{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Spherical null geodesics in Kerr spacetime\n",
"## Computation with `kerrgeodesic_gw`\n",
"\n",
"This Jupyter/SageMath notebook is relative to the lectures\n",
"[Geometry and physics of black holes](https://relativite.obspm.fr/blackholes/).\n",
"\n",
"It requires [SageMath](http://www.sagemath.org/) (version $\\geq$ 8.2), with the package [kerrgeodesic_gw](https://github.com/BlackHolePerturbationToolkit/kerrgeodesic_gw) (version $\\geq$ 0.3.2). To install the latter, simply run \n",
"```\n",
"sage -pip install kerrgeodesic_gw\n",
"```\n",
"in a terminal."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"'SageMath version 9.2, Release Date: 2020-10-24'"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"version()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"First, we set up the notebook to use LaTeX-formatted display:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"%display latex"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"and we ask for CPU demanding computations to be performed in parallel on 8 processes:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"Parallelism().set(nproc=8)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A Kerr black bole is entirely defined by two parameters $(m, a)$, where $m$ is the black hole mass and $a$ is the black hole angular momentum divided by $m$.\n",
"In this notebook, we shall set $m=1$ and we denote the angular momentum parameter $a$ by the symbolic variable `a`, using `a0` for a specific numerical value:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"a = var('a')\n",
"a0 = 0.95"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The spacetime object is created as an instance of the class `KerrBH`:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Kerr spacetime M\n"
]
}
],
"source": [
"from kerrgeodesic_gw import KerrBH\n",
"M = KerrBH(a)\n",
"print(M)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The Boyer-Lindquist coordinate $r$ of the event horizon:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{-a^{2} + 1} + 1\n",
"\\end{math}"
],
"text/plain": [
"sqrt(-a^2 + 1) + 1"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rH = M.event_horizon_radius()\n",
"rH"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}r_+ = 1.31224989991992\n",
"\\end{math}"
],
"text/plain": [
"r_+ = 1.31224989991992"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"rH0 = rH.subs({a: a0})\n",
"show(LatexExpr(r'r_+ = '), rH0)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}r_- = 0.687750100080080\n",
"\\end{math}"
],
"text/plain": [
"r_- = 0.687750100080080"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"show(LatexExpr(r'r_- = '),\n",
" M.inner_horizon_radius().subs({a: a0}))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The method `boyer_lindquist_coordinates()` returns the chart of Boyer-Lindquist coordinates `BL` and allows the user to instanciate the Python variables `(t, r, th, ph)` to the coordinates $(t,r,\\theta,\\phi)$:"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(M,(t, r, {\\theta}, {\\phi})\\right)\n",
"\\end{math}"
],
"text/plain": [
"Chart (M, (t, r, th, ph))"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"BL. = M.boyer_lindquist_coordinates()\n",
"BL"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The metric tensor is naturally returned by the method `metric()`:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}g = \\left( -\\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} - 2 \\, r}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} t + \\left( -\\frac{2 \\, a r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} t\\otimes \\mathrm{d} {\\phi} + \\left( \\frac{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}}{a^{2} + r^{2} - 2 \\, r} \\right) \\mathrm{d} r\\otimes \\mathrm{d} r + \\left( a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2} \\right) \\mathrm{d} {\\theta}\\otimes \\mathrm{d} {\\theta} + \\left( -\\frac{2 \\, a r \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} t + \\left( \\frac{2 \\, a^{2} r \\sin\\left({\\theta}\\right)^{4} + {\\left(a^{2} r^{2} + r^{4} + {\\left(a^{4} + a^{2} r^{2}\\right)} \\cos\\left({\\theta}\\right)^{2}\\right)} \\sin\\left({\\theta}\\right)^{2}}{a^{2} \\cos\\left({\\theta}\\right)^{2} + r^{2}} \\right) \\mathrm{d} {\\phi}\\otimes \\mathrm{d} {\\phi}\n",
"\\end{math}"
],
"text/plain": [
"g = -(a^2*cos(th)^2 + r^2 - 2*r)/(a^2*cos(th)^2 + r^2) dt*dt - 2*a*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dt*dph + (a^2*cos(th)^2 + r^2)/(a^2 + r^2 - 2*r) dr*dr + (a^2*cos(th)^2 + r^2) dth*dth - 2*a*r*sin(th)^2/(a^2*cos(th)^2 + r^2) dph*dt + (2*a^2*r*sin(th)^4 + (a^2*r^2 + r^4 + (a^4 + a^2*r^2)*cos(th)^2)*sin(th)^2)/(a^2*cos(th)^2 + r^2) dph*dph"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g = M.metric()\n",
"g.display()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Spherical photon orbits"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Functions $\\ell(r_0)$ and $q(r_0)$ for spherical photon orbits:"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( a, r \\right) \\ {\\mapsto} \\ -\\frac{a^{2} {\\left(r + 1\\right)} + {\\left(r - 3\\right)} r^{2}}{a {\\left(r - 1\\right)}}\n",
"\\end{math}"
],
"text/plain": [
"(a, r) |--> -(a^2*(r + 1) + (r - 3)*r^2)/(a*(r - 1))"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r = var('r')\n",
"lsph(a, r) = (r^2*(3 - r) - a^2*(r + 1))/(a*(r -1))\n",
"lsph"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( a, r \\right) \\ {\\mapsto} \\ -\\frac{{\\left({\\left(r - 3\\right)}^{2} r - 4 \\, a^{2}\\right)} r^{3}}{a^{2} {\\left(r - 1\\right)}^{2}}\n",
"\\end{math}"
],
"text/plain": [
"(a, r) |--> -((r - 3)^2*r - 4*a^2)*r^3/(a^2*(r - 1)^2)"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"qsph(a, r) = r^3 / (a^2*(r - 1)^2) * (4*a^2 - r*(r - 3)^2)\n",
"qsph"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$\\theta$-turning points:"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( a, l, q \\right) \\ {\\mapsto} \\ \\arccos\\left(\\sqrt{\\frac{1}{2} \\, \\sqrt{{\\left(\\frac{l^{2} + q}{a^{2}} - 1\\right)}^{2} + \\frac{4 \\, q}{a^{2}}} - \\frac{l^{2} + q}{2 \\, a^{2}} + \\frac{1}{2}}\\right)\n",
"\\end{math}"
],
"text/plain": [
"(a, l, q) |--> arccos(sqrt(1/2*sqrt(((l^2 + q)/a^2 - 1)^2 + 4*q/a^2) - 1/2*(l^2 + q)/a^2 + 1/2))"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"theta0(a, l, q) = arccos(sqrt(1/2*(1 - (l^2+q)/a^2 + sqrt((1 - (l^2+q)/a^2)^2 + 4*q/a^2))))\n",
"theta0"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left( a, l, q \\right) \\ {\\mapsto} \\ \\arccos\\left(\\sqrt{-\\frac{1}{2} \\, \\sqrt{{\\left(\\frac{l^{2} + q}{a^{2}} - 1\\right)}^{2} + \\frac{4 \\, q}{a^{2}}} - \\frac{l^{2} + q}{2 \\, a^{2}} + \\frac{1}{2}}\\right)\n",
"\\end{math}"
],
"text/plain": [
"(a, l, q) |--> arccos(sqrt(-1/2*sqrt(((l^2 + q)/a^2 - 1)^2 + 4*q/a^2) - 1/2*(l^2 + q)/a^2 + 1/2))"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"theta1(a, l, q) = arccos(sqrt(1/2*(1 - (l^2+q)/a^2 - sqrt((1 - (l^2+q)/a^2)^2 + 4*q/a^2))))\n",
"theta1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Spherical photon orbit at $r_0 = 3 m$ ($q = q_{\\rm max} = 27 m^2$)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(-1.90000000000000, 27.0000000000000\\right)\n",
"\\end{math}"
],
"text/plain": [
"(-1.90000000000000, 27.0000000000000)"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r0 = 3.\n",
"E = 1\n",
"L = lsph(a0, r0)\n",
"Q = qsph(a0, r0)\n",
"L, Q"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.345877348029357\n",
"\\end{math}"
],
"text/plain": [
"0.345877348029357"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"theta0(a0, L, Q)"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Point P on the Kerr spacetime M\n"
]
}
],
"source": [
"P = M.point((0, r0, pi/2, 0), name='P')\n",
"print(P)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A geodesic is constructed by providing the range $[\\lambda_{\\rm min},\\lambda_{\\rm max}]$ of the affine parameter $\\lambda$, the initial point and either \n",
" - (i) the Boyer-Lindquist components $(p^t_0, p^r_0, p^\\theta_0, p^\\phi_0)$ of the initial 4-momentum vector\n",
" $p_0 = \\left. \\frac{\\mathrm{d}x}{\\mathrm{d}\\lambda}\\right| _{\\lambda_{\\rm min}}$,\n",
" - (ii) the four integral of motions $(\\mu, E, L, Q)$\n",
" - or (iii) some of the components of $p_0$ along with with some integrals of motion. "
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [],
"source": [
"lmax = 100 # lambda_max"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 3.00000000000000 \\frac{\\partial}{\\partial t } + \\left( 9.36596633575423 \\times 10^{-9} \\right) \\frac{\\partial}{\\partial r } + 0.577350269189626 \\frac{\\partial}{\\partial {\\theta} }\n",
"\\end{math}"
],
"text/plain": [
"p = 3.00000000000000 d/dt + (9.36596633575423e-9) d/dr + 0.577350269189626 d/dth"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"Li = M.geodesic([0, lmax], P, mu=0, E=E, L=L, Q=Q, a_num=a0,\n",
" name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 3.00000000000000 \\frac{\\partial}{\\partial t } + 0.577350269189626 \\frac{\\partial}{\\partial {\\theta} }\n",
"\\end{math}"
],
"text/plain": [
"p = 3.00000000000000 d/dt + 0.577350269189626 d/dth"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"v0 = Li.initial_tangent_vector()\n",
"Li = M.geodesic([0, lmax], P, pt0=v0[0], pr0=0, pth0=v0[2], pph0=v0[3],\n",
" a_num=a0, name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The numerical integration of the geodesic equation is performed via `integrate()`, by providing the integration step $\\delta\\lambda$:"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
" quantity value initial value diff. relative diff.\n",
" $\\mu^2$ 1.1443998526594612e-11 -0.0 1.144e-11 -\n",
" $E$ 1.00000000000222 1.00000000000000 2.215e-12 2.215e-12\n",
" $L$ -1.89999999999583 -1.90000000000000 4.173e-12 -2.196e-12\n",
" $Q$ 27.0000000000166 27.0000000000000 1.663e-11 6.159e-13"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Li.integrate(step=0.001, method='dopri5')\n",
"Li.check_integrals_of_motion(0.999*lmax)"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Final point: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(t, r, {\\theta}, {\\phi}\\right) \\verb|=| \\left(291.1717434701057, 3.0, 2.144493861874287, -39.55667803565552\\right)\n",
"\\end{math}"
