def test_rotational_invariance(self):
j=4
k=1
avars = Andoyer.from_Simulation(self.sim, j, k, i1=1, i2=2)
rot = np.pi/3.
ps = self.sim.particles
simrot = rebound.Simulation()
simrot.G = self.sim.G
simrot.add(m=ps[0].m)
for p in ps[1:]:
simrot.add(m=p.m, a=p.a, e=p.e, inc=p.inc, Omega=p.Omega+rot, pomega=p.pomega+rot, l=p.l+rot)
simrot.move_to_com()
avarsrot = Andoyer.from_Simulation(simrot, j, k, i1=1, i2=2)
self.assertAlmostEqual(avars.X, avarsrot.X, delta=self.delta)
self.assertAlmostEqual(avars.Y, avarsrot.Y, delta=self.delta)
self.assertAlmostEqual(avars.Zcom, avarsrot.Zcom, delta=self.delta)
self.assertAlmostEqual(np.cos(avars.phiZcom+rot), np.cos(avarsrot.phiZcom), delta=self.delta)
self.assertAlmostEqual(avars.B, avarsrot.B, delta=self.delta)
self.assertAlmostEqual(avars.dKprime, avarsrot.dKprime, delta=self.delta)
self.assertAlmostEqual(np.cos(avars.theta+k*rot), np.cos(avarsrot.theta), delta=self.delta)
p = avars.params
fac = (p['m1']*p['sLambda10'] + p['m2']*p['sLambda20'])/p['K0']
self.assertAlmostEqual(np.cos(avars.theta1+fac*rot), np.cos(avarsrot.theta1), delta=self.delta)
def get_k(row):
sim = rebound.SimulationArchive(sim_names + "_sa_%d.bin" % (row[0]))[0]
# print(sim)
p2 = sim.particles[2]
row['h'] = p2.e * np.sin(p2.pomega)
row['k'] = p2.e * np.cos(p2.pomega)
avars = Andoyer.from_Simulation(sim,
a10=sim.particles[1].a,
j=5,
k=1,
i1=1,
i2=2,
average=False)
row['Z12'] = avars.Z
row['Zcom12'] = avars.Zcom
avars = Andoyer.from_Simulation(sim,
a10=sim.particles[1].a,
j=4,
k=1,
i1=2,
i2=3,
average=False)
row['Z23'] = avars.Z
row['Zcom23'] = avars.Zcom
row['e1'] = sim.particles[1].e
row['e2'] = sim.particles[2].e
row['e3'] = sim.particles[3].e
row['m1'] = sim.particles[1].m
row['m2'] = sim.particles[2].m
row['m3'] = sim.particles[3].m
return row
def test_from_elements(self):
j = 5
k = 2
Zstar = 0.1
libfac = 0.5
a10 = 3.
a1 = 3.1
G = 2.
m1 = 1.e-7
m2 = 1.e-4
ecom = 0.05
phiecom = 0.7
avars = Andoyer.from_elements(j,
k,
Zstar,
libfac,
a10,
a1,
G,
m1=m1,
m2=m2,
Zcom=ecom,
phiZcom=phiecom)
self.assertAlmostEqual(avars.Zstar, Zstar, delta=1.e-12)
self.assertAlmostEqual(avars.Zcom, ecom, delta=1.e-15)
self.assertAlmostEqual(avars.phiZcom, phiecom, delta=1.e-15)
sim = avars.to_Simulation()
self.assertAlmostEqual(
sim.particles[1].a, a1, delta=3 * ((a1 - a10) / a10)**
2) # should match to O(da/a)^2, atrue=1, a10=a10
def test_ecom(self):
j = 57
k = 2
sim = rebound.Simulation()
sim.G = 4 * np.pi**2
sim.add(m=1.)
