Orbit games
Contents
Orbit games#
We consider energy plots and orbital solutions in polar coordinates for the general potential energy
\(\begin{align} U(r) = k r^n \end{align}\)
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp
# Change the common font size
font_size = 14
plt.rcParams.update({'font.size': font_size})
class Orbit:
"""
Potentials and associated differential equations for central force motion
with the potential U(r) = k r^n.
"""
def __init__(self, ang_mom, n, k=1, mu=1):
self.ang_mom = ang_mom
self.n = n
self.k = k
self.mu = mu
def U(self, r):
"""Potential energy of the form U = kr^n."""
return self.k * r**self.n
def Ucf(self, r):
"""Centrifugal potential energy"""
return self.ang_mom**2 / (2. * self.mu * r**2)
def Ueff(self, r):
"""Effective potential energy"""
return self.U(r) + self.Ucf(r)
def U_deriv(self, r):
"""dU/dr"""
return self.n * self.k * r**(self.n - 1)
def Ucf_deriv(self, r):
"""dU_cf/dr"""
return -2. * self.ang_mom**2 / (2. * self.mu * r**3)
def Ueff_deriv(self, r):
"""dU_eff/dr"""
return self.U_deriv(r) + self.Ucf_deriv(r)
def dy_dt(self, t, y):
"""
This function returns the right-hand side of the diffeq:
[dr/dt d^2r/dt^2 dphi/dt]
Parameters
----------
t : float
time
y : float
3-component vector with y[0] = r(t), y[1] = dr/dt, y[2] = dphi/dt
"""
return [ y[1],
-1./self.mu * self.Ueff_deriv(y[0]),
self.ang_mom / (self.mu * y[0]**2) ]
def solve_ode(self, t_pts, r_0, r_dot_0, phi_0,
abserr=1.0e-8, relerr=1.0e-8):
"""
Solve the ODE given initial conditions.
For now use odeint, but we have the option to switch.
Specify smaller abserr and relerr to get more precision.
"""
y = [r_0, r_dot_0, phi_0]
solution = solve_ivp(self.dy_dt, (t_pts[0], t_pts[-1]),
y, t_eval=t_pts,
atol=abserr, rtol=relerr)
r, r_dot, phi = solution.y
return r, r_dot, phi
def energy(self, t_pts, r, r_dot):
"""Evaluate the energy as a function of time"""
return (self.mu/2.) * r_dot**2 + self.Ueff(r)
def start_stop_indices(t_pts, plot_start, plot_stop):
start_index = (np.fabs(t_pts-plot_start)).argmin() # index in t_pts array
stop_index = (np.fabs(t_pts-plot_stop)).argmin() # index in t_pts array
return start_index, stop_index
Gravity \(n = -1\)#
n = -1
k = -3.
ang_mom = 1.
o1 = Orbit(ang_mom, n=n, k=k, mu=1)
fig_2 = plt.figure(figsize=(7,5))
ax_2 = fig_2.add_subplot(1,1,1)
r_pts = np.linspace(0.001, 3., 200)
U_pts = o1.U(r_pts)
Ucf_pts = o1.Ucf(r_pts)
Ueff_pts = o1.Ueff(r_pts)
ax_2.plot(r_pts, U_pts, linestyle='dashed', color='blue', label='U(r)')
ax_2.plot(r_pts, Ucf_pts, linestyle='dotted', color='green', label='Ucf(r)')
ax_2.plot(r_pts, Ueff_pts, linestyle='solid', color='red', label='Ueff(r)')
ax_2.set_xlim(0., 3.)
ax_2.set_ylim(-5., 5.)
ax_2.set_xlabel('r')
ax_2.set_ylabel('U(r)')
ax_2.set_title(f'$n = {n},\ \ k = {k},\ \ l = {ang_mom}$')
ax_2.legend(loc='upper center')
ax_2.axhline(0., color='black', alpha=0.3)
fig_2.tight_layout()
fig_2.savefig('Gravitation_orbit_1.png')
Orbit (time dependence)#
We’ll directly solve the equations for r(t) and phi(t).
