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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 DSFP Object Oriented Programming Notebook
Incorporating classes, objects, and functions into your code will improve its efficiency, readability, and make it easier to extend to other programs down the road.  This notebook attempts to take you through many of the concepts associated with creating a class and have it do something useful and interesting.
By C Hummels (Caltech / NSF Fellow)
Step2:  Problem 1) Create a galaxy class
We will begin our experience with object-oriented code by creating a class to represent a galaxy.  In order to do this, we must define the class, give it some default methods, and then some attributes to describe each instance of a galaxy.
Problem 1a
What do you think should be the attributes and characteristics that should be tracked for a galaxy object?  Think critically and make a list of at least six different quantities that would make sense if this were a research project.  Please do this prior to starting (or looking at) the example below. Discuss with your code partner.
Possible Answers
Step3:  Problem 1c
Try creating a galaxy instance of your galaxy class!  Create a galaxy object, named after your favorite galaxy, with appropriate values for your arguments and keyword arguments (or as best as you can approximate).  There isn't much you can do with it right now, but at least you can be assured that your existing code works.  Try printing your object to see what python thinks of it.
Step5:  Problem 1d
So Python probably didn't do a very good internal representation of your galaxy.  Remember, classes have a few built-in class methods to be defined.  One is the __init__ method, so that we know how to initialize the class.  Another very useful class method is the __repr__, which tells us how to represent a class object when we try to print it out.  It's very useful for debugging.  Please copy your class definition from above, and add the __repr__ method to return a string somehow identifying a galaxy based on its attributes.  Now recreate your favorite galaxy, and print ut out again and see if it is a bit more readable than before.
Step7:  Problem 2) Make a more interesting galaxy class that can evolve with time
Now let's build on our basic galaxy class with some relevant class methods specific to galaxies that enables them to change with time.  Obviously, galaxies have a huge number of traits, and we don't currently understand all the ways in which these variables interrelate.  But we can make a toy model here just to get the idea of how they might evolve.  This is largely the premise of semi-analytic models that people use to study how galaxies might evolve with time.
Problem 2a
Think about what methods you would add to allow your galaxy to change with time using the galaxy attributes defined above and/or other galaxy attributes.  Come up with at least three and then discuss with your partner before continuing.
Possible Answers
Step10:  Problem 2c
Let's start by making an evolve method for our Galaxy class in order to evolve the galaxy forward in time.  evolve should expect an argument for time in years.  And at least initially, let's just make this a simple method that just changes the age of the galaxy by the appropriate amount.  We'll add more on to it in a second.
Because we're going to be following the evolution of this galaxy, we should create a current_state method, which returns a tuple of the attributes of the EvolvingGalaxy object at that moment.  Now as our EvolvingGalaxy changes with time, we will be able to track it and save that information somewhere.
Step13:  Problem 2d
Now that we can evolve our EvolvingGalaxy and check on its current_state over time, let's run it forward in time and see the results!  
Make a new function (not a class method) called integrate_time that accepts an EvolvingGalaxy object, a timestep (some number of years long), and a number of timesteps over which to evolve the galaxy.  Step the EvolvingGalaxy forward in time using your for loop, one timestep at a time, and store its current_state at each timestep.  You can store this current_state information however you'd like, but I find it easiest to create a NumPy array of size [5, n_timesteps], and fill it in using the for loop.
Similarly, make a new function called plot_galaxy_evolution that accepts the output from your integrate_time function and uses Matplotlib's pyplot module to plot up the evolution of the EvolvingGalaxy's properties versus its age.
Finally, create an instance of EvolvingGalaxy, and plug it into your new integrate_time and plot_galaxy_evolution functions to integrate for a gigayear, and plot how the EvolvingGalaxy quantities change with age.  Not much going on--sad!
