def doit():
# planet data
ecc_A = 0.0 # eccentricity of planet A
ecc_B = 0.4 # eccecntricity of planet B
a_A = 1.0 # semi-major axis of planet A
a_B = 4.0**(1. / 3.) # semi-major axis of planet B
# integration data
nsteps_year = 365 # number of steps per year
nyears = 4 # total integration time (years)
s = solar_system_integrator.SolarSystem()
s.add_planet(a_A, ecc_A, loc="perihelion")
s.add_planet(a_B, ecc_B, loc="aphelion")
sol = s.integrate(nsteps_year, nyears)
# plotting
for n in range(len(sol[0].x)):
pylab.clf()
# plot the foci
pylab.scatter([0], [0], s=250, marker=(5, 1), color="k")
pylab.scatter([0], [0], s=200, marker=(5, 1), color="y")
# plot planet A
pylab.plot(sol[0].x, sol[0].y, color="r")
pylab.scatter([sol[0].x[n]], [sol[0].y[n]], s=100, color="r")
# plot planet B
pylab.plot(sol[1].x, sol[1].y, color="b")
pylab.scatter([sol[1].x[n]], [sol[1].y[n]], s=100, color="b")
pylab.axis([-2.5, 1.5, -1.8, 1.8])
pylab.axis("off")
ax = pylab.gca()
ax.set_aspect("equal", "datalim")
f = pylab.gcf()
f.set_size_inches(9.6, 7.2)
pylab.text(0.05,
0.05,
"time = %6.3f yr" % sol[0].t[n],
transform=f.transFigure)
pylab.text(0.05,
0.9,
"a = %6.3f, e = %5.2f" % (a_A, ecc_A),
color="r",
transform=f.transFigure)
pylab.text(0.05,
0.85,
"a = %6.3f, e = %5.2f" % (a_B, ecc_B),
color="b",
transform=f.transFigure)
pylab.savefig("orbit_%04d.png" % n)
def doit():
# planet data
ecc_E = 0.016710 # eccentricity of planet Earth
ecc_M = 0.093315 # eccecntricity of planet Mars
a_E = 1.0 # semi-major axis of planet Earth
a_M = 1.523679 # semi-major axis of planet Mars
# integration data
nsteps_year = 365 * 2 # number of steps per year
nyears = 0.6 # total integration time (years)
s = solar_system_integrator.SolarSystem()
# Earth initialization
# set the initial conditions. The initial position is perihelion
#y[0] = a_E*(1.0 - ecc_E) # x
#y[1] = 0.0 # y
# at perihelion, all the veloicity is in the y-direction.
#y[2] = 0.0 # v_x
# v_y^2 = (GM/a) (1+e)/(1-e) (see C&O Eq. 2.33 for example)
# This is the perihelion velocity. For a = 1, e = 0, v = 2 pi
#y[3] = -math.sqrt( (G*M_sun/a_E) * (1.0 + ecc_E) / (1.0 - ecc_E))
# Mars initialization
# set the initial conditions. The initial position is perihelion
#y[4] = a_M*(1.0 - ecc_M) # x
#y[5] = 0.0 # y
# at perihelion, all the veloicity is in the y-direction.
#y[6] = 0.0 # v_x
# v_y^2 = (GM/a) (1+e)/(1-e) (see C&O Eq. 2.33 for example)
# This is the perihelion velocity. For a = 1, e = 0, v = 2 pi
#y[7] = -math.sqrt( (G*M_sun/a_M) * (1.0 + ecc_M) / (1.0 - ecc_M))