],
"text/plain": [
"(t, r, th, ph) '=' (291.1717434701057, 3.0, 2.144493861874287, -39.55667803565552)"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"print(\"Final point: \")\n",
"show(BL[:], \"=\", BL(Li(0.999*lmax)))"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"max lambda (plot): 32.0000000000000\n"
]
},
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lplot = 0.32*lmax\n",
"print(\"max lambda (plot): \", lplot)\n",
"Li.plot(prange=(0, lplot), plot_points=2000, thickness=3, color='green',\n",
" display_tangent=True, scale=0.15, width_tangent=2, color_tangent='green', \n",
" plot_points_tangent=20, horizon_color='lightgrey') \\\n",
" + line([(0,0,-3), (0,0,3)], color='black', thickness=2)"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"Graphics object consisting of 22 graphics primitives"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"graph = Li.plot(coordinates='xy', prange=(0, lplot), plot_points=2000, \n",
" thickness=3, color='green', display_tangent=True, scale=0.15, \n",
" width_tangent=2, color_tangent='green', plot_points_tangent=20, \n",
" horizon_color='lightgrey', axes_labels=[r'$x/m$', r'$y/m$'])\n",
"graph.save(\"gik_spher_3d_r_30_xy.pdf\")\n",
"graph"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"Graphics object consisting of 22 graphics primitives"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Li.plot(coordinates='xz', prange=(0, lplot), plot_points=2000, \n",
" thickness=3, color='green', display_tangent=True, scale=0.15, \n",
" width_tangent=2, color_tangent='green', plot_points_tangent=20, \n",
" horizon_color='lightgrey', axes_labels=[r'$x/m$', r'$z/m$'])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Prograde spherical photon orbit at $r_0 = 1.6m$"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(2.17105263157895, 5.97569713758080\\right)\n",
"\\end{math}"
],
"text/plain": [
"(2.17105263157895, 5.97569713758080)"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r0 = 1.6\n",
"L = lsph(a0, r0)\n",
"Q = qsph(a0, r0)\n",
"L, Q"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.705442812649839\n",
"\\end{math}"
],
"text/plain": [
"0.705442812649839"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"theta0(a0, L, Q)"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 7.66666666666666 \\frac{\\partial}{\\partial t } + \\left( 1.74608999691187 \\times 10^{-24} + 2.85158136717879 \\times 10^{-8}i \\right) \\frac{\\partial}{\\partial r } + 0.954892151626195 \\frac{\\partial}{\\partial {\\theta} } + 2.45614035087719 \\frac{\\partial}{\\partial {\\phi} }\n",
"\\end{math}"
],
"text/plain": [
"p = 7.66666666666666 d/dt + (1.74608999691187e-24 + 2.85158136717879e-8*I) d/dr + 0.954892151626195 d/dth + 2.45614035087719 d/dph"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"P = M.point((0, r0, pi/2, 0), name='P')\n",
"lmax = 70\n",
"Li = M.geodesic([0, lmax], P, mu=0, E=E, L=L, Q=Q, a_num=a0,\n",
" name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 7.66666666666666 \\frac{\\partial}{\\partial t } + 0.954892151626195 \\frac{\\partial}{\\partial {\\theta} } + 2.45614035087719 \\frac{\\partial}{\\partial {\\phi} }\n",
"\\end{math}"
],
"text/plain": [
"p = 7.66666666666666 d/dt + 0.954892151626195 d/dth + 2.45614035087719 d/dph"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"v0 = Li.initial_tangent_vector()\n",
"Li = M.geodesic([0, lmax], P, pt0=v0[0], pr0=0, pth0=v0[2], pph0=v0[3],\n",
" a_num=a0, name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
" quantity value initial value diff. relative diff.\n",
" $\\mu^2$ -1.6876999797688086e-11 -3.552713678800501e-15 -1.687e-11 4750.\n",
" $E$ 0.999999999991714 1.00000000000000 -8.286e-12 -8.286e-12\n",
" $L$ 2.17105263156127 2.17105263157895 -1.768e-11 -8.144e-12\n",
" $Q$ 5.97569713752106 5.97569713758080 -5.974e-11 -9.996e-12"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Li.integrate(step=0.0004, method='dopri5')\n",
"Li.check_integrals_of_motion(0.999*lmax)"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Final point: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(t, r, {\\theta}, {\\phi}\\right) \\verb|=| \\left(492.9599102672748, 1.599974600261711, 0.8276182451833887, 185.0198306183178\\right)\n",
"\\end{math}"
],
"text/plain": [
"(t, r, th, ph) '=' (492.9599102672748, 1.599974600261711, 0.8276182451833887, 185.0198306183178)"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"print(\"Final point: \")\n",
"show(BL[:], \"=\", BL(Li(0.999*lmax)))"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"max lambda (plot): 7.70000000000000\n"
]
},
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lplot = 0.11*lmax\n",
"print(\"max lambda (plot): \", lplot)\n",
"Li.plot(prange=(0, lplot), plot_points=2000, thickness=3, color='green',\n",
" display_tangent=True, scale=0.03, width_tangent=1, color_tangent='green', \n",
" plot_points_tangent=20, horizon_color='lightgrey') \\\n",
" + line([(0,0,-1.5), (0,0,1.5)], color='black', thickness=2)"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Li.plot(prange=(0, lmax), plot_points=2000, thickness=3, color='green',\n",
" display_tangent=True, scale=0.03, width_tangent=1, color_tangent='green', \n",
" plot_points_tangent=40, horizon_color='lightgrey') \\\n",
" + line([(0,0,-1.5), (0,0,1.5)], color='black', thickness=2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Spherical photon orbit at $r_0=2.8m$"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(-1.08859649122807, 26.2604206422489\\right)\n",
"\\end{math}"
],
"text/plain": [
"(-1.08859649122807, 26.2604206422489)"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r0 = 2.8\n",
"L = lsph(a0, r0)\n",
"Q = qsph(a0, r0)\n",
"L, Q"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.206050206829550\n",
"\\end{math}"
],
"text/plain": [
"0.206050206829550"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"theta0(a0, L, Q)"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 3.22222222222222 \\frac{\\partial}{\\partial t } + \\left( 4.65527024480212 \\times 10^{-25} + 7.60263326216718 \\times 10^{-9}i \\right) \\frac{\\partial}{\\partial r } + 0.653634213345212 \\frac{\\partial}{\\partial {\\theta} } + 0.116959064327486 \\frac{\\partial}{\\partial {\\phi} }\n",
"\\end{math}"
],
"text/plain": [
"p = 3.22222222222222 d/dt + (4.65527024480212e-25 + 7.60263326216718e-9*I) d/dr + 0.653634213345212 d/dth + 0.116959064327486 d/dph"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"P = M.point((0, r0, pi/2, 0), name='P')\n",
"lmax = 100\n",
"Li = M.geodesic([0, lmax], P, mu=0, E=E, L=L, Q=Q, a_num=a0,\n",
" name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 3.22222222222222 \\frac{\\partial}{\\partial t } + 0.653634213345212 \\frac{\\partial}{\\partial {\\theta} } + 0.116959064327486 \\frac{\\partial}{\\partial {\\phi} }\n",
"\\end{math}"
],
"text/plain": [
"p = 3.22222222222222 d/dt + 0.653634213345212 d/dth + 0.116959064327486 d/dph"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"v0 = Li.initial_tangent_vector()\n",
"Li = M.geodesic([0, lmax], P, pt0=v0[0], pr0=0, pth0=v0[2], pph0=v0[3],\n",
" a_num=a0, name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
" quantity value initial value diff. relative diff.\n",
" $\\mu^2$ 6.619788051054343e-12 2.536165721878092e-15 6.617e-12 2609.\n",
" $E$ 1.00000000000134 1.00000000000000 1.337e-12 1.337e-12\n",
" $L$ -1.08859649122876 -1.08859649122807 -6.950e-13 6.384e-13\n",
" $Q$ 26.2604206422658 26.2604206422489 1.695e-11 6.453e-13"
]
},
"execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Li.integrate(step=0.001, method='dopri5')\n",
"Li.check_integrals_of_motion(0.999*lmax)"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Final point: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(t, r, {\\theta}, {\\phi}\\right) \\verb|=| \\left(310.4228769082733, 2.8, 2.462502125969338, -39.07005458983995\\right)\n",
"\\end{math}"
],
"text/plain": [
"(t, r, th, ph) '=' (310.4228769082733, 2.8, 2.462502125969338, -39.07005458983995)"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"print(\"Final point: \")\n",
"show(BL[:], \"=\", BL(Li(0.999*lmax)))"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"max lambda (plot): 38.0000000000000\n"
]
},
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lplot = 0.38*lmax\n",
"print(\"max lambda (plot): \", lplot)\n",
"Li.plot(prange=(0, lplot), plot_points=2000, thickness=3, color='green',\n",
" display_tangent=True, scale=0.12, width_tangent=2, color_tangent='green', \n",
" plot_points_tangent=20, horizon_color='lightgrey')"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"Graphics object consisting of 22 graphics primitives"
]
},
"execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"graph = Li.plot(coordinates='xy', prange=(0, lplot), plot_points=2000, \n",
" thickness=3, color='green', display_tangent=True, scale=0.01, \n",
" width_tangent=1, color_tangent='green', \n",
" plot_points_tangent=20, horizon_color='lightgrey',\n",
" axes_labels=[r'$x/m$', r'$y/m$'])\n",
"graph.save(\"gik_spher_3d_r_28_xy.pdf\")\n",
"graph"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Li.plot(coordinates='txy', prange=(0, lplot), plot_points=2000, \n",
" thickness=3, color='green', display_tangent=True, scale=0.2, \n",
" width_tangent=2, color_tangent='green', \n",
" plot_points_tangent=20, horizon_color='lightgrey')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Retrograde spherical photon orbit at $r_0 = 3.9 m$"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(-6.57395644283122, 3.52474056409564\\right)\n",
"\\end{math}"
],
"text/plain": [
"(-6.57395644283122, 3.52474056409564)"
]
},
"execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r0 = 3.9\n",
"L = lsph(a0, r0)\n",
"Q = qsph(a0, r0)\n",
"L, Q"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.29003447185548\n",
"\\end{math}"
],
"text/plain": [
"1.29003447185548"
]
},
"execution_count": 44,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"theta0(a0, L, Q)"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {},
"outputs": [],
"source": [
"P = M.point((0, r0, pi/2, 0), name='P')"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 2.37931034482759 \\frac{\\partial}{\\partial t } + \\left( 2.22112222707613 \\times 10^{-8} \\right) \\frac{\\partial}{\\partial r } + 0.123433875307859 \\frac{\\partial}{\\partial {\\theta} } -0.326678765880218 \\frac{\\partial}{\\partial {\\phi} }\n",
