{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Reproduction of arXiv:1506.04175\n", "\n", "In this notebook, we'll go through [Burke et al. (2015)](http://arxiv.org/abs/1506.04175) and reproduce their findings." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": true }, "outputs": [], "source": [ "from __future__ import division, print_function\n", "\n", "%matplotlib inline\n", "%config InlineBackend.figure_format = \"retina\"\n", "\n", "from matplotlib import rcParams\n", "rcParams[\"savefig.dpi\"] = 100\n", "rcParams[\"font.size\"] = 20" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": true }, "outputs": [], "source": [ "import emcee\n", "import numpy as np\n", "import pandas as pd\n", "from itertools import izip\n", "from scipy.stats import gamma\n", "import matplotlib.pyplot as pl\n", "from scipy.optimize import minimize" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "# These helper functions let us download and cache\n", "# the catalogs from the Exoplanet Archive.\n", "from exopop2.data import get_kois, get_stlr" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Datasets\n", "\n", "To start, we'll download the stellar target catalog and the relevant KOI list." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Selected 91446 targets after cuts\n" ] } ], "source": [ "stlr = get_stlr()\n", "\n", "# Select G and K dwarfs.\n", "m = (4200 <= stlr.teff) & (stlr.teff <= 6100)\n", "m &= stlr.radius <= 1.15\n", "\n", "# Remove targets with missing mass estimates.\n", "m &= np.isfinite(stlr.mass)\n", "\n", "# Only include stars with sufficient data coverage.\n", "m &= stlr.dataspan > 365.25*2.\n", "m &= stlr.dutycycle > 0.6\n", "m &= stlr.rrmscdpp07p5 < 1000 \n", "\n", "stlr = stlr[m]\n", "\n", "print(\"Selected {0} targets after cuts\".format(len(stlr)))" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { 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"pl.xlim(6100, 4200)\n", "pl.ylim(pl.gca().get_ylim()[::-1]) # because astronomy...\n", "pl.ylabel(\"$\\log g$\");\n", "pl.xlabel(\"$T_\\mathrm{eff}$\");" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": true }, "outputs": [], "source": [ "period_rng = (50, 300)\n", "rp_rng = (0.75, 2.5)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Selected 155 KOIs after cuts\n" ] } ], "source": [ "kois = get_kois()\n", "\n", "# Join on the stellar list.\n", "kois = pd.merge(stlr[[\"kepid\"]], kois, on=\"kepid\")\n", "\n", "# Only select the KOIs in the relevant part of parameter space.\n", "m = (kois.koi_disposition == \"CANDIDATE\") | (kois.koi_disposition == \"CONFIRMED\")\n", "m &= (period_rng[0] <= kois.koi_period) & (kois.koi_period <= period_rng[1])\n", "m &= (rp_rng[0] <= kois.koi_prad) & (kois.koi_prad <= rp_rng[1])\n", "\n", "kois = kois[m]\n", "\n", "print(\"Selected {0} KOIs after