# These initial conditions were found by putting Mars and Earth
# both at their perihelion, but with their velocities in the
# opposite direction (i.e. we want them to go backwards). This
# configuration is opposition, and is setup by using the commented
# out initial conditions above. We then integrated backwards for 1/4
# year (91 steps) to get these starting coordinates (note, we reverse
# the direction of the velocity to get it orbitting in the forward
# direction.)
# Earth
x = -0.04631900088483218
y = -0.9994219951994862
vx = 6.277324691390798
vy = -0.185920887199495
vp0 = solar_system_integrator.PlanetPosVel(x, y, vx, vy)
s.add_planet(a_E, ecc_E, loc="specify", pos_vel=vp0)
# Mars
x = 0.7856599524256417
y = -1.203323492875661
vx = 4.280834571016523
vy = 3.272064392180777
vp0 = solar_system_integrator.PlanetPosVel(x, y, vx, vy)
s.add_planet(a_M, ecc_M, loc="specify", pos_vel=vp0)
# integrate
sol = s.integrate(nsteps_year, nyears)
# some background stars
N = 10
xpos = []
ypos = []
for s in range(N):
xpos.append(random.uniform(3.0, 3.9))
ypos.append(random.uniform(-1.4, 1.9))
if both_sides:
xpos_left = []
ypos_left = []
for s in range(N):
xpos_left.append(random.uniform(-2.4, -1.4))
ypos_left.append(random.uniform(-1.4, 1.9))
# ================================================================
# plotting
# ================================================================
for n in range(len(sol[0].x)):
f = plt.figure()
# plot the foci
plt.scatter([0], [0], s=1600, marker=(20, 1), color="k")
plt.scatter([0], [0], s=1500, marker=(20, 1), color="#FFFF00")
# plot Earth
plt.plot(sol[0].x, sol[0].y, color="b")
plt.scatter([sol[0].x[n]], [sol[0].y[n]], s=100, color="b")
# plot Mars
plt.plot(sol[1].x, sol[1].y, color="r")
plt.scatter([sol[1].x[n]], [sol[1].y[n]], s=100, color="r")
# draw a line connecting Earth and Mars and extending a bit
# further out
slope = (sol[1].y[n] - sol[0].y[n]) / (sol[1].x[n] - sol[0].x[n])
xpt = 3.5
ypt = sol[0].y[n] + slope * (xpt - sol[0].x[n])
plt.plot([sol[0].x[n], xpt], [sol[0].y[n], ypt], "b--")
# draw some random background stars
for s in range(N):
plt.scatter([xpos[s]], [ypos[s]], s=200, marker=(5, 1), color="c")
if both_sides:
xpt = -2.5
ypt = sol[0].y[n] + slope * (xpt - sol[0].x[n])
plt.plot([sol[0].x[n], xpt], [sol[0].y[n], ypt], "b--")
for s in range(N):
plt.scatter([xpos_left[s]], [ypos_left[s]],
s=200,
marker=(5, 1),
color="c")
plt.axis([-2.5, 4.0, -1.5, 2.0])
plt.axis("off")
ax = plt.gca()
ax.set_aspect("equal", "datalim")
f.set_size_inches(12.8, 7.2)
plt.xlabel("AU")
plt.ylabel("AU")
plt.text(-1.5, -1.5, "time = %6.3f yr" % sol[0].t[n])
plt.title("Retrograde Mars")
plt.savefig("retrograde_%04d.png" % n)
plt.close()
def sidereal_solar():
# set the semi-major axis and eccentricity
a = 1.0
e = 0.0
ss = ssi.SolarSystem()
ss.add_planet(a, e, loc="perihelion")
# compute the period of the orbit from Kepler's law and ...
P_orbital = ss.period(0)
# ... set the rotation period
P_rotation = 0.5 * P_orbital
omega = 2 * math.pi / P_rotation
# integrate
nsteps_per_year = 720
num_years = 2
sol = ss.integrate(nsteps_per_year, num_years)
# ================================================================
# plotting
# ================================================================
for n in range(len(sol[0].t)):
plt.clf()
# plot the foci
plt.scatter([0], [0], s=250, marker=(5, 1), color="k")
plt.scatter([0], [0], s=200, marker=(5, 1), color="y")
# plot the orbit
plt.plot(sol[0].x, sol[0].y, color="0.5", ls="--")
# plot planet
theta = np.radians(np.arange(360))
r = 0.05 # exaggerate the planet's size
x_surface = sol[0].x[n] + r * np.cos(theta)
y_surface = sol[0].y[n] + r * np.sin(theta)
plt.fill(x_surface, y_surface, "c", edgecolor="c", zorder=100)
# plot a point on the planet's surface
xpt = sol[0].x[n] + r * np.cos(omega * sol[0].t[n] + math.pi / 2.0)
ypt = sol[0].y[n] + r * np.sin(omega * sol[0].t[n] + math.pi / 2.0)
plt.scatter([xpt], [ypt], s=25, color="k", zorder=100)
plt.axis([-1.2, 1.2, -1.2, 1.2])
ax = plt.gca()
ax.set_aspect("equal", "datalim")
plt.subplots_adjust(left=0.05, right=0.98, bottom=0.05, top=0.98)
f = plt.gcf()
f.set_size_inches(7.2, 7.2)
plt.axis("off")
plt.savefig("sidereal_solar_{:04d}.png".format(n))
def doit():
s = solar_system_integrator.SolarSystem()
# set the semi-major axis and eccentricity
a = 1.5874
e = 0.5
# set the initial coordinates -- perihelion
s.add_planet(a, e, loc="perihelion")
# compute the period of the orbit from Kepler's law and make
# the timestep by 1/720th of a period
P = math.sqrt(a**3)
# put 720 steps per period (the input is steps per year)
sol = s.integrate(720 / P, P)