"\\end{math}"
],
"text/plain": [
"p = 2.37931034482759 d/dt + (2.22112222707613e-8) d/dr + 0.123433875307859 d/dth - 0.326678765880218 d/dph"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"lmax = 100\n",
"Li = M.geodesic([0, lmax], P, mu=0, E=E, L=L, Q=Q, a_num=a0,\n",
" name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 2.37931034482759 \\frac{\\partial}{\\partial t } + 0.123433875307859 \\frac{\\partial}{\\partial {\\theta} } -0.326678765880218 \\frac{\\partial}{\\partial {\\phi} }\n",
"\\end{math}"
],
"text/plain": [
"p = 2.37931034482759 d/dt + 0.123433875307859 d/dth - 0.326678765880218 d/dph"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"v0 = Li.initial_tangent_vector()\n",
"Li = M.geodesic([0, lmax], P, pt0=v0[0], pr0=0, pth0=v0[2], pph0=v0[3],\n",
" a_num=a0, name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 48,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
" quantity value initial value diff. relative diff.\n",
" $\\mu^2$ 1.5175131713295736e-11 6.661338147750939e-16 1.517e-11 22780.\n",
" $E$ 1.00000000000555 1.00000000000000 5.549e-12 5.549e-12\n",
" $L$ -6.57395644284764 -6.57395644283122 -1.643e-11 2.499e-12\n",
" $Q$ 3.52474056410325 3.52474056409564 7.611e-12 2.159e-12"
]
},
"execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Li.integrate(step=0.001, method='dopri5')\n",
"Li.check_integrals_of_motion(0.999*lmax)"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Final point: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(t, r, {\\theta}, {\\phi}\\right) \\verb|=| \\left(237.38270015383975, 3.9, 1.6881755637536546, -34.301693666892874\\right)\n",
"\\end{math}"
],
"text/plain": [
"(t, r, th, ph) '=' (237.38270015383975, 3.9, 1.6881755637536546, -34.301693666892874)"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"print(\"Final point: \")\n",
"show(BL[:], \"=\", BL(Li(0.999*lmax)))"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"max lambda (plot): 55.0000000000000\n"
]
},
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lplot = 0.55*lmax\n",
"print(\"max lambda (plot): \", lplot)\n",
"Li.plot(prange=(0, lplot), plot_points=2000, thickness=3, color='green',\n",
" display_tangent=True, scale=0.15, width_tangent=2, color_tangent='green', \n",
" plot_points_tangent=20, horizon_color='lightgrey')"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {},
"outputs": [
{
"data": {
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8BHrW6VkpRmVVtsRkxLIsdt3ahV5Le5X7Gg3CG2BR/0W4knkFE7ZOgIAR4Nob11AnrI4LIyVPUGICJSaPt/f2Xkz6ZxKuZl6t8LUaRTTC+NbjMaL5CITIQioenBeprImppLSCNLRZ3AYP8h6Uqz4DBixYiAQiDGs6DH8M/MO1ARKAEhMASkweK1uVjbe2v4W/zv9VoeuIBWK80OgFTGg9Ac/UfMannhs5wtMn2LpDpjITz654FkcfHC33NRgwuDb5GuqG1XVhZASUmABQYvJIG65uwIStE5BWkFbuawRJg/C/dv/D661epwfYoBaTNfcU9zBx60Rsvb7V6boMGLzQ8AWsGbKGg8gqNUpMoMTkUTKVmZi8bTJWXlxZ7mtUD6qOnrV7Yn6/+bT1QQmUmMqW/DAZ4zaPw7n0c07XndNjDt5u93ala4lziL6QoMTkMTZe3Yhxm8chQ5lRrvrP1HwG77Z/F33r9aUZ/FZQYrLNwBqw/MJyvLvrXadb6i80egEL+i1AlH8UR9FVKpSYQInJ7ZRaJd7Z8Q4WnlpYrvrda3fH9M7T0bFGRxdH5lsoMTmmQFOALw9+idlHZlvdSqUskX6R+PXZXzGgwQAOo6sUKDGBEpNbnU8/j5fXvozLGZedrtuzTk9M7zwd7WPbcxCZ76HE5DiWZdFveT9su7HN6bpT2k7BV92/glQk5SCySoESEwDq83EDlmXxw/EfkLA4wemk1D62PY68dgQ7Xt1BSYlw4tczv5YrKQHA98e/R9tf2uJmNndrNxLfRy0mnuWp8zBqw6hyrd6w89Wd6FGnBwdR+T5qMTnmQvoFtF7c2rQvVkWseHEFhjbx7I0sPRC1mEAtJl5dzriMhMUJTieln/r+BP3HekpK5TB//nzEx8ejTZs27g7FK7yy7hWXJCUAeHntyxiyZgi0eq1LrkcqD2ox8WT1pdUYvXE0CrWFDtdJjEvE9mHbqb/eBajF5JjPkj7DlmtbcPHxRah0KgCASCCCgTXAwBrKfd20d9JoPp1jqMUESkyc0xv0eH/3+5hzdI5T9c785wxtPeBClJicY2ANuJN7B+fTz+Nc2jmcSz+H049O457iHthy/pqvH7IeAxsOdG2gvocSEygxcSpfnY+X177s1Mz6/vX7Y8OQDRAKhBxGVvlQYnKNAk0BLj2+hHPp53Dk/hGsv7IeeZo8h+t3iO2Ag6MP0oTcstEXBpSYOHNPcQ8DVgzA+fTzDtdJGpWEZ2o+w2FU3mvBggVYsGAB7ty5AwBo3LgxPv74Y/Tp08eh+pSYuKE36PH5gc/x+YHPnerqS307FVUDq3IYmdeixAQa/MCJ5IfJSFic4HBSahrVFJnvZlJSsqF69eqYNWsWTp48iZMnT6Jr16547rnncOnSJXeHVqkJBUJ8kvgJ9o7Yi2qB1RyuF/NtDNZcpnX2iHXUYnKx9VfWY9i6YaYHx/a81+E9fNH1C+q6K4ewsDDMnj0bY8aMsVuWWkzcy1RmYvTG0dhybYvDdQbHD8bvz/0Of4k/h5F5FWoxgVpMLrX41GIMWj3I4aT096C/Mav7LEpKTtLr9Vi5ciUKCwvRrl07q2XUajXy8vLMXoRbEX4R2Dh0I77o+gUYB/++rr68GqFfheJcmvMLyBLfRYnJBViWxaxDs/D6ltcd6mevG1YXFyZcwODGg3mIzndcuHABAQEBkEqlGD9+PNavX4/4+HirZWfOnIng4GDTKzY2ludoKycBI8AHnT7AtmHbECoLdaiO1qBFi0Ut8MfZP7gNjngN6sqrIANrwLs738W3x751qHznmp2xbsg6hMnDOI7M92g0Gty7dw+5ublYu3YtfvnlFyQlJVlNTmq1Gmq12vRxXl4eYmNjqSuPR7dybuGFVS84tZ3GmJZjMK/PPMjFcg4j82jUlQdKTBWiM+gwbvM4h9/pjWg+AosHLIZEKOE2sEqie/fuqFOnDhYtWmS3LD1jcg+lVolxm8dh+YXlDtdpUaUF1gxegzphdTiMzGNRYgJ15ZWbzqDDiPUjHE5KM7rMwB/P/UFJyYVYljVrFRHP4yf2w9Lnl+KzxM8crnM27Sxa/dwKO2/u5DAy4skoMZWDzqDD8PXDseLiCrtlJUIJVry4Ah8+8yFNKqyADz74AAcPHsSdO3dw4cIFfPjhh9i/fz+GDRvm7tCIHQzD4KPOH2HFiysgFTq2vJZCrUDfZX0x7/g8uLhXh3gBkbsD8DY6gw7D1g3D35f+tls2UBKIjUM3okutLjxE5tvS09MxfPhwPHr0CMHBwWjWrBm2b9+OHj1oYVtvMbTJUNQMronnVj7n0E7NelaPN7e/iUsZlzCvzzyIhWIeoiSegJ4xOUGr1+L5Vc87tMRQpF8ktr+6HU9VfYqHyIg99IzJc9zOuY1+y/vhSuYVh+t0ieuC1YNXI9wvnMPIPAJ1q4C68hxmYA3otbSXQ0mpZnBNHHrtECUlQqyoFVoLB0cfRNtqbR2us+/OPnT4rQPu5t7lMDLiKSgxOYBlWfT4qwf23dlnt2x8ZDwOv3YY9cPr8xAZId4p3C8ce0bsQe+6vR2uk5KVgna/tnNq/UninSgxOaDdr+2w9/Zeu+WaRzdH0qgkVAtyfM0wwi3aKNBz+Uv8sWnoJgxr6vgAlkcFj9Dp907Yf2c/d4ERt6NnTFawLIvTj05jU8omzD4y26ElhuKC43Dy9ZOVoQ/cK9EzJs9lYA14e8fb+P749w7XkQglWPr8Ul9cPYWeMYFG5Zk59uAY/jj7B9ZfXY/HhY8drseAwaHXDlFSIqQcBIwAc3vNhb/YH18e+tKhOhq9BkPWDEGhthCjWoziNkDCO0pMJQxfPxw3sm84Xe+lxi9R9x0hFcAwDGZ0nQGxUIxPkz51qA4LFqM3joZGr8HrrV7nOELCJ3rGVMKSgUscngBY0tinxnIQDSGVC8Mw+CTxE3ze5XOn6v1ny3/w44kf7ZajibregxJTCW2rt0Xb6o4PYQWAKP8odImjCbSEuMr/PfN/+Kr7V07VmbxtMuYenWv1XKGmEMPXD0fVb6qiSFfkihAJx6gr7wm9QY9X1r2CA3cPOFxHyAgxsvlI2k+JEBeb2mEqDKwB0/ZMc7jO2zvfBgD8t91/TceuZFzB86ueR0pWCgDg4uOLaB3T2rXBEpejFhOKRwWN3TzWoWWGStKzegxvNpyjqAip3N7v+D4+7PShU3Xe3vk2fjn9CwBgxYUVeOrnp8yeG59NO+vKEAlHKn2LiWVZTNw6sVyblDWObIym0U1dHxRxmfnz52P+/PnQ6/XuDoWUw+ddPke+Oh8/nPjB4Tqvb34dqy6twu5bu82OiwVinHl0xtUhEg5U6nlMLMtiyvYpmHdintN1BYwAX3X/Cv9r/z8OIiOuRvOYvJeBNeD1za/j1zO/VvhaCdUScHzscRdExRmax4RK3JXHsiym7prqUFJirPyssCyLV5q+wkVohJASBIwAi/ovwkuNX6rwtc6nn4eBNbggKsKlSpuYZh2ahTlH5zhUdnaP2ZjafqrpYwEjQJdaXRATGMNVeISQEoQCIZYMXILEuMQKXadIV1SuuYqEX5UyMf1x9g98sPcDh8q+mfAm3m73NmZ1n4WJbSYCKO5aGNV8FIcREkJKk4qkWD9kPZpENanQdWgAhOerdInpn+v/YOwmxybE9q/fH9/2+hYMw4BhGMzrMw8jmo9AmDwMzzd6nuNICSGlhchCsG3YNlQPql6u+jQAwjtUqsR0/MFxDF49GHrW/git+Mh4LHthmdkcJQEjwJ8D/8T9/95HgCSAy1AJIWWoHlQd24ZtQ6Ak0Om6WoMWpx+d5iAq4kqVJjHdyL6Bfsv7QalV2i0bKgvFxqEbESS1PnrLT+zn6vAIIU5oEtUE64esh5BxfnL7qUenOIiIuFKlSEyKIgWeXfEsslRZdssKGSH+Hvw36obV5SEywjXaj8l3davdDd/3dnyrDKMsVRbSCtI4iIi4is8nJr1Bj5fXvowrmVccKv9Nz2/QvXZ3jqMifJk0aRIuX76M5ORkd4dCODCxzUSMbzXe6XrOrvJC+OXziWnqrqnYdmObQ2VfafoK3mz7JscREUJchWEY/NDnB6eHkX+872OkF6RzExSpMJ9OTL+e/hXfHvvWobINIxpiUf9FYBiaeE2INxELxVgzeA1qh9Z2uI5CrcCg1YOg1Ws5jIyUl88mpuSHyZiwdYJDZf3EflgzeA2NtCPES4X7hWPdS+sgE8nKLCMRSsw+PnTvED7a9xHXoZFy8MnElKXMKn43ZHDs3dDCfgvROKoxx1ERQrjUvEpz/NT3pzLPB0gCEOkXaXbsq8NfYceNHVyHRpzkc4nJwBowYsMI3FPcc6j82JZjMbw5bV1BiC8Y3XI0xrQcY/VctiobjSIbQSwQmx0fvn44UvNT+QiPOMjnEtOsQ7Pwz/V/HCrbMKIhvu/j/HBTQojnmtdnHlpUaWH13MG7B/Fsg2fNjmUoM/DqulehN9DWKJ7CpxLTvtv7HO4zFgvEWPbCMposS4iPkYvlWPvSWgRLgy3OsWBx8fFF9K7b2+z4vjv7MPPQTL5CJHb4TGLKUmbh1fWvOryk/YyuM/BU1ac4joq4G02wrZxqh9bGov6LrJ5LyUpBg/AGFuvtfZr0Ka2j5yF8YqNAlmUxePVgrL2y1qHyiXGJ2D18t9k6eMS30UaBldPojaOt7k4tFUqxoN8CjNs8zmztzKZRTZE8LhlSkZTHKM3QfBX4SItpybklDielEFkIlgxcQkmJkErgh94/oE5oHYvjar0af53/C+91eM/s+IXHF/Bp0qd8hUfK4PWJ6VbOLbyx7Q2Hy3/X6zvEBsdyGBEhxFMESgOx/MXlEAlEFuf23dmHWqG10Cy6mdnxrw5/heMPPHr7dZ/n1YlJb9Bj+PrhKNAUOFS+d93eGNF8BMdRES7MnDkTbdq0QWBgIKKiojBw4ECkpKS4OyziBRKqJeCzxM+snvtw74f4sc+PZkPIDawBIzeMhFqn5itEUopXJ6Yfjv+AI/ePOFQ2QBJASw55saSkJEyaNAnHjh3Drl27oNPp0LNnTxQWFro7NOIFpnaYirbV2locf1z4GJtSNmF65+lmx1OyUvDV4a/4Co+U4rWDH27n3EaTBU0c2l8JABb0W4DxrZ1fhZh4poyMDERFRSEpKQnPPPOM3fI0+IFcybiClotaQq23bAldnngZIzeMRHLqv6vQS4VSXJhwAfXC6/EZJr1zhpe2mFiWxetbXnc4KSXGJeL1Vq9zHBXhk0KhAACEhYVZPa9Wq5GXl2f2IpVbo8hG+DTR+sCG93a/h58H/AwB8++fRLVejQlbJ8DFb96JA7wyMf157k/svrXbobISoQSL+i8y+4Ej3o1lWbz99tvo2LEjmjRpYrXMzJkzERwcbHrFxtKAFwK80/4dtImxnNO2+dpmpBekY0rbKWbH99zegxUXV/AVHnnC67ry0gvS0Wh+I+QU5ThU/v86/R8+7/o5x1ERPk2aNAlbt27FoUOHUL16datl1Go11Op/u2zy8vIQGxtLXXkElx5fQpMFlm9oGkY0xLExx9BkQRM8yHtgOh7lH4Vrb1xDsMxyJQkOUFcevLDF9N7u9xxOSnEhcZjWaRrHERE+TZ48GZs2bcK+ffvKTEoAIJVKERQUZPYiBAAaRzXGx898bHH8auZVrL+6HvP6zDM7/rjwMb48+CVf4RF4WWI6ev8o/jz3p8Pl5/WZR2vh+QiWZfHGG29g3bp12Lt3L2rVquXukIgXm9ZpGmqFWP4MTdw6EX3q9kH/+v3Njn93/DvcyrnFV3iVntckJr1Bj8nbJjtc/tkGz1r8cBHvNWnSJCxduhTLly9HYGAg0tLSkJaWBpVK5e7QiBeSiWRY0G+BxXGVToWFJxfim57fmE3K1eg1eH/3+3yGWKl5TWL69cyvOPXolENlpUIpvuv1HbcBEV4tWLAACoUCiYmJqFq1qum1atUqd4dGvFSvur3wUuOXLI6/teMtVA2oijfamK8os/ryahy6d4iv8Co1r0hMlx5fcqq1NKXtFNQKpa4eX8KyrNXXqFGj3B0a8WJze821evybo9/g484fI0xuPh3h7R1v0/BxHnh0YjKwBiw8uRAtF7WERq8xOxcoCbRaJ8IvAh90+oCP8AghXi4mMAZzesyxOP5p0qdgGAafdP7E7HhyajI2XN3AT3CVmMcmpquZV9Hxt46YsHUCtAat2bko/yjka/Kt1vss8TO+hnUSL0D7MRF7Jre13hsz+/BsjG89HvXD65sd/3j/xw7v+0bKx+PmMWn0Gnx9+GvT0vM6g86iTN3QuriRc8PieKOIRjg/4bzVlYRJ5UZLEhFbNlzdgOdXPW9xPH9aPrZc24KX175sdnzliysxpMkQLkKheUzwsBbT8QfH0Xxhc3y872PoDDqrSQkAbuVaH7b5dY+vKSkRQpz2XIPnLJ4nAcAn+z/BS41fQpMo8wm5nyR9Ar1Bb1GeuIZHJKYCTQHe3PYm2v3aDtezroO10/Cy1oxuH9se/er14ypEQogPYxgGe0bssTj+zdFvoNapLdbYu5p5FcsvLOcrvErH7Ylp2/VtaPBjA8xPng8WrNk2x86Y0WUGbWlBCCm3FlVaoGlUU4vjH+/7GM83fB4tq7Q0O/7loS/pWRNH3JqYPtjzAfou74u0grQKfYO71uqKLrW6uDAyQkhltOWVLRbH5hydAwNrwGddzDcbvJp5FVuuWZYnFefWxBTlHwUhI6zwyt+fd6FFWgkhFVcjuAaaRze3OD77yGz0q9fPokX19eGv+QqtUnFrYnrr6bdwfsJ5dIjtAABgyjEgpW+9vmgf297VoRFCKqkdr+6wODZtzzQwDIOpHaaaHT98/zAO3zvMV2iVhtufMcVHxmPfyH1YNWgVovyjnE5O1lYJJoSQ8ooOiEaN4BoWx5ecW4IhjYcgNsh8b6/ZR2bzFVql4fbEBBSPiHmp8Uu48eYNi2GZtnSt1RVtq7flMDLi7WiCLSmP5HHJFsdGbhgJsVCMt9u9bXZ8Y8pGrLm8hq/QKgWPSExGhZpCpGSlOFx+Wkfaa4nYNmnSJFy+fBnJyZZ/aAgpS5R/FMQCscXxmQdn4tiDYxbHV1ygXW5dyaMS06JTi8zWxBMLxIj2j7ZatnVMa3Sr1Y2v0AghlcyliZcsjn2w9wOsvbLW4vie23ug1Cr5CKtS8JjEpNap8VPyT2bHXm32Kmb3sN5/O63jNJq3RAhxufSCdPx86mdM+meS1fPWVqRRqBVYdZG2YHEVj1m/5+9LfyO9MN3s2JS2U/DOzncsyjYIb4CBDQfyFBkhpDLILcrFcyufw8G7B8GCdXoay4KTCzC65WiOoqtcPKLFxLIsvj/+vdmxxLhEhMpDsee25TIhU9pOqfDcJ0IIKUnACJCS+e8zbmcn/SenJuNk6klXh1UpecRf9+TUZIvdaae0nWL1gaJUKMXw5sP5Co0QUkkESYNwYPQBhPuFQ8gIy3WN38/87uKoKiePSEy/nv7V7OOawTUxoP4ALDq1yKLspDaTECAJ4Cs0QkglUj+8PpJGJSFIGlSu5LTy0kqLTU2J89yemAo1hVhx0bxlNKblGFzOuIzbubctyk9KsP5AkhBraB4TcVZ8ZDz2j9qPAEmA08kpW5WNbde3cRRZ5eH2xLTm8hqz3WgZMBjVYhSWXVhmUbZ//f6oHVqbz/CIl6N5TKQ8mkU3w96Re+En9rP5PJsBY7Ek2pLzS7gOz+e5PTH9esa8G69X3V6oFlQNXx3+yqLs5ATrWyATAhQPotHr9dBoNCgqKoJSqURBQQEKCgqgUqnQvn17qFQqFBQUoLCwEEVFRdBoNNDpdDAYDHDxbs7Eyz1V9SnsHrEbMpGszKXSovyjMKr5KLNjW65tQY4qh4cIfZdbh4tfz7qOg/cOmh0b03IMkh9af3fbvXZ3PsIiXkCv10