Examples/Physics_applications/pulsar/Pulsar_scale_RstarPhysical.ipynb

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    "# Overview\n",
    "\n",
    "This a notebook that inspects the results of a WarpX simulation.\n",
    "\n",
    "# Instruction\n",
    "\n",
    "Enter the path of the data you wish to visualize below. Then execute the cells one by one, by selecting them with your mouse and typing `Shift + Enter`"
   ]
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    "# Import statements\n",
    "import yt ; yt.funcs.mylog.setLevel(50)\n",
    "import numpy as np\n",
    "import scipy.constants as scc\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib notebook"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#Define the Physical Constants, normalizations, and the scale-down parameter, r_scale, for the pulsar. "
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   "cell_type": "code",
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    "######################\n",
    "# physical constants #\n",
    "######################\n",
    "c = 299792458.0 # speed of light \n",
    "q_e = 1.602176634e-19  # elementary charge\n",
    "me=9.10938356*np.power(10.,-31) # electron mass\n",
    "epsilon=8.8541878128*np.power(10.,-12) # permittivity of free space\n",
    "mu_o = 4*3.14*1e-7 # permeability\n",
    "pi = 3.14159265358979323846\n",
    "SolarMass = 2e30\n",
    "G = 6.674e-11 # gravitational constant\n",
    "\n",
    "#############################################################################\n",
    "# Parameters for a real Pulsar                                              #\n",
    "# (Table 7 of J.Petri's article on Theory of Pulsar Magnetosphere and Wind) #\n",
    "#############################################################################\n",
    "Omega_real = 6283\n",
    "B_real = 7.4E4\n",
    "R_real = 12000\n",
    "n_real = 6.9e16\n",
    "omega_pe_real = (n_real*q_e*q_e/(me*epsilon))**0.5 #plasma frequency\n",
    "SkinDepth_real = c/omega_pe_real \n",
    "Mstar = 1.4*SolarMass\n",
    "Rstar_skinDepth_real = 6e5\n",
    "\n",
    "##################\n",
    "# Normalizations #\n",
    "##################\n",
    "Rstar_skinDepth = 6e0 # Ratio of radius of star to the skin Depth \n",
    "                      # For a real star, this ratio is 6e5\n",
    "exponent = np.arange(0,6,1)    \n",
    "Factor = np.array(10**exponent)\n",
    "Rstar_skinDepth = np.array(6*Factor)  \n",
    "\n",
    "RLC_Rstar = 4     # Ratio of light cylinder (where the particle speed ~ c) to Rstar\n",
    "                      # For a pulsar with 1ms period, this ratio is 4. \n",
    "                      # i.e., This ratio sets the omega value, since Omega*RLC = c\n",
    "\n",
    "# Choose skindepth as the free parameter for a choice of normalizations given above\n",
    "# The skin depth below is computed from the number density for a real pulsar\n",
    "# Keeping the SkinDepth constant across all the scales in our scaling study\n",
    "#SkinDepth = 0.02\n",
    "\n",
    "# Since skin depth is maintained across the scales, and RLC/Rstar is also maintained \n",
    "# lets define the decrease in the scale by comparing the value of \n",
    "# (Rstar/skinDepth)/(Rstar_real/skinDepth_real)\n",
    "#r_scale = Rstar_skinDepth/Rstar_skinDepth_real\n",
    "\n",
    "Rstar = np.ones(6)*12000\n",
    "SkinDepth = Rstar/Rstar_skinDepth\n",
    "r_scale = Rstar_skinDepth_real/Rstar_skinDepth"
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    "##################################################################\n",
    "# Derive other physical parameters from the above normalizations #\n",
    "##################################################################\n",
    "\n",
    "# 1. Lorentz Boost (dimensionless) #\n",
    "# Note that in Alex Chen's paper, gamma_o ~ (1/4)*normalized_values.\n",
    "# Instead here we have pi/2, since that leads to the closes gamma_o \n",
    "# value for a real 1ms pulsar. \n",
    "gamma_o = (pi/2)*(Rstar_skinDepth)**2/RLC_Rstar \n",
    "\n",
    "\n",
    "gamma_real = (pi/2)*6e5**2/RLC_Rstar;\n",
    "gamma_scaling = gamma_o/(gamma_real) # This is to see how the gamma value \n",
