15-Nov-2004

Temperature estimation of Saturn

First check how much the temperature of saturn changes due to maximum and minimum distance from sun.

From https://en.wikipedia.org/wiki/Black-body_radiation#Temperature_relation_between_a_planet_and_its_star

The fractional change in temperature is $\frac{T1}{T2} = \sqrt{\frac{D2}{D1}}$ ie.

In [1]:
import numpy as np
np.sqrt(10.11595804/9.04807635)
Out[1]:
1.0573660972166758

Which means, the temperature can change by up to 6% between appoge and perigee of Saturn from Sun. This seems to be just within the error bars of various temperature estimates.

Calculation of flux from Saturn

In [2]:
Temp_Saturn = 110  # K Ref: (1997) https://iopscience.iop.org/article/10.1086/310919/pdf ; (2004) http://link.springer.com/article/10.1007%2Fs11214-004-1454-9

from scipy.constants import h, c, k
Wavelength = 150e-6  # 150 micron
B_lambda = (2*h*c**2)/(Wavelength**5 *(np.exp(h*c/(Wavelength*k*Temp_Saturn)) -1))  #Unit: W·sr−1·m−3

#Size of the saturn ellispe seen from earth on Nov 15 2004 (got from Almanac): a = 19.37" and b= 17.73"
Size_Saturn_insr = 4*np.pi*19.37*(np.pi/(180*60*60)) * 17.73*(np.pi/(180*60*60))  #Unit: sr

Total_Saturn_Flux = B_lambda * Size_Saturn_insr #Unit: W·m−3
Total_Saturn_Flux_CGS = Total_Saturn_Flux*1e7/(1e2*1e2*1e6) # Unit: erg/sec/cm^2 / micron
print(Total_Saturn_Flux_CGS)

1.14323928035e-07
In [3]:
G = 1 # Lorenztian_Width in micron
# Area of the Lorenztian with amplitude = 1 and width G is pi*G
# Integrated Flux through the Lorenztian transmission curve
Net_Flux_CGS = Total_Saturn_Flux_CGS * np.pi*G # Unit: erg/sec/cm^2
print(Net_Flux_CGS)

3.59159212443e-07
In [4]:
# this corresponds to the peak FPS value in Saturn map. 21400 - 21170 = 230
PeakADC = 230.0
CountToFlux = Net_Flux_CGS/ PeakADC 
print('Line Amplitude Count to Flux (erg/sec/cm^2) conversion factor {0}'.format(CountToFlux) )

Line Amplitude Count to Flux (erg/sec/cm^2) conversion factor 1.56156179323e-09
In [5]:
# For continuum, since we have to integrate over the Lorentian, 
print('Continuum Count to Flux (erg/sec/cm^2/micron) conversion factor {0}'.format(CountToFlux/(np.pi*G)) )

Continuum Count to Flux (erg/sec/cm^2/micron) conversion factor 4.97060556672e-10
In [6]:
# The beam size telescope is
Diameter = 1.0 # m
Beam = 1.22*Wavelength/Diameter # Unit: radians
print('Beam size in rad :{0} arcsec:{1}'.format(Beam,Beam*(60*60*180/np.pi)))

Beam size in rad :0.000183 arcsec:37.7464595432
In [ ]: