def rho_param_INT_Rho(r0, rhoparam):
# use splines on variable transformed integral
# \Sigma(R) = \int_{r=R}^{R=\infty} \rho(r) d \sqrt(r^2-R^2)
# gh.checknan(rhoparam, 'rho_param_INT_Rho')
xmin = r0[0]/1e4
r0left = np.array([xmin, r0[0]*0.25, r0[0]*0.50, r0[0]*0.75])
r0nu = np.hstack([r0left, r0])
rhonu = phys.rho(r0nu, rhoparam)
Rho = 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. TODO: include in Rho[i] below
C = gh.quadinflog(xnew[-gp.nexp:], ynew[-gp.nexp:], xnew[-1], np.inf)
# tcknu = splrep(xnew, ynew, k=3)
# interpolation in real space, not log space
# problem: splint below could give negative values
# reason: for high radii (high i), have a spline that goes negative
# workaround: multiply by const/add const to keep spline positive @ all times
# or set to log (but then integral is not straightforward
# Rho[i] = splint(0., xnew[-1], tcknu) + C
Rho[i] = gh.quadinfloglog(xnew[1:], ynew[1:], xmin, xnew[-1]) + C
gh.checkpositive(Rho, 'Rho in rho_param_INT_Rho')
return Rho[4:] # @r0 (r0nu without r0left)
def rho(r0, arr):
rhoathalf = arr[0]
arr = arr[1:]
spline_n = nr(gp.xepol, arr) # fix on gp.xepol, as this is where definition is on
rs = np.log(r0/gp.rstarhalf) # have to integrate in d log(r)
# check assumption r_min < rstarhalf < r_max
if min(rs)>0. or max(rs)<0.:
print('assumption r_min < rstarhalf < r_max violated')
pdb.set_trace()
logrright = rs[(rs>=0.)]
logrleft = rs[(rs<0.)]
logrleft = logrleft[::-1]
logrhoright = []
for i in np.arange(0, len(logrright)):
logrhoright.append(np.log(rhoathalf) + \
splint(0., logrright[i], spline_n))
# TODO: integration along dlog(r) instead of dr?!
logrholeft = []
for i in np.arange(0, len(logrleft)):
logrholeft.append(np.log(rhoathalf) + \
splint(0., logrleft[i], spline_n))
tmp = np.exp(np.hstack([logrholeft[::-1], logrhoright])) # still defined on log(r)
gh.checkpositive(tmp, 'rho()')
return tmp
def calculate_surfdens(r, M):
r0 = np.hstack([0,r]) # [lunit]
M0 = np.hstack([0,M]) # [munit]
deltaM = M0[1:]-M0[:-1] # [munit]
gh.checkpositive(deltaM, 'unphysical negative mass increment encountered')
deltavol = np.pi*(r0[1:]**2 - r0[:-1]**2) # [lunit^2]
Rho = deltaM/deltavol # [munit/lunit^2]
gh.checkpositive(Rho, 'Rho in calculate_surfdens')
return Rho # [munit/lunit^2]
def calculate_rho(r, M):
r0 = np.hstack([0,r]) # [lunit]
M0 = np.hstack([0,M])
deltaM = M0[1:]-M0[:-1] # [munit]
gh.checkpositive(deltaM, 'unphysical negative mass increment encountered')
deltavol = 4./3.*np.pi*(r0[1:]**3-r0[:-1]**3) # [lunit^3]
rho = deltaM / deltavol # [munit/lunit^3]
gh.checkpositive(rho, 'rho in calculate_rho')
return rho # [munit/lunit^3]
def rho_INTDIRECT_Rho(r0, rho):
# use splines, spline derivative to go beyond first radial point
tck0 = splrep(r0,np.log(rho),k=3,s=0.1) # >= 0.1 against rising in last bin. previous: k=2, s=0.1
# more points to the left help, but not so much
r0ext = np.array([0.,r0[0]/6.,r0[0]/5.,r0[0]/4.,\
r0[0]/3.,r0[0]/2.,r0[0]/1.5])
# introduce new points in between
dR = r0[1:]-r0[:-1]
r0nu = np.hstack([r0ext,r0,dR/4.+r0[:-1],dR/2.+r0[:-1],.75*dR+r0[:-1]])
r0nu.sort()
rhonu = np.exp(splev(r0nu,tck0))
# extend to higher radii
tckr = splrep(r0[-3:],np.log(rho[-3:]),k=1,s=0.2) # previous: k=2
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,tckr))
r0nu = np.hstack([r0nu, r0ext]); rhonu = np.hstack([rhonu, rhoext])
Rho = np.zeros(len(r0nu)-4)
for i in range(len(r0nu)-4):
xint = r0nu[i:] # [lunit]
yint = np.ones(len(xint))
for j in range(i+1,len(r0nu)):
yint[j-i] = r0nu[j] * rhonu[j]/np.sqrt(r0nu[j]**2-r0nu[i]**2)
# [munit/lunit^3]
# Cubic-spline extrapolation to first bin
tck = splrep(xint[1:],np.log(yint[1:]),k=3,s=0.)
