{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Thermal Speed" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "%matplotlib inline\n", "\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "\n", "from astropy import units as u\n", "\n", "from plasmapy.formulary import (\n", " Maxwellian_speed_1D,\n", " Maxwellian_speed_2D,\n", " Maxwellian_speed_3D,\n", ")\n", "from plasmapy.formulary.parameters import thermal_speed" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "[thermal_speed]: ../../api/plasmapy.formulary.parameters.thermal_speed.rst#plasmapy.formulary.parameters.thermal_speed\n", "\n", "The [thermal_speed] function can be used to calculate the thermal velocity for a Maxwellian velocity distribution. There are three common definitions of the thermal velocity, which can be selected using the \"method\" keyword, which are defined for a 3D velocity distribution as\n", "\n", "- 'most_probable'
\n", "$v_{th} = \\sqrt{\\frac{2 k_B T}{m}}$\n", "\n", "- 'rms'
\n", "$v_{th} = \\sqrt{\\frac{3 k_B T}{m}}$\n", "\n", "- 'mean_magnitude'
\n", "$v_{th} = \\sqrt{\\frac{8 k_B T}{m\\pi}}$\n", "\n", "The differences between these velocities can be seen by plotitng them on a 3D Maxwellian speed distribution" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "nbsphinx-thumbnail": { "tooltip": "Thermal Speeds" } }, "outputs": [], "source": [ "T = 1e5 * u.K\n", "speeds = np.linspace(0, 8e6, num=600) * u.m / u.s\n", "\n", "pdf_3D = Maxwellian_speed_3D(speeds, T=T, particle=\"e-\")\n", "\n", "fig, ax = plt.subplots(figsize=(4, 3))\n", "\n", "v_most_prob = thermal_speed(T=T, particle=\"e-\", method=\"most_probable\", ndim=3)\n", "v_rms = thermal_speed(T=T, particle=\"e-\", method=\"rms\", ndim=3)\n", "v_mean_magnitude = thermal_speed(T=T, particle=\"e-\", method=\"mean_magnitude\", ndim=3)\n", "\n", "ax.plot(speeds / v_rms, pdf_3D, color=\"black\", label=\"Maxwellian\")\n", "\n", "ax.axvline(x=v_most_prob / v_rms, color=\"blue\", label=\"Most Probable\")\n", "ax.axvline(x=v_rms / v_rms, color=\"green\", label=\"RMS\")\n", "ax.axvline(x=v_mean_magnitude / v_rms, color=\"red\", label=\"Mean Magnitude\")\n", "\n", "ax.set_xlim(-0.1, 3)\n", "ax.set_ylim(0, None)\n", "ax.set_title(\"3D\")\n", "ax.set_xlabel(\"|v|/|v$_{rms}|$\")\n", "ax.set_ylabel(\"f(|v|)\")" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Similar speeds are defined for 1D and 2D distributions. The differences between these definitions can be illustrated by plotting them on their respective Maxwellian speed distributions." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "pdf_1D = Maxwellian_speed_1D(speeds, T=T, particle=\"e-\")\n", "pdf_2D = Maxwellian_speed_2D(speeds, T=T, particle=\"e-\")\n", "\n", "dim = [1, 2, 3]\n", "pdfs = [pdf_1D, pdf_2D, pdf_3D]\n", "\n", "plt.tight_layout()\n", "fig, ax = plt.subplots(ncols=3, figsize=(10, 3))\n", "\n", "for n, pdf in enumerate(pdfs):\n", " ndim = n + 1\n", " v_most_prob = thermal_speed(T=T, particle=\"e-\", method=\"most_probable\", ndim=ndim)\n", " v_rms = thermal_speed(T=T, particle=\"e-\", method=\"rms\", ndim=ndim)\n", " v_mean_magnitude = thermal_speed(\n", " T=T, particle=\"e-\", method=\"mean_magnitude\", ndim=ndim\n", " )\n", "\n", " ax[n].plot(speeds / v_rms, pdf, color=\"black\", label=\"Maxwellian\")\n", "\n", " ax[n].axvline(x=v_most_prob / v_rms, color=\"blue\", label=\"Most Probable\")\n", " ax[n].axvline(x=v_rms / v_rms, color=\"green\", label=\"RMS\")\n", " ax[n].axvline(x=v_mean_magnitude / v_rms, color=\"red\", label=\"Mean Magnitude\")\n", "\n", " ax[n].set_xlim(-0.1, 3)\n", " ax[n].set_ylim(0, None)\n", " ax[n].set_title(\"{:d}D\".format(ndim))\n", " ax[n].set_xlabel(\"|v|/|v$_{rms}|$\")\n", " ax[n].set_ylabel(\"f(|v|)\")\n", "\n", "\n", "ax[2].legend(bbox_to_anchor=(1.9, 0.8), loc=\"upper right\")" ] } ], "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": "3.7.6" } }, "nbformat": 4, "nbformat_minor": 4 }