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	<div class="section" id="generating-velocities">
	<h1>Generating velocities<a class="headerlink" href="#generating-velocities" title="Permalink to this headline">¶</a></h1>
	<p>Setting initial velocities may be trivial or very complex, depending
	on the aim. Of course, its allways possible to directely set the
	variable ‘’nb.vel’’ to some values. However, in galactic dynamics, this
	is usually nor the right way to do.</p>
	<p><strong>pNbody</strong> offers different methods to set the velocities in order to ensure
	the system to be in a resonably good equilibrium.</p>
	<table border="1" class="docutils">
	<colgroup>
	<col width="60%" />
	<col width="12%" />
	<col width="28%" />
	</colgroup>
	<thead valign="bottom">
	<tr><th class="head">method name</th>
	<th class="head">geometry</th>
	<th class="head">method</th>
	</tr>
	</thead>
	<tbody valign="top">
	<tr><td><a class="reference internal" href="MainModule.html#pNbody.main.NbodyDefault.Get_Velocities_From_AdaptativeSpherical_Grid" title="pNbody.main.NbodyDefault.Get_Velocities_From_AdaptativeSpherical_Grid"><tt class="xref py py-func docutils literal"><span class="pre">NbodyDefault.Get_Velocities_From_AdaptativeSpherical_Grid()</span></tt></a></td>
	<td>spherical</td>
	<td>jeans equation+self-adaptative grid</td>
	</tr>
	<tr><td><a class="reference internal" href="MainModule.html#pNbody.main.NbodyDefault.Get_Velocities_From_Spherical_Grid" title="pNbody.main.NbodyDefault.Get_Velocities_From_Spherical_Grid"><tt class="xref py py-func docutils literal"><span class="pre">NbodyDefault.Get_Velocities_From_Spherical_Grid()</span></tt></a></td>
	<td>spherical</td>
	<td>jeans equation</td>
	</tr>
	<tr><td><a class="reference internal" href="MainModule.html#pNbody.main.NbodyDefault.Get_Velocities_From_Cylindrical_Grid" title="pNbody.main.NbodyDefault.Get_Velocities_From_Cylindrical_Grid"><tt class="xref py py-func docutils literal"><span class="pre">NbodyDefault.Get_Velocities_From_Cylindrical_Grid()</span></tt></a></td>
	<td>cylindrical</td>
	<td>jeans equation</td>
	</tr>
	</tbody>
	</table>
	<p>Some examples showing how to set the initial velocities for spherical or axi-symetric models are provided in the
	example directory. To get it, type:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="n">pNbody_copy</span><span class="o">-</span><span class="n">examples</span>
	</pre></div>
	</div>
	<p>by default, this create a directory in your home <tt class="docutils literal"><span class="pre">~/pnbody_examples</span></tt>.
	The scripts are in the <tt class="docutils literal"><span class="pre">~/pnbody_examples/ic</span></tt> directory.</p>
	<div class="section" id="spherical-coordinate">
	<h2>Spherical coordinate<a class="headerlink" href="#spherical-coordinate" title="Permalink to this headline">¶</a></h2>
	<p>In spherical coordinates, supposing that the velocities are isotropics. the Jeans equations is reduced to one integral,
	giving the value of the velocity dispertion of a component <img class="math" src="../_images/math/34857b3ba74ce5cd8607f3ebd23e9015908ada71.png" alt="i"/> :</p>
	<div class="math">
	<p><img src="../_images/math/a5f2e7938b43b33c24e120d4766d15391227b717.png" alt="\sigma_{i}^2(r) = \frac{1}{\rho_{i}(r)}\int_r^\infty\! dr' \,\rho_{i}(r')\, \partial_{r'} \Phi(r')." /></p>
	</div></div>
	<div class="section" id="example-set-a-plummer-sphere-to-the-jeans-equilibrium">
	<h2>Example : Set a Plummer sphere to the jeans equilibrium<a class="headerlink" href="#example-set-a-plummer-sphere-to-the-jeans-equilibrium" title="Permalink to this headline">¶</a></h2>
	<p>First, generate the Plummer sphere:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">pNbody</span> <span class="kn">import</span> <span class="n">ic</span>
	<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="o">*</span>

	<span class="gp">>>> </span><span class="n">n</span> <span class="o">=</span> <span class="mi">100000</span>
