jarjonam
/
tidal_tails


			
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<header>
<h1 class="title"><code>simulation</code> module</h1>
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<section id="section-intro">
<p>Definition of the Simulation class and the Galaxy constructor.</p>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">&#34;&#34;&#34;Definition of the Simulation class and the Galaxy constructor.&#34;&#34;&#34;

import os
import pickle
import numpy as np
import matplotlib.pyplot as plt

from utils import random_unit_vectors, cascade_round
from distributions import PLUMMER, HERNQUIST, UNIFORM, EXP, NFW
import acceleration


##############################################################################
##############################################################################
class Simulation:
    &#34;&#34;&#34;&#34;Main class for the gravitational simulation.

    Attributes:
        r_vec (array): position of the particles in the current timestep.
            Shape: (number of particles, 3)
        rprev_vec (array): position of the particles in the previous timestep.
            Shape: (number of particles, 3)
        v_vec (array): velocity in the current timestep.
            Shape: (number of particles, 3)
        a_vec (array): acceleration in the current timestep.
            Shape: (number of particles, 3)
        mass (array): mass of each particle in the simulation.
            Shape: (number of particles,)
        type (array): non-unique identifier for each particle.
            Shape: (number of particles,)
        tracks (array): list of positions through the simulation for central
            masses. Shape: (tracked particles, n+1, 3).
        CONFIG (array): configuration used to create the simulation.
            It will be saved along the state of the simulation.

        dt (float): timestep of the simulation
        n (int): current timestep. Initialized as n=0.
        soft (float): softening length used by the simulation.
        verbose (boolean): When True progress statements will be printed.
    &#34;&#34;&#34;

    def __init__(self, dt, soft, verbose, CONFIG, method, **kwargs):
        &#34;&#34;&#34;Constructor for the Simulation class.

        Arguments:
            dt (float): timestep of the simulation
            n (int): current timestep. Initialized as n=0.
            soft (float): softening length used by the simulation.
            verbose (bool): When True progress statements will be printed.
            CONFIG (dict): configuration file used to create the simulation.
            method (string): Optional. Algorithm to use when computing the 
                gravitational forces. One of &#39;bruteForce&#39;, &#39;bruteForce_numba&#39;,
                &#39;bruteForce_numbaopt&#39;, &#39;bruteForce_CPP&#39;, &#39;barnesHut_CPP&#39;.
        &#34;&#34;&#34;
        self.n = 0
        self.t = 0
        self.dt = dt
        self.soft = soft
        self.verbose = verbose
        self.CONFIG = CONFIG
        # Initialize empty arrays for all necessary properties
        self.r_vec = np.empty((0, 3))
        self.v_vec = np.empty((0, 3))
        self.a_vec = np.empty((0, 3))
        self.mass = np.empty((0,))
        self.type = np.empty((0, 2))
        algorithms = {
            &#39;bruteForce&#39;: acceleration.bruteForce,
            &#39;bruteForceNumba&#39;: acceleration.bruteForceNumba,
            &#39;bruteForceNumbaOptimized&#39;: acceleration.bruteForceNumbaOptimized,
            &#39;bruteForceCPP&#39;: acceleration.bruteForceCPP,
            &#39;barnesHutCPP&#39;: acceleration.barnesHutCPP
        }
        try:
            self.acceleration = algorithms[method]
        except: raise Exception(&#34;Method &#39;{}&#39; unknown&#34;.format(method))

    def add(self, body):
        &#34;&#34;&#34;Add a body to the simulation. It must expose the public attributes
           body.r_vec, body.v_vec, body.a_vec, body.type, body.mass.

        Arguments:
            body: Object to be added to the simulation (e.g. a Galaxy object)
        &#34;&#34;&#34;
        # Extend all relevant attributes by concatenating the body
        for name in [&#39;r_vec&#39;, &#39;v_vec&#39;, &#39;a_vec&#39;, &#39;type&#39;, &#39;mass&#39;]:
            simattr, bodyattr = getattr(self, name), getattr(body, name)
            setattr(self, name, np.concatenate([simattr, bodyattr], axis=0))
        # Order based on mass
        order = np.argsort(-self.mass)
        for name in [&#39;r_vec&#39;, &#39;v_vec&#39;, &#39;a_vec&#39;, &#39;type&#39;, &#39;mass&#39;]: 
            setattr(self, name, getattr(self, name)[order])

        # Update the list of objects to keep track of
        self.tracks = np.empty((np.sum(self.type[:,0]==&#39;center&#39;), 0, 3))

    def step(self):
        &#34;&#34;&#34;Perform a single step of the simulation.
           Makes use of a 4th order Verlet integrator.
        &#34;&#34;&#34;
        # Calculate the acceleration
        self.a_vec = self.acceleration(self.r_vec, self.mass, soft=self.soft)
        # Update the state using the Verlet algorithm
        # (A custom algorithm is written mainly for learning purposes)
        self.r_vec, self.rprev_vec = (2*self.r_vec - self.rprev_vec
            + self.a_vec * self.dt**2, self.r_vec)
        self.n += 1
        # Update tracks
        self.tracks = np.concatenate([self.tracks,
            self.r_vec[self.type[:,0]==&#39;center&#39;][:,np.newaxis]], axis=1)

    def run(self, tmax, saveEvery, outputFolder, **kwargs):
        &#34;&#34;&#34;Run the galactic simulation.

        Attributes:
            tmax (float): Time to which the simulation will run to.
                This is measured here since the start of the simulation,
                not since pericenter.
            saveEvery (int): The state is saved every saveEvery steps.
            outputFolder (string): It will be saved to /data/outputFolder/
        &#34;&#34;&#34;
        # When the simulation starts, intialize self.rprev_vec
        self.rprev_vec = self.r_vec - self.v_vec * self.dt
        if self.verbose: print(&#39;Simulation starting. Bon voyage!&#39;)
        while(self.t &lt; tmax):
            self.step()
            if(self.n % saveEvery == 0):
                self.save(&#39;data/{}&#39;.format(outputFolder))

        print(&#39;Simulation complete.&#39;)

    def save(self, outputFolder):
        &#34;&#34;&#34;Save the state of the simulation to the outputFolder.
           Two files are saved:
                sim{self.n}.pickle: serializing the state.
                sim{self.n}.png: a simplified 2D plot of x, y.
        &#34;&#34;&#34;
        # Create the output folder if it doesn&#39;t exist
        if not os.path.exists(outputFolder): os.makedirs(outputFolder)

        # Compute some useful quantities
        # v_vec is not required by the integrator, but useful
        self.v_vec = (self.r_vec - self.rprev_vec)/self.dt
        self.t = self.n * self.dt # prevents numerical rounding errors

        # Serialize state
        file = open(outputFolder+&#39;/data{}.pickle&#39;.format(self.n), &#34;wb&#34;)
        pickle.dump({&#39;r_vec&#39;: self.r_vec, &#39;v_vec&#39;: self.v_vec,
                     &#39;type&#39;: self.type, &#39;mass&#39;: self.mass,
                     &#39;CONFIG&#39;: self.CONFIG, &#39;t&#39;: self.t,
                     &#39;tracks&#39;: self.tracks}, file)

        # Save simplified plot of the current state.
        # Its main use is to detect ill-behaved situations early on.
        fig = plt.figure()
        plt.xlim(-5, 5); plt.ylim(-5, 5); plt.axis(&#39;equal&#39;)
        # Dark halo is plotted in red, disk in blue, bulge in green
        PLTCON = [(&#39;dark&#39;, &#39;r&#39;, 0.3), (&#39;disk&#39;, &#39;b&#39;, 1.0), (&#39;bulge&#39;, &#39;g&#39;, 0.5)]
        for type_, c, a in PLTCON: 
            plt.scatter(self.r_vec[self.type[:,0]==type_][:,0],
                self.r_vec[self.type[:,0]==type_][:,1], s=0.1, c=c, alpha=a)
        # Central mass as a magenta star 
        plt.scatter(self.r_vec[self.type[:,0]==&#39;center&#39;][:,0],
            self.r_vec[self.type[:,0]==&#39;center&#39;][:,1], s=100, marker=&#34;*&#34;, c=&#39;m&#39;)
        # Save to png file
        fig.savefig(outputFolder+&#39;/sim{}.png&#39;.format(self.n), dpi=150)
        plt.close(fig)

    def project(self, theta, phi, view=0):
        &#34;&#34;&#34;Projects the 3D simulation onto a plane as viewed from the
           direction described by the (theta, phi, view). Angles in radians.
           (This is used by the simulated annealing algorithm)
        
        Parameters:
            theta (float): polar angle.
            phi (float): azimuthal angle.
            view (float): rotation along line of sight.
        &#34;&#34;&#34;
        M1 = np.array([[np.cos(phi), np.sin(phi), 0],
                       [-np.sin(phi), np.cos(phi), 0],
                       [0, 0, 1]])
        M2 = np.array([[1, 0, 0],
                       [0, np.cos(theta), np.sin(theta)],
                       [0, -np.sin(theta), np.cos(theta)]])
        M3 = np.array([[np.cos(view), np.sin(view), 0],
                       [-np.sin(view), np.cos(view), 0],
                       [0, 0, 1]])

        M = np.matmul(M1, np.matmul(M2, M3)) # combine rotations
        r = np.tensordot(self.r_vec, M, axes=[1, 0])

        return r[:,0:2]

    def setOrbit(self, galaxy1, galaxy2, e, rmin, R0):
        &#34;&#34;&#34;Sets the two galaxies galaxy1, galaxy2 in an orbit.

