def radial_velocity():
# set the masses
M_star1 = 4.0 * M_sun # star 1's mass
M_star2 = M_sun # star 2's mass
# set the semi-major axis of the star 2 (and derive that of star 1)
# M_star2 a_star2 = -M_star1 a_star1 (center of mass)
a_star2 = 10.0 * AU
a_star1 = (M_star2 / M_star1) * a_star2
# set the eccentricity
ecc = 0.4
# set the angle to rotate the semi-major axis wrt the observer
theta = np.pi / 6.0
# display additional information
annotate = True
# create the binary object
b = bi.Binary(M_star1,
M_star2,
a_star1 + a_star2,
ecc,
theta,
annotate=annotate)
# set the timestep in terms of the orbital period
dt = b.P / 360.0
tmax = 2.0 * b.P # maximum integration time
b.integrate(dt, tmax)
s1 = b.orbit1
s2 = b.orbit2
# ================================================================
# plotting
# ================================================================
iframe = 0
xmin = 1.05 * (s2.x - s1.x).min()
xmax = 1.05 * (s2.x - s1.x).max()
ymin = 1.05 * (s2.y - s1.y).min()
ymax = 1.05 * (s2.y - s1.y).max()
for n in range(len(s1.t)):
fig = plt.figure(1)
fig.clear()
plt.subplots_adjust(left=0.025, right=0.975, bottom=0.025, top=0.95)
axl = fig.add_subplot(121)
axr = fig.add_subplot(122)
for i in range(2):
if i == 0:
ax = axl
# offsets -- we want star 1 at the origin
xoffset = s1.x[n]
yoffset = s1.y[n]
else:
ax = axr
# no offsets
xoffset = 0.0
yoffset = 0.0
ax.set_aspect("equal", "datalim")
ax.set_axis_off()
# center of mass
ax.scatter([0 - xoffset], [0 - yoffset],
s=150,
marker="x",
color="k")
if i == 0:
# plot star 1's position
symsize = 200
ax.scatter([s1.x[n] - xoffset], [s1.y[n] - yoffset],
s=symsize,
color="C0")
# plot star 2's orbit and position
symsize = 200 * (b.M2 / b.M1)
ax.plot(s2.x - s1.x, s2.y - s1.y, color="C1")
ax.scatter([s2.x[n] - xoffset], [s2.y[n] - yoffset],
s=symsize,
color="C1",
zorder=100)
else:
# plot star 1's orbit position
symsize = 200
ax.scatter([s1.x[n]], [s1.y[n]],
s=symsize,
color="C0",
zorder=100)
ax.plot(s1.x, s1.y, color="C0")
# plot star 2's orbit and position
symsize = 200 * (b.M2 / b.M1)
ax.scatter([s2.x[n]], [s2.y[n]],
s=symsize,
color="C1",
zorder=100)
ax.plot(s2.x, s2.y, color="C1")
if i == 0 and annotate:
# display time
ax.text(0.05,
0.05,
"time = {:6.3f} yr".format(s1.t[n] / year),
transform=ax.transAxes)
# display information about stars
ax.text(0.025,
0.9,
r"mass ratio: {:3.2f}".format(b.M1 / b.M2),
transform=ax.transAxes,
color="k",
fontsize="medium")
ax.text(0.025,
0.86,
r"eccentricity: {:3.2f}".format(b.e),
transform=ax.transAxes,
color="k",
fontsize="medium")
if annotate:
# plot a reference line
ax.plot([0, 1 * AU], [0.93 * ymin, 0.93 * ymin], color="k")
ax.text(0.5 * AU,
0.975 * ymin,
"1 AU",
horizontalalignment="center",
verticalalignment="top")
ax.set_xlim(xmin, xmax)
ax.set_ylim(ymin, ymax)
if i == 0:
ax.set_title("massive star frame of reference")
else:
ax.set_title("center of mass frame of reference")
fig.set_size_inches(12.8, 7.2)
plt.savefig("binary_star_{:04d}.png".format(iframe))
iframe += 1
def doit():
# set the masses
M_star1 = M_sun # star 1's mass
M_star2 = 0.6 * M_sun # star 2's mass
# set the semi-major axis of the star 2 (and derive that of star 1)
# M_star2 a_star2 = -M_star1 a_star1 (center of mass)
a_star2 = 1.0 * AU