    "                      # decreases due to decrease in the ratio of R_star to skin depth\n",
    "\n",
    "\n",
    "\n",
    "# 3. Light cylinder location (m)\n",
    "RLC = Rstar*RLC_Rstar\n",
    "# 4. Angular Frequency (rad/s)\n",
    "#    Omega remains constant\n",
    "Omega = c/RLC\n",
    "# 5. Period (s)\n",
    "#    Period remains constant\n",
    "Period = 2*3.14/Omega\n",
    "# Moment of inertia for a sphere = (2/5)MR^2 (kg.m^2)\n",
    "# Note that when the Rstar is scaled by r_scale, \n",
    "# Mstar decreases as r_scale^3. Thus Mstar*r_scale**3 is the \n",
    "# mass of the scaled down star.\n",
    "# I remains constant across all scales\n",
    "I = (2/5)*(Mstar)*Rstar**2\n",
    "# 6. Rotation induced potential drop from pote to equator (V)\n",
    "#    Reference: Alex Chen's 2014 paper\n",
    "#    Scales as 1/r_scale**2\n",
    "phi_o = gamma_o * (me/q_e) * c**2\n",
    "\n",
    "# Braking is the rate of slow-down of the spin. \n",
    "# It is not relevant for the scaling. \n",
    "# However, 1e-15 is the P_dot for a pulsar with P=1s\n",
    "# and the rate of slowdown decreases proportional to the period. \n",
    "# So we were to compare two stars with same radius, but different Omega\n",
    "# then we can use Braking to determine the magnetic field strength at the surface as follows\n",
    "# Bo = (3*mu_o*c*c*c/(32*pi*pi*pi))**0.5 * (I*Braking*Period)**0.5/(Rstar**3) \n",
    "# (Reference : Table 7 of Jetri's article)\n",
    "# Remains constant across all scales\n",
    "Braking = 1e-15*Period\n",
    "\n",
    "# 7. Magnetic field strength (Tesla)\n",
    "#   Bo decreases as ~ 1/r_scale**2\n",
    "Bo = phi_o/(Rstar*Rstar*Omega)\n",
    "\n",
    "# 8. Volume charge density (C/m^3) for uniform magnetization insize the star \n",
    "#    Refer to Table 7 of Petri's article \n",
    "#    Since Omega increases as r_scale and Bo decreases as r_scale\n",
    "#    The product of Omega*Bo remains constant across all scales. \n",
    "#    Thus, rho_e decreases as (1/r_scale**2)\n",
    "rho_e = 2*epsilon*Omega*Bo #(Not adding the negative sign here)\n",
    "\n",
    "# 9. Electron number density (#/m^3)\n",
    "#    ne decreases as (1/r_scale**2)\n",
    "ne = rho_e/q_e\n",
    "# 9a. plasma frequency \n",
    "#     decreases as (1/r_scale)\n",
    "omega_pe = (ne*q_e*q_e/(me*epsilon))**0.5 #plasma frequency\n",
    "# 10. Magnetic moment (Am^2)\n",
    "#     decreases as (1/r_scale**2)\n",
    "magnetic_moment = Bo*Rstar*Rstar*Rstar*4*pi/(mu_o)\n",
    "# 11. E-field (V/m)\n",
    "#     Efield decreases as (1/r_scale**2)\n",
    "E = Omega*Bo*Rstar\n",
    "\n",
    "\n",
    "#########################################\n",
    "# How do the energies and forces scale? #\n",
    "#########################################\n",
    "# 12. Magnetic Energy (J)\n",
    "#     Magnetic energy decreases as (1/r_scale**4)\n",
    "magnetic_energy = (4*pi/3)*Bo**2*Rstar**3/(2*mu_o)\n",
    "\n",
    "# 13. Gravitational Potential Energy (J)\n",
    "#     G.pot energy remains constant \n",
    "GP_energy = (3/5)*G*Mstar*Mstar/(Rstar)\n",
    "\n",
    "# 14. Gravitational to Electric force (dimensionless)\n",
    "#     This ratio increases as r_scale**2\n",
    "G_EForce = G*Mstar*me/(Rstar*Rstar*q_e*E)\n",
    "\n",
    "# 15. Rotational kinetic energy \n",
    "#     Rot. KE remains constant\n",
    "rotational_KE = (1/2)*I*Omega**2\n",
    "\n",
    "# 16. From (12) and (13), we know the B.energy scales as (1/r_scale**4) and GP energy is constant \n",
    "#     Thus the ratio of GP_energy and B_energy increases as r_scale**4\n",
    "GB_energy_ratio = GP_energy/magnetic_energy\n",
    "\n",
    "# 17. Rate of change of Omega, or angular acceleration decreases as 1/r_scale**4\n",
    "Omega_dot = Bo*Bo*Rstar**6*Omega**3/(I) * (32*pi/(3*mu_o*c**3))* (1/(4*pi*pi))\n",
    "\n",
    "# 18. Torque decreases as 1/r_scale**4\n",
    "Torque = I * Omega_dot\n"
   ]
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   "cell_type": "code",
   "execution_count": 4,
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     "name": "stdout",
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     "text": [
