inline std::pair<typename property_traits<EccentricityMap>::value_type,
typename property_traits<EccentricityMap>::value_type>
all_eccentricities(const Graph& g, const DistanceMatrix& dist, EccentricityMap ecc)
{
function_requires< VertexListGraphConcept<Graph> >();
typedef typename graph_traits<Graph>::vertex_descriptor Vertex;
typedef typename graph_traits<Graph>::vertex_iterator VertexIterator;
function_requires< ReadablePropertyMapConcept<DistanceMatrix,Vertex> >();
typedef typename property_traits<DistanceMatrix>::value_type DistanceMap;
function_requires< WritablePropertyMapConcept<EccentricityMap,Vertex> >();
typedef typename property_traits<EccentricityMap>::value_type Eccentricity;
BOOST_USING_STD_MIN();
BOOST_USING_STD_MAX();
Eccentricity
r = numeric_values<Eccentricity>::infinity(),
d = numeric_values<Eccentricity>::zero();
VertexIterator i, end;
tie(i, end) = vertices(g);
for(tie(i, end) = vertices(g); i != end; ++i) {
DistanceMap dm = get(dist, *i);
Eccentricity e = eccentricity(g, dm);
put(ecc, *i, e);
// track the radius and diameter at the same time
r = min BOOST_PREVENT_MACRO_SUBSTITUTION (r, e);
d = max BOOST_PREVENT_MACRO_SUBSTITUTION (d, e);
}
return make_pair(r, d);
}
double theta3(double etaval, double q[2], double qdot[2]) {
double e = eccentricity(q, qdot);
double c = c_aux(q, qdot);
return etaval - e * c / (c + B_ISO) * sin(etaval);
}
inline typename property_traits<DistanceMap>::value_type
eccentricity(const Graph& g, DistanceMap dist)
{
function_requires< GraphConcept<Graph> >();
typedef typename graph_traits<Graph>::vertex_descriptor Vertex;
function_requires< ReadablePropertyMapConcept<DistanceMap,Vertex> >();
typedef typename property_traits<DistanceMap>::value_type Distance;
return eccentricity(g, dist, detail::maximize<Distance>());
}
double etaZero(double q[2], double qdot[2]) {
double e = eccentricity(q, qdot);
double c = c_aux(q, qdot);
double coseta = (1.0 - B_ISO / c * (sqrt(1.0 + q[0]*q[0]/(B_ISO * B_ISO))-1.0))/e;
return acos(coseta);
}
double theta2(double etaval, double phi, double q[2], double qdot[2]) {
double e = eccentricity(q, qdot);
double c = c_aux(q, qdot);
double J2 = Lmag(q, qdot);
double omega2 = Omega2(q,qdot);
double omega3 = Omega3(q,qdot);
double th3 = theta3(etaval, q, qdot);
double tan1 = atan(sqrt((1.0 + e)/(1.0 - e)) * tan(etaval/2.0)) + M_PI * floor((etaval+M_PI)/(2.0 * M_PI));
double tan2 = (atan(sqrt((1.0 + e + 2.0 * B_ISO / c)/(1.0 - e + 2.0 * B_ISO / c)) * tan(etaval/2.0)) + M_PI * floor((etaval+M_PI)/(2.0 * M_PI)))/sqrt(1 + 4.0 * M_PI * B_ISO / (J2 * J2));
return phi + omega2/omega3 * th3 - tan1 - tan2;
}
double etaZero(double q[2], double qdot[2]) {
double e = eccentricity(q, qdot);
double c = c_aux(q, qdot);
double coseta = (1.0 - B_ISO / c * (sqrt(1.0 + q[0]*q[0]/(B_ISO * B_ISO))-1.0))/e;
double etamag = acos(coseta);
if(qdot[0]<0)
return 2.0 * M_PI - etamag;
else
return etamag;
}
/// @return eccentricity squared (ellipsoid parameter).
