def parameter_selection():
'''
Selection of consistent parameters to describe the BH spin orientations, both at finite and infinitely large separation. Compute some quantities which characterize the spin-precession dynamics, such as morphology, precessional period and resonant angles.
All quantities are given in total-mass units c=G=M=1.
**Run using**
import precession.test
precession.test.parameter_selection()
'''
print "\n *Parameter selection at finite separations*"
q = 0.8 # Must be q<=1. Check documentation for q=1.
chi1 = 1. # Must be chi1<=1
chi2 = 1. # Must be chi2<=1
M, m1, m2, S1, S2 = precession.get_fixed(q, chi1,
chi2) # Total-mass units M=1
print "We study a binary with\n\tq=%.3f m1=%.3f m2=%.3f\n\tchi1=%.3f S1=%.3f\n\tchi2=%.3f S2=%.3f" % (
q, m1, m2, chi1, S1, chi2, S2)
r = 100 * M # Must be r>10M for PN to be valid
print "at separation\n\tr=%.3f" % r
xi_min, xi_max = precession.xi_lim(q, S1, S2)
Jmin, Jmax = precession.J_lim(q, S1, S2, r)
Sso_min, Sso_max = precession.Sso_limits(S1, S2)
print "The geometrical limits on xi,J and S are\n\t%.3f<=xi<=%.3f\n\t%.3f<=J<=%.3f\n\t%.3f<=S<=%.3f" % (
xi_min, xi_max, Jmin, Jmax, Sso_min, Sso_max)
J = (Jmin + Jmax) / 2.
print "We select a value of J\n\tJ=%.3f " % J
St_min, St_max = precession.St_limits(J, q, S1, S2, r)
print "This constrains the range of S to\n\t%.3f<=S<=%.3f" % (St_min,
St_max)
xi_low, xi_up = precession.xi_allowed(J, q, S1, S2, r)
print "The allowed range of xi is\n\t%.3f<=xi<=%.3f" % (xi_low, xi_up)
xi = (xi_low + xi_up) / 2.
print "We select a value of xi\n\txi=%.3f" % xi
test = (J >= min(precession.J_allowed(xi, q, S1, S2, r))
and J <= max(precession.J_allowed(xi, q, S1, S2, r)))
print "Is our couple (xi,J) consistent?", test
Sb_min, Sb_max = precession.Sb_limits(xi, J, q, S1, S2, r)
print "S oscillates between\n\tS-=%.3f\n\tS+=%.3f" % (Sb_min, Sb_max)
S = (Sb_min + Sb_max) / 2.
print "We select a value of S between S- and S+\n\tS=%.3f" % S
t1, t2, dp, t12 = precession.parametric_angles(S, J, xi, q, S1, S2, r)
print "The angles describing the spin orientations are\n\t(theta1,theta2,DeltaPhi)=(%.3f,%.3f,%.3f)" % (
t1, t2, dp)
xi, J, S = precession.from_the_angles(t1, t2, dp, q, S1, S2, r)
print "From the angles one can recovery\n\t(xi,J,S)=(%.3f,%.3f,%.3f)" % (
xi, J, S)
print "\n *Features of spin precession*"
t1_dp0, t2_dp0, t1_dp180, t2_dp180 = precession.resonant_finder(
xi, q, S1, S2, r)
print "The spin-orbit resonances for these values of J and xi are\n\t(theta1,theta2)=(%.3f,%.3f) for DeltaPhi=0\n\t(theta1,theta2)=(%.3f,%.3f) for DeltaPhi=pi" % (
t1_dp0, t2_dp0, t1_dp180, t2_dp180)
tau = precession.precession_period(xi, J, q, S1, S2, r)
print "We integrate dt/dS to calculate the precessional period\n\ttau=%.3f" % tau
alpha = precession.alphaz(xi, J, q, S1, S2, r)