	<span class="gp">>>> </span><span class="n">a</span> <span class="o">=</span> <span class="mf">1.</span>
	<span class="gp">>>> </span><span class="n">rmax</span> <span class="o">=</span> <span class="mf">10.</span>
	<span class="gp">>>> </span><span class="n">nb</span> <span class="o">=</span> <span class="n">ic</span><span class="o">.</span><span class="n">plummer</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">eps</span><span class="o">=</span><span class="n">a</span><span class="p">,</span><span class="n">rmax</span><span class="o">=</span><span class="n">rmax</span><span class="p">,</span><span class="n">name</span><span class="o">=</span><span class="s">'plummer.dat'</span><span class="p">,</span><span class="n">ftype</span><span class="o">=</span><span class="s">'gadget'</span><span class="p">)</span>
	</pre></div>
	</div>
	<p>Then, we need to create a 1D grid wraping the model:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">grmin</span> <span class="o">=</span> <span class="mi">0</span> <span class="c"># grid minimal radius</span>
	<span class="gp">>>> </span><span class="n">grmax</span> <span class="o">=</span> <span class="n">rmax</span><span class="o">*</span><span class="mf">1.05</span> <span class="c"># grid maximal radius</span>
	<span class="gp">>>> </span><span class="n">nr</span> <span class="o">=</span> <span class="mi">128</span> <span class="c"># number of radial bins</span>
	</pre></div>
	</div>
	<p>Now, in order to decrease the noise in outer regions, it is usefull to
	use a non linear grid. In this purpose, we need to define a function that
	defines the new nodes of the grid. Here, we set it logarithmic. The inverse function
	is also needed:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">rc</span> <span class="o">=</span> <span class="n">a</span>
	<span class="gp">>>> </span><span class="n">g</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">r</span><span class="p">:</span><span class="n">log</span><span class="p">(</span><span class="n">r</span><span class="o">/</span><span class="n">rc</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="c"># the function</span>
	<span class="gp">>>> </span><span class="n">gm</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">r</span><span class="p">:</span><span class="n">rc</span><span class="o">*</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">r</span><span class="p">)</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="c"># and its inverse</span>
	</pre></div>
	</div>
	<p>Now it possible to invoque the magic function:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">eps</span> <span class="o">=</span> <span class="mf">0.01</span> <span class="c"># gravitational softening</span>
	<span class="gp">>>> </span><span class="n">nb</span><span class="p">,</span><span class="n">phi</span><span class="p">,</span><span class="n">stats</span> <span class="o">=</span> <span class="n">nb</span><span class="o">.</span><span class="n">Get_Velocities_From_Spherical_Grid</span><span class="p">(</span><span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">,</span><span class="n">nr</span><span class="o">=</span><span class="n">nr</span><span class="p">,</span><span class="n">rmax</span><span class="o">=</span><span class="n">grmax</span><span class="p">,</span><span class="n">phi</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span><span class="n">g</span><span class="o">=</span><span class="n">g</span><span class="p">,</span><span class="n">gm</span><span class="o">=</span><span class="n">gm</span><span class="p">,</span><span class="n">UseTree</span><span class="o">=</span><span class="bp">True</span><span class="p">,</span><span class="n">ErrTolTheta</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
	</pre></div>
	</div>
	<p>If <tt class="docutils literal"><span class="pre">UseTree=True</span></tt> the function uses a <tt class="docutils literal"><span class="pre">treecode</span></tt> to compute the potential at each node.
