Animation: Planck's law

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1

Animation of how the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium varies at at different temperatures.

This is work in progress. Script is in bash.

Requires gnuplot.
Cookie for those who manages to do Planck’s law in gmt or bc.

#to-do:
# - SB-law - total energy
# - dominant Lambda - for each temperature
# - add some static curves, e.g. for the sun T=5800, and some others objects
# - legend becomes wider when T goes from two digits to three to four. Make wide enough or use printf??

cat <<eof > plancks-law.gpl
# input temperature via command line, e.g.:
# gnuplot -e "T=600" plancks-law.gpl
# constants
c=2.99792458e8
# Planck's constant
h=6.62606896e-34
# Boltzmann constant
k=1.3806504e-23
# Temperature - not in use. We get this from argument
#T=573
# Wavelength - not in use.
#L=1000e-9
# For Wien's displacement law
b=2.90e-3
# sigma, stefan-boltzmann constant
s=5.670374419e-8

set table

f(x)=((2*pi*h*c**2)/(x**5)*(1/(exp((h*c)/(x*k*T))-1)))

set samples 1000
set xrange [10e-9:15e-6]
plot f(x)
eof

cat <<eof > plancks-law-frame.sh
gmt set GMT_THEME minimal

echo "10e-9 ifg ultraviolet" > light.x
echo "400e-9 ifg visible" >> light.x
echo "750e-9 ifg infrared" >> light.x
echo "5e-6 ifg microwave" >> light.x

 #pwd
 gmt begin
  gnuplot -e "T=\${MOVIE_COL0}" $(pwd)/plancks-law.gpl | gmt plot -W -R4e-09/5e-06/1e1/5e+14 -JX20c/10c -L+yb -Ggray -l"Temperature [K]: \${MOVIE_COL0}"
  gmt psbasemap -Bxclight.x -BwesN --MAP_GRID_PEN_PRIMARY=black,- --FONT_ANNOT_PRIMARY=6
  gmt psbasemap -Bxafg+l"Wavelength" -Byafg+l"Watt/m@+2@+ m" -B+t"Radiated Power Density, Planck's law"
 gmt end

rm light.x

eof

# generate values for input
seq 20 20 10000 > time

# run movie script
gmt movie plancks-law-frame.sh -Nplancks-law-gmt -Ttime -Fmp4 -Cuhd -L -P -Z

# clean up
rm time plancks-law.gpl plancks-law-frame.sh
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2

Great!!!

Just a question. What does gnuplot -e do?

1 Like
3

A fine question! I’m to be considered a beginner in gnuplot - make no mistake.

gnuplot -e passes an argument (the temperature in this case) to the script.

Best with an example:

$ cat test.gpl 
print 5+T

$ gnuplot test.gpl
"test.gpl" line 1: undefined variable: T

$ gnuplot -e T=5 test.gpl
10
4

Ver. 2

#!/usr/bin/env bash
#Purpose: Animation of Planck’s law made in gmt from 20 K to 10.000 K
#Date: 20.12.2021
#Author: AnBj
#Requirements: gnuplot

#to-do:
# - add some static curves, e.g. for the sun T=5800, and some others objects
# - legend becomes wider when T goes from two digits to three to four. Make wide enough or use printf??
# - wanted to calculate Planck's law in bc, but bc does not supprt exp ^ (x/n) expressions - see https://stackoverflow.com/questions/28034126/cannot-get-complex-calculation-to-work-in-bc
# - include textual description of what kind of Lambda max we have; microwave, infrared, visible, UV, etc

#can do this with gmt?? gmt math -T20/10000/20 T 2 MUL =
#https://docs.generic-mapping-tools.org/latest/gmtmath.html
cat <<eof > plancks-law.gpl
# input temperature via command line, e.g.:
# gnuplot -e "T=600" plancks-law.gpl
# constants
c=2.99792458e8
# Planck's constant
h=6.62606896e-34
# Boltzmann constant
k=1.3806504e-23
# Temperature - not in use. We get this from argument
#T=573
# Wavelength - not in use.
#L=1000e-9
# For Wien's displacement law
b=2.90e-3
# sigma, stefan-boltzmann constant
s=5.670374419e-8

set table

f(x)=((2*pi*h*c**2)/(x**5)*(1/(exp((h*c)/(x*k*T))-1)))

set samples 1000
set xrange [10e-9:15e-6]
plot f(x)
eof

cat <<eof > plancks-law-frame.sh
gmt set GMT_THEME minimal

echo "10e-9 ifg ultraviolet" > light.x
echo "400e-9 ifg visible" >> light.x
echo "750e-9 ifg infrared" >> light.x
echo "5e-6 ifg microwave" >> light.x

 #pwd
 gmt begin
  gnuplot -e "T=\${MOVIE_COL0}" $(pwd)/plancks-law.gpl | gmt plot -W -R4e-09/5e-06/1e1/3.5e+14 -JX20c/10c -L+yb -Ggray
  gmt psbasemap -Bxclight.x -BwesN --MAP_GRID_PEN_PRIMARY=black,- --FONT_ANNOT_PRIMARY=6
  echo "\$(printf "%.20f" \$(echo 2.9*10^-3/\${MOVIE_COL0} | bc -l)) 3e15" | gmt plot -Sb0.02c -Gblack
  gmt psbasemap -Bxafg+l"Wavelength [m]" -Byafg+l"Radiated energy [Watt/m@+2@+ m]" -B+t"Planck's law"
  echo "4.9e-6 3.2e14 Temperature [K]: \${MOVIE_COL0}" | gmt pstext -W -Gwhite -F+jRM
  echo "4.9e-6 2.9e14 Total emitted energy [MW/m@+2@+]: \$(printf "%.0f" \$(echo 5.670374419*10^-8*\${MOVIE_COL0}^4/10^6 | bc -l))" | gmt pstext -W -Gwhite -F+jRM
  echo "4.9e-6 2.6e14 Dominant wavelength [nm]: \$(printf "%.0f" \$(echo 2.9*10^-3/\${MOVIE_COL0}*10^9 | bc -l))" | gmt pstext -W -Gwhite -F+jRM
 gmt end

rm light.x

eof

# generate values for input - these are the T-values; from 20 to 10.000 in steps of 20
#gmt version: gmt math -o0 -T20/10000/20 T = time
#(10000-20)/20=499 frames
seq -w 20 20 10000 > time

# run movie script
time gmt movie plancks-law-frame.sh -Nplancks-law-gmt-movie -Ttime -Fmp4 -Cuhd -L -P -Z

# clean up
rm time plancks-law.gpl plancks-law-frame.sh
2 Likes