def ln_posterior_initial(x, args):
""" Calculate the posterior probability for the initial mass
and birth time calculations.
Parameters
----------
x : M1, M2, t_obs
model parameters
args : M2_d
observed companion mass
Returns
-------
lp : float
posterior probability
"""
M1, M2, t_obs = x
M2_d = args
y = M1, M2, M2_d, t_obs
lp = ln_priors_initial(y)
if np.isinf(lp): return -np.inf
# Get observed mass, mdot
t_eff_obs = binary_evolve.func_get_time(M1, M2, t_obs)
M2_c = M1 + M2 - load_sse.func_sse_he_mass(M1)
M2_tmp, M2_dot, R_tmp, k_tmp = load_sse.func_get_sse_star(M2_c, t_eff_obs)
# Somewhat arbitrary definition of mass error
delta_M_err = 1.0
coeff = -0.5 * np.log( 2. * np.pi * delta_M_err*delta_M_err )
argument = -( M2_d - M2_tmp ) * ( M2_d - M2_tmp ) / ( 2. * delta_M_err*delta_M_err )
return coeff + argument + lp
def ln_priors_initial(x):
""" Calculate the prior probability for the initial mass
and birth time calculations.
Parameters
----------
x : M1, M2, M2_d, t_obs
Model parameters plus the observed companion mass
Returns
-------
ll : float
Prior probability of the model parameters
"""
M1, M2, M2_d, t_obs = x
# M1
if M1 < c.min_mass or M1 > c.max_mass: return -np.inf
# M2
if M2 < 0.3*M1 or M2 > M1: return -np.inf
# Add a prior so that the post-MT secondary is within the correct bounds
M2_c = M1 + M2 - load_sse.func_sse_he_mass(M1)
if M2_c > c.max_mass or M2_c < c.min_mass: return -np.inf
# Add a prior so the primary can go through a SN by t_obs
if load_sse.func_sse_tmax(M1) > t_obs: return -np.inf
# Add a prior so the effective time remains bounded
t_eff_obs = binary_evolve.func_get_time(M1, M2, t_obs)
if t_eff_obs < 0.0: return -np.inf
# Add a prior so that only those masses with a non-zero Mdot are allowed
M2_tmp, M2_dot, R_tmp, k_tmp = load_sse.func_get_sse_star(M2_c, t_eff_obs)
if M2_dot == 0.0: return -np.inf
return 0.0
def full_forward(M1, M2, A, ecc, v_k, theta, phi, t_obs):
""" Evolve a binary forward from its initial conditions
Parameters
----------
M1 : float
Initial primary mass (Msun)
M2 : float
Initial secondary mass (Msun)
A : float
Initial orbital separation (Rsun)
ecc : float
Initial orbital eccentricity (unitless)
v_k : float
SN kick velocity
theta : float
SN kick polar angle
phi : float
SN kick azimuthal angle
t_obs : float
observation time
Returns
-------
M_NS : float or ndarray
Array of final primary masses (Currently set to the NS mass, c.M_NS)
M_2 : float or ndarray
Array of final secondary masses (Msun)
L_x : float or ndarray
X-ray luminosity (erg/s)
v_sys : float or ndarray
Systemic velocity (km/s)
M2_dot : float or ndarray
Mass accretion rate (Msun/yr)
