def test_idl_tabulate_Err(self):
# Force an error by sending in x = [0]
self.assertEqual(utils.idl_tabulate(np.array([0]), f), 0)
def initial_sample(self,
M1min=0.08,
M2min=0.08,
M1max=150.0,
M2max=150.0,
porb_lo=0.15,
porb_hi=8.0,
rand_seed=0,
size=None,
nproc=1):
"""Sample initial binary distribution according to Moe & Di Stefano (2017)
<http://adsabs.harvard.edu/abs/2017ApJS..230...15M>`_
Parameters
----------
M1min : `float`
minimum primary mass to sample [Msun]
DEFAULT: 0.08
M2min : `float`
minimum secondary mass to sample [Msun]
DEFAULT: 0.08
M1max : `float`
maximum primary mass to sample [Msun]
DEFAULT: 150.0
M2max : `float`
maximum primary mass to sample [Msun]
DEFAULT: 150.0
porb_lo : `float`
minimum orbital period to sample [log10(days)]
porb_hi : `float`
maximum orbital period to sample [log10(days)]
rand_seed : int
random seed generator
DEFAULT: 0
size : int, optional
number of evolution times to sample
NOTE: this is set in cosmic-pop call as Nstep
Returns
-------
primary_mass_list : array
array of primary masses with size=size
secondary_mass_list : array
array of secondary masses with size=size
porb_list : array
array of orbital periods in days with size=size
ecc_list : array
array of eccentricities with size=size
mass_singles : `float`
Total mass in single stars needed to generate population
mass_binaries : `float`
Total mass in binaries needed to generate population
n_singles : `int`
Number of single stars needed to generate a population
n_binaries : `int`
Number of binaries needed to generate a population
binfrac_list : array
array of binary probabilities based on primary mass and period with size=size
"""
#Tabulate probably density functions of periods,
#mass ratios, and eccentricities based on
#analytic fits to corrected binary star populations.
numM1 = 101
#use binwidths to maintain structure of original array
# default size is: numlogP=158
bwlogP = 0.05
numq = 91
nume = 100
#; Vector of primary masses M1 (Msun), logarithmic orbital period P (days),
#; mass ratios q = Mcomp/M1, and eccentricities e
#
#; 0.8 < M1 < 40 (where we have statistics corrected for selection effects)
M1_lo = 0.8
M1_hi = 40
M1v = np.logspace(np.log10(M1_lo), np.log10(M1_hi), numM1)
#; 0.15 < log P < 8.0
#; or use user specified values
log10_porb_lo = porb_lo
log10_porb_hi = porb_hi
logPv = np.arange(log10_porb_lo, log10_porb_hi + bwlogP, bwlogP)
numlogP = len(logPv)
#; 0.10 < q < 1.00
q_lo = 0.1
q_hi = 1.0
qv = np.linspace(q_lo, q_hi, numq)
#; 0.0001 < e < 0.9901
#; set minimum to non-zero value to avoid numerical errors
e_lo = 0.0
e_hi = 0.99
ev = np.linspace(e_lo, e_hi, nume) + 0.0001
#; Note that companions outside this parameter space (e.g., q < 0.1,
#; log P (days) > 8.0 are not constrained in M+D16 and therefore
#; not considered.
#; Distribution functions - define here, but evaluate within for loops.
#; Frequency of companions with q > 0.1 per decade of orbital period.
#; Bottom panel in Fig. 37 of M+D17
flogP_sq = np.zeros([numlogP, numM1])
#; Given M1 and P, the cumulative distribution of mass ratios q
cumqdist = np.zeros([numq, numlogP, numM1])
#; Given M1 and P, the cumulative distribution of eccentricities e
cumedist = np.zeros([nume, numlogP, numM1])
#; Given M1 and P, the probability that the companion
#; is a member of the inner binary (currently an approximation).
#; 100% for log P < 1.5, decreases with increasing P
probbin = np.zeros([numlogP, numM1])
#; Given M1, the cumulative period distribution of the inner binary
#; Normalized so that max(cumPbindist) = total binary frac. (NOT unity)
cumPbindist = np.zeros([numlogP, numM1])
#; Slope alpha of period distribution across intermediate periods
#; 2.7 - DlogP < log P < 2.7 + DlogP, see Section 9.3 and Eqn. 23.
