def kappa(r0fine, Mrfine, nufine, sigr2nu, intbetasfine, gp):
# for the following: enabled calculation of kappa
# kappa_r^4
kapr4nu = np.ones(len(r0fine)-gp.nexp)
xint = r0fine # [pc]
yint = gu.G1__pcMsun_1km2s_2 * Mrfine/r0fine**2 # [1/pc (km/s)^2]
yint *= nufine # [Munit/pc^4 (km/s)^2]
yint *= sigr2nu # [Munit/pc^4 (km/s)^4
yint *= np.exp(intbetasfine) # [Munit/pc^4 (km/s)^4]
gh.checkpositive(yint, 'yint in kappa_r^4')
yscale = 10.**(1.-min(np.log10(yint[1:])))
yint *= yscale
# power-law extrapolation to infinity
C = max(0., gh.quadinflog(xint[-3:], yint[-3:], r0fine[-1], gp.rinfty*r0fine[-1]))
splpar_nu = splrep(xint, yint, k=3) # interpolation in real space
for k in range(len(r0fine)-gp.nexp):
# integrate from minimal radius to infinity
kapr4nu[k] = 3.*(np.exp(-intbetasfine[k])/nufine[k]) * \
(splint(r0fine[k], r0fine[-1], splpar_nu) + C) # [(km/s)^4]
kapr4nu /= yscale
gh.checkpositive(kapr4nu, 'kapr4nu in kappa_r^4')
splpar_kap = splrep(r0fine[:-gp.nexp], np.log(kapr4nu), k=3)
kapr4ext = np.exp(splev(r0ext, splpar_kap))
kapr4nu = np.hstack([kapr4nu, kapr4ext])
gh.checkpositive(kapr4nu, 'kapr4nu in extended kappa_r^4')
dbetafinedr = splev(r0fine, splrep(r0fine, betafine), der=1)
gh.checknan(dbetafinedr, 'dbetafinedr in kappa_r^4')
# kappa^4_los*surfdensity
kapl4s = np.zeros(len(r0fine)-gp.nexp)
for k in range(len(r0fine)-gp.nexp):
xnew = np.sqrt(r0fine[k:]**2-r0fine[k]**2) # [pc]
ynew = g(r0fine[k:], r0fine[k], betafine[k:], dbetafinedr[k:]) # [1]
ynew *= nufine[k:] * kapr4nu[k:]
C = max(0., gh.quadinflog(xnew[-3:], ynew[-3:], xnew[-1], gp.rinfty*xnew[-1]))
splpar_nu = splrep(xnew,ynew) # not s=0.1, this sometimes gives negative entries after int
kapl4s[k] = 2. * (splint(0., xnew[-1], splpar_nu) + C)
#kapl4s[k] /= yscale
# LOG('ynew = ',ynew,', kapl4s =', kapl4s[k])
gh.checkpositive(kapl4s, 'kapl4s in kappa_r^4')
# project kappa4_los as well
# only use middle values to approximate, without errors in center and far
kapl4s_out = np.exp(splev(r0, splrep(r0fine[4:-gp.nexp], kapl4s[4:], k=3))) # s=0.