xnew = xint
ynew = np.hstack([np.exp(splev(xnew[0],tck,der=0)),yint[1:]])
# gpl.start(); pdb.set_trace()
# power-law extension to infinity
tck = splrep(xint[-4:],np.log(yint[-4:]),k=1,s=1.)
invexp = lambda x: np.exp(splev(x,tck,der=0))
dropoffint = quad(invexp,r0nu[-1],np.inf)
tcknu = splrep(xnew,ynew,s=0) # interpolation in real space
Rho[i] = 2. * (splint(r0nu[i], r0nu[-1], tcknu) + dropoffint[0])
# gpl.plot(r0nu[:-4], Rho, '.', color='green')
tcke = splrep(r0nu[:-4],Rho)
Rhoout = splev(r0,tcke) # [munit/lunit^2]
# TODO: debug: raise Rho overall with factor 1.05?!
gh.checkpositive(Rhoout, 'Rhoout in rho_INTDIRECT_Rho')
# [munit/lunit^2]
return Rhoout
def Rho_INT_rho(R0, Rho):
# TODO: deproject with variable transformed, x = sqrt(r^2-R^2)
# TODO: any good? have 1/x in transformation, for x=0 to infty
pnts = len(Rho) # [1]
# use splines, spline derivative to go beyond first radial point
tck0 = splrep(R0,np.log(Rho),s=0.01)
R0ext = np.array([R0[0]/6.,R0[0]/5.,R0[0]/4.,R0[0]/3.,R0[0]/2.,R0[0]/1.5])
# introduce new points in between
R0nu = np.hstack([R0ext,R0,(R0[1:]-R0[:-1])/2.+R0[:-1]])
R0nu.sort()
Rhonu = np.exp(splev(R0nu,tck0))
# extend to higher radii
tckr = splrep(R0[-3:],np.log(Rho[-3:]),k=1,s=0.3) # previous: k=2, s=0.1
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,tckr))
R0nu = np.hstack([R0nu, R0ext])
Rhonu = np.hstack([Rhonu, Rhoext])
gh.checkpositive(Rhonu, 'Rhonu in Rho_INT_rho')
# gpl.start(); pdb.set_trace()
tck1 = splrep(R0nu,Rhonu,k=3, s=0.)