cuts\".format(len(kois)))" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAABIQAAAMuCAYAAABo8tgwAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAAewgAAHsIBbtB1PgAAIABJREFUeJzs3d1vZOedJ/Zv2zLQmZ2x2t3JBIKNxKI8ubZe/AfEkmaw\n", "ubQkOwnQd5bkCTDdWsyKtIEArrmySM9FdwvIWNLcNRDEUssXuVms1TJyb7041yNR2oUMI8Y21TIW\n", "C2IHVufiOVyy2WTVqapzqs7L5wMUWM1+qviweFis86vfSwIAAAAAAAAAAAAAAAAAAAAAAAAAAAAA\n", "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA\n", "AAAAAAAAAAAAAHCyc9Vl3d5N8nqSp2quP5fk6SRvJHmnrU0BAAAAyzuz7g2QJHkuyfNJHj72+c+S\n", "/DwlyPL2ivf0aZL7j/z7vSQfJbmVZDfJhZQg0EZ1efDI7R5N8vGqNgoAAADQJ0+nBFBupQR+/nWS\n", "71QfX0/y+ZHLB0keX+HePj329etc/m2SL69wjwAAAAC9spkSRPnJlDX3J/lZ7g66vNT+1pLMFxD6\n", "pyTfX9G+AAAAAHrpuSR7Sb5Zc/2zWX1QaC8lyPNsSrbSr1Mymf5YffynJP+Q5Nsr2AsAAABAr22k\n", 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get_delta(k, c=1.0874, s=1.0187):\n", " delta_max = k*k * (c + s*k)\n", " return 0.84 * delta_max\n", "\n", "cdpp_cols = [k for k in stlr.keys() if k.startswith(\"rrmscdpp\")]\n", "cdpp_vals = np.array([k[-4:].replace(\"p\", \".\") for k in cdpp_cols], dtype=float)\n", "def get_snr(star, tau, k):\n", " cdpp = np.array(star[cdpp_cols], dtype=float)\n", " sigma = np.interp(tau, cdpp_vals, cdpp)\n", " return get_delta(k) * 1e6 / sigma\n", "\n", "def get_mes(star, period, rp, tau, re=0.009171):\n", " k = rp * re / star.radius\n", " snr = get_snr(star, tau, k)\n", " ntrn = star.dataspan * star.dutycycle / period \n", " return snr * np.sqrt(ntrn)\n", "\n", "pgam = gamma(4.65, loc=0., scale=0.98)\n", "mesthres_cols = [k for k in stlr.keys() if k.startswith(\"mesthres\")]\n", "mesthres_vals = np.array([k[-4:].replace(\"p\", \".\") for k in mesthres_cols],\n", " dtype=float)\n", "def get_pdet(star, aor, period, rp, e):\n", " tau = get_duration(period, aor, e)\n", " mes = get_mes(star, period, rp, tau)\n", " mest = np.interp(tau, mesthres_vals,\n", " np.array(star[mesthres_cols],\n", " dtype=float))\n", " x = mes - 4.1 - (mest - 7.1)\n", " return pgam.cdf(x)\n", "\n", "def get_pwin(star, period):\n", " M = star.dataspan / period\n", " f = star.dutycycle\n", " omf = 1.0 - f\n", " pw = 1 - omf**M - M*f*omf**(M-1) - 0.5*M*(M-1)*f*f*omf**(M-2)\n", " msk = (pw >= 0.0) * (period < star.dataspan)\n", " return pw * msk\n", "\n", "def get_pgeom(aor, e):\n", " return 1. / (aor * (1 - e*e)) * (aor > 1.0) + (aor < 1.0)\n", "\n", "def get_completeness(star, period, rp, e, with_geom=True):\n", " aor = get_a(period, star.mass) / star.radius\n", " pdet = get_pdet(star, aor, period, rp, e)\n", " pwin = get_pwin(star, period)\n", " if not with_geom:\n", " return pdet * pwin\n", " pgeom = get_pgeom(aor, e)\n", " return pdet * pwin * pgeom" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ 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[R_\\oplus]$\");" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# Compute the mean completeness in the range.