def g(rvar, rfix, beta, dbetadr):
tmp = 1. # [1]
tmp -= 2*beta*rfix**2/rvar**2 # [1]
tmp += beta*(beta+1.)/2.*rfix**4/rvar**4 # [1]
tmp -= rfix**4/(4.*rvar**3)*dbetadr # last term of eq. 37 Richardson2012
gh.checknan(tmp, 'tmp in g')
return tmp
def ant_intbeta(r0, betapar, gp):
xint = 1.*r0
yint = phys.beta(xint, betapar, gp)[0]/xint
intbet = np.zeros(len(r0))
for k in range(len(r0)):
intbet[k] = gh.quadinf(xint, yint, r0[0], r0[k])
# assumption here is that integration goes down to min(r0)/1e5
gh.checknan(intbet, 'intbet in ant_intbeta')
return intbet
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_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 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 int_poly_inf(r0,poly):
f = -1/poly[0]*np.exp(poly[1]+poly[0]*r0)
gh.checknan(f, 'f in int_poly_inf')
return f