Step20:  Problem 2e
How do galaxies evolve with time?  They do in lots of ways, but let's make a few specific rules.  
Galaxies tend to accrete gas from the intergalactic medium over time.  We'll modify the code from above for your EvolvingGalaxy class to include a accrete_gas_from_IGM method, which adds to the cold_gas_mass and total_mass of the galaxy with time.  Let's give it a random component too.
Galaxies form stars, converting cold_gas_mass to stellar_mass according to their star formation rate (SFR).  But there must be sufficient cold_gas_mass to consume, so make sure there is enough for the SFR to operate for the alloted time, otherwise the SFR should drop to 0.
Finally, we haven't actually calculated a SFR, let's just create a method, calculate_star_formation_rate, choose a random value every time it is called.  We'll also scale it in some way by the total_mass of the galaxy and use the Milky Way as a guide.  The Milky Way forms about 1 Msun/year and it has a total_mass of 1e12 Msun.  
Include these functions in our evolve method, and re-generate a galaxy evolution plot like you did in Problem 2d.  Since there is a random component, you can try re-running it a few times and see how the behavior changes.
Step22:  Problem 2f
Based on your knowledge of how galaxies evolve, do these results make sense?  Discuss with your coding partner if so/not, and ways in which you might tweak it to work better.
Possible Answers
Step25:  Problem 3b
In order for our MovingGalaxy to move, we have to give it a few more methods.  Copy your class definition from above, and add three more methods.  
Create a move method which accepts a time argument and updates the MovingGalaxy's x_position, and y_position by accounting for the motion from its x_velocity and y_velocity.
Create a calculate_momentum method that returns a tuple of (x_momentum, y_momentum) for a galaxy.
Finally, create an evolve method, which accepts a time argument, and just executes your move method before calling the superclass's evolve method (i.e. EvolvingGalaxy's).
Step27:  Problem 3c
OK, so we have a MovingGalaxy that can move, but we need a Universe in which it can move!  
So let us create a new Universe class that can hold one or more galaxies.  We will give it x and y limits which define its borders to be 0 and 100, as well as an internal list to hold the galaxies themselves.  Make an evolve method for the Universe class that moves its time forward some timestep.  How does this affect the galaxies?  Are there boundaries to your Universe?  What happens when MovingGalaxy's run into each other?  Don't worry about gravity for now, but you could make them merge when they are close enough to each other, or just let them fly through each other.  After all, it's your Universe.
It may also be useful to make a method to add a random MovingGalaxy, so that you can add several to your Universe during initialization.  Start out only adding a few, as it might get expensive!
Make a command that plots the positions of the galaxies with time using matplotlib.scatter(), and save the corresponding outputs to a files called frame%04i.png.  The next steps allow you to make a movie of these frames.
Step28:  Problem 3d
Initialize a Universe with several galaxies in it. Run your Universe forward in time, and plot the resulting state of the Universe over time to files called frame%04i.png.
Step29:  Problem 3e
The command below will output a movie file called movie.mp4 by combining all of your frame%04i.png files and packaging them together at a framerate of 20 FPS.  It requires ffmpeg to be installed, but you can do this easily if you have homebrew installed by running brew install ffmpeg.  There are other means to download ffmpeg online.  Otherwise, you can flip through the frames manually to see how the Universe changes with time.  
If you were able to execute this step OK, execute to the next two fields and watch your movie!  Again, since there is a random component to these models, you can try running it a few times to see how the results change.
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Python Code:
import matplotlib.pyplot as plt
import random
import numpy as np
%matplotlib inline
Explanation: DSFP Object Oriented Programming Notebook
Incorporating classes, objects, and functions into your code will improve its efficiency, readability, and make it easier to extend to other programs down the road.  This notebook attempts to take you through many of the concepts associated with creating a class and have it do something useful and interesting.
By C Hummels (Caltech / NSF Fellow)
End of explanation
class Galaxy():
    
