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 setUp(self):
self.sim = get_sim()
self.pvars = Poincare.from_Simulation(self.sim)
self.pham = PoincareHamiltonian(self.pvars)
self.terms_list = SecularTermsList(4, 4)
for i in range(1, self.pvars.N):
for j in range(i + 1, self.pvars.N):
for term in self.terms_list:
k, z = term
self.pham.add_monomial_term(k,
z,
indexIn=i,
indexOut=j,
update=False)
self.pham._update()
qpsymbols = [S('eta{}'.format(i)) for i in range(1,self.pvars.N)] +\
[S('rho{}'.format(i)) for i in range(1,self.pvars.N)] +\
[S('kappa{}'.format(i)) for i in range(1,self.pvars.N)] +\
[S('sigma{}'.format(i)) for i in range(1,self.pvars.N)]
self.secular_variable_indices = [
self.pham.varsymbols.index(s) for s in qpsymbols
]
self.secular_sim = SecularSystemSimulation(self.pvars,
dtFraction=1 / 50.,
max_order=4)
def test_particles(self):
m = 1.e-5
M = 3.
G = 2.
a = 7.
e = 0.1
pomega = 1.3
l = 0.7
sLambda = np.sqrt(G * M * a)
sGamma = sLambda * (1. - np.sqrt(1. - e**2))
Lambda = m * sLambda
Gamma = m * sGamma
p = PoincareParticle(m=m,
M=M,
G=G,
Lambda=Lambda,
l=l,
Gamma=Gamma,
gamma=-pomega)
tp = PoincareParticle(m=0.,
M=M,
G=G,
sLambda=sLambda,
l=l,
sGamma=sGamma,
gamma=-pomega)
self.assertAlmostEqual(p.a, a, delta=1.e-15)
self.assertAlmostEqual(p.e, e, delta=1.e-15)
self.assertAlmostEqual(p.pomega, pomega, delta=1.e-15)
self.assertAlmostEqual(tp.a, a, delta=1.e-15)
self.assertAlmostEqual(tp.e, e, delta=1.e-15)
self.assertAlmostEqual(tp.pomega, pomega, delta=1.e-15)
pvars = Poincare(G, poincareparticles=[p, tp])
ps = pvars.particles
self.assertAlmostEqual(ps[1].a, a, delta=1.e-15)
self.assertAlmostEqual(ps[1].e, e, delta=1.e-15)
self.assertAlmostEqual(ps[1].pomega, pomega, delta=1.e-15)
self.assertAlmostEqual(ps[2].a, a, delta=1.e-15)
self.assertAlmostEqual(ps[2].e, e, delta=1.e-15)
self.assertAlmostEqual(ps[2].pomega, pomega, delta=1.e-15)
pvars = Poincare(G=G)
pvars.add(m=m, M=M, Lambda=Lambda, l=l, Gamma=Gamma, gamma=-pomega)
pvars.add(m=0.,
M=M,
sLambda=sLambda,
l=l,
sGamma=sGamma,
gamma=-pomega)
ps = pvars.particles
self.assertAlmostEqual(ps[1].a, a, delta=1.e-15)
self.assertAlmostEqual(ps[1].e, e, delta=1.e-15)
self.assertAlmostEqual(ps[1].pomega, pomega, delta=1.e-15)
self.assertAlmostEqual(ps[2].a, a, delta=1.e-15)
self.assertAlmostEqual(ps[2].e, e, delta=1.e-15)
self.assertAlmostEqual(ps[2].pomega, pomega, delta=1.e-15)
def test_orbelements(self):
pvars = Poincare.from_Simulation(self.sim, average=False)
ps = pvars.particles
o = self.sim.calculate_orbits(jacobi_masses=True)
for i in range(1, 4):
self.assertAlmostEqual(o[i - 1].a, ps[i].a, delta=1.e-15)
self.assertAlmostEqual(o[i - 1].e, ps[i].e, delta=1.e-15)
self.assertAlmostEqual(o[i - 1].l, ps[i].l, delta=1.e-15)
self.assertAlmostEqual(o[i - 1].pomega, ps[i].pomega, delta=1.e-15)
def test_particles(self):
m=1.e-5
M=3.
