def chip_averaged(q,chi1,chi2,theta1,theta2,deltaphi,r=None,fref=None, M_msun=None):
'''Averaged definition of chip. Eq (15) and Appendix A'''
# Convert frequency to separation, if necessary
if r is None and fref is None: raise ValueError
elif r is not None and fref is not None: raise ValueError
if r is None:
if M_msun is None: raise ValueError
# Eq A1
r = ftor_PN(fref, M_msun, q, chi1, chi2, theta1, theta2, deltaphi)
#Compute constants of motion
L = (r**0.5)*q/(1+q)**2
S1 = chi1/(1.+q)**2
S2 = (q**2)*chi2/(1.+q)**2
# Eq 5
chieff = (chi1*np.cos(theta1)+q*chi2*np.cos(theta2))/(1+q)
# Eq A2
J = (L**2 + S1**2 + S2**2 + 2*L*(S1*np.cos(theta1) + S2*np.cos(theta2)) + 2*S1*S2*(np.sin(theta1)*np.sin(theta2)*np.cos(deltaphi) + np.cos(theta1)*np.cos(theta2)))**0.5
# Solve dSdt=0. Details in arXiv:1506.03492
Sminus,Splus=precession.Sb_limits(chieff,J,q,S1,S2,r)
def integrand_numerator(S):
'''chip(S)/dSdt(S)'''
#Eq A3
theta1ofS = np.arccos( (1/(2*(1-q)*S1)) * ( (J**2-L**2-S**2)/L - 2*q*chieff/(1+q) ) )
#Eq A4
theta2ofS = np.arccos( (q/(2*(1-q)*S2)) * ( -(J**2-L**2-S**2)/L + 2*chieff/(1+q) ) )
# Eq A5
deltaphiofS = np.arccos( ( S**2-S1**2-S2**2 - 2*S1*S2*np.cos(theta1ofS)*np.cos(theta2ofS) ) / (2*S1*S2*np.sin(theta1ofS)*np.sin(theta2ofS)) )
# Eq A6 (prefactor cancels out)
dSdtofS = np.sin(theta1ofS)*np.sin(theta2ofS)*np.sin(deltaphiofS)/S
# Eq (15)
chipofS = chip_generalized(q,chi1,chi2,theta1ofS,theta2ofS,deltaphiofS)
return chipofS/dSdtofS
numerator = scipy.integrate.quad(integrand_numerator, Sminus, Splus)[0]
def integrand_denominator(S):
'''1/dSdt(S)'''
#Eq A3
theta1ofS = np.arccos( (1/(2*(1-q)*S1)) * ( (J**2-L**2-S**2)/L - 2*q*chieff/(1+q) ) )
#Eq A4
theta2ofS = np.arccos( (q/(2*(1-q)*S2)) * ( -(J**2-L**2-S**2)/L + 2*chieff/(1+q) ) )
# Eq A5
deltaphiofS = np.arccos( ( S**2-S1**2-S2**2 - 2*S1*S2*np.cos(theta1ofS)*np.cos(theta2ofS) ) / (2*S1*S2*np.sin(theta1ofS)*np.sin(theta2ofS)) )
# Eq A6 (prefactor cancels out)
dSdtofS = np.sin(theta1ofS)*np.sin(theta2ofS)*np.sin(deltaphiofS)/S
return 1/dSdtofS
denominator = scipy.integrate.quad(integrand_denominator, Sminus, Splus)[0]
return numerator/denominator
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 compare_evolutions():
'''
Compare precession averaged and orbit averaged integrations. Plot the
evolution of xi, J, S and their relative differences between the two
approaches. Since precession-averaged estimates of S require a random
sampling, this plot will look different every time this routine is executed.
Output is saved in ./spin_angles.pdf.
**Run using**
import precession.test
precession.test.compare_evolutions()
'''
fig = pylab.figure(figsize=(6, 6)) # Create figure object and axes
L, Ws, Wm, G = 0.85, 0.15, 0.3, 0.03 # Sizes
ax_Sd = fig.add_axes([0, 0, L, Ws]) # bottom-small
ax_S = fig.add_axes([0, Ws, L, Wm]) # bottom-main
ax_Jd = fig.add_axes([0, Ws + Wm + G, L, Ws]) # middle-small
ax_J = fig.add_axes([0, Ws + Ws + Wm + G, L, Wm]) # middle-main
ax_xid = fig.add_axes([0, 2 * (Ws + Wm + G), L, Ws]) # top-small
ax_xi = fig.add_axes([0, Ws + 2 * (Ws + Wm + G), L, Wm]) # top-main
q = 0.8 # 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
ri = 100. * M # Initial separation.
rf = 10. * M # Final separation.
