Section 39–3 Compressibility of radiation

Adiabatic system / Adiabatic law / Adiabatic index

 

In this section, there are three closely related concepts: the adiabatic system, the adiabatic law for photon gases, and the adiabatic index. Although titled “Compressibility of Radiation,” it is related to stellar structure and stability. These ideas originate in Arthur Eddington’s (1926) seminal work The Internal Constitution of the Stars, which laid the theoretical foundation for modern astrophysics.

       While Eddington’s model was groundbreaking, it was later refined by Subrahmanyan Chandrasekhar, whose 1933 theory of white dwarfs introduced a critical mass threshold—now known as the Chandrasekhar limit—and earned him the 1983 Nobel Prize in Physics. Initially, both Milne and Eddington praised Chandrasekhar’s thesis for resolving discrepancies in their models, but Chandrasekhar’s conclusion—that stars exceeding a certain mass cannot become white dwarfs—challenged Eddington’s predictions and reshaped our understanding of stellar evolution.

 

1. Adiabatic system

“We have a large number of photons in a box in which the temperature is very high. (The box is, of course, the gas in a very hot star. The sun is not hot enough; there are still too many atoms, but at still higher temperatures in certain very hot stars, we may neglect the atoms and suppose that the only objects that we have in the box are photons.) (Feynman et al., 1963, p. 39-6).”

 

Stars are often modeled as adiabatic systems, meaning that heat transfer with the surroundings is negligible. This approximation holds well in the stellar interior, where the high density inhibits significant energy loss. Within the stars, energy is transported primarily by radiative diffusion and convection (see below), but both processes operate over timescales much longer than those of local dynamical processes (Kippenhahn et al., 2012). Under conditions of extreme pressure and density, the photon behaves approximately adiabatic, especially in regions where radiation pressure dominates (Eddington, 1926). However, this approximation breaks down near the stellar surface, where densities decrease and photons can escape into space; near the photosphere, radiative losses becomes significant, and the adiabatic model no longer applies.

 

(Johnson et al., 2000, p. 311)

The Sun is composed primarily of hydrogen (» 71%) and helium (» 27%), with trace amounts of heavier elements such as oxygen, carbon, and iron (see below). Its energy is generated through nuclear fusion in the core, producing high-energy photons in the process. Due to the Sun’s extreme interior density, these photons undergo countless scatterings, taking thousands to millions of years to reach the surface. To illustrate how light behaves in such hot, dense environments, Feynman introduced a simplified model: a box filled with photons, representing an idealized photon gas. This model captures key concepts like radiation (photon) pressure, but it omits essential features of the real star—such as photon-matter interactions, the role of convection, and the star’s complex layered structure.

 

Source: (Wilkinson, 2012)

Note: The adiabatic assumption can be found in The Internal Constitution of The Stars, where Eddington (1926) mentions: “By hypothesis there is no appreciable gain or loss of heat by conduction or radiation it therefore expands without gain or loss of heat, i.e., adiabatically (p. 98).”

 

2. Adiabatic law:

For photons, then, since we have 1/3 in front, (γ−1) in (39.11) is 1/3, or γ=4/3, and we have discovered that radiation in a box obeys the law PV^4/3=C (Feynman et al., 1963, p. 39-6).”

 

It is more accurate to say that we idealize a system of photons as obeying the adiabatic law. This law can be expressed in various equivalent way: e.g., as a pressure-density relation (P = kρ^γ), a temperature-volume relation (TV^γ−1 = constant), and a pressure-the temperature relation (P^(1−γ)/γT = constant). In astrophysics, the pressure-density form is preferred because it directly relates two main variables without requiring knowledge of temperature profile. In short, Eddington’s (1926) work was a brilliant deduction—a logical consequence of applying known physics to stars, i.e., it was not a discovery whereby photons strictly obey the adiabatic law. By proposing the relation P = kρ^γ, a polytropic equation of state, he treated k and γ as adjustable parameters, thereby simplifying the stellar model by letting temperature as a dependent variable.

 

In Eddington’s model of stellar structure, the polytropic process serves as a powerful tool because it offers greater flexibility than the strict adiabatic assumption. A polytropic model introduces an adjustable index n, which is related to the adiabatic index by the relation γ=1+1/n. This allows the model to represent different types of energy transport, including both convection and radiation. Crucially, polytropic models allow intermediate values of n (e.g., n = 3 in Eddington’s model), making them suitable for modeling real stars in which both gas pressure and radiation pressure contribute significantly. In this way, Eddington’s use of polytropes provided a more general and adaptable framework, with adiabatic behavior emerging as a special case within a broader continuum.

 

In The Internal Constitution of The Stars, Eddington (1926) writes: “… we content ourselves with laying down an arbitrary connection between P and r and tracing the consequences. In general, whether the gas is perfect or imperfect, any value of the pressure can be made to correspond to given density by assigning an appropriate temperature our procedure thus amounts to imposing a particular temperature distribution on the star… The third relation is taken to be of the form P = kρ^g where k and g are disposable constants (p. 80).”