# Plotting time
t_start = 0.
t_end = 10.
delta_t = 0.01
t_pts = np.arange(t_start, t_end+delta_t, delta_t)
# Initial conditions
r_0 = 1.
r_dot_0 = 0.
phi_0 = 0.0
r_pts, r_dot_pts, phi_pts = o1.solve_ode(t_pts, r_0, r_dot_0, phi_0)
c = o1.ang_mom**2 / (np.abs(o1.k) * o1.mu)
epsilon = c / r_0 - 1.
energy_0 = o1.mu/2. * r_dot_0**2 + o1.Ueff(r_0)
print(f'energy = {energy_0:.2f}')
print(f'eccentricity = {epsilon:.2f}')
energy = -2.50
eccentricity = -0.67
fig_4 = plt.figure(figsize=(8,8))
overall_title = 'Gravitational orbit: ' + \
rf' $n = {o1.n},$' + \
rf' $k = {o1.k:.1f},$' + \
rf' $l = {o1.ang_mom:.1f},$' + \
rf' $r_0 = {r_0:.1f},$' + \
rf' $\dot r_0 = {r_dot_0:.2f},$' + \
rf' $\phi_0 = {phi_0:.2f}$' + \
'\n' # \n means a new line (adds some space here)
fig_4.suptitle(overall_title, va='baseline')
ax_4a = fig_4.add_subplot(2,2,1)
ax_4a.plot(t_pts, r_pts, color='black')
ax_4a.set_xlabel(r'$t$')
ax_4a.set_ylabel(r'$r$')
ax_4a.set_title('Time dependence of radius')
ax_4b = fig_4.add_subplot(2,2,2)
ax_4b.plot(t_pts, phi_pts/(2.*np.pi), color='black')
ax_4b.plot(t_pts, phi_pts/(2.*np.pi)%1, color='red')
ax_4b.set_xlabel(r'$t$')
ax_4b.set_ylabel(r'$\phi/2\pi$')
ax_4b.set_title(r'Time dependence of $\phi$')
ax_4c = fig_4.add_subplot(2,2,3)
ax_4c.plot(r_pts*np.cos(phi_pts), r_pts*np.sin(phi_pts), color='black')
ax_4c.set_xlabel(r'$x$')
ax_4c.set_ylabel(r'$y$')
ax_4c.set_aspect(1)
ax_4c.set_title('Cartesian plot')
ax_4d = fig_4.add_subplot(2,2,4, polar=True)
ax_4d.plot(phi_pts, r_pts, color='black')
ax_4d.set_title('Polar plot', pad=20.)
fig_4.tight_layout()
Energy versus time#
E_tot_pts = o1.energy(t_pts, r_pts, r_dot_pts)
E_tot_0 = E_tot_pts[0]
E_tot_rel_pts = np.abs((E_tot_pts - E_tot_0)/E_tot_0)
print(f' t E_tot rel. error')
for t, E_tot, E_tot_rel in zip(t_pts, E_tot_pts, E_tot_rel_pts):
print(f'{t:8.5f} {E_tot:8.5e} {E_tot_rel:12.4e}')
t E_tot rel. error
0.00000 -2.50000e+00 0.0000e+00
0.01000 -2.50000e+00 3.9790e-14
0.02000 -2.50000e+00 5.4987e-11
0.03000 -2.50000e+00 2.2135e-10
0.04000 -2.50000e+00 2.8173e-10
0.05000 -2.50000e+00 1.1803e-10
0.06000 -2.50000e+00 1.1083e-10
0.07000 -2.50000e+00 1.6178e-10
0.08000 -2.50000e+00 5.9066e-12
0.09000 -2.50000e+00 2.3637e-10
0.10000 -2.50000e+00 2.2980e-10
0.11000 -2.50000e+00 5.2911e-11
0.12000 -2.50000e+00 3.2821e-10
0.13000 -2.50000e+00 3.1704e-10
0.14000 -2.50000e+00 5.3087e-11
0.15000 -2.50000e+00 3.2847e-10
0.16000 -2.50000e+00 1.5683e-10
0.17000 -2.50000e+00 3.1309e-10
0.18000 -2.50000e+00 6.0824e-10
0.19000 -2.50000e+00 4.6531e-10
0.20000 -2.50000e+00 2.7574e-10
0.21000 -2.50000e+00 5.8478e-10
0.22000 -2.50000e+00 6.9095e-11
0.23000 -2.50000e+00 7.3493e-10
0.24000 -2.50000e+00 1.0180e-09
0.25000 -2.50000e+00 6.7889e-10
0.26000 -2.50000e+00 7.4282e-10