    Galaxy class for simply representing a galaxy.
    
    def __init__(self, total_mass, cold_gas_mass, stellar_mass, age=0):
        self.total_mass = total_mass
        self.cold_gas_mass = cold_gas_mass
        self.stellar_mass = stellar_mass
        self.age = age
        self.SFR = 0
        self.color = 'red'
Explanation: Problem 1) Create a galaxy class
We will begin our experience with object-oriented code by creating a class to represent a galaxy.  In order to do this, we must define the class, give it some default methods, and then some attributes to describe each instance of a galaxy.
Problem 1a
What do you think should be the attributes and characteristics that should be tracked for a galaxy object?  Think critically and make a list of at least six different quantities that would make sense if this were a research project.  Please do this prior to starting (or looking at) the example below. Discuss with your code partner.
Possible Answers: Mass, Luminosity, SFR, Metallicity, Color, Age, Central Black Hole Mass, Disk?, Type (Sab, E0, etc.), Spin
Problem 1b
Define a galaxy class, and give it an initializer class method.  Have your initializer accept arguments for total_mass, cold_gas_mass, and stellar_mass.  Have it also accept keyword arguments for age with a default set to 0.  Maybe also include a SFR and color in your attributes, but initialize it to 0 and 'red' respectively.  I'll include docstrings for you to help guide you along the way.
Hint: Don't forget to use self to indicate an object of this class.
End of explanation
milky_way = Galaxy(1e12, 1e8, 1e10, age=5e9)
print(milky_way)
Explanation: Problem 1c
Try creating a galaxy instance of your galaxy class!  Create a galaxy object, named after your favorite galaxy, with appropriate values for your arguments and keyword arguments (or as best as you can approximate).  There isn't much you can do with it right now, but at least you can be assured that your existing code works.  Try printing your object to see what python thinks of it.
End of explanation
class Galaxy():
    
    Galaxy class for simply representing a galaxy.
    
    def __init__(self, total_mass, cold_gas_mass, stellar_mass, age=0):
        self.total_mass = total_mass
        self.cold_gas_mass = cold_gas_mass
        self.stellar_mass = stellar_mass
        self.age = age
        self.SFR = 0
        self.color = 'red'
        
    def __repr__(self):
        return "Galaxy (m_total = %.1g; m_cold = %.1g; m_stars = %.1g; age = %.1g; SFR = %0.2f)" % \
                (self.total_mass, self.cold_gas_mass, self.stellar_mass, self.age, self.SFR)
milky_way = Galaxy(1e12, 1e8, 1e10, age=5e9)
print(milky_way)
Explanation: Problem 1d
So Python probably didn't do a very good internal representation of your galaxy.  Remember, classes have a few built-in class methods to be defined.  One is the __init__ method, so that we know how to initialize the class.  Another very useful class method is the __repr__, which tells us how to represent a class object when we try to print it out.  It's very useful for debugging.  Please copy your class definition from above, and add the __repr__ method to return a string somehow identifying a galaxy based on its attributes.  Now recreate your favorite galaxy, and print ut out again and see if it is a bit more readable than before.
End of explanation
class EvolvingGalaxy(Galaxy):
    
    Galaxy class for representing a galaxy that can evolve over time.
    
milky_way = EvolvingGalaxy(1e12, 1e8, 1e10, age=5e9)
print(milky_way)
Explanation: Problem 2) Make a more interesting galaxy class that can evolve with time
Now let's build on our basic galaxy class with some relevant class methods specific to galaxies that enables them to change with time.  Obviously, galaxies have a huge number of traits, and we don't currently understand all the ways in which these variables interrelate.  But we can make a toy model here just to get the idea of how they might evolve.  This is largely the premise of semi-analytic models that people use to study how galaxies might evolve with time.
Problem 2a
Think about what methods you would add to allow your galaxy to change with time using the galaxy attributes defined above and/or other galaxy attributes.  Come up with at least three and then discuss with your partner before continuing.
Possible Answers: evolve(time), cold_mode_accretion(time), hot_mode_accretion(time), absorb_galaxy(galaxy), star_burst(), agn_feedback(), change_color(), change_type()
Problem 2b
We want to enhance our general Galaxy class to enable galaxies to actually evolve with time.  Rather than just copying all of the code from our Galaxy class above, we can subclass the Galaxy class to inherit all of its attributes, methods, etc!  Create a new class named EvolvingGalaxy below, which uses the Galaxy class as its "parent" or "super" class.  Remember, because you already defined __init__ and __repr__ methods for this class, they are inherited and you don't need to redefine them, unless you want to.
Now try creating an instance of EvolvingGalaxy and printing it out, just like you did for your favorite galaxy in problem 1c.
End of explanation
class EvolvingGalaxy(Galaxy):
    