# set up some intervals (start time, end time) to shade.
# Note, the length of the intervals should be the same
intervals = []
intervals.append(interval(0, P / 12))
intervals.append(interval(P / 2, 7.0 * P / 12.0))
intervals.append(interval(9.5 * P / 12, 10.5 * P / 12.0))
for i in range(len(intervals)):
if intervals[i].start < 0 or intervals[i].end > P:
print "ERROR: interval {} not contained in a single orbit".format(
i)
print "interval {}, dt = {}".format(
i, intervals[i].end - intervals[i].start)
# plot the orbit
iframe = 0
for n in range(len(sol[0].x)):
pylab.clf()
pylab.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9)
ax = pylab.gca()
ax.set_aspect("equal", "datalim")
pylab.axis("off")
# plot the foci
pylab.scatter([0], [0], s=350, marker=(5, 1), color="k", zorder=10)
pylab.scatter([0], [0], s=300, marker=(5, 1), color="y", zorder=10)
# plot planet
pylab.plot(sol[0].x, sol[0].y, color="r")
pylab.scatter([sol[0].x[n]], [sol[0].y[n]],
s=100,
color="r",
zorder=10)
# shade the intervals
for i in range(len(intervals)):
# one vertex is the Sun
x = [0]
y = [0]
for m in range(len(sol[0].x)):
if sol[0].t[m] >= intervals[i].start and \
sol[0].t[m] <= min(intervals[i].end,sol[0].t[n]):
x.append(sol[0].x[m])
y.append(sol[0].y[m])
elif (sol[0].t[m] > intervals[i].end):
break
pylab.fill(x, y, alpha=0.20, facecolor="b")
xmin = -a * (1.0 + e)
xmax = a * (1.0 - e)
dx = xmax - xmin
pylab.axis([xmin - 0.1 * dx, xmax + 0.1 * dx, -1.5, 1.5])
f = pylab.gcf()
f.set_size_inches(9.6, 7.2)
pylab.savefig("second_law_%04d.png" % n)
def orbitalenergy():
# set the semi-major axis and eccentricity
a = 1.5874 * AU
e = 0.4
orbit = ss.SolarSystem(GM=G * M_sun, year=year)
orbit.add_planet(a, e, loc="perihelion")
# compute the period of the orbit from Kepler's law and make
# the timestep by 1/720th of a period
P = orbit.period(0)
print "period = ", P / year
sol = orbit.integrate(360, P / year)
# ================================================================
# plotting
# ================================================================
# plot the orbit
iframe = 0
# v1
for n in range(len(sol[0].t)):
plt.clf()
# plot the foci
plt.scatter([0], [0], s=250, marker=(5, 1), color="k")
plt.scatter([0], [0], s=200, marker=(5, 1), color="y")
# plot planet
plt.plot(sol[0].x, sol[0].y, color="r")
plt.scatter([sol[0].x[n]], [sol[0].y[n]], s=100, color="r")
# compute the kinetic energy / kg
KE = 0.5 * (sol[0].vx[n]**2 + sol[0].vy[n]**2)
# compute the potential energy / kg
r = math.sqrt(sol[0].x[n]**2 + sol[0].y[n]**2)
PE = -G * M_sun / r
plt.axis([-4 * AU, 2 * AU, -4 * AU, 2 * AU])
f = plt.gcf()
f.set_size_inches(7.2, 7.2)
ax = plt.gca()
ax.set_aspect("equal", "datalim")
plt.title("Orbital Energy")
plt.text(-3.5 * AU, -3 * AU, "KE / unit mass (J/kg): %10.5e" % (KE))
plt.text(-3.5 * AU, -3.3 * AU, "PE / unit mass (J/kg): %10.5e" % (PE))
plt.text(-3.5 * AU, -3.6 * AU,
"total energy / unit mass (J/kg): %10.5e" % (PE + KE))
plt.xlabel("x [m]")
plt.ylabel("y [m]")
ax.xaxis.set_major_formatter(plt.ScalarFormatter(useMathText=True))
ax.yaxis.set_major_formatter(plt.ScalarFormatter(useMathText=True))
plt.savefig("orbitalenergy_%04d.png" % n)
