SmoothlyBrokenPowerLaw1D

class astropy.modeling.powerlaws.SmoothlyBrokenPowerLaw1D(amplitude=1, x_break=1, alpha_1=-2, alpha_2=2, delta=1, **kwargs)

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

BrokenPowerLaw1D

Notes

Model formula (with \(A\) for amplitude, \(x_b\) for x_break, \(\alpha_1\) for alpha_1, \(\alpha_2\) for alpha_2 and \(\Delta\) for delta):

\[f(x) = A \left( \frac{x}{x_b} \right) ^ {-\alpha_1} \left\{ \frac{1}{2} \left[ 1 + \left( \frac{x}{x_b}\right)^{1 / \Delta} \right] \right\}^{(\alpha_1 - \alpha_2) \Delta}\]

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()

(png, svg, pdf)

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

Attributes Documentation

alpha_1

alpha_2

amplitude

delta

input_units

param_names = ('amplitude', 'x_break', 'alpha_1', 'alpha_2', 'delta')

x_break

Methods Documentation

static evaluate(x, amplitude, x_break, alpha_1, alpha_2, delta)

One dimensional smoothly broken power law model function

static fit_deriv(x, amplitude, x_break, alpha_1, alpha_2, delta)

One dimensional smoothly broken power law derivative with respect to parameters