    Galaxy class for representing a galaxy that can evolve over time.
    
    def evolve(self, time):
        
        Evolve this galaxy forward for a period time
        
        self.age += time
 
    def current_state(self):
        return (self.total_mass, self.cold_gas_mass, self.stellar_mass, self.age, self.SFR)
Explanation: Problem 2c
Let's start by making an evolve method for our Galaxy class in order to evolve the galaxy forward in time.  evolve should expect an argument for time in years.  And at least initially, let's just make this a simple method that just changes the age of the galaxy by the appropriate amount.  We'll add more on to it in a second.
Because we're going to be following the evolution of this galaxy, we should create a current_state method, which returns a tuple of the attributes of the EvolvingGalaxy object at that moment.  Now as our EvolvingGalaxy changes with time, we will be able to track it and save that information somewhere.
End of explanation
def integrate_time(galaxy, timestep, n_timesteps):
    
    Integrate the time forward for a galaxy and record its state at each timestep; return as array
        
    data_arr = np.empty([5, n_timesteps])
    for i in range(n_timesteps):
        galaxy.evolve(timestep)
        data_arr[:,i] = galaxy.current_state()
    return data_arr
def plot_galaxy_evolution(data_arr):
    
    Plot the evolution of a galaxy from its input data array
    
    plt.clf()
    plt.semilogy(data[3], data[0], color='k', label='Total')
    plt.semilogy(data[3], data[1], color='b', label='Gas')
    plt.semilogy(data[3], data[2], color='r', label='Stars')
    plt.semilogy(data[3], data[4], color='g', label='SFR')
    plt.xlabel('Age')
    plt.ylabel('Mass')
    plt.legend(loc=1)
    plt.show()
milky_way = EvolvingGalaxy(1e12, 1e8, 1e10, age=5e9)
data = integrate_time(milky_way, 1e6, 1000)
plot_galaxy_evolution(data)
Explanation: Problem 2d
Now that we can evolve our EvolvingGalaxy and check on its current_state over time, let's run it forward in time and see the results!  
Make a new function (not a class method) called integrate_time that accepts an EvolvingGalaxy object, a timestep (some number of years long), and a number of timesteps over which to evolve the galaxy.  Step the EvolvingGalaxy forward in time using your for loop, one timestep at a time, and store its current_state at each timestep.  You can store this current_state information however you'd like, but I find it easiest to create a NumPy array of size [5, n_timesteps], and fill it in using the for loop.
Similarly, make a new function called plot_galaxy_evolution that accepts the output from your integrate_time function and uses Matplotlib's pyplot module to plot up the evolution of the EvolvingGalaxy's properties versus its age.
Finally, create an instance of EvolvingGalaxy, and plug it into your new integrate_time and plot_galaxy_evolution functions to integrate for a gigayear, and plot how the EvolvingGalaxy quantities change with age.  Not much going on--sad!
End of explanation
class EvolvingGalaxy(Galaxy):
    
    Galaxy class for representing a galaxy that can evolve over time.
    
    def current_state(self):
        
        Return a tuple of the galaxy's total_mass, cold_gas_mass, stellar_mass, age, and SFR
        
        return (self.total_mass, self.cold_gas_mass, self.stellar_mass, self.age, self.SFR)
    
    def calculate_star_formation_rate(self):
        
        Calculate the star formation rate by taking a random number between 0 and 1 
        normalized by the galaxy total mass / 1e12; 
        
        Also updates the galaxy's color to blue if SFR > 0.01, otherwise color = red
        
        self.SFR = random.random() * (self.total_mass / 1e12)
        if self.SFR > 0.01: 
            self.color = 'blue'
        else:
            self.color = 'red'
            
    def accrete_gas_from_IGM(self, time):
        