# we work in units of AU, yr, and M_sun # in these units, G = 4 pi^2 # planet data ecc_A = 0.0 # eccentricity of planet A ecc_B = 0.4 # eccecntricity of planet B a_A = 1.0 # semi-major axis of planet A # we want the perihelion distance of planet B to be the same # as planet a r_p_A = a_A * (1.0 - ecc_A) a_B = r_p_A / (1.0 - ecc_B) ss = ssi.SolarSystem() ss.add_planet(a_A, ecc_A, loc="perihelion") ss.add_planet(a_B, ecc_B, loc="perihelion") P_B = ss.period(1) # integration data nsteps_year = 365 # number of steps per year sol = ss.integrate(nsteps_year, P_B) sol_A = sol[0] sol_B = sol[1] nsteps = len(sol_A.x)
def seasons():
# set the semi-major axis and eccentricity
a = 1.0 # AU
e = 0.0
s = ssi.SolarSystem()
s.add_planet(a, e, loc="perihelion")
P = s.period(0)
nsteps_per_year = 360
num_years = 1
sol = s.integrate(nsteps_per_year, num_years)
# apply a projection to account for the inclination
# wrt the observer
inc = 80
sol[0].y = sol[0].y * np.cos(np.radians(inc))
# ================================================================
# plotting
# ================================================================
# plot the orbit
for n in range(len(sol[0].t)):
plt.clf()
# plot the Sun at the foci
plt.scatter([0], [0], s=2600, marker=(20, 1), color="k", zorder=0)
plt.scatter([0], [0],
s=2500,
marker=(20, 1),
color="#FFFF00",
zorder=0)
# plot the orbit
plt.plot(sol[0].x, sol[0].y, color="0.5", ls="--", zorder=-50)
# plot planet -- hide it with zorder
if sol[0].y[n] > 0:
z = -20
else:
z = 20
theta = np.radians(np.arange(360))
r = 0.075 # exaggerate the planet's size
x_surface = sol[0].x[n] + r * np.cos(theta)
y_surface = sol[0].y[n] + r * np.sin(theta)
plt.fill(x_surface, y_surface, "b", edgecolor="b", zorder=z)
# axis
tilt = np.radians(23.5)
L = 0.1
x = [-L * np.sin(tilt), L * np.sin(tilt)]
y = [-L * np.cos(tilt), L * np.cos(tilt)]
plt.plot(sol[0].x[n] + x, sol[0].y[n] + y, color="k", lw=2, zorder=z)
# equator
y = [r * np.sin(tilt), -r * np.sin(tilt)]
x = [-r * np.cos(tilt), r * np.cos(tilt)]
plt.plot(sol[0].x[n] + x,
sol[0].y[n] + y,
color="k",
lw=1,
ls="--",
zorder=z)
plt.axis([-1.2, 1.2, -0.9, 0.9])
f = plt.gcf()
f.set_size_inches(9.6, 7.2)
plt.title("Seasons")
plt.axis("off")
ax = plt.gca()
ax.set_aspect("equal", "datalim")
plt.savefig("seasons_{:04d}".format(n))
def orbitalenergy():
# set the semi-major axis and eccentricity
a = 0.387
e = 0.2056
ss = ssi.SolarSystem()
ss.add_planet(a, e, loc="perihelion")
P_orbital = ss.period(0)
# set the rotation period
P_rotation = (2. / 3.) * P_orbital
omega = 2 * math.pi / P_rotation
print "orbital period = ", P_orbital
nsteps_per_year = int(360 / P_orbital)
nyears = 4 * P_orbital
sol = ss.integrate(nsteps_per_year, nyears)
# ================================================================
# plotting
# ================================================================
for n in range(len(sol[0].t)):
plt.clf()
# plot the foci
plt.scatter([0], [0], s=1600, marker=(20, 1), color="k")
plt.scatter([0], [0], s=1500, marker=(20, 1), color="#FFFF00")
# plot the orbit
plt.plot(sol[0].x, sol[0].y, color="0.5")
# plot planet
theta = np.radians(np.arange(360))
r = 0.03 # exaggerate the planet's size