      "Input parameters for WarpX for geometry scaled down by r_scale =  [1.e-05 1.e-04 1.e-03 1.e-02 1.e-01 1.e+00]\n",
      "Lorentz factor, (gamma_o) =  [1.41371669e+01 1.41371669e+03 1.41371669e+05 1.41371669e+07\n",
      " 1.41371669e+09 1.41371669e+11]\n",
      "Rstar (m) [12000. 12000. 12000. 12000. 12000. 12000.]\n",
      "Omega (rad/s) [6245.67620833 6245.67620833 6245.67620833 6245.67620833 6245.67620833\n",
      " 6245.67620833]\n",
      "Bo (Tesla) [8.03230942e-06 8.03230942e-04 8.03230942e-02 8.03230942e+00\n",
      " 8.03230942e+02 8.03230942e+04]\n",
      "ne (/m^3) [5.54482989e+06 5.54482989e+08 5.54482989e+10 5.54482989e+12\n",
      " 5.54482989e+14 5.54482989e+16]\n",
      "Size of the domain, (m) [360000. 360000. 360000. 360000. 360000. 360000.]\n",
      "Minimum cell size (m) [2.e+03 2.e+02 2.e+01 2.e+00 2.e-01 2.e-02]\n",
      "timestep (s) [3.76386776e-06 3.76386776e-07 3.76386776e-08 3.76386776e-09\n",
      " 3.76386776e-10 3.76386776e-11]\n",
      "Numver of cells assuming unif. grid,  [1.8e+02 1.8e+03 1.8e+04 1.8e+05 1.8e+06 1.8e+07]\n"
     ]
    }
   ],
   "source": [
    "############################################################\n",
    "# Print all the values that will be used as input in WarpX #\n",
    "############################################################\n",
    "print(\"Input parameters for WarpX for geometry scaled down by r_scale = \", Rstar_skinDepth/Rstar_skinDepth_real)\n",
    "print(\"Lorentz factor, (gamma_o) = \", gamma_o)\n",
    "print(\"Rstar (m)\", Rstar)\n",
    "print(\"Omega (rad/s)\", Omega)\n",
    "print(\"Bo (Tesla)\", Bo)\n",
    "print(\"ne (/m^3)\", ne)\n",
    "print(\"Size of the domain, (m)\", 30*Rstar)\n",
    "print(\"Minimum cell size (m)\", SkinDepth)\n",
    "print(\"timestep (s)\", 0.5*omega_pe**-1) # Or may be cfl criterion\n",
    "print(\"Numver of cells assuming unif. grid, \", 30*Rstar/SkinDepth)\n"
   ]
  },
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   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
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     "name": "stdout",
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     "text": [
      "Gravitational force to E-force :  [1.22562947e-02 1.22562947e-04 1.22562947e-06 1.22562947e-08\n",
      " 1.22562947e-10 1.22562947e-12]\n",
      "Gravitational energy to B-energy :  [1.40727411e+38 1.40727411e+34 1.40727411e+30 1.40727411e+26\n",
      " 1.40727411e+22 1.40727411e+18]\n",
      "B energy, (J) [1.85906071e+08 1.85906071e+12 1.85906071e+16 1.85906071e+20\n",
      " 1.85906071e+24 1.85906071e+28]\n",
      "Gravitational potential energy, (J) [2.616208e+46 2.616208e+46 2.616208e+46 2.616208e+46 2.616208e+46\n",
      " 2.616208e+46]\n",
      "Rotational Kinetic Energy (J) [3.14564313e+45 3.14564313e+45 3.14564313e+45 3.14564313e+45\n",
      " 3.14564313e+45 3.14564313e+45]\n"
     ]
    }
   ],
   "source": [
    "#############################################################\n",
    "# Print ratios of G.Pot.Energy/B_energy and G.Force/E_force #\n",
    "#############################################################\n",
    "print(\"Gravitational force to E-force : \", G_EForce)\n",
    "print(\"Gravitational energy to B-energy : \",GB_energy_ratio)\n",
    "\n",
    "\n",
    "# Print dimensional values of energies\n",
    "print(\"B energy, (J)\",magnetic_energy);\n",
    "print(\"Gravitational potential energy, (J)\", GP_energy)\n",
    "print(\"Rotational Kinetic Energy (J)\", rotational_KE )\n",
    "\n"
   ]
  },
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       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items) {\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) {\n",
       "            // put a spacer in here.\n",
       "            continue;\n",
       "        }\n",
       "        var button = $('<button/>');\n",
       "        button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n",
       "                        'ui-button-icon-only');\n",
       "        button.attr('role', 'button');\n",
       "        button.attr('aria-disabled', 'false');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "\n",