        Allow the galaxy to accrete cold gas from the IGM at a variable rate normalized to
        the galaxy's mass
        
        cold_gas_accreted = random.random() * 0.1 * time * (self.total_mass / 1e12)
        self.cold_gas_mass += cold_gas_accreted
        self.total_mass += cold_gas_accreted
        
    def form_stars(self, time):
        
        Form stars according to the current star formation rate and time available
        If unable cold gas, then shut off star formation
        
        if self.cold_gas_mass > self.SFR * time:
            self.cold_gas_mass -= self.SFR * time
            self.stellar_mass += self.SFR * time
        else:
            self.SFR = 0
            self.color = 'red'
            
    def evolve(self, time):
        
        Evolve this galaxy forward for a period time
        
        if random.random() < 0.01:
            self.calculate_star_formation_rate()
        self.accrete_gas_from_IGM(time)
        self.form_stars(time)
        self.age += time                
            
milky_way = EvolvingGalaxy(1e12, 1e8, 1e10, age=5e9)
data = integrate_time(milky_way, 1e6, 10000)
plot_galaxy_evolution(data)
Explanation: Problem 2e
How do galaxies evolve with time?  They do in lots of ways, but let's make a few specific rules.  
Galaxies tend to accrete gas from the intergalactic medium over time.  We'll modify the code from above for your EvolvingGalaxy class to include a accrete_gas_from_IGM method, which adds to the cold_gas_mass and total_mass of the galaxy with time.  Let's give it a random component too.
Galaxies form stars, converting cold_gas_mass to stellar_mass according to their star formation rate (SFR).  But there must be sufficient cold_gas_mass to consume, so make sure there is enough for the SFR to operate for the alloted time, otherwise the SFR should drop to 0.
Finally, we haven't actually calculated a SFR, let's just create a method, calculate_star_formation_rate, choose a random value every time it is called.  We'll also scale it in some way by the total_mass of the galaxy and use the Milky Way as a guide.  The Milky Way forms about 1 Msun/year and it has a total_mass of 1e12 Msun.  
Include these functions in our evolve method, and re-generate a galaxy evolution plot like you did in Problem 2d.  Since there is a random component, you can try re-running it a few times and see how the behavior changes.
End of explanation
class MovingGalaxy(EvolvingGalaxy):
    
    Galaxy class that can evolve and move in the x,y plane
    
    def __init__(self, total_mass, cold_gas_mass, stellar_mass, x_position, y_position, x_velocity, y_velocity, idnum, age=0):
        
        # Replace self with super to activate the superclass's methods
        super().__init__(total_mass, cold_gas_mass, stellar_mass)
        
        self.x_position = x_position
        self.y_position = y_position
        self.x_velocity = x_velocity
        self.y_velocity = y_velocity
        self.idnum = idnum
        
    def __repr__(self):
        return "Galaxy %i (x = %.0f; y = %.0f)" % (self.idnum, self.x_position, self.y_position)
milky_way = MovingGalaxy(1e12, 1e8, 1e10, 0, 0, 0, 0, 0)
print(milky_way)
Explanation: Problem 2f
Based on your knowledge of how galaxies evolve, do these results make sense?  Discuss with your coding partner if so/not, and ways in which you might tweak it to work better.
Possible Answers: For the most part, yes.  The galaxy consumes its cold gas and creates stars, and when it runs out of gas, it shuts off star formation for a time.  But after time, it accretes enough cold gas to allow it to form stars again, so it's in sort of loop.  But it's not really realistic because it needs to interact with its environment and other galaxies through galaxy mergers, gas accretion, gas stripping, etc.
Problem 3) Make a galaxy class that can interact with other galaxies (challenging!)
Most galaxies do not evolve in isolation, so let's make a galaxy class that enables galaxies to actually interact with each other.  We can re-use much of the code from our previous examples through inheritance!  Yeah!
Problem 3a
Let us create a new subclass of EvolveGalaxy, so that it inherits most of the traits of our previous work.  But we will add some new features including position and velocity values in the x-y plane, so that our galaxy can actually move over time.
Create a new class MovingGalaxy that is a subclass of EvolveGalaxy.  Make a new __init__ method with all of the arguments from it's parent class EvolveGalaxy, but add the additional arguments of x_position, y_position, x_velocity, y_velocity, and id_num.  
Note: all arguments must come before any keyword arguments, so the age kwarg has to come at the very end.
Also, make a new __repr__ method for this class that just includes it's idnum, x_position, and y_position.
To assure your code works, create an instance of a MovingGalaxy, and print it out.
End of explanation
class MovingGalaxy(EvolvingGalaxy):
    
