def fl_error(tpos, field, Bm):
"""
Return the max and mean relative error of calculated dipole field line
points, compared to the analytic result. For debugging.
Parameters
----------
tpos : array-like
4-element array of time and position.
field: Field object
The field object under which the integral is evaluated.
Bm : float
Mirror field value (T).
Returns
-------
e_max : float
The maximum error
e_avg : float
Th average error
"""
fl = Fieldline(tpos, field, Bmax=Bm)
fl.trace()
r0 = np.sqrt(np.dot(tpos[1:], tpos[1:]))
e_max = 0
e_avg = 0
for p in fl.curve:
r = np.sqrt(np.dot(p[1:], p[1:]))
l = np.arcsin(p[3] / r)
e = abs(1 - r / (r0 * np.cos(l)**2))
if e > e_max:
e_max = e
e_avg += e
e_avg /= (len(fl.curve) - 1)
return (e_max, e_avg)
def fl_error(tpos, field, Bm):
"""
Return the max and mean relative error of calculated dipole field line
points, compared to the analytic result. For debugging.
Parameters
----------
tpos : array-like
4-element array of time and position.
field: Field object
The field object under which the integral is evaluated.
Bm : float
Mirror field value (T).
Returns
-------
e_max : float
The maximum error
e_avg : float
Th average error
"""
fl = Fieldline(tpos, field, Bmax=Bm)
fl.trace()
r0 = np.sqrt(np.dot(tpos[1:],tpos[1:]))
e_max = 0
e_avg = 0
for p in fl.curve:
r = np.sqrt(np.dot(p[1:],p[1:]))
l = np.arcsin(p[3]/r)
e = abs(1 - r/(r0*np.cos(l)**2))
if e > e_max:
e_max = e
e_avg += e
e_avg /= (len(fl.curve)-1)
return (e_max, e_avg)
def criticalpoints(tpos, field, Bm):
"""
Return a list of critical points where the field strength is a maximum or a minimum along the field line.
Parameters
----------
tpos : array-like
Starting position and time for the field line.
field : Field
The field object that provides electric and magnetic field vectors and
related quantities.
Bm : float
The mirror field strength in Tesla (defines the endpoints of the field line).
Returns
-------
array
Array of (s, B, B'') triplets for each critical point, where s is the
location along the field line (with respect to the initial position),
B is the field strength at s, and B'' is the second derivative along
the field line.
"""
fl = Fieldline(tpos, field, Bmax = Bm)
fl.trace()
s = fl.gets()
b = fl.getB()
n = len(b)
slist = []
# first, make a list of approximate locations of minima or maxima.
for i in range(1,n-1):
if (b[i]-b[i-1])*(b[i+1]-b[i]) <= 0:
slist.append(s[i])
# Create an interpolated function B(s)
B = interp1d(s, b, kind='cubic', assume_sorted=True)
# Find the optimal point for each value in slist
res = []
for s0 in slist:
m = minimize(B, s0)
sm = m.x[0]
B2s = derivative(B, sm, dx=0.1*(s[1]-s[0]), n=2)
res.append((sm, float(B(sm)), B2s))
return res
def criticalpoints(tpos, field, Bm):
"""
Return a list of critical points where the field strength is a maximum or a minimum along the field line.
Parameters
----------
tpos : array-like
Starting position and time for the field line.
field : Field
The field object that provides electric and magnetic field vectors and
related quantities.
Bm : float
The mirror field strength in Tesla (defines the endpoints of the field line).
Returns
-------
array
Array of (s, B, B'') triplets for each critical point, where s is the
location along the field line (with respect to the initial position),
B is the field strength at s, and B'' is the second derivative along
the field line.
