docs/source/examples/Propagation using Cowells formulation.ipynb

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   "source": [
    "# Cowell's formulation\n",
    "\n",
    "For cases where we only study the gravitational forces, solving the Kepler's equation is enough to propagate the orbit forward in time. However, when we want to take perturbations that deviate from Keplerian forces into account, we need a more complex method to solve our initial value problem: one of them is **Cowell's formulation**.\n",
    "\n",
    "In this formulation we write the two body differential equation separating the Keplerian and the perturbation accelerations:\n",
    "\n",
    "$$\\ddot{\\mathbb{r}} = -\\frac{\\mu}{|\\mathbb{r}|^3} \\mathbb{r} + \\mathbb{a}_d$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-info\">For an in-depth exploration of this topic, still to be integrated in poliastro, check out <a href=\"https://github.com/Juanlu001/pfc-uc3m\">this Master thesis</a></div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-info\">An earlier version of this notebook allowed for more flexibility and interactivity, but was considerably more complex. Future versions of poliastro and plotly might bring back part of that functionality, depending on user feedback. You can still download the older version <a href=\"https://github.com/poliastro/poliastro/blob/0.8.x/docs/source/examples/Propagation%20using%20Cowell's%20formulation.ipynb\">here</a>.</div>"
   ]
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   "source": [
    "## First example\n",
    "\n",
    "Let's setup a very simple example with constant acceleration to visualize the effects on the orbit."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: AstropyDeprecationWarning: astropy.extern.six will be removed in 4.0, use the six module directly if it is still needed [astropy.extern.six]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from astropy import units as u\n",
    "from astropy import time\n",
    "\n",
    "from poliastro.bodies import Earth\n",
    "from poliastro.twobody import Orbit\n",
    "from poliastro.twobody.propagation import propagate\n",
    "from poliastro.examples import iss\n",
    "\n",
    "from poliastro.twobody.propagation import cowell\n",
    "from poliastro.plotting import OrbitPlotter3D\n",
    "from poliastro.util import norm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import plotly.io as pio\n",
    "pio.renderers.default = \"notebook_connected\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To provide an acceleration depending on an extra parameter, we can use **closures** like this one:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "accel = 2e-5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def constant_accel_factory(accel):\n",
    "    def constant_accel(t0, u, k):\n",
    "        v = u[3:]\n",
    "        norm_v = (v[0]**2 + v[1]**2 + v[2]**2)**.5\n",
    "        return accel * v / norm_v\n",
    "\n",
    "    return constant_accel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
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     "execution_count": 5,
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   ],
   "source": [
    "times = np.linspace(0, 10 * iss.period, 500)\n",
    "times"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "positions = propagate(\n",
    "    iss,\n",
    "    time.TimeDelta(times),\n",
    "    method=cowell,\n",
    "    rtol=1e-11,\n",
    "    ad=constant_accel_factory(accel),\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And we plot the results:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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\"white\", \"ticks\": \"\"}, \"bgcolor\": \"#E5ECF6\", \"caxis\": {\"gridcolor\": \"white\", \"linecolor\": \"white\", \"ticks\": \"\"}}, \"title\": {\"x\": 0.05}, \"xaxis\": {\"automargin\": true, \"gridcolor\": \"white\", \"linecolor\": \"white\", \"ticks\": \"\", \"zerolinecolor\": \"white\", \"zerolinewidth\": 2}, \"yaxis\": {\"automargin\": true, \"gridcolor\": \"white\", \"linecolor\": \"white\", \"ticks\": \"\", \"zerolinecolor\": \"white\", \"zerolinewidth\": 2}}}},\n",
       "                        {\"responsive\": true}\n",
       "                    ).then(function(){\n",
       "                            \n",
       "var gd = document.getElementById('8959f861-93a0-4082-aa59-4dcb3eeda155');\n",
       "var x = new MutationObserver(function (mutations, observer) {{\n",
       "        var display = window.getComputedStyle(gd).display;\n",
       "        if (!display || display === 'none') {{\n",
       "            console.log([gd, 'removed!']);\n",
       "            Plotly.purge(gd);\n",
       "            observer.disconnect();\n",
       "        }}\n",
       "}});\n",
       "\n",
       "// Listen for the removal of the full notebook cells\n",
       "var notebookContainer = gd.closest('#notebook-container');\n",
       "if (notebookContainer) {{\n",
       "    x.observe(notebookContainer, {childList: true});\n",
       "}}\n",
       "\n",
       "// Listen for the clearing of the current output cell\n",
       "var outputEl = gd.closest('.output');\n",
       "if (outputEl) {{\n",