r_vals = numpy.linspace(ri, rf, 1001) # Output requested
Ji = 2.24 # Magnitude of J: Jmin<J<Jmax as given by J_lim
xi = -0.5 # Effective spin: xi_low<xi<xi_up as given by xi_allowed
Jf_P = precession.evolve_J(xi, Ji, r_vals, q, S1, S2) # Pr.av. integration
Sf_P = [
precession.samplingS(xi, J, q, S1, S2, r)
for J, r in zip(Jf_P[0::10], r_vals[0::10])
] # Resample S (reduce output for clarity)
Sb_min, Sb_max = zip(*[
precession.Sb_limits(xi, J, q, S1, S2, r)
for J, r in zip(Jf_P, r_vals)
]) # Envelopes
S = numpy.average([precession.Sb_limits(xi, Ji, q, S1, S2,
ri)]) # Initialize S
Jf_O, xif_O, Sf_O = precession.orbit_averaged(Ji, xi, S, r_vals, q, S1,
S2) # Orb.av. integration
Pcol, Ocol, Dcol = 'blue', 'red', 'green'
Pst, Ost = 'solid', 'dashed'
ax_xi.axhline(xi, c=Pcol, ls=Pst, lw=2) # Plot xi, pr.av. (constant)
ax_xi.plot(r_vals, xif_O, c=Ocol, ls=Ost, lw=2) # Plot xi, orbit averaged
ax_xid.plot(r_vals, (xi - xif_O) / xi * 1e11, c=Dcol,
lw=2) # Plot xi deviations (rescaled)
ax_J.plot(r_vals, Jf_P, c=Pcol, ls=Pst, lw=2) # Plot J, pr.av.
ax_J.plot(r_vals, Jf_O, c=Ocol, ls=Ost, lw=2) # Plot J, orb.av
ax_Jd.plot(r_vals, (Jf_P - Jf_O) / Jf_O * 1e3, c=Dcol,
lw=2) # Plot J deviations (rescaled)
ax_S.scatter(r_vals[0::10], Sf_P, facecolor='none',
edgecolor=Pcol) # Plot S, pr.av. (resampled)
ax_S.plot(r_vals, Sb_min, c=Pcol, ls=Pst,
lw=2) # Plot S, pr.av. (envelopes)
ax_S.plot(r_vals, Sb_max, c=Pcol, ls=Pst,
lw=2) # Plot S, pr.av. (envelopes)
ax_S.plot(r_vals, Sf_O, c=Ocol, ls=Ost, lw=2) # Plot S, orb.av (evolved)
ax_Sd.plot(r_vals[0::10], (Sf_P - Sf_O[0::10]) / Sf_O[0::10], c=Dcol,
lw=2) # Plot S deviations
# Options for nice plotting
for ax in [ax_xi, ax_xid, ax_J, ax_Jd, ax_S, ax_Sd]:
ax.set_xlim(ri, rf)
ax.yaxis.set_label_coords(-0.16, 0.5)
ax.spines['left'].set_lw(1.5)
ax.spines['right'].set_lw(1.5)
for ax in [ax_xi, ax_J, ax_S]:
ax.spines['top'].set_lw(1.5)
for ax in [ax_xid, ax_Jd, ax_Sd]:
ax.axhline(0, c='black', ls='dotted')
ax.spines['bottom'].set_lw(1.5)
for ax in [ax_xid, ax_J, ax_Jd, ax_S]:
ax.set_xticklabels([])
ax_xi.set_ylim(-0.55, -0.45)
ax_J.set_ylim(0.4, 2.3)
ax_S.set_ylim(0.24, 0.41)
ax_xid.set_ylim(-0.2, 1.2)
ax_Jd.set_ylim(-3, 5.5)
ax_Sd.set_ylim(-0.7, 0.7)
ax_xid.yaxis.set_major_locator(matplotlib.ticker.MultipleLocator(0.5))
ax_Jd.yaxis.set_major_locator(matplotlib.ticker.MultipleLocator(2))
ax_S.yaxis.set_major_locator(matplotlib.ticker.MultipleLocator(0.05))
ax_Sd.yaxis.set_major_locator(matplotlib.ticker.MultipleLocator(0.5))
ax_xi.xaxis.set_ticks_position('top')
ax_xi.xaxis.set_label_position('top')
ax_Sd.set_xlabel("$r/M$")
ax_xi.set_xlabel("$r/M$")
ax_xi.set_ylabel("$\\xi$")
ax_J.set_ylabel("$J/M^2$")
ax_S.set_ylabel("$S/M^2$")
ax_xid.set_ylabel("$\\Delta\\xi/\\xi \;[10^{-11}]$")
ax_Jd.set_ylabel("$\\Delta J/J \;[10^{-3}]$")
ax_Sd.set_ylabel("$\\Delta S / S$")
fig.savefig("compare_evolutions.pdf", bbox_inches='tight') # Save pdf file
def phase_resampling():
'''
Precessional phase resampling. The magnidute of the total spin S is sampled
according to |dS/dt|^-1, which correspond to a flat distribution in t(S).