x_surface = sol[0].x[n] + r * np.cos(theta)
y_surface = sol[0].y[n] + r * np.sin(theta)
plt.fill(x_surface, y_surface, color="r", alpha=1.0, zorder=1000)
# plot a point on the planet's surface
xpt = sol[0].x[n] + r * np.cos(omega * sol[0].t[n] + math.pi)
ypt = sol[0].y[n] + r * np.sin(omega * sol[0].t[n] + math.pi)
plt.scatter([xpt], [ypt], s=25, color="k")
plt.axis([-0.5, 0.5, -0.5, 0.5])
plt.axis("off")
ax = plt.gca()
ax.set_aspect("equal", "datalim")
plt.subplots_adjust(left=0.05, right=0.98, bottom=0.05, top=0.98)
f = plt.gcf()
plt.text(0.5,
0.925,
r"Mercury: $P_\mathrm{rotation} = (2/3) P_\mathrm{orbital}$",
transform=f.transFigure,
horizontalalignment="center")
f.set_size_inches(7.2, 7.2)
plt.savefig("mercury_rotation_{:04d}.png".format(n))
def sidereal():
# set the semi-major axis and eccentricity
a = 1.0
e = 0.0
ss = ssi.SolarSystem()
x = 0.0
y = -a*(1.0 + e)
vx = math.sqrt((ss.GM/a)*(1.0+e)/(1.0-e))
vy = 0.0
pos = ssi.PlanetPosVel(x, y, vx, vy)
ss.add_planet(a, e, loc="specify", pos_vel=pos)
# compute the period of the orbit from Kepler's law
P_orbital = ss.period(0)
# set the rotation period -- this is the sidereal day
#P_rotation = 0.99726968*day # Earth
P_rotation = 0.05*P_orbital
omega = 2*math.pi/P_rotation
# compute the length of the solar day
P_solar = P_rotation/(1.0 - P_rotation/P_orbital)
print "period = ", P_orbital
print "sidereal day = ", P_rotation
print "solar day = ", P_solar
# We will maintain two trajectories: full_orbit is the full orbit
# of the planet. orbit is just a small segment carrying 2
# sidereal days
# evolve for 2 sidereral days with lots of frames to get a smooth
# animation
tmax = 2.0*P_rotation
nsteps = 1000
# fraction of a year we are doing
f = tmax/P_orbital
orbit = ss.integrate(nsteps/f, f)
dt = orbit[0].t[1] - orbit[0].t[0]
# evolve the full orbit for plotting purposes
nsteps_per_year = 720.0
full_orbit = ss.integrate(nsteps_per_year, 1.0)
# ================================================================
# plotting
# ================================================================
# plot the orbit
iframe = 0
for n in range(len(orbit[0].t)):
plt.clf()
# plot the foci
plt.scatter([0], [0], s=1600, marker=(20,1), color="k")
plt.scatter([0], [0], s=1500, marker=(20,1), color="#FFFF00")
# plot the orbit
plt.plot(full_orbit[0].x, full_orbit[0].y, color="0.5")
# plot planet
theta = np.radians(np.arange(360))
r = 0.075 # exaggerate the planet's size
x_surface = orbit[0].x[n] + r*np.cos(theta)
y_surface = orbit[0].y[n] + r*np.sin(theta)
plt.fill(x_surface, y_surface, "b", edgecolor="b", zorder=1000)
# plot a point on the planet's surface
xpt = orbit[0].x[n] + 1.2*r*np.cos(omega*orbit[0].t[n]+math.pi/2.0)
ypt = orbit[0].y[n] + 1.2*r*np.sin(omega*orbit[0].t[n]+math.pi/2.0)
plt.scatter([xpt], [ypt], s=15, color="k")
plt.axis([-1.2, 1.2, -1.2, 1.2])
f = plt.gcf()
f.set_size_inches(7.2, 7.2)
plt.title("Sidereal vs. Solar Day")
plt.axis("off")
ax = plt.gca()
ax.set_aspect("equal", "datalim")
plt.text(-1.0, -1.2, "Note: length of day exaggerated",
fontsize=9, color="0.5")
if (orbit[0].t[n] >= P_rotation - 0.5*dt and
orbit[0].t[n] <= P_rotation + 0.5*dt):
n_sidereal = n