On00Gr1UKn00Gn00GvN9/Ly5ioAECtVqN169YoLCyETqeDQCCAQGD+voxhGIjFYohEItO/QqGQ5stVYgnVErBr+C50W9INRboii/NSkRSDGw/G5G2TodarAQAavQarL6/G661e5ztcn+HWxLTy4kqzjyP8IvBsg2fxzg7LuUufJn5KQ8QrMZZlodFooFarodFoTAnHYDCYjmm1Wuj1etPLYPh3uK9Op0O1atWQlpYGkaj4x55hGAiFQtNLJBJBIpFAKpWalTEek0gkEArLN1qLeK/2se2x49Ud6L6kO7QGrdm5e4p7kAqlGNBggNl6eeuurKPEVAFuTUyrLpnPlB7SeAjEAjF+TP7RouzoFjRxrbIpmXTUajVYloVOp4NSqTQd1+mKZ+GzLGvqjiv9AgCNRoNjx47hqaeegkQiAcMwVl/GVpRAIIBEIoFEIoGfnx+kUikYhoFIJIJUKoVMJqMkVYk8U/MZbBu2Dd3/suy1WXtlLQbHDzZLTHtu70FuUS5CZCE8Ruk73JaYLj2+hEsZ5mtRDWk8BJczLluUjfCLQGxwrMVx4pt0Oh1UKhWKiopMyUmpVEKlUkGr1ZqSUMmXvedDer0eOTk5Fi0pa4zdfBqNBkKhEHl5eRAIBJDL5fDz84NcLkdhYSEkEgnkcrkp0RHf1q12N4xpOcbiufiCkwuwfdh2SIVSU3eezqDD1mtbMazZMHeE6vXc1jdWurVULbAaOtToYLECBAAs6LeAr7CIm7Asi6KiIuTk5CArKwv5+fnIzs7GgwcPkJaWBoVCAZVKBbVabfrX2HXn6kELBoMBOp0OGo3GlCDVajXy8/ORkZGB+/fvIyMjA/n5+cjNzUVWVhYKCwvtJjzi/T7o9IHFsSP3j+Bh/kP0qNPD7Pi6q+v4CsvnuKXFxLKsRWIaHD8YAkaAxacXW5R/tsGzfIVGeGZ8dlRQUACdToeioiLk5+dDqVSaBi8YR825i7FVptVqTc+l9Hq9qdUUGBho6mI0tqhKD6wgvqF2aG3ER8Zb9Oz8fuZ3vNDwBWy5tsV0bNv1bVBqlfAT+/EdptdzS2K6lHEJ17KumR0b0mQIHuU/sihbJ7QOJEIJX6ERHhkTklarhUqlQm5uLjQajanFYnx+VBEnTpxAcnKyy1pVxudcxpF9er0earUaOTk5CA4Ohl6vh0qlMiUo6uLzPUMbD8XH+813zl5+cTlOv34aAkZgWmNPpVNh7+296F+/vzvC9GpueVtX8l0FAMQGxaJttbb47th3FmWpG8/36PV65ObmIicnB4WFhUhPT8fjx49N3WZFRUUuSUoAkJCQgEmTJmHcuHEuuV5JxudfxvlQ2dnZSE1NhUKhMHVFqtVql9+XuNeL8S9aHHuQ9wBXMq+gY42OZsd33dzFV1g+xS0tpq3Xt5p9PKD+ADAMg6+PfG1RtmutrnyFRThmfI6Un58PrVaLnJwcKJVKUzdZ6XlI3sLYHckwjKnrMS8vD6GhodDpdJDL5QgICKDuPR/RKKIRGoQ3sNhte9n5ZehVpxcO3D1gOrbj5g6+w/MJvP+mZCmzcOT+EbNj/er3s7rEvL/YH0IBDcn1BcZWUl5eHvLy8pCamoqCggJTi8Nbk1JJxgRlbPU9fvzYNDCCWk++g2EYPNfgOYvjqy+vRmJcotmxlKwU3M29y1NkvoP3FtOOmzvMkpBcJEeXuC5Ycs5yfak/Bv7BY2SEK0VFRcjLy4NWq0VWVpapq06jcXwV5rt37+LIkSOmhDZkyBA0bNiQw6jLz9jFJxQKwbIsVCoVwsPDodfrIZfLERgYSM+evFyfen0senhyinLwuPAxwuXhyFJlmY7vurULY58ay3eIXo33FlPp50vda3eHXCzH+C3jLcq+2MiyL5d4D5ZlkZ+fb3rmkpqaCqVSaXom4wyNRoPo6Gj07duXo2hdT6/Xmz5XY+tJqVSa5lMR79U+tj0CJYEWx9dcXmOxdBp15zmP18TEsix239ptdqxfvX4AYJqYVhK9q/ReBoMBCoUCSqUS2dnZyMrKglarNU2adVa9evXQtWtXNGrUiINoucOyLNRqtWkeVFpammm+llartX8B4pEkQgm61e5mcXzLtS3oEtfF7NiBuwdogWAn8ZqYrmReQYYyw+xYzzo9cSvnlkXZNxPe5Css4mI6nQ45OTlQqVRIT09HXl4eNBqN062kisZgTAjGlzsZW09FRUV49OiRqeVUVGS5MCjxDn3q9rE4plArLI49LnyMmzk3+QjJZ/D6jGnf7X1mH9cIroG4kDiM2DDCoqy1GdbE82m1WuTm5kKtVuPx48fQarVQq9W8T5A9dOgQkpKSeL2nPcZRiSzLIj09HWFhYabllfz8aBKmt7GWmADgfPp5RPtHI70w3XTs0L1DqBtWl6/QvB6viWn/3f1mHyfGJYJhGCw9v9SibHRANE9REVcxJiWVSoXHjx+bWi3u6Mbo2LEj2rVrZ/pYrVZj7ty5vMdhjfFrkpWVZXrWxLIs/P393RwZcUZscCzqh9e3WCxgY8pGtI9tj/VX15uOHb53GKNajOI5Qu/FW2JiWRb77+w3O5ZYMxEqrcqibIRfBE9REVcpmZTS09NNScldRCKRaesKT6TRaMCyLHJzc02Jm2EYajl5mcSaiRaJ6WH+Q0hFUrNjh+8f5jMsr8fbM6bLGZeRqcw0O5YYl4iku5bdLZ8lfsZXWMQFdDqdRyUlb6HVaqHRaKBQKEwjF5VKpbvDIk7oHNfZ6nFFkfmzpiuZV5CtyuYjJJ/AW2I6/vC42cexQbGIC4nDjycs914a0GAAX2GRCjJOnDVOKOUqKWk0GqSlpSEtLQ0AkJOTY1p13JsZ53MZJx/n5+dTUvcinWtaT0yPCx9DKjRvNZ15dIaPkHwCb30dpbdLb1u9LRiGsVieCACqB1XnKyxSASzLQqFQmObpGAc6cCE1NRV//vmn6eOdO3cCAJo3b46BAwdyck++6HQ6MAyDnJwciEQiMAyDsLAwj+6KJMWqBVVDndA6FqPuzqSdQe3Q2riRfcN07GzaWatDzIkl/hJTqnliahPTxupq4pF+kXyFRCrAOHmWj6QEAHFxcZg+fTpn13c345YamZmZEIlEyM3NRVhYGK2v5wU61+xskZgMrAFZyiyzY2fTz/IYlXfj5ae+SFeEc+nnzI61iWmDPbf3WJSd2GYiHyGRClKpVFCpVMjKyvKIeUK+QKPRQK/XIyMjw/TsiSZmer4ONTpYPZ5TlGP28dm0szxE4xt4aTGdSzsHneHfbQwYMGgV0wpLt1sOE29XvZ3FMeJZjPsoKRQKFBYWmkaYeSJX78fENbVaDYZhkJGRgejoaBQWFiIgIMDdYREbEqolOFTuSsYVFOmKIBPJOI7I+/HSYjr16JTZxw0jGiJIGoTfzv5mUbZVTCs+QiLlZOzCKyoqQm5ursdvV8HlfkxcMC5hZPz6KpVKWrrIwzWKaAR/sf05aHpWj0uPL/EQkffjJTFdfHzR7OOWVVta9L8a0Rwmz2bccdY4OZT+aLqecX+qvLw8qNVq5OXleU2LrzISCoRoHdPaobKl93Ai1vGSmC5nXDb7uHFkY4s9mYCyl/ggnkGj0Zhtgc7n2neVjU6ng16vR2ZmJrRaLQoLC90dErHB0e6861nXOY7EN/DyjMlaYtp3Z59FuZjAGD7CIeVQsgvPuLcSvYvnlkajgUAgQG5uLgQCAaRSKcRisbvDIlY4mpiuZV+zX4hw32LKKMywWFG8UUQj/HX+L4uyhRp6V+ipVCoVdDqdqQtPp9PZr0QqhGVZsy69goICejPgoVpWaelQudLLFxHrOE9MVzKvmH0sFUqx8+ZOi+WJAOB6NjVzPZHBYEBhYaFp3hJ14fFHp9PBYDBQ96mHqxVaC35i++scXs+6Tm8uHMB9YsowT0w1gmvgrR1vWS17Nu0stZo8kFKphF6vh0KhgF6vp18snmm1WqhUKhQVFVGryUMJGAEaRza2W06hVlh9U07McZ6Y7uTeMfv4ft59GFjre/PoWb3FCuTEvfR6PZRKJfLy8qDT6WgUnhvo9XrTmoQ6nY42F/RQTaOaOlTuQd4DjiPxfpwPfriruGv2sVqnBgvr7/hEAhG239iOfvX7cR0WcZCxtWRMTN72bt3bJtiWxbjkk1KphFAohEwmA8Mw7g6LlNAkqolD5R7mP0TLqo49k6qseG8xlZWUAEBn0GHL9S0cR0QcZTAYoFKpkJeX57Vzlrxtgm1ZDAaDWXcqLQHleZpGO9Ziepj3kONIvB/vicmR8jezb9ovSDhn3Aa8oKCARuF5AOP+TWq1GiqV5QabxL0aRTRyqNzDfEpM9nCamNQ6NR4VWK4gbgsDBjtu7uAoIuIolmWhUqlMXXmUmNzPYDDAYDAgLy8PGo2GvicepmpgVYfWwaMWk32cJiZnW0tA8fbS225sc30wxClarRY6nQ75+fk0Es+D6HQ605sFajV5FgEjQJ3QOnbLOftmvTLiNDEdunfI6ToG1oA9t/ZAo6f5Gu6kUqmg1WpRVFRE78w9iHEASkFBgamrlXiOOmH2E1Pp7TCIJU4TU3RAtNnHYoEYDOyPJFLpVFbX0iP8MK5wbZwz48mrh1dGer0ehYWFMBgMNOHWwzjSYspRUWKyh9PElFFovhRRk6gmeLr60xblSiYrkaB4BPv2G9u5DI3YYNxfSaVSUVLyQMYRklzvGkycVzesrt0y1GKyj9N5TKXXyIv0j0Tyw2SLcl1rdcX8vvNxJu0MTj86jdOPTqNhREMuQyM2qNVq02Rab09MvjKPqSTjMz+VSgWZTAaWZWlOk4eIC4mzWyZHlQOtXmv1e2Z8Y+5uDMMMZll2tbvuz21iKtViEgvEVt8tVA2sigYRDdAgogGGNhnKZUjEDpZlodFooFQqfaIbLyEhAQkJCVCr1Zg1a5a7w3EZ44ocQUFB0Ol0tOq4h6gWWM1uGa1BC8kMidVz7HT3v4FiGKY9gJEA3JaYOO3KK52E8jX5VstVDajKZRjECcZ9gFQqFQwG60tHEfczTrI1GAzUnedBfGTrnmEAlrkzAE4TU6HWfEHWshYvpMTkOYz7LBUVFXl9a8mXGd80GEdPEs8Q7hcOsaB8rVcR4/5uPIZhRAD6Atjozjg4TUxKrdLs49Jde0a0nbrnMK4uAIASkwdjWdY0Ko82bfQcAkZQ7laTjvWIaRm9ARxiWVZptySHuG0xldrCovRgCKNQeSiXYRAn6HQ606g8+mPn2YyJyReeBfqS8iQmESNCrzq9OIjGaW7vxgN4bjGVJUQWwmUYxEHGhUKNzy6IZys5j4kmQXuOKgFVnK6jY3X4rMtnHETjOIZhAgA8DWC3WwMBx6PySj9jKgslJs9gXFXA+C6ceDbj2nnGof0ymf112gj3wuRhTpUXMSJ0q90NCdUSKnRfhmFCAUxH8d/1ugD+BrAcwGwADIBQAF+wLHu5jEu8AGAzy5r3KTIM0xHA6wDqAZgBYBuACQAaAhADaALgfyzLHmMY5hUAnZ5UbQrgY5Zl9zr7uXCamNQ6x0YLUWLyDMbEpNVqfabF5IvzmIyM3yNa0NWzhMqcezThitYSwzASAD8BeIdl2VSGYWoCuA3gOQBvoTipbAWQA+CNMi4zDMDHpa4rQHFSGoniBPc7gFUA1rEsO/9JmUUAljMM8weAiyzLTnhy/DMAqxmGiWTZMnaHLYP7h4GAEpOnMHblGf/vC3x1HpMRy7LQ6XQ+8/3yBc60mFzVWgIwHsDvLMumPvm4CMWtpDssy95mGKYhgOsAVlirzDBMNICaLMseL3WqDYAzLMuyDMPEAIgEsJVl2X0lyuQBqAUgnWXZdSWOpwMIe1In3ZlPxu2JiQEDf7G/u8MgME9MvtjC8EXGgQ+UmDzDPcU96AyOt15d+Gwph2XZnSU+bv3k3+0AwLLsNhR3wZVlKIpbQqVJAax/8v9OAHawLFt6vbhmKG6d/VzqeCMAKgBZdqMvhdPEZGu3WiOpSErLqXgISkzep2RioqWJ3CvpThIS/0x0ul7bX9qiZ+2e2DG8/PvQsSz7V6lDXQDoATi6xcMwAMOtXPcAADAMUxdAdQDflTz/ZN5TewCrWcs/Gj0BHCz9zMoRnO9ga49EaH1pDsI/vV5Pey95mZJDxanV5F4tq7ZEoCSwXHXjQuNcGwzQFcAplmWtL7dTAsMw9QEwLMum2LkeAOwvdbwNgIDSxxmGaYri51prHIzXjNsTk1QodXcI5Alji4kSk/comZhoLpN7BUmD8Ha7tx3a2qcksUCMj575yGVxPBmd1xyWyWJsGVUcmbvUBUAugDOljic++Xd/qeOvAFDjSWJiGGaMneub4TQxCRmh3TJSESUmT2FcTYB4F+P3jN5QuN+bbd90qhdIAAEmtJmA6kHVy31PhmEiGYY5wTDM9CeHeqP4b/uJkmVQ3OVmzRAAK+3cJhHAASuj67oAuMWy7L1Sx18AsIll2ZwnrSenPkFOE5Of2M9uGWoxEVJ+lIw8S5g8DMOaDnO4vEggwrSO0yp6284o7lJjGIaRozjRpKK4iw0Mw/gD+AHAJ6UrMgzzNIB7LMumlXVxhmEaAagCyxaYGMXJbr9lLUQASHoy3Pw9AD868wm5bPADwzCMQqEwOybRSYoHLdog0AiQl5fnqjBIBajVap/ZgM64SrqR8fPx9s+rNJZlIRaLTTsO+9rn542G1R+G34795lDZ8W3Hw8/gZ/obGBwcHAQg38pAAlt2APgVQBSARQCmAQgC8CXDMJ0BSADMtNKqARzrxgtHcaJbV+p4CIBCAEut1HkTwFgUt7QWsizr1Mg8xlXvuBiGCQKgsFuQeKz//Oc/iIyMBMuyOHz4sLvDIQ6oW7cuateujfv372Pr1q24d8/a3x7iZYJZluX83fqTEXU3ADR1ZJAEn1yZmJj79+8bYmNjcf/+fQQFBeHVda9ic8pmm/Vqh9bGmfHmz9PatGmD5GTLnW4dPW+vTF5eHkrGycU9XHHeFXE683koFArk5OQgLy8PhYX/Lie1ePFijBs3zmpdtVqNuXPn4r///S+k0rK7ZW1dwxXnS5cp3WLKz8/HL7/8gokTJ5b5tXRHnKU58vUsWV8ikcDf3x9VqlSBv7+/acNAX/vZLM95R2J0RZylz2cps1D7+9plljeSCCXImJphFmdsbGwwnG8xlQvDMH0BjGRZdgjX93KWy7ryWJZljc3RoKAgBAUFITg4GLCzfBcjYyx+aIRCoc0fJHvnHS1jjJOre7ji86honM58Hmq1GhKJBCKRyOyPIsMwNpMOAEilUptl7F2joudLlymrrKfFWRZbcZasLxaLIRaLIZVKERgYCImk+MG7r/1sVuQatmJ0xT1Kn9eINHb/7gFAgDzArF5QUBD4aCmV4BEriVvD6eCHIIn9H2yDlSWUJk2aZLOOvfOOluH6Hq74PLiOoWQZgUAAoVBoMUmzTZs25Q/QwWtU9LyjZSp6DU+Is2R9hmEgFBaPfhUI/v119rWfzYpcg+t7lD7vaGPHnXM4nwyI6Ajbq0G4jcu68gAgLy+PDQ4OhkKhQFBQEKbvm47PDthebiMuJA63p9x2WQyOyMvLQ8k4PRXfcebk5CAnJweZmZlQKh3bssS4Bt37779vtxXgTnl5eaYuMk/+njv79TS2lCIjIxEZGWmWnLjkDb9D7ooxozADUXOi7JarEVwDd9+6WzpOXpbuYBjmVQAdjAuuehqX/hRLpVJMnz7d9AvlyM60zqwr5Sql4/RUfMcpEAggEhX37jq6tI1QKETnzp1N79o9lTE+b4jTma+nscXEMAxvSQnwjt8hd8XoyFJswL8tJjfF2RnAEj5v6AyXtpgA8+/Iyosr8fLal21WCJeHI3NqpitjIOVUUFAAhUKBhw8foqioyKcm23pLy85ZcrkcYWFhCAsLQ3h4uLvDIQDSC9JR5Rv7mwW2qtoKJ18/WfowLXYIjp8xOdJicnQzQcI94zMmwPEWE3EvY4uJz9YSsc3RFhNt91M2TlcXj/SLtFumSFcEvUEPocCzu1gqA2OXkEgkglar9Ym113x5o0BjMpJIJB7fRVmZFOnsrCrwRLAsmONIvBenialKgP3mLAAotUoESsu3Ki9xHeMcGIlE4jMrCPjyRoECgQAMw5iGjBPPoNKqHCoXIg3hNhAvxmn7P9I/0qG18Kg7zzMYu/KkUil1DXkBgUBgmrdkHLRC3E+pdWxEK3XllY3Tvz4CRoAawTXslivQFHAZhk3PPvssatSoAZlMhqpVq2L48OFITU21X5FHd+7cwZgxY1CrVi3I5XLUqVMH06dPh0ajcfm9xGIxJBIJGIYp13OmAwcO4Ndff8UXX3zhcy0ULt29excrVqzAN998g08//RRXr161W8eYmIzdr1ybOXMm2rRpg8DAQERFRWHgwIFISbG1hY97LFiwAM2aNTNNrG3Xrh22beNvuo6jiSlUHmr6/8yZM42/c99xFJZX4fxtsSOJKUeVw3UYZerSpQv+/vtvpKSkYO3atbh58yYGDRrktnisuXr1KgwGAxYtWoRLly5h7ty5WLhwIT744AOX30skEpnehZen1WQwGBAfH++Sya6ViUajQXR0NPr27etwHYZhTCt18DFYJSkpCZMmTcKxY8ewa9cu6HQ69OzZ02z5Kk9QvXp1zJo1CydPnsTJkyfRtWtXPPfcc7h06RIv91fpHOvKiwmMAQAkJyfj559/RrNmzbgMy6tw/jZLmG//oWyWyukt4V3mv//9r+n/NWvWxPvvv4+BAwdCq9V6TL9979690bt3b9PHtWvXRkpKChYsWIA5c+a49F5isdj0Trz0enOOSExMBACcPXvWpXH5unr16qFevXoOlzcOVJHJZLz9nG7fvt3s499//x1RUVE4deoUnnnmGV5icMSAAQPMPv7iiy+wYMECHDt2DI0bN+b8/o62mGICY1BQUIBhw4Zh8eLFmDFjBseReQ9OW0zp6ek4uv2o3XKZSs+Yx5SdnY1ly5ahffv2HpOUyqJQKBAWFuby6xoTk1wup5FeHkwoFFpd15BPxm1uuPg5dBW9Xo+VK1eisLAQ7dq14+WejvYAxQTGYNKkSejXrx+6d+/OcVTehbPExLIsRo0ahZ5tetot6+7E9N5778Hf3x/h4eG4d+8eNm7c6NZ47Ll58ybmzZuH8ePHu/zaxu4hPz8/3lcTII4TCoWQy+UQCARueRPFsizefvttdOzYEU2aNOH9/vZcuHABAQEBkEqlGD9+PNavX4/4+Hhe7u1oD9Dx3cdx+vRpzJw5k+OIvI/TXXkMw3wCYLqtMsnJyThy5Ajy8vLw6WufYu2va21e09WJ6ZNPPsGnn35qs0xycjJat24NAHj33XcxZswY3L17F59++ilGjBiBLVu2cN5v72ycAJCamorevXtj8ODBGDt2LCdxSaVS09wYoVCIvXv3IikpyWadcePGISYmhpN4KsIX5zEZh4n7+fmZBj/w7Y033sD58+dx6NAh3u/tiAYNGuDs2bPIzc3F2rVrMXLkSCQlJfGSnLKU9hOTRCDBh29/iF07d0Emc2Ap8kqmPM+YfkQZ+8NfuXLlCgDExcVhxowZOHbsGNrVawe8b/uCGYUZ5QijbG+88QaGDh1qs0xcXJzp/xEREYiIiED9+vXRqFEjxMbGFsfOcdPf2ThTU1PRpUsXtGvXDj///DNncRn/2Mnlcmi1WiQkJNh9VxwSEsJZPBXhi/OYjG8YJBKJW7rxJk+ejE2bNuHAgQOoXr067/d3hEQiQd26dQEArVu3RnJyMr7//nssWrSI83s70mIKFYUi/XE6WrVqZTr25HnumwzDvAFAyrKs989wLyenExPLspkA7DZxfvjhB9PDvJ7beuKR8lGZZR8VlH2uPIyJpjyM76z5mGDqTJwPHz5Ely5d0KpVK/z++++cdrEZBz8EBASgoKAAAQEB8PPz4+x+xDkikQj+/v6mble+sCyLyZMnY/369di/fz9q1arF270rimVZ3iaNO5KYGldtjN0XdpsdGz16NE6ePLkMwFeVOSkBHI7Kq1Hj32Hizc81x6MbZSef+3n3uQrDphMnTuDEiRPo2LEjQkNDcevWLXz88ceoU6cObw9KHZGamorExETUqFEDc+bMQUbGvy3MKlUcW13DWTKZzLRxoE6nc3jOlEKhgEqlgkKhAMuySEtLA1D8gJzPP6LeRqPRIDs72/RxTk4O0tLSIJfLizfcfMI4Gi8wMJD3idCTJk3C8uXLsXHjRgQGBpq+t8HBwZDL5bzFYc8HH3yAPn36IDY2Fvn5+Vi5ciX2799vMaqQK4505cVHxVv0Qvj7+wNAFsuyF7mJzHvwMl28UUQjbL9R9g/FfYV7EpNcLse6deswffp0FBYWomrVqujduzdWrlzpUStQ79y5Ezdu3MCNGzcsuk