{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"\n",
"# Testing Piecewise Polytropic EOS\n",
"\n",
"## Author: Leo Werneck\n",
"\n",
"## This notebook assesses the piecewise polytropic Equation of State (EOS) relations used in neutron star models, following [J.C. Read *et al.* (2008)](https://arxiv.org/pdf/0812.2163.pdf) and [Takami *et al.* (2014)](https://arxiv.org/pdf/1412.3240v2.pdf). It evaluates $K$ and $\\Gamma$, formulating a function, and generates relevant plots for testing, concluding with an implementation of the EOS in C code."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"# Table of Contents\n",
"$$\\label{toc}$$\n",
"\n",
"This module is organized as follows\n",
"\n",
"1. [Step 1](#introduction): **Introduction**\n",
"1. [Step 2](#plugging_in_values): **Plugging in some values**\n",
"1. [Step 3](#p_cold): **Taking a look at $P_{\\rm cold}$**\n",
" 1. [Step 3.a](#p_cold__computing): *A function to evaluate $P_{\\rm cold}$ based on `IllinoisGRMHD`*\n",
" 1. [Step 3.b](#p_cold__plotting): *Plotting $P_{\\rm cold}\\left(\\rho\\right)$*\n",
"1. [Step 4](#eps_cold): **Taking a look at $\\epsilon_{\\rm cold}$**\n",
" 1. [Step 4.a](#eps_cold__computing): *A function to evaluate $\\epsilon_{\\rm cold}$ based on `IllinoisGRMHD`*\n",
" 1. [Step 4.b](#eps_cold__plotting): *Plotting $\\epsilon_{\\rm cold}\\left(\\rho\\right)$*\n",
"1. [Step 5](#ppeos__c_code): **The piecewise polytrope EOS C code**\n",
" 1. [Step 5.a](#ppeos__c_code__prelim): *Preliminary treatment of the input*\n",
" 1. [Step 5.a.i](#ppeos__c_code__prelim__computing_ktab): Determining $\\left\\{K_{1},K_{2},\\ldots,K_{\\rm neos}\\right\\}$\n",
" 1. [Step 5.a.ii](#ppeos__c_code__prelim__computing_eps_integ_consts): Determining $\\left\\{C_{0},C_{1},C_{2},\\ldots,C_{\\rm neos}\\right\\}$\n",
" 1. [Step 5.b](#ppeos__c_code__eos_struct_setup) *Setting up the `eos_struct`*\n",
" 1. [Step 5.c](#ppeos__c_code__find_polytropic_k_and_gamma_index) *The `find_polytropic_K_and_Gamma_index()` function*\n",
" 1. [Step 5.d](#ppeos__c_code__compute_P_cold__eps_cold__dPcold_drho__eps_th__h__Gamma_cold): *The new `compute_P_cold__eps_cold__dPcold_drho__eps_th__h__Gamma_cold()` function*\n",
"1. [Step n](#latex_pdf_output): **Output this notebook to $\\LaTeX$-formatted PDF file**"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"# Step 1: Introduction \\[Back to [top](#toc)\\]\n",
"$$\\label{introduction}$$\n",
"\n",
"We will test here a few piecewise polytrope (PP) equation of state (EOS) relations. The main reference we will be following is [J.C. Read *et al.* (2008)](https://arxiv.org/pdf/0812.2163.pdf), but we will also try out [the approach of `Whisky`, presented in Takami *et al.* (2014)](https://arxiv.org/pdf/1412.3240v2.pdf)\n",
"\n",
"First, we will start with the original one implemented by IllinoisGRMHD:\n",
"\n",
"$$\n",
"\\boxed{\n",
"P_{\\rm cold} =\n",
"\\left\\{\n",
"\\begin{matrix}\n",
"K_{0}\\rho^{\\Gamma_{0}} & , & \\rho \\leq \\rho_{0}\\\\\n",
"K_{1}\\rho^{\\Gamma_{1}} & , & \\rho_{0} \\leq \\rho \\leq \\rho_{1}\\\\\n",
"\\vdots & & \\vdots\\\\\n",
"K_{j}\\rho^{\\Gamma_{j}} & , & \\rho_{j-1} \\leq \\rho \\leq \\rho_{j}\\\\\n",
