Image 1 Caption: Blackbody spectra (in an specific intensity plot) for Kelvin temperatures 3000 K, 4000 K, and 5000 K. The classical physics Rayleigh-Jeans law for T = 5000 K is also shown in Image 1. The Rayleigh-Jeans law is only asymptotically correct as λ → ∞.

The horizontal axis is wavelength of electromagnetic radiation in microns (μm). The vertical axis is specific intensity (energy per unit time per unit steradian (sr) per unit area perpendicular to the beam path per wavelength) in appropriate units (i.e., kW/sr/m**2/nm).

Below, we explicate thermal radiation and its very important special case blackbody radiation.

Thermal Radiation:
1. All matter emits/absorbs electromagnetic radiation (EMR) from/into its own heat energy pool by various microscopic processes---which are transitions and charge accelerations brought about by thermal interactions. That matter always does this is due to a combination of physical laws, most prominently quantum mechanics and the 2nd law of thermodynamics.
  This kind of EMR is thermal radiation.
2. To qualify slightly the above item 1, to emit thermal radiation a body must have temperature above absolute zero (i.e., 0 kelvins (K)). However, this case is virtually always the case.
3. By "from/into its own heat energy pool", we mean that the EMR is NOT reflected EMR.
  Reflection is a process in which the EMR does NOT strongly interact with a body and only briefly and temporarily changes its surface state in the ideal case. In reflection, EMR just "bounces off" the body surface.
  Of course, bodies usually emit thermal radiation AND reflect EMR at the same time.
4. Dense materials (i.e., solids, liquids, and dense gases) have a continuum (insofar as we know) of emission and absorption channels that extend effectively over all wavelengths that are practically accessible: i.e., over all the electromagnetic spectrum that is practically accessible. So they radiate and absorb continuous spectra.
  Recall continuum means there are NO "missing points". The real numbers are a continuum of numbers. Integers and rational numbers are NOT continuums of numbers.
  Actually, crystalline solids like rocks can have broad emission and absorption bands (HI-92), and so do NOT necessarily radiate a continuum NOT counting a low-level background continuum of emission which alwmost always present for all radiators. We will NOT go into that detail about crystalline solids here.
5. Dilute gases have emission line spectra in emission and absorption line spectra in absorption. Actually, they always have weak (sometimes very weak) continuum emission and absorption too. We will NOT discuss line spectra further here.
Blackbody Radiation:
1. A sufficiently DENSE body at a SINGLE temperature will emit---NOT counting any reflection---blackbody radiation with a blackbody spectrum.
  The blackbody spectra for 3 temperatures are illustrated in Image 1.
  The blackbody spectra only depend on the temperature of the emitting body, NOT on any other properties: e.g., density (as long as sufficiently high), chemical composition, and shape. This remarkable fact is a result of thermodynamics.
2. The temperature of blackbody radiation is always specified on the Kelvin scale.
  The zero point of the Kelvin scale (i.e., T = 0 K) is absolute zero---which is the coldest possible state where all heat energy that can be removed has been removed.
  If true a blackbody radiator was at absolute zero (where there is no heat energy to radiate) and was NOT reflective, it would be "black" at all wavelengths.
3. A perfect blackbody spectrum is a smooth spectrum that is only zero at zero and infinite wavelength.
  As temperature increases, a blackbody spectrum increases at all wavelengths, except at zero and infinite wavelength. Thus, blackbody spectra of different temperatures only intersect at zero and infinite wavelength.
  A blackbody spectrum has single maximum and the maximum shifts blueward/redward in wavelength with increasing/decreasing temperature.
4. Blackbody radiation is the simple limiting case of thermal radiation when the emitter of the thermal radiation has a single temperature.
  In fact, the blackbody radiation itself has the temperature of the emitter. Below, we give the formula for blackbody radiation that shows the temperature dependence of blackbody radiation.
5. A perfect blackbody spectrum is an ideal limiting case, but very nearly perfect ones occur both in nature and in the laboratory and many bodies (e.g., stars) approximate blackbody radiators to one degree or another.
  
