/*
* Calculate Kepler position and velocity for given timestep _dt_
* for particle no. _pos_.
* _xp_ and _vp_ will be updated.
*/
void step_kepler_1(struct particle parts[], int pcount, int pos, double dt,
double *out_a, double *out_a_, double *out_a_2, double *out_a_3,
double *curr_a, double *curr_e)
{
_enter_function(_UL_KEPLER, _UL_KEPLER_STEP_KEPLER_1);
int i;
struct particle *p0 = parts, *p1 = parts + pos;
double r_[3], v_[3], j_[3], ecc_[3], a_[3], b_[3], _1_r2, afact, v_r_, v_v_, r_a_, v_a_, r_a__;
double ecc, a, r, v, b, omega, e, mean, cosp, sinp;
double m_c=p0->m, _cosp_ecc, e2, _1_ecc, _cosp_1, de_dt;//+p1->m;
// get relative position / motion
for(i = 0; i < 3; i++)
{
r_[i] = p1->xp[i] - p0->xp[i];
v_[i] = p1->vp[i] - p0->vp[i];
}
// calculate ellipse constants
get_constants(r_, v_, m_c, j_, ecc_, &a, &omega);
//printf("# [%d]:\t%e\t%e\t%e\n", pos, v_abs(ecc_), a, omega);
ecc = v_abs(ecc_);
// b_ = a * sqrt(|1-e²|) * (j_ x e_) / |j_ x e_|
vec_prod(j_, ecc_, b_);
b = a * sqrt(fabs(1-ecc*ecc)) / v_abs(b_);
for(i = 0; i < 3; i++)
{
a_[i] = a*ecc_[i]/ecc; // semi major vector
b_[i] *= b; // semi minor vector
}
if(curr_a != NULL) *curr_a = a;
if(curr_e != NULL) *curr_e = ecc;
if(ecc < 1)
// elliptical orbit
{
if(!p1->is_elliptical)
{
fprintf(get_file(FILE_WARNING),
"#### [t=%1.12e] Particle #%d captured onto elliptical orbit with e=%e ####\n",
t_total(p1->t), pos, ecc);
p1->is_elliptical = 1;
}
// calculate eccentric anomaly e at t+dt
e = (a - v_abs(r_)) / (ecc * a);
if(e >= 1.0) e = .0;
else if(e <= -1.0) e = M_PI;
else e = acos(e);
if(scal_prod(r_, b_) < 0)
e = 2*M_PI - e;
mean = (e - ecc*sin(e)) + dt * omega;
while(mean >= 2. * M_PI)
mean -= 2. * M_PI;
e = solve_kepler(mean, ecc);
cosp = cos(e);
sinp = sin(e);
_cosp_ecc = cosp - ecc;
de_dt = omega / (1. - ecc * cosp);
if(ecc > .99)
{
e2 = (e > 2. * M_PI - 1e-3) ? e - 2. * M_PI : e;
if(e2 < 1e-3)
{
e2 *= e2;
_1_ecc = scal_prod(j_, j_)/(p0->m*a*(1+ecc));
_cosp_1 = - .5 * e2 * (1 - e2 / 12. * (1 - e2 / 30.));
_cosp_ecc = _1_ecc + _cosp_1;
de_dt = omega / (_1_ecc - ecc * _cosp_1);
}
}
for(i = 0; i < DIMENSIONS; i++)
{
r_[i] = a_[i] * _cosp_ecc + b_[i] * sinp ; // new location
v_[i] = (-a_[i] * sinp + b_[i] * cosp) * de_dt; // direction of v only
}
}
else
// hyperbolic orbit // parabolic?
{
if(p1->is_elliptical)
{
fprintf(get_file(FILE_WARNING), "#### [t=%1.12e+%1.12e] Particle #%d thrown onto hyperbolic orbit with e=%e (E=%e, a=%e) ####\n",
t_total(p1->t), convert_time(dt, 0), pos, ecc, p1->energy, convert_length(a, 0));
p1->is_elliptical = 0;
}
if(ecc == 1)
fprintf(get_file(FILE_WARNING), "# # # %e\tParabolic orbit of m%d treated as hyperbolic: e=%e\t(x=%e)\n",
t_total(p1->t), pos, ecc, convert_length(v_abs(p1->xp), 0));
// calculate eccentric anomaly e at t+dt
e = (a + v_abs(r_)) / (ecc * a);
if(e < 1.0) e = .0;
else if(scal_prod(r_, v_) < 0) e = -acosh(e);
else e = acosh(e);
e = kepler(ecc, ecc * sinh(e) - e + dt * omega);
cosp = cosh(e);
sinp = sinh(e);
de_dt = omega / (ecc * cosp - 1.);
for(i = 0; i < DIMENSIONS; i++)
{
r_[i] = a_[i] * (ecc - cosp) + b_[i] * sinp; // new location
v_[i] = (-a_[i] * sinp + b_[i] * cosp) * de_dt; // direction of v only
}
}
// get |v_| from j_ = r_ x v_
v = v_abs(v_);
r = v_abs(r_);
v = v_abs(j_) / (r * v * sin(acos(scal_prod(r_, v_)/ (r * v))));
for(i = 0; i < DIMENSIONS; i++)
{
//v_[i] *= v;
// total motion relative to fix central mass
p1->xp[i] = p0->xp[i] + r_[i];
p1->vp[i] = p0->vp[i] + v_[i];
}
if(out_a != NULL)
{
_1_r2 = 1. / scal_prod(r_, r_);
afact = - m_c * _1_r2 * sqrt(_1_r2);
//printf("4 %e %e %e\n", *(out_a), *(out_a+1), *(out_a+2));
for(i = 0; i < DIMENSIONS; i++)
out_a[i] = afact * r_[i];
if(out_a_ != NULL)
{
