Import standard modules:

In [1]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from IPython.display import HTML 
HTML('../style/course.css') #apply general CSS
Out[1]:

Import section specific modules:

In [2]:
import scipy.constants
In [3]:
HTML('../style/code_toggle.html')
Out[3]:
The raw code for this notebook is by default hidden for easier reading. To toggle on/off the raw code, click here.

1.5 Thermal Emission:

There are two main types of continuum emission mechanisms we'll encounter in radio astronomy, thermal and non-thermal. As the names suggest, the thermal emission depends on the temperature of the black body - non-thermal emission depends on some other properties of the emitting body. In the next few sections, we'll have an overview of the emission mechanisms and some examples of each in radio astronomy.

We'll begin the overview with an example of one of the mechanisms of thermal emission, blackbody emission.

1.5.1 Blackbody Emission:

Blackbody emission or thermal emission is an important form of radiation in astronomy. This is in the cases when Thermodynamic Equilibrium exists between radiation and matter - where radiation and matter interact enough to produce the following expression for the specific intensity to be:

\begin{equation} B_{\nu}(T)=\frac{2h\nu^{3}}{c^{2}} \frac{1}{e^{\frac{h\nu}{kT}}-1}, \end{equation}

this is also known as the Planck Spectrum.

From the equation above, it can be inferred that in a given frequncy interval, the brightness depends only on the Temperature of the body. Also, one can see that a body at a higher temperature will produce higher brightness at all frequencies - which means that a body at a higher temperature has higher energy output via thermal emission than that of a body at a lower temperature. Yet another consequence is that the frequency at which maximum brightness is produced is directly proportional to the temperature - this is known as Wien's displacement law:

$$\nu_{max} = 58.789 T $$

,

where frequency is in GHz and temperature is in Kelvin. See, for example, Figure 1.5.2, where the black body spectrum for various temperatures is plotted. The displacement of the frequency at which the peak occures can be seen clearly.

At high frequencies where $h\nu >> kT$ , this will reduce to Wien's approximation :

$$B_{\nu}(T)= \frac{2h\nu^{3}}{c^{2}} e^{\frac{-h\nu}{kT}}$$

But in radio astronomy, a more typical situation is when the frequency of the radiation is so low that: $h\nu << kT$. In such a case, the Planck formula may be reduced to what is called the Rayleigh-Jeans approximation:

$$B_{\nu}(T)= \frac{2\nu^{2}}{c^{2}} kT $$
In [4]:
def B_v(T):
    '''Function to calculate specific intensity/brightness distribution of black
    body radiation at a given temperature. T is in Kelvins and frequency 
    is in Hertz'''
    #Use Wien's displacement law to find the frequency range for given Temperature.
    # nu_max(in GHz) = 58.789*T(in Kelvin)
    nu_max = 58.79 * T
    freq = np.arange(1,2000,1)
    freq = freq * nu_max / 500.0 # Scale so that the the peak value occures at 1/4 the length of the array.
    B = (2*scipy.constants.h/scipy.constants.c**2.0) * np.exp(-1. * scipy.constants.h * \
                                                              freq * 10**9/(scipy.constants.k*T)) * (freq*10**9)**3
    #Change units for plotting to Jansky
    # 1 Jansky = 10^-26 W/m^2 Hz
    B = B * 10**26
    return B
In [5]:
T = 2.275 #In Kelvin
freq = np.arange(1,2000,1)
freq = freq*58.79*T/500.0
B_CMB = B_v(T)
fig,ax = plt.subplots()
ax.plot(freq,B_CMB)
ax.set_xlabel("Frequency (GHz)")
ax.set_ylabel("Specific Intensity ($Jy/sr$)")
ax.set_title("Blackbody Spectral Distribution")
Out[5]:
<matplotlib.text.Text at 0x7fec84289e90>

Figure 1.5.1: Blackbody radiation spectral distribution for Cosmic Microwave Background .

The above distribution is for a temperature of 2.725 K, approximately the same as that of the Cosmic Microwave Background. If 'T' in the above script is changed to $~ 5700$ (K), we would get the plot for the emission from the Sun.

In [6]:
T = 4 #in Kelvin
freq = np.arange(1,2000,1)
freq = freq*58.79*T/500.0
B_2 = B_v(T)
B_3 = B_v(5)
fig,ax = plt.subplots()
ax.plot(freq,B_CMB, label='2.75 K')
ax.plot(freq,B_2, label='4.0 K')
ax.plot(freq,B_3, label='5.0 K')
ax.legend()
ax.set_xlabel("Frequency (GHz)")
ax.set_ylabel("Specific Intensity ($Jy/sr$)")
ax.set_title("Blackbody Spectral Distribution")
Out[6]:
<matplotlib.text.Text at 0x7fec840fe1d0>

Figure 1.5.2: Blackbody radiation spectral distribution.

As mentioned above, the brightness of the black body radiation depends only on the temperature of the body. This can be used to define a 'brightness temperature' for an object. Simply put, the brightness temperature for an object is given by : $$T_{b} = I_{\nu} \frac{c^{2}}{2k\nu^{2}}$$. It should be noted here that the actual specific intensity of an object is used here, even when it does not represent black body emission - the brightness temperature is simply the temperature which the object would have if the specific intensity or brightness was from black body emission. This allows characterization of any object in terms of a single parameter. This is used widely in radio astronomy, and will be encountered in next chapters of this book.

This was the example of thermal radiation; in the next chapter we'll see an example of a non-thermal emission, namely the Synchrotron emission.