  Actually, many examples of thermal radiation approximate blackbody radiation so closely that they are just considered blackbody radiation. In fact, thermal radiation can often be considered as consisting of a mixture blackbody radiations of different temperature.
  Also actually, thermal radiation and blackbody radiation are often used loosely as synonyms including by yours truly. Thermal radiation that is NOT blackbody radiation just being thought of as mixed-temperature blackbody radiation.
6. A qualification about nearly perfect blackbody spectra in nature is that usually they occur deep inside an overall opaque system with a low or zero temperature gradient. Of course, there must be regions in the system transparent enough for electromagnetic radiation (EMR) to propagate.
  On the other hand, a body with sufficiently high temperature gradient near its surface will NOT radiate like a blackbody radiator, but will radiate thermal radiation that is, loosely speaking, a mixture of blackbody spectra each arising from a layer with a different temperature. A temperature gradient is often high because a body is cooling down or heating up, or is in steady state of heat energy flow because of some permanent internal heat energy source or sink.
  And, of course, such bodies with near-surface temperature gradients are everywhere: e.g., the ground, bodies of water, and walls, all of which are heated up in the daytime and cool down at nighttime.
  In general, hot bodies in a surrounding colder environment will NOT radiate like blackbody radiators, but they do approximate blackbody radiators to one degree or another.
7. Stars are a special case of approximate blackbody radiators of great interest in astronomy. Deep in their interior where they are very opaque, the temperature gradient is very low and their radiation field approximates blackbody spectra at the local temperature to extremely high accuracy.
  However, near their surface where they are becoming transparent, stars have a steep temperature gradient and their emitted radiation comes from layers with a range of temperatures. However, by a conspiracy of nature, the emitted spectra of main-sequence stars (those nuclearly burning hydrogen (H) to helium (He) in their cores) usually approximate to one degree or another blackbody radiation at a temperature equal to that of the stellar photosphere (i.e., the surface layer from which most radiation escapes to infinity) (Hu-21--22). This is why we say that main-sequence stars (or just stars when we know what we mean) approximate blackbody radiators.
  The temperature of an exact blackbody radiator that gives the same emitted radiant flux integrated over all wavelength as an actual star is called the star's effective temperature. A main-sequence star's effective temperature usually approximates its photospheric temperature.
  Pre-main-sequence stars and post-main-sequence stars usually do NOT well approximate blackbody radiators (Hu-21--22).
  And, of course, many astro-bodies do NOT radiate at all like blackbody radiators overall: e.g., planets which usually emit mostly reflected light though they usually have fraction of their emission which has a blackbody spectrum.
8. In fact, the name blackbody radiation is NOT very good since a blackbody radiator is often NOT black. In fact, to the human eye, it can look very bright and have a color depending on temperature (as we discuss below in item Color of Blackbody Radiation). Of course, a blackbody radiator can also reflect EMR.
  However, if the temperature of the blackbody radiator is sufficiently low and there is NO reflection, then yes a blackbody radiator will look black to the human eye.
  In any case, tradition (beginning in the 19th century) has stuck us with the name blackbody radiation. Thermodynamic equilibrium thermal radiation would be a more accurately descriptive, if longwinded, name.
9. The formula for blackbody radiation is Planck's law (i.e., blackbody radiation specific intensity: Wikipedia: Planck's law: The law) and it is
```
           2hc**2      1         2hc**2         1
  B_λ(T) = ------ ----------  =  ------ -----------------  ,
           λ**5   [exp(x)-1]     λ**5   [exp(hc/(kTλ))-1] 
```
  where the Planck constant h = 6.62607015*10**(-34) J·s (exactly), the vacuum light speed c = 2.99792458*10**8 m/s (exactly) ≅ 3*10**8 m/s = 3*10**5 km/s ≅ 1 ft/ns, and the x = hc/(kTλ), where the Boltzmann contant k = 1.380649*10**(-23) J/K (exactly) and T is temperature and λ is wavelength. For reference for the fundamental physical constants, see also NIST: Fundamental Physical Constants.
  