\n", "period = np.linspace(period_rng[0], period_rng[1], 50)\n", "rp = np.linspace(rp_rng[0], rp_rng[1], 57)\n", "period_grid, rp_grid = np.meshgrid(period, rp, indexing=\"ij\")\n", "comp = np.zeros_like(period_grid)\n", "\n", "for _, star in stlr.iterrows():\n", " comp += get_completeness(star, period_grid, rp_grid, 0.0, with_geom=True)\n", "\n", "comp /= len(stlr)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAABIQAAAMuCAYAAABo8tgwAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAAewgAAHsIBbtB1PgAAIABJREFUeJzs3Xd8W9XZB/Df1faUV2I7O84mey+HkJ3YmbYz6Usg7FIK\n", "HXRR2gLlLbRv6YASVgmkJdPOtLNDyA5JyCCT7G07jveUrPH+IS/ZsiXZx9K1/ft+PhBH9+jRc+Vr\n", 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"pl.plot(kois.koi_period, kois.koi_prad, \"ok\", mec=\"none\", ms=4)\n", "\n", "pl.pcolor(period_grid, rp_grid, comp, cmap=\"BuGn\")\n", "c = pl.contour(period_grid, rp_grid, comp, [0.0003, 0.001, 0.003, 0.01],\n", " colors=\"k\", alpha=0.8)\n", "pl.clabel(c, fontsize=12, inline=1, fmt=\"%.4f\") \n", "pl.xlabel(\"period [days]\")\n", "pl.ylabel(\"$R_p \\, [R_\\oplus]$\");" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def population_model(theta, period, rp):\n", " lnf0, beta, alpha = theta\n", " v = np.exp(lnf0) * np.ones_like(period)\n", " for x, rng, n in izip((period, rp), (period_rng, rp_rng),\n", " (beta, alpha)):\n", " n1 = n + 1\n", " v *= x**n*n1 / (rng[1]**n1-rng[0]**n1)\n", " return v\n", "\n", "koi_periods = np.array(kois.koi_period)\n", "koi_rps = np.array(kois.koi_prad)\n", "vol = np.diff(period_grid, axis=0)[:, :-1] * np.diff(rp_grid, axis=1)[:-1, :]\n", "def lnlike(theta):\n", " pop = population_model(theta, period_grid, rp_grid)\n", " pop *= comp\n", " pop = 0.5 * (pop[:-1, :-1] + pop[1:, 1:])\n", " norm = np.sum(pop * vol)\n", " ll = np.sum(np.log(population_model(theta, koi_periods, koi_rps))) - norm\n", " return ll if np.isfinite(ll) else -np.inf\n", "\n", "def lnprob(theta):\n", " return lnlike(theta)\n", "\n", "def nll(theta):\n", " ll = lnlike(theta)\n", " return -ll if np.isfinite(ll) else 1e15" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [], "source": [ "theta_0 = np.array([np.log(0.75*len(stlr)), -0.53218, -0.5])\n", "r = minimize(nll, theta_0, method=\"L-BFGS-B\",\n", " bounds=[(None, None), (None, None), (None, None)])" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [], "source": [ "ndim, nwalkers = len(r.x), 16\n", "pos = [r.x + 1e-5 * np.random.randn(ndim) for i in range(nwalkers)]\n", "sampler = emcee.EnsembleSampler(nwalkers, ndim, lnprob)" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": true }, "outputs": [], "source": [ "pos, _, _ = sampler.run_mcmc(pos, 200)\n", "sampler.reset()\n", "pos, _, _ = sampler.run_mcmc(pos, 2000)" ] }, { "cell_type": "code", "execution_count": 78, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAABHQAAALoCAYAAAAZeKK0AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAAewgAAHsIBbtB1PgAAIABJREFUeJzsvWmwZcdZpvvsU3VqUM0qlVSaVaVZlqzZsoxHSZ4wNsby\n", "1BgbA7bBjBe6Dc0vmoh7AzvcEUTc6AYbmttBXC5cD0zR9KU9NDMN2JJtDBiDJdnGs2xr8qSpTt0f\n", "305W7tyZuXJNe699zvtEnDh7ryFX7rUyc2W++X1fghBCCCGEEEIIIYQQQgghhBBCCCGEEEIIIYQQ\n", "QgghhBBCCCGEEEIIIYQQQgghhBBCCCGEEEIIIYQQQgghhBBCCCGEEEIIIYQQQgghhBBCCCGEEEII\n", "IYQQQgghhBBCCCGEEEIIIYQQQgghhBBCCCGEEEIIIYQQQgghhBBCCCGEEEIIIYQQQgghhBBCCCGE\n", "EEIIIYQQQgghhBBCCCGEEEIIIYQQQgghhBBCCCGEEEIIIYQQQgghhBBCCCGEEEIIIYQQQgghhBBC\n", 