ISM_v4.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy import integrate\n",
    "%pylab inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\begin{equation}\n",
    "v_{c}^{2} = \\frac{4G\\Sigma_{g}R_{d}}{\\pi} y^{2}\\int_{0}^{\\pi/2}(\\cos(2u) + 1)du\\int_{0}^{\\infty}d\\beta exp(-y\\cos u\\cosh\\beta -\\frac{R_{m}}{2*R_{d}y\\cos u \\cosh\\beta})(\\frac{R_{m}}{2R_{d}y^{2}\\cos^{2}u\\cosh^{2}\\beta} -1)\n",
    "\\end{equation}$$\n",
    "$$\\begin{equation}\n",
    "\\frac{v_{c}^{2}\\pi}{4G\\Sigma_{g}R_{d}} =  y^{2}\\int_{0}^{\\pi/2}(\\cos(2u) + 1)du\\int_{0}^{\\infty}d\\beta exp(-y\\cos u\\cosh\\beta -\\frac{R_{m}}{2*R_{d}y\\cos u \\cosh\\beta})(\\frac{R_{m}}{2R_{d}y^{2}\\cos^{2}u\\cosh^{2}\\beta} -1)\n",
    "\\end{equation}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "donde $y = \\frac{R}{2R_{d}}$, por lo que $v_{c}^{2}(R)$ es \n",
    "$$\\begin{equation}\n",
    "v_{c}^{2} = \\frac{G\\Sigma_{g}R^{2}}{\\pi R_{d}}\\int_{0}^{\\pi/2}(\\cos2u + 1)du\\int_{0}^{\\infty} d\\beta exp(-\\frac{R\\cos u \\cosh\\beta}{2R_{d}} -\\frac{R_{m}}{R\\cos u \\cosh\\beta})(\\frac{2R_{m}R_{d}}{R^{2}\\cos^{2}u\\cosh^{2}\\beta} -1)\n",
    "\\end{equation}$$\n",
    "$$\\begin{equation}\n",
    "\\frac{v_{c}^{2}\\pi R_{d}}{G\\Sigma_{g}} = R^{2}\\int_{0}^{\\pi/2}(\\cos2u + 1)du\\int_{0}^{\\infty} d\\beta exp(-\\frac{R\\cos u \\cosh\\beta}{2R_{d}} -\\frac{R_{m}}{R\\cos u \\cosh\\beta})(\\frac{2R_{m}R_{d}}{R^{2}\\cos^{2}u\\cosh^{2}\\beta} -1)\n",
    "\\end{equation}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "R = np.linspace(0.1,15.,100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def func_R_I(u,beta,R,R_m=4.,R_d=2.):\n",
    "    exp1 = np.exp(-(R*np.cos(u)*np.cosh(beta))/(2.*R_d))\n",
    "    exp2 = np.exp(-R_m/(R*np.cos(u)*np.cosh(beta)))\n",
    "    t1 = (2.*R_m*R_d)/(R**2.*np.cos(u)**2.*np.cosh(beta)**2.) - 1.\n",
    "    t2 = np.cos(2.*u) + 1.\n",
    "    return exp1*t1*t2*exp2*R**2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/atalia/miniconda2/lib/python2.7/site-packages/ipykernel_launcher.py:3: RuntimeWarning: overflow encountered in exp\n",
      "  This is separate from the ipykernel package so we can avoid doing imports until\n",
      "/Users/atalia/miniconda2/lib/python2.7/site-packages/ipykernel_launcher.py:6: RuntimeWarning: overflow encountered in double_scalars\n",
      "  \n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, 9.380859177814076e+101, 5.0220435945109584e+98, 4.267138713792802e+95, 5.522337160423649e+92, 1.0495792608176021e+90, 2.836385884708926e+87, 1.058907149536895e+85, 5.322396865220267e+82, 3.5195356002583505e+80, 2.998995256822268e+78, 3.231835977198534e+76, 4.3306940355206266e+74, 7.106260263458468e+72, 1.408157291185326e+71, 3.327213385739584e+69, 9.266273860103171e+67, 3.009754188770873e+66, 1.1291468518288169e+65, 4.849549979673166e+63, 2.3650119812743113e+62, 1.299798066188126e+61, 7.994883011991178e+59, 5.468325114953398e+58, 4.134497343549117e+57, 3.4365790408712896e+56, 3.1242525790395393e+55, 3.0918597854675544e+54, 3.3161036871931323e+53, 3.8386878693469506e+52, 4.7776636185503546e+51, 6.370421256348525e+50, 9.069501712060987e+49, 