def test_actionConservation():
#_____initialize some KKSPot_____
Delta = 1.0
pot = KuzminKutuzovStaeckelPotential(ac=20., Delta=Delta, normalize=True)
#_____initialize an orbit (twice)_____
vxvv = [1., 0.1, 1.1, 0.01, 0.1]
o = Orbit(vxvv=vxvv)
#_____integrate the orbit with C_____
ts = numpy.linspace(0, 101, 100)
o.integrate(ts, pot, method='leapfrog_c')
#_____Setup ActionAngle object and calculate actions (Staeckel approximation)_____
aAS = actionAngleStaeckel(pot=pot, delta=Delta, c=True)
jrs, lzs, jzs = aAS(o.R(ts), o.vR(ts), o.vT(ts), o.z(ts), o.vz(ts))
assert numpy.all(numpy.fabs(jrs - jrs[0]) < 10.**-8.), \
'Radial action is not conserved along orbit.'
assert numpy.all(numpy.fabs(lzs - lzs[0]) < 10.**-8.), \
'Angular momentum is not conserved along orbit.'
assert numpy.all(numpy.fabs(jzs - jzs[0]) < 10.**-8.), \
'Vertical action is not conserved along orbit.'
return None
def test_estimateDelta():
#_____initialize some KKSPot_____
Delta = 1.0
pot = KuzminKutuzovStaeckelPotential(ac=20., Delta=Delta, normalize=True)
#_____initialize an orbit (twice)_____
vxvv = [1., 0.1, 1.1, 0.01, 0.1]
o = Orbit(vxvv=vxvv)
#_____integrate the orbit with C_____
ts = numpy.linspace(0, 101, 100)
o.integrate(ts, pot, method='leapfrog_c')
#____estimate Focal length Delta_____
#for each time step individually:
deltas_estimate = numpy.zeros(len(ts))
for ii in range(len(ts)):
deltas_estimate[ii] = estimateDeltaStaeckel(pot, o.R(ts[ii]),
o.z(ts[ii]))
assert numpy.all(numpy.fabs(deltas_estimate - Delta) < 10.**-8), \
'Focal length Delta estimated along the orbit is not constant.'
#for all time steps together:
delta_estimate = estimateDeltaStaeckel(pot, o.R(ts), o.z(ts))
assert numpy.fabs(delta_estimate - Delta) < 10.**-8, \
'Focal length Delta estimated from the orbit is not the same as the input focal length.'
return None
def test_actionAngleTorus_Staeckel_actions():
from galpy.potential import KuzminKutuzovStaeckelPotential
from galpy.actionAngle import actionAngleTorus, \
actionAngleStaeckel
delta = 1.2
kp = KuzminKutuzovStaeckelPotential(normalize=1., Delta=delta)
aAS = actionAngleStaeckel(pot=kp, delta=delta, c=True)
tol = -3.
aAT = actionAngleTorus(pot=kp, tol=tol)
jr, jphi, jz = 0.075, 1.1, 0.05
angler = numpy.array([0.])
anglephi = numpy.array([numpy.pi])
anglez = numpy.array([numpy.pi / 2.])
# Calculate position from aAT
RvR = aAT(jr, jphi, jz, angler, anglephi, anglez).T
# Calculate actions from aAI
ji = aAS(*RvR)
djr = numpy.fabs((ji[0] - jr) / jr)
dlz = numpy.fabs((ji[1] - jphi) / jphi)
djz = numpy.fabs((ji[2] - jz) / jz)
assert djr < 10.**tol, 'actionAngleTorus and actionAngleStaeckel applied to Staeckel potential disagree for Jr at %f%%' % (
djr * 100.)
assert dlz < 10.**tol, 'actionAngleTorus and actionAngleStaeckel applied to Staeckel potential disagree for Jr at %f%%' % (
dlz * 100.)
assert djz < 10.**tol, 'actionAngleTorus and actionAngleStaeckel applied to Staeckel potential disagree for Jr at %f%%' % (
djz * 100.)
return None
def test_actionAngleTorus_Staeckel_freqsAngles():
from galpy.potential import KuzminKutuzovStaeckelPotential
from galpy.actionAngle import actionAngleTorus, \
actionAngleStaeckel
delta = 1.2
kp = KuzminKutuzovStaeckelPotential(normalize=1., Delta=delta)
aAS = actionAngleStaeckel(pot=kp, delta=delta, c=True)
tol = -3.