        Parameters:
            galaxy1 (Galaxy): 1st galaxy (main)
            galaxy2 (Galaxy): 2nd galaxy (companion)
            e: eccentricity of the orbit
            rmin: distance of closest approach
            R0: initial separation
        &#34;&#34;&#34;
        m1, m2 = np.sum(galaxy1.mass), np.sum(galaxy2.mass)
        # Relevant formulae:
        # $r_0 = r (1 + e) \cos(\phi)$, where $r_0$ ($\neq R_0$) is the semi-latus rectum
        # $r_0 = r_\textup{min} (1 + e)$
        # $v^2 = G M (2/r - 1/a)$, where a is the semimajor axis

        # Solve the reduced two-body problem with reduced mass $\mu$ (mu)
        M = m1 + m2
        r0 = rmin * (1 + e)
        try:
            phi = np.arccos((r0/R0 - 1) / e) # inverting the orbit equation
            phi = -np.abs(phi) # Choose negative (incoming) angle
            ainv = (1 - e) / rmin # ainv = $1/a$, as a may be infinite
            v = np.sqrt(M * (2/R0 - ainv))
            # Finally, calculate the angle of motion. angle = tan(dy/dx)
            # $dy/dx = ((dr/d\phi) sin(\phi) + r \cos(\phi))/((dr/d\phi) cos(\phi) - r \sin(\phi))$
            dy = R0/r0 * e * np.sin(phi)**2 + np.cos(phi)
            dx = R0/r0 * e * np.sin(phi) * np.cos(phi) - np.sin(phi)
            vangle = np.arctan2(dy, dx)
        except: raise Exception(&#39;&#39;&#39;The orbital parameters cannot be satisfied.
            For elliptical orbits check that R0 is posible (&lt;rmax).&#39;&#39;&#39;)

        # We now need the actual motion of each body
        R_vec = np.array([[R0*np.cos(phi), R0*np.sin(phi), 0.]])
        V_vec = np.array([[v*np.cos(vangle), v*np.sin(vangle), 0.]])

        galaxy1.r_vec += m2/M * R_vec
        galaxy1.v_vec += m2/M * V_vec
        galaxy2.r_vec += -m1/M * R_vec
        galaxy2.v_vec += -m1/M * V_vec

        # Explicitely add the galaxies to the simulation
        self.add(galaxy1)
        self.add(galaxy2)

        if self.verbose: print(&#39;Galaxies set in orbit.&#39;)


##############################################################################
##############################################################################
class Galaxy():
    &#34;&#34;&#34;&#34;Helper class for creating galaxies.

    Attributes:
        r_vec (array): position of the particles in the current timestep.
            Shape: (number of particles, 3)
        v_vec (array): velocity in the current timestep.
            Shape: (number of particles, 3)
        a_vec (array): acceleration in the current timestep.
            Shape: (number of particles, 3)
        mass (array): mass of each particle in the simulation.
            Shape: (number of particles,)
        type (array): non-unique identifier for each particle.
            Shape: (number of particles,)    &#34;&#34;&#34;
    def __init__(self, orientation, centralMass, bulge, disk, halo, sim):
        &#34;&#34;&#34;Constructor for the Galaxy class.

           Parameters:
                orientation (tupple): (inclination i, argument of pericenter w)
                centralMass (float): mass at the center of the galaxy
                bulge (dict): passed to the addBulge method.
                disk (dict): passed to the addDisk method.
                halo (dict): passed to the addHalo method.
                sim (Simulation): simulation object
        &#34;&#34;&#34;
        if sim.verbose: print(&#39;Initializing galaxy&#39;)
        # Build the central mass
        self.r_vec = np.zeros((1, 3))
        self.v_vec = np.zeros((1, 3))
        self.a_vec = np.zeros((1, 3))
        self.mass = np.array([centralMass])
        self.type = np.array([[&#39;center&#39;, 0]])
        # Build the other components
        self.addBulge(**bulge)
        if sim.verbose: print(&#39;Bulge created.&#39;)
        self.addDisk(**disk)
        if sim.verbose: print(&#39;Disk created.&#39;)
        self.addHalo(**halo)
        if sim.verbose: print(&#39;Halo created.&#39;)
        # Correct particles velocities to generate circular orbits
        # $a_\textup{centripetal} = v^2/r$
        a_vec = sim.acceleration(self.r_vec, self.mass, soft=sim.soft)
        a = np.linalg.norm(a_vec, axis=-1, keepdims=True)
        r = np.linalg.norm(self.r_vec, axis=-1, keepdims=True)
        v = np.linalg.norm(self.v_vec[1:], axis=-1, keepdims=True)
        direction_unit = self.v_vec[1:]/v
        # Set orbital velocities (except for central mass)
        self.v_vec[1:] = np.sqrt(a[1:] * r[1:]) * direction_unit
        self.a_vec = np.zeros_like(self.r_vec)
        # Rotate the galaxy into its correct orientation
        self.rotate(*(np.array(orientation)/360 * 2*np.pi))
        if sim.verbose: print(&#39;Galaxy set in rotation and oriented.&#39;)

    def addBulge(self, model, totalMass, particles, l):
        &#34;&#34;&#34;Adds a bulge to the galaxy.

            Parameters:
                model (string): parametrization of the bulge.
                    &#39;plummer&#39; and &#39;hernquist&#39; are supported.
                totalMass (float): total mass of the bulge
                particles (int): number of particles in the bulge
                l (float): characteristic length scale of the model.
        &#34;&#34;&#34;
        if particles == 0: return None
        # Divide the mass equally among all particles
        mass = np.ones(particles) * totalMass/particles
        self.mass = np.concatenate([self.mass, mass], axis=0)
        # Create particles according to the radial distribution from model
        if model == &#39;plummer&#39;:
            r = PLUMMER.ppf(np.random.rand(particles), scale=l)
        elif model == &#39;hernquist&#39;:
            r = HERNQUIST.ppf(np.random.rand(particles), scale=l)
        else: raise Exception(&#34;&#34;&#34;Bulge distribution not allowed.
                    &#39;plummer&#39; and &#39;hernquist&#39; are the supported values&#34;&#34;&#34;)
        r_vec = r[:,np.newaxis] * random_unit_vectors(size=particles)
        self.r_vec = np.concatenate([self.r_vec, r_vec], axis=0)
        # Set them orbitting along random directions normal to r_vec
        v_vec = np.cross(r_vec, random_unit_vectors(size=particles))
        self.v_vec = np.concatenate([self.v_vec, v_vec], axis=0)
        # Label the particles
        type_ = [[&#39;bulge&#39;, 0]]*particles
        self.type = np.concatenate([self.type, type_], axis=0)

    def addDisk(self, model, totalMass, particles, l):
        &#34;&#34;&#34;Adds a disk to the galaxy.

            Parameters:
                model (string): parametrization of the disk.
                    &#39;rings&#39;, &#39;uniform&#39; and &#39;exp&#39; are supported.
                totalMass (float): total mass of the bulge
                particles (int): number of particles in the bulge
                l: fot &#39;exp&#39; and &#39;uniform&#39; characteristic length of the
                    model. For &#39;rings&#39; tupple of the form (inner radius,
                    outer radius, number of rings)
        &#34;&#34;&#34;
        if particles == 0: return None
        # Create particles according to the radial distribution from model
        if model == &#39;uniform&#39;:
            r = UNIFORM.ppf(np.random.rand(particles), scale=l)
            type_ = [[&#39;disk&#39;, 0]]*particles
            r_vec = r[:,np.newaxis] * random_unit_vectors(particles, &#39;2D&#39;)
            self.type = np.concatenate([self.type, type_], axis=0)
        elif model == &#39;rings&#39;:
            # l = [inner radius, outter radius, number of rings]
            distances = np.linspace(*l)
            # Aim for roughly constant areal density
            # Cascade rounding preserves the total number of particles
            perRing = cascade_round(particles * distances / np.sum(distances))
            particles = int(np.sum(perRing)) # prevents numerical errors
            r_vec = np.empty((0, 3))
            for d, n, i in zip(distances, perRing, range(l[2])):
                type_ = [[&#39;disk&#39;, i+1]]*int(n)
                self.type = np.concatenate([self.type, type_], axis=0)
                phi = np.linspace(0, 2 * np.pi, n, endpoint=False)
                ringr = d * np.array([[np.cos(i), np.sin(i), 0] for i in phi])
                r_vec = np.concatenate([r_vec, ringr], axis=0)
        elif model == &#39;exp&#39;:
            r = EXP.ppf(np.random.rand(particles), scale=l)
            r_vec = r[:,np.newaxis] * random_unit_vectors(particles, &#39;2D&#39;)
            type_ = [[&#39;disk&#39;, 0]]*particles
            self.type = np.concatenate([self.type, type_], axis=0)
        else:
            raise Exception(&#34;&#34;&#34;Disk distribution not allowed.
                    &#39;uniform&#39;, &#39;rings&#39; and &#39;exp&#39; are the supported values&#34;&#34;&#34;)
        self.r_vec = np.concatenate([self.r_vec, r_vec], axis=0)
        # Divide the mass equally among all particles
        mass = np.ones(particles) * totalMass/particles
        self.mass = np.concatenate([self.mass, mass], axis=0)
        # Set them orbitting along the spin plane
        v_vec = np.cross(r_vec, [0, 0, 1])
        self.v_vec = np.concatenate([self.v_vec, v_vec], axis=0)

    def addHalo(self, model, totalMass, particles, rs):
        &#34;&#34;&#34;Adds a halo to the galaxy.