# TODO: adapt s for problem at hand ==^
dnubydR = splev(R0nu,tck1,der=1)
# Rhofit = np.exp(splev(R0nu,tck1)) # debug for outer part
rho = np.zeros(len(R0nu)-4)
for i in range(len(R0nu)-4):
xint = R0nu[i:] # [lunit]
yint = np.ones(len(xint))
for j in range(i+1,len(R0nu)):
yint[j-i] = dnubydR[j]/np.sqrt(R0nu[j]**2-R0nu[i]**2)
# [munit/lunit^3/lunit]
# Cubic-spline extrapolation to first bin
tck = splrep(xint[1:],yint[1:],k=3,s=0.01)
xnew = xint
ynew = splev(xnew,tck,der=0)
tcknu = splrep(xnew,ynew,s=0) # interpolation in real space
rho[i] = -1./np.pi * splint(0.,max(R0)*2.,tcknu)
tcke = splrep(R0nu[:-4],rho)
rhoout = splev(R0,tcke) # [munit/lunit^3]
return rhoout # [munit/lunit^3]
def rho_INT_Rho(r0, rho):
# use splines on variable transformed integral
# \Sigma(R) = \int_{r=R}^{R=\infty} \rho(r) d \sqrt(r^2-R^2)
gh.checknan(rho, 'rho_INT_Rho')
# >= 0.1 against rising in last bin. previous: k=2, s=0.1
tck0 = splrep(r0,np.log(rho),k=3,s=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,tck0))
# extend to higher radii
tckr = 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,tckr))
r0nu = np.hstack([r0nu, r0ext])
rhonu = np.hstack([rhonu, rhoext])
gh.checkpositive(rhonu, 'rhonu in rho_INT_Rho')
Rho = 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],np.inf)
#print('C[',i,'] = ',C)
tcknu = splrep(xnew,ynew,k=3) # interpolation in real space. previous: k=2, s=0.1
Rho[i] = splint(0., xnew[-1], tcknu) + C
Rho /= yscale
gh.checkpositive(Rho, 'Rho in rho_INT_Rho')
# gpl.plot(r0nu[:-4],Rho,'.')
tcke = splrep(r0nu[:-4], Rho)
Rhoout = splev(r0, tcke) # [munit/lunit^2]
gh.checkpositive(Rhoout, 'Rhoout in rho_INT_Rho')
return Rhoout
def ant_sigkaplos2surf(r0, beta_param, rho_param, nu_param):
# TODO: check all values in ()^2 and ()^4 are >=0
minval = 1.e-30
r0nu = introduce_points_in_between(r0)
rhonu = phys.rho(r0nu, rho_param)
nunu = phys.rho(r0nu, nu_param)
betanu = phys.beta(r0nu, beta_param)
# calculate intbeta from beta approx directly
idnu = ant_intbeta(r0nu, beta_param)
# integrate enclosed 3D mass from 3D density
r0tmp = np.hstack([0.,r0nu])
rhoint = 4.*np.pi*r0nu**2*rhonu
# add point to avoid 0.0 in Mrnu(r0nu[0])
rhotmp = np.hstack([0.,rhoint])
tck1 = splrep(r0tmp, rhotmp, k=3, s=0.) # not necessarily monotonic
Mrnu = np.zeros(len(r0nu)) # work in refined model
for i in range(len(r0nu)): # get Mrnu
Mrnu[i] = splint(0., r0nu[i], tck1)
gh.checkpositive(Mrnu, 'Mrnu')
# (sigr2, 3D) * nu/exp(-idnu)
xint = r0nu # [pc]
yint = gp.G1 * Mrnu / r0nu**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(r0nu))
for i in range(len(r0nu)):
sigr2nu[i] = np.exp(-idnu[i])/nunu[i]*gh.quadinflog(xint, yint, r0nu[i], np.inf)
# project back to LOS values
# sigl2sold = np.zeros(len(r0nu)-gp.nexp)
sigl2s = np.zeros(len(r0nu)-gp.nexp)
dropoffintold = 1.e30
for i in range(len(r0nu)-gp.nexp): # get sig_los^2
xnew = np.sqrt(r0nu[i:]**2-r0nu[i]**2) # [pc]
ynew = 2.*(1-betanu[i]*(r0nu[i]**2)/(r0nu[i:]**2))
ynew *= nunu[i:] * sigr2nu[i:]
gh.checkpositive(ynew, 'ynew in sigl2s') # is hit several times..
# yscale = 10.**(1.-min(np.log10(ynew[1:])))
# ynew *= yscale
# gh.checkpositive(ynew, 'ynew sigl2s')
tcknu = splrep(xnew, ynew, k=1)
# interpolation in real space for int
# power-law approximation from last three bins to infinity
# tckex = splrep(xnew[-3:], np.log(ynew[-3:]),k=1,s=1.0) # fine
# invexp = lambda x: np.exp(splev(x,tckex,der=0))
# C = quad(invexp,xnew[-1],np.inf)[0]
# C = max(0.,gh.quadinflog(xnew[-2:],ynew[-2:],xnew[-1],np.inf))
# sigl2sold[i] = splint(xnew[0], xnew[-1], tcknu) + C
sigl2s[i] = gh.quadinflog(xnew[1:], ynew[1:], xnew[0], np.inf)
# sigl2s[i] /= yscale
# TODO: for last 3 bins, up to factor 2 off
# if min(sigl2s)<0.:
# pdb.set_trace()
gh.checkpositive(sigl2s, 'sigl2s')
# derefine on radii of the input vector
tck = splrep(r0nu[:-gp.nexp], np.log(sigl2s), k=3, s=0.)