# special lines
if (orbit[0].t[n] >= P_rotation - 0.5*dt and
orbit[0].t[n] <= P_solar + 0.5*dt):
plt.plot([orbit[0].x[n_sidereal], orbit[0].x[n_sidereal]],
[orbit[0].y[n_sidereal], 0], linestyle=":", color="0.5")
# special notes
if (orbit[0].t[n] < 1.5*dt):
plt.text(-1.0, -1.15, "Noon (Sun is on the meridian)")
for d in range(150):
plt.savefig("earth_{:04d}.png".format(iframe))
iframe += 1
elif (orbit[0].t[n] >= P_rotation - 0.5*dt and
orbit[0].t[n] <= P_rotation + 0.5*dt):
plt.text(-1.0, -1.15, "1 Sidereal period (Earth rotated 360 degrees)")
for d in range(150):
plt.savefig("earth_{:04d}.png".format(iframe))
iframe += 1
elif (orbit[0].t[n] >= P_solar - 0.5*dt and
orbit[0].t[n] <= P_solar + 0.5*dt):
plt.text(-1.0, -1.15, "1 Solar period (Sun is back on the meridian)")
for d in range(150):
plt.savefig("earth_{:04d}.png".format(iframe))
iframe += 1
plt.savefig("earth_{:04d}.png".format(iframe))
iframe += 1
def asteroids():
ss = ssi.SolarSystem()
# Jupiter
P_jupiter = 12 # years
e = 0.0
ss.add_planet_by_period(P_jupiter, e, loc="perihelion")
# an asteroid, with some eccentricity but 1/2 the period
e = 0.3
ss.add_planet_by_period(0.5*P_jupiter, e, loc="perihelion")
# add some background asteroids with random orbits
a_min = 2.0
a_max = 3.5
n_asteroids = 50
for n in range(n_asteroids):
a = random.uniform(a_min, a_max)
e = random.uniform(0.0, 0.4)
if n % 2 == 0:
loc="perihelion"
else:
loc="aphelion"
ss.add_planet(a, e, loc=loc, rot="random")
# integrate
nsteps_per_year = 45.0
sol = ss.integrate(nsteps_per_year, 2*P_jupiter)
# plots
iframe = 0
for n in range(len(sol[0].t)):
plt.clf()
# plot the Sun
plt.scatter([0], [0], s=1600, marker=(20,1), color="k")
plt.scatter([0], [0], s=1500, marker=(20,1), color="#FFFF00")
# plot Jupiter
plt.plot(sol[0].x, sol[0].y, color="k")
plt.scatter([sol[0].x[n]], [sol[0].y[n]], s=500, color="k")
# plot our asteroid
plt.plot(sol[1].x, sol[1].y, color="r")
plt.scatter([sol[1].x[n]], [sol[1].y[n]], s=50, color="r")
# and the background asteroids
for k in range(2, 2+n_asteroids):
plt.plot(sol[k].x, sol[k].y, color="0.85", zorder=-100)
plt.scatter([sol[k].x[n]], [sol[k].y[n]], s=25, color="0.5", zorder=-100)
f = plt.gcf()
f.set_size_inches(7.2, 7.2)
plt.axis("off")
ax = plt.gca()
ax.set_aspect("equal", "datalim")
# if jupiter is at perihelion, the pause and annotate
perihelion = False
if n == 0:
perihelion = True
elif n > 0 and n < len(sol[0].t)-1:
if sol[0].y[n]*sol[0].y[n+1] < 0.0 and sol[0].x[n] > 0.0:
perihelion = True
if perihelion:
plt.text(0.5, 0.96,
"Jupiter and our asteroid are at their closest point",
horizontalalignment="center", transform=f.transFigure,
fontsize="large")
plt.text(0.5, 0.92,
"The gravitational force on the asteroid is always",
horizontalalignment="center", transform=f.transFigure)
plt.text(0.5, 0.89,
"strongest and in the same direction here",
horizontalalignment="center", transform=f.transFigure)
plt.arrow(sol[1].x[n], sol[1].y[n], 1.0, 0.0, color="r",
length_includes_head=True,
head_width = 0.2, width=0.05, overhang=-0.1)
for k in range(150):
plt.savefig("asteroids_{:04d}.png".format(iframe))
iframe += 1
plt.savefig("asteroids_{:04d}.png".format(iframe))
iframe += 1