	A new <tt class="docutils literal"><span class="pre">nb</span></tt> object with velocities computed from the Jeans equation is returned. We also get the potential in each node
	and some statistics. The potential <tt class="docutils literal"><span class="pre">phi</span></tt> is usefull if we need to run the method for another component.</p>
	<p>Using the <tt class="docutils literal"><span class="pre">stats</span></tt> variable, it is possible to plot some interesting values used during the computation, like the velocity dispertion as a function of the radius:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">pylab</span> <span class="kn">as</span> <span class="nn">pt</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">stats</span><span class="p">[</span><span class="s">'r'</span><span class="p">],</span><span class="n">stats</span><span class="p">[</span><span class="s">'sigma'</span><span class="p">])</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
	</pre></div>
	</div>
	<img alt="../_images/r-sigma.png" src="../_images/r-sigma.png" />
	<p>or the density profile:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">stats</span><span class="p">[</span><span class="s">'r'</span><span class="p">],</span><span class="n">stats</span><span class="p">[</span><span class="s">'rho'</span><span class="p">])</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">loglog</span><span class="p">()</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
	</pre></div>
	</div>
	<img alt="../_images/r-rho.png" src="../_images/r-rho.png" />
	<p>It is possible to check the velocity dispertion along the line of sight directely
	by mesuring it using a grid object:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">pNbody</span> <span class="kn">import</span> <span class="n">libgrid</span>
	<span class="gp">>>> </span><span class="n">G</span> <span class="o">=</span> <span class="n">libgrid</span><span class="o">.</span><span class="n">Spherical_1d_Grid</span><span class="p">(</span><span class="n">rmin</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span><span class="n">rmax</span><span class="o">=</span><span class="n">rmax</span><span class="p">,</span><span class="n">nr</span><span class="o">=</span><span class="mi">128</span><span class="p">,</span><span class="n">g</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span><span class="n">gm</span><span class="o">=</span><span class="bp">None</span><span class="p">)</span>
	<span class="gp">>>> </span><span class="n">r</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">get_r</span><span class="p">()</span>
	<span class="gp">>>> </span><span class="n">sigma</span> <span class="o">=</span> <span class="n">G</span><span class="o">.</span><span class="n">get_SigmaValMap</span><span class="p">(</span><span class="n">nb</span><span class="p">,</span><span class="n">nb</span><span class="o">.</span><span class="n">vz</span><span class="p">())</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">r</span><span class="p">,</span><span class="n">sigma</span><span class="p">)</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">"radius"</span><span class="p">)</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">"velocity dispertion"</span><span class="p">)</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
	</pre></div>
	</div>
	<img alt="../_images/r-sigma-mes.png" src="../_images/r-sigma-mes.png" />
	</div>
	<div class="section" id="cylindrical-coordinate">
	<h2>Cylindrical coordinate<a class="headerlink" href="#cylindrical-coordinate" title="Permalink to this headline">¶</a></h2>
	<p>As in spherical geometry, we use the Jeans equations to get an approximation of the velocity dispertions.</p>
	<div class="section" id="vertical-velocity-dispersion">
	<h3>Vertical velocity dispersion<a class="headerlink" href="#vertical-velocity-dispersion" title="Permalink to this headline">¶</a></h3>
	<p>The resolution of these equations in cylindrical coordinates gives directely the dispersion along the z axis:</p>
	<div class="math">
	<p><img src="../_images/math/a6deaa7f455bbae6d9a3ed36b158801eed379434.png" alt="{\sigma_z}_i^2 = \frac{1}{\rho_i(z)}\int_z^\infty\! dz' \,\rho_i(z')\, \partial_{z'} \Phi(z')." /></p>
