def _accR(R, z, dens, sigma, qintr, bhMass, soft):
mgepot = np.empty_like(R)
pot = np.empty_like(dens)
e2 = 1. - qintr**2
s2 = sigma**2
r2 = R**2
z2 = z**2
d2 = r2 + z2
for k in range(R.size):
for j in range(dens.size):
if (d2[k] < s2[j] / 240.**2):
e = np.sqrt(
e2[j]
) # pot is Integral in {u,0,1} of -D[H[R,z,u],R]/R at (R,z)=0
pot[j] = (np.arcsin(e) / e - qintr[j]) / (
2 * e2[j] * s2[j]) # Cfr. equation (A5)
elif (d2[k] < s2[j] * 245**2):
pot[j] = quadva(_accelerationR_dRRcapitalh, [0., 1.],
args=(r2[k], z2[k], e2[j], s2[j]))[0]
else: # R acceleration in Keplerian limit (Cappellari et al. 2002)
pot[j] = np.sqrt(
np.pi / 2) * sigma[j] / d2[k]**1.5 # Cfr. equation (A4)
mgepot[k] = np.sum(s2 * qintr * dens * pot)
G = 0.00430237 # (km/s)**2 pc/Msun [6.674e-11 SI units (CODATA-10)]
return -R * (4 * np.pi * G * mgepot + G * bhMass / (d2 + soft**2)**1.5)
def _accR(R, z, dens, sigma, qintr, bhMass, soft):
mgepot = np.empty_like(R)
pot = np.empty_like(dens)
e2 = 1. - qintr**2
s2 = sigma**2
r2 = R**2
z2 = z**2
d2 = r2 + z2
for k in range(R.size):
for j in range(dens.size):
if (d2[k] < s2[j]/240.**2):
e = np.sqrt(e2[j]) # pot is Integral in {u,0,1} of -D[H[R,z,u],R]/R at (R,z)=0
pot[j] = (np.arcsin(e)/e - qintr[j])/(2*e2[j]*s2[j]) # Cfr. equation (A5)
elif (d2[k] < s2[j]*245**2):
pot[j] = quadva(_accelerationR_dRRcapitalh, [0.,1.],
args=(r2[k], z2[k], e2[j], s2[j]))[0]
else: # R acceleration in Keplerian limit (Cappellari et al. 2002)
pot[j] = np.sqrt(np.pi/2)*sigma[j]/d2[k]**1.5 # Cfr. equation (A4)
mgepot[k] = np.sum(s2*qintr*dens*pot)
G = 0.00430237 # (km/s)**2 pc/Msun [6.674e-11 SI units (CODATA-10)]
return -R*(4*np.pi*G*mgepot + G*bhMass/(d2 + soft**2)**1.5)
def _wvrms2(x_pc, y_pc, inc, lum3d_pc, pot3d_pc, beta, tensor):
'''
Calculate the surface brightness weighted vrms2 for all the points x, y
--input parameters
x_pc, y_pc: 1d array for x, y position in pc
inc: inclination in rad
lum3d_pc, pot3d_pc: N*3 array for mge coefficients, L in L_solar/pc^2,
sigma in pc
beta: Anisotropy paramter vactor for each luminous Gaussian component
tensor: moments to be calculated, usually zz (LOS)
--output paramters:
wvrms2: surface brightness weighted vrms2 (1d array with the same
length as x_pc)
'''
wvrms2 = np.empty_like(x_pc)
for i in range(len(x_pc)):
'''
lhy: Change _integrand function to enable scipy integration!
wvrms2[i] = quad(_integrand, 0.0, 1.0,
args=(lum3d_pc[:, 0], lum3d_pc[:, 1],
lum3d_pc[:, 2], pot3d_pc[:, 0],
pot3d_pc[:, 1], pot3d_pc[:, 2],
x_pc[i], y_pc[i], inc, beta, tensor))
'''
wvrms2[i] = quadva(_integrand, [0., 1.],
args=(lum3d_pc[:, 0], lum3d_pc[:, 1], lum3d_pc[:, 2],
pot3d_pc[:, 0], pot3d_pc[:, 1], pot3d_pc[:, 2],
x_pc[i], y_pc[i], inc, beta, tensor))[0]
# lhy This could be translated to c
return wvrms2
def _surf_vlos(x1, y1, inc_deg,
surf_lum, sigma_lum, qobs_lum,
surf_pot, sigma_pot, qobs_pot,
beta, kappa, gamma, component, nsteps):
"""
This routine gives the projected weighted first moment Sigma*<V_los>
"""
