Schwarzschild_Jacobi_Elliptic.ipynb

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    "## Trajectory of Light Signal using Jacobi’s Elliptic Function.\n",
    "\n",
    "\n",
    "This workbook describes the: Trajectory of Light Signal using Jacobi’s Elliptic Function. The light deflection arround the sun is determined. \n",
    "\n",
    "The theory behind the method is found in paper: 'Equations of Orbits, Deflection of Light, Mercury’s Perihelion Shift, Period of Revolution and Time Delay of Signals in Schwarzchild’s Geometry. Some closed-form solutions using Weinberg’s approach. New Version (19/08/2016) Solomon M. Antoniou'. [Researchgate](https://www.researchgate.net/publication/306379858_Equations_of_Orbits_Deflection_of_Light_Mercury%27s_Perihelion_Shift_Period_of_Revolution_and_Time_Delay_of_Signals_in_Schwarzchild%27s_Geometry_Some_closed-form_solutions_using_Weinberg%27s_approach)\n",
    "\n",
    "This project was started based upon a topic on: [Wetenschapsforum](https://www.wetenschapsforum.nl/viewtopic.php?f=85&t=212427). \n",
    "\n",
    "Here follows a summary of the result found on forum Wetenschapsforum. The mean set of formulas required are found in article. Where $r_s$ is Schwarzschild radius and $r_0$ radius lightbeam from center mass and $\\varphi$ is the angle following the light signal. The radius $r$ with angle $\\varphi$ are computed with $(6)$. Finally the startcondition $\\sigma$ and $\\varphi$ range are computed:\n",
    "\n",
    "$$e_1 = \\frac{r_0 - r_s + \\sqrt{(r_0 - r_s)(r_0 + 3r_s)}}{2 r_s r_0} \\tag{1}$$\n",
    "\n",
    "$$e_2 = \\frac{1}{r_0} \\tag{2}$$\n",
    "\n",
    "$$e_3 = \\frac{r_0 - r_s - \\sqrt{(r_0 - r_s)(r_0 + 3r_s)}}{2 r_s r_0}  \\tag{3}$$\n",
    "\n",
    "$$\\tau = \\sqrt{ \\frac{ r_s (e_1 - e_3)}{ 4 }}  \\tag{4}$$\n",
    "\n",
    "$$h = \\sqrt{ \\frac{ e_2 - e_3}{ e_1 - e_3 }}  \\tag{5}$$\n",
    "\n",
    "$$\\boxed{ r = \\frac{1}{ e_3 + (e_2 - e_3) \\mathrm{sn}^2( \\tau \\varphi + \\sigma ; h^2)}}  \\tag{6}$$\n",
    "\n",
    "Start conditions for $r=r_{0}$ at $\\varphi=\\pi/2$ determine $\\sigma$ substitude $(2)$:\n",
    "\n",
    "$$r_{0} = \\frac{1}{ e_3 + (e_2 - e_3) \\mathrm{sn}^2( \\tau \\frac{\\pi}{2} + \\sigma ; h^2)}=\\frac{1}{e_2}  \\tag{7}$$\n",
    "\n",
    "After simplification:\n",
    "\n",
    "$$\\mathrm{sn}( \\tau \\frac{\\pi}{2} + \\sigma ; h^2) = 1 \\tag{8}$$\n",
    "\n",
    "Property elliptic integrals:\n",
    "\n",
    "$$\\mathrm{sn}(K,h^2)=\\sin \\left( \\frac{\\pi}{2} \\right) \\tag{10} $$\n",
    "\n",
    "Where $\\tau$ and $h^2$,  are given and the startcondition $\\sigma$ can be solved with standard form complete elliptic integral [Wiki](https://en.wikipedia.org/wiki/Jacobi_elliptic_functions#Definition_as_inverses_of_elliptic_integralshttps://en.wikipedia.org/wiki/Jacobi_elliptic_functions#Definition_as_inverses_of_elliptic_integrals). In the simulation the following integral solver was used from scipy: [special.ellipk]. Definition of complete Jacobi elliptic integral:\n",
    "\n",
    "$$K(h^2)=\\int_{0}^{\\frac{\\pi}{2}} \\frac{d t}{\\sqrt{1-h^2 \\sin^2(t)   }}   =\\tau \\frac{\\pi}{2} + \\sigma   \\tag{9}  $$   \n",
    "\n",
    "Not all solutions are valid, the root of $(6)$ van have zeros. The $x$-range is found by searching the roots of $(6)$. \n",
    "\n",
    "$$e_3 + (e_2 - e_3) \\mathrm{sn}^2( \\tau \\varphi + \\sigma ; h^2) = 0   \\tag{12}$$\n",