G=2.
a=7.
e=0.1
pomega = 1.3
Omega = 2.1
inc = 0.07
l=0.7
sLambda = np.sqrt(G*M*a)
sGamma = sLambda*(1.-np.sqrt(1.-e**2))
sQ = (sLambda-sGamma)*(1-np.cos(inc))
Lambda = m*sLambda
Gamma = m*sGamma
Q = m*sQ
p = PoincareParticle(m=m, M=M, G=G, Lambda=Lambda, l=l, Gamma=Gamma, gamma=-pomega, Q=Q, q=-Omega)
tp = PoincareParticle(m=0., M=M, G=G, sLambda=sLambda, l=l, sGamma=sGamma, gamma=-pomega, sQ=sQ, q=-Omega)
self.assertAlmostEqual(p.a, a, delta=1.e-15)
self.assertAlmostEqual(p.e, e, delta=1.e-15)
self.assertAlmostEqual(p.inc, inc, delta=1.e-15)
self.assertAlmostEqual(p.pomega, pomega, delta=1.e-15)
self.assertAlmostEqual(p.Omega, Omega, delta=1.e-15)
self.assertAlmostEqual(tp.a, a, delta=1.e-15)
self.assertAlmostEqual(tp.e, e, delta=1.e-15)
self.assertAlmostEqual(tp.inc, inc, delta=1.e-15)
self.assertAlmostEqual(tp.pomega, pomega, delta=1.e-15)
self.assertAlmostEqual(tp.Omega, Omega, delta=1.e-15)
pvars = Poincare(G, poincareparticles=[p, tp])
ps = pvars.particles
self.assertAlmostEqual(ps[1].a, a, delta=1.e-15)
self.assertAlmostEqual(ps[1].e, e, delta=1.e-15)
self.assertAlmostEqual(ps[1].inc, inc, delta=1.e-15)
self.assertAlmostEqual(ps[1].pomega, pomega, delta=1.e-15)
self.assertAlmostEqual(ps[1].Omega, Omega, delta=1.e-15)
self.assertAlmostEqual(ps[2].a, a, delta=1.e-15)
self.assertAlmostEqual(ps[2].e, e, delta=1.e-15)
self.assertAlmostEqual(ps[2].inc, inc, delta=1.e-15)
self.assertAlmostEqual(ps[2].pomega, pomega, delta=1.e-15)
self.assertAlmostEqual(ps[2].Omega, Omega, delta=1.e-15)
pvars = Poincare(G=G)
pvars.add(m=m, M=M, Lambda=Lambda, l=l, Gamma=Gamma, gamma=-pomega, Q=Q, q=-Omega)
pvars.add(m=0., M=M, sLambda=sLambda, l=l, sGamma=sGamma, gamma=-pomega, sQ=sQ, q=-Omega)
ps = pvars.particles
self.assertAlmostEqual(ps[1].a, a, delta=1.e-15)
self.assertAlmostEqual(ps[1].e, e, delta=1.e-15)
self.assertAlmostEqual(ps[1].inc, inc, delta=1.e-15)
self.assertAlmostEqual(ps[1].pomega, pomega, delta=1.e-15)
self.assertAlmostEqual(ps[1].Omega, Omega, delta=1.e-15)
self.assertAlmostEqual(ps[2].a, a, delta=1.e-15)
self.assertAlmostEqual(ps[2].e, e, delta=1.e-15)
self.assertAlmostEqual(ps[2].inc, inc, delta=1.e-15)
self.assertAlmostEqual(ps[2].pomega, pomega, delta=1.e-15)
self.assertAlmostEqual(ps[2].Omega, Omega, delta=1.e-15)
def averaging_error(Nseed):
sim = packed_sim(Nseed)
ps = sim.particles
o = sim.calculate_orbits(jacobi_masses=True)
a10 = o[0].a
a20 = o[1].a
a30 = o[2].a
tsyn = 2 * np.pi / (o[0].n - o[1].n)
tmax = 30 * tsyn
Nout = 100
times = np.linspace(0, tmax, Nout)
pvars = Poincare.from_Simulation(sim)
Hsim = PoincareHamiltonian(pvars)
