{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## One Body Gravitational Problem\n",
    "#### We will consider the motion of an object under the influnce of another massive object whose motion we neglect."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The acceleration function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def aVec(rVec):\n",
    "    \n",
    "    # Physical Constants\n",
    "    G = 6.673e-11    #MKS units\n",
    "    M = 1.99e30      #Mass of the Sun in Kg\n",
    "    GM = G * M\n",
    "    \n",
    "    x = rVec[0]\n",
    "    y = rVec[1]\n",
    "   \n",
    "    # Calculate R cube\n",
    "    RCube = (x**2 + y**2)**(1.5)\n",
    "    \n",
    "    # acceleration\n",
    "    aX = -GM *(x/RCube)\n",
    "    aY = -GM * (y/RCube)\n",
    "    \n",
    "    # return the acceleration vector\n",
    "    return np.array([aX, aY]) \n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The energy function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def energy(rVec, vVec):\n",
    "    # Physical Constants\n",
    "    G = 6.673e-11    #MKS units\n",
    "    M = 1.99e30      #Mass of the Sun in Kg\n",
    "    GM = G * M\n",
    "    \n",
    "    # find the magnitude of vectors\n",
    "    r = np.linalg.norm(rVec)\n",
    "    v = np.linalg.norm(vVec)\n",
    "    \n",
    "    return 0.5 * v**2 - GM/r \n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The angular momentum function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def L(rVec, vVec):\n",
    "    x = rVec[0]\n",
    "    y = rVec[1]\n",
    "    vx = vVec[0]\n",
    "    vy = vVec[1]\n",
    "    return x*vy - y*vx"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The intial conditions for Earth"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Initial position"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "R = 2.0e11 #Earths distance from the Sun in meters\n",
    "r0 = R * np.array([1.0,0.0]) # intial position vector"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Initial Velocity"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "v0 =  0.1e4 # Magnitude of the Earth's velocity\n",
    "v0vec = v0 * np.array([0.0, 1.0]) # initally moving along the y axis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Time Interval over which we will observe the motion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "T =  (365 * 24 * 60 * 60) # we observe for 5 years - in sec\n",
    "deltaT = 10.0               # observe the motion every hour\n",
    "N = int(T/deltaT)            # Number of steps"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simulation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Array of vectors to store position vector, velocity vector and energy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "rVectors = np.zeros((N+1, 2), float)\n",
    "vVectors = np.zeros((N+1, 2), float)\n",
    "energyArray = np.zeros(N+1, float)\n",
    "Larray = np.zeros(N+1, float)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Array of time instants"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "tArray = np.zeros(N+1, float)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Calculate the motion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Set up the initial conditions for leap frog method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial energy =  -663463500.0  angular momentum =  200000000000000.0\n"
     ]
    }
   ],
   "source": [
    "rVectors[0] = r0\n",
    "vVectors[0] = v0vec + aVec(r0)*deltaT/2.0 # velocity at the midpoint \n",
    "initEnergy = energy(r0, v0vec)\n",
    "energyArray[0] = initEnergy\n",
    "initL = L(r0, v0vec)\n",
    "Larray[0] = L(r0, v0vec)\n",
    "print(\"Initial energy = \", energyArray[0], \" angular momentum = \", Larray[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "for n in range(1, N+1):\n",
    "    tArray[n] = n * deltaT\n",
    "    rVectors[n] = rVectors[n-1] + vVectors[n-1] * deltaT # using the velocity at the midpoint\n",
    "    vVectors[n] = vVectors[n-1] + aVec(rVectors[n]) * deltaT # using the position at the midpoint of velocity interval\n",
    "    \n",
    "    # conservaton\n",
    "    Larray[n] = L(rVectors[n], vVectors[n])\n",
    "    energyArray[n] = energy(rVectors[n], vVectors[n])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Plot the result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "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": [
    "#%matplotlib auto\n",
    "# x, y components of the vectors\n",
    "xArray = rVectors[:, 0]\n",
    "yArray = rVectors[:, 1]\n",
    "\n",
    "# plot the arrays \n",
    "plt.plot(xArray, yArray)\n",
    "\n",
    "# make the axis equals so circle looks like a circle\n",
    "plt.axis('equal')\n",
    "\n",
    "# label the axis\n",
    "plt.xlabel('X[m]')\n",
    "plt.ylabel('Y[m]')\n",
    "# show the plot\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, '$\\\\Delta L/L$')"
      ]
     },
     "execution_count": 13,
     "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(tArray, (Larray - initL)/initL)\n",
    "plt.xlabel(\"T[sec]\")\n",
    "plt.ylabel(\"$\\Delta L/L$\")\n",
    "#plt.ylim(0.5, 1.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-663463500.0 -663463702.8132645\n"
     ]
    },
    {
     "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(tArray, energyArray/initEnergy)\n",
    "plt.xlabel(\"T[sec]\")\n",
    "plt.ylabel(\"$\\Delta E/E$\")\n",
    "plt.ylim(0.5, 1.5)\n",
    "print(energyArray[0], energyArray[-1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}