       "        var icon_img = $('<span/>');\n",
       "        icon_img.addClass('ui-button-icon-primary ui-icon');\n",
       "        icon_img.addClass(image);\n",
       "        icon_img.addClass('ui-corner-all');\n",
       "\n",
       "        var tooltip_span = $('<span/>');\n",
       "        tooltip_span.addClass('ui-button-text');\n",
       "        tooltip_span.html(tooltip);\n",
       "\n",
       "        button.append(icon_img);\n",
       "        button.append(tooltip_span);\n",
       "\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    var fmt_picker_span = $('<span/>');\n",
       "\n",
       "    var fmt_picker = $('<select/>');\n",
       "    fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n",
       "    fmt_picker_span.append(fmt_picker);\n",
       "    nav_element.append(fmt_picker_span);\n",
       "    this.format_dropdown = fmt_picker[0];\n",
       "\n",
       "    for (var ind in mpl.extensions) {\n",
       "        var fmt = mpl.extensions[ind];\n",
       "        var option = $(\n",
       "            '<option/>', {selected: fmt === mpl.default_extension}).html(fmt);\n",
       "        fmt_picker.append(option);\n",
       "    }\n",
       "\n",
       "    // Add hover states to the ui-buttons\n",
       "    $( \".ui-button\" ).hover(\n",
       "        function() { $(this).addClass(\"ui-state-hover\");},\n",
       "        function() { $(this).removeClass(\"ui-state-hover\");}\n",
       "    );\n",
       "\n",
       "    var status_bar = $('<span class=\"mpl-message\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_message = function(type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function() {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function(fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1]);\n",
       "        fig.send_message(\"refresh\", {});\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n",
       "    var x0 = msg['x0'] / mpl.ratio;\n",
       "    var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
       "    var x1 = msg['x1'] / mpl.ratio;\n",
       "    var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0, 0, fig.canvas.width / mpl.ratio, fig.canvas.height / mpl.ratio);\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch(cursor)\n",
       "    {\n",
       "    case 0:\n",
       "        cursor = 'pointer';\n",
       "        break;\n",
       "    case 1:\n",
       "        cursor = 'default';\n",
       "        break;\n",
       "    case 2:\n",
       "        cursor = 'crosshair';\n",
       "        break;\n",
       "    case 3:\n",
       "        cursor = 'move';\n",
       "        break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message(\"ack\", {});\n",
       "}\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = \"image/png\";\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src);\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data);\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig[\"handle_\" + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "}\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function(e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e)\n",
       "        e = window.event;\n",
       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\n",
       "\n",
       "    return {\"x\": x, \"y\": y};\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys (original) {\n",
       "  return Object.keys(original).reduce(function (obj, key) {\n",
       "    if (typeof original[key] !== 'object')\n",
       "        obj[key] = original[key]\n",
       "    return obj;\n",
       "  }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function(event, name) {\n",
       "    var canvas_pos = mpl.findpos(event)\n",
       "\n",
       "    if (name === 'button_press')\n",
       "    {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * mpl.ratio;\n",
       "    var y = canvas_pos.y * mpl.ratio;\n",
       "\n",
       "    this.send_message(name, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.key_event = function(event, name) {\n",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, {key: value,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
