mattness_sec074

class ball: def __init__(self, m=1, r=np.array([0,0,0]), v=np.array([0,0,0]), a=np.array([0,0,0])): self.m = m self.r = r self.v = v self.a = a self.r_array = np.array([]) self.v_array = np.array([]) self.a_array = np.array([])

# Gravitational constant G = 6.674e-11 # N m^2 / kg^2 # Force function: inputs are ball_1 and ball_2 def force(ball_1, ball_2): ''' This is the force that ball_1 feels due to ball_2. This is dependent only on the position and mass of the balls, and the gravitational constant (G). The order in which we put the balls matters for calcualting the correct direction of the force. ''' output_force = np.zeros(3) # Initialze the returned force array with zeros r = ball_1.r - ball_2.r # Position vector from ball_2 to ball_1 # Quick check if r^2 is zero to avoid division by zero if np.sqrt(r.dot(r)) == 0: return output_force r_hat = r / np.sqrt(r.dot(r)) # Unit vector pointing from ball_2 to ball_1 ##### Begin Editing ##### # YOUR CODE HERE #note if you wish to use the magnitude of r^2, use r.dot(r) output_force = -(G * ball_2.m * ball_1.m * r_hat)/(r.dot(r)) ##### End Editing ##### return output_force

#Insert the initial velocity for the Earth ##### Begin Editing ##### v_initial_earth = 20000 # Your answer here ##### End Editing ##### # Physical constants: mass of the Sun, mass of the Earth, and average distance between the two M_Sun = 1.988e30 # kg M_Earth = 5.972e24 # kg R_Earth = 1.496e11 # m # Create instances of the ball class for the earth and sun earth = ball(m=M_Earth, r=np.array([R_Earth,0,0]), v=np.array([0, v_initial_earth, 0])) sun = ball(m=M_Sun)

##### Begin Editing ##### T = 30000000# Your answer here dt = 15000 # Your answer here ##### End Editing ##### times = np.arange(0, T+dt, dt) N = times.size earth.a_array = np.empty((N,3)) earth.a_array[0] = earth.a earth.v_array = np.empty((N,3)) earth.v_array[0] = earth.v earth.r_array = np.empty((N,3)) earth.r_array[0] = earth.r sun.a_array = np.empty((N,3)) sun.a_array[0] = sun.a sun.v_array = np.empty((N,3)) sun.v_array[0] = sun.v sun.r_array = np.empty((N,3)) sun.r_array[0] = sun.r

i = 1 for t in times[1:]: # first update the accelerations earth.a = force(earth, sun)/earth.m sun.a = np.zeros(3) # Here we assume no force on the sun (stays static) # then append the new value to the list earth.a_array[i] = earth.a sun.a_array[i] = sun.a # second update the velocities earth.v = earth.v + earth.a * dt sun.v = sun.v + sun.a * dt # then append the new value to the list earth.v_array[i] = earth.v sun.v_array[i] = sun.v # third update the positions earth.r = earth.r + earth.v * dt sun.r = sun.r + sun.v * dt # then append the new value to the list earth.r_array[i] = earth.r sun.r_array[i] = sun.r # update iteration count i = i + 1

plt.plot(earth.r_array[:,0], earth.r_array[:,1], '.') plt.plot(sun.r_array[:,0], sun.r_array[:,1], '.') plt.gca().set_aspect('equal') plt.show()

#Use this code cell to calculate the eccentricty of your orbit. #Here we use earth.r_array to determine the aphelion and perihelion distances of the orbit rMag = np.sqrt(earth.r_array[:,0]**2 + earth.r_array[:,1]**2) #turns earth.r_array to list of distances (magnitudes) A = max(rMag) #aphelion is maximum distance P = min(rMag) #perihelion is minimum distance ##### Begin Editing ##### #Code in the formula for eccentricity here in terms of A and P, then run the code cell to print out result ecc = (A-P)/(A+P) #Look at paragraph above for formula ##### End Editing ##### print('The eccentricity of our simulated orbit is {:.4f}.'.format(ecc))

The eccentricity of our simulated orbit is 0.5263.

def init(): line.set_data([], []) return(line) def animate_ball_list(ball_list): def animation_function(i): x = [] y = [] for j in range(0,len(ball_list)): x.append(ball_list[j].r_array[i, 0]) y.append(ball_list[j].r_array[i, 1]) line.set_data(x, y) return (line) return animation_function

fig3, axs3 = plt.subplots() axs3.set_xlim((np.min(earth.r_array[:,0]) - 0.1*np.abs(np.min(earth.r_array[:,0])), np.max(earth.r_array[:,0]) + 0.1*np.abs(np.max(earth.r_array[:,0])))) axs3.set_ylim((np.min(earth.r_array[:,1]) - 0.1*np.abs(np.min(earth.r_array[:,1])), np.max(earth.r_array[:,1]) + 0.1*np.abs(np.max(earth.r_array[:,1]))) ) axs3.set_aspect('equal') axs3.plot([-1,1], [0,0], color='C1', lw=2) line, = axs3.plot([], [], '.', markersize=24, lw=2) plt.close() anim_2 = animation.FuncAnimation(fig3, animate_ball_list([sun, earth]), init_func=init, frames=N, interval=40) anim_2

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rMag = np.sqrt(earth.r_array[:,0]**2 + earth.r_array[:,1]**2) rMaxIndex = np.argmax(rMag) rMinIndex = np.argmin(rMag) vMag = np.sqrt(earth.v_array[:,0]**2 + earth.v_array[:,1]**2) print('At its maximum distance, the speed of earth is {:.3f} m/s'.format(vMag[rMaxIndex])) print('At its minimum distance, the speed of earth is {:.3f} m/s'.format(vMag[rMinIndex]))