Creating a spiral structure in Python, using hyperbolic tangent

Question

I'm trying to create a spiral structure, like spiral arms of a galaxy, in a 2D array in Python. The first and easy way I did it, was using a simple log-spiral function, defined as in the image: log spiral function

The x and y values are created by

x,y=meshgrid(arange(0,M=400,1), arange(0,N=400,1))

M and N are the dimensions of the array. The radius coordinate is simple, like the equation of the last image,

r=(abs(x-gal_center[1])**(2.0)+((abs(y-gal_center[0]))/(q))**(2.0))**(0.5)

Creating the profile brightness of f(r), and ploting

plt.imshow((abs(galaxy_model))**0.2)

give me a commom spiral structure, like a spiral galaxy.

Another way to do this, is to use another function, the hyperbolic tangent . In the equations of the last image, unless r , that is defined like before, all the others parameters, are ajustable numbers.

For this function, I have problems to make a spiral structure in a 2D array. I don't know, if I need to use the hyperbolic tangent to make a coordinate transformation in the array, or a matrix/array distortion, to create a spiral structure. I tried it, but I could not.

How can I proced to make this spira/image, using the definitions above? Thanks for the help!

More information about the subject, in the references:

1. Peng, Y. Chien et al; Detailed Structural Decomposition of Galaxy Images, 2002
2. Peng, Y. Chien et al; Detailed Decomposition of Galaxy Images. II. Beyond Axisymmetric Models, 2009
3. Peng, Y. Chien, Galfit User's Manual, 2003
4. Rowe, Barnaby et al; GALSIM:The modular galaxy image simulation toolkit, 2015

Edited:

The code that I'm using is as follows:

from __future__ import division
import numpy as np
from numpy import*
import matplotlib.pyplot as pyplot
import scipy as sp
from scipy import*
import pylab as pl
from pylab import*
import math 
from math import*
import pyfits as pf
from pyfits import*

def exponential_profile(Io,ro,r):
    Iexp=0.5*Io*np.exp(-r/ro)
    return Iexp

def sersic_profile(Io,ro,r,n):
    Iser=Io*np.exp(-(r/ro)**(1/n))
    return Iser

def galaxy_model1(q,c,gal_center,Io,ro,n,M,N,xi,p,n1,n2,s1,s2,k):
    x,y=meshgrid(arange(-M/2,M/2,1), arange(-N/2,N/2,1))
    r=(abs(x-0*gal_center[1])**(c+2.0)+((abs(y-0*gal_center[0]))/(q))**(c+2.0))**(1.0/(c+2.0))
    power=2.0
    fr=(30-xi*np.log(1.0+r**power)+(1.0/p)*np.cos(n1*arctan2(x,y)+k*np.log(s1+r**power))+(1.0/p)*np.cos(n2*arctan2(x,y)+k*np.log(s2+r**power))  )
    I_exp=exponential_profile(Io,ro,r)
    I_ser=sersic_profile(Io,ro,r,n)
    galaxy_model_1=0.1*I_exp+0.1*I_ser+0.5*fr
    return galaxy_model_1

def galaxy_model2(q,c,Cb,rout,rin,Oout,a,M,N,Io,ro,n):
    gal_center=(M/2,N/2)
    x,y=meshgrid(arange(0,M,1), arange(0,N,1))
    r=(abs(x-0*gal_center[1])**(c+2.0)+((abs(y-0*gal_center[0]))/(q))**(c+2.0))**(1.0/(c+2.0))
    A=2*Cb/(abs(Oout)+Cb)-1.00001
    B=(2-np.arctanh(A))*((rout)/(rout-rin))
    T=0.5*(np.tanh(B*(r/rout-1)+2)+1)
    Or=Oout*T*(0.5*(r/rout+1))**a
    I_exp=exponential_profile(Io,ro,r)
    I_ser=sersic_profile(Io,ro,r,n)
    galaxy_model_2=0.1*I_exp+0.1*I_ser+0.5*Or
    return galaxy_model_2
galaxy_model_1=galaxy_model1(q,c,(M/2,N/2),Io,ro,n,M,N,xi,p,n1,n2,s1,s2,k)
galaxy_model_2=galaxy_model2(q,c,Cb,rout,rin,Oout,a,M,N,Io,ro,n)
fig=plt.figure()
ax1=fig.add_subplot(121)
ax1.imshow((abs(galaxy_model_1))**0.2)
pf.writeto('gal_1.fits', galaxy_model_1,  clobber=1)  
ax2=fig.add_subplot(122, axisbg='white')
ax2.imshow((abs(galaxy_model_2))**0.2)
plt.show()

A set of parameters can be:

M=400
N=400
q=0.8
c=0.0
Io=100.0
ro=10.0
n=3.0
xi=2.0
p=1.7
n1=3.0
n2=3.0
s1=0.05
s2=0.5
k=3.0
Cb=0.23
rout=100.0
rin=10.0
Oout=pi/2
a=0.0

Answer 1

I'm not sure this is exactly right but I think it is close, and produces results similar to the paper:

def galaxy_model2(q,c,Cb,rout,rin,Oout,a,M,N,Io,ro,n):
    gal_center=(0,0)
    x,y=meshgrid(arange(-M/2,M/2,1), arange(-N/2,N/2,1))
    r=(abs(x-gal_center[1])**(c+2.0)+((abs(y-gal_center[0]))/(q))**(c+2.0))**(1.0/(c+2.0))
    A=2*Cb/(abs(Oout)+Cb)-1.00001
    B=(2-np.arctanh(A))*((rout)/(rout-rin))
    T=0.5*(np.tanh(B*(r/rout-1)+2)+1)
    Or=Oout*T*(0.5*(r/rout+1))**a
    Or=30-np.log(1.0+r**2.0)+(2.0/p)*np.cos(n2*arctan2(x,y)+k*Or)
    I_exp=exponential_profile(Io,ro,r)
    I_ser=sersic_profile(Io,ro,r,n)
    #galaxy_model_2=0.5*Or
    return Or

The only change is that I use

Or=30-np.log(1.0+r**2.0)+(2.0/p)*np.cos(n2*arctan2(x,y)+k*Or)

to create a galaxy plot.

np.cos(n1*arctan2(x,y))

creates this plot:

And i spin it around by adding k*Or

Using this with n2=3 I get: