{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Calculating Transit Timing Variations (TTV) with REBOUND\n",
    "The following code finds the transit times in a two planet system. The transit times of the inner planet are not exactly periodic, due to planet-planet interactions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, let's import the REBOUND and numpy packages."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import rebound\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's set up a coplanar two planet system."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sim = rebound.Simulation()\n",
    "sim.add(m=1)\n",
    "sim.add(m=1e-5, a=1,e=0.1,omega=0.25)\n",
    "sim.add(m=1e-5, a=1.757)\n",
    "sim.move_to_com()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We're now going to integrate the system forward in time. We assume the observer of the system is in the direction of the positive x-axis. We want to measure the time when the inner planet transits. In this geometry, this happens when the y coordinate of the planet changes sign. Whenever we detect a change in sign between two steps, we try to find the transit time, which must lie somewhere within the last step, by bisection. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "N=174\n",
    "transittimes = np.zeros(N)\n",
    "p = sim.particles\n",
    "i = 0\n",
    "while i<N:\n",
    "    y_old = p[1].y - p[0].y  # (Thanks to David Martin for pointing out a bug in this line!)\n",
    "    t_old = sim.t\n",
    "    sim.integrate(sim.t+0.5) # check for transits every 0.5 time units. Note that 0.5 is shorter than one orbit\n",
    "    t_new = sim.t\n",
    "    if y_old*(p[1].y-p[0].y)<0. and p[1].x-p[0].x>0.:   # sign changed (y_old*y<0), planet in front of star (x>0)\n",
    "        while t_new-t_old>1e-7:   # bisect until prec of 1e-5 reached\n",
    "            if y_old*(p[1].y-p[0].y)<0.:\n",
    "                t_new = sim.t\n",
    "            else:\n",
    "                t_old = sim.t\n",
    "            sim.integrate( (t_new+t_old)/2.)\n",
    "        transittimes[i] = sim.t\n",
    "        i += 1\n",
    "        sim.integrate(sim.t+0.05)       # integrate 0.05 to be past the transit "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we do a linear least square fit to remove the linear trend from the transit times, thus leaving us with the transit time variations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "A = np.vstack([np.ones(N), range(N)]).T\n",
    "c, m = np.linalg.lstsq(A, transittimes, rcond=-1)[0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, let us plot the TTVs."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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N16tWFLdMi2c05kwSEed4gbyNPRARyZtQJ4W1Uq7ZzhVx1bHT9Bm1NOZMUhXneIE8jD0Q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\n",
      "text/plain": [
       "<Figure size 720x360 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "fig = plt.figure(figsize=(10,5))\n",
    "ax = plt.subplot(111)\n",
    "ax.set_xlim([0,N])\n",
    "ax.set_xlabel(\"Transit number\")\n",
    "ax.set_ylabel(\"TTV [hours]\")\n",
    "plt.scatter(range(N), (transittimes-m*np.array(range(N))-c)*(24.*365./2./np.pi));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "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": "3.9.1+"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}