The Physics
Opus in profectus


search icon



Heat radiation (as opposed to particle radiation) is the transfer of internal energy in the form of electromagnetic waves. For most bodies on the Earth, this radiation lies in the infrared region of the electromagnetic spectrum.

One of the first to recognize that heat radiation is related to light was the English astronomer William Herschel (1738–1822), who noticed in 1800 that if a thermometer was moved from one end of a prism produced spectrum to the other, the highest temperatures would register below the red band, where no light was visible. Because of this position, this form of radiation is called infrared (infra being the Latin word for below or within). Sometimes this kind of radiation is called "heat waves" but this is a misnomer. Recall that heat is the transfer of internal energy from one region to another. As all forms of electromagnetic radiation transfer internal energy, they could all be called "heat waves".

stefan-boltzmann law

Hot objects are "brighter" than cold objects. Dark objects lose and gain heat faster than light objects.

The Stefan-Boltzmann law relates the heat flow rate emitted or absorbed from an object to its temperature (and surface area and darkness). It was empirically derived by the Austrian physicist Joseph Stefan in 1879 and theoretically derived by the Austrian physicist Ludwig Boltzmann in 1884. It is now derived mathematically from Planck's law.

P = εσA(T4 − T04)


P =  net heat flow rate [W] emitted (+) or absorbed (−)
ε =  (epsilon) emissivity, a dimensionless (unitless) measure of a material's effective ability to emit or absorb thermal radiation from its surface; ranges from 0 (none) to 1 (maximal)
σ =  (sigma) Stefan's constant, 5.670 × 10−8 W/m2K4
A =  surface area [m2] of the object emitting or absorbing thermal radiation
T =  absolute temperature [K] of the object emitting or absorbing thermal radiation
T0 =  absolute temperature [K] of the environment

Connect Stefan-Boltzmann law to Planck's law.

σ = 
5k4  =  π2k4
15h3c2 60ℏ3c2
σ =  5(1.38 × 10−23 J/K)4
15(6.63 × 10−34 J s)3(3.00 × 108 m/s)2
σ =  5.670374419 × 10−8 W/m2K4  
Hiroshima burn victim
Source: US National Archives and Records Administration

Dark colors absorb more radiant energy than do light colors. The burns on this woman's skin mimic the pattern on her blouse. She was exposed to a monstrous dose of electromagnetic radiation from a nuclear blast.

Disconnected thoughts that aren't quotes.

Wien's displacement law

Warm objects are infrared, warmer objects are red hot, even warmer objects are white hot, even more warmerer objects are blue hot. Color and temperature are related or, more precisely, spectra and temperature are related. The absolute temperature of an object emitting thermal radiation is inversely proportional to the peak wavelength of its spectrum and directly proportional to the peak frequency of its spectrum.

T ∝1/λmax

T ∝fmax

These relationships became know as Wien's displacement law to honor the German physicist Wilhelm Wien who first formulated the law in 1893. Wien used a difficult thermodynamic argument that I will not pretend to understand. The law is now derived mathematically from Planck's law and is done so more formally in that section of this book.

The law was originally stated in terms of peak wavelength.

λmax = b


λmax =  the peak wavelength in the spectrum of the thermal radiation emitted by an object (read the symbol as "lambda max")
b =  Wien's displacement constant (sometimes called Wien's wavelength displacement constant). The choice of units depends on the nature of the radiation being studied. The SI unit of wavelength is the meter, but use whatever unit you prefer for whatever application you have to deal with.
  • 2.897772 × 10−3 m K
  • 2.897772 mm K
  • 2,897.772 μm K
  • 2,897,772 nm K
T =  the absolute surface temperature [K] of the radiating object

Wien's law now is sometimes also stated in terms of peak frequency.

fmax = bT


fmax =  the peak frequency in the spectrum of the thermal radiation emitted by an object
b′ =  Wien's frequency displacement constant (read the symbol as "bee prime"). Again, the choice of units depends on the situation. The SI units is the hertz, but I prefer the gigahertz for peaks in the microwave bands (like the cosmic microwave back ground) and terahertz for peaks in the infrared and visible bands (which is pretty much everything else).
  • 5.878926 × 1010 Hz/K
  • 58.78926 GHz/K
  • 0.05878926 THz/K
T =  the absolute surface temperature, in kelvin [K], of the radiating object

Please note that the two constants are not interconvertable using the wave speed equation. These equations are true…

f = c
 ⇔  c = fλ  ⇔ 
λ =  c

but these equations are inequalities…

fmax ≠  c
 ⇔  c ≠ fmaxλmax  ⇔ 
λmax ≠  c

The constants in Wien's displacement law are derived from spectral distributions over wavelength and frequency — basically, complicated probability distributions. Although wavelength and frequency are inversely proportional, their behavior as variables in spectral distributions do not transform so easily.

Try these simple comparisons. Determine the peak wavelength and frequency for the thermal radiation coming from the Sun (T = 5,772 K). Using your favorite reference source determine the type of radiation and its color if it is visible.

λmax = 
λmax = 
2,897,772 nm K
5772 K
λmax =  502 nm 
fmax = bT  
fmax = (0.0587926 THz/K)(5772 K)  
fmax = 339 THz  

The peak wavelength, 502 nm, is in the green part of the visible spectrum — although in some cultures this wavelength might be considered blue. The peak frequency, 339 THz, is in the infrared — that statement is true for all human cultures.