	</div><p>and the mean azimuthal velocity:</p>
	</div>
	<div class="section" id="radial-velocity-dispersion">
	<h3>Radial velocity dispersion<a class="headerlink" href="#radial-velocity-dispersion" title="Permalink to this headline">¶</a></h3>
	<ol class="arabic simple">
	<li>if <tt class="docutils literal"><span class="pre">mode_sigma_r['name']='epicyclic_approximation'</span></tt>, we use the epicyclic approximation which hold:</li>
	</ol>
	<div class="math">
	<p><img src="../_images/math/99c1f30d1f4c875d7fb755d4282085ea3818b1da.png" alt="\sigma_R^2 = \sigma_z^2 \frac{1}{\beta^2} \frac{\nu^2}{\kappa^2}" /></p>
	</div><p>the parameter <img class="math" src="../_images/math/04777e262a979166a6295fb67e24ca75bc42f3b6.png" alt="\beta^2"/> is set by the variable <tt class="docutils literal"><span class="pre">mode_sigma_r['param']</span></tt> and takes usually a value around 1.</p>
	<ol class="arabic simple" start="2">
	<li>if <tt class="docutils literal"><span class="pre">mode_sigma_r['name']='isothropic'</span></tt>, the value is simply:</li>
	</ol>
	<div class="math">
	<p><img src="../_images/math/26f488dbfe0e8985861ece799eaed8f37561717c.png" alt="\sigma_R^2 = \sigma_z^2" /></p>
	</div><ol class="arabic" start="3">
	<li><p class="first">if <tt class="docutils literal"><span class="pre">mode_sigma_r['name']='toomre'</span></tt>, we determine the velocity dispersion uses the Safronov-Toomre parameter <img class="math" src="../_images/math/9866e3a998d628ba0941eb4fea0666ac391d149a.png" alt="Q"/>:</p>
	<div class="math">
	<p><img src="../_images/math/ded125c9d4c060477de06e79b24b3d76661fc17e.png" alt="\sigma_R = \frac{3.36 Q G \Sigma_i}{\kappa}" /></p>
	</div><p><img class="math" src="../_images/math/9866e3a998d628ba0941eb4fea0666ac391d149a.png" alt="Q"/> is set with the variable <tt class="docutils literal"><span class="pre">mode_sigma_r['param']</span></tt>.</p>
	</li>
	<li><p class="first">if <tt class="docutils literal"><span class="pre">mode_sigma_r['name']='constant'</span></tt>, the value is simply constant, given by the variable <tt class="docutils literal"><span class="pre">mode_sigma_r['param']</span></tt>:</p>
	</li>
	</ol>
	<div class="math">
	<p><img src="../_images/math/b7b72642c21b9dc0c18bc48566b5a528387547b2.png" alt="\sigma_R = cte" /></p>
	</div></div>
	<div class="section" id="azimuthal-velocity-dispersion">
	<h3>Azimuthal velocity dispersion<a class="headerlink" href="#azimuthal-velocity-dispersion" title="Permalink to this headline">¶</a></h3>
	<ol class="arabic simple">
	<li>if <tt class="docutils literal"><span class="pre">mode_sigma_p['name']='epicyclic_approximation'</span></tt>, we use the epicyclic approximation which hold:</li>
	</ol>
	<div class="math">
	<p><img src="../_images/math/a70b008211f747612127d906de127eb49abd6f82.png" alt="\sigma_\phi^2 = \sigma_r^2 \frac{1}{4} \frac{\kappa^2}{\Omega^2}" /></p>
	</div><ol class="arabic simple" start="2">
	<li>if <tt class="docutils literal"><span class="pre">mode_sigma_p['name']='isothropic'</span></tt>, the value is simply:</li>
	</ol>
	<div class="math">
	<p><img src="../_images/math/ae1c33e8b543fea77262a8f055f39575e36e5a5e.png" alt="\sigma_\phi^2 = \sigma_z^2" /></p>
	</div></div>
	<div class="section" id="mean-azimuthal-velocity">
	<h3>Mean azimuthal velocity<a class="headerlink" href="#mean-azimuthal-velocity" title="Permalink to this headline">¶</a></h3>
	<p>Finally, the mean azimuthal velocity is also derived directely from the Jeans equations:</p>
	<div class="math">
	<p><img src="../_images/math/c6c6e55195f1df3be1847c9d7ae40ae6bfa4e70b.png" alt="\langle v_{\phi}^2 \rangle = R\,\partial_{R} \Phi(R,z) + \sigma_R^2 - \sigma_\phi^2 + \frac{R}{\rho_i} \partial_R \left( \rho_i \sigma_R^2 \right)," /></p>
	</div></div>
	</div>
	<div class="section" id="example-set-an-exponnential-disk-to-the-jeans-equilibrium">
	<h2>Example : Set an exponnential disk to the jeans equilibrium<a class="headerlink" href="#example-set-an-exponnential-disk-to-the-jeans-equilibrium" title="Permalink to this headline">¶</a></h2>
	<p>Lest try to put an exponnential disk at the Jeans equilibrium.
