if( mean_anom < 0.)
{
mean_anom = -mean_anom;
is_negative = 1;
}
curr = mean_anom;
thresh = THRESH * fabs( 1. - ecc);
if( thresh < 1e-15)
thresh = 1e-15;
if( ecc > .8 && mean_anom < PI / 3. || ecc > 1.) /* up to 60 degrees */
{
double trial = mean_anom / fabs( 1. - ecc);
if( trial * trial > 6. * fabs(1. - ecc)) /* cubic term is dominant */
{
if( mean_anom < PI)
trial = CUBE_ROOT( 6. * mean_anom);
else /* hyperbolic w/ 5th & higher-order terms predominant */
trial = asinh( mean_anom / ecc);
}
curr = trial;
}
if( ecc > 1. && mean_anom > 4.) /* hyperbolic, large-mean-anomaly case */
curr = log( mean_anom);
if( ecc < 1.)
while( fabs( delta_curr) > thresh)
{
if( n_iter++ > MAX_ITERATIONS)
err = near_parabolic( curr, ecc) - mean_anom;
else
err = curr - ecc * sin( curr) - mean_anom;
delta_curr = -err / (1. - ecc * cos( curr));
curr += delta_curr;
printf( "Iter %d: %.16lf %.16lf %.16lf\n",
n_iter, curr, err, delta_curr);
}
else
while( fabs( delta_curr) > thresh)
{
if( n_iter++ > MAX_ITERATIONS)
err = -near_parabolic( curr, ecc) - mean_anom;
else
err = ecc * sinh( curr) - curr - mean_anom;
delta_curr = -err / (ecc * cosh( curr) - 1.);
curr += delta_curr;
printf( "Iter %d: %.16lf %.16lf %.16lf\n",
n_iter, curr, err, delta_curr);
}
return( is_negative ? offset - curr : offset + curr);
}
void main( int argc, char **argv)
{
double ecc = atof( argv[1]);
double mean_anom = atof( argv[2]);
printf( "E=%lf\n", kepler( ecc, mean_anom));
}
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