Let's now do the naive transformations for a less subjective comparison. The numbers just don't work out.

λmax ≠  c
λmax ≠  3.00 × 108 m/s
339 × 1012 Hz
λmax ≠ 883 nm  
fmax ≠  c
fmax ≠  3.00 × 108 m/s
502 × 10−9 m
fmax ≠ 597 THz  
blackbody color by temperature
1,000 K6,500 K (daylight)10,000 K
Temperature (or effective temperature) of selected radiant sources
radiant energy source
2.73 cosmic background radiation
306 human skin
500 household oven at its hottest
660 minimum temperature for incandescence
770 dull red heat
1,400 glowing coals, electric stove, electric toaster
1,900 candle flame
2,000 kerosene lamp
2,800 incandescent light bulb, 75 W
2,900 incandescent light bulb, 100 W
3,000 incandescent light bulb, 200 W
3,100 sunrise or sunset (effective)
3,200 professional studio lights
3,600 one hour after sunrise or one hour before sunset (effective)
4,000 two hours after sunrise or two hours before sunset (effective)
5,500 direct midday sunlight
6,500 daylight (effective)
7,000 overcast sky (effective)
20–30,000 lightning bolt
Color scale of temperature

This table is the result of an effort to interpret in terms of thermometric readings, the common expressions used in describing temperatures. It is obvious that these values are only approximations.

Handbook of Chemistry & Physics, Ninth Edition, 1922

color temperature
°C K
incipient red heat 500–550 770–820
dark red heat 650–750 0920–1020
bright red heat 850–950 1120–1220
yellowish red heat 1050–1150 1320–1420
incipient white heat 1250–1350 1520–1620
white heat 1450–1550 1720–1820
Metal temperature by color Source: Process Associates of America
color approximate temperature
°F °C K
faint red 930 500 770
blood red 1075 580 855
dark cherry 1175 635 910
medium cherry 1275 0690 0965
cherry 1375 0745 1020
bright cherry 1450 0790 1060
salmon 1550 0845 1115
dark orange 1630 0890 1160
orange 1725 0940 1215
lemon 1830 1000 1270
light yellow 1975 1080 1355
white 2200 1205 1480
Spectral classification of stars
T (K) class λmax (nm) color name examples
30,000 O 100 blue Naos, Mintaka
20,000 B 150 blue-white Spica, Rigel
10,000 A 290 white Sirius, Vega
8000 F 360 yellow-white Adhafera, Procyon
6000 G 480 yellow Sun, Alpha Centauri
4000 K 720 orange Arcturus, Aldebaran
3000 M 970 red Betelgeuse, Rao

solar energy

greenhouse effect


The basic effect…

Cartoon illustration of the greenhouse effect on EarthMagnify

Global temperature and atmospheric carbon dioxide trends match. The very long graph made popular by Al Gore in An Inconvenient Truth.


Plot one against the other. The relation is approximately linear. Al Gore never did this one.


Naturally occurring greenhouse gases whose concentrations are increasing due to human activities

Other naturally occurring greenhouse gases of lesser concern.

Greenhouse gases that do not occur naturally.

Indirect greenhouse gases

key infrared absorption bands in the atmosphere correspond to H2O, CO2, O3

Graph of solar irradiance compared to an ideal blackbody and the effective irradiance after passing through the atmosphere

Global warming properties of selected greeenhouse gases Source: IPCC and others, * trifluoromethyl sulphur pentafluoride
molecule global warming
(CO2 = 1)
CO2 carbon dioxide 1 120 1.66 0.000014
CH4 methane 21 12 0.48 0.00037
N2O nitrous oxide 310 114 0.16 0.00303
CCl3F CFC-11 3,800 45 0.063 0.25
CF2Cl2 CFC-12 8,100 100 0.17 0.32
C2F3Cl3 CFC-113 4,800 85 0.024 0.3
CHClF2 HCFC-22 1,500 12 0.033 0.2
CCl4 carbon tetra­chloride 1,400 26 0.012 0.13
CH3CCl3 methyl chloro­form 146 5 0.0011 0.06
CHF3 HFC-23 11,700 270 0.0033 0.19
C2HF5 HFC-125 2,800 29 0.0009 0.23
C2H2F4 HFC-134a 1,300 14 0.0055 0.16
C2H4F2 HFC-152a 140 1.4 0.0004 0.09
SF6 sulfur hexa­fluoride 23,900 3,200 0.0029 0.52
SF5CF3 see note below* 19,000 1,000 ? 0.59
H2O water, tropo­spheric ? ? ? ?
H2O water, strato­spheric ? ? 0.02 ?
O3 ozone, tropo­spheric ? ? +0.35 ?
O3 ozone, strato­spheric ? ? −0.15 ?
CO carbon mon­oxide ? 0.25 ? ?
H2 hydrogen ? ? ? ?

Carbon dioxide levels are rising.

CO₂ vs. year, starting in 1958Magnify

Temperatures are rising across the globe.

Temperature anomaly vs. year, starting in 1880Magnify

But especially in and near the Arctic.