64enYilUohFAoREBAAtVoNhmEcute+fftw7tw508fGrpORI0eadUsSc6mpqfjzzz9NH+/cuRMA0Lx5cwwcONB0XCQSQS6Xm/7l04IFCwD8OyXA6Pfff8eoUaN4jcWW9PR0DB8+HI8ePUJwcDCaNWuG7du3o0ePHrzcP60gzW6ZeuGOTw2ojDjd9sLo9zO/47VNr9msWPhBIfzE1F3kSQoLC5Gfn4/79+9Do9FAq9W6O6Ry84VtLwQCAWQyGSIjIxEcHIzQ0FBaBd7DsCyLwJmBdpdZ++eVf9CnXh9rp+gbCh5WfgCAVjGt7JZxV6uJlE0mk0EgECAgIIDWYvMAIpEIQqEQfn5+kMvllJQ8kEKtcGjtT2ox2cZLYoqPjIdMZHtIpLueM5GyGRd0DQoK4m09NmKdcd+l4OBgCAQCr231+boHeQ/slpEKpYgLieM+GC/GS2ISCURoUaWFzTK3cm7xEQpxkr+/P8RiMQIDAz1+NQxfZtzawjhCkiY+eyZHElOTqCYQCehNni28fXVaVW2FYw+OlXn+WtY1vkIhThCJRJDJZAgODkZBQQHEYrFXPWvyhQm2AoEAIpEIISEhEIlENHTfgzmSmJpHN+chEu/Ga2KyhRKT5/L390dRURGCgoKQm5sLnU7nNX/ofWGCrXErEn9/f9P8JeKZHOn5sdd7RHjqygOAdrG25wVRYvJcxnXZgoODIRKJqEuPR8ZVHoytJVq+xrM58neMEpN9vCWmBuENbJ6/mXMTOoOOp2iIs/z9/SEUChEWFmYaHUa4ZVzZwd/fH3K5HAEBAdRa8nDXs6/bLdMsmvZdsoe3xMQwDOqH1y/zvM6gw+2c23yFQ5wkEAgQGBgIPz8/0+KhhFtisRgikQhhYWGQyWQ0Es/DGVgDrmfZTkz1wuohWBZsswzhMTEBwKjmo2yev5xxmZ9ASLnIZDLIZDKEh4ebFhEl3BAKhaakZBwVSTxban6q3d1rO9TowFM03o3XxNSrTi+b58+mneUnEFJugYGBEIlECA8Ppy49jpTswvPz80NgYCAND/cCjjxf6hBLickRvA6mV+ttr+57Lv2czfPE/YxdegaDAQEBAaZVmw0Gg7tD8xlSqRRisZi68LzMpceX7JZpH9ueh0i8H6+JadWlVTbPU4vJO8hkMmi1WrAsa5rTVFRU5JFDyL1tHpNxg8aoqChIJBLqwvMi9t5Yh8pC0TCiIU/ReDfe+gf0Bj2WXVhms8zt3NtQFCl4iohUhHHb6sjISIhEIo993pSQkIBJkyZh3Lhx7g7FLuNgh8jISEilUtPyQ8Q72EtM7WPbQ8DQ99MRvH2V9t/Z79AW6ufTz/MQDakohmEQHBwMiUSCqKgoj05O3kAoFEIsFiMkJAR+fn4ICgqitQk9mFKrxNH7R6E3FO/npzPocDL1pM06XeK68BGaT+AtMS27sMyh9aFOPDzBQzTEFQQCAUJCQiCVShEREUHJqZyMIxz9/f0RHBxsao0Sz7X60mq0/609anxXA18c+AJH7h+xW6dHHX72g/IFvLwlK9IVYfXl1Q5NoP3r/F94p/07PERFXEEkEpl2WGVZFpmZxa1iR3e8dcSBAwdw/fp1pKWlQSgU4v3333fZtd2tZFKKiIiAXC73qN1giXVSUfEbh9T8VHy8/2O7zzCj/aPRNKopH6H5BF5aTNuub0OBpsChsufSz9mdpEY8i3FrjICAANMwcle2nAwGA+Lj49GmTRuXXdMTlExKkZGRkMvlCAwMpNUdvEC4PNz0fwNrAGt9j1STnnV60vfVCbwkpqUXlkLEON44G7J6iKnvlngHmUyGoKAgBAYGurxbLzExEe3atUNUVJRLrucJjF+fgIAASkpeKMIvwqnyPWpTN54zOE9Meeo8bLm2BTrW8XXwzqSfwQ/Hf+AwKsIFY3IKCAhAVFQUxGIxZDKZW/7Y6nQ6qNVqs5enkEgkkEgkCA4ONnXfUVLyLuF+4fYLlfD14a/x25nfoNQqOYrIt3D+jGnD1Q3Q6J1/3jBtzzSMbz0ecjH1t3sTYyIy7nj7+PFjMAzD+yTcQ4cOISkpibf7OUoqlZqWGgoICKCtLLxUya48R1zKuIQxm8Zg2/VtWP3Sao6i8h2cJ6a/zv8FBoxZH6xYIIbWUPZmc5F+kZjVfZbpASPxLlKpFKGhoVAoFKhatSoyMjIAAFqtFjpdcct5//79dhPHuHHjEBMTU64YOnbsiHbt/t1qRa1WY+7cueW6lisIBAJIJBKIxWJERkaaWpe0jYV38hP72f07VpKAEcBP7Ic3277JcWS+gfPEdDL1JFiwEAvEqB5UHfXC6qF2aG0cf3gcZ9LOWK2TqczEwIYDaTKaFxOJRAgNDUVeXh6io6ORk5ODvLw8CIVCaDQaJCQkoEmTJjavERISUqH7e8o8IOPEWZlMhsjISFM3Hu1r5b0YhkGILAQZygy7ZQUQoEpAFewavguNIhvxEJ334/w39+qkqzCwBkQHRJslmvkn5uONbW9YrcOCxb7b+/Bi/Itch0c4JBAIEBwcjMLCQgCAXC5HVlYWBAIBhEKhz28RbmwlGTf6CwoKgkQiQVBQEC1+6wPC/cIdSkxNoptgx6s7UCWgCg9R+QbOmyTRAdGoGljVovXTOa6zzXp7b+/lMizCE4ZhEBAQgNDQUPj7+yMmJsb0B1oqlTq05I5CoUBaWhoUCgVYlkVaWhrS0tJcOlfK1cRiMaRSKeRyOapWrWqaOBsSEkJJyUdE+dsfJVontA4Ov3aYkpKT3NbX0TiyMSL9Ist8x7Hn9h6eIyJckkgkCAsLQ2FhIRiGgZ+fH7KzsyEUCqHT6UyLwlqzb98+nDv37zpkixYtAgCMHDkScXFxfITvMOPW80KhEMHBwQgKCoJYLKYlhnxQlJ/9xLTsxWUIkATwEI1vcdtvCsMw6Fqra5krjqdkpeBB3gNUD6rOc2SEK8YtM6RSKfLz8yGTyVBQUIDc3FxTgtLpdBYJauDAgRg4cKB7gnaQMSEZP8fg4GAIhULTnko06s73CAW2W74NIxoiISaBp2h8i1tHF3Sr1c3m+e03tvMUCeGTsfUUGBiIoKAgVKtWDWFhYZBKpZDJZJBIJF6xqjbDMBCLxZDL5abJsjExMQgLC4O/vz/Cw8NpKLgPu6e4Z/P8q01fpe99Obm1b6Frra42z2++thljnxrLUzSET8buPJlMBpVKBaFQiICAABQUFKCgoABarRYGg8HUivIkAoHA1DoytoqMXXUymQz+/v7Ubefj9AY9jj44arPMsGbDeIrG97j1t6d2aG3UDK6Ju4q7Vs/vurkLKq2KJtn6MIFAAH9/f8jlciiVSohEIgQFBaGoqAj5+flQKpUQi8UwGAzQ6/XQ6/VObfrnqo0ChUKh6WXc+jwwMBD+/v4QCASQSqXw8/OjIeCVhL3enE41OiEuJI6fYHyQWxMTwzDoV68ffjr5k9XzKp0K++7sQ996fXmOjPBNIBCYVkJQq9WQSCSQyWTQ6/VQKpVQqVSmXXKNScpgMMBgMNhMOgkJCUhISIBarcasWbMcioVhGAgEArOXMRnJ5XL4+fmZhoHL5XLIZDIaaVfJlPU3y+jVZq/yFIlvcnt/w4AGA2x+kzenbKbEVIkwDAOZTAaZTAadTgeVSgWpVIrAwECwLAuVSgWlUgm1Wm3q4jMmK2OSKvkyngdg1r1mXDap5KtkEgJgagkZk5FQKDTNTTI+C6NnCJXP7Zzb+Of6P2We9xP74aXGL/EYke9xe2JKjEuEv9gfhdpCq+c3X9uM+ex8WgWiEhKJRKaEpNfrTS0pf39/UzLSaDTQaDSmRGXs7itNIpGgR48eCAwMtHj+Y3xWZFzx2/gylhMKhZBKpZBKpRCLxZSMKrlFpxbZPP9yk5cRIgvhJxgf5fbEJBPJ0KtuL6y7ss7q+Yf5D3H8wXG0i21n9TzxfcYFYUUiEfz9/WEwGEzr7hnnQJVORsYEZXwmpVarkZGRgfDwcFM3XMlnRkYCgcA09Nt4z9JlSOWl1qnx65lfbZYZ33o8T9H4LrcnJgAYUH9AmYkJAFZdWkWJiZgYu9hKbj9e+rlTyf8DgEqlQkpKCoKCgiCXy8EwjKlrzvgyfkxJiJRl5cWVyFRmlnm+VdVWaB3TmseIfBNT0dFKpZTrYhmFGaj6TVXoWeubA4bLw/H43cfUnUfKLS8vD8HBwVAoFAgKCnJ3OMQLsSyLZgub4eLji2WWWTxgcUWnuNC7Irh5gq1RpH8kutUue7JtlioLh+8d5jEiQggxt+PmDptJKUgahKFNhvIYke/yiMQEAEMb2/6Grry4kqdIiC+ZP38+4uPj0aZNG3eHQrzc7COzbZ7/T6v/0Lp4LuIRXXkAkFuUi6jZUTY33tL8nwZiIU1gJM6jrjxSEacfnUarn1uVeV4kEOH2lNuuWNuTuvLgQS2mEFkI+tTrY7PMthvbeIqGEEL+NefIHJvnX27yMi047UIek5iA4m+uLUPWDOEpEkIIKZaSmVLmLghG/2v/P56iqRw8KjE92+BZBEnL7mYp0hUhvSCdx4gIIZXdjIMzYGANZZ7vWacnmkU34zEi3+dRiclP7IdXmrxis8zw9cN5ioYQUtmlZKZg+YXlNsu81+E9nqKpPDwqMQGwOwdg161dFV4pmhBCHGGvtfRMzWfQJa4LjxFVDh6XmJ6q+hRaVGlhs8wHez7gJxhCSKXlSGvp08RPaaUQDnhcYmIYBmNb2m41zTo8CzqDZ20eRzwTzWMi5fVJ0ic2W0uda3ZGYlwifwFVIh4zj6mk3KJcxHwTA5VOVWaZz7t8jv975v9ccTtSCdA8JuKM5IfJSPglwWaZfSP3cZGYqPkFD2wxAcVzmoY3sz3I4aN9H6FQY32rDEIIKS+WZTF191SbZbrEdaHWEoc8MjEBwJSnp9gt8/Xhr3mIhBBSmWy7sQ377+y3WebrHvS3h0sem5jiI+PRs05Pm2U+O/AZ0grSeIqIEOLr9AY9pu6y3Vp6ucnLtLUFxzw2MQHAW23fslvmuZXP2XxASQghjvrj7B+4lHHJZpkvu33JUzSVl0cnpl51e6FBeAObZU48PIGEnxPwIO8BT1ERQnxRblEupu2ZZrPM5ITJiAuJ4yegSsyjE5OAEeCddu/YLXcq7RQa/tgQy84vo8m3hJBymb5vOjKUGTbLfJr4KU/RVG4enZgAYETzEQ6t2luoLcSr61/Fi3+/aHPrY0IIKe1C+gXMT55vs8yniZ8iVB7KU0SVm8cnJqlI6tRaVJtSNqHhjw2x5doWDqMi3oIm2BJ7WJbF5G2ToWf1Nst92OlDniIiHjnBtjSVVoXaP9R2eAQeAwYsWLzW4jXM7T3X5orlpHKgCbakLCsurMAr62wvHr1m8Bq8GP8iH+HQBFt4QYsJAORiOf7XzvH9Ttgn+fHPc3+i8U+NceDuAa5CI4R4sWxVNt7a8ZbdcjwlJfKEVyQmABjfejwi/SKdqqNn9UjNT0XiH4n4cA81wwkh5t7d+S4eFz62Webwa4d5ioYYeU1i8pf446NnPnK6noE1gAWL5ReXQ2+w3YdMPM+dO3cwZswY1KpVC3K5HHXq1MH06dOh0WjcHRrxcntu7cFvZ3+zW659bHseoiEleU1iAoD/tP4PaoXUcri8kBFCJpThk86f4MKECxAKhBxGR7hw9epVGAwGLFq0CJcuXcLcuXOxcOFCfPABbX1Cyk+pVeI/W/5jt9zVSVd5iIaU5hWDH0pafmE5hq0b5lDZEc1GYFb3WagaWJXjqAifZs+ejQULFuDWrVsO16HBD6Sk93a9h6+P2F7vrkftHtg5fCdPEZnQ4Ad4WYsJAIY2GYoweZhDZaUiKSUlH6RQKBAWZvtnQK1WIy8vz+xFCAAcvX8Uc47OsVtu8YDFPERDrPG6xCRgBOhUo5NDZRefXozNKZs5jojw6ebNm5g3bx7Gjx9vs9zMmTMRHBxsesXGxvIUIfFkBZoCDF8/3O76mpMTJqNmSE2eoiKleV1iAoBGEY3AONjiHbt5rN1RN4R/n3zyCRiGsfk6efKkWZ3U1FT07t0bgwcPxtixtnc5njZtGhQKhel1//59Lj8d4iX+t/N/uJlz0245mkzrXiJ3B1AefmI/01wlex4XPsbYTWOxcehGMAx133qKN954A0OHDrVZJi4uzvT/1NRUdOnSBe3atcPPP/9s9/pSqRRSqbSiYRIfsvXaViw6tchuuS+6foHogGgeIiJl8brBDwCQkpmClRdXIledi++OfedQne97f483277JbWCEEw8fPkSXLl3QqlUrLF26FEKh86MrafBD5fb7md8xZtMYu29o40LicGXSFchEMp4is0DvnuGlicmoUFOIRvMb4X6e/W4asUCMw68dRptqtGaaN0lNTUXnzp1Ro0YNLFmyxCwpValSxeHrUGKqvFiWRdz3cbinuGe37OrBqzEofhAPUZWJEhO89BmTkb/EH3N62h9dAwBagxYvrXkJOaocjqMirrRz507cuHEDe/fuRfXq1VG1alXTixBHfHfsO4eS0jM1n8GLjWjpIU/g1S0moPjdUL/l/bDtxjaHyg9sOBDrXlpHz5sqGWoxVU4nHp5Ax986QmvQ2izHgMGp10+hZdWWPEVmIxTi3S0mAGAYBgv7L0SAJMCh8huubnD4uRQhxHvlFuViyJohdpMSAExKmOQJSYk84fWJCQBqBNfAV92/crj8u7vexd7bezmMiHgK2o+pcmJZFmM2jcGd3DsOlb+acRXZqmxugyIO8/quPCMDa0DnPzrj0L1DDpUPl4cjeVwyaoU6vvYe8V7UlVe5zD48G1N3T3W4vJARokpAFawfst7dA6SoKw8+0mICileE+GXAL5AKHZu7kqXKwsBVA1GoKeQ4MkIIn3be3In397zvVB09q0daQRra/9YePyX/BBe/YSdO8pnEBAANIhpgVvdZDpc/n34er216jX4ICfERN7NvYuiaoXaXHLJGz+qhM+gw6Z9JeH3z6xxERxzlU4kJAN5s+yZ61O7hcPm/L/2NLw5+wWFEhBA+FGoK8fyq55FTVP4pIUKmeJ5cWmGaq8Ii5eBziUnACPDHwD8QLg93uM5H+z7C8gvLOYyKEMIlA2vA6I2jceHxhXLVNyakTjU7Yf/I/dj8Mi3+7E4+l5gAICYwBr88+4tTdUZvHI2kO0kcRUQI4dL0fdOx+vJqp+sZE1K32t1w5LUj2DdyHzrHdXZ1eMRJPpmYgOKJtOOeGudweY1eg4GrBuJKxhUOoyKEuNofZ//AjIMznKojYorXr+5bry+SxyVjx6s70C62HRfhkXLwmeHi1ii1SrT7tR3Op593uE5cSByOjTlGqwv7GBou7pv23t6LXkt7QWfQOVSeAQMWLF5s9CI+7vwxmkU34zhCp9Fwcfhwiwko3h5j7UtrESR1/A/Rndw76LOsDxRFCg4jI3yhCba+60rGFbyw6gWHkxIAPN/weVyaeAlrXlrjiUmJPOHTLSajDVc34PlVzztVp1ONTtj+6nb4if04iorwiVpMviU1PxUdfuvg8MoODBisfHElXmryEreBVRy1mODjLSajgQ0H4t327zpV5+C9gxi8ejA0eg1HURFCyiNHlYNeS3s5nJQAYFb3Wd6QlMgTlSIxAcCX3b5EYlyiU3X+uf4PRm4YCb1Bz01QhBCnFGoK0X9Ff1x8fNHhOr3r9sb/2v+Pw6iIq1WaxCQSiLBm8BrUDavrVL2VF1di4taJ5ZpJTghxHY1eg0GrB+HI/SMO14kJjMGSgUsgYCrNnzqfUKm+W+F+4dj88mYES4PLLCMXyS3W2/v59M+YtHUSJSdC3MTAGjBqwyhsv7Hd4ToSoQTrXlqHSP9IDiMjXKhUiQkAGkY0xJqX1pgm1pWm0qkQFxIHsUBsdnzhqYWUnAhxAwNrwNhNY7Hi4gqn6i3stxBtq7flKCrCpUqXmACge+3umNdnXpnnU7JS0LFGR4gEIrPjlJwI4ZeBNeA/m/+D38/+7lS9yQmTMbrlaI6iIlyrlIkJACa0mWBzpF7S3SSMaDbCanKiZ07eg+YxeS8Da8CELRPwyxnnlhdLjEvENz2/4SgqwodKMY+pLCzL4rVNr+GPs39YPR8gCcDbT7+NLw99aTGJb3iz4fj12V8hFoqt1iWeheYxeReWZTHpn0lYcHKBU/Vqh9bGsTHHvPm5Es1jQiVuMQEAwzBYPGAx+tfvb/V8gaYAv539DXN6zLFoOf11/i8MWj0IRboiPkIlpNIwsAZM2DrB6aQUJg/DtmHbvDkpkScqdWICioeRrxq0Ch1iO1g9/yDvARaeWoif+v5kkZw2pWxCn2V9kKfO4yNUQnyeVq/F8PXDsejUIqfqSYVSbBq6CfXD63MUGeFTpU9MQPGaeptf3ozm0c2tnr+aeRWLTy/GsheWQSaSmZ3bf2c/ui3phkxlJh+hEuKzinRFGLR6kNN7ozFgsOyFZehQw/qbS+J9KDE9ESoPxe4Ru9E0qqnV88mpyVh8ejG2vLzFYlHYk6kn0eG3DriZfZOPUAnxOQWaAvRb3g+bUjaVWaZWSC2rE2W/6fkNXox/kcvwCM8oMZUQ4ReBPSP2oElUE6vnd9/ajYWnFmL38N2I9DPvx76WdQ3tfm2HYw+O8REqIT4jU5mJ7ku6Y+/tvWWW6VmnJx7mP7QYDftBxw/w33b/5TpEwjNKTKVE+kdiz4g9iI+Mt3p+zeU1mHtsLvaP2o/YoFizcxnKDHT5swvWXVnHR6iEeI1MZabVrWRuZt9E+1/b4/jD41briQQijG05FofuHbJYUPnNhDcxo6tzGwQS70CJyYoo/yjsHbG3zOS04uIKfH7gcxwcfdBiT5ciXREG/T0Ic4/OhYuH4pNyoHlM7qfSqtD659Zo/1t7aPVa0/ETD0+g3a/tcD37utV6obJQvNPuHSy9sBRKrdLs3GstXsPc3nPBMDS62hdV6nlM9mQqM9F3WV8kpyZbPd8oohEW9V+ELw5+gR03d1icH/fUOPzY90dIhBKuQyV20Dwm95m+bzo+P/A5AOCr7l/h3Q7vYlPKJrz494tlbvLXILwBxrcej/d2v2fRUhrSeAiWvbAMQoH1ZcW8HGVaUGKyK1+dj4GrBtrs/64XVg/B0mCcfHTS4lz72PZY+9JaVAmowmWYxA5KTO5xPes64n+KNyUgiUCCDzp9gE+SPimzTs86PTGk8RCM3zIeWoPW7NyLjV7EihdX+PLEdkpMoMTkkCJdEV5e+zI2XN1QZhkGDNgyPv1qgdWwfsh6tKlG3UnuQomJfyzLosdfPZB0Jwk6tjgx2fo9AYqfG7Ws2hJjN42FnjXfB+2Vpq/gz4F/Wswn9DGUmEDPmBwiE8mwevBqjGoxqswytn7ZHuY/RKffO2HJuSUcREeIZ1p9eTX23N5jSkpA2b8nQkaIX5/9FTGBMRi9cbRFUhrVYhSWDFzi60mJPEGJyUEigQi/Pfsb3n767XLVV+vVGLlhJMZvGU/LGDnp2WefRY0aNSCTyVC1alUMHz4cqamp7g6L2JCnzsMb/7wBxoEGQK2QWjg+9jjOPDqD9/e8b3F+fKvx+PXZX331mRKxghKTExiGwTe9vrEYJu6MRacW4elfnsb1LOsjkYilLl264O+//0ZKSgrWrl2LmzdvYtCgQe4Oi9gwfd90ZKmybPYkAMCA+gNwZMwRzDo8Cz8m/2hx/p127+Cnfj/RDrSVDD1jKofPkz7HJ/s/gQHl3/oiUBKIX579BS81fsmFkVUOmzZtwsCBA6FWqyEWO/YQnJ4x8edc2jm0XNTSblICgC0vb8HMQzNx+P5hi3Pf9vy2Mk6epWdMoBZTuQxsONCppPRcg+fgJ/YzO5avyceQNUMwaeskqLQqV4fos7Kzs7Fs2TK0b9/eZlJSq9XIy8szexHuGbdAdyQpCSDAwFUDLZKSWCDGihdXVMakRJ6gxFQOTaKaONyd1zSqKdYPWY/kcclWJ+z+dPIntPq5FU4/Ou3qMH3Ke++9B39/f4SHh+PevXvYuHGjzfIzZ85EcHCw6RUbW/7uV+K4iVsn4mz6WYfKGmCwmMcUKAnE9le3Y2iToRxER7wFJaZyYBgGg+MHOzRC6MLjCxB8JkCtkFo4MfYERjQfYVHmSuYVPP3L05h1aBb0Br2Vq/ieTz75BAzD2HydPPnvvLB3330XZ86cwc6dOyEUCjFixAibK2tMmzYNCoXC9Lp//z4fn1alVaQrwlOLnnJ6u4qSagTXwMHRB9G1VlcXRka8ET1jKqeDdw/imT+ecarOibEn0KZaG/x+5ndM+mcSVDrLLrxONTphyfNLEBcS56JIPVNmZiYyM21vFRIXFweZTGZx/MGDB4iNjcWRI0fQrl07h+5Hz5i4c+zBMbT71bHvQ1lCpCFImZyCKP8oF0XltegZEwCaFFBO7WPbI0QWgtyiXIfrJPySgMkJk/FDnx/wdPWn8er6Vy268A7eO4hmC5rh6x5f4/VWr/vsaKSIiAhERESUq67xzZRarXZlSMRJBtaArn92RdLdJLtlBRCU+VyWAYP4qHhKSsTEN//q8UAoEOL5hs+buvMYMBAyQvw96G9o/k+DhhENrdabd2IemE8ZRPhF4OiYo/ig4wcWySdfk48JWyeg659dK/2w8hMnTuDHH3/E2bNncffuXezbtw+vvPIK6tSp43BribjeoXuHIPxMaDMp+Yn9UDukNmqH1rY6GELACPBl1y+h+0iHw69ZjsojlRclpgp4vuHz0Bl0xUlJIMT6IesxuPFgiIViXJl0BT/0/qHMulFzovDH2T/wRbcvkDQqyWrXXdLdJDRb2AyzD88uc7FLXyeXy7Fu3Tp069YNDRo0wGuvvYYmTZogKSkJUqnU3eFVOkW6ItT9oS46/d6pzDKDGg1CwbQC3J5yG42jGuNWzi2LxBTtH41dw3dhWqdpEAjozxAxR8+YKkClVSHs6zCwLIvNL29Gjzo9LMrczL6JuvPqlnmNSL9InJ9wHn5iP/x3+3/x29nfrJZrVbUVFvVfhFYxrVwWf2VCz5gqhmVZzD02F+/sfMdmufPjz6NpdFNsStmEsZvGIkOZYVEmMS4RK15cQQsbW0fPmECJqcI2Xt2ImMAYmwu0GlgDGs1vhGtZ18os803Pb/Bm2zex7/Y+vL7lddzJvWNRhgGD8a3HY0bXGQiTh7ki/EqDElP5nUo9hdaLW9ssM6blGCwesBgFmgK8tf0tq2+wGDD4sNOHmJ44nda8KxslJlBi4tUfZ//A6I2jyzzvJ/Yzbe3+f3v/Dz8c/8Fq33yEXwS+6v4VRrUY5bODI1xl/vz5mD9/PvR6Pa5du0aJyQmp+anov7w/zqSdsVnu1pu3UCu0FpLuJGH0xtG4nXvbokykXyT+ev4v9Krbi6twfQUlJlBi4t2j/EeI+TbGZpnRLUZjVvdZuJl9E2M2jcGVzCtWyz1d/WnM6zMPrWNsv5sl1GJyRqYyEx/u+RA/n/7ZZrnPEj/DR50/Qo4qB1N3TcUvZ36xWu65Bs/h5wE/06g7x1BiAiUmtzCwBry/+33MPjLbZrkvun6BiW0mYkHyAnx+4HOr854AYFjTYfii6xeoGVKTi3B9AiUm+/LUefj26Lf4NOlTm+WCpcG489YdBEuDsfryary57U2kF6ZblAuQBOD73t9jdIvRtAW64+gLBUpMbuVI332UfxS+7v41Osd1xjs738G6K+uslpMKpZjSdgqmdZqGEFkIB9F6N0pMZSvQFGBB8gJM3T3VbtnNL29G//r9cTf3Lt7Y9ga2XNtitVzHGh2xZOAS1Aqt5epwfR0lJlBicrtCTSHe2/0e5ifPt1muRZUWmNFlBoQCISZvm4wb2TeslguXh+Pjzh/j9VavQyayXDWhsqLEZClHlYN5J+Zh+v7pdstOaD0B3/b6FnqDHl8d/gqzj8y2uq+Yv9gfX3b7EpPaTKL9k8qHEhMoMXmMw/cOo+fSnlBqlTbLJVRLwLSO03At6xq+PPglFGqF1XLVg6rjw04f4rWWr0EilHARslehxPSv9IJ0fHv0W8w+MtvuKuANwhtg6ytbUTu0NlZcXIGpu6biYf5Dq2X71++P+X3no0ZwDS7CriwoMYESk0dRaVX4aN9H+OboN3bLtq3WFpMTJuP4w+NYcHJBmRNwawbXxEfPfIQRzUdALHRs7yJfRIkJuJp5FT8c/wELTy50aFuKdS+tw8CGA3H84XG8s/MdHLl/xGq5aP9ozOszD4PiB9GzpIqjLyAoMXmk4w+OY+I/Ex3aCqNNTBs81+A5JKcmY2NK2VtB1A6tjWkdp2F4s+GQiirfigmVNTGxLIudN3fiu+PfYfuN7Q7V+bLrl3jr6bdwI/sG/m/f/2FTyiar5YSMEBPbTMSniZ8iVB7qyrArM0pMoMTksfQGPRadWoT3d7+PfE2+3fJxIXFoW60tbufexomHJ8osVzWgKv779H/xn9b/QZDU9/9AV9Z5TAWaAiw9vxQ/HP+hzOkGpY1vNR6fJH6CfE0+pu+fjhUXVpTZsupeuzu+6/UdGkc1dmXYhBITAEpMHu9x4WO8t/s9/HH2D4fKh8pC0SCiAR7kPcCDvAdllguWBmNC6wmY8vSUSrE0TGVoMbEsi5OpJ7H49GKsuLgCBZoCh+r1qdsHs3vMhkQowVeHv8Kf5/4ss2u4TmgdfNvrWwyoP4C67bhBX1RQYvIaR+4fwTs738GxB8ccKs+AgVgohkavsVlOIpRgSOMheCPhDSRUS3BFqB7JlxOTokiBZReWYfHpxTibdtbhep1rdsaMrjPgL/bHzEMzsebymjJbSBF+EZjWcRomtZlUKbuCeUSJCZSYvArLslh7ZS3e3/0+bubcdPn1Q2WhiPKPAgsWuUW5UGlV2Dh0I7rU6uLye/HN2xJTljILOoMO0QHRVs9r9Bpsv7Edyy4sw6aUTVaHbpfl6epPm6YefHX4K5vPngIlgfhf+//hv0//F4HSQKc/D+I0SkygxOSVNHoNFp5ciM+SPkOWKovTe12eeBmNIhtxeg8+eFNiOv7gOPou74u6oXVxfNxx03EDa8Dhe4ex7MIyrL68GtmqbKeu2zqmNaa2n4qcohzMT56P8+nnyywrE8kwOWEy3uvwHsL9wsv9uRCnUWICJSavpihS4Pvj32PusblO7aTrjA6xHfBay9cwOH6wV79j9pbEtO7KOry89mVo9BowYJDxbgbOp5/H+qvrseHqBtzPu+/0NbvX7o6X4l/C1cyr+O3sbzZ/VgIlgZjYZiLeevqtSvHs0QNRYgIlJp+gKFLgh+M/YO6xucgpyuHkHv5ifwxuPBivNn0ViXGJXjer39MTE8uy+Pbot3h317tmz3kCJYEOjcosjQGDPvX6oH5YfVx4fAF7b++1OXcp0i8Sbz39Fia2mUhLWrkXJSZQYvIpeeo8zDs+D3OPzeW0iy/aPxqD4gdhaJOhaB/b3iu23vDkxKTVazF8/XCsurSqwtfyF/ujXng9BEmDcObRGbtJrW5YXUxpOwWvtXwNfmK/Ct+fVBglJlBi8klKrRJ/nfsL3x771ubmhK5QPag6BscPxsCGA9E+tr3HbgDnaYkprSANB+8exD/X/8HKSyudGrxQFrlIDn+JPzKVmTbLGVtTkxMmo2ednl7xxqISocQESkw+zcAa8M/1f/Dt0W+x784+zu8XKgtF33p9MaD+APSu2xvBsmDO72mPJ0ywZVkWdxV3ceDuAdPrevZ1XmMAiueuvdbyNUxsMxF1w+ryfn/iEEpMoMRUaZxNO4ufkn/Cb2d+g57Vc34/kUCETjU6oUftHuheuzueqvqUW59L8dliyijMwMnUk8WvRyeR/DAZjwoecXrPsjBg0LNOT4xqMQrPNXgOcrHcLXEQh1FiAiWmSudB3gO0/rm11Y3duBQiC0FiXCK61+qOLrW6oGFEQ167kIyJ6dK9S0hKS0LPOj1RJ6xOha6p1qlxLesaLmVcwsXHF3Ep4xLOPDqDu4q7Loq6/OqH18eo5qMwvPlwVA+q7u5wiOMoMYESU6V0T3EPrX9ujSxlFgwwuC2O2qG1MaTxEHSt1RUJ1RI4W7svvSAdf534C+92exd4H4CseKHSaZ2m2a1bpCvC3dy7uJVz699X7i1czbyK61nXeWl9OqpOaB0Mjh+MwY0Ho2WVlrRkkHeibxooMVVaZx6dQfvf2kOtU5sNI5YKpVDr1W6MDPAT+6FnnZ7oUbsHGkc2RoRfBAKlgQiQBCBAEuDQ/lLZqmysv7Ieyy4sw/47+8EWscAsmBLTnB5z0DmuMxRFCuQW5SK9MB2P8h/hUcEjpBWk4VHBIzzKL/6/I1tEuEu9sHoYFD8Ig+MHo0WVFpSMvB99A0GJqVLbdn0b+q/oDwNb3GoSMkLcePMGTj86jfVX12Prta2czYuqqGBpMAKlgRAJRBAyQtPzK6VGCYVaYTlMughmiclbyUQyJMYlom/dvuhTrw8NYvA9lJhAianSW3xqMV7f8joYMBgUPwh/D/7bdE5n0OHwvcPYmLIRG1M24lbOLTdGWkFenpiaRTfDrG6zkBiXSAMYfBslJlBiIgA+3PMhvjz0JfaP3I/OcZ2tlmFZFlczr2LP7T3Yc3sP9t3eV+a27h7JyxNTvbB6uDaZ2zlpxCNQYgIlJoLipHMn9w5qhdZyuI7eoMfpR6ex5/YeJN1NwvEHxz222w+ARyYmISNE46jGaF21NVrFtEL98PpQaVU4/eg0jtw/gqMPjpp1SWZPzaadYn0fJSZQYiIuYmANuJ51HcceHMPxh8dx9MFRXHx8scwN53jnxsQkZISoG1YXjSIbIT4iHo0iG6FRRCPER8bb7JYr+TV9VPAIUztMpVUafB8lJlBiIhxS69RIyUrB+fTzuJB+ARceF79s7azLGQ4TU5g8DJF+kagaWBU1g2sWv0KK/60RXAM1Q2o6NJKQEFBiAgB45sJmxCdIRVI0i26GZtHNzI4XagpxO/c2bmbfxK2cW7iZcxM3c24iNT8Vj/IfIVOZ6TFDtFtUaYHm0c3RLLoZqgRUgb/YH8GyYET6RSLSPxJh8jCPXR+QEG9Fv1GEd/4SfzSJaoImUU2sntfqtchQZiCtIA1pBWnIUeUgX5OPPHWe2Uuj10Bn0EFn0EHP6lGoKcT59PPIUGbYvP8bbd5Ap5qdECILQYgsBNH+0agaWJVaNYR4CEpMxOOIhWLEBMYgJjDG7LharUbbtm1x7tw5nDlzBi1atLCoy7Is/jr/Fyb9MwlFuiKrz7gmJUxCw4iGXIVPCKkgepJKvMbUqVMRExNjswzDMBjRfAQuT7yMZ2o+Y7VMXEgcB9ERQlyFEhPxCtu2bcPOnTsxZ84ch8rHBsdi9/DdWNhvIWQimWlliAj/CMhEHjJenBBiFSUm4vHS09Mxbtw4/PXXX/Dzc2yXVbVajfz8fLxc/2UcG34MLaJbAABqhTg+V4sQ4h6uHi5OiEsxxauS/gPgMMuyMxiGiQNwG0BLlmXP2qj3CYDpVk4Fsyybx0GohBAXoRYTcQuGYT5hGIa182oNYDKAIAAznbzFTADBpV6RAPJtVSKEuB+1mIhbMAwTASDCTrE7AFYCGADzydtCAHoAy1iWHclJgIQQt6HERDwawzA1UNxiMooBsAPAIADHWZZ1wzIShBAu0Twm4tFYlr1X8mOGYQqe/PcmJSVCfBM9YyKEEOJRqCuPEEKIR6EWEyGEEI9CiYkQQohHocRECCHEo1BiIoQQ4lEoMRFCCPEolJgIIYR4FEpMhBBCPAolJkIIIR6FEhMhhBCPQomJEEKIR6HERAghxKP8P7iDJN0rUm/7AAAAAElFTkSuQmCC\n",
"text/plain": [
"Graphics object consisting of 22 graphics primitives"
]
},
"execution_count": 51,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"graph = Li.plot(coordinates='xy', prange=(0, lplot), plot_points=500, \n",
" thickness=3, color='green', display_tangent=True, scale=0.01, \n",
" width_tangent=1, color_tangent='green', \n",
" plot_points_tangent=20, horizon_color='lightgrey',\n",
" axes_labels=[r'$x/m$', r'$y/m$'])\n",
"graph.save(\"gik_spher_3d_r_39_xy.pdf\")\n",
"graph"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### A polar spherical photon orbit"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}a \\ {\\mapsto}\\ 2 \\, \\sqrt{-\\frac{1}{3} \\, a^{2} + 1} \\cos\\left(\\frac{1}{3} \\, \\arccos\\left(-\\frac{a^{2} - 1}{{\\left(-\\frac{1}{3} \\, a^{2} + 1\\right)}^{\\frac{3}{2}}}\\right)\\right) + 1\n",
"\\end{math}"
],
"text/plain": [
"a |--> 2*sqrt(-1/3*a^2 + 1)*cos(1/3*arccos(-(a^2 - 1)/(-1/3*a^2 + 1)^(3/2))) + 1"
]
},
"execution_count": 52,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rph_pol(a) = 1 + 2*sqrt(1 - a^2/3)*cos(1/3*arccos((1 - a^2)/(1 - a^2/3)^(3/2)))\n",
"rph_pol"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(2.49269429554008, -6.26333640524599 \\times 10^{-16}, 22.8640201857508\\right)\n",
"\\end{math}"
],
"text/plain": [
"(2.49269429554008, -6.26333640524599e-16, 22.8640201857508)"
]
},
"execution_count": 53,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r0 = rph_pol(a0)\n",
"L = lsph(a0, r0)\n",
"Q = qsph(a0, r0)\n",
"r0, L, Q"
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.000000000000000\n",
"\\end{math}"
],
"text/plain": [
"0.000000000000000"
]
},
"execution_count": 54,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"theta0(a0, L, Q)"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 3.67971815257238 \\frac{\\partial}{\\partial t } + \\left( 1.31343299672075 \\times 10^{-24} + 2.14499886438313 \\times 10^{-8}i \\right) \\frac{\\partial}{\\partial r } + 0.769552507817167 \\frac{\\partial}{\\partial {\\theta} } + 0.357746396090726 \\frac{\\partial}{\\partial {\\phi} }\n",
"\\end{math}"
],
"text/plain": [
"p = 3.67971815257238 d/dt + (1.31343299672075e-24 + 2.14499886438313e-8*I) d/dr + 0.769552507817167 d/dth + 0.357746396090726 d/dph"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"lmax = 30\n",
"P = M.point((0, r0, pi/2, 0), name='P')\n",
"Li = M.geodesic([0, lmax], P, mu=0, E=E, L=0, Q=Q, a_num=a0,\n",
" name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 3.67971815257238 \\frac{\\partial}{\\partial t } + 0.769552507817167 \\frac{\\partial}{\\partial {\\theta} } + 0.357746396090726 \\frac{\\partial}{\\partial {\\phi} }\n",
"\\end{math}"
],
"text/plain": [
"p = 3.67971815257238 d/dt + 0.769552507817167 d/dth + 0.357746396090726 d/dph"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"v0 = Li.initial_tangent_vector()\n",
"Li = M.geodesic([0, lmax], P, pt0=v0[0], pr0=0, pth0=v0[2], pph0=v0[3],\n",
" a_num=a0, name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"/home/eric/sage/9.2/local/lib/python3.8/site-packages/scipy/integrate/_ode.py:1179: UserWarning: dopri5: step size becomes too small\n",
" warnings.warn('{:s}: {:s}'.format(self.__class__.__name__,\n"
]
}
],
"source": [
"Li.integrate(step=0.002, method='dopri5')\n",
"#Li.check_integrals_of_motion(0.99*lmax)"
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"max lambda (plot): 15.0000000000000\n"
]
},
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 58,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lplot = 0.5*lmax\n",
"print(\"max lambda (plot): \", lplot)\n",
"Li.plot(prange=(0, lplot), plot_points=2000, thickness=3, color='green',\n",
" display_tangent=True, scale=0.2, width_tangent=2, color_tangent='green', \n",
" plot_points_tangent=20, horizon_color='lightgrey')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Inner spherical orbits"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"To plot the inner orbits, and in particular orbits with $r<0$, we use a map from Kerr spacetime to the Euclidean space based on the radial coordinate\n",
"$$\n",
" \\hat{r} := \\frac{1}{2} \\left( r + \\sqrt{r^2 + 4m^2} \\right) \n",
"$$\n",
"instead of the Boyer-Lindquist $r$. "
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\mathbb{E}^{4},(t, x, y, z)\\right)\n",
"\\end{math}"
],
"text/plain": [
"Chart (E^4, (t, x, y, z))"
]
},
"execution_count": 59,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"E4 = M.map_to_Euclidean().codomain()\n",
"X4 = E4.cartesian_coordinates()\n",
"t, x, y, z = X4[:]\n",
"X4"
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{2} \\, r + \\frac{1}{2} \\, \\sqrt{r^{2} + 4}\n",
"\\end{math}"
],
"text/plain": [
"1/2*r + 1/2*sqrt(r^2 + 4)"
]
},
"execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rhat = 1/2*(r + sqrt(r^2 + 4))\n",