    "\n",
    "$$ \\mathrm{sn}( \\tau \\varphi + \\sigma ; h^2) =\\sqrt{ -\\frac{e_3}{e_2 - e_3}}   \\tag{13}$$\n",
    "\n",
    "$$\\mathrm{sn}(K|h^2)=\\sin \\left( \\phi \\right) \\tag{14} $$\n",
    "\n",
    "$$\\phi=\\arcsin \\left( \\sqrt{ -\\frac{e_3}{e_2 - e_3}} \\right)  \\tag{15}$$\n",
    "\n",
    "Where $\\tau$, $h^2$, $\\phi$ and $\\sigma$ are given and can be solved with incomplete elliptic integral [scipy: special.ellipkinc]. The maximum angle $\\varphi$ can be calulated with:\n",
    "\n",
    "$$K(\\phi|h^2)=\\int_{0}^{\\phi} \\frac{d t}{\\sqrt{1-h^2 \\sin^2(t)   }} =\\tau \\varphi + \\sigma   \\tag{16}     $$"
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    "#Version 25.0 Schwarzschild, Jacobi Elliptic\n",
    "#https://www.wetenschapsforum.nl/viewtopic.php?f=85&t=212427\n",
    "\n",
    "#Open pyplot in separate interactive window\n",
    "#from IPython import get_ipython\n",
    "#get_ipython().run_line_magic('matplotlib', 'qt5')\n",
    "\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "#import matplotlib.gridspec as gridspec\n",
    "import scipy.special as sps\n",
    "from scipy import optimize\n",
    "from matplotlib.widgets import Slider, CheckButtons\n",
    "import matplotlib.ticker as ticker \n",
    "\n",
    "%matplotlib widget\n",
    "\n",
    "fig= plt.figure(figsize=(9, 5))  \n",
    "\n",
    "ax1 = plt.subplot2grid((3, 3), (0, 0),rowspan=2)\n",
    "ax2 = plt.subplot2grid((3, 3), (0, 1),rowspan=2)\n",
    "ax3 = plt.subplot2grid((3, 3), (0, 2),rowspan=2)\n",
    "ax4 = plt.subplot2grid((3, 3), (2, 0),colspan=3)\n",
    "ax4.axis('off')\n",
    "\n",
    "plt.suptitle('Schwarzschild, Jacobi Elliptic', fontsize=20)\n",
    "#fig.set_tight_layout(True)\n",
    "plt.tight_layout(pad=1.5, w_pad=1, h_pad=1)\n",
    "\n",
    "# Draw slider alpha\n",
    "axcolor = 'lightgoldenrodyellow'\n",
    "axalpha = plt.axes([0.25, 0.01,0.4, 0.04], facecolor=axcolor)\n",
    "s_zoom = Slider(axalpha, 'zoom: ', 0, 1, valinit=0.9)\n",
    "\n",
    "axsm = plt.axes([0.25, 0.06,0.4, 0.04], facecolor=axcolor)\n",
    "s_sm = Slider(axsm, 'solar masses: ', 1,157280, valinit=1)\n",
    "\n",
    "radius = plt.axes([0.25, 0.11,0.4, 0.04], facecolor=axcolor)\n",
    "s_sr = Slider(radius, 'radius: ', 0.1, 5, valinit=1)\n",
    "\n",
    "rax = plt.axes([0.75, 0.01, 0.075, 0.14])\n",
    "check = CheckButtons(rax, ('x=y', 'sun'), (True, True))\n",
    "\n",
    "#Set constants note G is divided by c^2    \n",
    "global M\n",
    "global G\n",
    "global Ro\n",
    "\n",
    "M=1.989e30\n",
    "G=6.67408e-11/((3e8)**2)\n",
    "Ro=696340000\n",
    "\n",
    "def e1(sm,sr):\n",
    "    esqrt=np.sqrt((sr*Ro-sm*2*M*G)*(sr*Ro+sm*6*M*G))\n",
    "    e1=(sr*Ro-sm*2*M*G+esqrt)/(sm*4*M*G*sr*Ro)\n",
    "    return e1\n",
    "\n",
    "def e2(sr):\n",
    "    e2=1/(sr*Ro)\n",
    "    return e2\n",
    "\n",
    "def e3(sm,sr):\n",
    "    esqrt=np.sqrt((sr*Ro-sm*2*M*G)*(sr*Ro+sm*6*M*G))\n",
    "    e3=(sr*Ro-sm*2*M*G-esqrt)/(sm*4*M*G*sr*Ro)\n",
    "    return e3\n",
    "  \n",
    "def r(phi):\n",
    "    sm = s_sm.val\n",
    "    sr = s_sr.val\n",
    "     \n",
    "    e1v=e1(sm,sr)\n",
    "    e2v=e2(sr)\n",
    "    e3v=e3(sm,sr)\n",
    "\n",
    "    tau=np.sqrt(sm*M*G*(e1v-e3v)/2)\n",
    "    h = (e2v-e3v)/(e1v-e3v)\n",
    "\n",