print "We integrate Omega*dt/dS to find\n\talpha=%.3f" % alpha
morphology = precession.find_morphology(xi, J, q, S1, S2, r)
if morphology == -1: labelm = "Librating about DeltaPhi=0"
elif morphology == 1: labelm = "Librating about DeltaPhi=pi"
elif morphology == 0: labelm = "Circulating"
print "The precessional morphology is: " + labelm
sys.stdout = os.devnull # Ignore warnings
phase, xi_transit_low, xi_transit_up = precession.phase_xi(J, q, S1, S2, r)
sys.stdout = sys.__stdout__ # Restore warnings
if phase == -1: labelp = "a single DeltaPhi~pi phase"
elif phase == 2: labelp = "two DeltaPhi~pi phases, a Circulating phase"
elif phase == 3:
labelp = "a DeltaPhi~0, a Circulating, a DeltaPhi~pi phase"
print "The coexisting phases are: " + labelp
print "\n *Parameter selection at infinitely large separation*"
print "We study a binary with\n\tq=%.3f m1=%.3f m2=%.3f\n\tchi1=%.3f S1=%.3f\n\tchi2=%.3f S2=%.3f" % (
q, m1, m2, chi1, S1, chi2, S2)
print "at infinitely large separation"
kappainf_min, kappainf_max = precession.kappainf_lim(S1, S2)
print "The geometrical limits on xi and kappa_inf are\n\t%.3f<=xi<=%.3f\n\t %.3f<=kappa_inf<=%.3f" % (
xi_min, xi_max, kappainf_min, kappainf_max)
print "We select a value of xi\n\txi=%.3f" % xi
kappainf_low, kappainf_up = precession.kappainf_allowed(xi, q, S1, S2)
print "This constrains the range of kappa_inf to\n\t%.3f<=kappa_inf<=%.3f" % (
kappainf_low, kappainf_up)
kappainf = (kappainf_low + kappainf_up) / 2.
print "We select a value of kappa_inf\n\tkappa_inf=%.3f" % kappainf
test = (xi >= min(precession.xiinf_allowed(kappainf, q, S1, S2))
and xi <= max(precession.xiinf_allowed(kappainf, q, S1, S2)))
print "Is our couple (xi,kappa_inf) consistent?", test
t1_inf, t2_inf = precession.thetas_inf(xi, kappainf, q, S1, S2)
print "The asymptotic (constant) values of theta1 and theta2 are\n\t(theta1_inf,theta2_inf)=(%.3f,%.3f)" % (
t1_inf, t2_inf)
xi, kappainf = precession.from_the_angles_inf(t1_inf, t2_inf, q, S1, S2)
print "From the angles one can recovery\n\t(xi,kappa_inf)=(%.3f,%.3f)" % (
xi, kappainf)
def spin_angles():
'''
Binary dynamics on the precessional timescale. The spin angles
theta1,theta2, DeltaPhi and theta12 are computed and plotted against the
time variable, which is obtained integrating dS/dt. The morphology is also
detected as indicated in the legend of the plot. Output is saved in
./spin_angles.pdf.
**Run using**
import precession.test
precession.test.spin_angles()
'''
fig = pylab.figure(figsize=(6, 6)) # Create figure object and axes
ax_t1 = fig.add_axes([0, 1.95, 0.9, 0.5]) # first (top)
ax_t2 = fig.add_axes([0, 1.3, 0.9, 0.5]) # second
ax_dp = fig.add_axes([0, 0.65, 0.9, 0.5]) # third
ax_t12 = fig.add_axes([0, 0, 0.9, 0.5]) # fourth (bottom)
q = 0.7 # Mass ratio. Must be q<=1.
