Assignment 1, brightness temperatures - solution

18. Assignment 1, brightness temperatures - solution#

For this assignment you’re asked to write a function to calculate the “brightness temperature”, defined as the temperature a blackbody would need to have to emit an observed monochromatic radiance \(I_\lambda\):

\[ B_\lambda(T_b) = I_{\lambda\,observed} \]

18.1. Question 1#

In the cell below, write a function that calculates the blackbody radiance. You can/should just adapt the code from Assignment 1, brightness temperatures so that it evaluates W&H 4.10:

\[ B_\lambda(T)=\frac{c_1 \lambda^{-5}}{\pi\left(e^{c_2 / \lambda T}-1\right)} \]

Your function should have the following name and signature

def calc_Blambda(wavel, Temp):
    """
    Calculate the blackbody radiance (W&H 4.10)

    Parameters
    ----------

      wavel: float or array
           wavelength (meters)

      Temp: float
           temperature (K)

    Returns
    -------

    Blambda:  float or arr
           monochromatic blackbody radiance (W/m^2/m/sr)
    """

###  Question 1
###  your Blambda function here
###

import numpy as np
def calc_Blambda(wavel, Temp):
    """
    Calculate the blackbody radiance (W&H 4.10)

    Parameters
    ----------

      wavel: float or array
           wavelength (meters)

      Temp: float
           temperature (K)

    Returns
    -------

    Blambda:  float or arr
           monochromatic blackbody radiance (W/m^2/m/sr)
    """
    c, h, k = 299792458.0, 6.62607004e-34, 1.38064852e-23
    c1 = 2.0 * h * c ** 2.0
    c2 = h * c / k
    Blambda = c1 / (wavel ** 5.0 * (np.exp(c2 / (wavel * Temp)) - 1))
    return Blambda

18.2. Question 2#

18.2.1. brightness temperature function#

In the cell below, write a new function to compute the brightness temperature \(T_{bright}\). It should have the following name and signature

def calc_Tbright(wavel, I):
    """
    Calculate the brightness temperature
    
    Parameters
    ----------
      wavel: float
           wavelength (meters)
      I: float or array
           radiance (W/m^2/m/sr)
    
    Returns
    -------
    Tbright:  float or arr
           brightness temperature (K)
    """

### Question 2 answer
### your version of calc_Tbright here
###
def calc_Tbright(wavel, Ilambda):
    """
    Calculate the brightness temperature
    
    Parameters
    ----------
      wavel: float
           wavelength (meters)
      Ilambda: float or array
           monochromatic radiance (W/m^2/m/sr)
    
    Returns
    -------
    Tbright:  float or arr
           brightness temperature (K)
    """
    c, h, k = 299792458.0, 6.62607004e-34, 1.38064852e-23
    c1 = 2.0 * h * c ** 2.0
    c2 = h * c / k
    Tbright = c2 / (wavel * np.log(c1 / (wavel ** 5.0 * Ilambda) + 1.0))
    return Tbright

18.3. Question 3#

Test your function by executing a round trip for some wavelength and temperature. In the cell below, define a wavelength wavelen and a temperture the_temp and use them to calculate blackbody radiance Iblack with your function calc_Blambda. Then calculate a brightness temperature Tbright using calc_Tbright and make sure it is equal to the_temp using the function numpy.testing.assert_almost_equal

Something like:

np.testing.assert_almost_equal(the_temp, Tbright)

### Question 3 answer
#
# your question 3 solution here
#
wavelen = 10.e-6
the_temp=250
Iblack = calc_Blambda(wavelen,the_temp)
Tbright = calc_Tbright(wavelen, Iblack)
np.testing.assert_almost_equal(the_temp, Tbright)
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