"\\vdots & & \\vdots\\\\\n",
"K_{N-1}\\rho^{\\Gamma_{N-1}} & , & \\rho_{N-2} \\leq \\rho \\leq \\rho_{N-1}\\\\\n",
"K_{N}\\rho^{\\Gamma_{N}} & , & \\rho \\geq \\rho_{N-1}\n",
"\\end{matrix}\n",
"\\right.\n",
"}\\ .\n",
"$$\n",
"\n",
"Notice that we have the following sets of variables:\n",
"\n",
"$$\n",
"\\left\\{\\underbrace{\\rho_{0},\\rho_{1},\\ldots,\\rho_{N-1}}_{N\\ {\\rm values}}\\right\\}\\ ;\\\n",
"\\left\\{\\underbrace{K_{0},K_{1},\\ldots,K_{N}}_{N+1\\ {\\rm values}}\\right\\}\\ ;\\\n",
"\\left\\{\\underbrace{\\Gamma_{0},\\Gamma_{1},\\ldots,\\Gamma_{N}}_{N+1\\ {\\rm values}}\\right\\}\\ .\n",
"$$\n",
"\n",
"Also, notice that $K_{0}$ and the entire sets $\\left\\{\\rho_{0},\\rho_{1},\\ldots,\\rho_{N-1}\\right\\}$ and $\\left\\{\\Gamma_{0},\\Gamma_{1},\\ldots,\\Gamma_{N}\\right\\}$ must be specified by the user. The values of $\\left\\{K_{1},\\ldots,K_{N}\\right\\}$, on the other hand, are determined by imposing that $P_{\\rm cold}$ be continuous, i.e.\n",
"\n",
"$$\n",
"P_{\\rm cold}\\left(\\rho_{0}\\right) = K_{0}\\rho_{0}^{\\Gamma_{0}} = K_{1}\\rho_{0}^{\\Gamma_{1}} \\implies\n",
"\\boxed{K_{1} = K_{0}\\rho_{0}^{\\Gamma_{0}-\\Gamma_{1}}}\\ .\n",
"$$\n",
"\n",
"Analogously,\n",
"\n",
"$$\n",
"\\boxed{K_{j} = K_{j-1}\\rho_{j-1}^{\\Gamma_{j-1}-\\Gamma_{j}}\\ ,\\ j\\in\\left[1,N\\right]}\\ .\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"# Step 2: Plugging in some values \\[Back to [top](#toc)\\]\n",
"$$\\label{plugging_in_values}$$\n",
"\n",
"Just so that we work with realistic values (i.e. values actually used by researchers), we will implement a simple check using the values from [Table II in J.C. Read *et al.* (2008)](https://arxiv.org/pdf/0812.2163.pdf):\n",
"\n",
"| $\\rho_{i}$ | $\\Gamma_{i}$ | $K_{\\rm expected}$ |\n",
"|------------|--------------|--------------------|\n",
"|2.44034e+07 | 1.58425 | 6.80110e-09 |\n",
"|3.78358e+11 | 1.28733 | 1.06186e-06 |\n",
"|2.62780e+12 | 0.62223 | 5.32697e+01 |\n",
"| $-$ | 1.35692 | 3.99874e-08 |"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"execution": {
"iopub.execute_input": "2021-05-17T23:43:05.097985Z",
"iopub.status.busy": "2021-05-17T23:43:05.097085Z",
"iopub.status.idle": "2021-05-17T23:43:05.246891Z",
"shell.execute_reply": "2021-05-17T23:43:05.247383Z"
}
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"K_ppoly_tab[1] = 1.0618444833278535e-06 | K_expected[1] = 1.06186e-06 | Diff = 1.5516672146434653e-11\n",
"K_ppoly_tab[2] = 53.2750845839615 | K_expected[2] = 53.2697 | Diff = 0.005384583961500766\n",
"K_ppoly_tab[3] = 3.999206917243191e-08 | K_expected[3] = 3.99874e-08 | Diff = 4.669172431909315e-12\n"
]
}
],
"source": [
"# Determining K_{i} i != 0\n",
"# We start by setting up all the values of rho_{i}, Gamma_{i}, and K_{0}\n",
"import numpy as np\n",
"rho_ppoly_tab = [2.44034e+07,3.78358e+11,2.62780e+12]\n",
"Gamma_ppoly_tab = [1.58425,1.28733,0.62223,1.35692]\n",
"K_ppoly_tab = [6.80110e-09,0,0,0]\n",
"K_expected = [6.80110e-09,1.06186e-06,5.32697e+01,3.99874e-08]\n",
"NEOS = len(rho_ppoly_tab)+1\n",
"\n",
"for j in range(1,NEOS):\n",