virtual double eccSquared() const throw()
{ return eccentricity() * eccentricity(); }
double eta(double q[2], double qdot[2], double t, double eta0) {
// calculates the auxiliary variable eta by solving Kepler's equation
// the equation is solved by searching for a root
// using the brent's algorithm root bracketer in GSL
// definitions for root solver
const gsl_root_fsolver_type *T = gsl_root_fsolver_brent;
gsl_root_fsolver *s;
gsl_function keq;
// iteration limits for root solver
double epsabs=0.0;
double epsrel = 1e-10;
size_t max_iter=100;
size_t iter=0;
int status;
double xlo,xhi;
double r;
//compute e, c, omega3, initial theta3
double e = eccentricity(q, qdot);
double c = c_aux(q, qdot);
double omega3 = Omega3(q,qdot);
double theta30 = theta3(eta0,q,qdot);
// set up function parameters
keplerEquationParamsType pkeq={ e * c / (c + B_ISO), omega3 * t + theta30};
//set up function for rootfinding
keq.function = &keplerEquation;
keq.params = &pkeq;
//initialize rootfinder
s=gsl_root_fsolver_alloc(T);
//set it to bracket root of kepler eqn. The initial guess should be near eta=Omega3*t + theta30
xlo = omega3 * t + theta30 - M_PI;
xhi = omega3 * t + theta30 + M_PI;
gsl_root_fsolver_set(s, &keq, xlo, xhi);
//iterate till root is solved
do {
iter++;
status=gsl_root_fsolver_iterate(s);
xlo=gsl_root_fsolver_x_lower(s);
xhi=gsl_root_fsolver_x_upper(s);
status=gsl_root_test_interval(xlo,xhi,epsabs,epsrel);
} while (status==GSL_CONTINUE && iter<max_iter);
r = gsl_root_fsolver_root(s);
gsl_root_fsolver_free(s);
return r;
}
void forwardMap(double q[2], double qdot[2], double t, double x[2], double v[2]) {
// maps from (q, qdot) -> (x(t), v(t)) (forward in time)
// to go backward, let t -> -t
// first step: find auxiliary variables
// eta0 is between 0 and pi but extends to between 0 and 2pi based on the sign of the initial r-velocity
double eta0 = etaZero(q,qdot);
if (qdot[0]<0.0) {
eta0 = 2.0 * M_PI - eta0;
}
// etabase is between 0 and pi
double etabase = eta(q,qdot,t, eta0);
double omega3 = Omega3(q,qdot);
double omega2 = Omega2(q,qdot);
double J2 = Lmag(q,qdot);
double ham = H(q,qdot);
double e = eccentricity(q, qdot);
double c = c_aux(q, qdot);
// compute number of pis to add for correct branch (theta2 and theta3 must increase continuously)
double Tr = 2.0 * M_PI / omega3;
double tperi = (theta3(0.0, q, qdot) - theta3(eta0,q, qdot))/omega3;
double tapo = (theta3(M_PI, q, qdot) - theta3(eta0,q, qdot))/omega3;
// both these should be larger than 0 (time of *next* pericenter and apocenter)
while (tperi<0) {
tperi += Tr;
}
while (tapo<0) {
tapo += Tr;
}
// calculate number of half-periods (no. of peri- or apocenter passages)
int nstar = ((int) floor((t - GSL_MIN(tapo, tperi))/(Tr/2.0))) + 1;
// get r
double base = (1.0 + c/B_ISO * (1.0 - e * cos(etabase)));
x[0] = B_ISO * sqrt(base * base - 1.0);
// get phi
double tan1 = atan(sqrt((1.0 + e)/(1.0 - e)) * tan(etabase/2.0)) + M_PI * floor((etabase+M_PI)/(2.0 * M_PI));
double tan2 = (atan(sqrt((1.0 + e + 2.0 * B_ISO / c)/(1.0 - e + 2.0 * B_ISO / c)) * tan(etabase/2.0)) + M_PI * floor((etabase+M_PI)/(2.0 * M_PI)))/sqrt(1 + 4.0 * M_PI * B_ISO / (J2 * J2));
double th20 = theta2(eta0, q[1], q, qdot);
double th3 = theta3(etabase, q, qdot);
// fprintf(stdout, "%lg %lg %lg %lg\n", t, tan1, tan2,th3);
x[1] = th20 + omega2 * t - omega2/omega3 * th3 + tan1 + tan2;
// conservation of angular momentum gives phidot
v[1] = J2 / (x[0] * x[0]);
// conservation of energy gives magnitude of rdot
v[0] = M_SQRT2 * sqrt(ham - J2 * J2 / (2.0 * x[0] * x[0]) - Phi(x[0]));
// figure out the sign - if it started heading toward pericenter
// and has gone thru an even number of half-periods since then, vr<0
// likewise if it started heading toward apo and has gone thru an odd nm
// of half-periods, vr<0
if((GSL_IS_EVEN(nstar) && (tperi<tapo)) || (GSL_IS_ODD(nstar) && (tapo<tperi)))
v[0] *= -1.0;
}