#; Slightly updated from version 1.
alpha = 0.018
DlogP = 0.7
#; Heaviside function for twins with 0.95 < q < 1.00
H = np.zeros(numq)
ind = np.where(qv >= 0.95)
H[ind] = 1.0
H = H / idl_tabulate(qv, H) #;normalize so that integral is unity
#; Relevant indices with respect to mass ratio
indlq = np.where(qv >= 0.3)
indsq = np.where(qv < 0.3)
indq0p3 = np.min(indlq)
# FILL IN THE MULTIDIMENSIONAL DISTRIBUTION FUNCTIONS
#; Loop through primary mass
for i in range(0, numM1):
myM1 = M1v[i]
#; Twin fraction parameters that are dependent on M1 only; section 9.1
FtwinlogPle1 = 0.3 - 0.15 * np.log10(myM1) #; Eqn. 6
logPtwin = 8.0 - myM1 #; Eqn. 7a
if (myM1 >= 6.5):
logPtwin = 1.5 #; Eqn. 7b
#; Frequency of companions with q > 0.3 at different orbital periods
#; and dependent on M1 only; section 9.3 (slightly modified since v1)
flogPle1 = 0.020 + 0.04 * np.log10(myM1) + \
0.07 * (np.log10(myM1))**2. #; Eqn. 20
flogPeq2p7 = 0.039 + 0.07 * np.log10(myM1) + \
0.01 * (np.log10(myM1))**2. #; Eqn. 21
flogPeq5p5 = 0.078 - 0.05 * np.log10(myM1) + \
0.04 * (np.log10(myM1))**2. #; Eqn. 22
#; Loop through orbital period P
for j in range(0, numlogP):
mylogP = logPv[j]
#; Given M1 and P, set excess twin fraction; section 9.1 and Eqn. 5
if (mylogP <= 1.0):
Ftwin = FtwinlogPle1
else:
Ftwin = FtwinlogPle1 * (1.0 - (mylogP - 1.0) /
(logPtwin - 1.0))
if (mylogP >= logPtwin):
Ftwin = 0.0
#; Power-law slope gamma_largeq for M1 < 1.2 Msun and various P; Eqn. 9
if (mylogP <= 5.0):
gl_1p2 = -0.5
if (mylogP > 5.0):
gl_1p2 = -0.5 - 0.3 * (mylogP - 5.0)
#; Power-law slope gamma_largeq for M1 = 3.5 Msun and various P; Eqn. 10
if (mylogP <= 1.0):
gl_3p5 = -0.5
if ((mylogP > 1.0) and (mylogP <= 4.5)):
gl_3p5 = -0.5 - 0.2 * (mylogP - 1.0)
if ((mylogP > 4.5) and (mylogP <= 6.5)):
gl_3p5 = -1.2 - 0.4 * (mylogP - 4.5)
if (mylogP > 6.5):
gl_3p5 = -2.0
#; Power-law slope gamma_largeq for M1 > 6 Msun and various P; Eqn. 11
if (mylogP <= 1.0):
gl_6 = -0.5
if ((mylogP > 1.0) and (mylogP <= 2.0)):
gl_6 = -0.5 - 0.9 * (mylogP - 1.0)
if ((mylogP > 2.0) and (mylogP <= 4.0)):
gl_6 = -1.4 - 0.3 * (mylogP - 2.0)
if (mylogP > 4.0):
gl_6 = -2.0
#; Given P, interpolate gamma_largeq w/ respect to M1 at myM1
if (myM1 <= 1.2):
gl = gl_1p2
if ((myM1 > 1.2) and (myM1 <= 3.5)):
gl = np.interp(np.log10(myM1), np.log10([1.2, 3.5]),
[gl_1p2, gl_3p5])
if ((myM1 > 3.5) and (myM1 <= 6.0)):
gl = np.interp(np.log10(myM1), np.log10([3.5, 6.0]),
[gl_3p5, gl_6])
if (myM1 > 6.0):
gl = gl_6
#; Power-law slope gamma_smallq for M1 < 1.2 Msun and all P; Eqn. 13
gs_1p2 = 0.3
#; Power-law slope gamma_smallq for M1 = 3.5 Msun and various P; Eqn. 14
if (mylogP <= 2.5):
gs_3p5 = 0.2
if ((mylogP > 2.5) and (mylogP <= 5.5)):
gs_3p5 = 0.2 - 0.3 * (mylogP - 2.5)
if (mylogP > 5.5):
gs_3p5 = -0.7 - 0.2 * (mylogP - 5.5)
#; Power-law slope gamma_smallq for M1 > 6 Msun and various P; Eqn. 15
if (mylogP <= 1.0):
gs_6 = 0.1
if ((mylogP > 1.0) and (mylogP <= 3.0)):
gs_6 = 0.1 - 0.15 * (mylogP - 1.0)
if ((mylogP > 3.0) and (mylogP <= 5.6)):
gs_6 = -0.2 - 0.50 * (mylogP - 3.0)
if (mylogP > 5.6):
gs_6 = -1.5
#; Given P, interpolate gamma_smallq w/ respect to M1 at myM1
if (myM1 <= 1.2):
gs = gs_1p2
if ((myM1 > 1.2) and (myM1 <= 3.5)):
gs = np.interp(np.log10(myM1), np.log10([1.2, 3.5]),
[gs_1p2, gs_3p5])
if ((myM1 > 3.5) and (myM1 <= 6.0)):
gs = np.interp(np.log10(myM1), np.log10([3.5, 6.0]),
[gs_3p5, gs_6])
if (myM1 > 6.0):
gs = gs_6
#; Given Ftwin, gamma_smallq, and gamma_largeq at the specified M1 & P,
#; tabulate the cumulative mass ratio distribution across 0.1 < q < 1.0
fq = qv**gl #; slope across 0.3 < q < 1.0
fq = fq / idl_tabulate(
qv[indlq], fq[indlq]) #; normalize to 0.3 < q < 1.0
fq = fq * (1.0 - Ftwin) + H * Ftwin #; add twins
fq[indsq] = fq[indq0p3] * (
qv[indsq] / 0.3)**gs #; slope across 0.1 < q < 0.3
cumfq = np.cumsum(fq) - fq[0] #; cumulative distribution
cumfq = cumfq / np.max(cumfq) #; normalize cumfq(q=1.0) = 1
cumqdist[:, j, i] = cumfq #; save to grid
#; Given M1 and P, q_factor is the ratio of all binaries 0.1 < q < 1.0
#; to those with 0.3 < q < 1.0
q_factor = idl_tabulate(qv, fq)
#; Given M1 & P, calculate power-law slope eta of eccentricity dist.
if (mylogP >= 0.7):
#; For log P > 0.7 use fits in Section 9.2.
#; Power-law slope eta for M1 < 3 Msun and log P > 0.7
eta_3 = 0.6 - 0.7 / (mylogP - 0.5) #; Eqn. 17
#; Power-law slope eta for M1 > 7 Msun and log P > 0.7
eta_7 = 0.9 - 0.2 / (mylogP - 0.5) #; Eqn. 18
else:
#; For log P < 0.7, set eta to fitted values at log P = 0.7
eta_3 = -2.9
eta_7 = -0.1
#; Given P, interpolate eta with respect to M1 at myM1
if (myM1 <= 3.):
eta = eta_3
if ((myM1 > 3.) and (myM1 <= 7.)):
eta = np.interp(np.log10(myM1), np.log10([3., 7.]),
[eta_3, eta_7])
if (myM1 > 7.):
eta = eta_7
#; Given eta at the specified M1 and P, tabulate eccentricity distribution
if (10**mylogP <= 2.):
#; For P < 2 days, assume all systems are close to circular
#; For adopted ev (spacing and minimum value), eta = -3.2 satisfies this
fe = ev**(-3.2)
else:
fe = ev**eta
e_max = 1.0 - (10**mylogP / 2.0)**(
-2.0 / 3.0) #; maximum eccentricity for given P
ind = np.where(ev >= e_max)
fe[ind] = 0.0 #; set dist. = 0 for e > e_max
#; Assume e dist. has power-law slope eta for 0.0 < e / e_max < 0.8 and
#; then linear turnover between 0.8 < e / e_max < 1.0 so that dist.
#; is continuous at e / e_max = 0.8 and zero at e = e_max
ind = np.where((ev >= 0.8 * e_max) & (ev <= 1.0 * e_max))
ind_cont = np.min(ind) - 1
fe[ind] = np.interp(ev[ind], [0.8 * e_max, 1.0 * e_max],
[fe[ind_cont], 0.])