a_star1 = (M_star2 / M_star1) * a_star2
# set the eccentricity
ecc = 0.4
# set the angle to rotate the semi-major axis wrt the observer
theta = math.pi / 6.0
# create the binary container
b = bi.Binary(M_star1, M_star2, a_star1 + a_star2, ecc, theta)
# set the timestep in terms of the orbital period
dt = b.P / 360.0
tmax = 2.0 * b.P # maximum integration time
b.integrate(dt, tmax)
s1 = b.orbit1
s2 = b.orbit2
# ================================================================
# plotting
# ================================================================
fig = plt.figure(1)
ax = fig.add_subplot(111)
plt.subplots_adjust(left=0.05, right=0.95, bottom=0.05, top=0.95)
ax.set_aspect("equal", "datalim")
ax.set_axis_off()
n = 0
ax.scatter([0], [0], s=150, marker="x", color="k")
# plot star 1's orbit and position
symsize = 200
ax.plot(s1.x, s1.y, color="C0")
# plot star 2's orbit and position
#symsize = 200*(M_star2/M_star1)
ax.plot(s2.x, s2.y, color="C1", linestyle="--")
ax.scatter([s2.x[n]], [s2.y[n]], s=symsize, color="C1", marker='h')
# reference points -- in terms of total number of steps integrated
f1 = 0.0943
f2 = 0.25
f3 = 0.5 - f1
npts = len(s2.t)
xc = 0.5 * (s2.x[0] + s2.x[npts // 4])
yc = 0.5 * (s2.y[0] + s2.y[npts // 4])
# compute the angle of f1 wrt the initial position (just for debugging)
a1 = math.atan2(s2.y[int(f1 * npts)] - yc, s2.x[int(f1 * npts)] - xc)
a0 = math.atan2(s2.y[0] - yc, s2.x[0] - xc)
print((a1 - a0) * 180. / math.pi)
ax.scatter([s2.x[int(f1 * npts)]], [s2.y[int(f1 * npts)]],
s=symsize,
color="C1",
marker='h')
ax.scatter([s2.x[int(f2 * npts)]], [s2.y[int(f2 * npts)]],
s=symsize,
color="C1",
marker='h')
ax.scatter([s2.x[int(f3 * npts)]], [s2.y[int(f3 * npts)]],
s=symsize,
color="C1",
marker='h')
# label the points
ax.text(s2.x[n] * 1.15, s2.y[n], "A4")
ax.text(s2.x[int(f1 * npts)] * 1.15, s2.y[int(f1 * npts)] * 1.15, "A1")
ax.text(s2.x[int(f2 * npts)] * 1.15, s2.y[int(f2 * npts)], "A2")
ax.text(s2.x[int(f3 * npts)] * 0.85, s2.y[int(f3 * npts)] * 1.15, "A3")
plt.axis([-1.4 * b.a2, 0.8 * b.a2, -1.4 * b.a2, 0.8 * b.a2])
fig.set_size_inches(7.2, 7.2)
plt.savefig("binary_fig.pdf", bbox_inches="tight")
def doit():
# set the masses
M_star1 = 50 * M_sun # star 1's mass
M_star2 = M_sun # star 2's mass
# set the semi-major axis of the star 2 (and derive that of star 1)
# M_star2 a_star2 = -M_star1 a_star1 (center of mass)
a_star2 = 1.0 * AU
a_star1 = (M_star2 / M_star1) * a_star2
# set the eccentricity
ecc = 0.0
# set the angle to rotate the semi-major axis wrt the observer
theta = math.pi / 6.0
# create the solar system container
b = bi.Binary(M_star1, M_star2, a_star1 + a_star2, ecc, theta)
# set the timestep in terms of the orbital period
dt = b.P / 360.0
tmax = 2.0 * b.P # maximum integration time
s1, s2 = b.integrate(dt, tmax)
# ================================================================
# plotting
# ================================================================
iframe = 0
for n in range(len(s1.t)):
plt.clf()
plt.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9)
a = plt.gca()
a.set_aspect("equal", "datalim")
plt.axis("off")