            Parameters:
                model (string): parametrization of the halo.
                    Only &#39;NFW&#39; is supported.
                totalMass (float): total mass of the halo
                particles (int): number of particles in the halo
                rs (float): characteristic length scale of the NFW profile.
        &#34;&#34;&#34;
        if particles == 0: return None
        # Divide the mass equally among all particles
        mass = np.ones(particles)*totalMass/particles
        self.mass = np.concatenate([self.mass, mass], axis=0)
        # Create particles according to the radial distribution from model
        if model == &#39;NFW&#39;:
            r = NFW.ppf(np.random.rand(particles), scale=rs)
        else: raise Exception(&#34;&#34;&#34;Bulge distribution not allowed.
                    &#39;plummer&#39; and &#39;hernquist&#39; are the supported values&#34;&#34;&#34;)
        r_vec = r[:,np.newaxis] * random_unit_vectors(size=particles)
        self.r_vec = np.concatenate([self.r_vec, r_vec], axis=0)
        # Orbit along random directions normal to the radial vector
        v_vec = np.cross(r_vec, random_unit_vectors(size=particles))
        self.v_vec = np.concatenate([self.v_vec, v_vec], axis=0)
        # Label the particles
        type_ = [[&#39;dark&#39;], 0]*particles
        self.type = np.concatenate([self.type, type_], axis=0)

    def rotate(self, theta, phi):
        &#34;&#34;&#34;Rotates the galaxy so that its spin is along the (theta, phi)
           direction.

        Parameters:
            theta (float): polar angle.
            phi (float): azimuthal angle.
        &#34;&#34;&#34;
        M1 = np.array([[1, 0, 0],
                       [0, np.cos(theta), np.sin(theta)],
                       [0, -np.sin(theta), np.cos(theta)]])
        M2 = np.array([[np.cos(phi), np.sin(phi), 0],
                       [-np.sin(phi), np.cos(phi), 0],
                       [0, 0, 1]])
        M = np.matmul(M1, M2) # combine rotations
        self.r_vec = np.tensordot(self.r_vec, M, axes=[1, 0])
        self.v_vec = np.tensordot(self.v_vec, M, axes=[1, 0])</code></pre>
</details>
</section>
<section>
</section>
<section>
</section>
<section>
</section>
<section>
<h2 class="section-title" id="header-classes">Classes</h2>
<dl>
<dt id="simulation.Galaxy"><code class="flex name class">
<span>class <span class="ident">Galaxy</span></span>
</code></dt>
<dd>
<section class="desc"><p>"Helper class for creating galaxies.</p>
<h2 id="attributes">Attributes</h2>
<dl>
<dt><strong><code>r_vec</code></strong> :&ensp;<code>array</code></dt>
<dd>position of the particles in the current timestep.
Shape: (number of particles, 3)</dd>
<dt><strong><code>v_vec</code></strong> :&ensp;<code>array</code></dt>
<dd>velocity in the current timestep.
Shape: (number of particles, 3)</dd>
<dt><strong><code>a_vec</code></strong> :&ensp;<code>array</code></dt>
<dd>acceleration in the current timestep.
Shape: (number of particles, 3)</dd>
<dt><strong><code>mass</code></strong> :&ensp;<code>array</code></dt>
<dd>mass of each particle in the simulation.
Shape: (number of particles,)</dd>
<dt><strong><code>type</code></strong> :&ensp;<code>array</code></dt>
<dd>non-unique identifier for each particle.
Shape: (number of particles,)</dd>
</dl></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">class Galaxy():
    &#34;&#34;&#34;&#34;Helper class for creating galaxies.

    Attributes:
        r_vec (array): position of the particles in the current timestep.
            Shape: (number of particles, 3)
        v_vec (array): velocity in the current timestep.
            Shape: (number of particles, 3)
        a_vec (array): acceleration in the current timestep.
            Shape: (number of particles, 3)
        mass (array): mass of each particle in the simulation.
            Shape: (number of particles,)
        type (array): non-unique identifier for each particle.
            Shape: (number of particles,)    &#34;&#34;&#34;
    def __init__(self, orientation, centralMass, bulge, disk, halo, sim):
        &#34;&#34;&#34;Constructor for the Galaxy class.

           Parameters:
                orientation (tupple): (inclination i, argument of pericenter w)
                centralMass (float): mass at the center of the galaxy
                bulge (dict): passed to the addBulge method.
                disk (dict): passed to the addDisk method.
                halo (dict): passed to the addHalo method.
                sim (Simulation): simulation object
        &#34;&#34;&#34;
        if sim.verbose: print(&#39;Initializing galaxy&#39;)
        # Build the central mass
        self.r_vec = np.zeros((1, 3))
        self.v_vec = np.zeros((1, 3))
        self.a_vec = np.zeros((1, 3))
        self.mass = np.array([centralMass])
        self.type = np.array([[&#39;center&#39;, 0]])
        # Build the other components
        self.addBulge(**bulge)
        if sim.verbose: print(&#39;Bulge created.&#39;)
        self.addDisk(**disk)
        if sim.verbose: print(&#39;Disk created.&#39;)
        self.addHalo(**halo)
        if sim.verbose: print(&#39;Halo created.&#39;)
        # Correct particles velocities to generate circular orbits
        # $a_\textup{centripetal} = v^2/r$
        a_vec = sim.acceleration(self.r_vec, self.mass, soft=sim.soft)
        a = np.linalg.norm(a_vec, axis=-1, keepdims=True)
        r = np.linalg.norm(self.r_vec, axis=-1, keepdims=True)
        v = np.linalg.norm(self.v_vec[1:], axis=-1, keepdims=True)
        direction_unit = self.v_vec[1:]/v
        # Set orbital velocities (except for central mass)
        self.v_vec[1:] = np.sqrt(a[1:] * r[1:]) * direction_unit
        self.a_vec = np.zeros_like(self.r_vec)
        # Rotate the galaxy into its correct orientation
        self.rotate(*(np.array(orientation)/360 * 2*np.pi))
        if sim.verbose: print(&#39;Galaxy set in rotation and oriented.&#39;)

    def addBulge(self, model, totalMass, particles, l):
        &#34;&#34;&#34;Adds a bulge to the galaxy.

            Parameters:
                model (string): parametrization of the bulge.
                    &#39;plummer&#39; and &#39;hernquist&#39; are supported.
                totalMass (float): total mass of the bulge
                particles (int): number of particles in the bulge
                l (float): characteristic length scale of the model.
        &#34;&#34;&#34;
        if particles == 0: return None
        # Divide the mass equally among all particles
        mass = np.ones(particles) * totalMass/particles
        self.mass = np.concatenate([self.mass, mass], axis=0)
        # Create particles according to the radial distribution from model
        if model == &#39;plummer&#39;:
            r = PLUMMER.ppf(np.random.rand(particles), scale=l)
        elif model == &#39;hernquist&#39;:
            r = HERNQUIST.ppf(np.random.rand(particles), scale=l)
        else: raise Exception(&#34;&#34;&#34;Bulge distribution not allowed.
                    &#39;plummer&#39; and &#39;hernquist&#39; are the supported values&#34;&#34;&#34;)
        r_vec = r[:,np.newaxis] * random_unit_vectors(size=particles)
        self.r_vec = np.concatenate([self.r_vec, r_vec], axis=0)
        # Set them orbitting along random directions normal to r_vec
        v_vec = np.cross(r_vec, random_unit_vectors(size=particles))
        self.v_vec = np.concatenate([self.v_vec, v_vec], axis=0)
        # Label the particles
        type_ = [[&#39;bulge&#39;, 0]]*particles
        self.type = np.concatenate([self.type, type_], axis=0)

    def addDisk(self, model, totalMass, particles, l):
        &#34;&#34;&#34;Adds a disk to the galaxy.

            Parameters:
                model (string): parametrization of the disk.
                    &#39;rings&#39;, &#39;uniform&#39; and &#39;exp&#39; are supported.
                totalMass (float): total mass of the bulge
                particles (int): number of particles in the bulge
                l: fot &#39;exp&#39; and &#39;uniform&#39; characteristic length of the
                    model. For &#39;rings&#39; tupple of the form (inner radius,
                    outer radius, number of rings)
        &#34;&#34;&#34;
        if particles == 0: return None
        # Create particles according to the radial distribution from model
        if model == &#39;uniform&#39;:
            r = UNIFORM.ppf(np.random.rand(particles), scale=l)
            type_ = [[&#39;disk&#39;, 0]]*particles
            r_vec = r[:,np.newaxis] * random_unit_vectors(particles, &#39;2D&#39;)
            self.type = np.concatenate([self.type, type_], axis=0)
        elif model == &#39;rings&#39;:
            # l = [inner radius, outter radius, number of rings]
            distances = np.linspace(*l)
            # Aim for roughly constant areal density
            # Cascade rounding preserves the total number of particles
            perRing = cascade_round(particles * distances / np.sum(distances))
            particles = int(np.sum(perRing)) # prevents numerical errors
            r_vec = np.empty((0, 3))
            for d, n, i in zip(distances, perRing, range(l[2])):
                type_ = [[&#39;disk&#39;, i+1]]*int(n)
                self.type = np.concatenate([self.type, type_], axis=0)
                phi = np.linspace(0, 2 * np.pi, n, endpoint=False)
                ringr = d * np.array([[np.cos(i), np.sin(i), 0] for i in phi])
                r_vec = np.concatenate([r_vec, ringr], axis=0)
        elif model == &#39;exp&#39;:
            r = EXP.ppf(np.random.rand(particles), scale=l)
            r_vec = r[:,np.newaxis] * random_unit_vectors(particles, &#39;2D&#39;)
            type_ = [[&#39;disk&#39;, 0]]*particles
            self.type = np.concatenate([self.type, type_], axis=0)
        else:
            raise Exception(&#34;&#34;&#34;Disk distribution not allowed.
                    &#39;uniform&#39;, &#39;rings&#39; and &#39;exp&#39; are the supported values&#34;&#34;&#34;)
        self.r_vec = np.concatenate([self.r_vec, r_vec], axis=0)
        # Divide the mass equally among all particles
        mass = np.ones(particles) * totalMass/particles
        self.mass = np.concatenate([self.mass, mass], axis=0)
        # Set them orbitting along the spin plane
        v_vec = np.cross(r_vec, [0, 0, 1])
        self.v_vec = np.concatenate([self.v_vec, v_vec], axis=0)

    def addHalo(self, model, totalMass, particles, rs):
        &#34;&#34;&#34;Adds a halo to the galaxy.