aAT = actionAngleTorus(pot=kp, tol=tol)
jr, jphi, jz = 0.075, 1.1, 0.05
angler = numpy.array([0.1]) + numpy.linspace(0., numpy.pi, 101)
angler = angler % (2. * numpy.pi)
anglephi = numpy.array([numpy.pi]) + numpy.linspace(0., numpy.pi, 101)
anglephi = anglephi % (2. * numpy.pi)
anglez = numpy.array([numpy.pi / 2.]) + numpy.linspace(0., numpy.pi, 101)
anglez = anglez % (2. * numpy.pi)
# Calculate position from aAT
RvRom = aAT.xvFreqs(jr, jphi, jz, angler, anglephi, anglez)
# Calculate actions, frequencies, and angles from aAI
ws = aAS.actionsFreqsAngles(*RvRom[0].T)
dOr = numpy.fabs((ws[3] - RvRom[1]))
dOp = numpy.fabs((ws[4] - RvRom[2]))
dOz = numpy.fabs((ws[5] - RvRom[3]))
dar = numpy.fabs((ws[6] - angler))
dap = numpy.fabs((ws[7] - anglephi))
daz = numpy.fabs((ws[8] - anglez))
dar[dar > numpy.pi] -= 2. * numpy.pi
dar[dar < -numpy.pi] += 2. * numpy.pi
dap[dap > numpy.pi] -= 2. * numpy.pi
dap[dap < -numpy.pi] += 2. * numpy.pi
daz[daz > numpy.pi] -= 2. * numpy.pi
daz[daz < -numpy.pi] += 2. * numpy.pi
assert numpy.all(
dOr < 10.**tol
), 'actionAngleTorus and actionAngleStaeckel applied to Staeckel potential disagree for Or at %f%%' % (
numpy.nanmax(dOr) * 100.)
assert numpy.all(
dOp < 10.**tol
), 'actionAngleTorus and actionAngleStaeckel applied to Staeckel potential disagree for Ophi at %f%%' % (
numpy.nanmax(dOp) * 100.)
assert numpy.all(
dOz < 10.**tol
), 'actionAngleTorus and actionAngleStaeckel applied to Staeckel potential disagree for Oz at %f%%' % (
numpy.nanmax(dOz) * 100.)
assert numpy.all(
dar < 10.**tol
), 'actionAngleTorus and actionAngleStaeckel applied to Staeckel potential disagree for ar at %f' % (
numpy.nanmax(dar))
assert numpy.all(
dap < 10.**tol
), 'actionAngleTorus and actionAngleStaeckel applied to Staeckel potential disagree for aphi at %f' % (
numpy.nanmax(dap))
assert numpy.all(
daz < 10.**tol
), 'actionAngleTorus and actionAngleStaeckel applied to Staeckel potential disagree for az at %f' % (
numpy.nanmax(daz))
return None
def test_orbitIntegrationC():
#_____initialize some KKSPot_____
Delta = 1.0
pot = KuzminKutuzovStaeckelPotential(ac=20., Delta=Delta, normalize=True)
#_____initialize an orbit (twice)_____
vxvv = [1., 0.1, 1.1, 0., 0.1]
o_P = Orbit(vxvv=vxvv)
o_C = Orbit(vxvv=vxvv)
#_____integrate the orbit with python and C_____
ts = numpy.linspace(0, 100, 101)
o_P.integrate(ts, pot, method='leapfrog') #python
o_C.integrate(ts, pot, method='leapfrog_c') #C
for ii in range(5):
exp3 = -1.7
if ii == 0:
Python, CC, string, exp1, exp2 = o_P.R(ts), o_C.R(
ts), 'R', -5., -10.
elif ii == 1:
Python, CC, string, exp1, exp2 = o_P.z(ts), o_C.z(
ts), 'z', -3.25, -4.
elif ii == 2:
Python, CC, string, exp1, exp2 = o_P.vR(ts), o_C.vR(
ts), 'vR', -3., -10.
elif ii == 3:
Python, CC, string, exp1, exp2, exp3 = o_P.vz(ts), o_C.vz(
ts), 'vz', -3., -4., -1.3
elif ii == 4:
Python, CC, string, exp1, exp2 = o_P.vT(ts), o_C.vT(
ts), 'vT', -5., -10.
rel_diff = numpy.fabs((Python - CC) / CC) < 10.**exp1
abs_diff = (numpy.fabs(Python - CC) < 10.**exp2) * (numpy.fabs(Python)
< 10.**exp3)
assert numpy.all(rel_diff+abs_diff), \
'Orbit integration for '+string+' coordinate different in ' + \
'C and Python implementation.'
return None
def test_density():
#test the density calculation of the KuzminKutuzovStaeckelPotential
#using parameters from Batsleer & Dejonghe 1994, tab. 2
#_____parameters_____
#table 2 in Batsleer & Dejonghe
ac_D = [
25., 25., 25., 25., 25., 25., 40., 40., 40., 40., 40., 50., 50., 50.,
50., 50., 50., 75., 75., 75., 75., 75., 75., 100., 100., 100., 100.,
100., 100., 150., 150., 150., 150., 150., 150.