# if e = 0 and M_star1 = M_star2, then the orbits lie on top of one
# another, so plot only a single orbital line.
# plot star 1's orbit and position
symsize = 500
if not (b.M1 == b.M2 and b.e == 0.0):
plt.plot(s1.x, s1.y, color="r")
else:
plt.plot(s1.x, s1.y, color="k")
plt.scatter([s1.x[n]], [s1.y[n]], s=symsize, color="r")
# plot star 2's orbit and position
symsize = 1000 * (b.M2 / b.M1)
if not (b.M1 == b.M2 and b.e == 0.0):
plt.plot(s2.x, s2.y, color="g")
plt.scatter([s2.x[n]], [s2.y[n]], s=symsize, color="g")
plt.scatter([0], [0], s=150, marker="x", color="k", zorder=1000)
plt.axis(
[-1.4 * a_star2, 1.4 * a_star2, -1.4 * a_star2, 1.4 * a_star2])
f = plt.gcf()
f.set_size_inches(7.2, 7.2)
plt.savefig("planetary_orbit_%04d.png" % iframe)
iframe += 1
def astrometric_binary():
# set the masses
M_star1 = M_sun # star 1's mass
M_star2 = 0.5 * M_sun # star 2's mass
# set the semi-major axis of the star 2 (and derive that of star 1)
# M_star2 a_star2 = -M_star1 a_star1 (center of mass)
a_star2 = 1.0 * AU
a_star1 = (M_star2 / M_star1) * a_star2
# set the eccentricity
ecc = 0.0
# set the angle to rotate the semi-major axis wrt the observer
theta = math.pi / 2.0
b = bi.Binary(M_star1, M_star2, a_star1 + a_star2, ecc, theta)
# velocity of the center of mass
v = 10 * a_star2 / b.P
# set the timestep in terms of the orbital period
dt = b.P / 360.0
tmax = 2.0 * b.P # maximum integration time
s1, s2 = b.integrate(dt, tmax)
# now move the center of mass according to the velocity defined
# above, and update the positions
x_cm = np.zeros(len(s1.t), np.float64)
y_cm = np.zeros(len(s2.t), np.float64)
for n in range(len(s1.t)):
x_cm[n] = v * s1.t[n]
y_cm[n] = 0.0
s1.x[n] += v * s1.t[n]
s2.x[n] += v * s2.t[n]
# ================================================================
# plotting
# ================================================================
plt.clf()
plt.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9)
a = plt.gca()
a.set_aspect("equal", "datalim")
plt.axis("off")
# plot the center of mass motion
plt.scatter([0], [0], s=50, marker="x", color="k")
plt.plot(x_cm, y_cm, color="0.5", linestyle="--")
# plot star 1's orbit and position
plt.plot(s1.x, s1.y, color="r")
plt.scatter([s1.x[0]], [s1.y[0]], s=200, color="r")
# plot star 2's orbit and position
plt.plot(s2.x, s2.y, color="g")
plt.scatter([s2.x[0]], [s2.y[0]], s=100, color="g")
plt.text(10 * a_star2,
3 * a_star2,
"mass ratio: %3.2f" % (b.M1 / b.M2),
color="k",
verticalalignment="center")
plt.text(10 * a_star2,
2.5 * a_star2,
"eccentricity: %3.2f" % (ecc),
color="k",
verticalalignment="center")
# labels
plt.plot([0.1 * a_star2, 1.1 * a_star2], [2.9 * a_star2, 2.9 * a_star2],
color="0.5",
linestyle="--")
plt.text(1.3 * a_star2,
2.9 * a_star2,
"path of center of mass",
verticalalignment="center",
color="0.5")
plt.plot([0.1 * a_star2, 1.1 * a_star2], [2.4 * a_star2, 2.4 * a_star2],
color="r")
plt.text(1.3 * a_star2,
2.4 * a_star2,
"path of primary star",
verticalalignment="center",
color="r")
plt.plot([0.1 * a_star2, 1.1 * a_star2], [1.9 * a_star2, 1.9 * a_star2],
color="g")
plt.text(1.3 * a_star2,
1.9 * a_star2,
"path of unseen companion",
verticalalignment="center",
color="g")
plt.axis([-0.3 * a_star2, 19.7 * a_star2, -3 * a_star2, 3 * a_star2])
f = plt.gcf()
f.set_size_inches(10.0, 3.0)
plt.savefig("astrometric_binary.png")
def radial_velocity():
# set the masses
M_star = M_sun # star's mass
M_p = 0.15 * M_sun # planet's mass
# set the semi-major axis of the planet (and derive that of the star)
a_p = 1.0 * AU
a_star = (M_p / M_star) * a_p # M_p a_p = -M_star a_star (center of mass)
ecc = 0.3
theta = math.pi / 6.0
# create the solar system container
b = bi.Binary(M_star, M_p, a_star + a_p, ecc, theta)
# set the timestep in terms of the orbital period
dt = b.P / 360.0
tmax = 2.0 * b.P # maximum integration time
s, p = b.integrate(dt, tmax)
# ================================================================
# plotting
# ================================================================
iframe = 0
# first plot a legend intro sequence
while iframe < 100:
plt.clf()
plt.subplot(121)
plt.subplots_adjust(left=0.1,
right=0.95,
bottom=0.15,
top=0.85,
wspace=0.45,
hspace=0.0)
a = plt.gca()
a.set_aspect("equal", "datalim")
plt.axis("off")
plt.scatter([0], [0], s=150, marker="x", color="k")