 

3. Adiabatic index:

“So we know the compressibility of radiation! That is what is used in an analysis of the contribution of radiation pressure in a star, that is how we calculate it, and how it changes when we compress it (Feynman et al., 1963, p. 39-6).”

 

In general, the adiabatic index γ depends on the microscopic structure of the gas, as it reflects how energy is distributed among translational, rotational, and vibrational degrees of freedom. In Eddington’s model, γ=4/3​ applies to the radiative core, where radiation pressure dominates, while γ=5/3​ is more appropriate for the outer convective layers, where gas pressure governs the dynamics. In Chandrasekhar’s theory of white dwarfs, the condition γ=4/3​ emerges as a critical threshold: when the effective γ falls below this value—due to relativistic electron degeneracy at high densities—the star becomes dynamically unstable and collapses under its own gravity. This threshold encapsulates the balance between internal pressure and gravitational force, shaped by the star’s mass, composition, and the relative contributions of gas and radiation pressure. In this sense, the deceptively simple value γ=4/3​ marks a critical boundary between stellar stability and gravitational collapse, and thus between the life and death of a star.

 

In The Internal Constitution of The Stars, Eddington (1926) writes: “The value of g for the stellar material must be estimated or guessed; but the range of uncertainty from this cause is not very great. It is impossible for g to exceed the value 5/3 which corresponds to a mon-atomic gas; and it can be shown that if g is less than 4/3 the distribution is unstable (p. 98).”

 

Chandrasekhar’s Breakthrough

Chandrasekhar extended Eddington’s model by incorporating electron degeneracy pressure, a concept that Eddington had largely dismissed. While Eddington’s polytropic approach effectively described stars with an adiabatic index γ ranging between 4/3 to 5/3, Chandrasekhar showed that white dwarfs—supported by degenerate electrons—require relativistic treatment. His analysis revealed that as a white dwarf's mass approaches a critical threshold—the Chandrasekhar limit (» 1.4 solar masses)—the pressure response weakens, and γ falls below 4/3​, triggering gravitational collapse. Beyond this limit, it may lead to the possible formation of supernovae, neutron stars, or black holes, depending on the mass of the progenitor star. In short, Chandrasekhar’s synthesis of quantum mechanics and special relativity overcame limitations of Eddington’s model and profoundly transformed our understanding of stellar evolution.

 

Review questions:

1. Why can a star be modeled as an adiabatic system in which photon (radiation) pressure dominates?

2. Why did Eddington prefer to use the polytropic equation of state P = kρ^γ in modeling stars, rather than limit himself to the strict adiabatic law?

3. How does the adiabatic index γ determine the stability of a star against gravitational collapse?

 

The moral of the lesson (in Feynman’s spirit): For years, Chandrasekhar’s model was dismissed—not because it was wrong, but because Eddington publicly ridiculed it. Even though physicists like Dirac*, Peierls, and Pryce refuted Eddington’s objections, many astrophysicists followed Eddington’s lead and ignored Chandrasekhar’s results. In a twist of irony—with humility—Chandrasekhar later described Eddington as “the most distinguished astrophysicist of his time,” a testament to science’s capacity for self-correction and grace, even when ideas clash. The warning? Brilliance is no protection against self-deception. As Feynman famously said, “The first principle is that you must not fool yourself—and you are the easiest person to fool.”

 

*Dirac, Peierls, and Pryce (1942) write: “Eddington raises an objection against the customary use of the Lorentz transformation in quantum mechanics, as for instance when applied to the theory of the hydrogen atom or the behaviour of a degenerate gas. This objection seems to us to be mainly based on a misunderstanding......”

 

Fun facts: Eddington, like Einstein, had a passion for cycling. In fact, the Eddington Number—named in his honor—is a metric used by cyclists to track their endurance accomplishments. The number E represents the largest value such that a cyclist has ridden at least E miles (or kilometers) on E different days. For example, an Eddington Number of 50 means the cyclist has completed 50 rides of at least 50 miles each on 50 separate days. Beyond its intellectual appeal, cycling provides significant physical benefits. It is a low-impact exercise that strengthens the muscles around the knee, improves joint mobility, and can alleviate knee pain without placing undue stress on the joints. However, individuals with conditions such as tendonitis, bursitis, or cartilage damage should approach cycling with caution, as improper form or intensity may aggravate existing issues.

 

References:

Dirac, P. A., Peierls, R., & Pryce, M. H. L. (1942). On Lorentz invariance in the quantum theory. In Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 38, No. 2, pp. 193-200). Cambridge University Press.

Eddington, A. S. (1926/1979). The internal constitution of the stars. In A Source Book in Astronomy and Astrophysics, 1900–1975 (pp. 281-290). Harvard University Press.

Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Johnson, K., Hewett, S., Holt, S., & Miller, J. (2000). Advanced Physics for You. Nelson Thornes.

Kippenhahn, R., Weigert, A., & Weiss, A. (2012). Stellar Structure and Evolution (2nd ed.). Springer.

Wilkinson, J. (2012). New Eyes on the Sun: A Guide to Satellite Images and Amateur Observation (p. 98). Springer.