def kws(self, offset, aZi, aXi):
'''
Finds 2D source terms to approximate a band-limited point source, based on
Hicks, Graham J. (2002) Arbitrary source and receiver positioning in finite-difference
schemes using Kaiser windowed sinc functions. Geophysics (67) 1, 156-166.
KaiserWindowedSinc(ireg, offset) --> 2D ndarray of size (2*ireg+1, 2*ireg+1)
Input offset is the 2D offsets in fractional gridpoints between the source location and
the nearest node on the modelling grid.
Args:
offset (tuple): Distance of the centre of the source region from the true source
point (in 2D coordinates, in units of cells).
Returns:
np.ndarray: Interpolated source region of size (2*ireg+1, 2*ireg+1)
'''
try:
b = self.HC_KAISER.get(self.ireg)
except KeyError:
print('Kaiser windowed sinc function not implemented for half-width of %d!'%(self.ireg,))
raise
freg = 2*self.ireg+1
xOffset, zOffset = offset
# Grid from 0 to freg-1
Zi, Xi = np.mgrid[:freg,:freg]
Zi, Xi = self.modifyGrid(Zi, Xi, aZi, aXi)
# Distances from source point
dZi = (zOffset + self.ireg - Zi)
dXi = (xOffset + self.ireg - Xi)
# Taper terms for decay function
with warnings.catch_warnings():
warnings.simplefilter('ignore')
tZi = np.nan_to_num(np.sqrt(1 - (dZi / self.ireg)**2))
tXi = np.nan_to_num(np.sqrt(1 - (dXi / self.ireg)**2))
tZi[tZi == np.inf] = 0
tXi[tXi == np.inf] = 0
# Actual tapers for Kaiser window
taperZ = bessi0(b*tZi) / bessi0(b)
taperX = bessi0(b*tXi) / bessi0(b)
# Windowed sinc responses in Z and X
responseZ = np.sinc(dZi) * taperZ
responseX = np.sinc(dXi) * taperX
# Combined 2D source response
result = responseX * responseZ
return result
def KaiserWindowedSinc(ireg, offset):
'''
Finds 2D source terms to approximate a band-limited point source, based on
Hicks, Graham J. (2002) Arbitrary source and receiver positioning in finite-difference
schemes using Kaiser windowed sinc functions. Geophysics (67) 1, 156-166.
KaiserWindowedSince(ireg, offset) --> 2D ndarray of size (2*ireg+1, 2*ireg+1)
Input offset is the 2D offsets in fractional gridpoints between the source location and
the nearest node on the modelling grid.
'''
from scipy.special import i0 as bessi0
import warnings
ireg = int(ireg)
try:
b = HC_KAISER.get(ireg)
except KeyError:
print(
'Kaiser windowed sinc function not implemented for half-width of %d!'
% (ireg, ))
raise
freg = 2 * ireg + 1
xOffset, zOffset = offset
# Grid from 0 to freg-1
Zi, Xi = numpy.mgrid[:freg, :freg]
# Distances from source point
dZi = (zOffset + ireg - Zi)
dXi = (xOffset + ireg - Xi)
# Taper terms for decay function
with warnings.catch_warnings():
warnings.simplefilter('ignore')
tZi = numpy.nan_to_num(numpy.sqrt(1 - (dZi / ireg)**2))
tXi = numpy.nan_to_num(numpy.sqrt(1 - (dXi / ireg)**2))
tZi[tZi == numpy.inf] = 0
tXi[tXi == numpy.inf] = 0
# Actual tapers for Kaiser window
taperZ = bessi0(b * tZi) / bessi0(b)
taperX = bessi0(b * tXi) / bessi0(b)
# Windowed sinc responses in Z and X
responseZ = numpy.sinc(dZi) * taperZ
responseX = numpy.sinc(dXi) * taperX
# Combined 2D source response
result = responseX * responseZ
return result
def KaiserWindowedSinc(ireg, offset):
'''
Finds 2D source terms to approximate a band-limited point source, based on
Hicks, Graham J. (2002) Arbitrary source and receiver positioning in finite-difference
schemes using Kaiser windowed sinc functions. Geophysics (67) 1, 156-166.
KaiserWindowedSinc(ireg, offset) --> 2D ndarray of size (2*ireg+1, 2*ireg+1)
Input offset is the 2D offsets in fractional gridpoints between the source location and
the nearest node on the modelling grid.
'''
from scipy.special import i0 as bessi0
import warnings
try:
b = HC_KAISER.get(ireg)
except KeyError:
print('Kaiser windowed sinc function not implemented for half-width of %d!'%(ireg,))
raise
freg = 2*ireg+1
xOffset, zOffset = offset
# Grid from 0 to freg-1
Zi, Xi = numpy.mgrid[:freg,:freg]
# Distances from source point
dZi = (zOffset + ireg - Zi)
dXi = (xOffset + ireg - Xi)
# Taper terms for decay function
with warnings.catch_warnings():
warnings.simplefilter('ignore')
tZi = numpy.nan_to_num(numpy.sqrt(1 - (dZi / ireg)**2))
tXi = numpy.nan_to_num(numpy.sqrt(1 - (dXi / ireg)**2))
tZi[tZi == numpy.inf] = 0
tXi[tXi == numpy.inf] = 0
# Actual tapers for Kaiser window
taperZ = bessi0(b*tZi) / bessi0(b)
taperX = bessi0(b*tXi) / bessi0(b)
# Windowed sinc responses in Z and X
responseZ = numpy.sinc(dZi) * taperZ
responseX = numpy.sinc(dXi) * taperX
# Combined 2D source response
result = responseX * responseZ
return result
u=1/x
Ae=e*(1+e/2)/(1+e)
Be=1+e
return 0.5-Ae*u+np.sqrt((0.5-Ae*u)**2+Be*u)
GMsun_kpc = 4.300917248e-6 # G*M_sun/kpc (km/s)^2
GMsun_kpc2 = 1.3938e-19 # G*M_sun/kpc^2 m/s^2
Md = 1.e11 # mass of the disk in solar masses
a0 = 1.2e-10 # MOND critical acceleration in m/s^2
Rd = 5. # disk scale length in kpc
R = np.logspace(-2.,1.7,100) # radius in units of kpc
M_R = Md*(1-np.exp(-R/Rd)*(1+R/Rd))
Vsphere = np.sqrt(GMsun_kpc*M_R/R)
y = R/(2*Rd)
bess = bessi0(y)*bessk0(y)-bessi1(y)*bessk1(y)
Vd = np.sqrt(2*GMsun_kpc*(Md/Rd)*y**2*bess)
Mb = 2.e10
R_H=1.5/(1.+np.sqrt(2.)) # Hernquist radius in kpc
Vb=1.e-3*np.sqrt(Mb)*np.sqrt(4.3)*(R/(R+R_H)**2)**0.5 # rotation velocity due to the bulge (km/s)
Vtot=np.sqrt(Vb**2+Vd**2)
gtot=(1.e3*Vtot)**2/(R*3.086e+19)
Vtot_mond = np.sqrt(fEFE([a0,0.],gtot/a0))*Vtot
fig = plt.figure(figsize=(15,10))
gs = gridspec.GridSpec(10,10)
plt.rc('xtick',labelsize=20)
plt.rc('ytick',labelsize=20)
plt.rc('axes',labelsize=25)