    "    sigma = -tau*np.pi/2 - sps.ellipk(h)  \n",
    "    sn,_,_,_=sps.ellipj(tau*phi+sigma, h)\n",
    "    r=e3v+(e2v-e3v)*sn**2\n",
    "        \n",
    "    return r\n",
    "\n",
    "def alpha():\n",
    "    #Determine the root of solution\n",
    "    sm = s_sm.val\n",
    "    sr = s_sr.val\n",
    "     \n",
    "    e1v=e1(sm,sr)\n",
    "    e2v=e2(sr)\n",
    "    e3v=e3(sm,sr)\n",
    "\n",
    "    h = (e2v-e3v)/(e1v-e3v)   \n",
    "    phi=np.arcsin(np.sqrt(-e3v/(e2v-e3v)))    \n",
    "    angle= sps.ellipkinc(phi,  h)  \n",
    "\n",
    "    tau=np.sqrt(sm*M*G*(e1v-e3v)/2)\n",
    "    sigma = -tau*np.pi/2 - sps.ellipk(h)  \n",
    "\n",
    "    alphav = (angle-sigma) /tau-2*np.pi\n",
    "\n",
    "    return alphav\n",
    "\n",
    "#The following function updateplot runs whenever slider changes value\n",
    "def updateplot(val):\n",
    "    \n",
    "    #Solar masses/radius and magnification alpha\n",
    "    zoom = s_zoom.val\n",
    "    sm = s_sm.val\n",
    "    sr = s_sr.val\n",
    "\n",
    "    #Find x range for all valid solutions  \n",
    "    alphamax=alpha()\n",
    "\n",
    "    #Set angle range and calculate radius\n",
    "    phi= np.linspace(alphamax/2,np.pi-alphamax/2,500)\n",
    "    phi=phi[int(0.5*(1-zoom)*500):int(0.5*(1+zoom)*500)]\n",
    "    #phi= np.linspace(0,np.pi,500)\n",
    "    \n",
    "    rv=1/r(phi)\n",
    "\n",
    "    clearsetplot()    \n",
    "    \n",
    "    #Check if results a biggen than photon sphere\n",
    "    if sr*Ro>sm*3*G*M:\n",
    "        x=rv*np.cos(phi)\n",
    "        y=rv*np.sin(phi)\n",
    "    \n",
    "        #plot lightpath curve\n",
    "        ax1.plot(x,y)\n",
    "        ax1.grid()\n",
    "    \n",
    "        #Angular Deflection\n",
    "        dx=np.diff(x)\n",
    "        dy=np.diff(y)\n",
    "        dydx=dy/dx\n",
    "        da=np.arctan(dydx)\n",
    "        dam=np.max(da)\n",
    "        ax2.plot(x[:-1],-da)\n",
    "        ax2.grid()\n",
    "    \n",
    "        #Angular Distribution\n",
    "        ddy=np.diff(da)\n",
    "        ddydx=ddy/dx[:-1]\n",
    "        ax3.plot(x[:-2],-ddydx)        \n",
    "        ax3.grid()       \n",
    "        \n",
    "        plotdatatext(sm,sr,dam)\n",
    "\n",
    "        #scaling method and draw sun\n",
    "        plotoptions(x,y,-da,-ddydx,sr,sm)            \n",
    "    \n",
    "    else:\n",
    "        #Plot Text No solution\n",
    "        plotnosolutiontext()\n",
    "   \n",
    "    fig.canvas.draw_idle()\n",
    "    plt.plot\n",
    "\n",
    "def plotdatatext(sm,sr,dam):\n",
    "    e1v=e1(sm,sr)\n",
    "    e2v=e2(sr)\n",
    "    e3v=e3(sm,sr)\n",
    "    \n",
    "    tau=np.sqrt(sm*M*G*(e1v-e3v)/2)\n",
    "    h = (e2v-e3v)/(e1v-e3v)\n",
    " \n",
    "    text=('$r / r_s$: ' + str('{:.4e}'.format(Ro/(sm*2*G*M))) +    '\\n' + \n",
    "    str( '$r / r_p$: ' +'{:.4e}'.format(Ro/(sm*3*G*M))) )        \n",
    "    ax1.text(0.98,0.98,text, horizontalalignment='right',verticalalignment='top', transform=ax1.transAxes, fontsize=10)   \n",
    "\n",
    "   \n",
    "\n",
    "\n",
    "    text=(str('{:.6f}'.format(dam))      +   ' $[rad]$\\n' +\n",
    "        str('{:.6f}'.format(np.degrees(dam)))      +   '$^{\\circ}$\\n' +\n",
    "        str('{:.6f}'.format(np.degrees(dam)*3600)) +  \"$''$\\n\")         \n",
