scipy.integrateZach Pace, Lia Corrales, Stephanie T. Douglas
astropy and scientific python contextastropy's built-in black-body curvesastropy's units interact with one another__call__ method worksmatplotlib figures using the latex_inline formattermodeling, units, synphot, OOP, LaTeX, astrostatistics, matplotlib, units, physics
In this tutorial, we will use the examples of the Planck function and the stellar initial mass function (IMF) to illustrate how to integrate numerically, using the trapezoidal approximation and Gaussian quadrature. We will also explore making a custom class, an instance of which is callable in the same way as a function. In addition, we will encounter astropy's built-in units, and get a first taste of how to convert between them. Finally, we will use $\LaTeX$ to make our figure axis labels easy to read.
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import numpy as np
from scipy import integrate
from astropy.modeling.blackbody import blackbody_lambda, blackbody_nu, BlackBody1D
from astropy import units as u, constants as c
import matplotlib.pyplot as plt
%matplotlib inline
Let's say we have a black-body at 5000 Kelvin. We can find out the total intensity (bolometric) from this object, by integrating the Planck function. The simplest way to do this is by approximating the integral using the trapezoid rule. Let's do this first using the frequency definition of the Planck function.
We'll define a photon frequency grid, and evaluate the Planck function at those frequencies. Those will be used to numerically integrate using the trapezoidal rule. By multiplying a numpy array by an astropy unit, we get a Quantity, which is effectively a combination of one or more numbers and a unit.
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nu = np.linspace(1., 3000., 1000) * u.THz
bb5000K_nu = blackbody_nu(in_x=nu, temperature=5000. * u.Kelvin)
plt.plot(nu, bb5000K_nu)
plt.xlabel(r'$\nu$, [{0:latex_inline}]'.format(nu.unit))
plt.ylabel(r'$I_{\nu}$, ' + '[{0:latex_inline}]'.format(bb5000K_nu.unit))
plt.title('Planck function in frequency')
plt.show()
Here, we've used $LaTeX$ markup to add nice-looking axis labels. To do that, we enclose $LaTeX$ markup text in dollar signs, within a string r'\$ ... \$'. The r before the open-quote denotes that the string is "raw," and backslashes are treated literally. This is the suggested format for axis label text that includes markup.
Now we numerically integrate using the trapezoid rule.
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np.trapz(x=nu, y=bb5000K_nu).to('erg s-1 cm-2 sr-1')
Now we can do something similar, but for a wavelength grid. We want to integrate over an equivalent wavelength range to the frequency range we did earlier. We can transform the maximum frequency into the corresponding (minimum) wavelength by using the .to() method, with the addition of an equivalency.
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lam = np.linspace(nu.max().to(u.AA, equivalencies=u.spectral()),
nu.min().to(u.AA, equivalencies=u.spectral()), 1000)
bb5000K_lam = blackbody_lambda(in_x=lam, temperature=5000. * u.Kelvin)
plt.plot(lam, bb5000K_lam)
plt.xlim([1.0e3, 5.0e4])
plt.xlabel(r'$\lambda$, [{0:latex_inline}]'.format(lam.unit))
plt.ylabel(r'$I_{\lambda}$, ' + '[{0:latex_inline}]'.format(bb5000K_lam.unit))
plt.title('Planck function in wavelength')
plt.show()
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np.trapz(x=lam, y=bb5000K_lam).to('erg s-1 cm-2 sr-1')
Notice this is within a couple percent of the answer we got in frequency space, despite our bad sampling at small wavelengths!
Many astropy functions use units and quantities directly. As you gain confidence working with them, consider incorporating them into your regular workflow. Read more here about how to use units.
As of Fall 2017, astropy does not explicitly support constructing synthetic observations of models like black-body curves. The synphot library does allow this. You can use synphot to perform tasks like turning spectra into visual magnitudes by convolving with a filter curve.
The stellar initial mass function tells us how many of each mass of stars are formed. In particular, low-mass stars are much more abundant than high-mass stars are. Let's explore more of the functionality of astropy using this concept.
People generally think of the IMF as a power-law probability density function. In other words, if you count the stars that have been born recently from a cloud of gas, their distribution of masses will follow the IMF. Let's write a little class to help us keep track of that:
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class PowerLawPDF(object):
def __init__(self, gamma, B=1.):
self.gamma = gamma
self.B = B
def __call__(self, x):
return x**self.gamma / self.B
__call__ methodBy defining the method __call__, we are telling the Python interpreter that an instance of the class can be called like a function. When called, an instance of this class, takes a single argument, x, but it uses other attributes of the instance, like gamma and B.
Classes are more advanced data structures, which can help you keep track of functionality within your code that all works together. You can learn more about classes in this tutorial.
In this section, we'll explore a method of numerical integration that does not require having your sampling grid set-up already. scipy.integrate.quad with reference here takes a function and both a lower and upper bound, and our PowerLawPDF class takes care of this just fine.
Now we can use our new class to normalize our IMF given the mass bounds. This amounts to normalizing a probability density function. We'll use Gaussian quadrature (quad) to find the integral. quad returns the numerical value of the integral and its uncertainty. We only care about the numerical value, so we'll pack the uncertainty into _ (a placeholder variable). We immediately throw the integral into our IMF object and use it for normalizing!
To read more about generalized packing and unpacking in Python, look at the original proposal, PEP 448, which was accepted in 2015.
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salpeter = PowerLawPDF(gamma=-2.35)
salpeter.B, _ = integrate.quad(salpeter, a=0.01, b=100.)
m_grid = np.logspace(-2., 2., 100)
plt.loglog(m_grid, salpeter(m_grid))
plt.xlabel(r'Stellar mass [$M_{\odot}$]')
plt.ylabel('Probability density')
plt.show()
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n_m, _ = integrate.quad(salpeter, a=.01, b=.6)
n_o, _ = integrate.quad(salpeter, a=15., b=100.)
print(n_m / n_o)
There are almost 21000 as many low-mass stars born as there are high-mass stars!
Now let's compute the relative total masses for all O stars and all M stars born. To do this, weight the Salpeter IMF by mass (i.e., add an extra factor of mass to the integral). To do this, we define a new function that takes the old power-law IMF as one of its arguments. Since this argument is unchanged throughout the integral, it is passed into the tuple args within quad. It's important that there is only one argument that changes over the integral, and that it is the first argument that the function being integrated accepts.
Mathematically, the integral for the M stars is
$$ m^M = \int_{.01 \, M_{\odot}}^{.6 \, M_{\odot}} m \, {\rm IMF}(m) \, dm $$and it amounts to weighting the probability density function (the IMF) by mass. More generally, you find the value of some property $\rho$ that depends on $m$ by calculating
$$ \rho(m)^M = \int_{.01 \, M_{\odot}}^{.6 \, M_{\odot}} \rho(m) \, {\rm IMF}(m) \, dm $$
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def IMF_m(m, imf):
return imf(m) * m
m_m, _ = integrate.quad(IMF_m, a=.01, b=.6, args=(salpeter, ))
m_o, _ = integrate.quad(IMF_m, a=15., b=100., args=(salpeter, ))
m_m / m_o
So about 20 times as much mass is tied up in M stars as in O stars.
PowerLawPDF in a way that allows both float and numpy.ndarray inputs.PowerLawPDF class to explicitly use astropy's units constructs.
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