	First, we generate the disk:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">pNbody</span> <span class="kn">import</span> <span class="n">ic</span>
	<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">numpy</span> <span class="kn">import</span> <span class="o">*</span>
	<span class="gp">>>> </span><span class="n">n</span> <span class="o">=</span> <span class="mi">100000</span>
	<span class="gp">>>> </span><span class="n">Hz</span> <span class="o">=</span> <span class="mf">0.3</span>
	<span class="gp">>>> </span><span class="n">Hr</span> <span class="o">=</span> <span class="mf">3.</span>
	<span class="gp">>>> </span><span class="n">Rmax</span> <span class="o">=</span> <span class="mf">30.</span>
	<span class="gp">>>> </span><span class="n">Zmax</span> <span class="o">=</span> <span class="mf">3.</span>
	<span class="gp">>>> </span><span class="n">nb</span> <span class="o">=</span> <span class="n">ic</span><span class="o">.</span><span class="n">expd</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">Hr</span><span class="p">,</span><span class="n">Hz</span><span class="p">,</span><span class="n">Rmax</span><span class="p">,</span><span class="n">Zmax</span><span class="p">,</span><span class="n">irand</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span><span class="n">name</span><span class="o">=</span><span class="s">'expd.dat'</span><span class="p">,</span><span class="n">ftype</span><span class="o">=</span><span class="s">'gadget'</span><span class="p">)</span>
	</pre></div>
	</div>
	<p>Then, we need to set the parameters for the cylindrical grid:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">grmin</span> <span class="o">=</span> <span class="mf">0.</span> <span class="c"># minimal grid radius</span>
	<span class="gp">>>> </span><span class="n">grmax</span> <span class="o">=</span> <span class="mf">1.1</span><span class="o">*</span><span class="n">Rmax</span> <span class="c"># maximal grid radius</span>
	<span class="gp">>>> </span><span class="n">gzmin</span> <span class="o">=</span> <span class="o">-</span><span class="mf">1.1</span><span class="o">*</span><span class="n">Zmax</span> <span class="c"># minimal grid z</span>
	<span class="gp">>>> </span><span class="n">gzmax</span> <span class="o">=</span> <span class="mf">1.1</span><span class="o">*</span><span class="n">Zmax</span> <span class="c"># maximal grid z</span>
	<span class="gp">>>> </span><span class="n">nr</span> <span class="o">=</span> <span class="mi">32</span> <span class="c"># number of bins in r</span>
	<span class="gp">>>> </span><span class="n">nt</span> <span class="o">=</span> <span class="mi">2</span> <span class="c"># number of bins in t</span>
	<span class="gp">>>> </span><span class="n">nz</span> <span class="o">=</span> <span class="mi">64</span><span class="o">+</span><span class="mi">1</span> <span class="c"># number of bins in z</span>
	</pre></div>
	</div>
	<dl class="docutils">
	<dt>Set some functions used to distort the grid along the radius::</dt>
	<dd><div class="first last highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">rc</span> <span class="o">=</span> <span class="n">Hr</span>
	<span class="gp">>>> </span><span class="n">g</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">r</span><span class="p">:</span><span class="n">log</span><span class="p">(</span><span class="n">r</span><span class="o">/</span><span class="n">rc</span><span class="o">+</span><span class="mf">1.</span><span class="p">)</span>
	<span class="gp">>>> </span><span class="n">gm</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">r</span><span class="p">:</span><span class="n">rc</span><span class="o">*</span><span class="p">(</span><span class="n">exp</span><span class="p">(</span><span class="n">r</span><span class="p">)</span><span class="o">-</span><span class="mf">1.</span><span class="p">)</span>
	</pre></div>
	</div>
	</dd>
	</dl>
	<p>Set some options on how to compute velocity dispertions:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">mode_sigma_z</span> <span class="o">=</span> <span class="p">{</span><span class="s">"name"</span><span class="p">:</span><span class="s">"jeans"</span><span class="p">,</span><span class="s">"param"</span><span class="p">:</span><span class="bp">None</span><span class="p">}</span>
	<span class="gp">>>> </span><span class="n">mode_sigma_r</span> <span class="o">=</span> <span class="p">{</span><span class="s">"name"</span><span class="p">:</span><span class="s">"toomre"</span><span class="p">,</span><span class="s">"param"</span><span class="p">:</span><span class="mf">1.0</span><span class="p">}</span>
