def ComapareSolvers(h, t_final=1.):
P_E = [Planet(3e-6, [1, 0, 0], [0, 2 * pi, 0], "Earth")]
P_V = [Planet(3e-6, [1, 0, 0], [0, 2 * pi, 0], "Earth")]
initial_en_E = P_E[0].total_energy() # initial energy P_E
initial_en_V = P_V[0].total_energy() # initial energy P_V
initial_l_E = P_E[0].angular_momentum() # initial angular momentum P_E
initial_l_V = P_V[0].angular_momentum() # initial angular momentum P_E
l = np.linalg.norm(np.cross(P_E[0].r, P_E[0].v * P_E[0].mass))
#print("before", initial_l_E/P_V[0].mass)
time0 = time.time()
x_verlet, y_verlet = solve(P_V, "VelocityVerlet", h,
t_final) # solve with Velocity Verlet
time_verlet = time.time() - time0
time0 = time.time()
x_euler, y_euler = solve(P_E, "EulerCromer", h,
t_final) # solve with Euler Cromer
time_euler = time.time() - time0
diff_en_E = (initial_en_E - P_E[0].total_energy(
)) / initial_en_E #relative difference in total energy Euler Cormer
diff_en_V = (initial_en_V - P_V[0].total_energy(
)) / initial_en_V #relatevi difference in total energy Velocity Verlet
diff_l_E = (initial_l_E - P_E[0].angular_momentum(
)) / initial_l_E #relative difference in angular momentum Euler Cromer
diff_l_V = (initial_l_V - P_V[0].angular_momentum(
)) / initial_l_V #relative difference in angular momentum Velocity Verlet
#print("after",P_E[0].angular_momentum()/P_V[0].mass)
l = np.linalg.norm(np.cross(P_E[0].r, P_E[0].v * P_E[0].mass))
#print(P_E[0].r, P_E[0].v)
# plt.plot(x_euler[0],y_euler[0],label="EulerCromer")
# plt.plot(x_verlet[0],y_verlet[0],label="VelocityVerlet")
# plt.title("Solutions of the Sun-Earh system, h=%.2f" %h)
# plt.xlabel("x/AU")
# plt.ylabel("y/AU")
# plt.axis('equal')
# plt.legend()
# plt.show()
# plt.savefig('VV_vs_EC2',dpi=225)
return (diff_en_E, diff_en_V, diff_l_E, diff_l_V, time_euler, time_verlet)
def GravitationalConstant(power, h, t_final=1.):
for p in power:
Earth = [Planet(3e-6, [1, 0, 0], [0, 2 * pi, 0], "Earth")]
x, y = solve(Earth, "VelocityVerlet", h, t_final, p)
x, y = x[0], y[0]
plt.plot(x, y, label="beta = %.1f" % (p - 1))
plt.legend()
plt.axis('equal')
plt.xlabel("x/AU")
plt.ylabel("y/AU")
plt.title(
"The orbit of Earth with different expressions for gravitational force"
)
#plt.savefig("gravitationalForce")
plt.show()
def InitialVelocity(
velocities,
h,
t_final=3.