]
ac_H = [
1.005, 1.005, 1.01, 1.01, 1.02, 1.02, 1.005, 1.005, 1.01, 1.01, 1.02,
1.005, 1.005, 1.01, 1.01, 1.02, 1.02, 1.005, 1.005, 1.01, 1.01, 1.02,
1.02, 1.005, 1.005, 1.01, 1.01, 1.02, 1.02, 1.005, 1.005, 1.01, 1.01,
1.02, 1.02
]
k = [
0.05, 0.08, 0.075, 0.11, 0.105, 0.11, 0.05, 0.08, 0.075, 0.11, 0.11,
0.05, 0.07, 0.07, 0.125, 0.1, 0.125, 0.05, 0.065, 0.07, 0.125, 0.10,
0.125, 0.05, 0.065, 0.07, 0.125, 0.10, 0.125, 0.05, 0.065, 0.075,
0.125, 0.11, 0.125
]
Delta = [
0.99, 1.01, 0.96, 0.99, 0.86, 0.88, 1.00, 1.01, 0.96, 0.99, 0.89, 1.05,
1.06, 1.00, 1.05, 0.91, 0.97, 0.98, 0.99, 0.94, 0.98, 0.85, 0.91, 1.06,
1.07, 1.01, 1.06, 0.94, 0.97, 1.06, 1.07, 0.98, 1.06, 0.94, 0.97
]
Mmin = [
7.49, 6.17, 4.08, 3.70, 2.34, 2.36, 7.57, 6.16, 4.08, 2.64, 2.38, 8.05,
6.94, 4.37, 3.70, 2.48, 2.50, 7.37, 6.66, 4.05, 3.46, 2.33, 2.36, 8.14,
7.27, 4.42, 3.72, 2.56, 2.50, 8.14, 7.26, 4.17, 3.72, 2.51, 2.50
]
Mmax = [
7.18, 6.12, 3.99, 3.69, 2.37, 2.40, 7.27, 6.11, 3.99, 2.66, 2.42, 7.76,
6.85, 4.26, 3.72, 2.51, 2.54, 7.07, 6.51, 3.95, 3.48, 2.36, 2.40, 7.85,
7.15, 4.30, 3.75, 2.58, 2.54, 7.85, 7.07, 4.08, 3.75, 2.53, 2.53
]
rhomin = [
0.04, 0.05, 0.04, 0.04, 0.03, 0.03, 0.06, 0.06, 0.05, 0.04, 0.04, 0.07,
0.08, 0.06, 0.07, 0.04, 0.05, 0.08, 0.09, 0.07, 0.09, 0.05, 0.06, 0.12,
0.13, 0.09, 0.13, 0.07, 0.09, 0.16, 0.19, 0.12, 0.18, 0.10, 0.12
]
rhomax = [
0.03, 0.03, 0.02, 0.03, 0.02, 0.02, 0.04, 0.04, 0.03, 0.03, 0.02, 0.05,
0.05, 0.04, 0.05, 0.03, 0.03, 0.05, 0.06, 0.04, 0.06, 0.03, 0.04, 0.07,
0.08, 0.06, 0.08, 0.04, 0.05, 0.09, 0.10, 0.07, 0.10, 0.06, 0.07
]
Sigmin = [
58, 52, 52, 49, 39, 40, 58, 55, 51, 44, 40, 59, 54, 53, 49, 41, 42, 58,
55, 51, 48, 39, 40, 59, 55, 53, 49, 42, 42, 59, 55, 52, 49, 42, 42
]
Sigmax = [
45, 41, 38, 37, 28, 28, 45, 32, 37, 32, 30, 46, 43, 40, 37, 30, 31, 45,
43, 38, 36, 28, 29, 46, 43, 40, 38, 31, 31, 46, 44, 39, 38, 30, 31
]
for ii in range(len(ac_D)):
if ac_D[ii] == 40.:
continue
#because I believe that there are typos in tab. 2 by Batsleer & Dejonghe...
for jj in range(2):
#_____parameters depending on solar position____
if jj == 0:
Rsun = 7.5
zsun = 0.004
GM = Mmin[ii] #units: G = 1, M in 10^11 solar masses
rho = rhomin[ii]
Sig = Sigmin[ii]
elif jj == 1:
Rsun = 8.5
zsun = 0.02
GM = Mmax[ii] #units: G = 1, M in 10^11 solar masses
rho = rhomax[ii]
Sig = Sigmax[ii]
outstr = 'ac_D='+str(ac_D[ii])+', ac_H='+str(ac_H[ii])+', k='+str(k[ii])+', Delta='+str(Delta[ii])+\
', Mtot='+str(GM)+'*10^11Msun, Rsun='+str(Rsun)+'kpc, rho(Rsun,zsun)='+str(rho)+'Msun/pc^3, Sig(Rsun,z<1.1kpc)='+str(Sig)+'Msun/pc^2'
#_____setup potential_____
amp_D = GM * k[ii]
V_D = KuzminKutuzovStaeckelPotential(amp=amp_D,
ac=ac_D[ii],
Delta=Delta[ii],
normalize=False)
amp_H = GM * (1. - k[ii])
V_H = KuzminKutuzovStaeckelPotential(amp=amp_H,
ac=ac_H[ii],
Delta=Delta[ii],
normalize=False)
pot = [V_D, V_H]
#_____local density_____
rho_calc = evaluateDensities(
pot, Rsun, zsun) * 100. #units: [solar mass / pc^3]
rho_calc = round(rho_calc, 2)
#an error of 0.01 corresponds to the significant digit
#given in the table, to which the density was rounded,
#to be wrong by one.