"""
fl = Fieldline(tpos, field, Bmax=Bm)
fl.trace()
s = fl.gets()
b = fl.getB()
n = len(b)
slist = []
# first, make a list of approximate locations of minima or maxima.
for i in range(1, n - 1):
if (b[i] - b[i - 1]) * (b[i + 1] - b[i]) <= 0:
slist.append(s[i])
# Create an interpolated function B(s)
B = interp1d(s, b, kind='cubic', assume_sorted=True)
# Find the optimal point for each value in slist
res = []
for s0 in slist:
m = minimize(B, s0)
sm = m.x[0]
B2s = derivative(B, sm, dx=0.1 * (s[1] - s[0]), n=2)
res.append((sm, float(B(sm)), B2s))
return res
def eye(tpos, field, Bm, usedipole=False):
"""
Return the I integral (second invariant) value.
Parameters
----------
tpos : array-like
4-element vector of time (s) and starting position of the field line
(meters).
Bm : float
Mirror field value in Tesla.
field: Field object
The field object under which the integral is evaluated.
Returns
-------
float
The second invariant I, evaluated over the field line passing through
the starting point at the given time.
Other Parameters
----------------
usedipole : bool, optional
If True, uses the formula for dipole, without following a field line.
"""
eqpa_threshold = 70 # Below this value, use Simpson's rule. Otherwise use interpolation.
# Return a more accurate form for I if field is a dipole.
if usedipole:
R0 = np.sqrt(np.dot(tpos[1:3],tpos[1:3]))
return dipoleI(R0,Bm)
# Follow a field line from the starting point up to Bm.
# The trace will slightly overshoot the mirror point.
fl = Fieldline(tpos, field, Bmax = Bm)
fl.trace()
s = fl.gets()
b = fl.getB()
n = len(b)
Bmin = np.min(b)
if Bmin > Bm:
return 0
if abs(Bmin-Bm)/Bm < 1e-12: # equatorial particle
return 0
# Determine the indices of points that satisfy B<Bm
inside = np.where(b<Bm)[0]
# Discard points so that only one point lies beyond the mirror locations.
b = np.delete(b, list(range(0,inside[0]-1)) + list(range(inside[-1]+2, n)))
s = np.delete(s, list(range(0,inside[0]-1)) + list(range(inside[-1]+2, n)))
assert b[0]>Bm and b[1]<Bm and b[-2]<Bm and b[-1]>Bm
eqpa = np.arcsin(np.sqrt(Bmin/Bm)) * 180 / np.pi # equatorial pitch angle (estimated)
if eqpa < eqpa_threshold: # use Simpson's rule to evaluate the integral
# Determine the lower (antiparallel direction) mirror point location
sm1 = (Bm-b[0]) * (s[1]-s[0]) / (b[1]-b[0]) + s[0]
# Determine the upper (parallel direction) mirror point location
sm2 = (Bm-b[-2]) * (s[-1]-s[-2]) / (b[-1]-b[-2]) + s[-2]
s[0], s[-1] = sm1, sm2
b[0], b[-1] = Bm, Bm
#Evaluate all but the end intervals with Simpson's rule
bi = np.sqrt(1-b[1:-1]/Bm)
eye = simps(bi,s[1:-1])
# Add the end intervals with special formula
d = s[-1]-s[-2]
eye += (2/3)*d*np.sqrt((Bm-b[-2])/Bm)
d = s[1]-s[0]
eye += (2/3)*d*np.sqrt((Bm-b[1])/Bm)
else: # use interpolation and adaptive quadrature for integration
# Generate an interpolating function for B(s)
B = interp1d(s, b, kind='quadratic', assume_sorted=True)
# Determine the lower (antiparallel direction) mirror point location
# with root finding.
sm1 = brentq(lambda x: B(x)-Bm, s[0],s[1])
# Determine the upper (parallel direction) mirror point location
if B(s[-2])==Bm:
sm2 = s[-2]
else:
sm2 = brentq(lambda x: B(x)-Bm, s[-2],s[-1])
eye = quad(lambda x: np.sqrt(1-B(x)/Bm), sm1, sm2, epsrel=1e-4)[0]
return eye
def halfbouncepath(tpos,field,Bm):
"""Return the half-bounce path length S_b.
Parameters
----------
tpos : array-like
4-element array of time and position.
field: Field object
The field object under which the integral is evaluated.