gh.checkpositive(kapl4s_out, 'kapl4s_out in kappa_r^4')
return sigl2s_out, kapl4s_out
def rho_param_INT_Sig(r0, rhodmpar, pop, gp):
# use splines on variable transformed integral
# \Sigma(R) = \int_{r=R}^{R=\infty} \rho(r) d \sqrt(r^2-R^2)
xmin = gp.xfine[0]/15. # needed, if not: loose on first 4 bins
r0nu = gp.xfine
rhonu = phys.rho(r0nu, rhodmpar, pop, gp)
Sig = np.zeros(len(r0nu)-gp.nexp)
for i in range(len(r0nu)-gp.nexp):
xnew = np.sqrt(r0nu[i:]**2-r0nu[i]**2) # [lunit]
ynew = 2.*rhonu[i:]
# power-law extension to infinity
C = gh.quadinflog(xnew[-gp.nexp:], ynew[-gp.nexp:], xnew[-1], gp.rinfty*xnew[-1])
Sig[i] = gh.quadinflog(xnew[1:], ynew[1:], xmin, xnew[-1]) + C # np.inf)
gh.checkpositive(Sig, 'Sig in rho_param_INT_Sig')
# interpolation onto r0
tck1 = splrep(np.log(gp.xfine[:-gp.nexp]), np.log(Sig))
Sigout = np.exp(splev(np.log(r0), tck1))
return Sigout
def rho_param_INT_Sig(r0, rhodmpar, pop, gp):
# use splines on variable transformed integral
# \Sigma(R) = \int_{r=R}^{R=\infty} \rho(r) d \sqrt(r^2-R^2)
xmin = gp.xfine[0] / 15. # needed, if not: loose on first 4 bins
r0nu = gp.xfine
rhonu = phys.rho(r0nu, rhodmpar, pop, gp)
Sig = np.zeros(len(r0nu) - gp.nexp)
for i in range(len(r0nu) - gp.nexp):
xnew = np.sqrt(r0nu[i:]**2 - r0nu[i]**2) # [lunit]
ynew = 2. * rhonu[i:]
# power-law extension to infinity
C = gh.quadinflog(xnew[-gp.nexp:], ynew[-gp.nexp:], xnew[-1],
gp.rinfty * xnew[-1])
Sig[i] = gh.quadinflog(xnew[1:], ynew[1:], xmin,
xnew[-1]) + C # np.inf)
gh.checkpositive(Sig, 'Sig in rho_param_INT_Sig')
# interpolation onto r0
tck1 = splrep(np.log(gp.xfine[:-gp.nexp]), np.log(Sig))
Sigout = np.exp(splev(np.log(r0), tck1))
return Sigout
def nu_param_INT_Sig_disc(z0, nupar, pop, gp):
# use splines on variable transformed integral
# \Sigma(R) = \int_{r=0}^{R} \rho(r) dr
z0nu = z0
nunu = phys.nu_decrease(z0nu, nupar, gp)
Sig = np.zeros(len(z0nu))
for i in range(len(z0nu)):
Sig[i] = gh.quadinflog(z0nu, nunu, z0nu[0], z0nu[i])
gh.checkpositive(Sig, 'Sig in nu_param_INT_Sig_disc')
return Sig
def nu_param_INT_Sig_disc(z0, nupar, pop, gp):
# use splines on variable transformed integral
# \Sigma(R) = \int_{r=0}^{R} \rho(r) dr
z0nu = z0
nunu = phys.nu_decrease(z0nu, nupar, gp)
Sig = np.zeros(len(z0nu))
for i in range(len(z0nu)):
Sig[i] = gh.quadinflog(z0nu, nunu, z0nu[0], z0nu[i])
gh.checkpositive(Sig, 'Sig in nu_param_INT_Sig_disc')
return Sig
def rho_param_INT_Sig_disc(z0, rhopar, pop, gp):
# use splines on variable transformed integral
# \Sigma(R) = \int_{r=0}^{R} \rho(r) dr
xmin = z0[0]/30. # tweaked. z0[0]/1e4 gives error in quad()