v_r_ = scal_prod(v_, r_);
for(i = 0; i < DIMENSIONS; i++)
out_a_[i] = afact * (v_[i] - 3 * _1_r2 * v_r_ * r_[i]);
if(out_a_2 != NULL)
{
v_v_ = scal_prod(v_, v_);
r_a_ = scal_prod(r_, out_a);
for(i = 0; i < DIMENSIONS; i++)
out_a_2[i] = afact * (out_a[i] - 3. * _1_r2 * (v_r_ * (2. * v_[i] - 5. * v_r_ * r_[i] * _1_r2)
+ (v_v_ + r_a_) * r_[i]));
if(out_a_3 != NULL)
{
v_a_ = scal_prod(v_, out_a);
r_a__ = scal_prod(r_, out_a_);
for(i = 0; i < DIMENSIONS; i++)
out_a_3[i] = afact * (out_a_[i]
- 3. * _1_r2 * (3. * v_r_ * out_a[i]
+ 3. * (v_v_ + r_a_)
* (v_[i] - 5. * v_r_ * _1_r2 * r_[i])
+ (3. * v_a_ + r_a__) * r_[i]
+ v_r_ * v_r_ * _1_r2
* (-15. * v_[i] + 35. * v_r_ * _1_r2 * r_[i])));
}
}
}
}
_exit_function();
}
double advance_orbit_set_up(BEMRI * b, double chi, double r_current, double dt1)
{
double t_current, t_new, psi_h_target, theta_h_target, psi_b_advanced, Mb, t_elapsed, new_true;
//find current hole eccentric anomaly
psi_h = acos( (eh + cos(theta_h)) / (1+eh*cos(theta_h)) );
//find current time in the orbit (since last periapse)
t_current = Ph/(2*PI) * (psi_h - eh*sin(psi_h));
//now find target (new) values
psi_h_target = 2*PI - acos( (1 - chi/ah) / eh); //hole eccentric anomaly value to skip to
theta_h_target = fix_angle( 2*atan2(sqrt(1+eh)*sin(psi_h_target/2.0) , sqrt(1-eh)*cos(psi_h_target/2)) ); //hole true anomaly value to skip to
t_new = Ph/(2*PI) * (psi_h_target - eh*sin(psi_h_target)); //new time value from Kepler's equation
//then the elapsed time is given by:
t_elapsed = t_new - t_current;
//printf("\nt_elapsed = %.5e",t_elapsed);
//_+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
//force t_elapsed to equal an integer multiple of dt1
t_elapsed = floor(t_elapsed/dt1) * dt1; //now t_elapsed is good
t_new = t_elapsed + t_current; //now we're recalculating the hole orbit anomalies
//printf("\nMh = %.15e",2*PI/Ph*t_new);
psi_h_target = solve_kepler(2*PI/Ph*t_new,eh); //here is the new target eccentric anomaly
theta_h_target = fix_angle( 2*atan2(sqrt(1+eh)*sin(psi_h_target/2.0) , sqrt(1-eh)*cos(psi_h_target/2.0)) ); //new hole true anomaly value to skip to
//printf("\npsi_h_target = %.3e\nsolving keplers eqn for t_new = %.3f",psi_h_target, Ph/(2*PI)*kepler(psi_h_target,eh));
//printf("\nt_elapsed / dt1 = %.20f\n",t_elapsed / dt1);
//_+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
//now the mean anomaly of the BEMRI orbit will have advanced by:
Mb = 2*PI/Pb*t_elapsed - 2*PI*floor(t_elapsed/Pb); //note this is not the current mean anomaly of the BEMRI, but how much it has advanced
//here is how much the eccentric anomaly of the BEMRI has advanced
psi_b_advanced = solve_kepler(Mb,eb);
//so the new value of the true anomaly of the BEMRI is given by: (current + advanced)
new_true = theta_b + fix_angle((2*atan2(sqrt(1+eb)*sin(psi_b_advanced/2.0),sqrt(1-eb)*cos(psi_b_advanced/2.0))) );
//_+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
//printf("\ncurrent hole anomaly is %.3f, target anomaly is %.3f\n",theta_h, theta_h_target);
//printf("chi = %.3e\nah = %.3e\neh=%.3e\n",chi,ah,eh);
//_+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
//before advancing the BH orbit, freeze the BEMRI CM motion so the new CM motion can be added on easily
for(int i=0; i < 3; i++)
{
V1[i] -= V4[i]; //remove CM motion for mass 1
V2[i] -= V4[i]; //remove CM motion for mass 2
R1[i] -= R4[i];
R2[i] -= R4[i];
}
//now that we know the new true anomaly of the hole AND the BEMRI, we can advance the two orbits to the new positions
advance_orbit(&(b->binary_b),lb,new_true);
//now advance the BH orbit
advance_orbit(&(b->binary_h),lh,theta_h_target);
//and now the BEMRI CM values need to be updated:
//BEMRI_CM_update(b);
return t_elapsed;
}
