Polytropic gas model and non-thermal pressure fraction with varying parameters

import jax.numpy as jnp
import matplotlib.pyplot as plt

from picasso.utils.plots import plot_impact_model_params, plot_Gamma_r

import seaborn as sns
sns.set_style("darkgrid")
sns.set_theme("notebook")

Polytropic gas model and non-thermal pressure fraction with varying parameters#

The polytropic gas model can be written as (see Kéruzoré+24):

\[ \rho(\phi, \, r) = \rho_0 \theta^{\Gamma(r) / (\Gamma(r) - 1)}(\phi), \\[10pt] P(\phi, \, r) = P_0 \theta^{1 / (\Gamma(r) - 1)}(\phi), \]

where \(\phi\) is the halo’s gravitational potential, and $\( \theta(\phi) = 1 - \theta_0 (\phi - \phi_0), \)$

The gas polytropic index, \(\Gamma\), is allowed to vary with radius as:

\[ \Gamma(r) = \begin{cases} \begin{aligned} & \; 1 + (\Gamma_0 - 1) \frac{1}{1 + e^{-x}} & c_\Gamma > 0; \\ & \; \Gamma_0 & c_\Gamma = 0; \\ & \; \Gamma_0 + (\Gamma_0 - 1) \left(1 - \frac{1}{1 + e^{x}}\right) & c_\Gamma < 0, \\ \end{aligned} \end{cases} \]

with \(x \equiv r / (c_\gamma R_{500c})\).

This model has five parameters: \((\rho_0, P_0)\) are the central value of gas density and pressure, \(\Gamma_0\) is the asymptotic value of the polytropic index as \(r \rightarrow \infty\), \(c_\gamma\) is the polytropic concentration (\(c_\gamma = 0\) implies \(\Gamma(r) = \Gamma_0\)), and \(\theta_0\) is a shape parameter.

We further write the fraction of non-thermal pressure as a power-law of radius, plus a constant plateau:

\[ f_{\rm nt}(r) = A_{\rm nt} + (B_{\rm nt} - A_{\rm nt}) \left(\frac{r}{2R_{500c}}\right)^{C_{\rm nt}} \]

This adds three parameters to our gas model: \(A_{\rm nt}\) is the central value of non-thermal pressure fraction, \(B_{\rm nt}\) is the non-thermal pressure fraction at \(r=2R_{500c}\), and \(C_{\rm nt}\) is the power law evolution with radius.

Let’s vary each parameter one at a time and look at the impact on the gas properties, for an NFW halo:

fig = plot_impact_model_params(n_curves=7, cmapname="magma")
fig.show()

/var/folders/3t/_vtqsjsx3hq7ktlc27qg58c00000gr/T/ipykernel_6895/1775941011.py:2: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown
  fig.show()
../_images/0ef00f35767fe2ffcf9ea3c71f4a0657457a192971a176394fdcade585f6ea00.png

Radius-dependent polytropic index#

fig = plot_Gamma_r(
    Gamma_0 = jnp.array([1.15, 1.20]),
    c_gamma = jnp.array([-1, -0.5, -0.2, -0.1, -0.05, 0.0, 0.05, 0.1, 0.2, 0.5, 1]),
    cmapname="magma"
)
fig.show()

/var/folders/3t/_vtqsjsx3hq7ktlc27qg58c00000gr/T/ipykernel_6895/71608787.py:6: UserWarning: FigureCanvasAgg is non-interactive, and thus cannot be shown
  fig.show()
../_images/2f0e9bd5d8498ade69ac3a3fcc23625212809282d06bd8dea0cf20d97d4615ae.png
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