# Axisymmetric deprojection of both luminous and total mass.
# See equation (12)-(14) of Cappellari (2008)
#
inc = np.radians(inc_deg)
qintr_lum = qobs_lum**2 - np.cos(inc)**2
if np.any(qintr_lum <= 0):
raise RuntimeError('Inclination too low q < 0')
qintr_lum = np.sqrt(qintr_lum)/np.sin(inc)
if np.any(qintr_lum < 0.05):
raise RuntimeError('q < 0.05 components')
dens_lum = surf_lum*qobs_lum / (sigma_lum*qintr_lum*np.sqrt(2*np.pi))
qintr_pot = qobs_pot**2 - np.cos(inc)**2
if np.any(qintr_pot <= 0):
raise RuntimeError('Inclination too low q < 0')
qintr_pot = np.sqrt(qintr_pot)/np.sin(inc)
if np.any(qintr_pot < 0.05):
raise RuntimeError('q < 0.05 components')
dens_pot = surf_pot*qobs_pot / (sigma_pot*qintr_pot*np.sqrt(2*np.pi))
s2_lum = sigma_lum**2
q2_lum = qintr_lum**2
s2_pot = sigma_pot**2
q2_pot = qintr_pot**2
bani = 1./(1. - beta) # Anisotropy ratio b = (sig_R/sig_z)**2
dens_lum = dens_lum[:, None, None, None]
s2_lum = s2_lum[:, None, None, None]
q2_lum = q2_lum[:, None, None, None]
bani = bani[:, None, None, None]
kappa = kappa[:, None, None, None]
gamma = gamma[:, None, None, None]
dens_pot = dens_pot[None, :, None, None]
s2_pot = s2_pot[None, :, None, None]
q2_pot = q2_pot[None, :, None, None]
# Restrict LOS integration to the range where the luminosity density
# is above 1/1000 of the peak value along that LOS
#
ms = 3*np.max(sigma_lum)
sh = np.sinh(np.linspace(0, 3, 50))
z1 = ms/sh[-1]*sh # sinh sampling of z1 in [0, ms]
R2 = (z1*np.sin(inc) - y1*np.cos(inc))**2 + x1**2 # R^2 Equation (25)
z2 = (z1*np.cos(inc) + y1*np.sin(inc))**2
nu = dens_lum * np.exp(-0.5/s2_lum * (R2 + z2/q2_lum))
nu = np.sum(nu, axis=(0, 1, 2))
j = np.argmax(nu)
w = (nu < nu[j]/1e3) & (z1 > z1[j]) # allow for negative Gaussians
if np.any(w):
ms = np.min(z1[w])
sb_mu1 = quadva(_los_integrand, [-ms, ms], epsrel=1e-3,
args=(dens_lum, s2_lum, q2_lum, dens_pot, s2_pot, q2_pot,
x1, y1, inc, bani, kappa, gamma, component, nsteps))
return sb_mu1[0] # Ignore the integral error
def _vrms2(x, y, inc_deg, surf_lum, sigma_lum, qobs_lum, surf_pot, sigma_pot,
qobs_pot, beta, tensor, sigmaPsf, normPsf, pixSize, pixAng, step,
nrad, nang):
"""
This routine gives the second V moment after convolution with a PSF.
The convolution is done using interpolation of the model on a
polar grid, as described in Appendix A of Cappellari (2008).