assert numpy.fabs(rho_calc - rho) <= 0.01+10.**-8, \
'Calculated density %f for KuzminKutuzovStaeckelPotential ' % rho_calc + \
'with model parameters:\n'+outstr+'\n'+ \
'does not agree with value from tab. 2 '+ \
'by Batsleer & Dejonghe (1994)'
#_____surface density_____
Sig_calc, err = scipy.integrate.quad(
lambda z: (evaluateDensities(pot, Rsun, z / 1000.) * 100.
), #units: [solar mass / pc^3]
0.,
1100.) #units: pc
Sig_calc = round(2. * Sig_calc)
#an error of 1 corresponds to the significant digit
#given in the table, to which the surface density was rounded,
#to be wrong by one.
assert numpy.fabs(Sig_calc - Sig) <= 1., \
'Calculated surface density %f for KuzminKutuzovStaeckelPotential ' % Sig_calc + \
'with model parameters:\n'+outstr+'\n'+ \
'does not agree with value from tab. 2 '+ \
'by Batsleer & Dejonghe (1994)'
return None
def test_vcirc():
#test the circular velocity of the KuzminKutuzovStaeckelPotential
#using parameters from Batsleer & Dejonghe 1994, fig. 1-3
#and their formula eq. (10)
#_____model parameters______
#surface ratios of disk and halo:
ac_Ds = [50., 50., 50., 50., 50., 50., 40., 40., 40.]
ac_Hs = [1.005, 1.005, 1.005, 1.01, 1.01, 1.01, 1.02, 1.02, 1.02]
#disk contribution to total mass:
ks = [0.05, 0.06, 0.07, 0.07, 0.1, 0.125, 0.1, 0.12, 0.125]
#focal distance:
Delta = 1.
for ii in range(9):
ac_D = ac_Ds[ii]
ac_H = ac_Hs[ii]
k = ks[ii]
#_____setup potential_____
#first try, not normalized:
V_D = KuzminKutuzovStaeckelPotential(amp=k,
ac=ac_D,
Delta=Delta,
normalize=False)
V_H = KuzminKutuzovStaeckelPotential(amp=(1. - k),
ac=ac_H,
Delta=Delta,
normalize=False)
pot = [V_D, V_H]
#normalization according to Batsleer & Dejonghe 1994:
V00 = evaluatePotentials(pot, 0., 0.)
#second try, normalized:
V_D = KuzminKutuzovStaeckelPotential(amp=k / (-V00),
ac=ac_D,
Delta=Delta,
normalize=False)
V_H = KuzminKutuzovStaeckelPotential(amp=(1. - k) / (-V00),
ac=ac_H,
Delta=Delta,
normalize=False)
pot = [V_D, V_H]
#_____calculate rotation curve_____
Rs = numpy.linspace(0., 20., 100)
z = 0.
vcirc_calc = numpy.sqrt(-Rs * evaluateRforces(pot, Rs, z))
#_____vcirc by Batsleer & Dejonghe eq. (10) (with proper Jacobian)_____
def vc2w(R):
g_D = Delta**2 / (1. - ac_D**2)
a_D = g_D - Delta**2
g_H = Delta**2 / (1. - ac_H**2)
a_H = g_H - Delta**2
l = R**2 - a_D
q = a_H - a_D
termD = numpy.sqrt(l) * (numpy.sqrt(l) + numpy.sqrt(-g_D))**2
termH = numpy.sqrt(l - q) * (numpy.sqrt(l - q) +
numpy.sqrt(-g_D - q))**2
return R**2 * (k / termD + (1. - k) / termH)
vcirc_formula = numpy.sqrt(vc2w(Rs) / (-V00))
assert numpy.all(numpy.fabs(vcirc_calc - vcirc_formula) < 10**-8.), \
'Calculated circular velocity for KuzminKutuzovStaeckelPotential '+ \
'does not agree with eq. (10) (corrected by proper Jacobian) '+ \
'by Batsleer & Dejonghe (1994)'
return None