added tables and description

Tutorials/three-body-stability.ipynb

@@ -33,7 +33,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## The old standard ([Holman & Wiegert 1999](https://ui.adsabs.harvard.edu/abs/1999AJ....117..621H/abstract))\n",
+    "## Updating the standard ([Holman & Wiegert 1999](https://ui.adsabs.harvard.edu/abs/1999AJ....117..621H/abstract))\n",
     "\n",
     "From ${\\sim}1980-2000$, the growth of computational power allowed researchers to better understand the motion of three bodies.  In the 1980s, most numerical simulations were limited to ${\\sim}10^3\\ T_{\\rm bin}$ (scaled by the binary orbital period $T_{\\rm bin}$) in simulation time due to the excessive wall time required for the computations.  Many researchers turned to dynamical indicators of chaos (e.g., [Benettin et al. 1980](https://ui.adsabs.harvard.edu/abs/1980Mecc...15....9B/abstract), [Lecar et al. 1992](https://ui.adsabs.harvard.edu/abs/1992AJ....104.1230L/abstract),  [Smith & Szebehely 1993](https://ui.adsabs.harvard.edu/abs/1993CeMDA..56..409S/abstract), [Froeschle et al. 1997](https://ui.adsabs.harvard.edu/abs/1997CeMDA..67...41F/abstract)) to identify *possible routes* to unstable orbits. \n",
     "\n",
@@ -55,7 +55,7 @@
     "   - $a_p > a_c$ are unbound within a time $t<10^4\\ T_{\\rm bin}$.\n",
     "6. The planetary and binary orbits are initially aligned $(\\omega_p = \\omega_{\\rm bin})$, where the binary begins at either periastron $(f_{\\rm bin} = 0^\\circ)$ or apastron $(f_{\\rm bin} = 180^\\circ)$.\n",
     "\n",
-    "These are not bad assumptions considering the computing power and knowledge of planetary systems during the late 1990s.  For a given mass ratio $\\mu$, and binary eccentricity $e_{\\rm bin}$, an initial condition $(\\rho,\\ f_p)$ is evaluated.  The semimajor axis ratio range differs for S-Type $(0.02\\leq \\rho \\leq 0.5)$ and P-Type $(1\\leq \\rho \\leq 5)$ (see Figure {numref}`{number}<planet_configs>`).  But eight equidistant values of the planetary true anomaly ($f_p = 0^\\circ - 315^\\circ;\\ 45^\\circ$ increments) are used.\n",
+    "These are not bad assumptions considering the computing power and knowledge of planetary systems during the late 1990s.  For a given mass ratio $\\mu$, and binary eccentricity $e_{\\rm bin}$, an initial condition $(\\rho,\\ f_p)$ is evaluated.  The semimajor axis ratio range differs for S-Type $(0.02\\leq \\rho \\leq 0.5)$ and P-Type $(1\\leq \\rho \\leq 5)$ (see Figure {numref}`{number}<planet_configs>`).  But eight equidistant values of the planetary true anomaly ($f_p = 0^\\circ - 315^\\circ;\\ 45^\\circ$ increments) are used.  *Note that the planetary mean anomaly $M$ and true anomaly $f$ are interchangeable for **circular** orbits.*\n",
     "\n",
     "```{figure-md} planet_ICs\n",
     "<img src=\"threebody_stability/HW_Table1.png\" alt=\"HW Table 1\"  width=\"400px\">\n",
@@ -69,7 +69,44 @@
     "<img src=\"threebody_stability/HW_4dim.png\" alt=\"HW dimensions\"  width=\"400px\">\n",
     "\n",
     "A 4D parameter space abstracted into two nested 2D grids.\n",
-    "```"
+    "```\n",
+    "\n",
