figures.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Goal:\n",
    "**Assuming** the tidal force of the Moon is exactly the same as present day\n",
    "\n",
    "Explore the (mass, density) space, look at impact on angular size\n",
    "\n",
    "Note: mass of the moon is directly tied to distance via the tidal force being fixed"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def _tidal_force(M2, d, M1=5.974e27):\n",
    "    return (M1*M2) / (d**3.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def newDist(M2, M2_0=7.36e25, M1=5.974e27, d0=3.844e10):\n",
    "    '''\n",
    "    compute the distance of the Moon given a new mass,\n",
    "    assuming a fixed Tidal Force\n",
    "    '''\n",
    "    # compute the original Tidal Force\n",
    "    t0 = (M1*M2_0) / (d0**3.)\n",
    "    \n",
    "    # invert eqn above for new mass\n",
    "    d = ((M1*M2) / t0)**(1./3.)\n",
    "    return d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def newRadius(mass, density):\n",
    "    '''\n",
    "    compute the radius of the Moon given a mass and mean density \n",
    "    '''\n",
    "    R = (mass / (4./3. * np.pi * density))**(1./3.)\n",
    "    return R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def AngularSize(radius, dist):\n",
    "    hyp = (radius**2. + dist**2.)**0.5\n",
    "    \n",
    "    # use Law of Sines\n",
    "    theta = np.arcsin(radius/hyp) # in radians\n",
    "    return theta * 180. / np.pi # return in deg"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# actual values\n",
    "M_moon = 7.36e25 # (g)\n",
    "R_moon = 1.7371e8 # (cm)\n",
    "D_moon = 3.844e10 # (cm)\n",
    "\n",
    "rho_moon = M_moon / (4./3. * np.pi * R_moon**3.)\n",
    "\n",
    "# parameter space to explore\n",
    "density = np.arange(2, 4.5, 0.1)\n",
    "mass = np.arange(0.1, 2.1, 0.1) * M_moon\n",
    "\n",
    "dist = newDist(mass) # mass yields distance, due to fixed Tidal Force"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x1059e5490>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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WLYI994RvfhN++cu8ozEz+7hqbkHsDDwfES9HxBJgFHBAk30OA0ZHxCyAUsmh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\nJCcIMzMryQnCzMxKcoIwM7OS/h8cswjORgUeQwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x103f66910>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(mass/M_moon, dist/D_moon)\n",
    "plt.xlabel('Mass')\n",
    "plt.ylabel('Distance')\n",
    "plt.title('Fixed Tidal Force')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "NewSize = np.zeros((len(mass), len(density)))\n",
    "\n",
    "for i in range(len(mass)):\n",
    "    for j in range(len(density)):\n",
    "        radius = newRadius(mass[i], density[j])\n",
    "        NewSize[i,j] = AngularSize(radius, dist[i])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x107d7d410>"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x107afab10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xx, yy = np.meshgrid(dist, density, indexing='ij')\n",
    "\n",
    "plt.contourf(xx / D_moon, yy, NewSize)\n",
    "plt.xlabel('$D$ / $D_{moon}$')\n",
    "plt.ylabel(r'$\\rho$ (g/cm$^3$)')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x107da5310>"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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aWmINHeqpa638FbvneM986yPileZcsBicOKwcPP44nHsudOyY+oPsv3+pIzKr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\n/fgAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x107d73c90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(density, NewSize[0,:])\n",
    "plt.xlabel('Density (g/cm$^3$)')\n",
    "plt.ylabel('Angular Size (deg)')\n",
    "plt.title('At one mass (aka distance)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So at a given mass (or equivalently distance) the angular size goes down with increasing density. This is pretty intuititve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x108099610>"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x107b60d50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(mass / M_moon, NewSize[:,0])\n",
    "plt.xlabel('relative mass')\n",
    "plt.ylabel('Angular Size (deg)')\n",
    "plt.title('At one density')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But at a given density, if we require the tidal force to be the same, the angular size is the same!\n",
    "\n",
    "Briefly, this is because \n",
    "\n",
    "$\\rho = M / (4\\pi/3 r^3) \\propto r^{1/3}$\n",
    "\n",
    "and \n",
    "\n",
    "angular size goes down as $d^2$\n",
    "\n",
    "and\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "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": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}