# plot the planet's orbit and position
plt.plot(p.x, p.y, color="g")
plt.scatter([p.x[0]], [p.y[0]], s=100, color="g")
# plot the star's orbit and position
plt.plot(s.x, s.y, color="r")
plt.scatter([s.x[0]], [s.y[0]], s=200, color="r")
# plot a reference line
plt.plot([0, 1 * AU], [-1.2 * a_p, -1.2 * a_p], color="k")
plt.text(0.5 * AU, -1.4 * a_p, "1 AU", horizontalalignment='center')
plt.arrow(0.0,
1.5 * a_p,
0.0,
0.2 * a_p,
width=0.025 * a_p,
facecolor="b",
edgecolor="k",
alpha=1.0,
clip_on=0,
length_includes_head=True,
head_width=0.075 * a_p,
head_length=0.075 * a_p)
plt.text(0.0, 1.4 * a_p, "observer", horizontalalignment='center')
plt.text(-1.8 * a_p, -1.6 * a_p, "time = %6.3f yr" % (p.t[0] / year))
plt.axis([-1.5 * a_p, 1.5 * a_p, -1.5 * a_p, 1.5 * a_p])
plt.subplot(122)
plt.scatter([0.1], [0.8], s=200, color="r")
plt.text(0.2, 0.8, "star", color="r")
plt.scatter([0.1], [0.6], s=100, color="g")
plt.text(0.2, 0.6, "planet", color="g")
plt.text(0.2,
0.5,
"(note: planet's mass exaggerated",
size="small",
color="k")
plt.text(0.2, 0.45, " to enhance effect)", size="small", color="k")
plt.scatter([0.1], [0.3], s=150, marker="x", color="k")
plt.text(0.2, 0.3, "center of mass", color="k")
plt.axis([0, 1, 0, 1])
plt.axis("off")
f = plt.gcf()
f.set_size_inches(12.8, 7.2)
plt.savefig("radial_velocity_%04d.png" % iframe)
iframe += 1
# now plot the actual data
for n in range(len(s.t)):
plt.clf()
plt.subplot(121)
plt.subplots_adjust(left=0.1,
right=0.95,
bottom=0.15,
top=0.85,
wspace=0.45,
hspace=0.0)
a = plt.gca()
a.set_aspect("equal", "datalim")
plt.axis("off")
plt.scatter([0], [0], s=150, marker="x", color="k")
# plot the planet's orbit and position
plt.plot(p.x, p.y, color="g")
plt.scatter([p.x[n]], [p.y[n]], s=100, color="g")
# plot the star's orbit and position
plt.plot(s.x, s.y, color="r")
plt.scatter([s.x[n]], [s.y[n]], s=200, color="r")
# plot a reference line
plt.plot([0, 1 * AU], [-1.2 * a_p, -1.2 * a_p], color="k")
plt.text(0.5 * AU, -1.4 * a_p, "1 AU", horizontalalignment='center')
plt.arrow(0.0,
1.5 * a_p,
0.0,
0.2 * a_p,
width=0.025 * a_p,
facecolor="b",
edgecolor="k",
alpha=1.0,
clip_on=0,
length_includes_head=True,
head_width=0.075 * a_p,
head_length=0.075 * a_p)
plt.text(0.0, 1.4 * a_p, "observer", horizontalalignment='center')
plt.text(-1.8 * a_p, -1.6 * a_p, "time = %6.3f yr" % (s.t[n] / year))
plt.axis([-1.5 * a_p, 1.5 * a_p, -1.5 * a_p, 1.5 * a_p])
plt.subplot(122)
# plot km/s vs years
# note: radial velocity convention is that if it is moving
# toward us, then the velocity is negative. Since the observer
# is at +Y, we want to plot -vy as the radial velocity.
plt.plot(s.t / b.P, -s.vy / 1.e5, color="r")
plt.scatter([s.t[n] / b.P], [-s.vy[n] / 1.e5], color="r", s=50)
# plot a reference line at vy = 0
plt.plot([0.0, tmax / b.P], [0.0, 0.0], "k--")
plt.axis([
0.0, tmax / b.P, -int(numpy.max(s.vy)) / 1.e5 - 1,
-int(numpy.min(s.vy)) / 1.e5 + 1
])
plt.xlabel("t/period")
plt.ylabel("radial velocity [km/s]")
f = plt.gcf()
f.set_size_inches(12.8, 7.2)
plt.savefig("radial_velocity_%04d.png" % iframe)
iframe += 1
def radial_velocity(M1=1, M2=1, a2=10, e=0.0, annotate=False):
# set the masses
M_star1 = M1*M_sun # star 1's mass
M_star2 = M2*M_sun # star 2's mass
# set the semi-major axis of the star 2 (and derive that of star 1)
# M_star2 a_star2 = -M_star1 a_star1 (center of mass)
a_star2 = a2*AU
a_star1 = (M_star2/M_star1)*a_star2
# set the eccentricity
ecc = e
# set the angle to rotate the semi-major axis wrt the observer
theta = np.pi/6.0
# create the binary object
b = bi.Binary(M_star1, M_star2, a_star1 + a_star2, ecc, theta, annotate=annotate)
# set the timestep in terms of the orbital period
dt = b.P/360.0
tmax = 2.0*b.P # maximum integration time
b.integrate(dt, tmax)
s1 = b.orbit1
s2 = b.orbit2
# ================================================================
# plotting
# ================================================================
iframe = 0
KE1, KE2 = b.kinetic_energies()
PE = b.potential_energy()
bar = progressbar.ProgressBar()
for n in bar(range(len(s1.t))):
fig = plt.figure(1)
fig.clear()
ax = fig.add_subplot(111)
plt.subplots_adjust(left=0.025, right=0.975, bottom=0.025, top=0.975)
ax.set_aspect("equal", "datalim")
ax.set_axis_off()
ax.scatter([0], [0], s=150, marker="x", color="k")
# if e = 0 and M_star1 = M_star2, then the orbits lie on top of one
# another, so plot only a single orbital line.
# plot star 1's orbit and position
symsize = 200
if not (b.M1 == b.M2 and b.e == 0.0):
ax.plot(s1.x, s1.y, color="C0")
else:
ax.plot(s1.x, s1.y, color="k")
ax.scatter([s1.x[n]], [s1.y[n]], s=symsize, color="C0", zorder=100)
# plot star 2's orbit and position
symsize = 200*(b.M2/b.M1)
if not (b.M1 == b.M2 and b.e == 0.0):
ax.plot(s2.x, s2.y, color="C1")
ax.scatter([s2.x[n]], [s2.y[n]], s=symsize, color="C1", zorder=100)
if annotate:
# display time
ax.text(0.05, 0.05, "time = {:6.3f} yr".format(s1.t[n]/year),
transform=ax.transAxes)
# display information about stars
ax.text(0.05, 0.95, r"mass ratio: {:3.2f}".format(b.M1/b.M2),
transform=ax.transAxes, color="k", fontsize="large")
ax.text(0.05, 0.9, r"eccentricity: {:3.2f}".format(b.e),
transform=ax.transAxes, color="k", fontsize="large")
# energies
if annotate:
# KE 1
sig, ex = find_scinotat(KE1[n])
ax.text(0.05, 0.4, r"$K_1 = {:+4.2f} \times 10^{{{:2d}}}$ erg".format(sig, ex),
transform=ax.transAxes, color="C0")
sig, ex = find_scinotat(KE2[n])
ax.text(0.05, 0.35, r"$K_2 = {:+4.2f} \times 10^{{{:2d}}}$ erg".format(sig, ex),
transform=ax.transAxes, color="C1")
sig, ex = find_scinotat(PE[n])
ax.text(0.05, 0.3, r"$U = {:+4.2f} \times 10^{{{:2d}}}$ erg".format(sig, ex),
transform=ax.transAxes)
sig, ex = find_scinotat(KE1[n] + KE2[n] + PE[n])
ax.text(0.05, 0.25, r"$E = {:+4.2f} \times 10^{{{:2d}}}$ erg".format(sig, ex),
transform=ax.transAxes)
xmin = 1.05*min(s1.x.min(), s2.x.min())
xmax = 1.05*max(s1.x.max(), s2.x.max())
ymin = 1.05*min(s1.y.min(), s2.y.min())
ymax = 1.05*max(s1.y.max(), s2.y.max())
if annotate:
# plot a reference line
ax.plot([0, 1*AU], [0.93*ymin, 0.93*ymin], color="k")
ax.text(0.5*AU, 0.975*ymin, "1 AU",
horizontalalignment="center", verticalalignment="top")
ax.set_xlim(xmin, xmax)
ax.set_ylim(ymin, ymax)
fig.set_size_inches(12.8, 7.2)
plt.tight_layout()
if annotate:
plt.savefig("binary_star_mratio={:3.2f}_e={:3.2f}_{:04d}_energy.png".format(M_star1/M_star2, e, iframe))
else:
plt.savefig("binary_star_mratio={:3.2f}_e={:3.2f}_{:04d}.png".format(M_star1/M_star2, e, iframe))
iframe += 1
def radial_velocity():
# set the masses
M_star1 = 4.0 * M_sun # star 1's mass
M_star2 = M_sun # star 2's mass
# set the semi-major axis of the star 2 (and derive that of star 1)
# M_star2 a_star2 = -M_star1 a_star1 (center of mass)
a_star2 = 10.0 * AU
a_star1 = (M_star2 / M_star1) * a_star2
# set the eccentricity
ecc = 0.4
# set the angle to rotate the semi-major axis wrt the observer
theta = math.pi / 6.0
# display additional information
annotate = True
# create the binary object
b = bi.Binary(M_star1,
M_star2,
a_star1 + a_star2,
ecc,
theta,
annotate=annotate)
# set the timestep in terms of the orbital period
dt = b.P / 360.0
tmax = 2.0 * b.P # maximum integration time
s1, s2 = b.integrate(dt, tmax)
# ================================================================
# plotting
# ================================================================
plt.rc('text', usetex=True)
plt.rc('font', family='serif')
iframe = 0
for n in range(len(s1.t)):
plt.clf()
plt.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9)