): # Input: a vector of initial x-velocities in multiples of pi, h: step length, t_final
for v in velocities:
Earth = [Planet(3e-6, [1, 0, 0], [0, v * pi, 0], "Earth")]
print(Earth[0].total_energy())
x, y = solve(Earth, "VelocityVerlet", h, t_final)
x, y = x[0], y[0]
plt.plot(x, y, label="v= %.2f pi" % v)
plt.legend()
plt.axis('equal')
plt.xlabel("x/AU")
plt.ylabel("y/AU")
plt.title(
"The orbit of Earth with different inital velocities. t_final = %d yr "
% t_final)
#plt.savefig('%d year' %t_final, dpi=225)
plt.show()
for j in range(N_p):
for k in range(N_plot): #samples N_plot points from the solution vectors x[j] and y[j]
X[k] = x[j,k*delta_t]
Y[k] = y[j,k*delta_t]
plt.plot(x[j],y[j],label=Planets[j].name)
plt.title("The solar system. t_final = 2" )
plt.xlabel("x/AU")
plt.ylabel("y/AU")
plt.axis('equal')
plt.legend()
plt.show()
#plt.savefig('solar_system',dpi=225)
# Initial values of the planets from 2017-Oct-04:
Mercury = Planet(1.65e-7, [-0.3789210,2.56223e-02,0],[-7.32704e-03*days, -2.68813e-02*days,0], "Mercury")
MercuryNewtonian = Planet(1.65e-7, [-0.3789210,2.56223e-02,0],[-7.32704e-03*days, -2.68813e-02*days,0], "MercuryNewtonian")
Venus = Planet(2.45e-6, [-.483111, .534117,0],[-1.49738e-02*days, -1.37846e-02*days,0 ], "Venus")
Earth = Planet(3e-6,[.985157,.191737e-01,0],[-3.48636e-3*days,1.68385e-02*days,0],"Earth")
Mars = Planet(3.3e-7, [-1.49970, 0.724618,0], [-5.52182e-03*days, -1.14214e-02*days,0],"Mars")
Jupiter = Planet(9.5e-4,[-4.63556,-2.8425,0],[3.85608e-03*days,-6.075643e-03*days,0],"Jupiter")
Saturn = Planet(2.75e-4,[-0.421224,-10.0462,0],[5.26820e-03*days,-2.52167e-04*days,0], "Saturn")
Uranus = Planet(4.4e-5,[17.8825, 8.76632,0],[-1.75983e-03*days,3.34820e-03*days],"Uranus")
Neptune = Planet(0.515e-4, [28.6020, -8.86057, 0],[9.08471e-04*days, 3.01748e-03*days,0],"Neptune")
# A list of all the planets
Planets = [Earth, Jupiter, Mercury, Venus, Mars, Saturn, Uranus, Neptune]
x,y = solve(Planets, "VelocityVerlet", 1/100, 2)
plot(Planets, x,y,10)
h=1/(360*60*60*8)
from project3b import Planet # attributes: mass, r, v, a_s, a, name, total_energy(), angular_momentum()
from project3b import Acceleration # attributes: N_p, Update(Planets, beta = 3.)
from project3b import DiffEqSolver # attributes: h, EulerCromer(Planets), VelcityVerlet(Planets)
from project3b import solve # input: (Planets, method, h, t_final=1., beta = 3.)
# Initial values of the planets from 2017-Oct-04:
Mercury = Planet(1.65e-7, [-0.3789210,2.56223e-02,0],[-7.32704e-03*days, -2.68813e-02*days,0], "Mercury")
MercuryNewtonian = Planet(1.65e-7, [-0.3789210,2.56223e-02,0],[-7.32704e-03*days, -2.68813e-02*days,0], "MercuryNewtonian")
# A list of all the planets
OnlyMercury = [Mercury]
OnlyMercury_Newtonian = [MercuryNewtonian]
x_Newtonian, y_Newtonian = solve(OnlyMercury_Newtonian, "VelocityVerlet",h, 100) #The newtonian solution of the Mercury-Sun system
#plt.plot(x_Newtonian[0], y_Newtonian[0])
x_Relatvistic, y_Relativistic = solve(OnlyMercury, "VelocityVerlet",h, 100) #The relativistic sloution of the Mercury-Sun system
for i in range(len(x_Newtonian[0])):#
if (x_Newtonian[0,i]**2+y_Newtonian[0,i]**2 - 0.3075**2)<1.e-7: # search for the perihelion (perihelion distance : 0.3075 AU)
print(np.arctan(y_Newtonian[0,i]/x_Newtonian[0,i]), np.arctan(y_Relativistic[0,i]/x_Relatvistic[0,i])) # calculate perihelion angle
plt.plot(x_Relatvistic[0], y_Relativistic[0])
plt.axis('equal')
plt.title("Mercury")
plt.xlabel("x/AU")
plt.ylabel("y/AU")
plt.show()