SmoothlyBrokenPowerLaw1D¶
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class
astropy.modeling.powerlaws.
SmoothlyBrokenPowerLaw1D
[source] [edit on github]¶ Bases:
astropy.modeling.Fittable1DModel
One dimensional smoothly broken power law model.
Parameters: amplitude : float
Model amplitude at the break point.
x_break : float
Break point.
alpha_1 : float
Power law index for
x << x_break
.alpha_2 : float
Power law index for
x >> x_break
.delta : float
Smoothness parameter.
See also
Notes
Model formula (with \(A\) for
amplitude
, \(x_b\) forx_break
, \(\alpha_1\) foralpha_1
, \(\alpha_2\) foralpha_2
and \(\Delta\) fordelta
):\[f(x) = A \left( \frac{x}{x_b} \right) ^ {-\alpha_1} \left\{ \frac{1}{2} \left[ 1 + \left( \frac{x}{x_b}\right)^{1 / \Delta} \right] \right\}^{(\alpha_1 - \alpha_2) \Delta}\]The change of slope occurs between the values \(x_1\) and \(x_2\) such that:
\[\log_{10} \frac{x_2}{x_b} = \log_{10} \frac{x_b}{x_1} \sim \Delta\]At values \(x \lesssim x_1\) and \(x \gtrsim x_2\) the model is approximately a simple power law with index \(\alpha_1\) and \(\alpha_2\) respectively. The two power laws are smoothly joined at values \(x_1 < x < x_2\), hence the \(\Delta\) parameter sets the “smoothness” of the slope change.
The
delta
parameter is bounded to values greater than 1e-3 (corresponding to \(x_2 / x_1 \gtrsim 1.002\)) to avoid overflow errors.The
amplitude
parameter is bounded to positive values since this model is typically used to represent positive quantities.Examples
import numpy as np import matplotlib.pyplot as plt from astropy.modeling import models x = np.logspace(0.7, 2.3, 500) f = models.SmoothlyBrokenPowerLaw1D(amplitude=1, x_break=20, alpha_1=-2, alpha_2=2) plt.figure() plt.title("amplitude=1, x_break=20, alpha_1=-2, alpha_2=2") f.delta = 0.5 plt.loglog(x, f(x), '--', label='delta=0.5') f.delta = 0.3 plt.loglog(x, f(x), '-.', label='delta=0.3') f.delta = 0.1 plt.loglog(x, f(x), label='delta=0.1') plt.axis([x.min(), x.max(), 0.1, 1.1]) plt.legend(loc='lower center') plt.grid(True) plt.show()
Attributes Summary
alpha_1
alpha_2
amplitude
delta
input_units
param_names
x_break
Methods Summary
evaluate
(x, amplitude, x_break, alpha_1, …)One dimensional smoothly broken power law model function fit_deriv
(x, amplitude, x_break, alpha_1, …)One dimensional smoothly broken power law derivative with respect to parameters Attributes Documentation
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alpha_1
¶
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alpha_2
¶
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amplitude
¶
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delta
¶
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input_units
¶
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param_names
= ('amplitude', 'x_break', 'alpha_1', 'alpha_2', 'delta')¶
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x_break
¶
Methods Documentation
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static
evaluate
(x, amplitude, x_break, alpha_1, alpha_2, delta)[source] [edit on github]¶ One dimensional smoothly broken power law model function
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static
fit_deriv
(x, amplitude, x_break, alpha_1, alpha_2, delta)[source] [edit on github]¶ One dimensional smoothly broken power law derivative with respect to parameters
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