1.374349965112003e+49, 2.21019997450111e+48, 3.761724092232859e+47, 6.758298106491106e+46, 1.278569371224934e+46, 2.5412764618015218e+45, 5.295199963942715e+44, 1.1543345640666183e+44, 2.627641033069387e+43, 6.234448773985272e+42, 1.5391602318877614e+42, 3.9474790406078794e+41, 1.0501231947877372e+41, 2.8934401466004858e+40, 8.246030285753177e+39, 2.427530293884969e+39, 7.372854864144407e+38, 2.3075282219799026e+38, 7.433837106296481e+37, 2.4624754981061114e+37, 8.378845696031367e+36, 2.9257110290943e+36, 1.0474058009170815e+36, 3.8410928625246506e+35, 1.441745000723655e+35, 5.534393029433356e+34, 2.1710465521472653e+34, 8.69701688648792e+33, 3.555264428829521e+33, 1.4821271695959566e+33, 6.297019050957136e+32, 2.724936148369382e+32, 1.2003182703252652e+32, 5.379139829276212e+31, 2.451168625208273e+31, 1.135153560574347e+31, 5.340024997163514e+30, 2.5505507955146043e+30]\n"
     ]
    }
   ],
   "source": [
    "fun = []\n",
    "t = np.pi/2.-0.1\n",
    "for i in range (0,len(R)):\n",
    "    integ = lambda u,beta: func_R_I(u,beta,R[i])\n",
    "    resul = integrate.dblquad(integ,0.1,np.pi/2.,lambda u: 0.1, lambda u: np.inf)[0]\n",
    "    fun.append(resul)\n",
    "print fun"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Constants\n",
    "G = 4.302e-6#kpc/SM(km/s)^2\n",
    "q_b = 0.6\n",
    "r_b = 1.9#kpc\n",
    "a_b = 1.#kpc\n",
    "q_h = 0.8\n",
    "alpha_b = 1.8\n",
    "R_m = 4.#kpc\n",
    "#parameters model I\n",
    "rho_b_I = 0.427e9#sM/kpc^3\n",
    "a_h_I = 3.83#kpc\n",
    "rho_h_I = 0.711e9#SM/kpc^3\n",
    "alpha_h_I = -2.\n",
    "beta_h_I = 2.96\n",
    "R_d_I = 2.0#kpc\n",
    "R_d_II = 3.2#kpc\n",
    "Sigma_d_SD_I = (1905.0e6)*0.75#SM/kpc^2\n",
    "Sigma_g_ISM_I = (1905.0e6)*0.25#SM/kpc^2\n",
    "#parameters model II\n",
    "rho_b_II = 0.3e9#sM/kpc^3\n",
    "a_h_II = 1.9#kpc\n",
    "rho_h_II = 0.266e9#SM/kpc^3\n",
    "alpha_h_II = 1.63\n",
    "beta_h_II = 2.17\n",
    "R_d_II = 3.2#kpc\n",
    "Sigma_d_SD_II = (536.0e6)*0.75#SM/kpc^2\n",
    "Sigma_g_ISM_II = (536.0e6)*0.25#SM/kpc^2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "vel_integral_I = np.array(fun)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[           inf            inf            inf            inf\n",
      "            inf            inf            inf            inf\n",
      "            inf            inf            inf            inf\n",
      "            inf            inf            inf            inf\n",
      "            inf            inf            inf            inf\n",
      "            inf            inf            inf            inf\n",
      "            inf            inf            inf            inf\n",
      "            inf            inf 1.74897689e+52 4.04671857e+50\n",
      " 1.17959017e+49 4.24350016e+47 1.84999427e+46 9.61712854e+44\n",
      " 5.87613417e+43 4.16597242e+42 3.38770390e+41 3.12716396e+40\n",
      " 3.24629078e+39 3.75786794e+38 4.81374763e+37 6.77623326e+36\n",
      " 1.04160508e+36 1.73826237e+35 3.13276831e+34 6.06789388e+33\n",
      " 1.25751589e+33 2.77702274e+32 6.51029557e+31 1.61461439e+31\n",