0.27000 -2.50000e+00 1.1964e-09
0.28000 -2.50000e+00 1.0926e-10
0.29000 -2.50000e+00 1.3503e-09
0.30000 -2.50000e+00 1.7085e-09
0.31000 -2.50000e+00 1.2202e-09
0.32000 -2.50000e+00 1.1001e-09
0.33000 -2.50000e+00 2.2014e-09
0.34000 -2.50000e+00 7.6467e-10
0.35000 -2.50000e+00 1.6446e-09
0.36000 -2.50000e+00 2.6799e-09
0.37000 -2.50000e+00 2.2847e-09
0.38000 -2.50000e+00 1.1544e-09
0.39000 -2.50000e+00 6.3315e-10
0.40000 -2.50000e+00 1.7217e-09
0.41000 -2.50000e+00 1.2599e-09
0.42000 -2.50000e+00 9.7891e-10
0.43000 -2.50000e+00 3.4940e-09
0.44000 -2.50000e+00 6.4154e-09
0.45000 -2.50000e+00 6.4049e-09
0.46000 -2.50000e+00 5.9422e-11
0.47000 -2.50000e+00 1.9396e-09
0.48000 -2.50000e+00 3.9348e-09
0.49000 -2.50000e+00 1.3715e-08
0.50000 -2.50000e+00 7.6172e-09
0.51000 -2.50000e+00 7.5057e-09
0.52000 -2.50000e+00 1.5841e-09
0.53000 -2.50000e+00 1.9661e-08
0.54000 -2.50000e+00 1.0743e-08
0.55000 -2.50000e+00 1.5008e-08
0.56000 -2.50000e+00 4.3178e-09
0.57000 -2.50000e+00 3.3344e-08
0.58000 -2.50000e+00 1.6230e-08
0.59000 -2.50000e+00 2.8139e-10
0.60000 -2.50000e+00 4.6405e-08
0.61000 -2.50000e+00 2.5538e-08
0.62000 -2.50000e+00 3.5538e-09
0.63000 -2.50000e+00 4.7292e-08
0.64000 -2.50000e+00 4.2732e-08
0.65000 -2.50000e+00 7.0327e-08
0.66000 -2.50000e+00 5.2806e-08
0.67000 -2.50000e+00 6.6673e-08
0.68000 -2.50000e+00 2.7369e-08
0.69000 -2.50000e+00 1.8511e-10
0.70000 -2.50000e+00 6.3437e-08
0.71000 -2.50000e+00 1.0228e-08
0.72000 -2.50000e+00 5.7222e-08
0.73000 -2.50000e+00 4.6488e-09
0.74000 -2.50000e+00 1.0201e-07
0.75000 -2.50000e+00 5.5432e-08
0.76000 -2.50000e+00 4.1167e-08
0.77000 -2.50000e+00 2.3906e-08
0.78000 -2.50000e+00 2.6859e-08
0.79000 -2.50000e+00 2.4018e-08
0.80000 -2.50000e+00 1.6807e-08
0.81000 -2.50000e+00 2.4489e-08
0.82000 -2.50000e+00 3.1727e-08
0.83000 -2.50000e+00 2.6533e-08
0.84000 -2.50000e+00 2.5021e-08
0.85000 -2.50000e+00 2.4827e-08
0.86000 -2.50000e+00 2.4990e-08
0.87000 -2.50000e+00 2.2530e-08
0.88000 -2.50000e+00 1.8485e-08
0.89000 -2.50000e+00 1.4529e-08
0.90000 -2.50000e+00 1.4165e-08
0.91000 -2.50000e+00 1.8196e-08
0.92000 -2.50000e+00 9.4591e-09
0.93000 -2.50000e+00 1.6221e-08
0.94000 -2.50000e+00 2.8593e-08
0.95000 -2.50000e+00 2.4612e-08
0.96000 -2.50000e+00 8.1724e-09
0.97000 -2.50000e+00 5.2331e-09
0.98000 -2.50000e+00 1.5965e-07
0.99000 -2.50000e+00 9.7842e-08
1.00000 -2.50000e+00 1.5077e-07
1.01000 -2.50000e+00 6.3371e-08
1.02000 -2.50000e+00 9.2148e-08
1.03000 -2.50000e+00 1.1760e-08
1.04000 -2.50000e+00 3.8406e-08
1.05000 -2.50000e+00 5.1112e-08
1.06000 -2.50000e+00 7.8091e-09
1.07000 -2.50000e+00 7.8441e-08
1.08000 -2.50000e+00 1.0661e-08
1.09000 -2.50000e+00 2.6395e-08
1.10000 -2.50000e+00 6.9259e-08
1.11000 -2.50000e+00 1.8172e-08