    "    ax2.text(0.02,0.98,text, horizontalalignment='left',verticalalignment='top', transform=ax2.transAxes, fontsize=10)\n",
    "    \n",
    "    text=('$e1$: ' +   str('{:.4e}'.format(e1v))      +   '\\n' +\n",
    "         str('$e2$: ' + '{:.4e}'.format(e2v))     +    '\\n' +\n",
    "          str('$e3$: ' + '{:.4e}'.format(e3v))  + '\\n' +\n",
    "        str('$h^2$: '  + '{:.4e}'.format(h)) )         \n",
    "    ax3.text(0.98,0.98,text, horizontalalignment='right',verticalalignment='top', transform=ax3.transAxes, fontsize=10)\n",
    "\n",
    "    \n",
    "def plotnosolutiontext():   \n",
    "    text=('No solution')         \n",
    "    ax2.annotate(text,xy=(0.5,0.5), xycoords='axes fraction',ha='center',color='red', fontsize=20)\n",
    "      \n",
    "#This function set scaling axis and drawes sun\n",
    "def plotoptions(x,y,dy,ddy,sr,sm):\n",
    "    i=0\n",
    "    for r in check.get_status():\n",
    "       \n",
    "        if i==0:\n",
    "            #Set scaling auto or equal\n",
    "            if r==False:\n",
    "                ax1.axis('auto')   \n",
    "                \n",
    "                #Set y scaling plot ax1 \n",
    "                range=np.max(y)-np.min(y)\n",
    "                binsy=np.linspace(np.min(y)-range*2/100,np.max(y)+range*2/100, 5)\n",
    "                ax1.yaxis.set_ticks(binsy) \n",
    "                ax1.set_yticklabels(binsy, rotation=90, va=\"center\")  \n",
    "                ax1.yaxis.set_major_formatter(ticker.FormatStrFormatter('%.1e'))\n",
    "                #plt.setp(ax2.get_yticklabels()[-1], visible=False) \n",
    "\n",
    "            else:\n",
    "                ax1.axis('equal')\n",
    "                \n",
    "                #Set y scaling plot ax1 (equal x=y and 5 bins) \n",
    "                a=ax1.get_yticks().tolist()\n",
    "                ax1.yaxis.set_major_locator(ticker.FixedLocator(a))\n",
    "                ax1.set_yticklabels(a,rotation=90, va=\"center\")  \n",
    "                ax1.yaxis.set_major_locator(ticker.MaxNLocator(5))  \n",
    "                ax1.yaxis.set_major_formatter(ticker.FormatStrFormatter('%.1e'))  \n",
    "                               \n",
    "        if i==1:\n",
    "            #Draw Sun yes/no\n",
    "              if r==True:\n",
    "                circle=plt.Circle((0,0),Ro, color='orange',label='$r_{sun}$: sun')\n",
    "                ax1.add_patch(circle)\n",
    "                \n",
    "                circle=plt.Circle((0,0),sr*Ro,fill=False, ls='--')\n",
    "                ax1.add_patch(circle)\n",
    "\n",
    "                circle=plt.Circle((0,0),sm*3*G*M, color='gray',label='$r_p$: photon sphere')\n",
    "                ax1.add_patch(circle)\n",
    "             \n",
    "                circle=plt.Circle((0,0),sm*2*G*M, color='black',label='$r_s$: Schwarzschild radius')\n",
    "                ax1.add_patch(circle)    \n",
    "                \n",
    "                ax1.legend(loc=3,fontsize='8',markerscale=0.25,facecolor='none',frameon=False,prop={'size': 8})\n",
    "         \n",
    "        i=i+1              \n",
    "    #Set x axis ticks\n",
    "    range=np.max(x)-np.min(x)  \n",
    "    max=np.max(x)          \n",
    "    binsx=np.linspace(-max-range*2/100,max+range*2/100, 5)                  \n",
    "    \n",
    "    ax1.xaxis.set_ticks(binsx)   \n",