" K_ppoly_tab[j] = K_ppoly_tab[j-1] * rho_ppoly_tab[j-1]**(Gamma_ppoly_tab[j-1] - Gamma_ppoly_tab[j])\n",
" print(\"K_ppoly_tab[\"+str(j)+\"] = \"+str(K_ppoly_tab[j])+\\\n",
" \" | K_expected[\"+str(j)+\"] = \"+str(K_expected[j])+\\\n",
" \" | Diff = \"+str(np.fabs(K_ppoly_tab[j] - K_expected[j])))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"# Step 3: Taking a look at $P_{\\rm cold}$ \\[Back to [top](#toc)\\]\n",
"$$\\label{p_cold}$$\n",
"\n",
"\n",
"\n",
"## Step 3.a: A function to evaluate $P_{\\rm cold}$ based on `IllinoisGRMHD` \\[Back to [top](#toc)\\]\n",
"$$\\label{p_cold__computing}$$\n",
"\n",
"The results above look reasonable to the expected ones. Let us then see what the plot of $P_{\\rm cold}\\times\\rho$ looks like. For the case of our specific example, which has $N_{\\rm EOS}=4$ (where $N_{\\rm EOS}$ stands for the number of polytropic EOSs used), we have\n",
"\n",
"$$\n",
"\\boxed{\n",
"P_{\\rm cold} =\n",
"\\left\\{\n",
"\\begin{matrix}\n",
"K_{0}\\rho^{\\Gamma_{0}} & , & \\rho \\leq \\rho_{0}\\\\\n",
"K_{1}\\rho^{\\Gamma_{1}} & , & \\rho_{0} \\leq \\rho \\leq \\rho_{1}\\\\\n",
"K_{2}\\rho^{\\Gamma_{2}} & , & \\rho_{1} \\leq \\rho \\leq \\rho_{2}\\\\\n",
"K_{3}\\rho^{\\Gamma_{3}} & , & \\rho \\geq \\rho_{2}\n",
"\\end{matrix}\n",
"\\right.\n",
"}\\ .\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"execution": {
"iopub.execute_input": "2021-05-17T23:43:05.252941Z",
"iopub.status.busy": "2021-05-17T23:43:05.252228Z",
"iopub.status.idle": "2021-05-17T23:43:05.254115Z",
"shell.execute_reply": "2021-05-17T23:43:05.254589Z"
}
},
"outputs": [],
"source": [
"# Set the pressure through the polytropic EOS: P_cold = K * rho^{Gamma}\n",
"def P_cold(rho,rho_ppoly_tab,Gamma_ppoly_tab,K_ppoly_tab,NEOS):\n",
" if rho < rho_ppoly_tab[0] or NEOS==1:\n",
" return K_ppoly_tab[0] * rho**Gamma_ppoly_tab[0]\n",
"\n",
" if NEOS > 2:\n",
" for j in range(1,NEOS-1):\n",
" if rho_ppoly_tab[j-1] <= rho and rho < rho_ppoly_tab[j]:\n",
" return K_ppoly_tab[j] * rho**Gamma_ppoly_tab[j]\n",
"\n",
" if rho >= rho_ppoly_tab[NEOS-2]:\n",
" return K_ppoly_tab[NEOS-1] * rho**Gamma_ppoly_tab[NEOS-1]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"## Step 3.b: Plotting $P_{\\rm cold}\\left(\\rho\\right)$ \\[Back to [top](#toc)\\]\n",
"$$\\label{p_cold__plotting}$$\n",
"\n",
"To create a reasonably interesting plot, which will also test all regions of our piecewise polytrope EOS, let us plot $P_{\\rm cold}\\left(\\rho\\right)$ with $\\rho\\in\\left[\\frac{\\rho_{0}}{10},10\\rho_{2}\\right]$.\n",
"\n",
"**Remember**: We *don't* expect $P_{\\rm cold}$ to be *smooth*, but we *do* expect it to be *continuous* (by construction)."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
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"iopub.execute_input": "2021-05-17T23:43:05.264933Z",
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"shell.execute_reply": "2021-05-17T23:43:06.146722Z"
}
},
"outputs": [
{
"data": {
"text/plain": [
"[]"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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\n",
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