sim.add(m=1.e-6,
e=0.01,
P=1.,
pomega=-np.pi / 6,
f=np.pi,
jacobi_masses=True)
sim.add(m=3.e-6,
e=0.03,
pomega=np.pi / 3,
P=float(j) / (j - k),
jacobi_masses=True) #float(j)/(j-k), theta=3.14)
sim.move_to_com()
ps = sim.particles
e1x = ps[1].e * np.cos(ps[1].pomega)
e1y = ps[1].e * np.sin(ps[1].pomega)
e2x = ps[2].e * np.cos(ps[2].pomega)
e2y = ps[2].e * np.sin(ps[2].pomega)
avars = Andoyer.from_Simulation(sim, j, k, a10=1.02, average=True)
m1 = avars.params['m1']
m2 = avars.params['m2']
ecomx = (m1 * e1x + m2 * e2x) / (m1 + m2)
ecomy = (m1 * e1y + m2 * e2y) / (m1 + m2)
ecomsim = np.sqrt(ecomx**2 + ecomy**2)
self.assertAlmostEqual(avars.Zcom, ecomsim, delta=1.e-3)
phiecomsim = np.arctan2(ecomy, ecomx)
self.assertAlmostEqual(avars.phiZcom, phiecomsim, delta=1.e-3)
def test_ZsGammaConversion(self):
avars = Andoyer.from_Simulation(self.sim, 4, 1)
Z = avars.Z
phiZ = avars.psi1
Zcom = avars.Zcom
phiZcom = avars.phiZcom
sGamma1, gamma1, sGamma2, gamma2 = avars.Zs_to_sGammas(Z, phiZ, Zcom, phiZcom)
nZ, nphiZ, nZcom, nphiZcom = avars.sGammas_to_Zs(sGamma1, gamma1, sGamma2, gamma2)
self.assertAlmostEqual(nZ,Z, delta=1.e-15)
self.assertAlmostEqual(np.mod(nphiZ, 2*np.pi), np.mod(phiZ, 2*np.pi), delta=1.e-15)
self.assertAlmostEqual(nZcom, Zcom, delta=1.e-15)
self.assertAlmostEqual(np.mod(nphiZcom, 2*np.pi), np.mod(phiZcom, 2*np.pi), delta=1.e-15)
# should also be equivalent for 2 massive particles to rotation
pvars = Poincare.from_Simulation(self.sim)
ps = pvars.particles
p = avars.params
f, g = p['f'], p['g']
ff = f*np.sqrt(p['eta']/p['m1']/p['sLambda10'])
gg = g*np.sqrt(p['eta']/p['m2']/p['sLambda20'])
norm = np.sqrt(ff*ff + gg*gg)
psirotmatrix = np.array([[ff,gg],[-gg,ff]]) / norm
invpsirotmatrix = np.array([[ff,-gg],[gg,ff]]) / norm
Psi1,psi1,Psi2,psi2 = self.rotate_actions(ps[1].Gamma/p['eta'],ps[1].gamma,ps[2].Gamma/p['eta'],ps[2].gamma, psirotmatrix)
self.assertAlmostEqual(Psi1, avars.Psi1, delta=1.e-15)
self.assertAlmostEqual(np.mod(psi1, 2*np.pi), np.mod(avars.psi1, 2*np.pi), delta=1.e-15)
self.assertAlmostEqual(Psi2, avars.Psi2, delta=1.e-15)
self.assertAlmostEqual(np.mod(psi2, 2*np.pi), np.mod(avars.psi2, 2*np.pi), delta=1.e-15)
def test_dP(self):
j=5
k=2
sim = rebound.Simulation()
sim.add(m=1.)
sim.add(m=1.e-6, P=1.)
sim.add(m=3.e-6, P=1.68)
sim.move_to_com()
avars = Andoyer.from_Simulation(sim,j,k, a10=0.32, average=False) # real a0 ~0.29, 10% err
self.assertAlmostEqual(1.68-float(j)/(j-k), avars.dP, delta=0.01) # err is da^2 or smaller, so 1%
def test_masses(self):
# turn off averaging for all transformation tests since averaging
# is not symmetric back and forth (there's diff of O(s^2))
masses = [1., 1.e-5, 1.e-7, 1.e-3]
pairs = [[1,2], [2,3], [1,3]]
for i1, i2 in pairs:
m = [masses[0], masses[i1], masses[i2]]
avars = Andoyer.from_Simulation(self.sim, 4, 1, average=False, i1=i1, i2=i2)
sim = avars.to_Simulation(masses=m, average=False)
self.compare_particles(self.sim, sim, i1, 1, self.delta)
self.compare_particles(self.sim, sim, i2, 2, self.delta)
def test_scale_invariance(self):
avars = Andoyer.from_Simulation(self.sim, 4, 1, i1=1, i2=2)
mfac = 3.