    "    ax1.set_xticklabels(binsx, rotation=0, ha=\"left\") \n",
    "    ax1.xaxis.set_major_formatter(ticker.FormatStrFormatter('%.1e'))\n",
    "    plt.setp(ax1.get_xticklabels()[-1], visible=False) \n",
    "\n",
    "    ax2.xaxis.set_ticks(binsx)   \n",
    "    ax2.set_xticklabels(binsx, rotation=0, ha=\"left\") \n",
    "    ax2.xaxis.set_major_formatter(ticker.FormatStrFormatter('%.1e'))\n",
    "    plt.setp(ax2.get_xticklabels()[-1], visible=False) \n",
    "    \n",
    "    ax3.xaxis.set_ticks(binsx)   \n",
    "    ax3.set_xticklabels(binsx, rotation=0, ha=\"left\") \n",
    "    ax3.xaxis.set_major_formatter(ticker.FormatStrFormatter('%.1e'))\n",
    "    plt.setp(ax3.get_xticklabels()[-1], visible=False) \n",
    "\n",
    "\n",
    "    #Set y axis plots ax2 and ax3    \n",
    "    range=np.max(dy)-np.min(dy)\n",
    "    binsy=np.linspace(np.min(dy)-range*2/100,np.max(dy)+range*2/100, 5)\n",
    "    ax2.yaxis.set_ticks(binsy) \n",
    "    ax2.set_yticklabels(binsy, rotation=90, va=\"center\")  \n",
    "    ax2.yaxis.set_major_formatter(ticker.FormatStrFormatter('%.1e'))\n",
    "    #plt.setp(ax2.get_yticklabels()[-1], visible=False) \n",
    "\n",
    "    range=np.max(ddy)-np.min(ddy)\n",
    "    binsy=np.linspace(np.min(ddy)-range*2/100,np.max(ddy)+range*2/100, 5)\n",
    "    ax3.yaxis.set_ticks(binsy) \n",
    "    ax3.set_yticklabels(binsy, rotation=90, va=\"center\")  \n",
    "    ax3.yaxis.set_major_formatter(ticker.FormatStrFormatter('%.1e'))\n",
    "    #plt.setp(ax3.get_yticklabels()[-1], visible=False)  \n",
    "    \n",
    "   \n",
    "    \n",
    "def clearsetplot():\n",
    "    #plot lightpath curve\n",
    "    \n",
    "    ax1.clear()\n",
    "    ax1.set_title('Light path', fontsize=12, pad=10)\n",
    "    ax1.set_xlabel('x [m]', fontsize=12)\n",
    "    ax1.set_ylabel('y [m]', fontsize=12)\n",
    "    ax1.tick_params(axis='both', which='major', labelsize=8)\n",
    "    \n",
    "    ax2.clear()\n",
    "    ax2.set_title('Angular Deflection', fontsize=12, pad=10)\n",
    "    ax2.set_xlabel('x [m]', fontsize=12)\n",
    "    ax2.set_ylabel('deflection angle  [rad]', fontsize=12)\n",
    "    ax2.tick_params(axis='both', which='major', labelsize=8)\n",
    "    \n",
    "    ax3.clear()\n",
    "    ax3.set_title('Angular Distribution', fontsize=12, pad=10)\n",
    "    ax3.set_xlabel('x [m]', fontsize=12)\n",
    "    ax3.set_ylabel('angular change [rad/m]', fontsize=12)\n",
    "    ax3.tick_params(axis='both', which='major', labelsize=8)\n",
    "    \n",
    "#Display plot n startup.\n",
    "updateplot(1)\n",
    "\n",
    "#The following code checkes if slider changes. This line is looped automatic by pyplot\n",
    "s_sm.on_changed(updateplot);\n",
    "s_sr.on_changed(updateplot);\n",
    "s_zoom.on_changed(updateplot);\n",
    "check.on_clicked(updateplot);"
   ]
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       "            width=\"400\"\n",
       "            height=\"300\"\n",
       "            src=\"https://www.youtube.com/embed/0xngLoNqM1g\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.YouTubeVideo at 0x1575fe902b0>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import YouTubeVideo\n",
    "YouTubeVideo('0xngLoNqM1g')"
   ]
  },
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   "execution_count": null,
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