Review: Basic checks for Spheroidal harmonic function



In [1]:

    
# Setup ipython environment
%load_ext autoreload
%autoreload 2
%matplotlib inline

# Import useful things from kerr
from kerr.pttools import slm
from kerr import sYlm

# Other things  
from numpy import pi,cos,sin,log,exp,linalg,linspace,array,sqrt
from scipy.integrate import trapz

# Setup plotting backend
import matplotlib as mpl
mpl.rcParams['lines.linewidth'] = 0.8
mpl.rcParams['font.family'] = 'serif'
mpl.rcParams['font.size'] = 12
mpl.rcParams['axes.labelsize'] = 20
from matplotlib.pyplot import *









    



The highest level init for nrutils is located at: /Users/book/JOKI/Libs/KOALA/nrutils_dev/nrutils/__init__.pyc

>> Initiating nrutils ...

>> Sub-Packages to be imported:
   -> core
   -> generate
   -> manipulate
   -> tools
>> Please note style conventions:                  
   * lower case function/method/variable names                  
   * no underscore in names unless there are repeated letters, or counfounded syllables                  
   * information is implicitely in time domain unless explicitely stated.                  
   * frequency domain information will start with "fd".

nrutils:

  .core: 
      .basics*
      .basics
      .nrsc
  .generate: 
  .manipulate: 
      .bundlers
      .rotate
  .tools: 
    .unit: 
      .conversion

kerr## Found formula module "ksm2_cw"
kerr## Found formula module "ksm2_sc"
kerr## Found formula module "ksm2_slm_norm"
kerr## Found formula module "mmrdnp_090117v2"
kerr## Found formula module "mmrdnp_100117"
kerr## Found formula module "mmrdns_amplitudes"
kerr## Found formula module "mmrdns_Mfjf"

Evaluate the spheroidal harmonic function, and plot: convergence and theta dependence (note that phi dependence is $exp(i\phi)$). Other useful things to note are:

slm calls leaver, which return QNM frequencies and separation constants
The consistency of the values used with leaver's equation is an internal check
Ideally the spheroidal harmonic fucntions would be normalized automatically, but this is not the case. Currently the scaling is such that there is correspondence with spherical hamonic conventions at $j=0$
Normalization must be toggle via the norm keyword.



In [6]:

    
# Define which mode to use 
l,m,n = 3,3,1
jf = 0.68

# Define which angles to use 
ph = pi/4
th = linspace(0,pi,1024)

# Call the spheroidal fucntion: 
f = slm(jf,l,m,n,th,ph,plot=True,verbose=True)









    



(slm)>> Checking consistency of QNM frequncy and separation constant used against Leavers constraint equations:
	*  leaver_workfunction(jf=0.6800,l,m,[cw,sc]) = 1.75004887587e-08
	*  cw = (0.824358032575-0.252038259428j)
	*  sc = (8.72996164884+0.446985546772j)
(slm)>> The constant needed to recover the spherical harmonic scaling at zero spin is: (1.00000000003+1.05399515596e-16j)

Demonstrate Recovery of Spherical Harmonics at zero spin



In [13]:

    
# NOTE that spin weights for slm and sYlm are set to -2 by defualt.

f = slm(0,l,m,n,th,ph,verbose=True)
g = sYlm(-2,l,m,th,ph)

figure( figsize=3*array([5,2]) )
plot(th,abs(f),'c-')
plot(th,abs(g),'--k')
xlabel(r'$\theta$')
ylabel(r'$|S_{lmn}|=|_{-2}Y_{lm}|$')









    



(slm)>> Checking consistency of QNM frequncy and separation constant used against Leavers constraint equations:
	*  leaver_workfunction(jf=0.0000,l,m,[cw,sc]) = 2.83872022771e-15
	*  cw = (0.373671684418-0.0889623156889j)
	*  sc = (4-1.34478918208e-19j)
(slm)>> The constant needed to recover the spherical harmonic scaling at zero spin is: (1.00001970396-2.33648056627e-20j)






    Out[13]:





<matplotlib.text.Text at 0x119d30b90>

Verify internal Normalization



In [14]:

    
for jf_ in linspace(0,0.99,11):
    f = slm(jf_,l,m,n,th,ph)
    print sqrt(trapz(f.conj()*f*sin(th),x=th)*(2*pi))

g = sYlm(-2,l,m,th,ph)

print sqrt(trapz(g.conj()*g*sin(th),x=th)*(2*pi))









    



(0.999999017624+0j)
(0.999998815796+0j)
(1.00000206322+0j)
(1.00000998674+0j)
(1.00002351194+0j)
(1.00004251998+0j)
(1.00006419146+0j)
(1.00007920525+0j)
(1.00006269369+0j)
(0.999956615736+0j)
(1.00014766272+0j)
(0.999999017624+0j)

Make plots for many examples



In [8]:

    
#
lmn = [ (2,2,0), (2,2,1), (2,1,0), (3,3,0), (3,3,1), (3,2,0), (4,4,0), (4,3,0), (5,5,0) ]

#
jf_range = linspace(0,0.995,4)

# Define which angles to use 
ph = pi/4
th = linspace(0,pi,180)

#
fig,axarr = subplots(len(lmn), 2*len(jf_range),figsize=4*array( [2*len(jf_range),len(lmn)] ))
tight_layout(pad=3, w_pad=3, h_pad=3)

#
for k,(l,m,n) in enumerate(lmn):
    for h,jf in enumerate(jf_range):
        #
        ax = [ axarr[k,2*h],axarr[k,2*h+1] ]
        slm(jf,l,m,n,th,ph,plot=True,norm=True,ax=ax)

savefig('test_slm_composite.pdf')

View how phase of Slm function change with spin (?)



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