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.

Section 39–2 The pressure of a gas

Force per unit area / Energy per unit volume / Quasi-static adiabatic compression

 

In this section, Feynman relates the pressure of a gas to force per unit area, energy per unit volume, and quasi-static adiabatic compression. Most of these concepts trace back to Clausius’ (1857) paper, “On the Nature of the Motion which we call Heat.” In that seminal work, Clausius laid the foundation for the kinetic theory of gases, linking macroscopic properties like pressure and temperature to the microscopic motion of molecules. However, Feynman’s discussion goes beyond a basic explanation of gas pressure and includes a derivation of the adiabatic law.

 

1. Force per unit area

“We define the pressure, then, as equal to the force that we have to apply on a piston, divided by the area of the piston: P = F/A……. So we see that the force, which we already have said is the pressure times the area, is equal to the momentum per second delivered to the piston by the colliding molecules (Feynman et al., 1963, p. 39-3).”

 

The pressure of a gas is a macroscopic property that arises from the collective motion of microscopic particles. Its physical origin can be understood from three perspectives:

1. Macroscopic definition: Pressure (P) is defined as the force (F) exerted perpendicularly on a surface, divided by the area (A) of that surface: P = F / A.

2. Microscopic Origin: At the molecular level, gas particles move randomly at high speeds. When they collide with the walls of a container, they transfer momentum to the surface—producing a measurable force.

3. Statistical Average: The net pressure arises from averaging the momentum changes of many molecular collisions, as described by kinetic theory.

Feynman shows how macroscopic quantities like force and pressure arise from the statistical behavior of microscopic particles, bridging Newtonian mechanics with kinetic theory of gases and linking individual molecular motion to thermodynamic properties.

 

“… but eventually, when equilibrium has set in, the net result is that the collisions are effectively perfectly elastic. On the average, every particle that comes in leaves with the same energy. So we shall imagine that the gas is in a steady condition, and we lose no energy to the piston because the piston is standing still (Feynman et al., 1963, p. 39-3).”

 

In his 1857 paper, Clausius writes: “In order that Mariotte's and Gay-Lussac's laws, as well as others in connexion with the same, may be strictly fulfilled, the gas must satisfy the following conditions with respect to its molecular condition:

(1) The space actually filled by the molecules of the gas must be infinitesimal in comparison to the whole space occupied by the gas itself.

(2) The duration of an impact, that is to say, the time required to produce the actually occurring change in the motion of a molecule when it strikes another molecule or a fixed surface, must be infinitesimal in comparison to the interval of time between two successive collisions.

(3) The influence of the molecular forces must be infinitesimal (p. 116).”

In essence, Clausius stated three key simplifying assumptions to model ideal gases:

1. Infinitesimal Molecular Volume: The volume occupied by gas molecules is negligible compared to the container’s volume.
2. Infinitesimal Collision Duration: The time a collision takes is negligible compared to the time between collisions.
3. Infinitesimal Molecular Forces: Intermolecular forces are negligible except during collisions.

While Clausius restricted his model to these three assumptions, later developments in kinetic theory expanded the framework to account for real-gas effects (e.g., van der Waals forces). However, Clausius's framework remains a milestone in the development of statistical mechanics.

 

The kinetic theory of gases is built on a set of simplifying assumptions that ensure analytical solvability while offering reasonable agreement with experimental observations for gases:

1. Point Particles: Gas molecules are idealized as point masses since their individual volumes are negligible compared to the container size (L). Thus, the time between collisions with the same wall can be simplified as Δt = 2L/v_x, where vₓ is the average velocity component in the x-direction.

2. No Intermolecular Forces (Except During Collisions): Molecules are assumed not to exert forces on each other except during brief, elastic collisions. Between collisions, they move in straight lines at constant speeds.

3. Short Collision Duration: Collisions are assumed to occur instantaneously, allowing the momentum change to be treated as abrupt without the need of modeling the detailed interaction over time.

4. Perfectly Elastic Collisions: All collisions—whether between molecules or with the container walls—are assumed to be perfectly elastic. This implies:

(a) Kinetic energy is conserved, with no energy loss to heat or deformation.

(b) When a molecule collides with a wall, its momentum changes from +p​ to –p​, but its speed remains unchanged.