a = plt.gca()
a.set_aspect("equal", "datalim")
plt.axis("off")
# offsets -- we want star 1 at the origin
xoffset = s1.x[n]
yoffset = s1.y[n]
plt.scatter([0 - xoffset], [0 - yoffset], s=150, marker="x", color="k")
# plot star 1's position
symsize = 200
plt.scatter([s1.x[n] - xoffset], [s1.y[n] - yoffset],
s=symsize,
color="r")
# plot star 2's orbit and position
symsize = 200 * (b.M2 / b.M1)
plt.plot(s2.x - s1.x, s2.y - s1.y, color="g")
plt.scatter([s2.x[n] - xoffset], [s2.y[n] - yoffset],
s=symsize,
color="g")
if annotate:
# plot a reference line
plt.plot([0, 1 * AU], [-1.6 * b.a2, -1.6 * b.a2], color="k")
plt.text(0.5 * AU,
-1.8 * b.a2,
"1 AU",
horizontalalignment='center')
# display time
plt.text(-1.8 * b.a2, -1.7 * b.a2,
"time = %6.3f yr" % (s1.t[n] / year))
# display information about stars
plt.text(-1.8 * b.a2,
0.9 * b.a2,
r"mass ratio: %3.2f" % (b.M1 / b.M2),
color="k")
plt.text(-1.8 * b.a2,
0.7 * b.a2,
r"eccentricity: %3.2f" % (b.e),
color="k")
plt.axis([-1.8 * b.a2, 1.0 * b.a2, -1.8 * b.a2, 1.0 * b.a2])
f = plt.gcf()
f.set_size_inches(7.2, 7.2)
plt.savefig("binary_star_%04d.png" % iframe)
iframe += 1
def radial_velocity():
# set the masses
M_star = M_sun # star's mass
M_p = 0.15 * M_sun # planet's mass
# set the semi-major axis of the planet (and derive that of the star)
a_p = 1.0 * AU
a_star = (M_p / M_star) * a_p # M_p a_p = -M_star a_star (center of mass)
ecc = 0.0
theta = 0.0
# create the solar system container
b = bi.Binary(M_star, M_p, a_star + a_p, ecc, theta)
# set the timestep in terms of the orbital period
dt = b.P / 360.0
tmax = 2.0 * b.P # maximum integration time
b.integrate(dt, tmax)
s = b.orbit1
p = b.orbit2
# ================================================================
# plotting
# ================================================================
title_frames = 70
iframe = 0
# first plot a legend intro sequence
for iframe in range(title_frames):
fig = plt.figure(1)
fig.clear()
plt.subplots_adjust(left=0.1,
right=0.95,
bottom=0.15,
top=0.85,
wspace=0.45,
hspace=0.0)
ax = fig.add_subplot(121)
ax.set_aspect("equal", "datalim")
ax.set_axis_off()
ax.scatter([0], [0], s=150, marker="x", color="k")
# plot the planet's orbit and position
ax.plot(p.x, p.y, color="C0")
ax.scatter([p.x[0]], [p.y[0]], s=100, color="C0")
# plot the star's orbit and position
ax.plot(s.x, s.y, color="C1")
ax.scatter([s.x[0]], [s.y[0]], s=200, color="C1")
# plot a reference line
ax.plot([0, 1 * AU], [-1.2 * a_p, -1.2 * a_p], color="k")
ax.text(0.5 * AU, -1.4 * a_p, "1 AU", horizontalalignment='center')
ax.arrow(0.0,
1.5 * a_p,
0.0,
0.2 * a_p,
width=0.025 * a_p,
facecolor="0.5",
edgecolor="k",
alpha=1.0,
clip_on=0,
length_includes_head=True,
head_width=0.075 * a_p,
head_length=0.075 * a_p)
ax.text(0.0,
1.4 * a_p,
"to the observer",
horizontalalignment='center')
ax.text(-1.8 * a_p, -1.6 * a_p, "time = %6.3f yr" % (p.t[0] / year))
ax.set_xlim(-1.5 * a_p, 1.5 * a_p)
ax.set_ylim(-1.5 * a_p, 1.5 * a_p)
ax = fig.add_subplot(122)
ax.scatter([0.1], [0.8], s=200, color="C1")
ax.text(0.2, 0.8, "star", color="C1")
ax.scatter([0.1], [0.6], s=100, color="C0")
ax.text(0.2, 0.6, "planet", color="C0")
ax.text(0.2,
0.5,
"(note: planet's mass exaggerated",