"rhat"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\begin{array}{llcl} F:& M & \\longrightarrow & \\mathbb{E}^{4} \\\\ & \\left(t, r, {\\theta}, {\\phi}\\right) & \\longmapsto & \\left(t, x, y, z\\right) = \\left(t, \\frac{1}{2} \\, {\\left(r + \\sqrt{r^{2} + 4}\\right)} \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), \\frac{1}{2} \\, {\\left(r + \\sqrt{r^{2} + 4}\\right)} \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right), \\frac{1}{2} \\, {\\left(r + \\sqrt{r^{2} + 4}\\right)} \\cos\\left({\\theta}\\right)\\right) \\end{array}\n",
"\\end{math}"
],
"text/plain": [
"F: M --> E^4\n",
" (t, r, th, ph) |--> (t, x, y, z) = (t, 1/2*(r + sqrt(r^2 + 4))*cos(ph)*sin(th), 1/2*(r + sqrt(r^2 + 4))*sin(ph)*sin(th), 1/2*(r + sqrt(r^2 + 4))*cos(th))"
]
},
"execution_count": 61,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"F = M.diff_map(E4, {(BL, X4): [t, rhat*sin(th)*cos(ph),\n",
" rhat*sin(th)*sin(ph),\n",
" rhat*cos(th)]}, name='F')\n",
"F.display()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Marginally stable spherical photon orbit"
]
},
{
"cell_type": "code",
"execution_count": 62,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.539741795874205\n",
"\\end{math}"
],
"text/plain": [
"0.539741795874205"
]
},
"execution_count": 62,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r0 = 1 - (1 - a0^2)^(1/3)\n",
"r0"
]
},
{
"cell_type": "code",
"execution_count": 63,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1.53893384905757, 0.282109845505914\\right)\n",
"\\end{math}"
],
"text/plain": [
"(1.53893384905757, 0.282109845505914)"
]
},
"execution_count": 63,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L = lsph(a0, r0)\n",
"Q = qsph(a0, r0)\n",
"L, Q"
]
},
{
"cell_type": "code",
"execution_count": 64,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.17336440458345\n",
"\\end{math}"
],
"text/plain": [
"1.17336440458345"
]
},
"execution_count": 64,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"theta0(a0, L, Q)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Since all orbits with $0< r_0 \\leq r_{\\rm ph}^*$ (such as the marginally stable spherical orbit) have $E<0$, we set `E = -1` and `L = -L`:"
]
},
{
"cell_type": "code",
"execution_count": 65,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 7.69077392677338 \\frac{\\partial}{\\partial t } + \\left( 3.13205131833926 \\times 10^{-24} + 5.11502797462896 \\times 10^{-8}i \\right) \\frac{\\partial}{\\partial r } + 1.82321137638829 \\frac{\\partial}{\\partial {\\theta} } + 5.62672311935440 \\frac{\\partial}{\\partial {\\phi} }\n",
"\\end{math}"
],
"text/plain": [
"p = 7.69077392677338 d/dt + (3.13205131833926e-24 + 5.11502797462896e-8*I) d/dr + 1.82321137638829 d/dth + 5.62672311935440 d/dph"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"lmax = 50\n",
"P = M.point((0, r0, pi/2, 0), name='P')\n",
"Li = M.geodesic([0, lmax], P, mu=0, E=-1, L=-L, Q=Q, a_num=a0,\n",
" name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 66,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 7.69077392677338 \\frac{\\partial}{\\partial t } + 1.82321137638829 \\frac{\\partial}{\\partial {\\theta} } + 5.62672311935440 \\frac{\\partial}{\\partial {\\phi} }\n",
"\\end{math}"
],
"text/plain": [
"p = 7.69077392677338 d/dt + 1.82321137638829 d/dth + 5.62672311935440 d/dph"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"v0 = Li.initial_tangent_vector()\n",
"Li = M.geodesic([0, lmax], P, pt0=v0[0], pr0=0, pth0=v0[2], pph0=v0[3],\n",
" a_num=a0, name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 67,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
" quantity value initial value diff. relative diff.\n",
" $\\mu^2$ 2.1695647015040898e-11 -2.842170943040401e-14 2.172e-11 -764.3\n",
" $E$ -0.999999999955740 -1.00000000000000 4.426e-11 -4.426e-11\n",
" $L$ -1.53893384899354 -1.53893384905757 6.403e-11 -4.161e-11\n",
" $Q$ 0.282109845487987 0.282109845505914 -1.793e-11 -6.355e-11"
]
},
"execution_count": 67,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Li.integrate(step=0.001, method='dopri5')\n",
"Li.check_integrals_of_motion(0.999*lmax)"
]
},
{
"cell_type": "code",
"execution_count": 68,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Final point: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(t, r, {\\theta}, {\\phi}\\right) \\verb|=| \\left(302.91453819062644, 0.5397417961010423, 1.5090634888129522, 210.47540919172215\\right)\n",
"\\end{math}"
],
"text/plain": [
"(t, r, th, ph) '=' (302.91453819062644,\n",
" 0.5397417961010423,\n",
" 1.5090634888129522,\n",
" 210.47540919172215)"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"print(\"Final point: \")\n",
"show(BL[:], \"=\", BL(Li(0.999*lmax)))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Plot with respect to $\\hat{r}$ coordinate:"
]
},
{
"cell_type": "code",
"execution_count": 69,
"metadata": {},
"outputs": [],
"source": [
"sing_ring = circle((0,0,0), 1, thickness=3, color='orangered')\n",
"r_minf = sphere((0,0,0), 0.03, color='black') \\\n",
" + text3d(\"r=\\u2212\\u221E\", (0,0,0.07), fontfamily='serif', \n",
" fontstyle='italic', fontsize=20)"
]
},
{
"cell_type": "code",
"execution_count": 70,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"max lambda (plot): 3.00000000000000\n"
]
},
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 70,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lplot = 0.06*lmax\n",
"print(\"max lambda (plot): \", lplot)\n",
"graph = Li.plot_integrated(chart=X4, mapping=F,\n",
" ambient_coords=(x,y,z), prange=(0, lplot),\n",
" thickness=3, plot_points=2000, color='green',\n",
" display_tangent=True, scale=0.012, width_tangent=1, \n",
" color_tangent='green', plot_points_tangent=20,\n",
" label_axes=False)\n",
"graph += sing_ring + r_minf \\\n",
" + line([(0,0,-0.53), (0,0,0.53)], color='black', thickness=2)\n",
"graph._extra_kwds['axes_labels'] = ['\\u0302x', '\\u0302y', '\\u0302z']\n",
"graph"
]
},
{
"cell_type": "code",
"execution_count": 71,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"max lambda (plot): 20.0000000000000\n"
]
},
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 71,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lplot = 0.4*lmax\n",
"print(\"max lambda (plot): \", lplot)\n",
"graph = Li.plot_integrated(chart=X4, mapping=F,\n",
" ambient_coords=(x,y,z), prange=(0, lplot),\n",
" thickness=3, plot_points=2000, color='green',\n",
" display_tangent=True, scale=0.012, width_tangent=1, \n",
" color_tangent='green', plot_points_tangent=20,\n",
" label_axes=False)\n",
"graph += sing_ring + r_minf \\\n",
" + line([(0,0,-0.53), (0,0,0.53)], color='black', thickness=2)\n",
"graph._extra_kwds['axes_labels'] = ['\\u0302x', '\\u0302y', '\\u0302z']\n",
"graph"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Stable inner spherical photon orbit at $r_0 = 0.5 m$"
]
},
{
"cell_type": "code",
"execution_count": 72,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(1.53421052631579, 0.268698060941828\\right)\n",
"\\end{math}"
],
"text/plain": [
"(1.53421052631579, 0.268698060941828)"
]
},
"execution_count": 72,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r0 = 0.5\n",
"L = lsph(a0, r0)\n",
"Q = qsph(a0, r0)\n",
"L, Q"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}1.17991588772518\n",
"\\end{math}"
],
"text/plain": [
"1.17991588772518"
]
},
"execution_count": 73,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"theta0(a0, L, Q)"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 7.00000000000000 \\frac{\\partial}{\\partial t } + 2.07344374774655 \\frac{\\partial}{\\partial {\\theta} } + 5.26315789473685 \\frac{\\partial}{\\partial {\\phi} }\n",
"\\end{math}"
],
"text/plain": [
"p = 7.00000000000000 d/dt + 2.07344374774655 d/dth + 5.26315789473685 d/dph"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"lmax = 30\n",
"P = M.point((0, r0, pi/2, 0), name='P')\n",
"Li = M.geodesic([0, lmax], P, mu=0, E=-1, L=-L, Q=Q, a_num=a0,\n",
" name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 75,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 7.00000000000000 \\frac{\\partial}{\\partial t } + 2.07344374774655 \\frac{\\partial}{\\partial {\\theta} } + 5.26315789473685 \\frac{\\partial}{\\partial {\\phi} }\n",
"\\end{math}"
],
"text/plain": [
"p = 7.00000000000000 d/dt + 2.07344374774655 d/dth + 5.26315789473685 d/dph"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"v0 = Li.initial_tangent_vector()\n",
"Li = M.geodesic([0, lmax], P, pt0=v0[0], pr0=0, pth0=v0[2], pph0=v0[3],\n",
" a_num=a0, name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 76,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
" quantity value initial value diff. relative diff.\n",
" $\\mu^2$ 1.0061125543803229e-11 1.4210854715202004e-14 1.005e-11 707.0\n",
" $E$ -0.999999999989093 -0.999999999999993 1.090e-11 -1.090e-11\n",
" $L$ -1.53421052630196 -1.53421052631578 1.382e-11 -9.008e-12\n",
" $Q$ 0.268698060941096 0.268698060941828 -7.321e-13 -2.725e-12"
]
},
"execution_count": 76,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Li.integrate(step=0.001, method='dopri5')\n",
"Li.check_integrals_of_motion(0.999*lmax)"
]
},
{
"cell_type": "code",
"execution_count": 77,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Final point: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(t, r, {\\theta}, {\\phi}\\right) \\verb|=| \\left(160.26568413055006, 0.499999999999837, 1.4610686498265664, 113.33060855013213\\right)\n",
"\\end{math}"
],
"text/plain": [
"(t, r, th, ph) '=' (160.26568413055006, 0.499999999999837, 1.4610686498265664, 113.33060855013213)"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"print(\"Final point: \")\n",
"show(BL[:], \"=\", BL(Li(0.999*lmax)))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Plot in (Cartesian) Boyer-Lindquist coordinates:"
]
},
{
"cell_type": "code",
"execution_count": 78,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 78,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Li.plot(prange=(0, lmax), plot_points=2000, thickness=3, color='green',\n",
" display_tangent=True, scale=0.01, width_tangent=1, color_tangent='green', \n",
" plot_points_tangent=20, plot_horizon=False)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Plot with respect to $\\hat{r}$ coordinate:"
]
},
{
"cell_type": "code",
"execution_count": 79,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"max lambda (plot): 7.50000000000000\n"
]
},
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 79,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lplot = 0.25*lmax\n",