	<span class="gp">>>> </span><span class="n">mode_sigma_p</span> <span class="o">=</span> <span class="p">{</span><span class="s">"name"</span><span class="p">:</span><span class="s">"epicyclic_approximation"</span><span class="p">,</span><span class="s">"param"</span><span class="p">:</span><span class="bp">None</span><span class="p">}</span>
	<span class="gp">>>> </span><span class="n">params</span> <span class="o">=</span> <span class="p">[</span><span class="n">mode_sigma_z</span><span class="p">,</span><span class="n">mode_sigma_r</span><span class="p">,</span><span class="n">mode_sigma_p</span><span class="p">]</span>
	</pre></div>
	</div>
	<p>Set the gravitational softening:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">eps</span><span class="o">=</span><span class="mf">0.1</span>
	</pre></div>
	</div>
	<p>And finally, lunch the magic function:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">nb</span><span class="p">,</span><span class="n">phi</span><span class="p">,</span><span class="n">stats</span> <span class="o">=</span> <span class="n">nb</span><span class="o">.</span><span class="n">Get_Velocities_From_Cylindrical_Grid</span><span class="p">(</span><span class="n">select</span><span class="o">=</span><span class="s">'0'</span><span class="p">,</span><span class="n">disk</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span><span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">,</span><span class="n">nR</span><span class="o">=</span><span class="n">nr</span><span class="p">,</span><span class="n">nz</span><span class="o">=</span><span class="n">nz</span><span class="p">,</span><span class="n">nt</span><span class="o">=</span><span class="n">nt</span><span class="p">,</span><span class="n">Rmax</span><span class="o">=</span><span class="n">grmax</span><span class="p">,</span><span class="n">zmin</span><span class="o">=</span><span class="n">gzmin</span><span class="p">,</span><span class="n">zmax</span><span class="o">=</span><span class="n">gzmax</span><span class="p">,</span><span class="n">params</span><span class="o">=</span><span class="n">params</span><span class="p">,</span><span class="n">Phi</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span><span class="n">g</span><span class="o">=</span><span class="n">g</span><span class="p">,</span><span class="n">gm</span><span class="o">=</span><span class="n">gm</span><span class="p">)</span>
	</pre></div>
	</div>
	<p>The parameter <tt class="docutils literal"><span class="pre">select='0'</span></tt> tells that we want to compute the velocities on particles of type 0, while
	<tt class="docutils literal"><span class="pre">disk=0</span></tt> tells that what we considere as the disk is only the particles 0. This is usefull when dealing with
	multi-component models.</p>
	<p>The latter function return <tt class="docutils literal"><span class="pre">nb</span></tt> with the new velocities, a matrix <tt class="docutils literal"><span class="pre">phi</span></tt> containing the potential at each node of the grid,
	and a dictrionary <tt class="docutils literal"><span class="pre">stats</span></tt> containing some physical usefull quantities. Lets plot some of them:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">pylab</span> <span class="kn">as</span> <span class="nn">pt</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">stats</span><span class="p">[</span><span class="s">'R'</span><span class="p">],</span><span class="n">stats</span><span class="p">[</span><span class="s">'vc'</span><span class="p">])</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">stats</span><span class="p">[</span><span class="s">'R'</span><span class="p">],</span><span class="n">stats</span><span class="p">[</span><span class="s">'vm'</span><span class="p">])</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">stats</span><span class="p">[</span><span class="s">'R'</span><span class="p">],</span><span class="n">stats</span><span class="p">[</span><span class="s">'sr'</span><span class="p">])</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">stats</span><span class="p">[</span><span class="s">'R'</span><span class="p">],</span><span class="n">stats</span><span class="p">[</span><span class="s">'sz'</span><span class="p">])</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">stats</span><span class="p">[</span><span class="s">'R'</span><span class="p">],</span><span class="n">stats</span><span class="p">[</span><span class="s">'sp'</span><span class="p">])</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">'Radius'</span><span class="p">)</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">'velocity'</span><span class="p">)</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
	</pre></div>
	</div>
	<img alt="../_images/jean_cyl1.png" src="../_images/jean_cyl1.png" />
	<p>Lets try instead of fixing the Tomre parameter, to use for the radial velocity dispertion
	the epicyclic approximation.