    This galaxy can move over time in the x,y plane
    
    def __init__(self, total_mass, cold_gas_mass, stellar_mass, x_position, y_position, x_velocity, y_velocity, idnum, age=0):
        
        # Replace self with super to activate the superclass's methods
        super().__init__(total_mass, cold_gas_mass, stellar_mass)
        
        self.x_position = x_position
        self.y_position = y_position
        self.x_velocity = x_velocity
        self.y_velocity = y_velocity
        self.idnum = idnum
        
    def __repr__(self):
        return "Galaxy %i (x = %.0f; y = %.0f)" % (self.idnum, self.x_position, self.y_position)
        
    def move(self, time):
        
        
        self.x_position += self.x_velocity * time
        self.y_position += self.y_velocity * time
        
    def calculate_momentum(self):
        return (self.total_mass * self.x_velocity, self.total_mass * self.y_velocity)
    def evolve(self, time):
        self.move(time)
        super().evolve(time)
Explanation: Problem 3b
In order for our MovingGalaxy to move, we have to give it a few more methods.  Copy your class definition from above, and add three more methods.  
Create a move method which accepts a time argument and updates the MovingGalaxy's x_position, and y_position by accounting for the motion from its x_velocity and y_velocity.
Create a calculate_momentum method that returns a tuple of (x_momentum, y_momentum) for a galaxy.
Finally, create an evolve method, which accepts a time argument, and just executes your move method before calling the superclass's evolve method (i.e. EvolvingGalaxy's).
End of explanation
def distance(galaxy1, galaxy2):
    x_diff = galaxy1.x_position - galaxy2.x_position
    y_diff = galaxy1.y_position - galaxy2.y_position
    return (x_diff**2 + y_diff**2)**0.5
class Universe():
    
    
    def __init__(self):
        self.xrange = (0,100)
        self.yrange = (0,100)
        self.galaxies = []
        self.added_galaxies = []
        self.removed_galaxies = []
        self.time = 0
        pass
    
    def __repr__(self):
        out = 'Universe: t=%.2g\n' % self.time
        for galaxy in self.galaxies:
            out = "%s%s\n" % (out, galaxy)
        return out
        
    def add_galaxy(self, galaxy=None):
        if galaxy is None:
            stellar_mass = 10**(4*random.random()) * 1e6
            cold_gas_mass = 10**(4*random.random()) * 1e6
            total_mass = (cold_gas_mass + stellar_mass)*1e2
            galaxy = MovingGalaxy(total_mass,
                                  cold_gas_mass,
                                  stellar_mass,
                                  x_position=random.random()*100,
                                  y_position=random.random()*100,
                                  x_velocity=random.uniform(-1,1)*1e-7,
                                  y_velocity=random.uniform(-1,1)*1e-7,
                                  idnum=len(self.galaxies))
        self.galaxies.append(galaxy)
        
    def remove_galaxy(self, galaxy):
        if galaxy in self.galaxies:
            del self.galaxies[self.galaxies.index(galaxy)]
        
    def evolve(self, time):
        for galaxy in self.galaxies:
            galaxy.evolve(time)
            galaxy.x_position %= 100
            galaxy.y_position %= 100
        self.check_for_mergers()
        for galaxy in self.removed_galaxies:
            self.remove_galaxy(galaxy)
        for galaxy in self.added_galaxies:
            self.add_galaxy(galaxy)
        self.removed_galaxies = []
        self.added_galaxies = []
        self.time += time
            