            Parameters:
                model (string): parametrization of the halo.
                    Only &#39;NFW&#39; is supported.
                totalMass (float): total mass of the halo
                particles (int): number of particles in the halo
                rs (float): characteristic length scale of the NFW profile.
        &#34;&#34;&#34;
        if particles == 0: return None
        # Divide the mass equally among all particles
        mass = np.ones(particles)*totalMass/particles
        self.mass = np.concatenate([self.mass, mass], axis=0)
        # Create particles according to the radial distribution from model
        if model == &#39;NFW&#39;:
            r = NFW.ppf(np.random.rand(particles), scale=rs)
        else: raise Exception(&#34;&#34;&#34;Bulge distribution not allowed.
                    &#39;plummer&#39; and &#39;hernquist&#39; are the supported values&#34;&#34;&#34;)
        r_vec = r[:,np.newaxis] * random_unit_vectors(size=particles)
        self.r_vec = np.concatenate([self.r_vec, r_vec], axis=0)
        # Orbit along random directions normal to the radial vector
        v_vec = np.cross(r_vec, random_unit_vectors(size=particles))
        self.v_vec = np.concatenate([self.v_vec, v_vec], axis=0)
        # Label the particles
        type_ = [[&#39;dark&#39;], 0]*particles
        self.type = np.concatenate([self.type, type_], axis=0)

    def rotate(self, theta, phi):
        &#34;&#34;&#34;Rotates the galaxy so that its spin is along the (theta, phi)
           direction.

        Parameters:
            theta (float): polar angle.
            phi (float): azimuthal angle.
        &#34;&#34;&#34;
        M1 = np.array([[1, 0, 0],
                       [0, np.cos(theta), np.sin(theta)],
                       [0, -np.sin(theta), np.cos(theta)]])
        M2 = np.array([[np.cos(phi), np.sin(phi), 0],
                       [-np.sin(phi), np.cos(phi), 0],
                       [0, 0, 1]])
        M = np.matmul(M1, M2) # combine rotations
        self.r_vec = np.tensordot(self.r_vec, M, axes=[1, 0])
        self.v_vec = np.tensordot(self.v_vec, M, axes=[1, 0])</code></pre>
</details>
<h3>Methods</h3>
<dl>
<dt id="simulation.Galaxy.__init__"><code class="name flex">
<span>def <span class="ident">__init__</span></span>(<span>self, orientation, centralMass, bulge, disk, halo, sim)</span>
</code></dt>
<dd>
<section class="desc"><p>Constructor for the Galaxy class.</p>
<h2 id="parameters">Parameters</h2>
<dl>
<dt><strong><code>orientation</code></strong> :&ensp;<code>tupple</code></dt>
<dd>(inclination i, argument of pericenter w)</dd>
<dt><strong><code>centralMass</code></strong> :&ensp;<code>float</code></dt>
<dd>mass at the center of the galaxy</dd>
<dt><strong><code>bulge</code></strong> :&ensp;<code>dict</code></dt>
<dd>passed to the addBulge method.</dd>
<dt><strong><code>disk</code></strong> :&ensp;<code>dict</code></dt>
<dd>passed to the addDisk method.</dd>
<dt><strong><code>halo</code></strong> :&ensp;<code>dict</code></dt>
<dd>passed to the addHalo method.</dd>
<dt><strong><code>sim</code></strong> :&ensp;<a title="simulation.Simulation" href="#simulation.Simulation"><code>Simulation</code></a></dt>
<dd>simulation object</dd>
</dl></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">def __init__(self, orientation, centralMass, bulge, disk, halo, sim):
    &#34;&#34;&#34;Constructor for the Galaxy class.

       Parameters:
            orientation (tupple): (inclination i, argument of pericenter w)
            centralMass (float): mass at the center of the galaxy
            bulge (dict): passed to the addBulge method.
            disk (dict): passed to the addDisk method.
            halo (dict): passed to the addHalo method.
            sim (Simulation): simulation object
    &#34;&#34;&#34;
    if sim.verbose: print(&#39;Initializing galaxy&#39;)
    # Build the central mass
    self.r_vec = np.zeros((1, 3))
    self.v_vec = np.zeros((1, 3))
    self.a_vec = np.zeros((1, 3))
    self.mass = np.array([centralMass])
    self.type = np.array([[&#39;center&#39;, 0]])
    # Build the other components
    self.addBulge(**bulge)
    if sim.verbose: print(&#39;Bulge created.&#39;)
    self.addDisk(**disk)
    if sim.verbose: print(&#39;Disk created.&#39;)
    self.addHalo(**halo)
    if sim.verbose: print(&#39;Halo created.&#39;)
    # Correct particles velocities to generate circular orbits
    # $a_\textup{centripetal} = v^2/r$
    a_vec = sim.acceleration(self.r_vec, self.mass, soft=sim.soft)
    a = np.linalg.norm(a_vec, axis=-1, keepdims=True)
    r = np.linalg.norm(self.r_vec, axis=-1, keepdims=True)
    v = np.linalg.norm(self.v_vec[1:], axis=-1, keepdims=True)
    direction_unit = self.v_vec[1:]/v
    # Set orbital velocities (except for central mass)
    self.v_vec[1:] = np.sqrt(a[1:] * r[1:]) * direction_unit
    self.a_vec = np.zeros_like(self.r_vec)
    # Rotate the galaxy into its correct orientation
    self.rotate(*(np.array(orientation)/360 * 2*np.pi))
    if sim.verbose: print(&#39;Galaxy set in rotation and oriented.&#39;)</code></pre>
</details>
</dd>
<dt id="simulation.Galaxy.addBulge"><code class="name flex">
<span>def <span class="ident">addBulge</span></span>(<span>self, model, totalMass, particles, l)</span>
</code></dt>
<dd>
<section class="desc"><p>Adds a bulge to the galaxy.</p>
<h2 id="parameters">Parameters</h2>
<dl>
<dt><strong><code>model</code></strong> :&ensp;<code>string</code></dt>
<dd>parametrization of the bulge.
'plummer' and 'hernquist' are supported.</dd>
<dt><strong><code>totalMass</code></strong> :&ensp;<code>float</code></dt>
<dd>total mass of the bulge</dd>
<dt><strong><code>particles</code></strong> :&ensp;<code>int</code></dt>
<dd>number of particles in the bulge</dd>
<dt><strong><code>l</code></strong> :&ensp;<code>float</code></dt>
<dd>characteristic length scale of the model.</dd>
</dl></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">def addBulge(self, model, totalMass, particles, l):
    &#34;&#34;&#34;Adds a bulge to the galaxy.

        Parameters:
            model (string): parametrization of the bulge.
                &#39;plummer&#39; and &#39;hernquist&#39; are supported.
            totalMass (float): total mass of the bulge
            particles (int): number of particles in the bulge
            l (float): characteristic length scale of the model.
    &#34;&#34;&#34;
    if particles == 0: return None
    # Divide the mass equally among all particles
    mass = np.ones(particles) * totalMass/particles
    self.mass = np.concatenate([self.mass, mass], axis=0)
    # Create particles according to the radial distribution from model
    if model == &#39;plummer&#39;:
        r = PLUMMER.ppf(np.random.rand(particles), scale=l)
    elif model == &#39;hernquist&#39;:
        r = HERNQUIST.ppf(np.random.rand(particles), scale=l)
    else: raise Exception(&#34;&#34;&#34;Bulge distribution not allowed.
                &#39;plummer&#39; and &#39;hernquist&#39; are the supported values&#34;&#34;&#34;)
    r_vec = r[:,np.newaxis] * random_unit_vectors(size=particles)
    self.r_vec = np.concatenate([self.r_vec, r_vec], axis=0)
    # Set them orbitting along random directions normal to r_vec
    v_vec = np.cross(r_vec, random_unit_vectors(size=particles))
    self.v_vec = np.concatenate([self.v_vec, v_vec], axis=0)
    # Label the particles
    type_ = [[&#39;bulge&#39;, 0]]*particles
    self.type = np.concatenate([self.type, type_], axis=0)</code></pre>
</details>
</dd>
<dt id="simulation.Galaxy.addDisk"><code class="name flex">
<span>def <span class="ident">addDisk</span></span>(<span>self, model, totalMass, particles, l)</span>
</code></dt>
<dd>
<section class="desc"><p>Adds a disk to the galaxy.</p>
<h2 id="parameters">Parameters</h2>
<dl>
<dt><strong><code>model</code></strong> :&ensp;<code>string</code></dt>
<dd>parametrization of the disk.
'rings', 'uniform' and 'exp' are supported.</dd>
<dt><strong><code>totalMass</code></strong> :&ensp;<code>float</code></dt>
<dd>total mass of the bulge</dd>
<dt><strong><code>particles</code></strong> :&ensp;<code>int</code></dt>
<dd>number of particles in the bulge</dd>
<dt><strong><code>l</code></strong></dt>
<dd>fot 'exp' and 'uniform' characteristic length of the
model. For 'rings' tupple of the form (inner radius,
outer radius, number of rings)</dd>
</dl></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">def addDisk(self, model, totalMass, particles, l):
    &#34;&#34;&#34;Adds a disk to the galaxy.