	This is done by using the following parameters:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">mode_sigma_z</span> <span class="o">=</span> <span class="p">{</span><span class="s">"name"</span><span class="p">:</span><span class="s">"jeans"</span><span class="p">,</span><span class="s">"param"</span><span class="p">:</span><span class="bp">None</span><span class="p">}</span>
	<span class="gp">>>> </span><span class="n">mode_sigma_r</span> <span class="o">=</span> <span class="p">{</span><span class="s">"name"</span><span class="p">:</span><span class="s">"epicyclic_approximation"</span><span class="p">,</span><span class="s">"param"</span><span class="p">:</span><span class="mi">1</span><span class="p">}</span>
	<span class="gp">>>> </span><span class="n">mode_sigma_p</span> <span class="o">=</span> <span class="p">{</span><span class="s">"name"</span><span class="p">:</span><span class="s">"epicyclic_approximation"</span><span class="p">,</span><span class="s">"param"</span><span class="p">:</span><span class="bp">None</span><span class="p">}</span>
	<span class="gp">>>> </span><span class="n">params</span> <span class="o">=</span> <span class="p">[</span><span class="n">mode_sigma_z</span><span class="p">,</span><span class="n">mode_sigma_r</span><span class="p">,</span><span class="n">mode_sigma_p</span><span class="p">]</span>
	</pre></div>
	</div>
	<p>again, run the magic function:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">nb</span><span class="p">,</span><span class="n">phi</span><span class="p">,</span><span class="n">stats</span> <span class="o">=</span> <span class="n">nb</span><span class="o">.</span><span class="n">Get_Velocities_From_Cylindrical_Grid</span><span class="p">(</span><span class="n">select</span><span class="o">=</span><span class="s">'0'</span><span class="p">,</span><span class="n">disk</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">),</span><span class="n">eps</span><span class="o">=</span><span class="n">eps</span><span class="p">,</span><span class="n">nR</span><span class="o">=</span><span class="n">nr</span><span class="p">,</span><span class="n">nz</span><span class="o">=</span><span class="n">nz</span><span class="p">,</span><span class="n">nt</span><span class="o">=</span><span class="n">nt</span><span class="p">,</span><span class="n">Rmax</span><span class="o">=</span><span class="n">grmax</span><span class="p">,</span><span class="n">zmin</span><span class="o">=</span><span class="n">gzmin</span><span class="p">,</span><span class="n">zmax</span><span class="o">=</span><span class="n">gzmax</span><span class="p">,</span><span class="n">params</span><span class="o">=</span><span class="n">params</span><span class="p">,</span><span class="n">Phi</span><span class="o">=</span><span class="bp">None</span><span class="p">,</span><span class="n">g</span><span class="o">=</span><span class="n">g</span><span class="p">,</span><span class="n">gm</span><span class="o">=</span><span class="n">gm</span><span class="p">)</span>
	</pre></div>
	</div>
	<p>Now lets plot the Tommre parameter:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">stats</span><span class="p">[</span><span class="s">'R'</span><span class="p">],</span><span class="n">stats</span><span class="p">[</span><span class="s">'Q'</span><span class="p">])</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">'Radius'</span><span class="p">)</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">'Q'</span><span class="p">)</span>
	<span class="gp">>>> </span><span class="n">pt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
	</pre></div>
	</div>
	<img alt="../_images/jean_cyl2.png" src="../_images/jean_cyl2.png" />
	<div class="section" id="multi-component-systems">
	<h3>Multi component systems<a class="headerlink" href="#multi-component-systems" title="Permalink to this headline">¶</a></h3>
	<p>Examples using multicomponents systems are provided in the <tt class="docutils literal"><span class="pre">pnbody_examples/ic</span></tt> directory obtained
	with the command:</p>
	<div class="highlight-python"><div class="highlight"><pre><span class="n">pNbody_copy</span><span class="o">-</span><span class="n">examples</span>
	</pre></div>
	</div>
	</div>
	</div>
	</div>


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	<h3><a href="../index.html">Table Of Contents</a></h3>
	<ul>
	<li><a class="reference internal" href="#">Generating velocities</a><ul>
	<li><a class="reference internal" href="#spherical-coordinate">Spherical coordinate</a></li>
	<li><a class="reference internal" href="#example-set-a-plummer-sphere-to-the-jeans-equilibrium">Example : Set a Plummer sphere to the jeans equilibrium</a></li>
	<li><a class="reference internal" href="#cylindrical-coordinate">Cylindrical coordinate</a><ul>
	<li><a class="reference internal" href="#vertical-velocity-dispersion">Vertical velocity dispersion</a></li>
	<li><a class="reference internal" href="#radial-velocity-dispersion">Radial velocity dispersion</a></li>
	<li><a class="reference internal" href="#azimuthal-velocity-dispersion">Azimuthal velocity dispersion</a></li>
	<li><a class="reference internal" href="#mean-azimuthal-velocity">Mean azimuthal velocity</a></li>
	</ul>
	</li>
	<li><a class="reference internal" href="#example-set-an-exponnential-disk-to-the-jeans-equilibrium">Example : Set an exponnential disk to the jeans equilibrium</a><ul>
	<li><a class="reference internal" href="#multi-component-systems">Multi component systems</a></li>
	</ul>
	</li>
	</ul>
	</li>
	</ul>

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