z0left = np.array([xmin, z0[0]*0.25, z0[0]*0.50, z0[0]*0.75])
z0nu = np.hstack([z0left, z0])
rhonu = phys.rho(z0nu, rhopar, pop, gp) # rho takes rho(rhalf) and n(r) parameters
Sig = np.zeros(len(z0nu)-gp.nexp)
for i in range(len(z0nu)-gp.nexp):
Sig[i] = gh.quadinflog(z0nu, rhonu, xmin, z0nu[i])
gh.checkpositive(Sig, 'Sig in rho_param_INT_Sig_disc')
return Sig[len(z0left):] # @z0 (z0nu without z0left, and without 3 extension bins)
def rho_param_INT_Sig_disc(z0, rhopar, pop, gp):
# use splines on variable transformed integral
# \Sigma(R) = \int_{r=0}^{R} \rho(r) dr
xmin = z0[0] / 30. # tweaked. z0[0]/1e4 gives error in quad()
z0left = np.array([xmin, z0[0] * 0.25, z0[0] * 0.50, z0[0] * 0.75])
z0nu = np.hstack([z0left, z0])
rhonu = phys.rho(z0nu, rhopar, pop,
gp) # rho takes rho(rhalf) and n(r) parameters
Sig = np.zeros(len(z0nu) - gp.nexp)
for i in range(len(z0nu) - gp.nexp):
Sig[i] = gh.quadinflog(z0nu, rhonu, xmin, z0nu[i])
gh.checkpositive(Sig, 'Sig in rho_param_INT_Sig_disc')
return Sig[len(
z0left):] # @z0 (z0nu without z0left, and without 3 extension bins)
def Jpar(R0, Sig, gp):
if Sig[0] < 1e-100:
return np.ones(len(R0)-gp.nexp)
splpar_Sig = splrep(R0, np.log(Sig),k=1,s=0.1) # get spline in log space to circumvent flattening
J = np.zeros(len(Sig)-gp.nexp)
for i in range(len(Sig)-gp.nexp):
xint = np.sqrt(R0[i:]**2-R0[i]**2)
yint = np.exp(splev(np.sqrt(xint**2+R0[i]**2), splpar_Sig))
J[i] = gh.quadinflog(xint, yint, 0., gp.rinfty*max(gp.xepol))
#yf = lambda r: np.exp(splev(np.sqrt(r**2+R0[i]**2), splpar_Sig))
#J[i] = quad(yf, 0, np.inf)[0]
# OLD: direct integration with integrand as fct of x
#xint = R0[i:]
#yint = Sig[i:]*R0[i:]/np.sqrt(R0[i:]**2-R0[i]**2)
#J[i] = gh.quadinflog(xint[1:], yint[1:], R0[i], np.inf)
gh.checkpositive(J)
return J
def rho_INT_Sig(r0, rho, gp):
# use splines on variable transformed integral
# \Sigma(R) = \int_{r=R}^{R=\infty} \rho(r) d \sqrt(r^2-R^2)
if rho[0] <= 1e-100:
return np.zeros(gp.nepol)
gh.checknan(rho, 'rho_INT_Sig')
# >= 0.1 against rising in last bin. previous: k=2, s=0.1
# TODO: check use of np.log(r0) here
splpar_rho = splrep(r0, np.log(rho), k=3, s=0.01) # 0.01?
r0ext = np.array([0., r0[0] * 0.25, r0[0] * 0.50, r0[0] * 0.75])
#dR = r0[1:]-r0[:-1]
r0nu = np.hstack([r0ext, r0])
# points in between possible, but not helping much:
# ,dR*0.25+r0[:-1],dR*0.50+r0[:-1],dR*0.75+r0[:-1]])
r0nu.sort()
rhonu = np.exp(splev(r0nu, splpar_rho))
# extend to higher radii
splpar_lrhor = splrep(r0[-3:], np.log(rho[-3:]), k=1,
s=1.) # k=2 gives NaN!
dr0 = (r0[-1] - r0[-2]) / 8.