1.12000 -2.50000e+00 2.3060e-08
1.13000 -2.50000e+00 5.4972e-08
1.14000 -2.50000e+00 4.3295e-08
1.15000 -2.50000e+00 1.3891e-08
1.16000 -2.50000e+00 2.9865e-08
1.17000 -2.50000e+00 4.8707e-08
1.18000 -2.50000e+00 3.9716e-08
1.19000 -2.50000e+00 2.1374e-08
1.20000 -2.50000e+00 2.8416e-08
1.21000 -2.50000e+00 3.5244e-08
1.22000 -2.50000e+00 3.8388e-08
1.23000 -2.50000e+00 3.1432e-08
1.24000 -2.50000e+00 2.6794e-08
1.25000 -2.50000e+00 3.0628e-08
1.26000 -2.50000e+00 3.1624e-08
1.27000 -2.50000e+00 3.1109e-08
1.28000 -2.50000e+00 3.0274e-08
1.29000 -2.50000e+00 3.0224e-08
1.30000 -2.50000e+00 3.1105e-08
1.31000 -2.50000e+00 3.1598e-08
1.32000 -2.50000e+00 2.9588e-08
1.33000 -2.50000e+00 2.8757e-08
1.34000 -2.50000e+00 2.9968e-08
1.35000 -2.50000e+00 3.1428e-08
1.36000 -2.50000e+00 3.1666e-08
1.37000 -2.50000e+00 3.1202e-08
1.38000 -2.50000e+00 2.9435e-08
1.39000 -2.50000e+00 2.9077e-08
1.40000 -2.50000e+00 3.0222e-08
1.41000 -2.50000e+00 3.1305e-08
1.42000 -2.50000e+00 3.1346e-08
1.43000 -2.50000e+00 3.1013e-08
1.44000 -2.50000e+00 3.0006e-08
1.45000 -2.50000e+00 2.9737e-08
1.46000 -2.50000e+00 3.0309e-08
1.47000 -2.50000e+00 3.0871e-08
1.48000 -2.50000e+00 3.0913e-08
1.49000 -2.50000e+00 3.0759e-08
1.50000 -2.50000e+00 3.0347e-08
1.51000 -2.50000e+00 3.0184e-08
1.52000 -2.50000e+00 3.0392e-08
1.53000 -2.50000e+00 3.0570e-08
1.54000 -2.50000e+00 3.0530e-08
1.55000 -2.50000e+00 3.0443e-08
1.56000 -2.50000e+00 3.0335e-08
1.57000 -2.50000e+00 3.0314e-08
1.58000 -2.50000e+00 3.0423e-08
1.59000 -2.50000e+00 3.0425e-08
1.60000 -2.50000e+00 3.0274e-08
1.61000 -2.50000e+00 3.0160e-08
1.62000 -2.50000e+00 3.0138e-08
1.63000 -2.50000e+00 3.0212e-08
1.64000 -2.50000e+00 3.0352e-08
1.65000 -2.50000e+00 3.0330e-08
1.66000 -2.50000e+00 3.0116e-08
1.67000 -2.50000e+00 2.9924e-08
1.68000 -2.50000e+00 2.9902e-08
1.69000 -2.50000e+00 3.0020e-08
1.70000 -2.50000e+00 3.0211e-08
1.71000 -2.50000e+00 3.0217e-08
1.72000 -2.50000e+00 2.9977e-08
1.73000 -2.50000e+00 2.9718e-08
1.74000 -2.50000e+00 2.9680e-08
1.75000 -2.50000e+00 2.9859e-08
1.76000 -2.50000e+00 3.0107e-08
1.77000 -2.50000e+00 3.0113e-08
1.78000 -2.50000e+00 2.9818e-08
1.79000 -2.50000e+00 2.9498e-08
1.80000 -2.50000e+00 2.9462e-08
1.81000 -2.50000e+00 2.9805e-08
1.82000 -2.50000e+00 3.0166e-08
1.83000 -2.50000e+00 3.0116e-08
1.84000 -2.50000e+00 2.9657e-08
1.85000 -2.50000e+00 2.9217e-08
1.86000 -2.50000e+00 2.9153e-08
1.87000 -2.50000e+00 2.9797e-08
1.88000 -2.50000e+00 3.0416e-08
1.89000 -2.50000e+00 3.0252e-08
1.90000 -2.50000e+00 2.9440e-08
1.91000 -2.50000e+00 2.8764e-08
1.92000 -2.50000e+00 2.8719e-08
1.93000 -2.50000e+00 2.9682e-08
1.94000 -2.50000e+00 3.0977e-08
1.95000 -2.50000e+00 3.0806e-08
1.96000 -2.50000e+00 2.9335e-08
