By Andy May

This is part 3 of a series on defining temperature. In part one I covered various temperature definitions used and pointed out that “temperature,” unmodified is only a measurement of an emergent statistical property with little meaning beyond that. It is important in physics and daily life, but not a primitive well defined property like mass or energy. In part two I covered kinetic temperature, which is “what we measure with a thermometer” (Schroeder, 2000). In the original X discussion that spurred me to write these posts, some argued that the “real” definition of “temperature” was the thermodynamic equilibrium temperature. Of course this is not true. For nearly everyone on Earth, thermometers measure temperature.

There are a number of unique conditions required to measure a thermodynamic equilibrium temperature and a separate set of conditions to measure a kinetic (or thermometer) temperature, and they are not the same. While the kinetic temperature may be the same as the thermodynamic equilibrium temperature in a system at equilibrium, they are not the same thing. There is also another category of temperatures that are very different from kinetic or thermodynamic temperatures, called non-equilibrium temperatures. Non-equilibrium temperatures do not meet all the conditions required for kinetic and thermodynamic temperatures. Examples:

• The particle velocity distribution may not be Maxwellian
  
• Different directions in the system may have different temperatures (anisotropic)
  
• Different system components may have different temperatures (electron vs ion temperature in plasmas)
  
• System pressure may not be isotropic
  

Thermodynamic equilibrium temperature

Let’s briefly review what thermodynamic equilibrium temperature is, some illustrative plots are presented in figure 1.

At the top of figure 1 I show the formal mathematical definition of thermodynamic equilibrium temperature; it applies to the middle and right-hand plots. It is described in detail in post 1. In the equation, T is temperature, U is the system internal energy, S is system entropy, ∂U/∂S is the partial derivative of internal energy with respect to entropy, N is the number of particles, and V is the volume.

Figure 1 illustrates how thermodynamic temperature arises from the geometry of the equilibrium state space. The left panel shows the equilibrium manifold in the thermodynamic state space. The horizontal axis is entropy , the vertical axis is volume , and the shading and contour lines represent internal energy . For this plot N (the number of particles) is held fixed. Entropy may be interpreted as the number of microscopic configurations compatible with the macroscopic constraints, although in equilibrium thermodynamics it is treated as a state variable. For a system with fixed internal energy, volume, and particle number (a “microcanonical” ensemble), the equilibrium state is the point on this manifold where entropy is maximized. This point is the red dot. The blue arrow projects this equilibrium point down to the -axis, indicating how the full two‑dimensional manifold reduces to a one‑dimensional slice when volume is held fixed.

This projection shows how the manifold reduces to the slice shown in the middle and right-hand illustrations. The middle panel shows the curveat fixed and . The equilibrium point lies on this curve. The right panel zooms in on the equilibrium point and shows the tangent line to the curve at equilibrium. The slope of this tangent, , is the thermodynamic equilibrium temperature. Thus, temperature is not a coordinate in the state space but a geometric property.

Kinetic Temperature

Kinetic temperature was discussed in the last post, it is the conventional thermometer measured temperature. Whereas the thermodynamic temperature comes from the slope of the equilibrium surface U(S,V,N), Kinetic temperature comes from the average translational kinetic energy of the particles in the system being measured. It is defined through the velocity distribution of the particles, not through an equilibrium manifold. Figure 2 illustrates the development of kinetic temperature.

Comparing figure 1 to figure 2 shows that, while thermodynamic equilibrium temperature and kinetic temperature of a system at equilibrium might be equal, it is a superficial equality, the two temperatures are defined differently.

Thermodynamic temperature

• Defined on the macroscopic equilibrium manifold .
  
• Temperature is the slope .
  
• Requires equilibrium and a well‑defined entropy.
  

Kinetic temperature

• Defined from microscopic particle motion.
  
• Temperature is proportional to the average translational kinetic energy.
  
• Does not require entropy or equilibrium surfaces.
  
• Emerges from the velocity distribution.
  

Kinetic temperature arises from statistical mechanics rather than equilibrium thermodynamics.

Non-equilibrium temperature

There are number of techniques for measuring temperatures in non-equilibrium systems, but the commonly used translational temperature in a shock front is a very good example. It illustrates how important it can be to measure non-equilibrium temperatures. It is defined from the non‑Maxwellian velocity distribution immediately behind a shock front. The shock front temperature is widely used in aerospace, combustion, and atmospheric entry and it differs from thermodynamic and kinetic temperature in almost every way.

In a shock front, the particle velocity distribution is distorted. It often has a high‑energy tail, is anisotropic, and contains mode‑dependent temperatures, like translational, rotational, vibrational, and electron. The development of a shock front temperature is illustrated in figure 3. A shock front is the advancing edge of a shock wave, a propagating disturbance that moves faster than the speed of sound in a fluid and causes an abrupt, nearly discontinuous change in pressure, temperature, density, and other flow properties (Wikipedia).

The left illustration in figure 3 shows an example shock wave velocity space, it is clearly not at equilibrium. The middle illustration compares this distribution to a Maxwellian distribution in blue. The right illustration computes the shock front temperature using only the core region of the distribution.

The shock-front non-equilibrium temperature is only one of many. There are others, for example the brightness temperature used by Spencer and Christy (Spencer & Christy, 1990) at the UAH to determine atmospheric temperature for several intervals.

