03-UnitsQuantities/Astropy_Units_Solutions.ipynb

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   "cell_type": "markdown",
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   "source": [
    "# Astropy Units and Quantities\n",
    "\n",
    "## Exercise Solutions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1\n",
    "\n",
    "The *James Webb Space Telescope (JWST)* will be located at the second Sun-Earth Lagrange (L2) point:\n",
    "\n",
    "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;☀️ &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 🌎 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; L2 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; *(not to scale)*\n",
    "\n",
    "\n",
    "L2 is located at a distance from the Earth (opposite the Sun) of approximately:\n",
    "\n",
    "$$ r \\approx R \\left(\\frac{M_{earth}}{3 M_{sun}}\\right) ^{(1/3)} $$\n",
    "\n",
    "where $R$ is the Sun-Earth distance.\n",
    "\n",
    "Calculate the Earth-L2 distance in kilometers and miles.\n",
    "\n",
    "*Hints*:\n",
    "\n",
    "* $M_{earth}$ and $M_{sun}$ are defined [constants](http://docs.astropy.org/en/stable/constants/#reference-api) \n",
    "\n",
    "* the mile unit is defined as ``u.imperial.mile`` (see [imperial units](http://docs.astropy.org/en/v0.2.1/units/index.html#module-astropy.units.imperial))"
   ]
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  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import astropy.units as u"
   ]
  },
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   "execution_count": 2,
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$1496555.1 \\; \\mathrm{km}$"
      ],
      "text/plain": [
       "<Quantity 1496555.0858807534 km>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d_earth = u.au * (1. * u.M_earth / (3. * u.M_sun))**(1./3)\n",
    "d_earth.to(u.km)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$929916.22 \\; \\mathrm{mi}$"
      ],
      "text/plain": [
       "<Quantity 929916.2179625693 mi>"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d_earth.to(u.imperial.mile)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2\n",
    "\n",
    "The L2 point is about 1.5 million kilometers away from the Earth opposite the Sun.\n",
    "The total mass of the *James Webb Space Telescope (JWST)* is about 6500 kg.\n",
    "\n",
    "Using the value you obtained above for the Earth-L2 distance, calculate the gravitational force in Newtons between \n",
    "\n",
    "* *JWST* (at L2) and the Earth\n",
    "* *JWST* (at L2) and the Sun\n",
    "\n",
    "*Hint*: the gravitational force between two masses separated by a distance *r* is:\n",
    "\n",
    "$$ F_g = \\frac{G m_1 m_2}{r^2} $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from astropy.constants import G"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$1.1571336 \\; \\mathrm{N}$"
      ],
      "text/plain": [
       "<Quantity 1.157133562380145 N>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F = (G * 1. * u.M_earth * 6500. * u.kg) / d_earth**2\n",
    "F.to(u.N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$37.796265 \\; \\mathrm{N}$"
      ],
      "text/plain": [
       "<Quantity 37.796264500778264 N>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d_sun = d_earth + 1. * u.au\n",
    "F = (G * 1. * u.M_sun * 6500. * u.kg) / d_sun**2\n",
    "F.to(u.N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3\n",
    "\n",
    "Calculate the Schwarzschild radius in units of solar radii of the Sgr A*, the Milky Way's supermassive black hole with $M = 4.31 \\times 10^6 M_\\odot$, given\n",
    "\n",
    "$$r_\\mathrm{s} = \\frac{2 G M}{c^2}$$\n",
    "\n",
    "Also calculate the angular size of the event horizon on the sky in microarcseconds, given the distance to the galactic center $d_{center} = 7.94$ kpc."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Schwarzschild radius = 18.295972154652976 solRad\n"
     ]
    }
   ],
   "source": [
    "import astropy.units as u\n",
    "from astropy.constants import c\n",
    "\n",
    "# Schwarzschild radius:\n",
    "r_s = 2*G*4.31e6*u.Msun/c**2\n",
    "print(\"Schwarzschild radius = {0}\".format(r_s.to(u.Rsun)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$21.431944 \\; \\mathrm{\\mu arcsec}$"
      ],
      "text/plain": [
       "<Quantity 21.43194446 uarcsec>"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Size on the sky given small angle approximation\n",
    "sgr_a_distance = 7940 * u.pc\n",
    "angular_diameter = np.arctan(2*r_s/sgr_a_distance)\n",
    "angular_diameter.to(u.uarcsec)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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