Bm : float
Mirror field value (T).
Returns
-------
float
The half-bounce path length (m) : The total helical distance traveled by a
particle around the guiding line passing through the given location and
mirroring at B=Bm.
"""
# Follow a field line from the starting point up to Bm.
# The trace will slightly overshoot the mirror points.
fl = Fieldline(tpos, field, Bmax = Bm)
fl.trace()
s = fl.gets()
b = fl.getB()
n = len(b)
# Determine the indices of points that satisfy B<Bm
inside = np.where(b<=Bm)[0]
# there are 0 or 1 such points, we have an equatorial particle.
# Discard points so that only one point lies beyond each mirror location.
# i1,i2 are the indices of points to keep.
if len(inside) == 0:
i1 = int((n-3)/2)
i2 = int((n+1)/2)
else:
i1,i2 = inside[0]-1,inside[-1]+1
b = np.delete(b, list(range(0,i1)) + list(range(i2+1, n)))
s = np.delete(s, list(range(0,i1)) + list(range(i2+1, n)))
n = len(b)
if n == 3: # Equatorial particle
# Evaluate Sb = pi * sqrt(2Bm/B2s)
# where B2s is the second derivative at the minimum.
# Estimate it using Lagrange polynomials of degree 2.
s1, s2, s3 = s[0], s[1], s[2]
B1, B2, B3 = b[0], b[1], b[2]
s12 = s1-s2
s23 = s2-s3
s13 = s1-s3
B2s = 2*(B1*s23 - B2*s13 + B3*s12) / (s12*s13*s23)
res = np.pi * np.sqrt(2*Bm/B2s)
else:
# Get an interpolating function for B(s)
B = interp1d(s, b, kind='quadratic', assume_sorted=True)
# Determine the lower (antiparallel direction) mirror point location
# with root finding.
sm1 = brentq(lambda x: B(x)-Bm, s[0],s[1])
# Determine the upper (parallel direction) mirror point location
sm2 = brentq(lambda x: B(x)-Bm, s[-2],s[-1])
res = quad(lambda x: 1/np.sqrt(1-B(x)/Bm), sm1, sm2, epsrel=1e-4)[0]
return res
def eye(tpos, field, Bm, usedipole=False):
"""
Return the I integral (second invariant) value.
Parameters
----------
tpos : array-like
4-element vector of time (s) and starting position of the field line
(meters).
Bm : float
Mirror field value in Tesla.
field: Field object
The field object under which the integral is evaluated.
Returns
-------
float
The second invariant I, evaluated over the field line passing through
the starting point at the given time.
Other Parameters
----------------
usedipole : bool, optional
If True, uses the formula for dipole, without following a field line.
"""
eqpa_threshold = 70 # Below this value, use Simpson's rule. Otherwise use interpolation.
# Return a more accurate form for I if field is a dipole.
if usedipole:
R0 = np.sqrt(np.dot(tpos[1:3], tpos[1:3]))
return dipoleI(R0, Bm)
# Follow a field line from the starting point up to Bm.
# The trace will slightly overshoot the mirror point.
fl = Fieldline(tpos, field, Bmax=Bm)
fl.trace()
s = fl.gets()
b = fl.getB()
n = len(b)
Bmin = np.min(b)
if Bmin > Bm:
return 0
if abs(Bmin - Bm) / Bm < 1e-12: # equatorial particle
return 0
# Determine the indices of points that satisfy B<Bm
inside = np.where(b < Bm)[0]