A : float or ndarray
Orbital separation (Rsun)
ecc : float or ndarray
Orbital eccentricity (unitless)
theta : float or ndarray
Projected angular distance traveled from birth location (radians)
k_type : int
k-type of HMXB donor
"""
if load_sse.func_sse_mass is None:
load_sse.load_sse()
if isinstance(M1, np.ndarray):
dtypes = [('M_NS','<f8'), \
('M_2','<f8'), \
('L_x','<f8'), \
('v_sys','<f8'), \
('M2_dot','<f8'), \
('A','<f8'), \
('ecc','<f8'), \
('theta','<f8'), \
('k_type','<i8')]
HMXB = np.recarray(len(M1), dtype=dtypes)
for i in np.arange(len(M1)):
if isinstance(t_obs, np.ndarray):
if t_obs[i] < load_sse.func_sse_ms_time(M1[i]):
HMXB["M_NS"][i] = M1[i]
HMXB["M_2"][i] = M2[i]
HMXB["A"][i] = A[i]
continue
else:
if t_obs < load_sse.func_sse_ms_time(M1[i]):
HMXB["M_NS"][i] = M1[i]
HMXB["M_2"][i] = M2[i]
HMXB["A"][i] = A[i]
continue
# First MT phase
M_1_b, M_2_b, A_b = binary_evolve.func_MT_forward(M1[i], M2[i], A[i], ecc[i])
if isinstance(t_obs, np.ndarray):
if t_obs[i] < load_sse.func_sse_tmax(M1[i]):
HMXB["M_NS"][i] = M_1_b
HMXB["M_2"][i] = M_2_b
HMXB["A"][i] = A_b
continue
else:
if t_obs < load_sse.func_sse_tmax(M1[i]):
HMXB["M_NS"][i] = M_1_b
HMXB["M_2"][i] = M_2_b
HMXB["A"][i] = A_b
continue
# SN
A_tmp, v_sys_tmp, e_tmp = binary_evolve.func_SN_forward(M_1_b, M_2_b, A_b, v_k[i], theta[i], phi[i])
# XRB
if isinstance(t_obs, np.ndarray):
M_2_tmp, L_x_tmp, M2_dot_out, A_out = binary_evolve.func_Lx_forward(M1[i], M2[i], M_2_b, A_tmp, e_tmp, t_obs[i])
theta_out = (t_obs[i] - load_sse.func_sse_tmax(M1[i])) * v_sys_tmp / c.dist_SMC * c.yr_to_sec * 1.0e6 * np.sin(get_theta(1))
tobs_eff = binary_evolve.func_get_time(M1[i], M2[i], t_obs[i])
else:
M_2_tmp, L_x_tmp, M2_dot_out, A_out = binary_evolve.func_Lx_forward(M1[i], M2[i], M_2_b, A_tmp, e_tmp, t_obs)
theta_out = (t_obs - load_sse.func_sse_tmax(M1[i])) * v_sys_tmp / c.dist_SMC * c.yr_to_sec * 1.0e6 * np.sin(get_theta(1))
tobs_eff = binary_evolve.func_get_time(M1[i], M2[i], t_obs)
# To get k-type of HMXB donor
if M_2_b > c.max_mass:
k_type = -999
else:
M_tmp, M_dot_tmp, R_tmp, k_type = load_sse.func_get_sse_star(M_2_b, tobs_eff)
HMXB["M_NS"][i] = c.M_NS
HMXB["M_2"][i] = M_2_tmp
HMXB["L_x"][i] = L_x_tmp
HMXB["v_sys"][i] = v_sys_tmp
HMXB["M2_dot"][i] = M2_dot_out
HMXB["A"][i] = A_out
HMXB["ecc"][i] = e_tmp
HMXB["theta"][i] = theta_out
HMXB["k_type"][i] = int(k_type)
return HMXB["M_NS"], HMXB["M_2"], HMXB["L_x"], HMXB["v_sys"], HMXB["M2_dot"], HMXB["A"], HMXB["ecc"], HMXB["theta"], HMXB["k_type"]
else:
# Star does not make it to MT phase
if t_obs < load_sse.func_sse_ms_time(M1): return M1, M2, 0.0, 0.0, 0.0, A, ecc, 0.0