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"AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA\n", "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADABP0LJMGy\n", "j2cAH+4AAAAASUVORK5CYII=\n" ], "text/plain": [ "" ] }, "metadata": { "image/png": { "height": 372, "width": 570 } }, "output_type": "display_data" } ], "source": [ "pl.plot(sampler.chain[:, :, 2].T, \"k\", alpha=0.5);" ] }, { "cell_type": "code", "execution_count": 79, "metadata": { "collapsed": false }, "outputs": [], "source": [ "samps = sampler.chain[:, ::5, :]\n", "samps = samps.reshape((-1, samps.shape[-1]))\n", "pop = np.empty((len(samps), period_grid.shape[0], period_grid.shape[1]))\n", "gamma_earth = np.empty((len(samps)))\n", "for i, p in enumerate(samps):\n", " pop[i] = population_model(p, period_grid, rp_grid)\n", " gamma_earth[i] = population_model(p, 365.25, 1.0)" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": true }, "outputs": [], "source": [ "pop_comp = pop * comp[None, :, :]" 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563 } }, "output_type": "display_data" } ], "source": [ "dx = 0.25\n", "x = np.arange(rp_rng[0], rp_rng[1] + dx, dx)\n", "\n", "pop_rp = 0.5 * (pop_comp[:, 1:] + pop_comp[:, :-1])\n", "pop_rp = np.sum(pop_rp * np.diff(period)[None, :, None], axis=1)\n", "a, b, c, d, e = np.percentile(pop_rp * dx, [2.5, 16, 50, 84, 97.5], axis=0)\n", "\n", "pl.fill_between(rp, a, e, color=\"k\", alpha=0.1, edgecolor=\"none\")\n", "pl.fill_between(rp, b, d, color=\"k\", alpha=0.3, edgecolor=\"none\")\n", "pl.plot(rp, c, \"k\", lw=2)\n", "\n", "n, _ = np.histogram(koi_rps, x)\n", "pl.errorbar(0.5*(x[:-1]+x[1:]), n, yerr=np.sqrt(n), fmt=\".k\", capsize=0)\n", "pl.xlim(rp_rng[0], rp_rng[1])\n", "pl.xlabel(\"$R_p\\,[R_\\oplus]$\")\n", "pl.ylabel(\"number of planets\");" ] }, { "cell_type": "code", "execution_count": 74, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAABG0AAAMuCAYAAACjBf6LAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", 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"pl.xlabel(\"$P\\,[\\mathrm{days}]$\")\n", "pl.ylabel(\"number of planets\");" ] }, { "cell_type": "code", "execution_count": 84, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAABEIAAALoCAYAAAB8lVDLAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAAewgAAHsIBbtB1PgAAIABJREFUeJzs3W2MXedhJ/Y/bWmRpAE5kb3eGFlgScpA0c22sGi5W8Ao\n", "FhCpBF6k7a5JSUE36Jc1ySb7JYuIlNWitQy0kZgY/VKkIiWjxQZoN+JI2q7brSOSVj90EWAtUnKK\n", "LPqBL1JaNsa6ISkqrbMbhTP98JzBHF7embnn3HvuuXPP7wccnDOc523kK4/On89LAgAAAAAAAAAA\n", "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA\n", "AAAAAAAAAAAAAAAAAAAAAAAAAIvvSpLjLeqcT3J0wvIrSY4lWU1yuWFfG45U9a8luV3dL6T52AEA\n", "AICBOpFkLcmLDevdqeptXJdTQoqzSU4nOZPkXJKLSa7Xyt1Ksr9hX4dqbTxbq78vJYi5Vn1PIAIA\n", "AABsaSWbAcW0Qcgk11tJ9jbs51hV950d6p6vyp1t2D4AAAAwEKuZTxByNclXW4zvUDZnkUwSoFyu\n", "yp9q0RcAAACwxI7k/rCiaRByOyXcOJ4yG+OdlMDiXnW/muTlJE9MMcbrVXtfmbD8vmz+PI9N0S8A\n", 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