size="small",
color="k")
ax.text(0.2, 0.45, " to enhance effect)", size="small", color="k")
ax.scatter([0.1], [0.3], s=150, marker="x", color="k")
ax.text(0.2, 0.3, "center of mass", color="k")
ax.set_xlim(0, 1)
ax.set_ylim(0, 1)
ax.set_axis_off()
fig.set_size_inches(12.8, 7.2)
plt.savefig("radial_velocity_{:04d}.png".format(iframe))
# now plot the actual data
print("starting orbit part, iframe = ", iframe)
for n in range(len(s.t)):
print("iframe = ", iframe)
fig = plt.figure(1)
fig.clear()
plt.subplots_adjust(left=0.1,
right=0.95,
bottom=0.15,
top=0.85,
wspace=0.45,
hspace=0.0)
ax = fig.add_subplot(121)
ax.set_aspect("equal", "datalim")
ax.set_axis_off()
ax.scatter([0], [0], s=150, marker="x", color="k")
# plot the planet's orbit and position
ax.plot(p.x, p.y, color="C0")
ax.scatter([p.x[n]], [p.y[n]], s=100, color="C0")
# plot the star's orbit and position
ax.plot(s.x, s.y, color="C1")
ax.scatter([s.x[n]], [s.y[n]], s=200, color="C1")
# plot a reference line
ax.plot([0, 1 * AU], [-1.2 * a_p, -1.2 * a_p], color="k")
ax.text(0.5 * AU, -1.4 * a_p, "1 AU", horizontalalignment='center')
ax.arrow(0.0,
1.5 * a_p,
0.0,
0.2 * a_p,
width=0.025 * a_p,
facecolor="0.5",
edgecolor="k",
alpha=1.0,
clip_on=0,
length_includes_head=True,
head_width=0.075 * a_p,
head_length=0.075 * a_p)
ax.text(0.0,
1.4 * a_p,
"to the observer",
horizontalalignment='center')
ax.text(-1.8 * a_p, -1.6 * a_p, "time = %6.3f yr" % (s.t[n] / year))
ax.set_xlim(-1.5 * a_p, 1.5 * a_p)
ax.set_ylim(-1.5 * a_p, 1.5 * a_p)
ax = fig.add_subplot(122)
# plot km/s vs years
# note: radial velocity convention is that if it is moving
# toward us, then the velocity is negative. Since the observer
# is at +Y, we want to plot -vy as the radial velocity.
ax.plot(s.t / b.P, -s.vy / 1.e5, color="C1")
ax.scatter([s.t[n] / b.P], [-s.vy[n] / 1.e5], color="C1", s=50)
# plot a reference line at vy = 0
ax.plot([0.0, tmax / b.P], [0.0, 0.0], "k--")
ax.axis([
0.0, tmax / b.P,
int(np.min(s.vy)) / 1.e5 - 1,
int(np.max(s.vy)) / 1.e5 + 1
])
ax.set_xlabel("t/period")
ax.set_ylabel("radial velocity [km/s]")
fig.set_size_inches(12.8, 7.2)
plt.savefig("radial_velocity_%04d.png" % iframe)
iframe += 1
def doit():
# set the masses
M_star1 = 50 * M_sun # star 1's mass
M_star2 = M_sun # star 2's mass
# set the semi-major axis of the star 2 (and derive that of star 1)
# M_star2 a_star2 = -M_star1 a_star1 (center of mass)
a_star2 = 1.0 * AU
a_star1 = (M_star2 / M_star1) * a_star2
# set the eccentricity
ecc = 0.0
# set the angle to rotate the semi-major axis wrt the observer
theta = np.pi / 6.0
# create the solar system container
b = bi.Binary(M_star1, M_star2, a_star1 + a_star2, ecc, theta)
# set the timestep in terms of the orbital period
dt = b.P / 360.0
tmax = 2.0 * b.P # maximum integration time
b.integrate(dt, tmax)
s1 = b.orbit1
s2 = b.orbit2
# ================================================================
# plotting
# ================================================================
iframe = 0
for n in range(len(s1.t)):
fig = plt.figure(1)
fig.clear()
ax = fig.add_subplot(111)
plt.subplots_adjust(left=0.05, right=0.95, bottom=0.05, top=0.95)
ax.set_aspect("equal", "datalim")
ax.set_axis_off()