cumfe = np.cumsum(fe) - fe[0] #; cumulative distribution
cumfe = cumfe / np.max(cumfe) #; normalize cumfe(e=e_max) = 1
cumedist[:, j, i] = cumfe #; save to grid
#; Given constants alpha and DlogP and
#; M1 dependent values flogPle1, flogPeq2p7, and flogPeq5p5,
#; calculate frequency flogP of companions with q > 0.3 per decade
#; of orbital period at given P (Section 9.3 and Eqn. 23)
if (mylogP <= 1.):
flogP = flogPle1
if ((mylogP > 1.0) and (mylogP <= 2.7 - DlogP)):
flogP = flogPle1 + (mylogP - 1.0) / (1.7 - DlogP) * \
(flogPeq2p7 - flogPle1 - alpha*DlogP)
if ((mylogP > 2.7 - DlogP) and (mylogP <= 2.7 + DlogP)):
flogP = flogPeq2p7 + alpha * (mylogP - 2.7)
if ((mylogP > 2.7 + DlogP) and (mylogP <= 5.5)):
flogP = flogPeq2p7 + alpha*DlogP + \
(mylogP - 2.7 - DlogP)/(2.8 - DlogP) * \
(flogPeq5p5 - flogPeq2p7 - alpha*DlogP)
if (mylogP > 5.5):
flogP = flogPeq5p5 * np.exp(-0.3 * (mylogP - 5.5))
#; Convert frequency of companions with q > 0.3 to frequency of
#; companions with q > 0.1 according to q_factor; save to grid
flogP_sq[j, i] = flogP * q_factor
#; Calculate prob. that a companion to M1 with period P is the
#; inner binary. Currently this is an approximation.
#; 100% for log P < 1.5
#; For log P > 1.5 adopt functional form that reproduces M1 dependent
#; multiplicity statistics in Section 9.4, including a
#; 41% binary star faction (59% single star fraction) for M1 = 1 Msun and
#; 96% binary star fraction (4% single star fraction) for M1 = 28 Msun
if (mylogP <= 1.5):
probbin[j, i] = 1.0
else:
probbin[j,
i] = 1.0 - 0.11 * (mylogP -
1.5)**1.43 * (myM1 / 10.0)**0.56
if (probbin[j, i] <= 0.0):
probbin[j, i] = 0.0
#; Given M1, calculate cumulative binary period distribution
mycumPbindist = np.cumsum(flogP_sq[:,i] * probbin[:,i]) - \
flogP_sq[0,i] * probbin[0,i]
#; Normalize so that max(cumPbindist) = total binary star fraction (NOT 1)
mycumPbindist = mycumPbindist / np.max(mycumPbindist) * \
idl_tabulate(logPv, flogP_sq[:,i]*probbin[:,i])
cumPbindist[:, i] = mycumPbindist #;save to grid
#; Step 2
#; Implement Monte Carlo method / random number generator to select
#; single stars and binaries from the grids of distributions
#; Create vector for PRIMARY mass function, which is the mass distribution
#; of single stars and primaries in binaries.
#; This is NOT the IMF, which is the mass distribution of single stars,
#; primaries in binaries, and secondaries in binaries.
primary_mass_list = []
secondary_mass_list = []
porb_list = []
ecc_list = []
def _sample_initial_pop(M1min, M2min, M1max, M2max, size, nproc, seed,
output):
# get unique and replicatable seed for each process
process = mp.Process()
mp_seed = (process._identity[0] - 1) + (nproc *
(process._identity[1] - 1))
np.random.seed(seed + mp_seed)
mass_singles = 0.0
mass_binaries = 0.0
n_singles = 0
n_binaries = 0
primary_mass_list = []
secondary_mass_list = []
porb_list = []
ecc_list = []
binfrac_list = []
#; Full primary mass vector across 0.08 < M1 < 150
M1 = np.linspace(0, 150, 150000) + 0.08
#; Slope = -2.3 for M1 > 1 Msun
fM1 = M1**(-2.3)
#; Slope = -1.6 for M1 = 0.5 - 1.0 Msun
ind = np.where(M1 <= 1.0)
fM1[ind] = M1[ind]**(-1.6)
#; Slope = -0.8 for M1 = 0.15 - 0.5 Msun
ind = np.where(M1 <= 0.5)
fM1[ind] = M1[ind]**(-0.8) / 0.5**(1.6 - 0.8)
#; Cumulative primary mass distribution function
cumfM1 = np.cumsum(fM1) - fM1[0]
cumfM1 = cumfM1 / np.max(cumfM1)
#; Value of primary mass CDF where M1 = M1min
#; Minimum primary mass to generate (must be >0.080 Msun)
cumf_M1min = np.interp(0.08, M1, cumfM1)
while len(primary_mass_list) < size:
#; Select primary M1 > M1min from primary mass function
myM1 = np.interp(
cumf_M1min + (1.0 - cumf_M1min) * np.random.rand(), cumfM1,
M1)