dfac = 0.05
vfac = np.sqrt(mfac/dfac)
for p in self.sim.particles:
p.m *= mfac
p.vx *= vfac
p.vy *= vfac
p.vz *= vfac
p.x *= dfac
p.y *= dfac
p.z *= dfac
avarsscaled = Andoyer.from_Simulation(self.sim, 4, 1, i1=1, i2=2)
self.assertAlmostEqual(avars.X, avarsscaled.X, delta=self.delta)
self.assertAlmostEqual(avars.Y, avarsscaled.Y, delta=self.delta)
self.assertAlmostEqual(avars.Zcom, avarsscaled.Zcom, delta=self.delta)
self.assertAlmostEqual(np.cos(avars.phiZcom), np.cos(avarsscaled.phiZcom), delta=self.delta)
self.assertAlmostEqual(avars.B, avarsscaled.B, delta=self.delta)
self.assertAlmostEqual(avars.dKprime, avarsscaled.dKprime, delta=self.delta)
self.assertAlmostEqual(np.cos(avars.theta), np.cos(avarsscaled.theta), delta=self.delta)
self.assertAlmostEqual(np.cos(avars.theta1), np.cos(avarsscaled.theta1), delta=self.delta)
def test_H(self):
j=3
k=1
a10 = 1.02
sim = rebound.Simulation()
sim.G = 4*np.pi**2
sim.add(m=1.)
sim.add(m=1.e-6, e=0.01, P=1., pomega=-np.pi/2, f=np.pi, jacobi_masses=True)
sim.add(m=3.e-6, e=0.03, pomega=np.pi/2, P=float(j)/(j-k), jacobi_masses=True)#float(j)/(j-k), theta=3.14)
sim.move_to_com()
avars = Andoyer.from_Simulation(sim,j,k, a10=a10, average=True)
p = avars.params
pvars = Poincare.from_Simulation(sim, average=True)
sGamma1 = pvars.particles[1].sGamma
sGamma2 = pvars.particles[2].sGamma
sLambda1 = pvars.particles[1].sLambda
sLambda2 = pvars.particles[2].sLambda
lambda1 = pvars.particles[1].l
lambda2 = pvars.particles[2].l
gamma1 = pvars.particles[1].gamma
gamma2 = pvars.particles[2].gamma
n10 = np.sqrt(p['G']*p['M1']/p['a10']**3)
n20 = np.sqrt(p['G']*p['M2']/p['a10']**3*p['alpha']**3)
z1 = np.sqrt(2*sGamma1/p['sLambda10'])
z2 = np.sqrt(2*sGamma2/p['sLambda20'])
Hkep = -0.5*(n10*p['m1']*p['sLambda10']**3/sLambda1**2 + n20*p['m2']*p['sLambda20']**3/sLambda2**2)
sL10 = p['sLambda10']
sL20 = p['sLambda20']
Hkepexpanded = -n10*p['m1']*sL10/2*(1. - 2*avars.dL1hat + 3*avars.dL1hat**2)-n20*p['m2']*sL20/2*(1. - 2*avars.dL2hat + 3*avars.dL2hat**2)
Hresprefac = -p['G']*p['m1']*p['m2']/p['a10']*p['alpha']
Hres = Hresprefac*(p['f']*z1*np.cos(j*lambda2 - (j-k)*lambda1 + gamma1)+p['g']*z2*np.cos(j*lambda2 - (j-k)*lambda1 + gamma2))
H0 = -p['n0']*p['K0']/2
H1 = p['eta']*p['n0']*avars.dK*(1-1.5*p['eta']/p['K0']*avars.dK)
H2 = p['eta']*p['a']*avars.dP**2
Hkeptransformed = H0 + H1 + H2
Hrestransformed = p['eta']*p['c']*(2*avars.Psi1)**(k/2.)*np.cos(avars.theta+k*avars.psi1)
self.assertAlmostEqual(Hkeptransformed, Hkepexpanded, delta=1.e-15) # should be exact
self.assertAlmostEqual(Hrestransformed, Hres, delta=1.e-15) # should be exact for first order resonance (k=1)
self.assertAlmostEqual(Hkepexpanded, Hkep, delta=(a10-1.)**2) # should match to O(da/a)^2, atrue=1, a10=a10
#Hfinal = -p['eta']*p['Phi0']/p['tau']*(4.*avars.Phi**2 - 3.*avars.Phiprime*avars.Phi + 9./16.*avars.Phiprime**2 + (2.*avars.Phi)**(k/2.)*np.cos(avars.phi) - p['n0']*p['tau']/p['Phi0']*avars.dK*(1.-1.5*p['eta']/p['K0']*avars.dK))+ H0
Hfinal = -p['eta']*p['Phi0']/p['tau']*(4.*(avars.Phi-avars.B)**2 + (2.*avars.Phi)**(k/2.)*np.cos(avars.phi) - p['n0']*p['tau']/p['Phi0']*avars.dK*(1.-1.5*p['eta']/p['K0']*avars.dK))+ H0
self.assertAlmostEqual(Hfinal, Hkepexpanded+Hres, delta=1.e-15) # should be exact
def populate_trio(sim, trio, pairs, jk, a10, tseries, i):
Ns = 3
ps = sim.particles
for q, [label, i1, i2] in enumerate(pairs):