# MT phase
M_1_b, M_2_b, A_b = binary_evolve.func_MT_forward(M1, M2, A, ecc)
# Star does not make it to SN
if t_obs < load_sse.func_sse_tmax(M1): return M_1_b, M_2_b, 0.0, 0.0, 0.0, A_b, ecc, 0.0
# SN
A_tmp, v_sys_tmp, e_tmp = binary_evolve.func_SN_forward(M_1_b, M_2_b, A_b, v_k, theta, phi)
# XRB
M_2_tmp, L_x_tmp, M2_dot_out, A_out = binary_evolve.func_Lx_forward(M1, M2, M_2_b, A_tmp, e_tmp, t_obs)
theta_out = (t_obs - load_sse.func_sse_tmax(M1)) * v_sys_tmp / c.dist_SMC * c.yr_to_sec * 1.0e6 * np.sin(get_theta(1))
# To get k-type of HMXB donor
tobs_eff = binary_evolve.func_get_time(M1, M2, t_obs)
M_tmp, M_dot_tmp, R_tmp, k_type = load_sse.func_get_sse_star(M_2_b, tobs_eff)
return c.M_NS, M_2_tmp, L_x_tmp, v_sys_tmp, M2_dot_out, A_out, e_tmp, theta_out, int(k_type)
def get_initial_values(M2_d, nwalkers=32):
""" Calculate an array of initial masses and birth times
Parameters
----------
M2_d : float
Observed secondary mass
Returns
-------
pos : ndarray, shape=(nwalkers,3)
Array of (M1, M2, t_b)
"""
# Start by using MCMC on just the masses to get a distribution of M1 and M2
args = [[M2_d]]
sampler = emcee.EnsembleSampler(nwalkers=nwalkers, dim=3, lnpostfn=ln_posterior_initial, args=args)
# Picking the initial masses and birth time will need to be optimized
t_b = 1000.0
M1_tmp = max(0.6*M2_d, c.min_mass)
M2_tmp = 1.1*M2_d - M1_tmp
p_i = [M1_tmp, M2_tmp,t_b]
tmp = binary_evolve.func_get_time(*p_i) - 1000.0
t_b = 0.9 * (load_sse.func_sse_tmax(p_i[0] + p_i[1] - load_sse.func_sse_he_mass(p_i[0])) - tmp)
p_i[2] = t_b
t_eff_obs = binary_evolve.func_get_time(*p_i)
M_b_prime = p_i[0] + p_i[1] - load_sse.func_sse_he_mass(p_i[0])
M_tmp, Mdot_tmp, R_tmp, k_tmp = load_sse.func_get_sse_star(M_b_prime, t_eff_obs)
min_M = load_sse.func_sse_min_mass(t_b)
n_tries = 0
while t_eff_obs < 0.0 or Mdot_tmp == 0.0:
p_i[0] = (c.max_mass - min_M) * np.random.uniform() + min_M
p_i[1] = (0.7 * np.random.uniform() + 0.3) * p_i[0]
p_i[2] = (np.random.uniform(5.0) + 1.2) * load_sse.func_sse_tmax(M2_d*0.6)
t_eff_obs = binary_evolve.func_get_time(*p_i)
if t_eff_obs < 0.0: continue
M_b_prime = p_i[0] + p_i[1] - load_sse.func_sse_he_mass(p_i[0])
if M_b_prime > c.max_mass: continue
M_tmp, Mdot_tmp, R_tmp, k_tmp = load_sse.func_get_sse_star(M_b_prime, t_eff_obs)
# Exit condition
n_tries += 1
if n_tries > 100: break
# initial positions for walkers
p0 = np.zeros((nwalkers,3))
a, b = (min_M - p_i[0]) / 0.5, (c.max_mass - p_i[0]) / 0.5
p0[:,0] = truncnorm.rvs(a, b, loc=p_i[0], scale=1.0, size=nwalkers) # M1
p0[:,1] = np.random.normal(p_i[1], 0.5, size=nwalkers) # M2
p0[:,2] = np.random.normal(p_i[2], 0.2, size=nwalkers) # t_b