1.97000 -2.50000e+00 2.8027e-08
1.98000 -2.50000e+00 2.8011e-08
1.99000 -2.50000e+00 2.8931e-08
2.00000 -2.50000e+00 3.1367e-08
2.01000 -2.50000e+00 3.1871e-08
2.02000 -2.50000e+00 2.9921e-08
2.03000 -2.50000e+00 2.7504e-08
2.04000 -2.50000e+00 2.6852e-08
2.05000 -2.50000e+00 2.7363e-08
2.06000 -2.50000e+00 2.9128e-08
2.07000 -2.50000e+00 3.0888e-08
2.08000 -2.50000e+00 3.1161e-08
2.09000 -2.50000e+00 2.9821e-08
2.10000 -2.50000e+00 2.7572e-08
2.11000 -2.50000e+00 2.5998e-08
2.12000 -2.50000e+00 1.9258e-08
2.13000 -2.50000e+00 1.6918e-08
2.14000 -2.50000e+00 2.8844e-08
2.15000 -2.50000e+00 4.2448e-08
2.16000 -2.50000e+00 3.9541e-08
2.17000 -2.50000e+00 2.4564e-08
2.18000 -2.50000e+00 6.7603e-09
2.19000 -2.50000e+00 1.0100e-08
2.20000 -2.50000e+00 4.1866e-08
2.21000 -2.50000e+00 4.2528e-08
2.22000 -2.50000e+00 1.9076e-08
2.23000 -2.50000e+00 1.1391e-08
2.24000 -2.50000e+00 3.7500e-08
2.25000 -2.50000e+00 5.2163e-08
2.26000 -2.50000e+00 8.3625e-09
2.27000 -2.50000e+00 1.1108e-08
2.28000 -2.50000e+00 6.8170e-08
2.29000 -2.50000e+00 2.3860e-08
2.30000 -2.50000e+00 2.9712e-08
2.31000 -2.50000e+00 7.9435e-08
2.32000 -2.50000e+00 4.4674e-09
2.33000 -2.50000e+00 8.6412e-09
2.34000 -2.50000e+00 5.1929e-08
2.35000 -2.50000e+00 6.6709e-08
2.36000 -2.50000e+00 1.0055e-07
2.37000 -2.50000e+00 1.9473e-07
2.38000 -2.50000e+00 2.1942e-07
2.39000 -2.50000e+00 2.2803e-07
2.40000 -2.50000e+00 2.8652e-07
2.41000 -2.50000e+00 2.1659e-07
2.42000 -2.50000e+00 1.8433e-07
2.43000 -2.50000e+00 8.7791e-08
2.44000 -2.50000e+00 5.3790e-08
2.45000 -2.50000e+00 6.2019e-08
2.46000 -2.50000e+00 5.0856e-08
2.47000 -2.50000e+00 5.8734e-08
2.48000 -2.50000e+00 5.7665e-08
2.49000 -2.50000e+00 5.5052e-08
2.50000 -2.50000e+00 6.4470e-08
2.51000 -2.50000e+00 5.3802e-08
2.52000 -2.50000e+00 5.4336e-08
2.53000 -2.50000e+00 5.3307e-08
2.54000 -2.50000e+00 5.2329e-08
2.55000 -2.50000e+00 4.9606e-08
2.56000 -2.50000e+00 5.0846e-08
2.57000 -2.50000e+00 4.6301e-08
2.58000 -2.50000e+00 5.1003e-08
2.59000 -2.50000e+00 4.4954e-08
2.60000 -2.50000e+00 4.8803e-08
2.61000 -2.50000e+00 3.7095e-08
2.62000 -2.50000e+00 3.2212e-08
2.63000 -2.50000e+00 3.1163e-08
2.64000 -2.50000e+00 5.2749e-09
2.65000 -2.50000e+00 4.4275e-08
2.66000 -2.50000e+00 1.7910e-07
2.67000 -2.50000e+00 5.6989e-08
2.68000 -2.50000e+00 2.1647e-07
2.69000 -2.50000e+00 4.6915e-08
2.70000 -2.50000e+00 1.5710e-07
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fig_5 = plt.figure(figsize=(6,6))
overall_title = 'Gravitational orbit: ' + \
rf' $n = {o1.n},$' + \
rf' $k = {o1.k:.1f},$' + \
rf' $l = {o1.ang_mom:.1f},$' + \
rf' $r_0 = {r_0:.1f},$' + \
rf' $\dot r_0 = {r_dot_0:.2f},$' + \
rf' $\phi_0 = {phi_0:.2f}$' + \
'\n' # \n means a new line (adds some space here)