        Parameters:
            model (string): parametrization of the disk.
                &#39;rings&#39;, &#39;uniform&#39; and &#39;exp&#39; are supported.
            totalMass (float): total mass of the bulge
            particles (int): number of particles in the bulge
            l: fot &#39;exp&#39; and &#39;uniform&#39; characteristic length of the
                model. For &#39;rings&#39; tupple of the form (inner radius,
                outer radius, number of rings)
    &#34;&#34;&#34;
    if particles == 0: return None
    # Create particles according to the radial distribution from model
    if model == &#39;uniform&#39;:
        r = UNIFORM.ppf(np.random.rand(particles), scale=l)
        type_ = [[&#39;disk&#39;, 0]]*particles
        r_vec = r[:,np.newaxis] * random_unit_vectors(particles, &#39;2D&#39;)
        self.type = np.concatenate([self.type, type_], axis=0)
    elif model == &#39;rings&#39;:
        # l = [inner radius, outter radius, number of rings]
        distances = np.linspace(*l)
        # Aim for roughly constant areal density
        # Cascade rounding preserves the total number of particles
        perRing = cascade_round(particles * distances / np.sum(distances))
        particles = int(np.sum(perRing)) # prevents numerical errors
        r_vec = np.empty((0, 3))
        for d, n, i in zip(distances, perRing, range(l[2])):
            type_ = [[&#39;disk&#39;, i+1]]*int(n)
            self.type = np.concatenate([self.type, type_], axis=0)
            phi = np.linspace(0, 2 * np.pi, n, endpoint=False)
            ringr = d * np.array([[np.cos(i), np.sin(i), 0] for i in phi])
            r_vec = np.concatenate([r_vec, ringr], axis=0)
    elif model == &#39;exp&#39;:
        r = EXP.ppf(np.random.rand(particles), scale=l)
        r_vec = r[:,np.newaxis] * random_unit_vectors(particles, &#39;2D&#39;)
        type_ = [[&#39;disk&#39;, 0]]*particles
        self.type = np.concatenate([self.type, type_], axis=0)
    else:
        raise Exception(&#34;&#34;&#34;Disk distribution not allowed.
                &#39;uniform&#39;, &#39;rings&#39; and &#39;exp&#39; are the supported values&#34;&#34;&#34;)
    self.r_vec = np.concatenate([self.r_vec, r_vec], axis=0)
    # Divide the mass equally among all particles
    mass = np.ones(particles) * totalMass/particles
    self.mass = np.concatenate([self.mass, mass], axis=0)
    # Set them orbitting along the spin plane
    v_vec = np.cross(r_vec, [0, 0, 1])
    self.v_vec = np.concatenate([self.v_vec, v_vec], axis=0)</code></pre>
</details>
</dd>
<dt id="simulation.Galaxy.addHalo"><code class="name flex">
<span>def <span class="ident">addHalo</span></span>(<span>self, model, totalMass, particles, rs)</span>
</code></dt>
<dd>
<section class="desc"><p>Adds a halo to the galaxy.</p>
<h2 id="parameters">Parameters</h2>
<dl>
<dt><strong><code>model</code></strong> :&ensp;<code>string</code></dt>
<dd>parametrization of the halo.
Only 'NFW' is supported.</dd>
<dt><strong><code>totalMass</code></strong> :&ensp;<code>float</code></dt>
<dd>total mass of the halo</dd>
<dt><strong><code>particles</code></strong> :&ensp;<code>int</code></dt>
<dd>number of particles in the halo</dd>
<dt><strong><code>rs</code></strong> :&ensp;<code>float</code></dt>
<dd>characteristic length scale of the NFW profile.</dd>
</dl></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">def addHalo(self, model, totalMass, particles, rs):
    &#34;&#34;&#34;Adds a halo to the galaxy.

        Parameters:
            model (string): parametrization of the halo.
                Only &#39;NFW&#39; is supported.
            totalMass (float): total mass of the halo
            particles (int): number of particles in the halo
            rs (float): characteristic length scale of the NFW profile.
    &#34;&#34;&#34;
    if particles == 0: return None
    # Divide the mass equally among all particles
    mass = np.ones(particles)*totalMass/particles
    self.mass = np.concatenate([self.mass, mass], axis=0)
    # Create particles according to the radial distribution from model
    if model == &#39;NFW&#39;:
        r = NFW.ppf(np.random.rand(particles), scale=rs)
    else: raise Exception(&#34;&#34;&#34;Bulge distribution not allowed.
                &#39;plummer&#39; and &#39;hernquist&#39; are the supported values&#34;&#34;&#34;)
    r_vec = r[:,np.newaxis] * random_unit_vectors(size=particles)
    self.r_vec = np.concatenate([self.r_vec, r_vec], axis=0)
    # Orbit along random directions normal to the radial vector
    v_vec = np.cross(r_vec, random_unit_vectors(size=particles))
    self.v_vec = np.concatenate([self.v_vec, v_vec], axis=0)
    # Label the particles
    type_ = [[&#39;dark&#39;], 0]*particles
    self.type = np.concatenate([self.type, type_], axis=0)</code></pre>
</details>
</dd>
<dt id="simulation.Galaxy.rotate"><code class="name flex">
<span>def <span class="ident">rotate</span></span>(<span>self, theta, phi)</span>
</code></dt>
<dd>
<section class="desc"><p>Rotates the galaxy so that its spin is along the (theta, phi)
direction.</p>
<h2 id="parameters">Parameters</h2>
<dl>
<dt><strong><code>theta</code></strong> :&ensp;<code>float</code></dt>
<dd>polar angle.</dd>
<dt><strong><code>phi</code></strong> :&ensp;<code>float</code></dt>
<dd>azimuthal angle.</dd>
</dl></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">def rotate(self, theta, phi):
    &#34;&#34;&#34;Rotates the galaxy so that its spin is along the (theta, phi)
       direction.

    Parameters:
        theta (float): polar angle.
        phi (float): azimuthal angle.
    &#34;&#34;&#34;
    M1 = np.array([[1, 0, 0],
                   [0, np.cos(theta), np.sin(theta)],
                   [0, -np.sin(theta), np.cos(theta)]])
    M2 = np.array([[np.cos(phi), np.sin(phi), 0],
                   [-np.sin(phi), np.cos(phi), 0],
                   [0, 0, 1]])
    M = np.matmul(M1, M2) # combine rotations
    self.r_vec = np.tensordot(self.r_vec, M, axes=[1, 0])
    self.v_vec = np.tensordot(self.v_vec, M, axes=[1, 0])</code></pre>
</details>
</dd>
</dl>
</dd>
<dt id="simulation.Simulation"><code class="flex name class">
<span>class <span class="ident">Simulation</span></span>
</code></dt>
<dd>
<section class="desc"><p>"Main class for the gravitational simulation.</p>
<h2 id="attributes">Attributes</h2>
<dl>
<dt><strong><code>r_vec</code></strong> :&ensp;<code>array</code></dt>
<dd>position of the particles in the current timestep.
Shape: (number of particles, 3)</dd>
<dt><strong><code>rprev_vec</code></strong> :&ensp;<code>array</code></dt>
<dd>position of the particles in the previous timestep.
Shape: (number of particles, 3)</dd>
<dt><strong><code>v_vec</code></strong> :&ensp;<code>array</code></dt>
<dd>velocity in the current timestep.
Shape: (number of particles, 3)</dd>
<dt><strong><code>a_vec</code></strong> :&ensp;<code>array</code></dt>
<dd>acceleration in the current timestep.
Shape: (number of particles, 3)</dd>
<dt><strong><code>mass</code></strong> :&ensp;<code>array</code></dt>
<dd>mass of each particle in the simulation.
Shape: (number of particles,)</dd>
<dt><strong><code>type</code></strong> :&ensp;<code>array</code></dt>
<dd>non-unique identifier for each particle.
Shape: (number of particles,)</dd>
<dt><strong><code>tracks</code></strong> :&ensp;<code>array</code></dt>
<dd>list of positions through the simulation for central
masses. Shape: (tracked particles, n+1, 3).</dd>
<dt><strong><code>CONFIG</code></strong> :&ensp;<code>array</code></dt>
<dd>configuration used to create the simulation.
It will be saved along the state of the simulation.</dd>
<dt><strong><code>dt</code></strong> :&ensp;<code>float</code></dt>
<dd>timestep of the simulation</dd>
<dt><strong><code>n</code></strong> :&ensp;<code>int</code></dt>
<dd>current timestep. Initialized as n=0.</dd>
<dt><strong><code>soft</code></strong> :&ensp;<code>float</code></dt>
<dd>softening length used by the simulation.</dd>
<dt><strong><code>verbose</code></strong> :&ensp;<code>boolean</code></dt>
<dd>When True progress statements will be printed.</dd>
</dl></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">class Simulation:
    &#34;&#34;&#34;&#34;Main class for the gravitational simulation.