      " 4.22269707e+30 1.16111202e+30 3.34754121e+29 1.00933616e+29\n",
      " 3.17521150e+28 1.03986465e+28 3.53797014e+27 1.24816083e+27\n",
      " 4.55771158e+26 1.71970707e+26 6.69439649e+25 2.68459358e+25\n",
      " 1.10753187e+25 4.69441452e+24 2.04185506e+24 9.10308756e+23\n",
      " 4.15531494e+23 1.94012009e+23 9.25647816e+22 4.50880854e+22\n",
      " 2.24029227e+22 1.13454749e+22 5.85171126e+21 3.07163788e+21\n",
      " 1.63977864e+21 8.89703077e+20 4.90321126e+20 2.74306611e+20\n",
      " 1.55693070e+20 8.96083999e+19 5.22702840e+19 3.08871955e+19\n",
      " 1.84807780e+19 1.11915481e+19 6.85657120e+18 4.24812964e+18\n",
      " 2.66070871e+18 1.68402262e+18 1.07670993e+18 6.95193171e+17\n",
      " 4.53137774e+17 2.98085553e+17 1.97838573e+17 1.32440002e+17\n",
      " 8.94024371e+16 6.08401213e+16 4.17286571e+16 2.88389704e+16]\n"
     ]
    }
   ],
   "source": [
    "C = (G*Sigma_g_ISM_I)/(np.pi*R_d_I)\n",
    "vel = np.sqrt(C*vel_integral_I)\n",
    "print vel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11c8b2cd0>]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(R,vel)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Probar con el integrando con def function en vez de lambda"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "def func_R_II(u,beta,R,R_m=4.,R_d=3.2):\n",
    "    exp1 = np.exp(-(R*np.cos(u)*np.cosh(beta))/(2.*R_d))\n",
    "    exp2 = np.exp(-R_m/(R*np.cos(u)*np.cosh(beta)))\n",
    "    t1 = (2.*R_m*R_d)/(R**2.*np.cos(u)**2.*np.cosh(beta)**2.) - 1.\n",
    "    t2 = np.cos(2.*u) + 1.\n",
    "    return exp1*t1*t2*exp2*R**2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/atalia/miniconda2/lib/python2.7/site-packages/ipykernel_launcher.py:3: RuntimeWarning: overflow encountered in exp\n",
      "  This is separate from the ipykernel package so we can avoid doing imports until\n",
      "/Users/atalia/miniconda2/lib/python2.7/site-packages/ipykernel_launcher.py:6: RuntimeWarning: overflow encountered in double_scalars\n",
      "  \n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, inf, 1.4985895158219783e+102, 8.022290851962525e+98, 6.816046421909554e+95, 8.820568004927571e+92, 1.6763578485454737e+90, 4.5299631177722095e+87, 1.6910836564904835e+85, 8.499479473307015e+82, 5.62015511489428e+80, 4.7886884777575795e+78, 5.1602172557748175e+76, 6.91439280541339e+74, 1.1345288090721067e+73, 2.248037037500882e+71, 5.3114222113558605e+69, 1.4791532807125521e+68, 4.8041544164888536e+66, 1.802246710940355e+65, 7.740038807929787e+63, 3.774443519666655e+62, 2.074308569344921e+61, 1.2758143406472852e+60, 8.725846865415774e+58, 6.597112232630717e+57, 5.483216450779473e+56, 4.984631869151697e+55, 4.932699345008181e+54, 5.290185008218368e+53, 6.123552780817975e+52, 7.62103827524891e+51, 1.0161191499855798e+51, 1.4465645715675433e+50, 2.1919449769329363e+49, 3.524859406050844e+48, 5.9989478668791454e+47, 1.0777137291006713e+47, 2.0387702676108736e+46, 4.052041037688342e+45, 8.44271666716306e+44, 1.8403883708601014e+44, 4.189109848025663e+43, 9.938748701697647e+42, 2.453552867093732e+42, inf, 1.6738160185461896e+41, 4.611687983084643e+40, 1.3142206460084998e+40, 3.868708142101239e+39, 