sigl2s_out = np.exp(splev(r0, tck))
gh.checkpositive(sigl2s_out, 'sigl2s_out')
if not gp.usekappa:
# print('not using kappa')
return sigl2s_out, np.ones(len(sigl2s_out))
# for the following: enabled calculation of kappa
# TODO: include another set of anisotropy parameters beta_'
# kappa_r^4
kapr4nu = np.ones(len(r0nu)-gp.nexp)
xint = r0nu # [pc]
yint = gp.G1 * Mrnu/r0nu**2 # [1/pc (km/s)^2]
yint *= nunu # [munit/pc^4 (km/s)^2]
yint *= sigr2nu # [munit/pc^4 (km/s)^4
yint *= np.exp(idnu) # [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:], r0nu[-1], np.inf))
# tckexp = splrep(xint[-3:],np.log(yint[-3:]),k=1,s=0.) # fine, exact interpolation
# invexp = lambda x: np.exp(splev(x,tckexp,der=0))
# C = quad(invexp,r0nu[-1],np.inf)[0]
tcknu = splrep(xint, yint, k=3) # interpolation in real space # TODO:
for i in range(len(r0nu)-gp.nexp):
# integrate from minimal radius to infinity
kapr4nu[i] = 3.*(np.exp(-idnu[i])/nunu[i]) * \
(splint(r0nu[i], r0nu[-1], tcknu) + C) # [(km/s)^4]
kapr4nu /= yscale
gh.checkpositive(kapr4nu, 'kapr4nu in kappa_r^4')
tcke = splrep(r0nu[:-gp.nexp], np.log(kapr4nu), k=3)
kapr4ext = np.exp(splev(r0ext, tcke))
kapr4nu = np.hstack([kapr4nu, kapr4ext])
gh.checkpositive(kapr4nu, 'kapr4nu in extended kappa_r^4')
tckbet = splrep(r0nu, betanu)
dbetanudr = splev(r0nu, tckbet, der=1)
gh.checknan(dbetanudr, 'dbetanudr in kappa_r^4')
# kappa^4_los*surfdensity
kapl4s = np.zeros(len(r0nu)-gp.nexp)
# gpl.start(); gpl.yscale('linear')
for i in range(len(r0nu)-gp.nexp):
xnew = np.sqrt(r0nu[i:]**2-r0nu[i]**2) # [pc]
ynew = g(r0nu[i:], r0nu[i], betanu[i:], dbetanudr[i:]) # [1]
ynew *= nunu[i:] * kapr4nu[i:] # [TODO]
# TODO: ynew could go negative here.. fine?
#gpl.plot(xnew, ynew)
#gh.checkpositive(ynew, 'ynew in kapl4s')
#yscale = 10.**(1.-min(np.log10(ynew[1:])))
#ynew *= yscale
# gpl.plot(xnew,ynew)
C = max(0., gh.quadinflog(xnew[-3:], ynew[-3:], xnew[-1], np.inf))
tcknu = splrep(xnew,ynew) # not s=0.1, this sometimes gives negative entries after int
kapl4s[i] = 2. * (splint(0., xnew[-1], tcknu) + C)
#kapl4s[i] /= yscale
# print('ynew = ',ynew,', kapl4s =', kapl4s[i])
# TODO: sometimes the last value of kapl4s is nan: why?
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
tck = splrep(r0nu[4:-gp.nexp], kapl4s[4:], k=3) # s=0.
kapl4s_out = np.exp(splev(r0, tck))
gh.checkpositive(kapl4s_out, 'kapl4s_out in kappa_r^4')
return sigl2s_out, kapl4s_out