5. Large Number of Particles: The gas consists a large number of molecules (e.g., 10²³ or more), allowing statistical averaging. This enables definitions of macroscopic quantities such as pressure and temperature.

6. Random Motion: Molecules move randomly, following the Maxwell-Boltzmann distribution. At any moment, molecules are equally likely to move in any direction. The mean square velocity is distributed evenly among the three spatial dimensions: ⟨v²⟩=⟨v_x²⟩+⟨v_y²⟩+⟨v_z²⟩ where ⟨·⟩ denotes an ensemble averaging.

7. Negligible Gravitational Effects: Gravitational forces are considered too weak to significantly influence molecular motion. As a result, the velocity distribution remains isotropic: ⟨v_x²⟩ = ⟨v_y²⟩ = ⟨v_z²⟩ = ⟨v²⟩/3.

Some physicists introduce additional simplifying assumptions, such as identical particle masses (m), negligible relativistic effects (valid at low to moderate temperatures), and the absence of quantum effects (valid at high temperatures and low densities). These assumptions underpin the derivation of the ideal gas law and help connect microscopic particle dynamics to macroscopic thermodynamic observables. While real gases deviate from ideal behavior—especially at high densities or low temperatures—the kinetic theory remains a foundational framework for understanding gas behavior under most practical conditions.

 

2. Energy per unit volume

“For a monatomic gas we will suppose that the total energy U is equal to a number of atoms times the average kinetic energy of each, because we are disregarding any possibility of excitation or motion inside the atoms themselves. Then, in these circumstances, we would have PV= (2/3)U (Feynman et al., 1963, p. 39-5).”

 

Feynman showed that the product PV of a monatomic ideal gas corresponds is directly proportional to the internal energy U. This does not mean PV represents the work done by compressing a gas to zero volume at constant pressure—such a process would be physically unattainable. Rather, pressure and volume are interdependent, governed by the ideal gas law, and cannot be varied independently without altering other state variables. Notably, kinetic theory offers a more fundamental view of pressure—not merely as force per unit area, but as the rate of momentum transfer per unit area due to molecular collisions. This perspective also allows pressure to be interpreted as an energy density (energy per unit volume), revealing a deep connection between the mechanical origin of pressure and its thermodynamic role.

 

Note: A single particle cannot exert pressure in the thermodynamic sense—this pressure is a statistical property that emerges only from the collective behavior of many particles.

 

“It is only a matter of rather tricky mathematics to notice, therefore, that they are each equal to one-third of their sum, which is of course the square of the magnitude of the velocity: ⟨v_x²⟩= (1/3)⟨v_x²+ v_y²+ v_z²⟩=⟨v²⟩/3 (Feynman et al., 1963, p. 39-4).”

 

The "trick" Feynman highlights goes beyond the familiar Pythagorean theorem (or famous theorem of Greek*); it lies in bridging geometric symmetry with statistical reasoning. First, the equation v²=v_x²​+v_y²​+v_z²​, which relies on the three-dimensional Pythagorean theorem, applies to the velocity of a single atom or molecule. Second, the expression <v²> = <v_x²​ + v_y² + v_z²> may resemble the Pythagorean theorem, but its physical meaning is about statistical averaging over a large number of atoms. Third, a key insight comes from exploiting spherical symmetry. In an idealized isotropic system—where no direction is preferred and external forces such as gravity are absent—the average kinetic energy is evenly distributed across all spatial dimensions. This symmetry allows us to reduce the complexity of three-directional motion to a simple relation: <v_x²​> = <v²>/3. The one-third factor, a consequence of isotropy, underpins the derivation of the equipartition theorem and ultimately leads to the ideal gas law.

 

*In the audio recording [18:00], Feynman mentions the 'famous theorem of the Greeks,' more commonly known today as the Pythagorean theorem—though in higher dimensions, it is sometimes associated with de Gua’s theorem.

 

3. Quasi-static adiabatic compression

“A compression in which there is no heat energy added or removed is called an adiabatic compression, from the Greek a (not) + dia (through) + bainein (to go). (The word adiabatic is used in physics in several ways, and it is sometimes hard to see what is common about them.) (Feynman et al., 1963, p. 39-5).”