r0ext = np.hstack(
[r0[-1] + dr0, r0[-1] + 2 * dr0, r0[-1] + 3 * dr0, r0[-1] + 4 * dr0])
rhoext = np.exp(splev(r0ext, splpar_lrhor))
r0nu = np.hstack([r0nu, r0ext])
rhonu = np.hstack([rhonu, rhoext])
gh.checkpositive(rhonu, 'rhonu in rho_INT_Sig')
Sig = np.zeros(len(r0nu) - 4)
for i in range(len(r0nu) - 4):
xnew = np.sqrt(r0nu[i:]**2 - r0nu[i]**2) # [lunit]
ynew = 2. * rhonu[i:]
yscale = 10.**(1. - min(np.log10(ynew)))
ynew *= yscale
# power-law extension to infinity
C = gh.quadinflog(xnew[-4:], ynew[-4:], xnew[-1],
gp.rinfty * max(gp.xepol))
splpar_nu = splrep(
xnew, ynew,
k=3) # interpolation in real space. previous: k=2, s=0.1
Sig[i] = splint(0., xnew[-1], splpar_nu) + C
Sig /= yscale
gh.checkpositive(Sig, 'Sig in rho_INT_Sig')
Sigout = splev(r0, splrep(r0nu[:-4], Sig)) # [Munit/lunit^2]
gh.checkpositive(Sigout, 'Sigout in rho_INT_Sig')
return Sigout
def Jpar(R0, Sig, gp):
if Sig[0] < 1e-100:
return np.ones(len(R0) - gp.nexp)
splpar_Sig = splrep(
R0, np.log(Sig), k=1,
s=0.1) # get spline in log space to circumvent flattening
J = np.zeros(len(Sig) - gp.nexp)
for i in range(len(Sig) - gp.nexp):
xint = np.sqrt(R0[i:]**2 - R0[i]**2)
yint = np.exp(splev(np.sqrt(xint**2 + R0[i]**2), splpar_Sig))
J[i] = gh.quadinflog(xint, yint, 0., gp.rinfty * max(gp.xepol))
#yf = lambda r: np.exp(splev(np.sqrt(r**2+R0[i]**2), splpar_Sig))
#J[i] = quad(yf, 0, np.inf)[0]
# OLD: direct integration with integrand as fct of x
#xint = R0[i:]
#yint = Sig[i:]*R0[i:]/np.sqrt(R0[i:]**2-R0[i]**2)
#J[i] = gh.quadinflog(xint[1:], yint[1:], R0[i], np.inf)
gh.checkpositive(J)
return J
def zeta(r0fine, nufine, Sigfine, Mrfine, betafine, sigr2nu, gp):
# common parameters
N = gh.Ntot(r0fine, Sigfine, gp)
# vr2 = sigr2nu
# dPhidr = gu.G1__pcMsun_1km2s_2*Mrfine/r0fine**2
# zetaa scalar
#xint = r0fine
#yint = nufine*(5-2*betafine)*vr2*dPhidr*r0fine**3
#nom = gh.quadinflog(xint, yint, 0., gp.rinfty*max(gp.xepol), False)
#yint = nufine*dPhidr*r0fine**3
#denom = (gh.quadinflog(xint, yint, 0., gp.rinfty*max(gp.xepol), False))**2
theta = np.arccos(r0min/r0fine)
cth = np.cos(theta)
sth = np.sin(theta)
# TODO calculate nuinterp, sigr2interp, Minterp, betainterp
yint = gu.G1__pcMsun_1km2s_2*(5-2*betainterp)*sigr2
yint *= Minterp*rmin**2/cth**3*sth
nom = quad(theta, yint, 0, np.pi/2)
yint = gu.G1__pcMsun_1km2s_2**2*nuinterp*Mrinterp
yint *= rmin**2/cth**3*sth
denom = quad(theta, yint, 0, np.pi/2)
zetaa = 9*N/10. * nom/denom
# zetab scalar
#------------------------------------------------------------
#yint = nufine*(7-6*betafine)*vr2*dPhidr*r0fine**5
#nom=gh.quadinflog(xint, yint, 0., gp.rinfty*max(gp.xepol), False)
yint = gu.G1__pcMsun_1km2s_2*nuinterp*(7-6*betainterp)*sigr2interp*Mrinterp
yint *= rmin**2/cth**3*sth
nom = quad(theta, yint, 0, np.pi/2)
yint = Sigfine*r0fine**3
denom *= gh.quadinflog(xint, yint, 0., gp.rinfty*max(gp.xepol), False)
yint = Sigmainterp*Rmin**3/cth**6*sth
denom *= quad(theta, yint, 0, np.pi/2)
zetab = 9*N**2/35 * nom / denom
return zetaa, zetab
def rho_INT_Sig(r0, rho, gp):
# use splines on variable transformed integral
# \Sigma(R) = \int_{r=R}^{R=\infty} \rho(r) d \sqrt(r^2-R^2)
if rho[0] <= 1e-100:
return np.zeros(gp.nepol)
gh.checknan(rho, 'rho_INT_Sig')
# >= 0.1 against rising in last bin. previous: k=2, s=0.1
# TODO: check use of np.log(r0) here
splpar_rho = splrep(r0, np.log(rho), k=3, s=0.01) # 0.01?