"""
# Axisymmetric deprojection of both luminous and total mass.
# See equation (12)-(14) of Cappellari (2008)
#
inc = np.radians(inc_deg)
qintr_lum = qobs_lum**2 - np.cos(inc)**2
if np.any(qintr_lum <= 0):
raise RuntimeError('Inclination too low q < 0')
qintr_lum = np.sqrt(qintr_lum) / np.sin(inc)
if np.any(qintr_lum < 0.05):
raise RuntimeError('q < 0.05 components')
dens_lum = surf_lum * qobs_lum / (sigma_lum * qintr_lum *
np.sqrt(2 * np.pi))
qintr_pot = qobs_pot**2 - np.cos(inc)**2
if np.any(qintr_pot <= 0):
raise RuntimeError('Inclination too low q < 0')
qintr_pot = np.sqrt(qintr_pot) / np.sin(inc)
if np.any(qintr_pot < 0.05):
raise RuntimeError('q < 0.05 components')
dens_pot = surf_pot * qobs_pot / (sigma_pot * qintr_pot *
np.sqrt(2 * np.pi))
# Define parameters of polar grid for interpolation
#
w = sigma_lum < np.max(
np.abs(x)) # Characteristic MGE axial ratio in observed range
if w.sum() < 3:
qmed = np.median(qobs_lum)
else:
qmed = np.median(qobs_lum[w])
rell = np.sqrt(x**2 + (y / qmed)**2) # Elliptical radius of input (x, y)
psfConvolution = (np.max(sigmaPsf) > 0) and (pixSize > 0)
# Kernel step is 1/4 of largest value between sigma(min) and 1/2 pixel side.
# Kernel half size is the sum of 3*sigma(max) and 1/2 pixel diagonal.
#
if (nrad * nang >
x.size) and (not psfConvolution): # Just calculate values
xPol = x
yPol = y
else: # Interpolate values on polar grid
if psfConvolution: # PSF convolution
if step == 0:
step = max(pixSize / 2., np.min(sigmaPsf)) / 4.
mx = 3 * np.max(sigmaPsf) + pixSize / np.sqrt(2)
else: # No convolution
step = np.min(rell.clip(1)) # Minimum radius of 1pc
mx = 0
# Make linear grid in log of elliptical radius RAD and eccentric anomaly ANG
# See Appendix A
#
rmax = np.max(
rell
) + mx # Major axis of ellipse containing all data + convolution
logRad = np.linspace(np.log(step), np.log(rmax),
nrad) # Linear grid in np.log(rell)
ang = np.linspace(0, np.pi / 2,
nang) # Linear grid in eccentric anomaly
radGrid, angGrid = map(np.ravel, np.meshgrid(np.exp(logRad), ang))
xPol = radGrid * np.cos(angGrid)
yPol = radGrid * np.sin(angGrid) * qmed
# The model Vrms computation is only performed on the polar grid
# which is then used to interpolate the values at any other location
#
wm2Pol = np.empty_like(xPol)
mgePol = np.empty_like(xPol)
for j in range(xPol.size):
wm2Pol[j] = quadva(_integrand, [0., 1.],
args=(dens_lum, sigma_lum, qintr_lum, dens_pot,
sigma_pot, qintr_pot, xPol[j], yPol[j], inc,
beta, tensor))[0]
mgePol[j] = np.sum(surf_lum * np.exp(-0.5 / sigma_lum**2 *
(xPol[j]**2 +
(yPol[j] / qobs_lum)**2)))
if psfConvolution: # PSF convolution
nx = np.ceil(rmax / step)
ny = np.ceil(rmax * qmed / step)
x1 = np.linspace(-nx, nx, 2 * nx) * step
y1 = np.linspace(-ny, ny, 2 * ny) * step
xCar, yCar = np.meshgrid(x1, y1) # Cartesian grid for convolution
# Interpolate MGE model and Vrms over cartesian grid
#
r1 = 0.5 * np.log(
xCar**2 +
(yCar / qmed)**2) # Log elliptical radius of cartesian grid
e1 = np.arctan2(np.abs(yCar / qmed),
np.abs(xCar)) # Eccentric anomaly of cartesian grid
wm2Car = bilinear_interpolate(logRad, ang, wm2Pol.reshape(nang, nrad),
r1, e1)
mgeCar = bilinear_interpolate(logRad, ang, mgePol.reshape(nang, nrad),
r1, e1)
nk = np.ceil(mx / step)
kgrid = np.linspace(-nk, nk, 2 * nk) * step
xgrid, ygrid = np.meshgrid(kgrid, kgrid) # Kernel is square
if pixAng != 0:
xgrid, ygrid = rotate_points(xgrid, ygrid, pixAng)
# Compute kernel with equation (A6) of Cappellari (2008).
# Normaliztion is irrelevant here as it cancels out.
#
kernel = np.zeros_like(xgrid)
dx = pixSize / 2
sp = np.sqrt(2) * sigmaPsf
for j in range(len(sigmaPsf)):
kernel += normPsf[j] \
* (special.erf((dx-xgrid)/sp[j]) + special.erf((dx+xgrid)/sp[j])) \
* (special.erf((dx-ygrid)/sp[j]) + special.erf((dx+ygrid)/sp[j]))
kernel /= np.sum(kernel)
# Seeing and aperture convolution with equation (A3)
#
muCar = signal.fftconvolve(wm2Car, kernel, mode='same') \
/ signal.fftconvolve(mgeCar, kernel, mode='same')
# Interpolate convolved image at observed apertures.
# Aperture integration was already included in the kernel.
#
mu = bilinear_interpolate(x1, y1, muCar, x, y)
else: # No PSF convolution
muPol = wm2Pol / mgePol
if nrad * nang > x.size: # Just returns values
mu = muPol
else: # Interpolate values
r1 = 0.5 * np.log(
x**2 + (y / qmed)**2) # Log elliptical radius of input (x,y)
e1 = np.arctan2(np.abs(y / qmed),
np.abs(x)) # Eccentric anomaly of input (x,y)
mu = bilinear_interpolate(logRad, ang, muPol.reshape(nang, nrad),
r1, e1)
return mu
def _surf_vlos(x1, y1, inc_deg, surf_lum, sigma_lum, qobs_lum, surf_pot,
sigma_pot, qobs_pot, beta, kappa, gamma, component, nsteps):
"""
This routine gives the projected weighted first moment Sigma*<V_los>
"""
# Axisymmetric deprojection of both luminous and total mass.
# See equation (12)-(14) of Cappellari (2008)
#
inc = np.radians(inc_deg)
qintr_lum = qobs_lum**2 - np.cos(inc)**2
if np.any(qintr_lum <= 0):
raise RuntimeError('Inclination too low q < 0')
qintr_lum = np.sqrt(qintr_lum) / np.sin(inc)
if np.any(qintr_lum < 0.05):
raise RuntimeError('q < 0.05 components')
dens_lum = surf_lum * qobs_lum / (sigma_lum * qintr_lum *
np.sqrt(2 * np.pi))
qintr_pot = qobs_pot**2 - np.cos(inc)**2
if np.any(qintr_pot <= 0):
raise RuntimeError('Inclination too low q < 0')
qintr_pot = np.sqrt(qintr_pot) / np.sin(inc)
if np.any(qintr_pot < 0.05):
raise RuntimeError('q < 0.05 components')
dens_pot = surf_pot * qobs_pot / (sigma_pot * qintr_pot *
np.sqrt(2 * np.pi))
s2_lum = sigma_lum**2
q2_lum = qintr_lum**2
s2_pot = sigma_pot**2
q2_pot = qintr_pot**2
bani = 1. / (1. - beta) # Anisotropy ratio b = (sig_R/sig_z)**2
dens_lum = dens_lum[:, None, None, None]
s2_lum = s2_lum[:, None, None, None]
q2_lum = q2_lum[:, None, None, None]
bani = bani[:, None, None, None]
kappa = kappa[:, None, None, None]
gamma = gamma[:, None, None, None]
dens_pot = dens_pot[None, :, None, None]
s2_pot = s2_pot[None, :, None, None]
q2_pot = q2_pot[None, :, None, None]
# Restrict LOS integration to the range where the luminosity density
# is above 1/1000 of the peak value along that LOS
#
ms = 3 * np.max(sigma_lum)
sh = np.sinh(np.linspace(0, 3, 50))
z1 = ms / sh[-1] * sh # sinh sampling of z1 in [0, ms]
R2 = (z1 * np.sin(inc) - y1 * np.cos(inc))**2 + x1**2 # R^2 Equation (25)
z2 = (z1 * np.cos(inc) + y1 * np.sin(inc))**2
nu = dens_lum * np.exp(-0.5 / s2_lum * (R2 + z2 / q2_lum))
nu = np.sum(nu, axis=(0, 1, 2))
j = np.argmax(nu)
w = (nu < nu[j] / 1e3) & (z1 > z1[j]) # allow for negative Gaussians
if np.any(w):
ms = np.min(z1[w])
sb_mu1 = quadva(_los_integrand, [-ms, ms],
epsrel=1e-3,
args=(dens_lum, s2_lum, q2_lum, dens_pot, s2_pot, q2_pot,
x1, y1, inc, bani, kappa, gamma, component, nsteps))
return sb_mu1[0] # Ignore the integral error
def _second_moment(R, sig_l, sig_m, lum, mass, Mbh, beta, tensor,
sigmaPsf, normPsf, step, nrad, surf_l, pixSize):
"""
This routine gives the second V moment after convolution with a PSF.
