SmoothlyBrokenPowerLaw1D¶

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

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)

../_images/astropy-modeling-powerlaws-SmoothlyBrokenPowerLaw1D-1.png

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)[source] [edit on github]¶: One dimensional smoothly broken power law model function

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

Navigation

SmoothlyBrokenPowerLaw1D¶