    Attributes:
        r_vec (array): position of the particles in the current timestep.
            Shape: (number of particles, 3)
        rprev_vec (array): position of the particles in the previous timestep.
            Shape: (number of particles, 3)
        v_vec (array): velocity in the current timestep.
            Shape: (number of particles, 3)
        a_vec (array): acceleration in the current timestep.
            Shape: (number of particles, 3)
        mass (array): mass of each particle in the simulation.
            Shape: (number of particles,)
        type (array): non-unique identifier for each particle.
            Shape: (number of particles,)
        tracks (array): list of positions through the simulation for central
            masses. Shape: (tracked particles, n+1, 3).
        CONFIG (array): configuration used to create the simulation.
            It will be saved along the state of the simulation.

        dt (float): timestep of the simulation
        n (int): current timestep. Initialized as n=0.
        soft (float): softening length used by the simulation.
        verbose (boolean): When True progress statements will be printed.
    &#34;&#34;&#34;

    def __init__(self, dt, soft, verbose, CONFIG, method, **kwargs):
        &#34;&#34;&#34;Constructor for the Simulation class.

        Arguments:
            dt (float): timestep of the simulation
            n (int): current timestep. Initialized as n=0.
            soft (float): softening length used by the simulation.
            verbose (bool): When True progress statements will be printed.
            CONFIG (dict): configuration file used to create the simulation.
            method (string): Optional. Algorithm to use when computing the 
                gravitational forces. One of &#39;bruteForce&#39;, &#39;bruteForce_numba&#39;,
                &#39;bruteForce_numbaopt&#39;, &#39;bruteForce_CPP&#39;, &#39;barnesHut_CPP&#39;.
        &#34;&#34;&#34;
        self.n = 0
        self.t = 0
        self.dt = dt
        self.soft = soft
        self.verbose = verbose
        self.CONFIG = CONFIG
        # Initialize empty arrays for all necessary properties
        self.r_vec = np.empty((0, 3))
        self.v_vec = np.empty((0, 3))
        self.a_vec = np.empty((0, 3))
        self.mass = np.empty((0,))
        self.type = np.empty((0, 2))
        algorithms = {
            &#39;bruteForce&#39;: acceleration.bruteForce,
            &#39;bruteForceNumba&#39;: acceleration.bruteForceNumba,
            &#39;bruteForceNumbaOptimized&#39;: acceleration.bruteForceNumbaOptimized,
            &#39;bruteForceCPP&#39;: acceleration.bruteForceCPP,
            &#39;barnesHutCPP&#39;: acceleration.barnesHutCPP
        }
        try:
            self.acceleration = algorithms[method]
        except: raise Exception(&#34;Method &#39;{}&#39; unknown&#34;.format(method))

    def add(self, body):
        &#34;&#34;&#34;Add a body to the simulation. It must expose the public attributes
           body.r_vec, body.v_vec, body.a_vec, body.type, body.mass.

        Arguments:
            body: Object to be added to the simulation (e.g. a Galaxy object)
        &#34;&#34;&#34;
        # Extend all relevant attributes by concatenating the body
        for name in [&#39;r_vec&#39;, &#39;v_vec&#39;, &#39;a_vec&#39;, &#39;type&#39;, &#39;mass&#39;]:
            simattr, bodyattr = getattr(self, name), getattr(body, name)
            setattr(self, name, np.concatenate([simattr, bodyattr], axis=0))
        # Order based on mass
        order = np.argsort(-self.mass)
        for name in [&#39;r_vec&#39;, &#39;v_vec&#39;, &#39;a_vec&#39;, &#39;type&#39;, &#39;mass&#39;]: 
            setattr(self, name, getattr(self, name)[order])

        # Update the list of objects to keep track of
        self.tracks = np.empty((np.sum(self.type[:,0]==&#39;center&#39;), 0, 3))

    def step(self):
        &#34;&#34;&#34;Perform a single step of the simulation.
           Makes use of a 4th order Verlet integrator.
        &#34;&#34;&#34;
        # Calculate the acceleration
        self.a_vec = self.acceleration(self.r_vec, self.mass, soft=self.soft)
        # Update the state using the Verlet algorithm
        # (A custom algorithm is written mainly for learning purposes)
        self.r_vec, self.rprev_vec = (2*self.r_vec - self.rprev_vec
            + self.a_vec * self.dt**2, self.r_vec)
        self.n += 1
        # Update tracks
        self.tracks = np.concatenate([self.tracks,
            self.r_vec[self.type[:,0]==&#39;center&#39;][:,np.newaxis]], axis=1)

    def run(self, tmax, saveEvery, outputFolder, **kwargs):
        &#34;&#34;&#34;Run the galactic simulation.

        Attributes:
            tmax (float): Time to which the simulation will run to.
                This is measured here since the start of the simulation,
                not since pericenter.
            saveEvery (int): The state is saved every saveEvery steps.
            outputFolder (string): It will be saved to /data/outputFolder/
        &#34;&#34;&#34;
        # When the simulation starts, intialize self.rprev_vec
        self.rprev_vec = self.r_vec - self.v_vec * self.dt
        if self.verbose: print(&#39;Simulation starting. Bon voyage!&#39;)
        while(self.t &lt; tmax):
            self.step()
            if(self.n % saveEvery == 0):
                self.save(&#39;data/{}&#39;.format(outputFolder))

        print(&#39;Simulation complete.&#39;)

    def save(self, outputFolder):
        &#34;&#34;&#34;Save the state of the simulation to the outputFolder.
           Two files are saved:
                sim{self.n}.pickle: serializing the state.
                sim{self.n}.png: a simplified 2D plot of x, y.
        &#34;&#34;&#34;
        # Create the output folder if it doesn&#39;t exist
        if not os.path.exists(outputFolder): os.makedirs(outputFolder)

        # Compute some useful quantities
        # v_vec is not required by the integrator, but useful
        self.v_vec = (self.r_vec - self.rprev_vec)/self.dt
        self.t = self.n * self.dt # prevents numerical rounding errors

        # Serialize state
        file = open(outputFolder+&#39;/data{}.pickle&#39;.format(self.n), &#34;wb&#34;)
        pickle.dump({&#39;r_vec&#39;: self.r_vec, &#39;v_vec&#39;: self.v_vec,
                     &#39;type&#39;: self.type, &#39;mass&#39;: self.mass,
                     &#39;CONFIG&#39;: self.CONFIG, &#39;t&#39;: self.t,
                     &#39;tracks&#39;: self.tracks}, file)

        # Save simplified plot of the current state.
        # Its main use is to detect ill-behaved situations early on.
        fig = plt.figure()
        plt.xlim(-5, 5); plt.ylim(-5, 5); plt.axis(&#39;equal&#39;)
        # Dark halo is plotted in red, disk in blue, bulge in green
        PLTCON = [(&#39;dark&#39;, &#39;r&#39;, 0.3), (&#39;disk&#39;, &#39;b&#39;, 1.0), (&#39;bulge&#39;, &#39;g&#39;, 0.5)]
        for type_, c, a in PLTCON: 
            plt.scatter(self.r_vec[self.type[:,0]==type_][:,0],
                self.r_vec[self.type[:,0]==type_][:,1], s=0.1, c=c, alpha=a)
        # Central mass as a magenta star 
        plt.scatter(self.r_vec[self.type[:,0]==&#39;center&#39;][:,0],
            self.r_vec[self.type[:,0]==&#39;center&#39;][:,1], s=100, marker=&#34;*&#34;, c=&#39;m&#39;)
        # Save to png file
        fig.savefig(outputFolder+&#39;/sim{}.png&#39;.format(self.n), dpi=150)
        plt.close(fig)

    def project(self, theta, phi, view=0):
        &#34;&#34;&#34;Projects the 3D simulation onto a plane as viewed from the
           direction described by the (theta, phi, view). Angles in radians.
           (This is used by the simulated annealing algorithm)
        
        Parameters:
            theta (float): polar angle.
            phi (float): azimuthal angle.
            view (float): rotation along line of sight.
        &#34;&#34;&#34;
        M1 = np.array([[np.cos(phi), np.sin(phi), 0],
                       [-np.sin(phi), np.cos(phi), 0],
                       [0, 0, 1]])
        M2 = np.array([[1, 0, 0],
                       [0, np.cos(theta), np.sin(theta)],
                       [0, -np.sin(theta), np.cos(theta)]])
        M3 = np.array([[np.cos(view), np.sin(view), 0],
                       [-np.sin(view), np.cos(view), 0],
                       [0, 0, 1]])

        M = np.matmul(M1, np.matmul(M2, M3)) # combine rotations
        r = np.tensordot(self.r_vec, M, axes=[1, 0])

        return r[:,0:2]

    def setOrbit(self, galaxy1, galaxy2, e, rmin, R0):
        &#34;&#34;&#34;Sets the two galaxies galaxy1, galaxy2 in an orbit.