The convolution is done using interpolation of the model on a
polar grid, as described in Appendix A of Cappellari (2008).
"""
if (max(sigmaPsf) > 0) and (pixSize > 0): # PSF convolution
# Kernel step is 1/4 of largest value between sigma(min) and 1/2 pixel side.
# Kernel half size is the sum of 3*sigma(max) and 1/2 pixel diagonal.
#
if step == 0:
step = max(pixSize/2., np.min(sigmaPsf))/4.
mx = 3*np.max(sigmaPsf) + pixSize/np.sqrt(2)
# Make grid linear in log of radius RR
#
rmax = np.max(R) + mx # Radius of circle containing all data + convolution
logRad = np.linspace(np.log(step), np.log(rmax), nrad) # Linear grid in log(RR)
rr = np.exp(logRad)
# The model Vrms computation is only performed on the radial grid
# which is then used to interpolate the values at any other location
#
wm2Pol = np.empty_like(rr)
mgePol = np.empty_like(rr)
rup = 3*np.max(sig_l)
for j in range(rr.size): # Integration of equation (50)
wm2Pol[j] = quadva(_integrand, [rr[j], rup],
args=(sig_l, sig_m, lum, mass, Mbh, rr[j], beta, tensor))[0]
mgePol[j] = np.sum(surf_l * np.exp(-0.5*(rr[j]/sig_l)**2))
nx = np.ceil(rmax/step)
x1 = np.linspace(-nx, nx, 2*nx)*step
xCar, yCar = np.meshgrid(x1, x1) # Cartesian grid for convolution
# Interpolate MGE model and Vrms over cartesian grid
#
r1 = 0.5*np.log(xCar**2 + yCar**2) # Log radius of cartesian grid
wm2Car = np.interp(r1, logRad, wm2Pol)
mgeCar = np.interp(r1, logRad, mgePol)
nk = np.ceil(mx/step)
kgrid = np.linspace(-nk, nk, 2*nk)*step
xgrid, ygrid = np.meshgrid(kgrid, kgrid) # Kernel is square
# Compute kernel with equation (A6) of Cappellari (2008).
# Normalization is irrelevant here as it cancels out.
#
kernel = np.zeros_like(xgrid)
dx = pixSize/2
sp = np.sqrt(2)*sigmaPsf
for j in range(len(sigmaPsf)):
kernel += normPsf[j] \
* (special.erf((dx-xgrid)/sp[j]) + special.erf((dx+xgrid)/sp[j])) \
* (special.erf((dx-ygrid)/sp[j]) + special.erf((dx+ygrid)/sp[j]))
kernel /= np.sum(kernel)
# Seeing and aperture convolution with equation (A3)
#
muCar = np.sqrt(signal.fftconvolve(wm2Car, kernel, mode='same')
/ signal.fftconvolve(mgeCar, kernel, mode='same'))
# Interpolate convolved image at observed apertures.
# Aperture integration was already included in the kernel.
#
mu = bilinear_interpolate(x1, x1, muCar, R/np.sqrt(2), R/np.sqrt(2))
else: # No PSF convolution: just compute values
mu = np.empty_like(R)
rmax = 3*np.max(sig_l)
for j in range(R.size):
wm2Pol = quadva(_integrand, [R[j], rmax],
args=(sig_l, sig_m, lum, mass, Mbh, R[j], beta, tensor))[0]
mgePol = np.sum( surf_l * np.exp(-0.5*(R[j]/sig_l)**2) )
mu[j] = np.sqrt(wm2Pol/mgePol)
return mu