  Note, Planck's law can be regarded as a function of wavelength λ with temperature T as a parameter or as a function of both wavelength λ and temperature T.
10. To recapitulate from the preamble, the classical physics Rayleigh-Jeans law (published 1900, 1905) (Wikipedia: Rayleigh-Jeans law: Historical development) for T = 5000 K is also shown in Image 1.
  Rayleigh-Jeans law asymptotically agrees with the correct Planck's law in the limit that wavelength λ goes to infinity. But as λ goes to zero, the Rayleigh-Jeans law predicts infinite energy energy. This is the famous ultraviolet catastrophe which, noticed immediately in 1900, was a clue that classical physics failed outside of what we now call the classical limit.
  In fact, Planck's law, which gives correct formula for blackbody radiation, was also published in 1900 (Wikipedia: Planck's law: Finding the empirical law), but its understanding from quantum mechanics developed later.
11. Image 2 Caption: Blackbody spectra (in an specific intensity plot) for temperatures between 100 K and 10**4 K plotted on a log-log plot.
  The blackbody spectrum for T = 5777 K approximates the solar spectrum as seen above the Earth's atmosphere since T = 5777 K approximates the solar photosphere effective temperature = 5772 K (current best value). Effective temperature of a body (e.g., star or planet) is the temperature a perfect blackbody radiator (with a radius equal to the defined radius of the body) that radiates exactly as much radiant flux integrated over all wavelength Effective temperatures for stars are usually good average temperatures for their stellar photospheres.
  The blackbody spectrum for T = 300 K approximates the average Earth atmospheric temperature ≅ 14 C ≅ 287 K and the average human body temperature ≅ 37 C ≅ 310 K.
12. Note, the maximizing wavelength of the blackbody spectrum on the log-log plot in Image 2 shifts with temperature in a systematic way. The maximizing wavelength, in fact, obeys Wien's law which is explicated in detail file Blackbody file: wien_law.html (see also Wikipedia: Wien's displacement law: Derivation from Planck's law). However, to give the short explication, for a blackbody spectrum, Wien's law states that the maximizing wavelength λ_max = hc/(x_max*kT) = (2897.771955185172661 ... μm-K)/T, and so is inversely proportional to temperature T: i.e., λ_max ∝ 1/T. Thus, temperature as a function of maximizing wavelength obeys the inverse Wien's law T = hc/(x_max*λ_max*k)=(2897.771955185172661 ... μm-K)/λ_max. Therefore, if we constrain Planck's law by inverse Wien's law and so input the temperature for which any wavelength gives a maximum, we get a curve which is the maximum of blackbody spectrum as just a function of maximizing wavelength: i.e.,
```
                 2hc**2      1
  B_λ[T(λ)]^max = ------ --------------  ∝ λ**(-5) ,
                 λ**5  [exp(x_max)-1]   
```
  where we constrain x = x_max = hc/(kTλ) = (4.965114231744276303 ...) which is just a constant. Note that B_λ[T(λ)]^max is power law with power p = -5, and so gives straight line on the log-log plot in Image 2 with slope = -5. A rough calculation from the log-log plot values gives
```
  slope = (-1-9)/(1.5-0.5) = -5 
```
  using the fact that log(0.3) ≅ -1/2 and log(30) ≅ 3/2. We get the exact result -5 fortuitously.
  We need to remark that location of the maximum of a spectrum of any kind is actually representation dependent. For example, for the solar spectrum, the maximum in the wavelength representation is 0.502 μm, in the logarithmic representation is 0.635 μm, and in the frequency representation is 0.883 μm (Wikimedia Commons: File:Spectral Distribution of Sunlight.svg; Wikipedia: Sunlight: Composition and power). For more on spectrum representations, see Blackbody file: representation.html and J. Marr 2012, "A Better Presentation of Planck's Radiation Law" (also at J. Marr 2012, "A Better Presentation of Planck's Radiation Law").
13. As mentioned above, blackbody radiation itself has a temperature and this is what appears in Planck's law. Usually, this temperature will be the temperature of the emitting body. However, adiabatic expansion/contraction of the space occupied by the blackbody radiation can cause the blackbody radiation temperature to change independent of matter. A very important special case of this kind of change is that the cosmic microwave background (CMB) (formed in recombination era cosmic time t = 377,770(3200) Jyr = 1.192*10**13 s (z = 1089.80(21)) after the Big Bang with temperature ≅ 3000 K) cooled with the expansion of the universe and now is the cosmic microwave background (CMB) with CMB temperature = 2.72548(57) K (Fixsen 2009).
  If blackbody radiation and the matter it is in thermal contact with have the same temperature, then the two are in thermodynamic equilibrium with respect to each other.
  The CMB has nearly NOT been in thermal contact with matter since the recombination era. Occasionally, CMB photons scatter off free electrons or get absorbed by a star, planet, or an observational detector, but ∼ 95 % of them have NOT been in thermal contact with matter since the recombination era (Wikipedia: Λ-CDM model: Parameters). The temperatures of matter and non-CMB electromagnetic radiation (EMR) are extremely diverse.
