def time_dependent(t_rad, t_kin, nc_h):
# set the initial abundances for t = 0
n_levels = len(species_data.energy_levels.data)
#initial_fractional_abundances_wrathmall = numpy.ones(n_levels_wrathmall) \
# / n_levels_wrathmall
initial_fractional_abundances = numpy.random.rand(n_levels)
#initial_fractional_abundances[0] = 0.8
#initial_fractional_abundances[1] = 1.e-3
initial_fractional_abundances = \
initial_fractional_abundances/initial_fractional_abundances.sum()
y_0 = initial_fractional_abundances
solver = ode(ode_rhs, jac=jac).set_integrator(
'lsoda',
#method='bdf',
with_jacobian=True,
rtol=1e-14,
atol=1e-14)
solver.set_initial_value(y_0, t_0)
step_counter = 0
t_all = []
x_all = []
while solver.t < t_f:
solver.integrate(solver.t + dt)
t_all.append(solver.t)
x_all.append(solver.y)
rel_diff = None
if len(x_all) >= 2:
y_2 = x_all[-2]
y_1 = x_all[-1]
rel_diff = numpy.abs((y_2 - y_1) / y_1)
if (rel_diff < 1e-8).all():
break
if step_counter % 100 == 0:
print('{} t={:e} 1-x.sum()={:e}'.format(
step_counter, solver.t, 1.0 - solver.y.sum()))
if rel_diff is not None:
print(
numpy.where((rel_diff < 1e-6).all())[0].size,
rel_diff.mean())
step_counter += 1
t_all, x_all = numpy.array(t_all), numpy.vstack(x_all)
pop_dens_eq = population_density_at_steady_state(
species_data, t_kin, t_rad, nc_h)
return t_all, x_all, pop_dens_eq
def test_that_two_level_analytic_pop_dens_ratios_are_computed_correctly():
#
# load the two level species data
#
species_data = DataLoader().load('two_level_1')
n_c = 1e6 * u.meter ** -3
t_rad = 1000.0 * u.Kelvin
#
# for each value in the temperature range
# - compute the population density ration numerically
# - compute the population density analytically
#
# plot the comaprision
#
ratio_numeric_vs_t_kin = []
ratio_analytic_vs_t_kin = []
t_kin_range = numpy.logspace(1.0, 10.0, 50) * u.Kelvin
for t_kin in t_kin_range:
pop_dens_eq_numeric = population_density_at_steady_state(
species_data,
t_kin,
t_rad,
n_c)
pop_dens_eq_ratio_numeric = pop_dens_eq_numeric[1] / pop_dens_eq_numeric[0]
ratio_numeric_vs_t_kin.append(pop_dens_eq_ratio_numeric)
pop_dens_eq_ratio_analytic = analytic.population_density_ratio_two_level(
species_data.energy_levels.data['g'],
species_data.energy_levels.data['E'],
species_data.k_dex_matrix_interpolator(t_kin)[1, 0],
species_data.a_matrix[1, 0],
n_c,
t_kin,
t_rad
)
ratio_analytic_vs_t_kin.append(pop_dens_eq_ratio_analytic)
print(t_kin)
ratio_numeric_vs_t_kin = numpy.array(ratio_numeric_vs_t_kin).flatten()
ratio_analytic_vs_t_kin = numpy.array(ratio_analytic_vs_t_kin).flatten()
relative_error = numpy.abs(
1.0 - numpy.array(ratio_numeric_vs_t_kin) / numpy.array(
ratio_analytic_vs_t_kin)
)
assert relative_error.max() < 1.2e-15
assert relative_error.std() < 5.0e-16
for nc_index, nc_h in enumerate(nc_h_rng):
lambda_vs_t_kin = []
lambda_vs_t_kin_wrathmall = []
pop_dens_vs_t_kin = []
for i, t_kin in enumerate(t_rng):
print(t_kin, nc_h)
lambda_vs_t_kin += [
cooling_rate_at_steady_state(species_data, t_kin, t_rad, nc_h)
]
pop_dens_vs_t_kin += [
population_density_at_steady_state(species_data, t_kin, t_rad,
nc_h)
]
lambda_vs_t_kin_wrathmall += [
cooling_rate_at_steady_state(species_data_wrathmall, t_kin, t_rad,
nc_h)
]
print(t_kin, pop_dens_vs_t_kin[i][0], pop_dens_vs_t_kin[i][2],
pop_dens_vs_t_kin[i][4], pop_dens_vs_t_kin[i][6],
pop_dens_vs_t_kin[i][8], pop_dens_vs_t_kin[i][10],
pop_dens_vs_t_kin[i][12], pop_dens_vs_t_kin[i][14],
pop_dens_vs_t_kin[i][16], pop_dens_vs_t_kin[i][1],
pop_dens_vs_t_kin[i][3], pop_dens_vs_t_kin[i][5],
pop_dens_vs_t_kin[i][7], pop_dens_vs_t_kin[i][9],
pop_dens_vs_t_kin[i][11], pop_dens_vs_t_kin[i][13],
pop_dens_vs_t_kin[i][15], pop_dens_vs_t_kin[i][17])
#
# for each value in the temperature range
# - compute the population density ration numerically
# - compute the population density analytically
#
# plot the comaprision
#
ratio_numeric_vs_t_kin = []
ratio_analytic_vs_t_kin = []
t_kin_range = numpy.logspace(1.0, 10.0, 50) * u.Kelvin
for t_kin in t_kin_range:
# population densities using matrix inversion
pop_dens_eq_numeric = population_density_at_steady_state(
species_data,
t_kin,
t_rad,
n_c
)
pop_dens_eq_ratio_numeric = pop_dens_eq_numeric[1] / pop_dens_eq_numeric[0]
ratio_numeric_vs_t_kin.append(pop_dens_eq_ratio_numeric)
# popultation densities using analytic expressions
pop_dens_eq_ratio_analytic = analytic.population_density_ratio_two_level(
species_data.energy_levels.data['g'],
species_data.energy_levels.data['E'],
species_data.k_dex_matrix_interpolator(t_kin)[1, 0],
species_data.a_matrix[1, 0],
n_c,
t_kin,
t_rad
)
# the kinetic temperature of the gas
t_kin = 3000.0 * u.Kelvin
t_rad = 30.0 * u.Kelvin
species_data = DataLoader().load('two_level_1')
t_kin_range = logspace(2.0, 6.0, 100) * u.K
x_exact = array([
population_denisty_ratio_two_level_no_radiation(species_data, T_kin, nc)
for T_kin in t_kin_range
], 'f8')
x_numerical = array([
population_density_at_steady_state(species_data, T_kin, 0.0, nc)
for T_kin in t_kin_range
], 'f8')[:, :, 0]
# plot the relative population densities as a function of temperatures
pylab.figure()
pylab.loglog(t_kin_range, x_exact[:, 0], 'r-', label='x_0 exact')
pylab.loglog(t_kin_range, x_numerical[:, 0], 'ro', label='x_0 numerical')
pylab.loglog(t_kin_range, x_exact[:, 1], 'b-', label='x_1 exact')
pylab.loglog(t_kin_range, x_numerical[:, 1], 'bo', label='x_1 numerical')
pylab.legend()
pylab.title('fractional population densities of levels 0 and level 1')
pylab.show()
rel_errors = fabs(1.0 - x_numerical / x_exact)
