{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Cloud evolution" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Evolution equations\n", "\n", "The evolution of the cloud (boson mass $m_b = \\hbar /(\\mu c^2)$) is described within the quasiadiabatic approximation by a system of coupled differential equations (see e.g. [arXiv:1411.0686](http://stacks.iop.org/0264-9381/32/i=13/a=134001?key=crossref.8a4e2e36ca36fb28e1d0a6868573e646)).\n", "These determine the rate of change of the cloud's mass ($\\dot{M}_c= \\dot{E}_c/c^2$) and angular momentum ($\\dot{J}_c$) in terms of the power ($P_{gw}$) and angular momentum ($\\dot{J}_{gw} = P_{gw}/\\omega_R$) radiated in GWs and the black-hole parameters ($M, J$), which is possibly affected by accretion ($\\dot{M}_{\\rm Acc}, \\dot{J}_{\\rm Acc}$).\n", "\n", "\\begin{align}\n", "\\dot{M}_c^{\\rm (sr)} &= 2 M_c \\omega_I\\\\\n", "\\dot{M} &= - \\dot{M}_c^{\\rm (sr)} + \\dot{M}_{\\rm Acc}\\\\\n", "\\dot{M} + \\dot{M}_c &= - P_{gw}/c^2 + \\dot{M}_{\\rm Acc} \\\\\n", "\\dot{J} &= - m c^2 \\dot{M}_c^{\\rm (sr)} / \\omega_R + \\dot{J}_{\\rm Acc}\\\\\n", "\\dot{J} + \\dot{J}_c &= - m P_{gw}/\\omega_R + \\dot{J}_{\\rm Acc}\\, ,\n", "\\end{align}\n", "\n", "where $\\dot{M}_c^{\\rm (sr)}$ is the change in the cloud mass due to superradiance only ($\\dot{M}_c = \\dot{M}_c^{\\rm (sr)} - P_{gw}/c^2$), and $\\omega = \\omega_R + i \\omega_I$ is the boson field frequency. In the case of no accretion, this simplifies to\n", "\n", "\\begin{align}\n", "\\dot{M} &= - 2 M_c \\omega_I\\\\\n", "\\dot{M} + \\dot{M}_c &= - P_{gw}/c^2 \\\\\n", "\\dot{J} &=- 2 m c^2 M_c \\omega_I / \\omega_R \\\\\n", "\\dot{J} + \\dot{J}_c &= - m P_{gw}/\\omega_R \\, .\n", "\\end{align}\n", "\n", "we can solve these equations numerically, starting from some initial parameters $(M_0, J_0, M_{c,0}, J_{c,0})$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Numerical setup\n", "\n", "Set up the above system of DE's in a way ammenable to numerical treatment." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### No accretion\n", "\n", "Define dimensionless $(x, y, j^x, j^y, {\\cal P}, w, \\tau)$ quantities such that\n", "\\begin{align}\n", "M &= \\alpha x\\\\\n", "M_c &= \\alpha y \\\\\n", "J &= \\beta j^x\\\\\n", "J_c &= \\beta j^y\\\\\n", "P_{gw} &= \\gamma {\\cal P}\\\\\n", "\\omega &= w \\mu\\\\\n", "t &= \\epsilon \\tau\\, .\n", "\\end{align}\n", "We will make sure these are dimensionless by defining the dimensionful factors\n", "\\begin{align}\n", "\\alpha &= M_0 \\\\\n", "\\beta &= M_0 c^2 \\mu^{-1} \\\\\n", "\\gamma &= M_0 c^2 \\mu \\\\\n", "\\epsilon &= \\mu^{-1}\\, ,\n", "\\end{align}\n", "and letting $\\mu = \\hbar/(m_b c^2)$ be the Compton frequency associated with the boson rest mass, $m_b$.