        Parameters:
            galaxy1 (Galaxy): 1st galaxy (main)
            galaxy2 (Galaxy): 2nd galaxy (companion)
            e: eccentricity of the orbit
            rmin: distance of closest approach
            R0: initial separation
        &#34;&#34;&#34;
        m1, m2 = np.sum(galaxy1.mass), np.sum(galaxy2.mass)
        # Relevant formulae:
        # $r_0 = r (1 + e) \cos(\phi)$, where $r_0$ ($\neq R_0$) is the semi-latus rectum
        # $r_0 = r_\textup{min} (1 + e)$
        # $v^2 = G M (2/r - 1/a)$, where a is the semimajor axis

        # Solve the reduced two-body problem with reduced mass $\mu$ (mu)
        M = m1 + m2
        r0 = rmin * (1 + e)
        try:
            phi = np.arccos((r0/R0 - 1) / e) # inverting the orbit equation
            phi = -np.abs(phi) # Choose negative (incoming) angle
            ainv = (1 - e) / rmin # ainv = $1/a$, as a may be infinite
            v = np.sqrt(M * (2/R0 - ainv))
            # Finally, calculate the angle of motion. angle = tan(dy/dx)
            # $dy/dx = ((dr/d\phi) sin(\phi) + r \cos(\phi))/((dr/d\phi) cos(\phi) - r \sin(\phi))$
            dy = R0/r0 * e * np.sin(phi)**2 + np.cos(phi)
            dx = R0/r0 * e * np.sin(phi) * np.cos(phi) - np.sin(phi)
            vangle = np.arctan2(dy, dx)
        except: raise Exception(&#39;&#39;&#39;The orbital parameters cannot be satisfied.
            For elliptical orbits check that R0 is posible (&lt;rmax).&#39;&#39;&#39;)

        # We now need the actual motion of each body
        R_vec = np.array([[R0*np.cos(phi), R0*np.sin(phi), 0.]])
        V_vec = np.array([[v*np.cos(vangle), v*np.sin(vangle), 0.]])

        galaxy1.r_vec += m2/M * R_vec
        galaxy1.v_vec += m2/M * V_vec
        galaxy2.r_vec += -m1/M * R_vec
        galaxy2.v_vec += -m1/M * V_vec

        # Explicitely add the galaxies to the simulation
        self.add(galaxy1)
        self.add(galaxy2)

        if self.verbose: print(&#39;Galaxies set in orbit.&#39;)</code></pre>
</details>
<h3>Methods</h3>
<dl>
<dt id="simulation.Simulation.__init__"><code class="name flex">
<span>def <span class="ident">__init__</span></span>(<span>self, dt, soft, verbose, CONFIG, method, **kwargs)</span>
</code></dt>
<dd>
<section class="desc"><p>Constructor for the Simulation class.</p>
<h2 id="arguments">Arguments</h2>
<dl>
<dt><strong><code>dt</code></strong> :&ensp;<code>float</code></dt>
<dd>timestep of the simulation</dd>
<dt><strong><code>n</code></strong> :&ensp;<code>int</code></dt>
<dd>current timestep. Initialized as n=0.</dd>
<dt><strong><code>soft</code></strong> :&ensp;<code>float</code></dt>
<dd>softening length used by the simulation.</dd>
<dt><strong><code>verbose</code></strong> :&ensp;<code>bool</code></dt>
<dd>When True progress statements will be printed.</dd>
<dt><strong><code>CONFIG</code></strong> :&ensp;<code>dict</code></dt>
<dd>configuration file used to create the simulation.</dd>
<dt><strong><code>method</code></strong> :&ensp;<code>string</code></dt>
<dd>Optional. Algorithm to use when computing the
gravitational forces. One of 'bruteForce', 'bruteForce_numba',
'bruteForce_numbaopt', 'bruteForce_CPP', 'barnesHut_CPP'.</dd>
</dl></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">def __init__(self, dt, soft, verbose, CONFIG, method, **kwargs):
    &#34;&#34;&#34;Constructor for the Simulation class.

    Arguments:
        dt (float): timestep of the simulation
        n (int): current timestep. Initialized as n=0.
        soft (float): softening length used by the simulation.
        verbose (bool): When True progress statements will be printed.
        CONFIG (dict): configuration file used to create the simulation.
        method (string): Optional. Algorithm to use when computing the 
            gravitational forces. One of &#39;bruteForce&#39;, &#39;bruteForce_numba&#39;,
            &#39;bruteForce_numbaopt&#39;, &#39;bruteForce_CPP&#39;, &#39;barnesHut_CPP&#39;.
    &#34;&#34;&#34;
    self.n = 0
    self.t = 0
    self.dt = dt
    self.soft = soft
    self.verbose = verbose
    self.CONFIG = CONFIG
    # Initialize empty arrays for all necessary properties
    self.r_vec = np.empty((0, 3))
    self.v_vec = np.empty((0, 3))
    self.a_vec = np.empty((0, 3))
    self.mass = np.empty((0,))
    self.type = np.empty((0, 2))
    algorithms = {
        &#39;bruteForce&#39;: acceleration.bruteForce,
        &#39;bruteForceNumba&#39;: acceleration.bruteForceNumba,
        &#39;bruteForceNumbaOptimized&#39;: acceleration.bruteForceNumbaOptimized,
        &#39;bruteForceCPP&#39;: acceleration.bruteForceCPP,
        &#39;barnesHutCPP&#39;: acceleration.barnesHutCPP
    }
    try:
        self.acceleration = algorithms[method]
    except: raise Exception(&#34;Method &#39;{}&#39; unknown&#34;.format(method))</code></pre>
</details>
</dd>
<dt id="simulation.Simulation.add"><code class="name flex">
<span>def <span class="ident">add</span></span>(<span>self, body)</span>
</code></dt>
<dd>
<section class="desc"><p>Add a body to the simulation. It must expose the public attributes
body.r_vec, body.v_vec, body.a_vec, body.type, body.mass.</p>
<h2 id="arguments">Arguments</h2>
<dl>
<dt><strong><code>body</code></strong></dt>
<dd>Object to be added to the simulation (e.g. a Galaxy object)</dd>
</dl></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">def add(self, body):
    &#34;&#34;&#34;Add a body to the simulation. It must expose the public attributes
       body.r_vec, body.v_vec, body.a_vec, body.type, body.mass.

    Arguments:
        body: Object to be added to the simulation (e.g. a Galaxy object)
    &#34;&#34;&#34;
    # Extend all relevant attributes by concatenating the body
    for name in [&#39;r_vec&#39;, &#39;v_vec&#39;, &#39;a_vec&#39;, &#39;type&#39;, &#39;mass&#39;]:
        simattr, bodyattr = getattr(self, name), getattr(body, name)
        setattr(self, name, np.concatenate([simattr, bodyattr], axis=0))
    # Order based on mass
    order = np.argsort(-self.mass)
    for name in [&#39;r_vec&#39;, &#39;v_vec&#39;, &#39;a_vec&#39;, &#39;type&#39;, &#39;mass&#39;]: 
        setattr(self, name, getattr(self, name)[order])

    # Update the list of objects to keep track of
    self.tracks = np.empty((np.sum(self.type[:,0]==&#39;center&#39;), 0, 3))</code></pre>
</details>
</dd>
<dt id="simulation.Simulation.project"><code class="name flex">
<span>def <span class="ident">project</span></span>(<span>self, theta, phi, view=0)</span>
</code></dt>
<dd>
<section class="desc"><p>Projects the 3D simulation onto a plane as viewed from the
direction described by the (theta, phi, view). Angles in radians.
(This is used by the simulated annealing algorithm)</p>
<h2 id="parameters">Parameters</h2>
<dl>
<dt><strong><code>theta</code></strong> :&ensp;<code>float</code></dt>
<dd>polar angle.</dd>
<dt><strong><code>phi</code></strong> :&ensp;<code>float</code></dt>
<dd>azimuthal angle.</dd>
<dt><strong><code>view</code></strong> :&ensp;<code>float</code></dt>
<dd>rotation along line of sight.</dd>
</dl></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">def project(self, theta, phi, view=0):
    &#34;&#34;&#34;Projects the 3D simulation onto a plane as viewed from the
       direction described by the (theta, phi, view). Angles in radians.
       (This is used by the simulated annealing algorithm)
    
    Parameters:
        theta (float): polar angle.
        phi (float): azimuthal angle.
        view (float): rotation along line of sight.
    &#34;&#34;&#34;
    M1 = np.array([[np.cos(phi), np.sin(phi), 0],
                   [-np.sin(phi), np.cos(phi), 0],
                   [0, 0, 1]])
    M2 = np.array([[1, 0, 0],
                   [0, np.cos(theta), np.sin(theta)],
                   [0, -np.sin(theta), np.cos(theta)]])
    M3 = np.array([[np.cos(view), np.sin(view), 0],
                   [-np.sin(view), np.cos(view), 0],
                   [0, 0, 1]])

    M = np.matmul(M1, np.matmul(M2, M3)) # combine rotations
    r = np.tensordot(self.r_vec, M, axes=[1, 0])

    return r[:,0:2]</code></pre>
</details>
</dd>
<dt id="simulation.Simulation.run"><code class="name flex">
<span>def <span class="ident">run</span></span>(<span>self, tmax, saveEvery, outputFolder, **kwargs)</span>
</code></dt>
<dd>
<section class="desc"><p>Run the galactic simulation.</p>
<h2 id="attributes">Attributes</h2>
<dl>
<dt><strong><code>tmax</code></strong> :&ensp;<code>float</code></dt>
<dd>Time to which the simulation will run to.
This is measured here since the start of the simulation,
not since pericenter.</dd>
<dt><strong><code>saveEvery</code></strong> :&ensp;<code>int</code></dt>
<dd>The state is saved every saveEvery steps.</dd>
<dt><strong><code>outputFolder</code></strong> :&ensp;<code>string</code></dt>
<dd>It will be saved to /data/outputFolder/</dd>
</dl></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">def run(self, tmax, saveEvery, outputFolder, **kwargs):
    &#34;&#34;&#34;Run the galactic simulation.