Radiation (brightness) temperature

This temperature measurement is ubiquitous in the atmospheric sciences, astrophysics, and remote sensing. Whereas the previous examples of temperature measurements used either entropy or particle velocity to determine temperature, this measure uses brightness or radiation intensity (I) to determine temperature. In essence, brightness temperature is the temperature you would infer if you assumed the radiation intensity was coming from a blackbody. You simply take the measured intensity at a given frequency (I_v), plug it into Planck’s law and solve for temperature. Crucially, it does not require the radiation field to be Planckian.

Brightness temperature is not necessarily the real temperature of the emitting body. A real radiation field may be a mixture of temperatures, and it may be far from equilibrium and if so, the spectrum is not a Planck curve. But, at any given frequency, radiation intensity always increases with temperature (see the right-hand plot in figure 4) and you can always find a unique temperature for any brightness. So, brightness temperature is valuable because it expresses radiance on a temperature scale, making radiative transfer relationships intuitive even when the radiation field is far from equilibrium. This is convenient in microwave remote sensing, such as that done at UAH.

However, that said, brightness temperature can equal physical temperature, or be very close to it, when the emitting medium is optically thick and in local thermodynamic equilibrium (LTE) at that frequency. The O₂ microwave bands from the atmosphere are close to this ideal, which is why UAH can treat brightness temperature as physical temperature after instrument corrections and calibration. UAH made a very good choice when they picked the oxygen microwave bands (~50-60 GHz) to use in their work. Oxygen molecules (O₂) in the atmosphere are well mixed and behave like a blackbody in local thermodynamic equilibrium. Thus, the brightness temperatures in the O₂ frequencies are very close to the real atmospheric temperature.

The satellite radiation measurements must be corrected for various instrument effects and orbital drift, and the satellites contain a two-point internal calibration to make their calculation of brightness temperature more accurate. Periodically the instrument is pointed to space (about 2.7K) and then to a warm target inside the satellite with a known temperature and these readings are used as part of the process of calibrating the brightness temperature to real temperatures (Spencer & Christy, 1990). These calibration points and the instrument corrections are good enough that both UAH and RSS use the final brightness temperature as is, they don’t try and calibrate the brightness temperatures to any ground-based data, but they do use the ground-based data to estimate the accuracy at chosen points, mainly weather balloon launch sites. The accuracy is quite good (Christy et al., 2018). The development of brightness temperature is illustrated in figure 4.

The left panel in figure 4 shows an equilibrium Planck curve versus frequency in black and a distorted brightness (I) versus frequency (v) curve in red. The distortion could be due to the lack of equilibrium or other complicating factors, like optical thinness, mixtures of temperatures, or anisotropic radiation fields. Real spectrums are more like the red distorted curve than the black Planckian curve. The middle panel zooms into a portion of the left plot and computes two brightness temperatures at two different frequencies. The two estimated temperatures are far from the Planckian curve. Finally, the right-hand plot inverts the Planck curve and shows a new plot of brightness versus temperature for one frequency. The curve is monotonic, with temperature increasing as brightness increases.

While the brightness temperature is not always equal to actual temperature, it can be very close, as it is in the UAH oxygen brightness measurements discussed above.

Besides brightness temperature, there are other non-equilibrium temperature measurements, these include:

• Vibrational temperature: Used for combustion and plasmas. It depends upon internal state populations and is defined with Boltzmann plots of excited populations of particles.
  
• Electron temperature: Used in plasmas, determined from the slope of a distorted, often non-Maxwellian energy distribution. It is defined by the high energy tail of the electron energy distribution.
  

Discussion

The key point is that the term “temperature” is just a measurement arbitrarily scaled with increasing energy. It has little meaning, until the type of temperature is specified with a modifier, like “thermodynamic,” “kinetic,” “shock-front,” “brightness,” etc. In everyday usage, “temperature” is assumed to be a temperature measured with a thermometer, which is a kinetic temperature, not a thermodynamic equilibrium temperature as some argued in the subject X thread.

Temperature does not have one definition or one meaning, either in physics or everyday use of the word. The examples discussed and illustrated above all have very different meanings and definitions. All are useful temperature measurements, but they do not all fit into one definition.

The illustrations were all made with R, the programs can be downloaded here.

H/T to Mike Chillit who suggested I make the illustrations to make the definitions clearer.

Works Cited

Christy, J. R., Herman, B., Sr., R. P., Klotzbach, P., McNider, R. T., Hnilo, J. J., . . . Douglass, D. (2010). What Do Observational Datasets Say about Modeled Tropospheric Temperature Trends since 1979? Remote Sensing, 9, 2148-2169. https://doi.org/10.3390/rs2092148

Christy, J. R., Spencer, R. W., Braswell, W. D., & Junod, R. (2018). Examination of space-based bulk atmospheric temperatures used in climate research. International Journal of Remote Sensing, 39(11), 3580–3607. https://doi.org/10.1080/01431161.2018.1444293

Schroeder, D. V. (2000). Thermal Physics. San Francisco: Addison Wesley Longman.

Spencer, R., & Christy, J. (1990). Precise Monitoring of Global Temperature Trends from Satellites. Science, 247. Retrieved from https://science.sciencemag.org/content/247/4950/1558.abstract