static float
kepler(const float ecc, float mean_anom)
{
float curr, err, thresh;
int is_negative = 0, n_iter = 0;
if( !mean_anom) {
return( 0.);
}
if( ecc < .3) { /* low-eccentricity formula from Meeus, p. 195 */
curr = atan2f( sinf( mean_anom), cosf( mean_anom) - ecc);
/* one correction step, and we're done */
err = curr - ecc * sinf( curr) - mean_anom;
curr -= err / (1. - ecc * cosf( curr));
} else {
curr = mean_anom;
}
if( mean_anom < 0.) {
mean_anom = -mean_anom;
curr = - curr;
is_negative = 1;
}
thresh = THRESH * fabsf( 1. - ecc);
if( ecc > .8 && mean_anom < PI / 3. || ecc > 1.) /* up to 60 degrees */ {
float trial = mean_anom / fabsf( 1. - ecc);
if( trial * trial > 6. * fabsf(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 = asinhf( mean_anom / ecc);
}
}
curr = trial;
}
if( ecc < 1.) {
err = curr - ecc * sinf( curr) - mean_anom;
while( fabsf( err) > thresh) {
n_iter++;
curr -= err / (1. - ecc * cosf( curr));
err = curr - ecc * sinf( curr) - mean_anom;
}
} else {
err = ecc * sinhf( curr) - curr - mean_anom;
while( fabsf( err) > thresh) {
n_iter++;
curr -= err / (ecc * coshf( curr) - 1.);
err = ecc * sinhf( curr) - curr - mean_anom;
}
}
return( is_negative ? -curr : curr);
}
void comet_posn_part_ii( const ELEMENTS DLLPTR *elem, const double t,
double DLLPTR *loc, double DLLPTR *vel)
{
double true_anom, r, x, y, r0;
if( elem->ecc == 1.) /* parabolic */
{
double g = elem->w0 * t * .5;
y = CUBE_ROOT( g + sqrt( g * g + 1.));
true_anom = 2. * atan( y - 1. / y);
}
else /* got the mean anomaly; compute eccentric, then true */
{
double ecc_anom;
ecc_anom = kepler( elem->ecc, elem->mean_anomaly);
if( elem->ecc > 1.) /* hyperbolic case */
{
x = (elem->ecc - cosh( ecc_anom));
y = sinh( ecc_anom);
}
else /* elliptical case */
{
x = (cos( ecc_anom) - elem->ecc);
y = sin( ecc_anom);
}
y *= elem->minor_to_major;
true_anom = atan2( y, x);
}
r0 = elem->q * (1. + elem->ecc);
r = r0 / (1. + elem->ecc * cos( true_anom));
x = r * cos( true_anom);
y = r * sin( true_anom);
loc[0] = elem->perih_vec[0] * x + elem->sideways[0] * y;
loc[1] = elem->perih_vec[1] * x + elem->sideways[1] * y;
loc[2] = elem->perih_vec[2] * x + elem->sideways[2] * y;
loc[3] = r;
if( vel && (elem->angular_momentum != 0.))
{
double angular_component = elem->angular_momentum / (r * r);
double radial_component = elem->ecc * sin( true_anom) *
elem->angular_momentum / (r * r0);
double x1 = x * radial_component - y * angular_component;
double y1 = y * radial_component + x * angular_component;
unsigned i;
for( i = 0; i < 3; i++)
vel[i] = elem->perih_vec[i] * x1 + elem->sideways[i] * y1;
}
}
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);
}
double kepler(double ecc, double mean_anom)
{
double curr, err, thresh;
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);
/* one correction step, and we're done */
err = curr - ecc * sin(curr) - mean_anom;
curr -= err / (1. - ecc * cos(curr));
return (curr);
}
if (mean_anom < 0.)
{
mean_anom = -mean_anom;
is_negative = 1;
}
curr = mean_anom;
thresh = THRESH * fabs(1. - ecc);
if (ecc > .8 && mean_anom < M_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 < M_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.)
{
err = curr - ecc * sin(curr) - mean_anom;
while (fabs(err) > thresh)
{
n_iter++;
curr -= err / (1. - ecc * cos(curr));
err = curr - ecc * sin(curr) - mean_anom;
}
}
else
{
err = ecc * sinh(curr) - curr - mean_anom;
while (fabs(err) > thresh)
{
n_iter++;
curr -= err / (ecc * cosh(curr) - 1.);
err = ecc * sinh(curr) - curr - mean_anom;
}
}
return (is_negative ? -curr : curr);
}
static double kepler( const double ecc, double mean_anom)
{
double curr, err, thresh, offset = 0.;
double delta_curr = 1.;
bool is_negative = false;
unsigned n_iter = 0;
if( !mean_anom)
return( 0.);
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( ecc < .9) /* low-eccentricity formula from Meeus, p. 195 */
{
curr = atan2( sin( mean_anom), cos( mean_anom) - ecc);
do
{
err = (curr - ecc * sin( curr) - mean_anom) / (1. - ecc * cos( curr));
curr -= err;
}
while( fabs( err) > THRESH);
return( curr + offset);
}
}
if( mean_anom < 0.)
{
mean_anom = -mean_anom;
is_negative = true;
}
curr = mean_anom;
thresh = THRESH * fabs( 1. - ecc);
/* Due to roundoff error, there's no way we can hope to */
/* get below a certain minimum threshhold anyway: */
if( thresh < MIN_THRESH)
thresh = MIN_THRESH;
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( thresh > THRESH) /* happens if e > 2. */
thresh = THRESH;
}
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;
}
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;
}
return( is_negative ? offset - curr : offset + curr);
}