# if e = 0 and M_star1 = M_star2, then the orbits lie on top of one
# another, so plot only a single orbital line.
# plot star 1's orbit and position
symsize = 500
if not (b.M1 == b.M2 and b.e == 0.0):
ax.plot(s1.x, s1.y, color="C1")
else:
ax.plot(s1.x, s1.y, color="k")
ax.scatter([s1.x[n]], [s1.y[n]], s=symsize, color="C1", zorder=100)
# plot star 2's orbit and position
symsize = 1000 * (b.M2 / b.M1)
if not (b.M1 == b.M2 and b.e == 0.0):
plt.plot(s2.x, s2.y, color="C0")
plt.scatter([s2.x[n]], [s2.y[n]], s=symsize, color="C0", zorder=100)
plt.scatter([0], [0], s=150, marker="x", color="k", zorder=1000)
plt.axis(
[-1.1 * a_star2, 1.1 * a_star2, -1.1 * a_star2, 1.1 * a_star2])
fig.set_size_inches(12.8, 7.2)
plt.savefig("planetary_orbit_{:04d}.png".format(iframe))
iframe += 1
def radial_velocity():
# set the masses
M_star1 = 2.0 * M_sun # star 1's mass
M_star2 = M_sun # star 2's mass
# set the semi-major axis of the star 2 (and derive that of star 1)
# M_star2 a_star2 = -M_star1 a_star1 (center of mass)
a_star2 = 10.0 * AU
a_star1 = (M_star2 / M_star1) * a_star2
# set the eccentricity
ecc = 0.4
# set the angle to rotate the semi-major axis wrt the observer
theta = np.pi / 6.0
# display additional information
annotate = False
# create the binary object
b = bi.Binary(M_star1,
M_star2,
a_star1 + a_star2,
ecc,
theta,
annotate=annotate)
# set the timestep in terms of the orbital period
dt = b.P / 360.0
tmax = 2.0 * b.P # maximum integration time
s1, s2 = b.integrate(dt, tmax)
# ================================================================
# plotting
# ================================================================
plt.rc('text', usetex=True)
plt.rc('font', family='serif')
iframe = 0
for n in range(len(s1.t)):
plt.clf()
plt.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9)
a = plt.gca()
a.set_aspect("equal", "datalim")
plt.axis("off")
plt.scatter([0], [0], s=150, marker="x", color="k")
# if e = 0 and M_star1 = M_star2, then the orbits lie on top of one
# another, so plot only a single orbital line.
# plot star 1's orbit and position
symsize = 200
if not (b.M1 == b.M2 and b.e == 0.0):
plt.plot(s1.x, s1.y, color="r")
else:
plt.plot(s1.x, s1.y, color="k")
plt.scatter([s1.x[n]], [s1.y[n]], s=symsize, color="r")
# plot star 2's orbit and position
symsize = 200 * (b.M2 / b.M1)
if not (b.M1 == b.M2 and b.e == 0.0):
plt.plot(s2.x, s2.y, color="g")
plt.scatter([s2.x[n]], [s2.y[n]], s=symsize, color="g")
if annotate:
# plot a reference line
plt.plot([0, 1 * AU], [-1.2 * b.a2, -1.2 * b.a2], color="k")
plt.text(0.5 * AU,
-1.4 * b.a2,
"1 AU",
horizontalalignment='center')
# display time
plt.text(-1.4 * b.a2, -1.3 * b.a2,
"time = %6.3f yr" % (s1.t[n] / year))
# display information about stars
plt.text(-1.4 * b.a2,
1.3 * b.a2,
r"mass ratio: %3.2f" % (b.M1 / b.M2),
color="k")
plt.text(-1.4 * b.a2,
1.1 * b.a2,
r"eccentricity: %3.2f" % (b.e),
color="k")
# energies
if annotate:
KE1 = 0.5 * b.M1 * (s1.vx[n]**2 + s1.vy[n]**2)
KE2 = 0.5 * b.M2 * (s2.vx[n]**2 + s2.vy[n]**2)
PE = -G*b.M1*b.M2/ \
np.sqrt((s1.x[n] - s2.x[n])**2 + (s1.y[n] - s2.y[n])**2)
print(KE1, KE2, PE, KE1 + KE2 + PE)
# KE 1
sig, ex = find_scinotat(KE1)
plt.text(0, 1.3 * b.a2, r"$K_1 =$")
plt.text(0.3 * b.a2, 1.3 * b.a2,
r"$%+4.2f \times 10^{%2d}$ erg" % (sig, ex))
sig, ex = find_scinotat(KE2)
plt.text(0, 1.15 * b.a2, r"$K_2 =$")
plt.text(0.3 * b.a2, 1.15 * b.a2,
r"$%+4.2f \times 10^{%2d}$ erg" % (sig, ex))
sig, ex = find_scinotat(PE)
plt.text(0, 1.0 * b.a2, r"$U =$")
plt.text(0.3 * b.a2, 1.0 * b.a2,
r"$%+4.2f \times 10^{%2d}$ erg" % (sig, ex))
sig, ex = find_scinotat(KE1 + KE2 + PE)
plt.text(0, 0.85 * b.a2, r"$E =$")
plt.text(0.3 * b.a2, 0.85 * b.a2,
r"$%+4.2f \times 10^{%2d}$ erg" % (sig, ex))
plt.axis([-1.4 * b.a2, 1.4 * b.a2, -1.4 * b.a2, 1.4 * b.a2])
f = plt.gcf()
f.set_size_inches(7.2, 7.2)
plt.savefig("binary_star_%04d.png" % iframe)
iframe += 1