# ; Find index of M1v that is closest to myM1.
# ; For M1 = 40 - 150 Msun, adopt binary statistics of M1 = 40 Msun.
# ; For M1 = 0.08 - 0.8 Msun, adopt P and e dist of M1 = 0.8Msun,
# ; scale and interpolate the companion frequencies so that the
# ; binary star fraction of M1 = 0.08 Msun primaries is zero,
# ; and truncate the q distribution so that q > q_min = 0.08/M1
indM1 = np.where(abs(myM1 - M1v) == min(abs(myM1 - M1v)))
indM1 = indM1[0]
# ; Given M1, determine cumulative binary period distribution
mycumPbindist_flat = (cumPbindist[:, indM1]).flatten()
#; If M1 < 0.8 Msun, rescale to appropriate binary star fraction
if (myM1 <= 0.8):
mycumPbindist_flat = mycumPbindist_flat * np.interp(
np.log10(myM1), np.log10([0.08, 0.8]), [0.0, 1.0])
# ; Given M1, determine the binary star fraction
mybinfrac = np.max(mycumPbindist_flat)
# ; Generate random number myrand between 0 and 1
myrand = np.random.rand()
#; If random number < binary star fraction, generate a binary
if (myrand < mybinfrac):
#; Given myrand, select P and corresponding index in logPv
mylogP = np.interp(myrand, mycumPbindist_flat, logPv)
indlogP = np.where(
abs(mylogP - logPv) == min(abs(mylogP - logPv)))
indlogP = indlogP[0]
#; Given M1 & P, select e from eccentricity distribution
mye = np.interp(np.random.rand(),
cumedist[:, indlogP, indM1].flatten(), ev)
#; Given M1 & P, determine mass ratio distribution.
#; If M1 < 0.8 Msun, truncate q distribution and consider
#; only mass ratios q > q_min = 0.08 / M1
mycumqdist = cumqdist[:, indlogP, indM1].flatten()
if (myM1 < 0.8):
q_min = 0.08 / myM1
#; Calculate cumulative probability at q = q_min
cum_qmin = np.interp(q_min, qv, mycumqdist)
#; Rescale and renormalize cumulative distribution for q > q_min
mycumqdist = mycumqdist - cum_qmin
mycumqdist = mycumqdist / max(mycumqdist)
#; Set probability = 0 where q < q_min
indq = np.where(qv <= q_min)
mycumqdist[indq] = 0.0
#; Given M1 & P, select q from cumulative mass ratio distribution
myq = np.interp(np.random.rand(), mycumqdist, qv)
if myM1 > M1min and myq * myM1 > M2min and myM1 < M1max and myq * myM1 < M2max and mylogP < porb_hi and mylogP > porb_lo:
primary_mass_list.append(myM1)
secondary_mass_list.append(myq * myM1)
porb_list.append(10**mylogP)
ecc_list.append(mye)
binfrac_list.append(mybinfrac)
mass_binaries += myM1
mass_binaries += myq * myM1
n_binaries += 1
else:
mass_singles += myM1
n_singles += 1
output.put([
primary_mass_list, secondary_mass_list, porb_list, ecc_list,
mass_singles, mass_binaries, n_singles, n_binaries,
binfrac_list
])
return
output = mp.Queue()
processes = [mp.Process(target = _sample_initial_pop,\
args = (M1min, M2min, M1max, M2max, size/nproc, nproc, rand_seed, output))\
for x in range(nproc)]
for p in processes:
p.daemon = True
p.start()
results = [output.get() for p in processes]
for p in processes:
p.join()
primary_mass_list = []
secondary_mass_list = []
porb_list = []
ecc_list = []
mass_singles = []
mass_binaries = []
n_singles = []
n_binaries = []
binfrac_list = []
dat_lists = [[], [], [], [], [], [], [], [], []]
for output_list in results:
ii = 0
for dat_list in output_list:
dat_lists[ii].append(dat_list)
ii += 1
primary_mass_list = np.hstack(dat_lists[0])
secondary_mass_list = np.hstack(dat_lists[1])
porb_list = np.hstack(dat_lists[2])
ecc_list = np.hstack(dat_lists[3])
mass_singles = np.sum(dat_lists[4])
mass_binaries = np.sum(dat_lists[5])
n_singles = np.sum(dat_lists[6])
n_binaries = np.sum(dat_lists[7])
binfrac_list = np.hstack(dat_lists[8])
return primary_mass_list, secondary_mass_list, porb_list, ecc_list, mass_singles, mass_binaries, n_singles, n_binaries, binfrac_list
def test_idl_tabulate(self):
# Give this custom integrator a simple integration
# of a line from x = 0 to 1 and y= 0 to 1
self.assertAlmostEqual(utils.idl_tabulate(x, f), IDL_TABULATE_ANSWER)