e1x, e1y = ps[i1].e * np.cos(ps[i1].pomega), -ps[i1].e * np.sin(
ps[i1].pomega)
e2x, e2y = ps[i2].e * np.cos(ps[i2].pomega), -ps[i2].e * np.sin(
ps[i2].pomega)
try:
# it's unlikely but possible for bodies to land nearly on top of each other midtimestep and get a big kick that doesn't get caught by collisions post timestep. All these cases are unstable, so flag them as above
# average only affects a (Lambda) Z and I think Zcom don't depend on a. Zsep and Zstar slightly, but several sig figs in even when close at conjunction
j, k = jk[label]
avars = Andoyer.from_Simulation(sim,
a10=a10[label],
j=j,
k=k,
i1=i1,
i2=i2,
average=False)
tseries[i, Ns * q + 1] = avars.Z * np.sqrt(2) # EM = Z*sqrt(2)
tseries[i, Ns * q + 2] = avars.Zcom # no sqrt(2) factor
except: # no nearby resonance, use EM and ecom
tseries[i, Ns * q + 1] = np.sqrt((e2x - e1x)**2 + (e2y - e1y)**2)
tseries[i, Ns * q + 2] = np.sqrt(
(ps[i1].m * e1x + ps[i2].m * e2x)**2 +
(ps[i1].m * e1y + ps[i2].m * e2y)**2) / (ps[i1].m + ps[i2].m)
try:
j, k, tseries[i, Ns * q + 3] = find_strongest_MMR(
sim, i1, i2
) # will fail if any osculating orbit is hyperbolic, flag as unstable
except:
return False
tseries[i, 7] = sim.calculate_megno() # megno
return True
def get_resonant(seed, Nplanets=3):
r = random.Random()
r.seed(seed)
Phiprimecrits = [0, 1., -2. / 3.]
pairs = ['inner', 'outer', 'split']
k = r.randint(1, 2)
pairindex = r.randint(0, 2)
pair = pairs[pairindex]
m1 = logunif(r, 1.e-7, 1.e-4)
m2 = logunif(r, 1.e-7, 1.e-4)
eH = ((m1 + m2) / 3.)**(1. / 3.)
ehillstable = 3.5 * eH
jmax = k / (1 - 1 / (1 + 3.5 * eH)**1.5)
if pair == 'split':
if Nplanets == 2:
return # don't want 2planet systems 60 hill radii apart
maxHillradii = 60. # 3rd planet will go in middle so draw up to 60
else:
maxHillradii = 30.
jmin = max(k + 1, k / (1 - 1 / (1 + maxHillradii * eH)**1.5))
jmin = int(np.ceil(jmin))
jmax = int(np.floor(jmax))
if k == 2: # if k = 2, want odd j so we don't get e.g. 8:6 = 4:3
if jmin % 2 == 0: # even
jmin += 1
if jmax % 2 == 0:
jmax -= 1
j = r.randrange(
jmin, jmax + 1,
k) # choose randomly between limits in steps of k e.g. (3,5,7,9)
a1 = 1.
a2 = (float(j) / (j - k))**(2. / 3.)
ecross1 = (a2 - a1) / a1
ecross2 = (a2 - a1) / a2
emin1 = m2 / ecross1**2
emin2 = m1 / ecross2**2
emin = max(
emin1, emin2
) # take as min Z the larger of the kicks a planet gets at conjunction
emin = max(
emin, (m1 + m2)**(1. / k)
) # below mtot^1/k, the resonant term is smaller than the second order mass terms we ignore
emax = min(ecross1, ecross2)
avars = Andoyer(j=j, k=k, X=0, Y=0, m1=m1, m2=m2)
Phiprimecrit = Phiprimecrits[k]
Xcrit = get_Xstarres(k, Phiprimecrit)
Phicrit = 0.5 * Xcrit**2
emin = max(
avars.Phi_to_Z(Phicrit), emin
) # first quantity is value of Z at bifurcation when res first appears
Zstar = logunif(r, emin, emax)
libfac = logunif(r, 3.e-3, 3)
negative = r.randint(0, 1)
if negative:
libfac *= -1
Zcom = logunif(r, emin, emax)
avars = Andoyer.from_elements(j=j,
k=k,
Zstar=Zstar,
libfac=libfac,
m1=m1,
m2=m2,
Zcom=Zcom,
phiZcom=r.uniform(0, 2 * np.pi),
theta=r.uniform(0, 2 * np.pi),
theta1=r.uniform(0, 2 * np.pi))
tmax = r.uniform(0, 10 * avars.tlib)
H = AndoyerHamiltonian(avars)
H.integrate(tmax)
pvars = avars.to_Poincare()
ps = pvars.particles
if Nplanets == 3:
m3 = logunif(r, 1.e-7, 1.e-4)
pvarssorted = Poincare(G=pvars.G)
if pair == "inner":
eH = ((m2 + m3) / 3.)**(1. / 3.)
