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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 [171109 - AMS] Created notebook
This notebook re-creates Fig 1 from Brentjens & de Bruyn (2005)
Step1:  Make a function for the Galactic foreground
Step2:  We're going to specify that $\phi_{\rm fg}= 2\,{\rm rad\,m^{-2}}$. We can then compute the Galactic contribution at each value of $\lambda^2$
Step3:  Now make a function for the radio galaxy lobe
Step4:  The total polarized signal will be the sum of the radio galaxy contribution and the Galactic contribution
Step5:  Now let's re-create Fig. 1 from Brentjens & de Bruyn (2005; https
Step6:  Now let's plot on the magnitude of the total polarization as well
Step7:  Now let's calculate the polarization angle
Step8:  Now plot it all together
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Python Code:
# this embeds plots in the notebook
%matplotlib inline  
import numpy as np  # for arrays
import pylab as pl  # for plotting
Explanation: [171109 - AMS] Created notebook
This notebook re-creates Fig 1 from Brentjens & de Bruyn (2005)
End of explanation
lam_sq = np.arange(0.01,1,0.01)
Explanation: Make a function for the Galactic foreground:
$$
F_{\rm gal}(\phi) = (2\phi_{\rm fg})^{-1}~~~~-\phi_{\rm fg} \lt \phi \lt \phi_{\rm fg}
$$
and zero elsewhere. Therefore:
$$
P_{\rm gal}(\lambda^2) = \frac{\sin (2\phi_{\rm fg} \lambda^2)}{2\phi_{\rm fg}\lambda^2}
$$
Make an array of $\lambda^2$ values:
End of explanation
phi_fg = 2.
P_gal = np.sin(2*phi_fg*lam_sq)/(2*phi_fg*lam_sq) + 0*1j
Explanation: We're going to specify that $\phi_{\rm fg}= 2\,{\rm rad\,m^{-2}}$. We can then compute the Galactic contribution at each value of $\lambda^2$:
End of explanation
phi_1 = 10.
P_rg = 0.25*np.cos(2*phi_1*lam_sq) + 1j*0.25*np.sin(2*phi_1*lam_sq)
Explanation: Now make a function for the radio galaxy lobe:
$$
F_{\rm rg}(\phi) = 0.25\delta(\phi - \phi_1)
$$
therefore:
$$
P_{\rm rg}(\lambda^2) = 0.25 \exp (2i\phi_1 \lambda^2) 
$$
which is equivalent to:
$$
P_{\rm rg}(\lambda^2) = 0.25 \cos (2\phi_1 \lambda^2) + 0.25 i \sin (2\phi_1 \lambda^2)
$$
so,
$$
Q_{\rm rg}(\lambda^2) = 0.25 \cos (2\phi_1 \lambda^2) 
$$
and 
$$
U_{\rm rg}(\lambda^2) = 0.25 \sin (2\phi_1 \lambda^2) 
$$
We're going to specify that $\phi_1= 10\,{\rm rad\,m^{-2}}$. We can then compute the contribution from the radio galaxy lobe at each value of $\lambda^2$:
End of explanation
P_tot = P_gal + P_rg
Explanation: The total polarized signal will be the sum of the radio galaxy contribution and the Galactic contribution:
End of explanation
pl.subplot(111)
pl.plot(lam_sq,P_gal.real,ls='--')
pl.xlabel(r"$\lambda^2$ [m$^2$]")
pl.ylabel("Flux [Jy]")
pl.axis([0,1,-0.2,1.4])
pl.show()
Explanation: Now let's re-create Fig. 1 from Brentjens & de Bruyn (2005; https://arxiv.org/pdf/astro-ph/0507349.pdf)
First let's plot $Q_{\rm gal}$ (called $Q_{\rm fg}$ in the paper):
End of explanation
pl.subplot(111)
pl.plot(lam_sq,P_gal.real,ls='--')
pl.plot(lam_sq,np.absolute(P_tot),ls=':')
pl.xlabel(r"$\lambda^2$ [m$^2$]")
pl.ylabel("Flux [Jy]")
pl.axis([0,1,-0.2,1.4])
pl.show()
Explanation: Now let's plot on the magnitude of the total polarization as well:
End of explanation
chi = 0.5*np.arctan2(P_tot.imag,P_tot.real)
chi*= (180./np.pi)   # convert radians to degrees
# hack to unwrap the arctangent [-pi/2,pi/2] wrap:
for i in range(1,len(chi)):
    delta_chi = np.abs(chi[i]-chi[i-1])
    if (delta_chi>45.):
        chi[i:]+=180.
pl.subplot(111)
pl.plot(lam_sq,chi)
pl.xlabel(r"$\lambda^2$ [m$^2$]")
pl.ylabel(r"$\chi$ [deg]")
pl.axis([0,1,-50,350])
pl.show()
Explanation: Now let's calculate the polarization angle:
$$
\chi = 0.5\tan^{-1}\left(\frac{U}{Q}\right)
$$
where $U$ is the imaginary part of the complex polarization, $P$, and $Q$ is the real part.
End of explanation
fig, ax1 = pl.subplots()
ln1 = ax1.plot(lam_sq, chi, 'b-',label=r"$\chi$")
ax1.set_xlabel(r"$\lambda^2$ [m$^2$]")
ax1.set_ylabel(r"$\chi$ [deg]")
ax1.set_ylim(-50, 350)
ax2 = ax1.twinx()
ln2 = ax2.plot(lam_sq,np.absolute(P_tot),ls=':',label=r"$|P|$")
ln3 = ax2.plot(lam_sq,P_gal.real,ls='--',label=r"$Q_{\rm fg}$")
ax2.set_ylabel("Flux [Jy]")
ax2.set_ylim(-0.2, 1.4)
# figure legend:
lns = ln1+ln2+ln3
labs = [l.get_label() for l in lns]
ax2.legend(lns, labs, loc=1)
fig.tight_layout()
pl.savefig("Fig1.png")
pl.show()
Explanation: Now plot it all together:
End of explanation
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