\n", "We will also let $M_0$ be the *initial* black-hole mass.\n", "Note that then we have:\n", "\\begin{align}\n", "a &= \\frac{c j_x \\epsilon}{x} \\\\\n", "\\chi &= \\frac{j_x}{x^2} \\frac{\\epsilon}{T_0}\\, ,\n", "\\end{align}\n", "where we have defined the horizon crossing time $T_0 = G M_0/c^3$.\n", "Also note that the SR condition is saturated when\n", "\\begin{equation}\n", "\\left( \\frac{2}{m} \\frac{T_0}{\\epsilon}\\right)^2 = \\frac{j_x}{x^2\\left(mx-j_x\\right)}\n", "\\end{equation}\n", "\n", "With the above notation, the BHB evolution equations become:\n", "\\begin{align}\n", "x' &= -2 w_I y\\\\\n", "x' + y' &= -{\\cal P}\\\\\n", "(j^x)' &= -2 m w_I w_R^{-1} y\\\\\n", "(j^x)' + (j^y)' &= -m w_R^{-1} {\\cal P}\n", "\\end{align}\n", "for $m$ the boson magnetic quantum number ($m=1$ for dominant scalar level), and where the prime denotes ${\\rm d}/{\\rm d}\\tau$.\n", "We may turn these into finite-difference equations to obtain:\n", "\\begin{align}\n", "x_{i+1} &= x_i -2 w^I_i y_i \\delta \\tau \\\\\n", "y_{i+1} &= y_i + 2 w^I_i y_i \\delta\\tau -{\\cal P}_i \\delta \\tau\\\\\n", "j^x_{i+1} &= j^x_i -2 m (w^I_i/ w^R_i) y_i \\delta \\tau\\\\\n", "j^y_{i+1} &= j^y_i + m \\left( 2 w^I_i y_i - {\\cal P}_i\\right)\\delta \\tau / w^R_i\n", "\\end{align}\n", "These can be evolved from $(x_0, y_0, j^x_0, j^y_0)$ until the SR spin threshold is reached.\n", "The final state can then be converted into physical units using the scalings defined above." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Adding back accretion" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the presence of accretion, we may define the dimensioless accretion mass and angular momentum rates:\n", "\n", "\\begin{align}\n", "{\\cal M}_a ' &\\equiv c^2 \\gamma^{-1} \\dot{M}_{\\rm Acc} = (M_0 \\mu)^{-1} \\, \\dot{M}_{\\rm Acc} \\\\\n", "{\\cal J}_a ' &\\equiv\\, \\epsilon \\beta^{-1} \\dot{J}_{\\rm Acc} = (M_0 c^2)^{-1} \\dot{J}_{\\rm Acc} \\, ,\n", "\\end{align}\n", "\n", "so that the evolution equations become:\n", "\n", "\\begin{align}\n", "x' &= -2 w_I y + {\\cal M}_a '\\\\\n", "x' + y' &= -{\\cal P} + {\\cal M}_a '\\\\\n", "(j^x)' &= -2 m w_I w_R^{-1} y + {\\cal J}_a' \\\\\n", "(j^x)' + (j^y)' &= -m w_R^{-1} {\\cal P} + {\\cal J}_a' \\, .\n", "\\end{align}\n", "\n", "The corresponding finite-difference equations are:\n", "\\begin{align}\n", "x_{i+1} &= x_i -2 w^I_i y_i \\delta \\tau + {\\cal M}_a' \\delta\\tau \\\\\n", "y_{i+1} &= y_i + 2 w^I_i y_i \\delta\\tau -{\\cal P}_i \\delta \\tau \\\\\n", "j^x_{i+1} &= j^x_i -2 m (w^I_i/ w^R_i) y_i \\delta \\tau + {\\cal J}_a' \\delta\\tau\\\\\n", "j^y_{i+1} &= j^y_i + m \\left( 2 w^I_i y_i - {\\cal P}_i\\right)\\delta \\tau / w^R_i \\, ,\n", "\\end{align}\n", "\n", "assuming that the accretion rate is constant.\n", "Note that accretion only affects the BH quantities ($x, j^x$) by construction, because we assumed no direct interaction between the cloud and the accretion disk." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Evolving forward" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "jupyter": { "outputs_hidden": false } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Populating the interactive namespace from numpy and matplotlib\n" ] } ], "source": [ "%pylab inline\n", "import gwaxion" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Final state" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The `gwaxion` package can use the above equations (ignoring cloud angular momentum and GW power) to automatically obtain the final (post-superradiance) properities of a given black-hole--boson (BHB) system. Simply make sure this feature is turned on through the `evolve` argument when creating a `BosonCloud` object." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here's an example for the $\\ell=m=1$ cloud from pre-superradiance BH with $60\\, M_\\odot$ and $\\chi=0.7$, and a boson such that $\\alpha=0.176$." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "c = gwaxion.BosonCloud.from_parameters(1, 1, 0, evolve=True, m_bh=60,\n", " chi_bh=0.7, alpha=0.176)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Print some post-superradiance parameters, which will be internally computed using the above DEs." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Final BH mass: 58.89 Msun\n", "Final BH spin: 0.617\n", "Final BH J: 1.89e+45\n", "Final alpha: 0.173\n", "Final cloud mass: 1.11 Msun\n" ] } ], "source": [ "l = [\n", " \"Final BH mass: %.2f Msun\" % c.bhb_final.bh.mass_msun,\n", " \"Final BH spin: %.3f\" % c.bhb_final.bh.chi,\n", " \"Final BH J: %.2e\" % c.bhb_final.bh.angular_momentum,\n", " \"Final alpha: %.3f\" % c.bhb_final.alpha,\n", " \"Final cloud mass: %.2f Msun\" % c.mass_msun\n", "]\n", "print('\\n'.join(l))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If the DE evolution is turned off with `evolve=False`, then the code approximates the final BH parameters using Eqs. (25) and (26) in [arXiv:1706.06311](https://arxiv.org/abs/1706.06311), with constant $\\alpha$ corresponding to the initial BH parameters.\n", "\n", "The result is slightly different:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "jupyter": { "outputs_hidden": false }, "scrolled": true }, "outputs": [], "source": [ "c_approx = gwaxion.BosonCloud.from_parameters(1, 1, 0, evolve=False, m_bh=60,\n", " chi_bh=0.7, alpha=0.176)" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Final BH mass: 59.21 Msun\n", "Final BH spin: 0.624\n", "Final BH J: 1.93e+45\n", "Final alpha: 0.174\n", "Final cloud mass: 0.79 Msun\n" ] } ], "source": [ "l = [\n", " \"Final BH mass: %.2f Msun\" % c_approx.bhb_final.bh.mass_msun,\n", " \"Final BH spin: %.3f\" % c_approx.bhb_final.bh.chi,\n", " \"Final BH J: %.2e\" % c_approx.bhb_final.bh.angular_momentum,\n", " \"Final alpha: %.3f\" % c_approx.bhb_final.alpha,\n", " \"Final cloud mass: %.2f Msun\" % c_approx.mass_msun\n", "]\n", "print('\\n'.join(l))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Evolution" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can study the evolution of the system during the superrandiance process by looking at the properties of the BHB system over time. To do this with `gwaxion` we need to call the DE solver directly." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can start from the `BosonCloud` object we defined above, and call the internal method `_evolve_instability()`" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "bhb_final, (xs, jxs, ys, inv_wRs, wIs, sr_conds, times) = c._evolve_instability()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The `bhb_final` object contains the final state of the evolution, and should give the same results as above, e.g." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Final BH mass: 58.89 Msun\n", "Final BH spin: 0.617\n", "Final BH J: 1.89e+45\n", "Final alpha: 0.173\n" ] } ], "source": [ "l = [\n", " \"Final BH mass: %.2f Msun\" % bhb_final.bh.mass_msun,\n", " \"Final BH spin: %.3f\" % bhb_final.bh.chi,\n", " \"Final BH J: %.2e\" % bhb_final.bh.angular_momentum,\n", " \"Final alpha: %.3f\" % bhb_final.alpha,\n", "]\n", "print('\\n'.join(l))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The other outputs contain the quantities computed at each step of the DE evolution: $x$, $j^x$, $y$, $w_R^{-1}$, $w_I$, defined [above](#No-accretion); as well as the \"SR factor\", which vanishes when SR stops; and time." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For example, as time goes by, the superradiance quickly saturates until it stops:" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "image/png": 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Vm63ANcnZVV3AoYjY36DvD0h2k0l6O9AJvJRiPGZmNgrqHeNoj4iXk9d/CGyMiHuBeyU91mjFEVGUtBbYDrQDmyJit6Q1Sf0GYBtwGdAHHAauq9c3WfUmYJOkXZSD7drq3VRmZpafusEhqSMiisDFJAeaU/Q7JiK2UQ6HyrINFa8D+NO0fZPyIeA/pnl/s1PVu3eQ7v6DdC2cztJ508Z7OGZNoV4AfAf4qaSXKN869m8BJL2N8u4qs0mtd+8gV9/RzVCxRGdHG1tu6HJ4mFH/rKqvSHoImAk8ULE7qA341FgMzmw8dfcfZKhYohRwpFiiu/+gg8OMBrucIqK7RtnP8huOWfPoWjidzo42jhRLTOloo2vh9PEekllTSHWswqwVLZ03jS03dPkYh1kVB4dZHUvnTXNgmFXxjZzMzCwTB4eZmWXi4DAzs0wcHGZmlomDw8zMMnFwmJlZJg4OMzPLxMFhZmaZODjMzCwTB4eZmWXi4DAzs0wcHGZmlomDw8zMMsk1OCStkLRHUp+kdTXqJem2pP4JSUsy9L1RUkg6O89tMDOz4+UWHJLagfXASuA84CpJ51U1WwksSh6rgdvT9JU0B7gEeC6v8ZuZWW15zjiWAX0R0R8RQ8DdwKqqNquAzVHWDUyVNDNF368DNwGBmZmNqTyDYxbwfMXyQFKWps2IfSVdAbwQEY+P9oDNzKyxPO8AqBpl1TOEkdrULJd0OvB54NKGby6tprz7i7lz5zZqbmZmKeU54xgA5lQszwb2pWwzUvlbgQXA45KeTcp3Snpz9ZtHxMaIKEREYcaMGae4KWZmNizP4NgBLJK0QFIncCWwtarNVuCa5OyqLuBQROwfqW9EPBkR50TE/IiYTzlglkTEz3PcDjMzq5DbrqqIKEpaC2wH2oFNEbFb0pqkfgOwDbgM6AMOA9fV65vXWM3MLD1FTP4TkwqFQvT09Iz3MMzMJhRJvRFRqC73N8fNzCwTB4eZmWXi4DAzs0wcHGZmlomDw8zMMnFwmJlZJg4OMzPLxMFhZmaZODjMzCwTB4eZmWXi4DAzs0wcHGZmlomDw8zMMnFwmJlZJg4OMzPLxMFhZmaZODjMzCwTB4fZSejdO8j6h/vo3Ts43kMxG3O53XPcbLLq3TvI1Xd0M1Qs0dnRxpYbulg6b9p4D8tszOQ645C0QtIeSX2S1tWol6TbkvonJC1p1FfSLZL+KWn/fUlT89wGs2rd/QcZKpYoBRwplujuPzjeQzIbU7kFh6R2YD2wEjgPuErSeVXNVgKLksdq4PYUfR8ELoiIC4GfAZ/LaxvMaulaOJ3OjjbaBVM62uhaOH28h2Q2pvLcVbUM6IuIfgBJdwOrgKcq2qwCNkdEAN2SpkqaCcwfqW9EPFDRvxv4SI7bYHaCpfOmseWGLrr7D9K1cLp3U1nLyTM4ZgHPVywPAMtTtJmVsi/Ax4G/qfXmklZTnsUwd+7cLOM2a2jpvGkODGtZeR7jUI2ySNmmYV9JnweKwJZabx4RGyOiEBGFGTNmpBiumZmlkeeMYwCYU7E8G9iXsk1nvb6SrgUuBy5OdnOZmdkYyXPGsQNYJGmBpE7gSmBrVZutwDXJ2VVdwKGI2F+vr6QVwGeBKyLicI7jNzOzGnKbcUREUdJaYDvQDmyKiN2S1iT1G4BtwGVAH3AYuK5e32TV3wBOAx6UBNAdEWvy2g4zMzueWmFPT6FQiJ6envEehpnZhCKpNyIK1eW+5IiZmWXi4DAzs0wcHGZmlomDw8zMMnFwmJlZJg4OMzPLxMFhZmaZODjMzCwTB4eZmWXi4DAzs0wcHGZmlomDw2wU9e4dZP3DffTuHRzvoZjlJs/7cZi1lN69g1x9RzdDxRKdHW1suaHLdwm0SckzDrNR0t1/kKFiiVLAkWKJ7v6D4z0ks1w4OMxGSdfC6XR2tNEumNLRRtfC6eM9JLNceFeV2ShZOm8aW27oorv/IF0Lp3s3lU1aDg6zUbR03jQHhk163lVlljOfaWWTTa7BIWmFpD2S+iStq1EvSbcl9U9IWtKor6SzJD0o6Znk2f+9s6Y1fKbVrQ/s4eo7uo+Fh8PEJrLcdlVJagfWA5cAA8AOSVsj4qmKZiuBRcljOXA7sLxB33XAQxFxcxIo64DP5rUdZqdipDOtqk/bHW47fGykd+/gseXKulqvp53eyeDhobrPu/Yd4qVXX2fGGadx/lvOZNe+Q/T94lVe/tchprS38cqvj/D60RJHjpb49VCJUgSlCCKgTSDpuOWAEeuyLmdZV3VbYNTWlec2jDROUX5dvY5G29ne1saUNlECOttFCH49VKJYKh1bbymgrQ0Wn3sGX/793x7VXah5HuNYBvRFRD+ApLuBVUBlcKwCNkdEAN2SpkqaCcyv03cV8P6k/13AT3BwWJMaPtPqSLF07Eyr6jC5d+cA39s5cCxIvnD5+Xzph7sZKpboKP/LQfFo7ddHiiWC3/wDNNLzqSgFx62lFI3qsi6fetvRXFee21DdfuR11O9fLJV4PVk+fOLqjjlagqf2v8pHN/wD313znlELjzyDYxbwfMXyAOVZRaM2sxr0PTci9gNExH5J59R6c0mrgdUAc+fOPclNMDs1I51pVRkmguOC5P5d+3+zfDSAIBj5NdDw2VpbKcqz04kQHKpRVv05HqlNmr51RcRGYCNAoVDw34+Nm+ozrarDBODenQPHgmTlBTPZ8ezLHCmWaE9mFkeP1n5dLJYoke+Mwya+NjGq3yvKMzgGgDkVy7OBfSnbdNbp+wtJM5PZxkzgxVEdtdkYqA6T6lnJ4jef4WMcPsbRtMc4VD68MPokdQA/Ay4GXgB2AB+LiN0VbT4IrAUuo7wr6raIWFavr6RbgIMVB8fPioib6o2lUChET0/P6G+kmdkkJqk3IgrV5bnNOCKiKGktsB1oBzYl//CvSeo3ANsoh0Yf5WM819Xrm6z6ZuAeSdcDzwEfzWsbzMzsRLnNOJqJZxxmZtmNNOPwN8fNzCwTB4eZmWXi4DAzs0wcHGZmlklLHByXdADYe5LdzwZeGsXhTESt/jNo9e0H/wxadfvnRcSM6sKWCI5TIamn1lkFraTVfwatvv3gn0Grb38176oyM7NMHBxmZpaJg6OxjeM9gCbQ6j+DVt9+8M+g1bf/OD7GYWZmmXjGYWZmmTg4zMwsEwdHQtIKSXsk9SWXa6+ul6TbkvonJC0Zj3HmJcX2v1/SIUmPJY8vjMc48yJpk6QXJe0aoX5S//4h1c9gsn8G5kh6WNLTknZL+rMabSb95yCViGj5B+VLt/8zsJDyTaQeB86ranMZcD/le6R0Af843uMe4+1/P/DD8R5rjj+D9wFLgF0j1E/a33+Gn8Fk/wzMBJYkr8+gfE+glvl3IMvDM46yZUBfRPRHxBBwN7Cqqs0qYHOUdQNTkzsQTgZptn9Si4hHgJfrNJnMv38g1c9gUouI/RGxM3n9KvA0MKuq2aT/HKTh4CibBTxfsTzAiR+YNG0mqrTb9m5Jj0u6X9L5YzO0pjGZf/9ZtMRnQNJ84J3AP1ZV+XNAvvccn0hUo6z6POU0bSaqNNu2k/J1a34l6TLgB8CivAfWRCbz7z+tlvgMSHojcC/w6Yh4pbq6RpdW+xx4xpEYAOZULM8G9p1Em4mq4bZFxCsR8avk9TZgiqSzx26I424y//5TaYXPgKQplENjS0R8r0aTlv8cgINj2A5gkaQFkjqBK4GtVW22AtckZ1V0AYciYv9YDzQnDbdf0pslKXm9jPJn5+CYj3T8TObffyqT/TOQbNu3gKcj4msjNGv5zwF4VxUAEVGUtBbYTvkMo00RsVvSmqR+A7CN8hkVfcBh4LrxGu9oS7n9HwE+IakIvAZcGclpJpOBpO9QPmvobEkDwJ8DU2Dy//6HpfgZTOrPAPBe4I+AJyU9lpT9F2AutM7nIA1fcsTMzDLxriozM8vEwWFmZpk4OMzMLBMHh5mZZeLgMDOboBpdmLKq7fsk7ZRUlPSRqrprJT2TPK5ttC4Hh1kGkqZXXB3255JeSF7/StL/zOk9Py3pmjr1l0v6Yh7vbU3vTmBFyrbPAX8MfLuyUNJZlE+9Xk75unV/LmlavRU5OMwyiIiDEfGOiHgHsAH4erL8xoj45Gi/n6QO4ONU/bFX+RFwhaTTR/v9rbnVujClpLdK+rGkXkl/K+m3krbPRsQTQKlqNb8HPBgRL0fEIPAgDcLIwWE2CpJ7Vfwwef1fJd0l6QFJz0r6kKSvSnoy+YOekrRbKumnyR/49hGusvoBYGdEFJM+/0nSU8m9IO4GSL6E9xPg8jHZWGt2G4FPRcRS4Eag0Uw484Ub/c1xs3y8FbgIOA94FPhwRNwk6fvAByX9CPgfwKqIOCDpD4GvUJ5dVHov0FuxvA5YEBGvS5paUd4D/C5wTy5bYxNCcoHG9wDfTa4OA3Bao241yup+M9zBYZaP+yPiiKQnKV/G5cdJ+ZPAfGAxcAHwYPIH3g7UuubRTMr3hRj2BLBF0g8oX5122IvAW0Zv+DZBtQG/THalpjVA+VIzw2ZTnsHWfRMzG32vA0RECThScU2nEuX/sAnYPXy8JCJ+OyIurbGe14A3VCx/EFgPLAV6k2MgJG1ey2E7bAJJLgP/L5I+Csdudfs7DbptBy6VNC05KH5pUjYiB4fZ+NgDzJD0bihfznuEGyM9DbwtadMGzImIh4GbgKnAG5N2bwcanpJpk0tyYcpHgcWSBiRdD1wNXC/pcWA3yd08Jb0ruXjlR4G/krQbICJeBr5M+SrZO4AvJWUj8q4qs3EQEUPJufS3STqT8t/if6f8h17pfuB/Ja/bgf+dtBflM7p+mdRdBHwu73Fbc4mIq0aoOuGsqIjYQXk3VK31bAI2pX1fXx3XrMklB9RviohnRqg/F/h2RFw8tiOzVuXgMGtykhYD5ybn7Neqfxfl4yiPjenArGU5OMzMLBMfHDczs0wcHGZmlomDw8zMMnFwmJlZJg4OMzPL5P8D+PAdBWf8zfAAAAAASUVORK5CYII=\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "scatter(times, sr_conds, marker='.')\n", "ylabel('SR factor');\n", "xlabel('Time (s)');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "As this happens, the cloud grows in mass at the expense of the BH:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plot(times, ys/xs)\n", "axvline(6E9, ls='--', c='k')\n", "axvline(6E9+1500*c.number_growth_time, ls='--', c='k')\n", "xlabel('Time (s)');\n", "ylabel(r'$M_c/M_{\\rm BH}$');" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that the timesteps are not equally separated, since the solver adapts the interval according to the evolution of the system:" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "jupyter": { "outputs_hidden": false } }, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "plot(times)\n", "xlabel('Steps');\n", "ylabel('Time (s)');" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "gwaxion", "language": "python", "name": "gwaxion" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.9" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": false, "toc_position": {}, "toc_section_display": true, "toc_window_display": true } }, "nbformat": 4, "nbformat_minor": 4 }