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Did you know you can predict a stellar explosion’s energy by counting bits of information that reorganize during this catastrophic phase transition? Type Ia supernovae release approximately $10^{44}$ J with consistency that makes them cosmological standard candles. Standard astrophysical models invoke carbon detonation cascades through white dwarf cores. An alternative mathematical framework counts phase space reorganization from electron to neutron degeneracy—thermodynamics and geometry.

The Chandrasekhar Boundary

White dwarfs maintain structure through electron degeneracy pressure ¹—quantum mechanical resistance arising from Pauli exclusion when electrons pack tightly enough that their wavefunctions overlap. For a white dwarf of mass $M$ and radius $R$, the gravitational compression scales with the Schwarzschild radius $r_s = 2GM/c^2$.

At the Chandrasekhar limit $M_{Ch} = 1.4 M_{\odot}$ ², white dwarfs reach $R/r_s \approx 10^3$. The ratio of Schwarzschild radius to actual radius provides a dimensionless measure of gravitational stress. For a 1.4 $M_{\odot}$ white dwarf with $R \approx 5000$ km,

$$\frac{r_s}{R} = \frac{4.15 \text{ km}}{5000 \text{ km}} = 8.3 \times 10^{-4}.$$

This boundary coincides with electron relativistic effects becoming important. The Fermi momentum approaches $p_F \sim m_e c$, at which point degeneracy pressure no longer scales steeply enough with density to resist further compression.

The electron number density at this limit is,

$$n_e = \frac{\rho}{m_p} \frac{Z}{A} \approx \frac{10^9 \text{ kg/m}^3}{2 \times 1.67 \times 10^{-27} \text{ kg}} = 3 \times 10^{35} \text{ m}^{-3},$$

where we used $Z/A \approx 0.5$ for carbon-oxygen composition. The corresponding Fermi wavelength,

$$\lambda_F = \frac{2\pi}{k_F} = \frac{2\pi}{(3\pi^2 n_e)^{1/3}} = 1.2 \times 10^{-11} \text{ m},$$

yields quantum overlap parameter $n_e \lambda_F^3 = 0.52$, approaching the critical threshold where quantum statistics dominate completely.

Information Content of Degenerate States

Consider the phase space volume accessible to a degenerate system. For $N$ particles in volume $V$ with momentum states up to Fermi momentum $p_F$, the phase space volume scales as,

$$\Omega \sim \left(\frac{V p_F^3}{h^3}\right)^N.$$

The information content—number of bits required to specify the microstate—follows from Boltzmann’s entropy,

$$N_b = \frac{S}{k_B \ln 2} = \frac{k_B \ln \Omega}{k_B \ln 2} = \frac{\ln \Omega}{\ln 2}.$$

For the white dwarf, electrons occupy momentum states up to $p_e \sim m_e c$ in volume $V_{WD} \sim (5000 \text{ km})^3$. For the neutron star, neutrons occupy states up to $p_n \sim m_n c$ in volume $V_{NS} \sim (10 \text{ km})^3$. The number of particles is approximately,

$$N_p = \frac{M_{Ch}}{m_p} = \frac{1.4 \times 2 \times 10^{30} \text{ kg}}{1.67 \times 10^{-27} \text{ kg}} \approx 1.7 \times 10^{57}.$$

The information difference between states is,

$$\Delta N_b = N_p \log_2\left(\frac{\Omega_{WD}}{\Omega_{NS}}\right) = N_p \log_2\left(\frac{V_{WD}}{V_{NS}}\right).$$

The volume ratio is,

$$\frac{V_{WD}}{V_{NS}} = \left(\frac{5000}{10}\right)^3 = 1.25 \times 10^8.$$

Therefore,

$$\Delta N_b = 1.7 \times 10^{57} \times \log_2(1.25 \times 10^8) = 1.7 \times 10^{57} \times 26.6 \approx 4.5 \times 10^{58} \text{ bits}.$$

This represents the information reorganization during the transition from electron to neutron degeneracy.

The Energy Calculation

Landauer’s principle establishes that erasing or reorganizing information costs energy $E = k_B T \ln 2$ per bit at temperature $T$ ³. For the phase transition occurring at shock temperature $T_s$, the total energy associated with information reorganization is,

$$E = \Delta N_b k_B T_s \ln 2.$$

Observations of supernova shock breakout give characteristic shock temperatures $T_s \sim 10^9$ K during the explosion ⁴. Substitute the calculated information reorganization of $\Delta N_b = 4.5 \times 10^{58}$ bits along with Boltzmann’s constant $k_B = 1.38 \times 10^{-23}$ J/K and $\ln 2 = 0.693$,

$$E = 4.5 \times 10^{58} \times 1.38 \times 10^{-23} \times 10^9 \times 0.693.$$

Evaluate the product step by step. First, combine the constants,

$$k_B T_s \ln 2 = (1.38 \times 10^{-23}) \times 10^9 \times 0.693 = 9.56 \times 10^{-15} \text{ J/bit}.$$

Multiply by the number of bits,

$$E = 4.5 \times 10^{58} \times 9.56 \times 10^{-15} = 4.3 \times 10^{44} \text{ J}.$$

This matches the observed Type Ia supernova energy scale of $\sim 10^{44}$ J ⁵. The framework requires four inputs: Chandrasekhar mass (1.4 $M_\odot$), white dwarf radius (5000 km), neutron star radius (10 km), and observed shock temperature (10^9 K).