Output is saved in ./phase_resampling.pdf and data stored in
`precession.storedir'/phase_resampling_.dat
**Run using**
import precession.test
precession.test.phase_resampling()
'''
fig = pylab.figure(figsize=(6, 6)) #Create figure object and axes
ax_tS = fig.add_axes([0, 0, 0.6, 0.6]) #bottom-left
ax_td = fig.add_axes([0.65, 0, 0.3, 0.6]) #bottom-right
ax_Sd = fig.add_axes([0, 0.65, 0.6, 0.3]) #top-left
q = 0.5 # Mass ratio. Must be q<=1.
chi1 = 0.3 # Primary spin. Must be chi1<=1
chi2 = 0.9 # Secondary spin. Must be chi2<=1
M, m1, m2, S1, S2 = precession.get_fixed(q, chi1,
chi2) # Total-mass units M=1
r = 200. * M # Separation. Must be r>10M for PN to be valid
J = 3.14 # Magnitude of J: Jmin<J<Jmax as given by J_lim
xi = -0.01 # Effective spin: xi_low<xi<xi_up as given by xi_allowed
Sb_min, Sb_max = precession.Sb_limits(xi, J, q, S1, S2, r) # Limits in S
tau = precession.precession_period(xi, J, q, S1, S2,
r) # Precessional period
d = 2000 # Size of the statistical sample
precession.make_temp() # Create store directory, if necessary
filename = precession.storedir + "/phase_resampling.dat" # Output file name
if not os.path.isfile(filename): # Compute and store data if not present
out = open(filename, "w")
out.write("# q chi1 chi2 r J xi d\n") # Write header
out.write("# " +
' '.join([str(x)
for x in (q, chi1, chi2, r, J, xi, d)]) + "\n")
# S and t values for the S(t) plot
S_vals = numpy.linspace(Sb_min, Sb_max, d)
t_vals = numpy.array([
abs(
precession.t_of_S(Sb_min, S, Sb_min, Sb_max, xi, J, q, S1, S2,
r)) for S in S_vals
])
# Sample values of S from |dt/dS|. Distribution should be flat in t.
S_sample = numpy.array(
[precession.samplingS(xi, J, q, S1, S2, r) for i in range(d)])
t_sample = numpy.array([
abs(
precession.t_of_S(Sb_min, S, Sb_min, Sb_max, xi, J, q, S1, S2,
r)) for S in S_sample
])
# Continuous distributions (normalized)
S_distr = numpy.array([
2. * abs(precession.dtdS(S, xi, J, q, S1, S2, r) / tau)
for S in S_vals
])
t_distr = numpy.array([2. / tau for t in t_vals])
out.write("# S_vals t_vals S_sample t_sample S_distr t_distr\n")
for Sv, tv, Ss, ts, Sd, td in zip(S_vals, t_vals, S_sample, t_sample,
S_distr, t_distr):
out.write(' '.join([str(x)
for x in (Sv, tv, Ss, ts, Sd, td)]) + "\n")
out.close()
else: # Read
S_vals, t_vals, S_sample, t_sample, S_distr, t_distr = numpy.loadtxt(
filename, unpack=True)
# Rescale all time values by 10^-6, for nicer plotting
tau *= 1e-6
t_vals *= 1e-6
t_sample *= 1e-6
t_distr /= 1e-6
ax_tS.plot(S_vals, t_vals, c='blue', lw=2) # S(t) curve
ax_td.plot(t_distr, t_vals, lw=2., c='red') # Continous distribution P(t)
ax_Sd.plot(S_vals, S_distr, lw=2., c='red') # Continous distribution P(S)
ax_td.hist(t_sample,
bins=60,
range=(0, tau / 2.),
normed=True,
histtype='stepfilled',
color="blue",
alpha=0.4,
orientation="horizontal") # Histogram P(t)
ax_Sd.hist(S_sample,
bins=60,
range=(Sb_min, Sb_max),
normed=True,
histtype='stepfilled',
color="blue",
alpha=0.4) # Histogram P(S)
# Options for nice plotting
ax_tS.set_xlim(Sb_min, Sb_max)
ax_tS.set_ylim(0, tau / 2.)
ax_tS.set_xlabel("$S/M^2$")
ax_tS.set_ylabel("$t/(10^6 M)$")
ax_td.set_xlim(0, 0.5)
ax_td.set_ylim(0, tau / 2.)
ax_td.set_xlabel("$P(t)$")
ax_td.set_yticklabels([])
ax_Sd.set_xlim(Sb_min, Sb_max)
ax_Sd.set_ylim(0, 20)
ax_Sd.set_xticklabels([])
ax_Sd.set_ylabel("$P(S)$")
fig.savefig("phase_resampling.pdf", bbox_inches='tight') # Save pdf file
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