"print(\"max lambda (plot): \", lplot)\n",
"graph1 = Li.plot_integrated(chart=X4, mapping=F,\n",
" ambient_coords=(x,y,z), prange=(0, lplot),\n",
" thickness=3, plot_points=2000, color='green',\n",
" display_tangent=True, scale=0.015, width_tangent=1, \n",
" color_tangent='green', plot_points_tangent=20,\n",
" label_axes=False)\n",
"graph1 += sing_ring + r_minf\n",
"graph1._extra_kwds['axes_labels'] = ['\\u0302x', '\\u0302y', '\\u0302z']\n",
"graph1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Vortical circular photon orbit at $r_0 = r_{\\rm ph}^{**}$"
]
},
{
"cell_type": "code",
"execution_count": 80,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}-0.477673658836338\n",
"\\end{math}"
],
"text/plain": [
"-0.477673658836338"
]
},
"execution_count": 80,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rph_ss = n(1/2 + cos(2/3*asin(a0) + 2*pi/3))\n",
"rph_ss"
]
},
{
"cell_type": "code",
"execution_count": 81,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(-0.229456449433387, -0.519183008263141\\right)\n",
"\\end{math}"
],
"text/plain": [
"(-0.229456449433387, -0.519183008263141)"
]
},
"execution_count": 81,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L = lsph(a0, rph_ss)\n",
"Q = qsph(a0, rph_ss)\n",
"L, Q "
]
},
{
"cell_type": "code",
"execution_count": 82,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0.513765577758655, 0.513765602376411\\right)\n",
"\\end{math}"
],
"text/plain": [
"(0.513765577758655, 0.513765602376411)"
]
},
"execution_count": 82,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"th0 = theta0(a0, L, Q)\n",
"th1 = theta1(a0, L, Q)\n",
"th0, th1"
]
},
{
"cell_type": "code",
"execution_count": 83,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.513765590067533\n",
"\\end{math}"
],
"text/plain": [
"0.513765590067533"
]
},
"execution_count": 83,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"th_ss = arcsin(sqrt(abs(L)/a0))\n",
"th_ss"
]
},
{
"cell_type": "code",
"execution_count": 84,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}0.789289150745023\n",
"\\end{math}"
],
"text/plain": [
"0.789289150745023"
]
},
"execution_count": 84,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rh_ss = rhat.subs({r: rph_ss})\n",
"rh_ss"
]
},
{
"cell_type": "code",
"execution_count": 85,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 0.323260590036169 \\frac{\\partial}{\\partial t } + \\left( 1.63266670244580 \\times 10^{-8} \\right) \\frac{\\partial}{\\partial r } + \\left( 1.99960017051231 \\times 10^{-8} \\right) \\frac{\\partial}{\\partial {\\theta} } -1.40881021577044 \\frac{\\partial}{\\partial {\\phi} }\n",
"\\end{math}"
],
"text/plain": [
"p = 0.323260590036169 d/dt + (1.63266670244580e-8) d/dr + (1.99960017051231e-8) d/dth - 1.40881021577044 d/dph"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"lmax = 10\n",
"P = M.point((0, rph_ss, th_ss, 0), name='P')\n",
"Li = M.geodesic([0, lmax], P, mu=0, E=E, L=L, Q=Q, a_num=a0,\n",
" name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 86,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 0.323260590036169 \\frac{\\partial}{\\partial t } -1.40881021577044 \\frac{\\partial}{\\partial {\\phi} }\n",
"\\end{math}"
],
"text/plain": [
"p = 0.323260590036169 d/dt - 1.40881021577044 d/dph"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"v0 = Li.initial_tangent_vector()\n",
"Li = M.geodesic([0, lmax], P, pt0=v0[0], pr0=0, pth0=0, pph0=v0[3],\n",
" a_num=a0, name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 87,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
" quantity value initial value diff. relative diff.\n",
" $\\mu^2$ 1.30395694242225e-13 9.71445146547012e-16 1.294e-13 133.2\n",
" $E$ 1.00000000000031 1.00000000000000 3.078e-13 3.078e-13\n",
" $L$ -0.229456449433411 -0.229456449433387 -2.470e-14 1.077e-13\n",
" $Q$ -0.519183008263439 -0.519183008263142 -2.971e-13 5.722e-13"
]
},
"execution_count": 87,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Li.integrate(step=0.001, method='dopri5')\n",
"Li.check_integrals_of_motion(0.999*lmax)"
]
},
{
"cell_type": "code",
"execution_count": 88,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\verb|Final|\\phantom{\\verb!x!}\\verb|point:| \\left(t, r, {\\theta}, {\\phi}\\right) \\verb|=| \\left(3.229373294461296, -0.47767365883633794, 0.513765590067533, -14.074014055546579\\right)\n",
"\\end{math}"
],
"text/plain": [
"'Final point: ' (t, r, th, ph) '=' (3.229373294461296,\n",
" -0.47767365883633794,\n",
" 0.513765590067533,\n",
" -14.074014055546579)"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"show(\"Final point: \", BL[:], \"=\", BL(Li(0.999*lmax)))"
]
},
{
"cell_type": "code",
"execution_count": 89,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"max lambda (plot): 4.44000000000000\n"
]
},
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 89,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"lplot = 0.444*lmax\n",
"print(\"max lambda (plot): \", lplot)\n",
"vort_circ = Li.plot_integrated(chart=X4, mapping=F,\n",
" ambient_coords=(x,y,z), prange=(0, lplot),\n",
" thickness=3, plot_points=2000, color='lightgreen',\n",
" display_tangent=True, scale=0.1, width_tangent=1, \n",
" color_tangent='lightgreen', plot_points_tangent=5,\n",
" label_axes=False)\n",
"graph = vort_circ + sing_ring + r_minf \\\n",
" + line([(0,0,0), (0,0,1.3*rh_ss)], color='black', thickness=2)\n",
"graph._extra_kwds['axes_labels'] = ['\\u0302x', '\\u0302y', '\\u0302z']\n",
"graph"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Vortical inner spherical photon orbit at $r_0 = -0.46 m$"
]
},
{
"cell_type": "code",
"execution_count": 90,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(-0.176485940879596, -0.461285155596124\\right)\n",
"\\end{math}"
],
"text/plain": [
"(-0.176485940879596, -0.461285155596124)"
]
},
"execution_count": 90,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r0 = -0.46\n",
"L = lsph(a0, r0)\n",
"Q = qsph(a0, r0)\n",
"L, Q"
]
},
{
"cell_type": "code",
"execution_count": 91,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(0.281889824619659, 0.731306339672501\\right)\n",
"\\end{math}"
],
"text/plain": [
"(0.281889824619659, 0.731306339672501)"
]
},
"execution_count": 91,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"th0 = theta0(a0, L, Q)\n",
"th1 = theta1(a0, L, Q)\n",
"th0, th1"
]
},
{
"cell_type": "code",
"execution_count": 92,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 0.357024161116205 \\frac{\\partial}{\\partial t } + \\left( 1.84776601065215 \\times 10^{-8} \\right) \\frac{\\partial}{\\partial r } + 0.396167049160545 \\frac{\\partial}{\\partial {\\theta} } -1.22114489356954 \\frac{\\partial}{\\partial {\\phi} }\n",
"\\end{math}"
],
"text/plain": [
"p = 0.357024161116205 d/dt + (1.84776601065215e-8) d/dr + 0.396167049160545 d/dth - 1.22114489356954 d/dph"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"lmax = 20\n",
"P = M.point((0, r0, (th0+th1)/2, 0), name='P')\n",
"Li = M.geodesic([0, lmax], P, mu=0, E=E, L=L, Q=Q, a_num=a0,\n",
" name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 93,
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Initial tangent vector: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}p = 0.357024161116205 \\frac{\\partial}{\\partial t } + 0.396167049160545 \\frac{\\partial}{\\partial {\\theta} } -1.22114489356954 \\frac{\\partial}{\\partial {\\phi} }\n",
"\\end{math}"
],
"text/plain": [
"p = 0.357024161116205 d/dt + 0.396167049160545 d/dth - 1.22114489356954 d/dph"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"name": "stdout",
"output_type": "stream",
"text": [
"The curve was correctly set.\n",
"Parameters appearing in the differential system defining the curve are [a].\n"
]
}
],
"source": [
"v0 = Li.initial_tangent_vector()\n",
"Li = M.geodesic([0, lmax], P, pt0=v0[0], pr0=0, pth0=v0[2], pph0=v0[3],\n",
" a_num=a0, name='Li', latex_name=r'\\mathcal{L}', verbose=True)"
]
},
{
"cell_type": "code",
"execution_count": 94,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
" quantity value initial value diff. relative diff.\n",
" $\\mu^2$ 3.315542285164952e-13 8.465450562766819e-16 3.307e-13 390.7\n",
" $E$ 1.00000000000081 1.00000000000000 8.118e-13 8.118e-13\n",
" $L$ -0.176485940879686 -0.176485940879596 -9.026e-14 5.114e-13\n",
" $Q$ -0.461285155596888 -0.461285155596125 -7.633e-13 1.655e-12"
]
},
"execution_count": 94,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Li.integrate(step=0.001, method='dopri5')\n",
"Li.check_integrals_of_motion(0.999*lmax)"
]
},
{
"cell_type": "code",
"execution_count": 95,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Final point: \n"
]
},
{
"data": {
"text/html": [
""
],
"text/latex": [
"\\begin{math}\n",
"\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(t, r, {\\theta}, {\\phi}\\right) \\verb|=| \\left(6.6100475583006695, -0.45999868463861343, 0.39359482789056505, -28.895385729902912\\right)\n",
"\\end{math}"
],
"text/plain": [
"(t, r, th, ph) '=' (6.6100475583006695,\n",
" -0.45999868463861343,\n",
" 0.39359482789056505,\n",
" -28.895385729902912)"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"print(\"Final point: \")\n",
"show(BL[:], \"=\", BL(Li(0.999*lmax)))"
]
},
{
"cell_type": "code",
"execution_count": 96,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"max lambda (plot): 20\n"
]
}
],
"source": [
"lplot = lmax\n",
"print(\"max lambda (plot): \", lplot)\n",
"graph = Li.plot_integrated(chart=X4, mapping=F,\n",
" ambient_coords=(x,y,z), prange=(0, lplot),\n",
" thickness=3, plot_points=2000, color='green',\n",
" display_tangent=True, scale=0.1, width_tangent=1, \n",
" color_tangent='green', plot_points_tangent=20,\n",
" label_axes=False)"
]
},
{
"cell_type": "code",
"execution_count": 97,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 97,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"graph += sing_ring + r_minf + vort_circ \\\n",
" + line([(0,0,0), (0,0,0.9)], color='black', thickness=2)\n",
"graph._extra_kwds['axes_labels'] = ['\\u0302x', '\\u0302y', '\\u0302z']\n",
"graph"
]
},
{
"cell_type": "code",
"execution_count": 98,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 98,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"graph + graph1"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "SageMath 9.2",
"language": "sage",
"name": "sagemath"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.8.5"
}
},
"nbformat": 4,
"nbformat_minor": 2
}