 

The term adiabatic derives from the Greek a- (not) and diabatos (passable), meaning “not passable”—in this context, referring to the absence of thermal energy transfer. In thermodynamics, a process is considered adiabatic if there is no heat transfer between a system and its surroundings, formally expressed as Q = 0 where Q is the heat transfer. This condition can be achieved—or more accurately, approximated—in two main ways:

1.      Perfect insulation – The system is thermally isolated, preventing any heat flow.

2.      Rapid process – The process occurs so quickly that there is insufficient time for significant heat transfer (e.g., a sudden gas expansion).

In general, adiabatic conditions are idealized approximations—used to simplify the analysis of systems in which heat transfer is minimal or intentionally ignored. In reality, perfectly adiabatic processes are physically unattainable; even under highly controlled conditions, some degree of thermal interaction inevitably occurs. As such, the term adiabatic can be somewhat misleading when applied to complex real-world systems—such as quantum systems—where complete isolation from thermal exchange is practically impossible.

 

“That is, for an adiabatic compression all the work done goes into changing the internal energy. That is the key—that there are no other losses of energy—for then we have PdV=−dU (Feynman et al., 1963, p. 39-5).”

 

Specifically, it is a quasi-static adiabatic compression, which has the following key features:

1. Quasi-static (Reversible): The process is carried out infinitely slowly, allowing the system to remain in thermal equilibrium at every stage. This ensures the process is reversible and that pressure P and volume V are well-defined throughout, allowing work to be calculated as ∫P dV.

2. Adiabatic condition (no heat transfer): The system is perfectly insulated, so no heat transfer with the surroundings. This implies ΔQ=0, and all energy transfer occurs only through mechanical work.

3. Compression (external work): Work is done on the gas by compressing it, which increases its internal energy. Since ΔQ = 0, the first law of thermodynamics reduces to dU = −PdV, where the change in internal energy dU results entirely from volume change under pressure.

This idealized model is fundamental in thermodynamics, providing insight into processes such as the temperature rise of a gas during compression and forming the theoretical basis for thermodynamic cycles, including those in heat engines.

 

Everyday Connection

“In banging against the eardrums they make an irregular tattoo—boom, boom, boom—which we do not hear because the atoms are so small, and the sensitivity of the ear is not quite enough to notice it. The result of this perpetual bombardment is to push the drum away, but of course there is an equal perpetual bombardment of atoms on the other side of the eardrum, so the net force on it is zero…... We sometimes feel this uncomfortable effect when we go up too fast in an elevator or an airplane…... (Feynman et al., 1963, p. 39-3).”

 

Another notable example is swimming-induced vertigo, often caused by air pressure imbalances in the ear—a condition known as alternobaric vertigo (swimming-induced vertigo). This type of vertigo can occur in several scenarios:

Shallow-water diving: Even small changes in depth (1–2 meters) can cause discomfort due to unequal pressure in the ears.

Uneven pressure equalization: Clearing pressure in one ear but not the other, sometimes due to nasal congestion or poor technique.

Tight swim goggles: Excessive external pressure on the outer ear may worsen air pressure imbalances.

While swimming may improve posture (e.g., reverse hunchback), relieve neck discomfort or reduce back pain, addressing one issue can sometimes introduce another.

 

Review Questions:

1. What is the minimum number of assumptions needed for the kinetic theory of (ideal) gases?

2. How does the mechanical definition of pressure as force per unit area (F/A) relate to its thermodynamic interpretation as energy per unit volume (U/V)?

3. Can a Zen master releasing intestinal gas be considered an example of an adiabatic process? What conditions (e.g., rapid expansion, thermal isolation) would be necessary for it to approximate an adiabatic process?

 

The moral of the lesson (in Feynman’s spirit): While one might theoretically liken flatulence to a quasi-static process, biological reality imposes strict constraints—no human can regulate the release slowly enough to maintain thermal equilibrium. In practice, achieving 'no heat transfer' requires either near-perfect insulation or a process so rapid that heat exchange is negligible—such as an adiabatic compression.

Note: The term 'adiabatic' was first introduced by William John Macquorn Rankine in his 1866 publication as shown below.

(Rankine, 1866)

References:

Clausius, R. (1857). Ueber die Art der Bewegung, welche wir Wärme nennen [On the nature of the motion which we call heat]. Annalen der Physik und Chemie, 176(3), 353–380.

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

Rankine, W. J. M. (1866). On the theory of explosive gas engines. Proceedings of the Institution of Civil Engineers, 25, 509–539.