    Attributes:
        tmax (float): Time to which the simulation will run to.
            This is measured here since the start of the simulation,
            not since pericenter.
        saveEvery (int): The state is saved every saveEvery steps.
        outputFolder (string): It will be saved to /data/outputFolder/
    &#34;&#34;&#34;
    # When the simulation starts, intialize self.rprev_vec
    self.rprev_vec = self.r_vec - self.v_vec * self.dt
    if self.verbose: print(&#39;Simulation starting. Bon voyage!&#39;)
    while(self.t &lt; tmax):
        self.step()
        if(self.n % saveEvery == 0):
            self.save(&#39;data/{}&#39;.format(outputFolder))

    print(&#39;Simulation complete.&#39;)</code></pre>
</details>
</dd>
<dt id="simulation.Simulation.save"><code class="name flex">
<span>def <span class="ident">save</span></span>(<span>self, outputFolder)</span>
</code></dt>
<dd>
<section class="desc"><p>Save the state of the simulation to the outputFolder.
Two files are saved:
sim{self.n}.pickle: serializing the state.
sim{self.n}.png: a simplified 2D plot of x, y.</p></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">def save(self, outputFolder):
    &#34;&#34;&#34;Save the state of the simulation to the outputFolder.
       Two files are saved:
            sim{self.n}.pickle: serializing the state.
            sim{self.n}.png: a simplified 2D plot of x, y.
    &#34;&#34;&#34;
    # Create the output folder if it doesn&#39;t exist
    if not os.path.exists(outputFolder): os.makedirs(outputFolder)

    # Compute some useful quantities
    # v_vec is not required by the integrator, but useful
    self.v_vec = (self.r_vec - self.rprev_vec)/self.dt
    self.t = self.n * self.dt # prevents numerical rounding errors

    # Serialize state
    file = open(outputFolder+&#39;/data{}.pickle&#39;.format(self.n), &#34;wb&#34;)
    pickle.dump({&#39;r_vec&#39;: self.r_vec, &#39;v_vec&#39;: self.v_vec,
                 &#39;type&#39;: self.type, &#39;mass&#39;: self.mass,
                 &#39;CONFIG&#39;: self.CONFIG, &#39;t&#39;: self.t,
                 &#39;tracks&#39;: self.tracks}, file)

    # Save simplified plot of the current state.
    # Its main use is to detect ill-behaved situations early on.
    fig = plt.figure()
    plt.xlim(-5, 5); plt.ylim(-5, 5); plt.axis(&#39;equal&#39;)
    # Dark halo is plotted in red, disk in blue, bulge in green
    PLTCON = [(&#39;dark&#39;, &#39;r&#39;, 0.3), (&#39;disk&#39;, &#39;b&#39;, 1.0), (&#39;bulge&#39;, &#39;g&#39;, 0.5)]
    for type_, c, a in PLTCON: 
        plt.scatter(self.r_vec[self.type[:,0]==type_][:,0],
            self.r_vec[self.type[:,0]==type_][:,1], s=0.1, c=c, alpha=a)
    # Central mass as a magenta star 
    plt.scatter(self.r_vec[self.type[:,0]==&#39;center&#39;][:,0],
        self.r_vec[self.type[:,0]==&#39;center&#39;][:,1], s=100, marker=&#34;*&#34;, c=&#39;m&#39;)
    # Save to png file
    fig.savefig(outputFolder+&#39;/sim{}.png&#39;.format(self.n), dpi=150)
    plt.close(fig)</code></pre>
</details>
</dd>
<dt id="simulation.Simulation.setOrbit"><code class="name flex">
<span>def <span class="ident">setOrbit</span></span>(<span>self, galaxy1, galaxy2, e, rmin, R0)</span>
</code></dt>
<dd>
<section class="desc"><p>Sets the two galaxies galaxy1, galaxy2 in an orbit.</p>
<h2 id="parameters">Parameters</h2>
<dl>
<dt><strong><code>galaxy1</code></strong> :&ensp;<a title="simulation.Galaxy" href="#simulation.Galaxy"><code>Galaxy</code></a></dt>
<dd>1st galaxy (main)</dd>
<dt><strong><code>galaxy2</code></strong> :&ensp;<a title="simulation.Galaxy" href="#simulation.Galaxy"><code>Galaxy</code></a></dt>
<dd>2nd galaxy (companion)</dd>
<dt><strong><code>e</code></strong></dt>
<dd>eccentricity of the orbit</dd>
<dt><strong><code>rmin</code></strong></dt>
<dd>distance of closest approach</dd>
<dt><strong><code>R0</code></strong></dt>
<dd>initial separation</dd>
</dl></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">def setOrbit(self, galaxy1, galaxy2, e, rmin, R0):
    &#34;&#34;&#34;Sets the two galaxies galaxy1, galaxy2 in an orbit.

    Parameters:
        galaxy1 (Galaxy): 1st galaxy (main)
        galaxy2 (Galaxy): 2nd galaxy (companion)
        e: eccentricity of the orbit
        rmin: distance of closest approach
        R0: initial separation
    &#34;&#34;&#34;
    m1, m2 = np.sum(galaxy1.mass), np.sum(galaxy2.mass)
    # Relevant formulae:
    # $r_0 = r (1 + e) \cos(\phi)$, where $r_0$ ($\neq R_0$) is the semi-latus rectum
    # $r_0 = r_\textup{min} (1 + e)$
    # $v^2 = G M (2/r - 1/a)$, where a is the semimajor axis

    # Solve the reduced two-body problem with reduced mass $\mu$ (mu)
    M = m1 + m2
    r0 = rmin * (1 + e)
    try:
        phi = np.arccos((r0/R0 - 1) / e) # inverting the orbit equation
        phi = -np.abs(phi) # Choose negative (incoming) angle
        ainv = (1 - e) / rmin # ainv = $1/a$, as a may be infinite
        v = np.sqrt(M * (2/R0 - ainv))
        # Finally, calculate the angle of motion. angle = tan(dy/dx)
        # $dy/dx = ((dr/d\phi) sin(\phi) + r \cos(\phi))/((dr/d\phi) cos(\phi) - r \sin(\phi))$
        dy = R0/r0 * e * np.sin(phi)**2 + np.cos(phi)
        dx = R0/r0 * e * np.sin(phi) * np.cos(phi) - np.sin(phi)
        vangle = np.arctan2(dy, dx)
    except: raise Exception(&#39;&#39;&#39;The orbital parameters cannot be satisfied.
        For elliptical orbits check that R0 is posible (&lt;rmax).&#39;&#39;&#39;)

    # We now need the actual motion of each body
    R_vec = np.array([[R0*np.cos(phi), R0*np.sin(phi), 0.]])
    V_vec = np.array([[v*np.cos(vangle), v*np.sin(vangle), 0.]])

    galaxy1.r_vec += m2/M * R_vec
    galaxy1.v_vec += m2/M * V_vec
    galaxy2.r_vec += -m1/M * R_vec
    galaxy2.v_vec += -m1/M * V_vec

    # Explicitely add the galaxies to the simulation
    self.add(galaxy1)
    self.add(galaxy2)

    if self.verbose: print(&#39;Galaxies set in orbit.&#39;)</code></pre>
</details>
</dd>
<dt id="simulation.Simulation.step"><code class="name flex">
<span>def <span class="ident">step</span></span>(<span>self)</span>
</code></dt>
<dd>
<section class="desc"><p>Perform a single step of the simulation.
Makes use of a 4th order Verlet integrator.</p></section>
<details class="source">
<summary>Source code</summary>
<pre><code class="python">def step(self):
    &#34;&#34;&#34;Perform a single step of the simulation.
       Makes use of a 4th order Verlet integrator.
    &#34;&#34;&#34;
    # Calculate the acceleration
    self.a_vec = self.acceleration(self.r_vec, self.mass, soft=self.soft)
    # Update the state using the Verlet algorithm
    # (A custom algorithm is written mainly for learning purposes)
    self.r_vec, self.rprev_vec = (2*self.r_vec - self.rprev_vec
        + self.a_vec * self.dt**2, self.r_vec)
    self.n += 1
    # Update tracks
    self.tracks = np.concatenate([self.tracks,
        self.r_vec[self.type[:,0]==&#39;center&#39;][:,np.newaxis]], axis=1)</code></pre>
</details>
</dd>
</dl>
</dd>
</dl>
</section>
</article>
<nav id="sidebar">
<h1>Index</h1>
<div class="toc">
<ul></ul>
</div>
<ul id="index">
<li><h3><a href="#header-classes">Classes</a></h3>
<ul>
<li>
<h4><code><a title="simulation.Galaxy" href="#simulation.Galaxy">Galaxy</a></code></h4>
<ul class="">
<li><code><a title="simulation.Galaxy.__init__" href="#simulation.Galaxy.__init__">__init__</a></code></li>
<li><code><a title="simulation.Galaxy.addBulge" href="#simulation.Galaxy.addBulge">addBulge</a></code></li>
<li><code><a title="simulation.Galaxy.addDisk" href="#simulation.Galaxy.addDisk">addDisk</a></code></li>
<li><code><a title="simulation.Galaxy.addHalo" href="#simulation.Galaxy.addHalo">addHalo</a></code></li>
<li><code><a title="simulation.Galaxy.rotate" href="#simulation.Galaxy.rotate">rotate</a></code></li>
</ul>
</li>
<li>
<h4><code><a title="simulation.Simulation" href="#simulation.Simulation">Simulation</a></code></h4>
<ul class="two-column">
<li><code><a title="simulation.Simulation.__init__" href="#simulation.Simulation.__init__">__init__</a></code></li>
<li><code><a title="simulation.Simulation.add" href="#simulation.Simulation.add">add</a></code></li>
<li><code><a title="simulation.Simulation.project" href="#simulation.Simulation.project">project</a></code></li>
<li><code><a title="simulation.Simulation.run" href="#simulation.Simulation.run">run</a></code></li>
<li><code><a title="simulation.Simulation.save" href="#simulation.Simulation.save">save</a></code></li>
<li><code><a title="simulation.Simulation.setOrbit" href="#simulation.Simulation.setOrbit">setOrbit</a></code></li>
<li><code><a title="simulation.Simulation.step" href="#simulation.Simulation.step">step</a></code></li>
</ul>
</li>
</ul>
</li>
</ul>
</nav>
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