+    "The results of each nested grid can be summarized by identifying a critical semimajor axis ratio $\\rho_{\\rm cr}$ for a given $(\\mu,\\ e_{\\rm bin})$ combination.  The critical semimajor axis ratio $\\rho_{\\rm cr}$ is defined as the boundary between stable and unstable initial conditions $(\\rho,\\ f_p)$:\n",
+    "\n",
+    "- one side of the boundary has a planet that survives the full simulation time for all values of $f_p$\n",
+    "- the other side has simulations that terminate early due to close approaches (i.e., scatters) with either star or an escape from the system.\n",
+    "\n",
+    "Table {numref}`{number}<stability_grid1>` shows the results for $\\mu = e_{\\rm bin} = 0.5$ from [Holman & Wiegert 1999](https://ui.adsabs.harvard.edu/abs/1999AJ....117..621H/abstract) for S-Type planets, where the $\"+\"$ symbols denote stable (i.e., no scatters or escapes).\n",
+    "\n",
+    "```{figure-md} stability_grid1\n",
+    "<img src=\"threebody_stability/HW_Table2.png\" alt=\"HW Table 2\"  width=\"400px\">\n",
+    "\n",
+    "Planet survival results for $\\mu = e_{\\rm bin} = 0.5$ from [Holman & Wiegert 1999](https://ui.adsabs.harvard.edu/abs/1999AJ....117..621H/abstract) for S-Type planets.\n",
+    "```\n",
+    "\n",
+    "Table {numref}`{number}<stability_grid2>` shows data that has been reduced, where each cell in the table represents the critical semimajor axis $\\rho_{\\rm cr}$ determined through a smaller grid similar to Table {numref}`{number}<stability_grid1>`.\n",
+    "\n",
+    "```{figure-md} stability_grid2\n",
+    "<img src=\"threebody_stability/HW_Table3.png\" alt=\"HW Table 3\"  width=\"400px\">\n",
+    "\n",
+    "Planetary critical semimajor axis $\\rho_{\\rm cr}$ from [Holman & Wiegert 1999](https://ui.adsabs.harvard.edu/abs/1999AJ....117..621H/abstract) for S-Type planets.\n",
+    "```\n",
+    "\n",
+    "From Table {numref}`{number}<stability_grid2>`, [Holman & Wiegert 1999](https://ui.adsabs.harvard.edu/abs/1999AJ....117..621H/abstract) perform a least-squares fit given the trial function:\n",
+    "\n",
+    "\\begin{align}\n",
+    "\\rho_{\\rm cr} = c_1 + c_2\\mu + c_3e_{\\rm bin} + c_4\\mu e_{\\rm bin} + c_5 e_{\\rm bin}^2 + c_6 \\mu e_{\\rm bin}^2.\n",
+    "\\end{align}\n",
+    "\n",
+    "The least-squares fitting procedure identifies values of $c_1-c_6$ that minimizes $\\chi^2$ (e.g., the residuals or cost function).  \n",
+    "\n",
+    "The above procedure has merits in that it will accurately reproduce the critical semimajor axis $\\rho_{\\rm cr}$ for other parts of the $(\\mu,\\ e_{\\rm bin})$ parameter space through an easy-to-use polynomial.  However, larger discrepancies appear when extrapolating to broad regimes of mass ration $\\mu$ (e.g., $\\mu \\leq 0.1$ and $\\mu \\geq 0.9$).  There can be significant differences if the planetary orbit is misaligned with the host binary, which is expected from disk observations in binaries (e.g., [Monin et al. (2006)](https://ui.adsabs.harvard.edu/abs/2006A%26A...446..201M/abstract)).\n",
+    "\n",
+    "[Quarles et al. (2020)](https://ui.adsabs.harvard.edu/abs/2020AJ....159...80Q/abstract) addresses these issues in two ways:\n",
+    "\n",
+    "1. expand the simulations to cover a wider range in $\\mu$ and increase the grid resolution,\n",
+    "2. evaluate the simulations for a range of planet inclinations.\n",
+    "\n"
    ]
   },
   {