Nsma = np.zeros((3, Nout))
Hsma = np.zeros((3, Nout))
for i, t in enumerate(times):
# Store N-body data
o = sim.calculate_orbits(jacobi_masses=True)
Nsma[0, i] = o[0].a
Nsma[1, i] = o[1].a
Nsma[2, i] = o[2].a
ps = Hsim.particles
Hsma[0, i] = ps[1].a
Hsma[1, i] = ps[2].a
Hsma[2, i] = ps[3].a
sim.integrate(t)
Hsim.integrate(t)
Nmad1 = mad((Nsma[0] - a10) / a10)
Nmad2 = mad((Nsma[1] - a20) / a20)
Nmad3 = mad((Nsma[2] - a30) / a30)
Nmed1 = np.median((Nsma[0] - a10) / a10)
Pmed1 = np.median((Hsma[0] - a10) / a10)
err1 = abs(Pmed1 - Nmed1) / Nmad1
Nmed2 = np.median((Nsma[1] - a20) / a20)
Pmed2 = np.median((Hsma[1] - a20) / a20)
err2 = abs(Pmed2 - Nmed2) / Nmad2
Nmed3 = np.median((Nsma[2] - a30) / a30)
Pmed3 = np.median((Hsma[2] - a30) / a30)
err3 = abs(Pmed3 - Nmed3) / Nmad3
return max(err1, err2, err3)
def test_jacobi_masses(self):
pvars = Poincare.from_Simulation(self.sim, average=False)
sim = pvars.to_Simulation(average=False)
self.compare_simulations_orb(self.sim, sim, delta=1.e-14)
sim = pvars.to_Simulation(masses=[1., 1.e-3, 1.e-7, 1.e-5],
average=False)
self.compare_simulations_orb(self.sim, sim, delta=1.e-14)
# use default jacobi masses but pass it an inconsistent set of jacobi masses
pvars.particles[1].M = 1.
pvars.particles[2].M = 1.
pvars.particles[3].M = 1.
sim = pvars.to_Simulation(average=False)
with self.assertRaises(AssertionError):
self.compare_simulations_orb(self.sim, sim, delta=1.e-14)
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 calc_tau(sim):
if np.isnan(
sim.dt
): # init_sim_parameters sets timestep to nan if any orbit is hyperbolic. Return tau=inf, i.e. chaotic/unstable
tau = np.inf
return tau
lsys = LaplaceLagrangeSystem.from_Simulation(sim)
pvars = Poincare.from_Simulation(sim)
Lambda = np.array([p.Lambda for p in pvars.particles[1:]])
tau_max = 0
for i in range(1, sim.N_real):
tau = 0
if i - 1 >= 1:
tau += calc_tau_pair(sim, lsys, Lambda, i - 1, i)
if i + 1 < sim.N_real:
tau += calc_tau_pair(sim, lsys, Lambda, i, i + 1)
if tau > tau_max:
tau_max = tau
return tau_max
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
def test_rebound_transformations(self):
# averaging operation is not symmetric because have to use current
# Lambdas to calculate correction. So turn off averaging for test
pvars = Poincare.from_Simulation(self.sim, average=False)
sim = pvars.to_Simulation(average=False)
self.compare_simulations_orb(self.sim, sim, delta=1.e-14)
def test_copy(self):
pvars = Poincare.from_Simulation(self.sim)
pvars2 = pvars.copy()
self.compare_poincare_particles(
pvars.particles, pvars2.particles) # ignore nans in particles[0]