def _rectForce(x, pot, t=0.):
"""
NAME:
_rectForce
PURPOSE:
returns the force in the rectangular frame
INPUT:
x - current position
t - current time
pot - (list of) Potential instance(s)
OUTPUT:
force
HISTORY:
2011-02-02 - Written - Bovy (NYU)
"""
#x is rectangular so calculate R and phi
R = nu.sqrt(x[0]**2. + x[1]**2.)
phi = nu.arccos(x[0] / R)
sinphi = x[1] / R
cosphi = x[0] / R
if x[1] < 0.: phi = 2. * nu.pi - phi
#calculate forces
Rforce = _evaluateRforces(pot, R, x[2], phi=phi, t=t)
phiforce = _evaluatephiforces(pot, R, x[2], phi=phi, t=t)
return nu.array([
cosphi * Rforce - 1. / R * sinphi * phiforce,
sinphi * Rforce + 1. / R * cosphi * phiforce,
_evaluatezforces(pot, R, x[2], phi=phi, t=t)
])
def _FullEOM(y, t, pot):
"""
NAME:
_FullEOM
PURPOSE:
implements the EOM, i.e., the right-hand side of the differential
equation
INPUT:
y - current phase-space position
t - current time
pot - (list of) Potential instance(s)
OUTPUT:
dy/dt
HISTORY:
2010-04-16 - Written - Bovy (NYU)
"""
l2 = (y[0]**2. * y[3])**2.
return [
y[1], l2 / y[0]**3. + _evaluateRforces(
pot, y[0], y[4], phi=y[2], t=t, v=[y[1], y[0] * y[3], y[5]]), y[3],
1. / y[0]**2. * (_evaluatephiforces(
pot, y[0], y[4], phi=y[2], t=t, v=[y[1], y[0] * y[3], y[5]]) -
2. * y[0] * y[1] * y[3]), y[5],
_evaluatezforces(pot,
y[0],
y[4],
phi=y[2],
t=t,
v=[y[1], y[0] * y[3], y[5]])
]
def estimateDeltaStaeckel(pot, R, z, no_median=False):
"""
NAME:
estimateDeltaStaeckel
PURPOSE:
Estimate a good value for delta using eqn. (9) in Sanders (2012)
INPUT:
pot - Potential instance or list thereof
R,z- coordinates (if these are arrays, the median estimated delta is returned, i.e., if this is an orbit)
no_median - (False) if True, and input is array, return all calculated values of delta (useful for quickly
estimating delta for many phase space points)
OUTPUT:
delta
HISTORY:
2013-08-28 - Written - Bovy (IAS)
2016-02-20 - Changed input order to allow physical conversions - Bovy (UofT)
"""
if isinstance(R, nu.ndarray):
delta2 = nu.array([
(
z[ii]**2. - R[ii]**2. #eqn. (9) has a sign error
+
(3. * R[ii] * _evaluatezforces(pot, R[ii], z[ii]) - 3. *
z[ii] * _evaluateRforces(pot, R[ii], z[ii]) + R[ii] * z[ii] *
(evaluateR2derivs(pot, R[ii], z[ii], use_physical=False) -
evaluatez2derivs(pot, R[ii], z[ii], use_physical=False))) /
evaluateRzderivs(pot, R[ii], z[ii], use_physical=False))
for ii in range(len(R))
])
indx = (delta2 < 0.) * (delta2 > -10.**-10.)
delta2[indx] = 0.
if not no_median:
delta2 = nu.median(delta2[True ^ nu.isnan(delta2)])
else:
delta2 = (
z**2. - R**2. #eqn. (9) has a sign error
+ (3. * R * _evaluatezforces(pot, R, z) -
3. * z * _evaluateRforces(pot, R, z) + R * z *
(evaluateR2derivs(pot, R, z, use_physical=False) -
evaluatez2derivs(pot, R, z, use_physical=False))) /
evaluateRzderivs(pot, R, z, use_physical=False))
if delta2 < 0. and delta2 > -10.**-10.: delta2 = 0.
return nu.sqrt(delta2)
def estimateDeltaStaeckel(pot,R,z, no_median=False):
"""
NAME:
estimateDeltaStaeckel
PURPOSE:
Estimate a good value for delta using eqn. (9) in Sanders (2012)
INPUT:
pot - Potential instance or list thereof
R,z- coordinates (if these are arrays, the median estimated delta is returned, i.e., if this is an orbit)
no_median - (False) if True, and input is array, return all calculated values of delta (useful for quickly
estimating delta for many phase space points)
OUTPUT:
delta
HISTORY:
2013-08-28 - Written - Bovy (IAS)
2016-02-20 - Changed input order to allow physical conversions - Bovy (UofT)
"""
if isinstance(R,nu.ndarray):
delta2= nu.array([(z[ii]**2.-R[ii]**2. #eqn. (9) has a sign error
+(3.*R[ii]*_evaluatezforces(pot,R[ii],z[ii])
-3.*z[ii]*_evaluateRforces(pot,R[ii],z[ii])
+R[ii]*z[ii]*(evaluateR2derivs(pot,R[ii],z[ii],
use_physical=False)
-evaluatez2derivs(pot,R[ii],z[ii],
use_physical=False)))/evaluateRzderivs(pot,R[ii],z[ii],use_physical=False)) for ii in range(len(R))])
indx= (delta2 < 0.)*(delta2 > -10.**-10.)
delta2[indx]= 0.
if not no_median:
delta2= nu.median(delta2[True^nu.isnan(delta2)])
else:
delta2= (z**2.-R**2. #eqn. (9) has a sign error
+(3.*R*_evaluatezforces(pot,R,z)
-3.*z*_evaluateRforces(pot,R,z)
+R*z*(evaluateR2derivs(pot,R,z,use_physical=False)
-evaluatez2derivs(pot,R,z,use_physical=False)))/evaluateRzderivs(pot,R,z,use_physical=False))
if delta2 < 0. and delta2 > -10.**-10.: delta2= 0.
return nu.sqrt(delta2)
def FZStaeckel(u, v, pot, delta): #pragma: no cover because unused
"""
NAME:
FZStaeckel
PURPOSE:
return the vertical force
INPUT:
u - confocal u
v - confocal v
pot - potential
delta - focus
OUTPUT:
FZ(u,v)
HISTORY:
2012-11-30 - Written - Bovy (IAS)
"""
R, z = bovy_coords.uv_to_Rz(u, v, delta=delta)
return _evaluatezforces(pot, R, z)
def FZStaeckel(u,v,pot,delta): #pragma: no cover because unused
"""
NAME:
FZStaeckel
PURPOSE:
return the vertical force
INPUT:
u - confocal u
v - confocal v
pot - potential
delta - focus
OUTPUT:
FZ(u,v)
HISTORY:
2012-11-30 - Written - Bovy (IAS)
"""
R,z= bovy_coords.uv_to_Rz(u,v,delta=delta)
return _evaluatezforces(pot,R,z)
def _RZEOM(y,t,pot,l2):
"""
NAME:
_RZEOM
PURPOSE:
implements the EOM, i.e., the right-hand side of the differential
equation
INPUT:
y - current phase-space position
t - current time
pot - (list of) Potential instance(s)
l2 - angular momentum squared
OUTPUT:
dy/dt
HISTORY:
2010-04-16 - Written - Bovy (NYU)
"""
return [y[1],
l2/y[0]**3.+_evaluateRforces(pot,y[0],y[2],t=t),
y[3],
_evaluatezforces(pot,y[0],y[2],t=t)]
def _RZEOM(y, t, pot, l2):
"""
NAME:
_RZEOM
PURPOSE:
implements the EOM, i.e., the right-hand side of the differential
equation
INPUT:
y - current phase-space position
t - current time
pot - (list of) Potential instance(s)
l2 - angular momentum squared
OUTPUT:
dy/dt
HISTORY:
2010-04-16 - Written - Bovy (NYU)
"""
return [
y[1], l2 / y[0]**3. + _evaluateRforces(pot, y[0], y[2], t=t), y[3],
_evaluatezforces(pot, y[0], y[2], t=t)
]