       "    x.observe(outputEl, {childList: true});\n",
       "}}\n",
       "\n",
       "                        })\n",
       "                };\n",
       "                });\n",
       "            </script>\n",
       "        </div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "frame = OrbitPlotter3D()\n",
    "\n",
    "frame.set_attractor(Earth)\n",
    "frame.plot_trajectory(positions, label=\"ISS\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Error checking"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def state_to_vector(ss):\n",
    "    r, v = ss.rv()\n",
    "    x, y, z = r.to(u.km).value\n",
    "    vx, vy, vz = v.to(u.km / u.s).value\n",
    "    return np.array([x, y, z, vx, vy, vz])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "k = Earth.k.to(u.km ** 3 / u.s ** 2).value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "rtol = 1e-13\n",
    "full_periods = 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([ 8.59072560e+02, -4.13720368e+03,  5.29556871e+03,  7.37289205e+00,\n",
       "         2.08223573e+00,  4.39999794e-01]), <Quantity 13892.42425291 s>)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u0 = state_to_vector(iss)\n",
    "tf = ((2 * full_periods + 1) * iss.period / 2)\n",
    "\n",
    "u0, tf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "iss_f_kep = iss.propagate(tf, rtol=1e-18)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "r, v = cowell(iss.attractor.k, iss.r, iss.v, [tf] * u.s, rtol=rtol)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "iss_f_num = Orbit.from_vectors(Earth, r[0], v[0], iss.epoch + tf)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(<Quantity [ -835.92108005,  4151.60692532, -5303.60427969] km>,\n",
       " <Quantity [ -835.92108005,  4151.60692532, -5303.60427969] km>)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "iss_f_num.r, iss_f_kep.r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "assert np.allclose(iss_f_num.r, iss_f_kep.r, rtol=rtol, atol=1e-08 * u.km)\n",
    "assert np.allclose(iss_f_num.v, iss_f_kep.v, rtol=rtol, atol=1e-08 * u.km / u.s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "assert np.allclose(iss_f_num.a, iss_f_kep.a, rtol=rtol, atol=1e-08 * u.km)\n",
    "assert np.allclose(iss_f_num.ecc, iss_f_kep.ecc, rtol=rtol)\n",
    "assert np.allclose(iss_f_num.inc, iss_f_kep.inc, rtol=rtol, atol=1e-08 * u.rad)\n",
    "assert np.allclose(iss_f_num.raan, iss_f_kep.raan, rtol=rtol, atol=1e-08 * u.rad)\n",
    "assert np.allclose(iss_f_num.argp, iss_f_kep.argp, rtol=rtol, atol=1e-08 * u.rad)\n",
    "assert np.allclose(iss_f_num.nu, iss_f_kep.nu, rtol=rtol, atol=1e-08 * u.rad)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numerical validation\n",
    "\n",
    "According to [Edelbaum, 1961], a coplanar, semimajor axis change with tangent thrust is defined by:\n",
    "\n",
    "$$\\frac{\\operatorname{d}\\!a}{a_0} = 2 \\frac{F}{m V_0}\\operatorname{d}\\!t, \\qquad \\frac{\\Delta{V}}{V_0} = \\frac{1}{2} \\frac{\\Delta{a}}{a_0}$$\n",
    "\n",
    "So let's create a new circular orbit and perform the necessary checks, assuming constant mass and thrust (i.e. constant acceleration):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "ss = Orbit.circular(Earth, 500 * u.km)\n",
    "tof = 20 * ss.period\n",
    "\n",
    "ad = constant_accel_factory(1e-7)\n",
    "\n",
    "r, v = cowell(ss.attractor.k, ss.r, ss.v, [tof] * u.s, ad=ad)\n",
    "\n",
    "ss_final = Orbit.from_vectors(Earth, r[0], v[0], ss.epoch + tof)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$2.989621 \\times 10^{-6} \\; \\mathrm{\\frac{km}{m}}$"
      ],
      "text/plain": [
       "<Quantity 2.98962096e-06 km / m>"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "da_a0 = (ss_final.a - ss.a) / ss.a\n",
    "da_a0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$0.0029960538 \\; \\mathrm{}$"
      ],
      "text/plain": [
       "<Quantity 0.00299605>"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dv_v0 = abs(norm(ss_final.v) - norm(ss.v)) / norm(ss.v)\n",
    "2 * dv_v0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.allclose(da_a0, 2 * dv_v0, rtol=1e-2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This means **we successfully validated the model against an extremely simple orbit transfer with approximate analytical solution**. Notice that the final eccentricity, as originally noticed by Edelbaum, is nonzero:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$6.6621427 \\times 10^{-6} \\; \\mathrm{}$"
      ],
      "text/plain": [
       "<Quantity 6.66214272e-06>"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ss_final.ecc"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## References\n",
    "\n",
    "* [Edelbaum, 1961] \"Propulsion requirements for controllable satellites\""
   ]
  }
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        "encoding": "base64",
        "path": [
         "_data",
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        ]
       },
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        "encoding": "base64",
        "path": [
         "_data",
         0,
         "y",
         "value"
        ]
       },
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