1.17493798779186e+39, 3.67708979791496e+38, 1.1845358553281924e+38, 3.923602903143463e+37, 1.3349816088888244e+37, 4.6612294569992426e+36, 1.6686374789966472e+36, 6.118990711886352e+35, 2.2966319998675193e+35, 8.815580753170526e+34, 3.458024304701961e+34, 1.3851831389990477e+34, 5.662220409638485e+33, 2.360360369723611e+33, 1.002780402126692e+33, 4.339154840603573e+32, 1.911275198120086e+32, 8.564808035234252e+31, 3.9026174611583534e+31, 1.8072382386453573e+31, 8.501236161345593e+30, 4.0602312551179727e+30]\n"
     ]
    }
   ],
   "source": [
    "fun_II = []\n",
    "t = np.pi/2.-0.1\n",
    "for i in range (0,len(R)):\n",
    "    integral_II = lambda u,beta: func_R_II(u,beta,R[i])\n",
    "    resultado_II = integrate.dblquad(integral_II,0.1,np.pi/2.,lambda u: 0.1, lambda u: np.inf)[0]\n",
    "    fun_II.append(resultado_II)\n",
    "print fun_II"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[           inf            inf            inf            inf\n",
      "            inf            inf            inf            inf\n",
      "            inf            inf            inf            inf\n",
      "            inf            inf            inf            inf\n",
      "            inf            inf            inf            inf\n",
      "            inf            inf            inf            inf\n",
      "            inf            inf            inf            inf\n",
      "            inf            inf 9.26998479e+51 2.14480038e+50\n",
      " 6.25178374e+48 2.24898186e+47 9.80440048e+45 5.09665234e+44\n",
      " 3.11401158e+43 2.20766840e+42 1.79519582e+41 1.65708952e+40\n",
      " 1.72017119e+39 1.99119916e+38 2.55061812e+37 3.59037166e+36\n",
      " 5.51878010e+35 9.20967421e+34 1.65976329e+34 3.21473218e+33\n",
      " 6.66207083e+32 1.47117434e+32 3.44885041e+31 8.55325493e+30\n",
      " 2.23687374e+30 6.15055934e+29 1.77319040e+29 5.34631104e+28\n",
      " 1.68182188e+28 5.50773589e+27 1.87386979e+27 6.61065889e+26\n",
      " 2.41384842e+26 9.10765537e+25 3.54529600e+25 1.42170201e+25\n",
      " 5.86509729e+24 2.48593280e+24 1.08123930e+24 4.82030584e+23\n",
      " 2.20028422e+23 1.02728848e+23 4.90115628e+22 2.38728095e+22\n",
      " 1.18613846e+22            inf 3.09807221e+21 1.62617633e+21\n",
      " 8.68104124e+20 4.70999732e+20 2.59564413e+20 1.45207750e+20\n",
      " 8.24160474e+19 4.74329571e+19 2.76678440e+19 1.63488767e+19\n",
      " 9.78179763e+18 5.92348883e+18 3.62897004e+18 2.24834597e+18\n",
      " 1.40815901e+18 8.91233033e+17 5.69811306e+17 3.67897498e+17\n",
      " 2.39795261e+17 1.57739424e+17 1.04688575e+17 7.00803873e+16\n",
      " 4.73059391e+16 3.21918088e+16 2.20789653e+16 1.52585429e+16]\n"
     ]
    }
   ],
   "source": [
    "vel2arrayII = np.array(fun_II)\n",
    "C = (G*Sigma_g_ISM_II)/(np.pi*R_d_II)\n",
    "vel_II = np.sqrt(C*vel2arrayII)\n",
    "print vel_II"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 2e+30)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(R,vel_II)\n",
    "plt.xlim(6,15)\n",
    "plt.ylim(0,2e30)"
   ]
  },
  {
   "cell_type": "code",
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