# burn-in
pos,prob,state = sampler.run_mcmc(p0, N=100)
return pos
def ln_priors(y):
""" Priors on the model parameters
Parameters
----------
y : ra, dec, M1, M2, A, ecc, v_k, theta, phi, ra_b, dec_b, t_b
Current HMXB location (ra, dec) and 10 model parameters
Returns
-------
lp : float
Natural log of the prior
"""
# M1, M2, A, v_k, theta, phi, ra_b, dec_b, t_b = y
ra, dec, M1, M2, A, ecc, v_k, theta, phi, ra_b, dec_b, t_b = y
lp = 0.0
# P(M1)
if M1 < c.min_mass or M1 > c.max_mass: return -np.inf
norm_const = (c.alpha+1.0) / (np.power(c.max_mass, c.alpha+1.0) - np.power(c.min_mass, c.alpha+1.0))
lp += np.log( norm_const * np.power(M1, c.alpha) )
# M1 must be massive enough to evolve off the MS by t_obs
if load_sse.func_sse_tmax(M1) > t_b: return -np.inf
# P(M2)
# Normalization is over full q in (0,1.0)
q = M2 / M1
if q < 0.3 or q > 1.0: return -np.inf
lp += np.log( (1.0 / M1 ) )
# P(ecc)
if ecc < 0.0 or ecc > 1.0: return -np.inf
lp += np.log(2.0 * ecc)
# P(A)
if A*(1.0-ecc) < c.min_A or A*(1.0+ecc) > c.max_A: return -np.inf
norm_const = np.log(c.max_A) - np.log(c.min_A)
lp += np.log( norm_const / A )
# A must avoid RLOF at ZAMS, by a factor of 2
r_1_roche = binary_evolve.func_Roche_radius(M1, M2, A*(1.0-ecc))
if 2.0 * load_sse.func_sse_r_ZAMS(M1) > r_1_roche: return -np.inf
# P(v_k)
if v_k < 0.0: return -np.inf
lp += np.log( maxwell.pdf(v_k, scale=c.v_k_sigma) )
# P(theta)
if theta <= 0.0 or theta >= np.pi: return -np.inf
lp += np.log(np.sin(theta) / 2.0)
# P(phi)
if phi < 0.0 or phi > np.pi: return -np.inf
lp += -np.log( np.pi )
# Get star formation history
sf_history.load_sf_history()
sfh = sf_history.get_SFH(ra_b, dec_b, t_b, sf_history.smc_coor, sf_history.smc_sfh)
if sfh <= 0.0: return -np.inf
# P(alpha, delta)
# From spherical geometric effect, we need to care about cos(declination)
lp += np.log(np.cos(c.deg_to_rad * dec_b) / 2.0)
##################################################################
# We add an additional prior that scales the RA and Dec by the
# area available to it, i.e. pi theta^2, where theta is the angle
# of the maximum projected separation over the distance.
#
# Still under construction
##################################################################
M1_b, M2_b, A_b = binary_evolve.func_MT_forward(M1, M2, A, ecc)
A_c, v_sys, ecc = binary_evolve.func_SN_forward(M1_b, M2_b, A_b, v_k, theta, phi)
if ecc < 0.0 or ecc > 1.0 or np.isnan(ecc): return -np.inf
# Ensure that we only get non-compact object companions
tobs_eff = binary_evolve.func_get_time(M1, M2, t_b)
M_tmp, M_dot_tmp, R_tmp, k_type = load_sse.func_get_sse_star(M2_b, tobs_eff)
if int(k_type) > 9: return -np.inf
# t_sn = (t_b - func_sse_tmax(M1)) * 1.0e6 * yr_to_sec # The time since the primary's core collapse
# theta_max = (v_sys * t_sn) / dist_LMC # Unitless
# area = np.pi * rad_to_dec(theta_max)**2
# lp += np.log(1.0 / area)
##################################################################
# Instead, let's estimate the number of stars formed within a cone
# around the observed position, over solid angle and time.
# Prior is in Msun/Myr/steradian
##################################################################
t_min = load_sse.func_sse_tmax(M1) * 1.0e6 * c.yr_to_sec
t_max = (load_sse.func_sse_tmax(M2_b) - binary_evolve.func_get_time(M1, M2, 0.0)) * 1.0e6 * c.yr_to_sec
if t_max-t_min < 0.0: return -np.inf
theta_C = (v_sys * (t_max - t_min)) / c.dist_SMC
stars_formed = get_stars_formed(ra, dec, t_min, t_max, v_sys, c.dist_SMC)
if stars_formed == 0.0: return -np.inf
volume_cone = (np.pi/3.0 * theta_C**2 * (t_max - t_min) / c.yr_to_sec / 1.0e6)
lp += np.log(sfh / stars_formed / volume_cone)
##################################################################
# # P(t_b | alpha, delta)
# sfh_normalization = 1.0e-6
# lp += np.log(sfh_normalization * sfh)
# Add a prior so that the post-MT secondary is within the correct bounds
M2_c = M1 + M2 - load_sse.func_sse_he_mass(M1)
if M2_c > c.max_mass or M2_c < c.min_mass: return -np.inf
# Add a prior so the effective time remains bounded
t_eff_obs = binary_evolve.func_get_time(M1, M2, t_b)
if t_eff_obs < 0.0: return -np.inf
if t_b * 1.0e6 * c.yr_to_sec < t_min: return -np.inf
if t_b * 1.0e6 * c.yr_to_sec > t_max: return -np.inf
return lp
def ln_priors_population(y):
""" Priors on the model parameters
Parameters
----------
y : M1, M2, A, ecc, v_k, theta, phi, ra_b, dec_b, t_b
10 model parameters
Returns
-------
lp : float
Natural log of the prior
"""
M1, M2, A, ecc, v_k, theta, phi, ra_b, dec_b, t_b = y
lp = 0.0
# P(M1)
if M1 < c.min_mass or M1 > c.max_mass: return -np.inf
norm_const = (c.alpha+1.0) / (np.power(c.max_mass, c.alpha+1.0) - np.power(c.min_mass, c.alpha+1.0))
lp += np.log( norm_const * np.power(M1, c.alpha) )
# M1 must be massive enough to evolve off the MS by t_obs
if load_sse.func_sse_tmax(M1) > t_b: return -np.inf
# P(M2)
# Normalization is over full q in (0,1.0)
q = M2 / M1
if q < 0.3 or q > 1.0: return -np.inf
lp += np.log( (1.0 / M1 ) )
# P(ecc)
if ecc < 0.0 or ecc > 1.0: return -np.inf
lp += np.log(2.0 * ecc)
# P(A)
if A*(1.0-ecc) < c.min_A or A*(1.0+ecc) > c.max_A: return -np.inf
norm_const = np.log(c.max_A) - np.log(c.min_A)
lp += np.log( norm_const / A )
# A must avoid RLOF at ZAMS, by a factor of 2
r_1_roche = binary_evolve.func_Roche_radius(M1, M2, A*(1.0-ecc))
if 2.0 * load_sse.func_sse_r_ZAMS(M1) > r_1_roche: return -np.inf
# P(v_k)
if v_k < 0.0: return -np.inf
lp += np.log( maxwell.pdf(v_k, scale=c.v_k_sigma) )
# P(theta)
if theta <= 0.0 or theta >= np.pi: return -np.inf
lp += np.log(np.sin(theta) / 2.0)
# P(phi)
if phi < 0.0 or phi > np.pi: return -np.inf
lp += -np.log( np.pi )
# Get star formation history
sfh = sf_history.get_SFH(ra_b, dec_b, t_b, sf_history.smc_coor, sf_history.smc_sfh)
if sfh <= 0.0: return -np.inf
lp += np.log(sfh)
# P(alpha, delta)
# From spherical geometric effect, scale by cos(declination)
lp += np.log(np.cos(c.deg_to_rad*dec_b) / 2.0)
M1_b, M2_b, A_b = binary_evolve.func_MT_forward(M1, M2, A, ecc)
A_c, v_sys, ecc = binary_evolve.func_SN_forward(M1_b, M2_b, A_b, v_k, theta, phi)
if ecc < 0.0 or ecc > 1.0 or np.isnan(ecc): return -np.inf
# Ensure that we only get non-compact object companions
tobs_eff = binary_evolve.func_get_time(M1, M2, t_b)
M_tmp, M_dot_tmp, R_tmp, k_type = load_sse.func_get_sse_star(M2_b, tobs_eff)
if int(k_type) > 9: return -np.inf
# Add a prior so that the post-MT secondary is within the correct bounds
M2_c = M1 + M2 - load_sse.func_sse_he_mass(M1)
if M2_c > c.max_mass or M2_c < c.min_mass: return -np.inf
# Add a prior so the effective time remains bounded
t_eff_obs = binary_evolve.func_get_time(M1, M2, t_b)
if t_eff_obs < 0.0: return -np.inf
t_max = (load_sse.func_sse_tmax(M2_b) - binary_evolve.func_get_time(M1, M2, 0.0))
if t_b > t_max: return -np.inf
return lp
def func_Lx_forward(M_1_a, M_2_a, M_2_in, A_in, ecc_in, t_obs):
""" Calculate the X-ray luminosity from accretion for a binary
Parameters
----------
M_1_a : float
ZAMS mass of the primary, now a NS (Msun)
M_2_a : float
ZAMS mass of the secondary (Msun)
M_2_in : float
Post-mass transfer mass of the secondary (Msun)
A_in : float
Post-SN orbital separation (Rsun)
ecc_in : float
Post-SN eccentricity (unitless)
t_obs : float
Time which the binary is being observed
Returns
-------
M_2_out : float
Current mass of the secondary
L_x : float
X_ray luminosity
M_dot_out : float
Mass accretion rate
A_out : float
Current orbital separation
"""
t_eff_obs = func_get_time(M_1_a, M_2_a, t_obs)
if isinstance(t_eff_obs, np.ndarray):
M_2_out = np.array([])
M_dot_wind = np.array([])
R_out = np.array([])
k_out = np.array([])
for i in np.arange(len(t_eff_obs)):
if (t_eff_obs[i] < 0.0 or ecc_in[i] < 0.0 or ecc_in[i] >= 1.0):
ecc_in[i] = 0.0
if isinstance(M_2_in, np.ndarray):
M_2_out = np.append(M_2_out, M_2_in[i])
else:
M_2_out = np.append(M_2_out, M_2_in)
M_dot_wind = np.append(M_dot_wind, 0.0)
R_out = np.append(R_out, 0.0)
else:
if isinstance(M_2_in, np.ndarray):
if M_2_in[i] > c.max_mass:
aa, bb, cc = 0.0, 0.0, 0.0
else:
aa, bb, cc, dd = load_sse.func_get_sse_star(M_2_in[i], t_eff_obs[i])
else:
if M_2_in > c.max_mass:
aa, bb, cc = 0.0, 0.0, 0.0
else:
aa, bb, cc, dd = load_sse.func_get_sse_star(M_2_in, t_eff_obs[i])
M_2_out = np.append(M_2_out, aa)
M_dot_wind = np.append(M_dot_wind, bb)
R_out = np.append(R_out, cc)
k_out = np.append(k_out, dd)
else:
if (t_eff_obs < 0.0 or M_2_in > c.max_mass or ecc_in < 0.0 or ecc_in > 1.0 or t_eff_obs > load_sse.func_sse_tmax(M_2_in)):
M_2_out = M_2_in
M_dot_wind = 0.0
R_out = 0.0
ecc_in = 0.0
else:
M_2_out, M_dot_wind, R_out, k_out = load_sse.func_get_sse_star(M_2_in, t_eff_obs)
# Get wind velocity
v_wind = get_v_wind(M_2_out, R_out)
if isinstance(v_wind, np.ndarray):
v_wind[np.where(v_wind <= 0.0)] = 1.0e50 # To eliminate "bad" winds
else:
if v_wind <= 0.0: v_wind = 1.0e50
# Get final orbital separation
if isinstance(A_in, np.ndarray):
A_in[np.where(A_in <= 0.0)] = 1.0e50 # To eliminate "bad" separations
else:
if A_in <= 0.0: A_in = 1.0e50
A_out = (c.M_NS + M_2_in) / (c.M_NS + M_2_out) * A_in
# Capture fraction takes into account eccentricity
f_capture = (c.GGG*c.M_NS / (v_wind*v_wind*A_out))**2 / np.sqrt(1.0 - ecc_in**2)
M_dot_out = f_capture * M_dot_wind
L_bol = c.GGG * c.M_NS * M_dot_out / c.R_NS * c.km_to_cm * c.Msun_to_g * c.Rsun_to_cm / c.yr_to_sec
L_x = c.eta_bol * L_bol
return M_2_out, L_x, M_dot_out, A_out