14. Color of Blackbody Radiation:
  What color does the human eye with its peculiar psychophysical response see for a blackbody spectrum?
  Well, if the blackbody spectrum has sufficiently low temperature (i.e., its maximum in any representation is far redward from the visible band (fiducial range 0.400--0.700 μm = 400--700 nm)), the human eye sees nothing. (Recall, for spectrum representation, see Blackbody file: representation.html.)
  However, above "sufficiently low temperature" the psychophysical response to blackbody spectrum is given below in Table: Psychophysical Response Color to the Human Eye to Blackbody Radiation.
```
  Table:  Psychophysical Response Color to the Human Eye to Blackbody Radiation
      
   T (K)   T (C)   Psychophysical Response Color
      
  ⪅ 750   ⪅ 480   invisibile 
    753     480   faint red glow (red hot)
    853     580   dark red (red hot)
   1003     730   bright red, slightly orange (orange red hot)
   1203     930   bright orange (orange hot) 
   1373    1100   pale yellowish orange (yellow orange hot)
   1573    1300   yellowish white (yellow hot)
   1673    1400   white (yellowish if seen from a distance through atmosphere, (white hot)
   6500           white (white hot)
   8000           pale blue (white-pale-blue hot) 
   9000           pale blue (white-pale-blue hot) 
 >10000           blue-white (white-blue hot)
      
```
  Note: The table is NOT an authoritative reference. It was compiled and conflated from the references given below which were NOT entirely consistent and do NOT give exact specifications.
  References:
  1. Google AI: Not an authoritative source.
  2. Carine Fang: Temperature of a "White Hot Object": See the bottom of the page.
  3. Wikipedia: Thermal radiation: Frequency.
  In fact, from Image 2, it is clear that the maximum in the wavelength representation for blackbody spectra for temperature ⪅ 3000 K is always in the infrared band (fiducial range 0.7 μm -- 0.1 cm) or redward. Thus, even blackbody spectra that look whitish hot at temperature ⪅ 1500 K are, in fact, dominated by light from the redward end of the visible band (fiducial range 0.400--0.700 μm = 400--700 nm).
  Note there is NO "green hot" or "violet hot" in Table: Psychophysical Response Color to the Human Eye to Blackbody Radiation. Of course, we see green and violet, but that is NEVER our psychophysical response to the mixture of colors in blackbody spectra. The lack of "green hot" is particularly strking since the psychophysical luminosity function (photopic peak 0.555 μm, green band, scotopic peak 0.498 μm, green band) (i.e., human eye response to visible band light) peaks for both photopic vision (bright light) and scotopic vision (dim light) in the green band (fiducial range 0.495--0.570 μm).
15. Star Colors:
  At least for yours truly at this moment in 2025, the colors of stars from both ground-based astronomy and space-based astronomy is an unclear subject.
  Explication of the origins of the unclearness in points:
  1. Stars only approximate blackbody radiators, and so the applicability of the above Table: Psychophysical Response Color to the Human Eye to Blackbody Radiation is limited, but how limited is the question.
  2. Effective temperatures are usually what is cited for stars as a kind of average for their photospheric temperature, but the effective temperature may NOT be the best average photospheric temperature for describing star colors.
  3. Online AI tells yours truly that the Earth's atmosphere reddening effect on starlight is small, but does NOT give an authoritative discussion.
  4. All astronomical images are digital nowadays, and therefore the colors are whatever the image maker wants. Even if the image is described as true color, the colors may have been enhanced in some way to bring out features.
  5. What is true color anyway? Just what the average human eye see in certain average illumination.
  The upshot of this discussion is that provisionally yours truly accepts that apparent star colors in ground-based astronomy obey the following results with the cited temperatures probably being effective temperatures:
  Reference: Star Colors Explained, Brian Ventrudo, 2008 December 23.
  The explication given above explains why the star colors for given effective temperatures do NOT agree exactly with the blackbody radiator colors for given blackbody radiation temperatures in Table: Psychophysical Response Color to the Human Eye to Blackbody Radiation.
Blackbody Keywords:
See blackbody keywords below (local link / general link: blackbody_keywords.html).
Blackbody Radiation Videos:
See the Blackbody radiation videos below (local link / general link: blackbody_videos.html).

Images:

Credit/Permission: User:Darth Kule, 2010 / Public domain.
Image link: Wikipedia: File:Black body.svg.

Local file: local link: blackbody_spectra.html.
File: Blackbody file: blackbody_spectra.html.