       "    if (name == 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message(\"toolbar_button\", {name: name});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\"];\n",
       "\n",
       "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function() {\n",
       "        comm.close()\n",
       "    };\n",
       "    ws.send = function(m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function(msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overridden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data'])\n",
       "    });\n",
       "    return ws;\n",
       "}\n",
       "\n",
       "mpl.mpl_figure_comm = function(comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element.get(0);\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error(\"Failed to find cell for figure\", id, fig);\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    var width = fig.canvas.width/mpl.ratio\n",
       "    fig.root.unbind('remove')\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable()\n",
       "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
       "    fig.close_ws(fig, msg);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.close_ws = function(fig, msg){\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width/mpl.ratio\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message(\"ack\", {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>');\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items){\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
       "    titlebar.prepend(buttongrp);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(el){\n",
       "    var fig = this\n",
       "    el.on(\"remove\", function(){\n",
       "\tfig.close_ws(fig, {});\n",
       "    });\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(el){\n",
       "    // this is important to make the div 'focusable\n",
       "    el.attr('tabindex', 0)\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    }\n",
       "    else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager)\n",
       "        manager = IPython.keyboard_manager;\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which == 13) {\n",
       "        this.canvas_div.blur();\n",
       "        // select the cell after this one\n",
       "        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n",
       "        IPython.notebook.select(index + 1);\n",
       "    }\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.find_output_cell = function(html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i=0; i<ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code'){\n",
       "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] == html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel != null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<img 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\" width=\"640\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot Gravitational force to electric force -- should be constant across all scales\n",
    "\n",
    "plt.plot(Rstar_skinDepth/6e5,G_EForce,'ro')\n",
    "plt.xlabel(\"Log( Normalized (Rstar/Lambda) ) Scaling\")\n",
    "plt.ylabel(\"G.Force/E.Force\")\n",
    "plt.xscale(\"log\")\n",
    "plt.yscale(\"log\")\n",
    "plt.savefig(\"G_Eforce_scaling.png\",bbox_inches='tight')\n",
    "#plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# plot Gravitational to magnetic energy -- should be constant across all scales\n",
    "\n",
    "plt.plot(Rstar_skinDepth/6e5,GB_energy_ratio,'ro')\n",
    "plt.rcParams.update({'font.size': 18, 'font.family': 'serif'})\n",
    "plt.xlabel(\"Log( Normalized (Rstar/Lambda) ) Scaling\")\n",
    "plt.ylabel(\"G.Energy/B.energy\")\n",
    "plt.xscale(\"log\")\n",
    "plt.yscale(\"log\")\n",
    "plt.savefig(\"G_Benergy_scaling.png\",bbox_inches='tight')\n",
    "#plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Trends of all energies\n",
    "plt.plot(Rstar_skinDepth/6e5,magnetic_energy,color=\"red\",ls='--',lw=2,marker=\"^\",markersize=8)\n",
    "plt.plot(Rstar_skinDepth/6e5,GP_energy,color=\"blue\",lw=2,marker='o',markersize=8,markerfacecolor='blue')\n",
    "plt.plot(Rstar_skinDepth/6e5,rotational_KE,color=\"purple\",lw=2,marker='s',markersize=8,markeredgecolor='purple')\n",
    "plt.plot(Rstar_skinDepth/6e5,Torque,color=\"cyan\",lw=2,marker='>',markersize=8,markeredgecolor='purple')\n",
    "plt.rcParams.update({'font.size': 18, 'font.family': 'serif'})\n",
    "plt.xlabel(\"Log( Normalized (Rstar/Lambda) ) Scaling\")\n",
    "plt.ylabel(\"Energy (J)\")\n",
    "plt.xscale(\"log\")\n",
    "plt.yscale(\"log\")\n",
    "plt.rcParams.update({'font.size': 18, 'font.family': 'serif'})\n",
    "plt.legend([\"Magnetic\",\"G. Potential\",\"Rotational KE\",\"Torque\"],loc=2,fontsize='small',frameon=\"false\",borderpad=0.1)\n",
    "plt.ylim([1e0,1e90])\n",
    "plt.savefig(\"Energy_scaling.png\",bbox_inches='tight')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# plot Gravitational to magnetic energy -- should be constant across all scales\n",
    "plt.plot(Rstar_skinDepth/6e5,Bo/Bo[5],color=\"blue\",lw=2,marker='s',markersize=8)\n",
    "plt.ylabel(\"Normalized B. field\",color=\"blue\")\n",
    "plt.xlabel(\"Log( Normalized (Rstar/Lambda) ) Scaling\")\n",
    "plt.yticks(color=\"blue\")\n",
    "plt.xscale(\"log\")\n",
    "plt.yscale(\"log\")\n",
    "plt.twinx()\n",
    "plt.plot(Rstar_skinDepth/6e5,Omega/Omega[5],color=\"red\",lw=2,marker='o',markersize=8)\n",
    "plt.ylabel(\"Normalized \\u03A9\",color=\"red\")\n",
    "plt.yticks(color=\"red\")\n",
    "plt.xscale(\"log\")\n",
    "plt.ylim([0,2])\n",
    "plt.savefig(\"Normalized_B_Omega.png\",bbox_inches='tight')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(Rstar_skinDepth/6e5,gamma_o,'ro')\n",
    "plt.rcParams.update({'font.size': 18, 'font.family': 'serif'})\n",
    "plt.xlabel(\"Log( Normalized (Rstar/Lambda) ) Scaling\")\n",
    "plt.ylabel(\"Normalized G.Energy/B.energy\")\n",
    "plt.xscale(\"log\")\n",
    "plt.yscale(\"log\")\n",
    "plt.savefig(\"G_Benergy_scaling.png\",bbox_inches='tight')\n",
    "#plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(Rstar_skinDepth/6e5,Omega_dot,'ro')\n",
    "plt.rcParams.update({'font.size': 18, 'font.family': 'serif'})\n",
    "plt.xlabel(\"Log( Normalized (Rstar/Lambda) ) Scaling\")\n",
    "plt.ylabel(\"Omega_dot\")\n",
    "plt.xscale(\"log\")\n",
    "plt.yscale(\"log\")\n",
    "plt.savefig(\"G_Benergy_scaling.png\",bbox_inches='tight')\n",
    "#plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(Rstar_skinDepth/6e5,Torque,'ro')\n",
    "plt.rcParams.update({'font.size': 18, 'font.family': 'serif'})\n",
    "plt.xlabel(\"Log( Normalized (Rstar/Lambda) ) Scaling\")\n",
    "plt.ylabel(\"Torque\")\n",
    "plt.xscale(\"log\")\n",
    "plt.yscale(\"log\")\n",
    "plt.savefig(\"Torque.png\",bbox_inches='tight')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(Rstar_skinDepth/6e5,phi_o,'ro')\n",
    "plt.rcParams.update({'font.size': 18, 'font.family': 'serif'})\n",
    "plt.xlabel(\"Log( Normalized (Rstar/Lambda) ) Scaling\")\n",
    "plt.ylabel(\"Normalized G.Energy/B.energy\")\n",
    "plt.xscale(\"log\")\n",
    "plt.yscale(\"log\")\n",
    "plt.savefig(\"phi_o.png\",bbox_inches='tight')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.020230407144897423"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SkinDepth_real"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([48000., 48000., 48000., 48000., 48000., 48000.])"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RLC\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "140.625"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "36000/256"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "particle_weight = 140.625**3*5.544e6/64"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "240896701812.74414"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "particle_weight"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.0010055, 0.0010055, 0.0010055, 0.0010055, 0.0010055, 0.0010055])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Period"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([48000., 48000., 48000., 48000., 48000., 48000.])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RLC\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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