chi1 = 0.6 # Primary spin. Must be chi1<=1
chi2 = 1. # Secondary spin. Must be chi2<=1
M, m1, m2, S1, S2 = precession.get_fixed(q, chi1,
chi2) # Total-mass units M=1
r = 20 * M # Separation. Must be r>10M for PN to be valid
J = 0.94 # Magnitude of J: Jmin<J<Jmax as given by J_lim
xi_vals = [-0.41, -0.3,
-0.22] # Effective spin: xi_low<xi<xi_up as given by xi_allowed
for xi, color in zip(xi_vals,
['blue', 'green', 'red']): # Loop over three binaries
tau = precession.precession_period(xi, J, q, S1, S2, r) # Period
morphology = precession.find_morphology(xi, J, q, S1, S2,
r) # Morphology
if morphology == -1: labelm = "${\\rm L}0$"
elif morphology == 1: labelm = "${\\rm L}\\pi$"
elif morphology == 0: labelm = "${\\rm C}$"
Sb_min, Sb_max = precession.Sb_limits(xi, J, q, S1, S2,
r) # Limits in S
S_vals = numpy.linspace(Sb_min, Sb_max,
1000) # Create array, from S- to S+
S_go = S_vals # First half of the precession cycle: from S- to S+
t_go = map(lambda x: precession.t_of_S(
S_go[0], x, Sb_min, Sb_max, xi, J, q, S1, S2, r, 0, sign=-1.),
S_go) # Compute time values. Assume t=0 at S-
t1_go, t2_go, dp_go, t12_go = zip(*[
precession.parametric_angles(S, J, xi, q, S1, S2, r) for S in S_go
]) # Compute the angles.
dp_go = [-dp
for dp in dp_go] # DeltaPhi<=0 in the first half of the cycle
S_back = S_vals[::
-1] # Second half of the precession cycle: from S+ to S-
t_back = map(
lambda x: precession.t_of_S(S_back[0],
x,
Sb_min,
Sb_max,
xi,
J,
q,
S1,
S2,
r,
t_go[-1],
sign=1.), S_back
) # Compute time, start from the last point of the first half t_go[-1]
t1_back, t2_back, dp_back, t12_back = zip(*[
precession.parametric_angles(S, J, xi, q, S1, S2, r)
for S in S_back
]) # Compute the angles. DeltaPhi>=0 in the second half of the cycle
for ax, vec_go, vec_back in zip(
[ax_t1, ax_t2, ax_dp, ax_t12], [t1_go, t2_go, dp_go, t12_go],
[t1_back, t2_back, dp_back, t12_back]): # Plot all curves
ax.plot([t / tau for t in t_go],
vec_go,
c=color,
lw=2,
label=labelm)
ax.plot([t / tau for t in t_back], vec_back, c=color, lw=2)
# Options for nice plotting
for ax in [ax_t1, ax_t2, ax_dp, ax_t12]:
ax.set_xlim(0, 1)
ax.set_xlabel("$t/\\tau$")
ax.set_xticks(numpy.linspace(0, 1, 5))
for ax in [ax_t1, ax_t2, ax_t12]:
ax.set_ylim(0, numpy.pi)
ax.set_yticks(numpy.linspace(0, numpy.pi, 5))
ax.set_yticklabels(
["$0$", "$\\pi/4$", "$\\pi/2$", "$3\\pi/4$", "$\\pi$"])
ax_dp.set_ylim(-numpy.pi, numpy.pi)
ax_dp.set_yticks(numpy.linspace(-numpy.pi, numpy.pi, 5))
ax_dp.set_yticklabels(
["$-\\pi$", "$-\\pi/2$", "$0$", "$\\pi/2$", "$\\pi$"])
ax_t1.set_ylabel("$\\theta_1$")
ax_t2.set_ylabel("$\\theta_2$")
ax_t12.set_ylabel("$\\theta_{12}$")
ax_dp.set_ylabel("$\\Delta\\Phi$")
ax_t1.legend(
loc='lower right',
fontsize=18) # Fill the legend with the precessional morphology
fig.savefig("spin_angles.pdf", bbox_inches='tight') # Save pdf file