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August 5, 2015 09:41
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Formula for the surface density of NFW profile
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| { | |
| "cells": [ | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "##Formula for the projected NFW profile##" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "For a NFW (Navarro 1996) density profile given by,\n", | |
| "\n", | |
| "\\begin{equation}\n", | |
| "\\rho(r) = \\frac{\\rho_s}{r/r_s (1 + r/r_s)^2}\n", | |
| "\\end{equation}\n", | |
| "\n", | |
| "a surface density can be computed using,\n", | |
| "\n", | |
| "\\begin{equation}\n", | |
| "\\Sigma(x) = 2 \\int^\\infty_0 \\rho(r/r_s) dz,\\ \\text{where}\\ r^2 = R^2 +z^2 \\text{and}\\ x = R/r_s.\n", | |
| "\\end{equation}\n", | |
| "\n", | |
| "I have found 3 different mathematical forms in Bartelmann et al. 1996, Yang et al. 2007 and Coe et al. 2010. \n", | |
| "This excercise is just to convince myself that they are all same. It should be a trivial trigonometric excercise to prove their equality as well. " | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 102, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "import numpy as np\n", | |
| "import matplotlib.pyplot as plt\n", | |
| "%matplotlib inline" | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 103, | |
| "metadata": { | |
| "collapsed": true | |
| }, | |
| "outputs": [], | |
| "source": [ | |
| "#Coe et al. 2010\n", | |
| "def coe(x):\n", | |
| " #incase x is scalar/list convert to array\n", | |
| " #incase x is already an array don't copy it\n", | |
| " x = np.array( x, copy=False, ndmin=1 )\n", | |
| " fx = np.zeros(x.shape[0])\n", | |
| "\n", | |
| " #x<1 case \n", | |
| " xlt1_cut = (x<1)\n", | |
| " xlt1 = x[xlt1_cut]\n", | |
| " fx[xlt1_cut] = ( 1- np.arccosh(1./xlt1)/np.sqrt(1-xlt1**2))/(xlt1**2-1)\n", | |
| " \n", | |
| " #x>1 case\n", | |
| " xgt1_cut = (x>1)\n", | |
| " xgt1 = x[xgt1_cut]\n", | |
| " fx[xgt1_cut] = ( 1- np.arccos(1./xgt1)/np.sqrt(xgt1**2-1))/(xgt1**2-1)\n", | |
| " \n", | |
| " #x==1 case\n", | |
| " fx[x==1] = 1/3.\n", | |
| " return fx\n", | |
| "\n", | |
| "#Bartelmann et al. 1996\n", | |
| "def bartelmann(x):\n", | |
| " x = np.array( x, copy=False, ndmin=1 )\n", | |
| " fx = np.zeros(x.shape[0])\n", | |
| "\n", | |
| " #x<1 case \n", | |
| " xlt1_cut = (x<1)\n", | |
| " xlt1 = x[xlt1_cut]\n", | |
| " fx[xlt1_cut] = ( 1- 2*np.arctanh(np.sqrt((1-xlt1)/(1+xlt1)))/np.sqrt(1-xlt1**2))/(xlt1**2-1)\n", | |
| " \n", | |
| " #x>1 case\n", | |
| " xgt1_cut = (x>1)\n", | |
| " xgt1 = x[xgt1_cut]\n", | |
| " fx[xgt1_cut] = ( 1- 2*np.arctan( np.sqrt((xgt1-1)/(xgt1+1)) )/np.sqrt(xgt1**2-1))/(xgt1**2-1)\n", | |
| " \n", | |
| " #x==1 case\n", | |
| " fx[x==1] = 0.\n", | |
| " return fx" | |
| ] | |
| }, | |
| { | |
| "cell_type": "markdown", | |
| "metadata": {}, | |
| "source": [ | |
| "Now, let's plot the profiles. Equation 9 in Yang et al. returns the same profile as the above 2. Hence, not repeating here agian. " | |
| ] | |
| }, | |
| { | |
| "cell_type": "code", | |
| "execution_count": 108, | |
| "metadata": { | |
| "collapsed": false | |
| }, | |
| "outputs": [ | |
| { | |
| "data": { | |
| "text/plain": [ | |
| "<matplotlib.text.Text at 0x7f51f6cbeed0>" | |
| ] | |
| }, | |
| "execution_count": 108, | |
| "metadata": {}, | |
| "output_type": "execute_result" | |
| }, | |
| { | |
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| "RK5CYII=\n" | |
| ], | |
| "text/plain": [ | |
| "<matplotlib.figure.Figure at 0x7f51f6ce13d0>" | |
| ] | |
| }, | |
| "metadata": {}, | |
| "output_type": "display_data" | |
| } | |
| ], | |
| "source": [ | |
| "x = np.logspace(-5,5,2000)\n", | |
| "plt.loglog( x, coe(x), '--', label=\"Coe'10\" )\n", | |
| "plt.loglog( x, bartelmann(x), 'r-.', label=\"Bartelmann'96\" )\n", | |
| "plt.legend( fontsize = 15)\n", | |
| "plt.ylabel(r'$\\Sigma(x)/(2 \\rho_s r_s)$')\n", | |
| "plt.xlabel(r'$x(=R/r_s)$')" | |
| ] | |
| } | |
| ], | |
| "metadata": { | |
| "kernelspec": { | |
| "display_name": "Python 2", | |
| "language": "python", | |
| "name": "python2" | |
| }, | |
| "language_info": { | |
| "codemirror_mode": { | |
| "name": "ipython", | |
| "version": 2 | |
| }, | |
| "file_extension": ".py", | |
| "mimetype": "text/x-python", | |
| "name": "python", | |
| "nbconvert_exporter": "python", | |
| "pygments_lexer": "ipython2", | |
| "version": "2.7.10" | |
| } | |
| }, | |
| "nbformat": 4, | |
| "nbformat_minor": 0 | |
| } |
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