    def merge_galaxies(self, galaxy1, galaxy2):
        print('Merging:\n%s\n%s' % (galaxy1, galaxy2))
        x_mom1, y_mom1 = galaxy1.calculate_momentum()
        x_mom2, y_mom2 = galaxy2.calculate_momentum()
        new_total_mass = galaxy1.total_mass + galaxy2.total_mass
        new_galaxy = MovingGalaxy(total_mass = new_total_mass,
                                  cold_gas_mass = galaxy1.cold_gas_mass + galaxy2.cold_gas_mass,
                                  stellar_mass = galaxy1.stellar_mass + galaxy2.stellar_mass,
                                  x_position = galaxy1.x_position,
                                  y_position = galaxy1.y_position,
                                  x_velocity = (x_mom1 + x_mom2) / new_total_mass,
                                  y_velocity = (y_mom1 + y_mom2) / new_total_mass,
                                  idnum = galaxy1.idnum)
        self.added_galaxies.append(new_galaxy)
        self.removed_galaxies.append(galaxy1)
        self.removed_galaxies.append(galaxy2)
        
    def check_for_mergers(self):
        for i, galaxy1 in enumerate(self.galaxies):
            for j, galaxy2 in enumerate(self.galaxies[i+1:]):
                if distance(galaxy1, galaxy2) <= 2:
                    self.merge_galaxies(galaxy1, galaxy2)
                
    def plot_state(self, frame_id):
        plt.clf()
        x = [galaxy.x_position for galaxy in self.galaxies]
        y = [galaxy.y_position for galaxy in self.galaxies]
        color = [galaxy.color for galaxy in self.galaxies]
        size = [galaxy.total_mass / 1e9 for galaxy in self.galaxies]
        plt.scatter(x,y, color=color, s=size)
        plt.xlim(uni.xrange)
        plt.ylim(uni.yrange)
        plt.savefig('frame%04i.png' % frame_id)
Explanation: Problem 3c
OK, so we have a MovingGalaxy that can move, but we need a Universe in which it can move!  
So let us create a new Universe class that can hold one or more galaxies.  We will give it x and y limits which define its borders to be 0 and 100, as well as an internal list to hold the galaxies themselves.  Make an evolve method for the Universe class that moves its time forward some timestep.  How does this affect the galaxies?  Are there boundaries to your Universe?  What happens when MovingGalaxy's run into each other?  Don't worry about gravity for now, but you could make them merge when they are close enough to each other, or just let them fly through each other.  After all, it's your Universe.
It may also be useful to make a method to add a random MovingGalaxy, so that you can add several to your Universe during initialization.  Start out only adding a few, as it might get expensive!
Make a command that plots the positions of the galaxies with time using matplotlib.scatter(), and save the corresponding outputs to a files called frame%04i.png.  The next steps allow you to make a movie of these frames.
End of explanation
uni = Universe()
n_timesteps = 2e2
n_galaxies = 25
for i in range(n_galaxies):
    uni.add_galaxy()
for i in range(int(n_timesteps)):
    uni.evolve(2e9/n_timesteps)
    uni.plot_state(i)
Explanation: Problem 3d
Initialize a Universe with several galaxies in it. Run your Universe forward in time, and plot the resulting state of the Universe over time to files called frame%04i.png.
End of explanation
%%bash
ffmpeg -r 20 -f image2 -i frame%04d.png -vcodec libx264 -pix_fmt yuv420p -crf 25 -y movie.mp4
%%HTML
<video width="1000" height="1000" controls>
  <source src="movie.mp4" type="video/mp4">
</video>
Explanation: Problem 3e
The command below will output a movie file called movie.mp4 by combining all of your frame%04i.png files and packaging them together at a framerate of 20 FPS.  It requires ffmpeg to be installed, but you can do this easily if you have homebrew installed by running brew install ffmpeg.  There are other means to download ffmpeg online.  Otherwise, you can flip through the frames manually to see how the Universe changes with time.  
If you were able to execute this step OK, execute to the next two fields and watch your movie!  Again, since there is a random component to these models, you can try running it a few times to see how the results change.
End of explanation
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