fig_5.suptitle(overall_title, va='baseline')
ax_5a = fig_5.add_subplot(1,1,1)
ax_5a.semilogy(t_pts, np.abs(E_tot_pts), color='black', label=r'$E(t)$')
ax_5a.semilogy(t_pts, E_tot_rel_pts, color='red', label=r'$\Delta E(t)$')
ax_5a.set_xlabel(r'$t$')
ax_5a.set_ylabel(r'Energy')
ax_5a.set_title('Change in energy with time')
ax_5a.legend(loc='lower right')
fig_5.tight_layout()
Verifying the virial theorem numerically#
First a quick calculation with the current orbit:
t_range = t_pts[-1] - t_pts[0]
U_pts = o1.U(r_pts)
T_pts = o1.energy(t_pts, r_pts, r_dot_pts) - U_pts
from scipy.integrate import simps
T_avg = simps(T_pts, t_pts, delta_t) / t_range
U_avg = simps(U_pts, t_pts, delta_t) / t_range
print(f'<T> = {T_avg:.6f}, <U> = {U_avg:.6f}, n<U>/2 = {o1.n*U_avg/2:.6f}')
<T> = 2.522990, <U> = -5.022990, n<U>/2 = 2.511495
Now a more complete study with different \(n\) values and varying integration times:
from scipy.integrate import simps
# For simplicity, set the angular momentum to 1 and k to +2 or -2 with n*k > 0
n_values = (-1, 2, 3, 4, 7) # different n values to test
t_end_values = (10., 100., 1000.) # integration max values to test
k_magnitude = 2.
# Time start and spacing
t_start = 0.
delta_t = 0.001
# Initial conditions
r_0 = 1.
r_dot_0 = 0.
phi_0 = 0.0
ang_mom = 1.
print(f' n t_end <T> <U> n<U>/2 |<T>-n<U>/2|/<T>')
for n in n_values:
k = np.sign(n) * k_magnitude # ensure that n*k > 0
o_now = Orbit(ang_mom, n=n, k=k, mu=1.) # current orbit
for t_end in t_end_values:
t_pts = np.arange(t_start, t_end+delta_t, delta_t)
r_pts, r_dot_pts, phi_pts = o_now.solve_ode(t_pts, r_0,
r_dot_0, phi_0)
t_range = t_pts[-1] - t_pts[0]
# Generate the integrands to be averaged
U_pts = o_now.U(r_pts)
T_pts = o_now.energy(t_pts, r_pts, r_dot_pts) - U_pts
# Calculate the time averages from 0 to t_end
T_avg = simps(T_pts, t_pts, delta_t) / t_range
U_avg = simps(U_pts, t_pts, delta_t) / t_range
rel_error = np.fabs((T_avg - n*U_avg/2.) / T_avg)
print(f'{n:2.0f} {t_end:5.0f} {T_avg:9.6f} {U_avg:9.6f} ',
f' {n*U_avg/2.:9.6f} {rel_error:9.3e}')
n t_end <T> <U> n<U>/2 |<T>-n<U>/2|/<T>
-1 10 1.468564 -2.968564 1.484282 1.070e-02
-1 100 1.494275 -2.994275 1.497137 1.915e-03
-1 1000 1.499960 -2.999956 1.499978 1.197e-05
2 10 1.236029 1.263971 1.263971 2.261e-02
2 100 1.251595 1.248404 1.248404 2.550e-03
2 1000 1.250122 1.249865 1.249865 2.050e-04
3 10 1.512491 0.987509 1.481264 2.065e-02
3 100 1.498214 1.001784 1.502676 2.978e-03
3 1000 1.500149 0.999830 1.499745 2.695e-04
4 10 1.665613 0.834386 1.668773 1.897e-03
4 100 1.668070 0.831928 1.663856 2.526e-03
4 1000 1.666828 0.833147 1.666295 3.196e-04
7 10 1.954318 0.545682 1.909887 2.273e-02
7 100 1.943101 0.556897 1.949141 3.109e-03
7 1000 1.944343 0.555637 1.944731 1.994e-04