beta = r.uniform(3.5, 30)
a3 = a2 * (1 + beta * eH)
ecross3 = (a3 - a2) / a3
emin3 = m2 / ecross3**2
e3 = logunif(r, emin3, ecross3)
pvarssorted.add(m=ps[1].m,
M=ps[1].M,
a=ps[1].a,
e=ps[1].e,
gamma=ps[1].gamma,
l=ps[1].l)
pvarssorted.add(m=ps[2].m,
M=ps[2].M,
a=ps[2].a,
e=ps[2].e,
gamma=ps[2].gamma,
l=ps[2].l)
pvarssorted.add(m=m3,
M=1,
a=a3,
e=e3,
gamma=r.uniform(0, 2 * np.pi),
l=r.uniform(0, 2 * np.pi))
elif pair == "outer":
eH = ((m1 + m3) / 3.)**(1. / 3.)
beta = r.uniform(3.5, 30)
a3 = a1 / (1 + beta * eH)
ecross3 = (a1 - a3) / a3
emin3 = m1 / ecross3**2
e3 = logunif(r, emin3, ecross3)
pvarssorted.add(m=m3,
M=1,
a=a3,
e=e3,
gamma=r.uniform(0, 2 * np.pi),
l=r.uniform(0, 2 * np.pi))
pvarssorted.add(m=ps[1].m,
M=ps[1].M,
a=ps[1].a,
e=ps[1].e,
gamma=ps[1].gamma,
l=ps[1].l)
pvarssorted.add(m=ps[2].m,
M=ps[2].M,
a=ps[2].a,
e=ps[2].e,
gamma=ps[2].gamma,
l=ps[2].l)
elif pair == "split":
eH1 = ((m1 + m3) / 3.)**(1. / 3.)
eH2 = ((m2 + m3) / 3.)**(1. / 3.)
amin = a1 * (1 + 3.5 * eH1)
amax = min(a1 * (1 + 30 * eH1), a2 / (1 + 3.5 * eH2))
if amin > amax:
seed += 900000
return get_resonant(
seed=seed) # draw new sample (may not be 'split')
a3 = r.uniform(amin, amax)
ecross3 = min((a3 - a1) / a1, (a2 - a3) / a3)
emin3 = max(m1 / ((a3 - a1) / a3)**2, m2 / ((a2 - a3) / a3)**2)
e3 = logunif(r, emin3, ecross3)
pvarssorted.add(m=ps[1].m,
M=ps[1].M,
a=ps[1].a,
e=ps[1].e,
gamma=ps[1].gamma,
l=ps[1].l)
pvarssorted.add(m=m3,
M=1,
a=a3,
e=e3,
gamma=r.uniform(0, 2 * np.pi),
l=r.uniform(0, 2 * np.pi))
pvarssorted.add(m=ps[2].m,
M=ps[2].M,
a=ps[2].a,
e=ps[2].e,
gamma=ps[2].gamma,
l=ps[2].l)
sim = pvarssorted.to_Simulation()
else:
sim = avars.to_Simulation()
# add inclinations and scale s.t. a1 = 1, Mprimary=1. and G = 4*pi**2
ps = sim.particles
siminc = rebound.Simulation()
siminc.G = 4 * np.pi**2
dscale = ps[1].a
tscale = ps[1].P
mscale = siminc.G * dscale**3 / tscale**2 # ps[0].m + ps[1].m
siminc.add(m=ps[0].m / mscale,
x=ps[0].x / dscale,
y=ps[0].y / dscale,
vx=ps[0].vx / dscale * tscale,
vy=ps[0].vy / dscale * tscale)
for p in ps[1:]:
siminc.add(m=p.m / mscale,
a=p.a / dscale,
e=p.e,
inc=logunif(r, 1.e-3, 1.e-1),
Omega=r.uniform(0, 2 * np.pi),
pomega=p.pomega,
l=p.l)
rH = siminc.particles[-1].a * (siminc.particles[-1].m / 3. /
siminc.particles[0].m)**(1. / 3.)
siminc.particles[-1].r = rH
siminc.move_to_com()
siminc.integrator = "whfast"
siminc.dt = 2. * np.sqrt(3) / 100. * siminc.particles[1].P
siminc.ri_whfast.safe_mode = 0
siminc.collision = "line"
return siminc, j, k, pairindex, Zstar, libfac, Zcom