# Discard points so that only one point lies beyond the mirror locations.
b = np.delete(
b,
list(range(0, inside[0] - 1)) + list(range(inside[-1] + 2, n)))
s = np.delete(
s,
list(range(0, inside[0] - 1)) + list(range(inside[-1] + 2, n)))
assert b[0] > Bm and b[1] < Bm and b[-2] < Bm and b[-1] > Bm
eqpa = np.arcsin(np.sqrt(
Bmin / Bm)) * 180 / np.pi # equatorial pitch angle (estimated)
if eqpa < eqpa_threshold: # use Simpson's rule to evaluate the integral
# Determine the lower (antiparallel direction) mirror point location
sm1 = (Bm - b[0]) * (s[1] - s[0]) / (b[1] - b[0]) + s[0]
# Determine the upper (parallel direction) mirror point location
sm2 = (Bm - b[-2]) * (s[-1] - s[-2]) / (b[-1] - b[-2]) + s[-2]
s[0], s[-1] = sm1, sm2
b[0], b[-1] = Bm, Bm
#Evaluate all but the end intervals with Simpson's rule
bi = np.sqrt(1 - b[1:-1] / Bm)
eye = simps(bi, s[1:-1])
# Add the end intervals with special formula
d = s[-1] - s[-2]
eye += (2 / 3) * d * np.sqrt((Bm - b[-2]) / Bm)
d = s[1] - s[0]
eye += (2 / 3) * d * np.sqrt((Bm - b[1]) / Bm)
else: # use interpolation and adaptive quadrature for integration
# Generate an interpolating function for B(s)
B = interp1d(s, b, kind='quadratic', assume_sorted=True)
# Determine the lower (antiparallel direction) mirror point location
# with root finding.
sm1 = brentq(lambda x: B(x) - Bm, s[0], s[1])
# Determine the upper (parallel direction) mirror point location
if B(s[-2]) == Bm:
sm2 = s[-2]
else:
sm2 = brentq(lambda x: B(x) - Bm, s[-2], s[-1])
eye = quad(lambda x: np.sqrt(1 - B(x) / Bm), sm1, sm2, epsrel=1e-4)[0]
return eye
def halfbouncepath(tpos, field, Bm):
"""Return the half-bounce path length S_b.
Parameters
----------
tpos : array-like
4-element array of time and position.
field: Field object
The field object under which the integral is evaluated.
Bm : float
Mirror field value (T).
Returns
-------
float
The half-bounce path length (m) : The total helical distance traveled by a
particle around the guiding line passing through the given location and
mirroring at B=Bm.
"""
# Follow a field line from the starting point up to Bm.
# The trace will slightly overshoot the mirror points.
fl = Fieldline(tpos, field, Bmax=Bm)
fl.trace()
s = fl.gets()
b = fl.getB()
n = len(b)
# Determine the indices of points that satisfy B<Bm
inside = np.where(b <= Bm)[0]
# there are 0 or 1 such points, we have an equatorial particle.
# Discard points so that only one point lies beyond each mirror location.
# i1,i2 are the indices of points to keep.
if len(inside) == 0:
i1 = int((n - 3) / 2)
i2 = int((n + 1) / 2)
else:
i1, i2 = inside[0] - 1, inside[-1] + 1
b = np.delete(b, list(range(0, i1)) + list(range(i2 + 1, n)))
s = np.delete(s, list(range(0, i1)) + list(range(i2 + 1, n)))
n = len(b)
if n == 3: # Equatorial particle
# Evaluate Sb = pi * sqrt(2Bm/B2s)
# where B2s is the second derivative at the minimum.
# Estimate it using Lagrange polynomials of degree 2.
s1, s2, s3 = s[0], s[1], s[2]
B1, B2, B3 = b[0], b[1], b[2]
s12 = s1 - s2
s23 = s2 - s3
s13 = s1 - s3
B2s = 2 * (B1 * s23 - B2 * s13 + B3 * s12) / (s12 * s13 * s23)
res = np.pi * np.sqrt(2 * Bm / B2s)
else:
# Get an interpolating function for B(s)
B = interp1d(s, b, kind='quadratic', assume_sorted=True)
# Determine the lower (antiparallel direction) mirror point location
# with root finding.
sm1 = brentq(lambda x: B(x) - Bm, s[0], s[1])
# Determine the upper (parallel direction) mirror point location
sm2 = brentq(lambda x: B(x) - Bm, s[-2], s[-1])
res = quad(lambda x: 1 / np.sqrt(1 - B(x) / Bm), sm1, sm2,
epsrel=1e-4)[0]
return res