Physical Mechanism

The calculation suggests that explosion energy corresponds to thermodynamic cost of reorganizing phase space information during the transition between degenerate states. This interpretation complements rather than replaces nuclear burning models.

Standard models emphasize carbon fusion releasing binding energy,

$$^{12}\text{C} + ^{12}\text{C} \to ^{24}\text{Mg} + \gamma,$$

with binding energy per nucleon increasing from 7.7 MeV to 8.3 MeV, releasing approximately 0.6 MeV per nucleon. For $10^{57}$ nucleons,

$$E_{\text{fusion}} \sim 10^{57} \times 0.6 \text{ MeV} \times 1.6 \times 10^{-13} \text{ J/MeV} \approx 10^{44} \text{ J}.$$

The information framework gives similar order of magnitude through different accounting. Nuclear fusion provides the energy source. Phase space reorganization determines how that energy distributes during the collapse and explosion. Both perspectives describe the same event through different mathematical structures.

The electron capture process $e^- + p \to n + \nu_e$ during collapse releases neutrinos carrying away binding energy. Each capture reduces the electron degeneracy contribution while converting protons to neutrons. The neutrino energy loss is approximately,

$$E_\nu \sim N_c (m_n - m_p - m_e)c^2 \sim 10^{57} \times 0.78 \text{ MeV} \approx 1.2 \times 10^{44} \text{ J},$$

where $N_c$ is the number of electron captures.

This neutrino energy represents information leaving the system—bits that were maintaining electron degeneracy now carry away as unrecoverable neutrino phase space.

Comparison with Detonation Models

Type Ia supernovae exhibit observational diversity requiring multiple explosion mechanisms. The information calculation provides a thermodynamic upper bound on energy available from reorganization, independent of specific ignition and burning patterns.

Deflagration models invoke subsonic burning fronts propagating through the white dwarf. These produce asymmetric explosions with lower peak luminosity, matching subluminous Type Iax events. The information framework suggests incomplete phase space reorganization—only partial transition from electron to neutron degeneracy before disruption.

Detonation models involve supersonic shocks that completely unbind the white dwarf. These match normal Type Ia luminosities. Complete phase space reorganization releases the full $\Delta N_b$ calculated above.

Delayed detonation combines both: deflagration transitions to detonation after sufficient expansion. The observed diversity in peak luminosity correlates with how much mass undergoes complete versus partial reorganization.

The information perspective suggests that variation in Type Ia brightness reflects variation in what fraction of available phase space actually reorganizes during the explosion, rather than just variation in nuclear fuel composition or ignition location.

Implications

The calculation demonstrates that thermodynamic cost of phase space reorganization yields explosion energy matching observations to within an order of magnitude. This correspondence suggests information-theoretic accounting may capture essential aspects of stellar collapse energetics.

The framework makes no physical claims about mechanism—it simply counts accessible states before and after transition, multiplies by temperature, and obtains energy. That this matches observed supernova energies indicates phase space geometry constrains explosion energetics independent of detailed nuclear burning models.

Three numbers determine the result: Chandrasekhar mass ($1.4 M_{\odot}$), white dwarf radius (5000 km), neutron star radius (10 km). The transition temperature uses observed shock temperatures. The mathematics is straightforward thermodynamics and counting.

Whether this represents the “true” mechanism or simply a different mathematical perspective on established physics remains open. The correspondence is suggestive. The framework provides an alternative calculational approach that may offer insights into Type Ia diversity and the role of degeneracy pressure in setting explosion energetics.

Footnotes

1. Pathria, R. K., & Beale, P. D. (2011). Statistical Mechanics (3rd ed.). Academic Press. ↩

2. Chandrasekhar, S. (1931). The Maximum Mass of Ideal White Dwarfs. Astrophysical Journal, 74, 81-82. ↩

3. Landauer, R. (1961). Irreversibility and Heat Generation in the Computing Process. IBM Journal of Research and Development, 5(3), 183-191. ↩

4. Nomoto, K., Thielemann, F.-K., & Yokoi, K. (1984). Accreting White Dwarf Models of Type I Supernovae. Astrophysical Journal, 286, 644-658. ↩

5. Hillebrandt, W., & Niemeyer, J. C. (2000). Type Ia Supernova Explosion Models. Annual Review of Astronomy and Astrophysics, 38(1), 191-230. ↩