r0ext = np.array([0., r0[0]*0.25, r0[0]*0.50, r0[0]*0.75])
#dR = r0[1:]-r0[:-1]
r0nu = np.hstack([r0ext,r0])
# points in between possible, but not helping much:
# ,dR*0.25+r0[:-1],dR*0.50+r0[:-1],dR*0.75+r0[:-1]])
r0nu.sort()
rhonu = np.exp(splev(r0nu, splpar_rho))
# extend to higher radii
splpar_lrhor = splrep(r0[-3:],np.log(rho[-3:]),k=1,s=1.) # k=2 gives NaN!
dr0 = (r0[-1]-r0[-2])/8.
r0ext = np.hstack([r0[-1]+dr0, r0[-1]+2*dr0, r0[-1]+3*dr0, r0[-1]+4*dr0])
rhoext = np.exp(splev(r0ext, splpar_lrhor))
r0nu = np.hstack([r0nu, r0ext])
rhonu = np.hstack([rhonu, rhoext])
gh.checkpositive(rhonu, 'rhonu in rho_INT_Sig')
Sig = np.zeros(len(r0nu)-4)
for i in range(len(r0nu)-4):
xnew = np.sqrt(r0nu[i:]**2-r0nu[i]**2) # [lunit]
ynew = 2.*rhonu[i:]
yscale = 10.**(1.-min(np.log10(ynew)))
ynew *= yscale
# power-law extension to infinity
C = gh.quadinflog(xnew[-4:],ynew[-4:],xnew[-1], gp.rinfty*max(gp.xepol))
splpar_nu = splrep(xnew,ynew,k=3) # interpolation in real space. previous: k=2, s=0.1
Sig[i] = splint(0., xnew[-1], splpar_nu) + C
Sig /= yscale
gh.checkpositive(Sig, 'Sig in rho_INT_Sig')
Sigout = splev(r0, splrep(r0nu[:-4], Sig)) # [Munit/lunit^2]
gh.checkpositive(Sigout, 'Sigout in rho_INT_Sig')
return Sigout
gh.checkpositive(Mrnu, 'Mrnu')
Mr_hern = ga.M_hern(rfine, 1, 1)
# (sigr2, 3D) * nu/exp(-idnu)
xint = rfine # [pc]
yint = G1 * Mrnu / rfine**2 # [1/pc (km/s)^2]
yint *= nunu # [Munit/pc^4 (km/s)^2]
yint *= np.exp(idnu) # [Munit/pc^4 (km/s)^2]
gh.checkpositive(yint, 'yint sigr2')
# use quadinflog or quadinfloglog here
sigr2nu = np.zeros(len(rfine))
for i in range(len(rfine)):
sigr2nu[i] = np.exp(-idnu[i])/nunu[i]*\
gh.quadinflog(xint, yint, rfine[i], gp.rinfty*rfine[-1], True)
#if sigr2nu[i] == np.inf:
# sigr2nu[i] = 1e-100
# last arg: warn if no convergence found
gh.checkpositive(sigr2nu, 'sigr2nu in sigl2s')
sigr2anf = ga.sigr2_hern(rfine, 1, 1, G1)
# project back to LOS values
# sigl2sold = np.zeros(len(rfine)-gp.nexp
sigl2s = np.zeros(len(rfine)-gp.nexp)
for i in range(len(rfine)-gp.nexp): # get sig_los^2
xnew = np.sqrt(rfine[i:]**2-rfine[i]**2) # [pc]
ynew = 2.*(1-betanu[i]*(rfine[i]**2)/(rfine[i:]**2))
