- 762
- 899
New method I tried once queried on how to calculate the World Engine’s end goal of compressing the Earth’s core to such a dense state that it collapses into a neutron star.
It’s explicitly said that Krypton’s destruction was caused by terraforming attempts which changed the density of the planet’s core such that it eventually transformed to a neutron star, replicated again by the Kryptonians in using the World Engine on Earth. Hence the derived astrophys method.
Any input would be appreciated.
The calculation begins by anchoring to PREM-derived core mass M_c ≈ 1.867 × 10²⁴ kg, converts this to total nucleon number N via the neutron rest mass (with electron and proton masses carried forward for later charge-balance terms), and then imposes the target baryon density ρ_ns = 5 × 10¹⁷ kg m⁻³ (corresponding to n ≈ 0.298 fm⁻³) to obtain the final volume and radius; this geometry immediately yields the neutron number density n = N/V_f that sets the Fermi wave number k_F = (3π²n)^{1/3} and hence the dimensionless relativity parameter x_F = p_F/(m_n c) ≈ 0.434.
Because x_F lies in the mildly relativistic regime, the Fermi momentum p_F is obtained from the exact relativistic dispersion, after which the total kinetic energy of the degenerate neutron gas is evaluated not via the crude non-relativistic 3/5 N E_F but through the exact integral expression E_k = (m_n c²/π²) (m_n c/ℏ)³ V_f I(x_F), where I(x) is the standard Fermi integral ∫₀ˣ √(1+t²) t² dt; the series expansion I(x) = x³/3 − x⁵/5×(3/2) + … truncated at x¹¹/11×(35/2) suffices because higher-order coefficients remain negligible (<10⁻⁶ relative) at this x_F, delivering a 3.5% downward correction relative to the non-relativistic path and confirming internal consistency.
The nuclear contributions are isolated next because the initial state consists of Fe-56 nuclei (≈ 97% in the inner core), whose binding must be fully overcome before the system can reach the uniform neutron fluid: the dissociation term is therefore N × 8.7906 MeV per nucleon (NNDC value), while the subsequent neutronization p + e⁻ → n + ν_e is costed at the exact Q-value 0.7823 MeV per proton (arising from the n–p–e mass difference), multiplied by the proton fraction Z/A = 26/56; both are converted to SI units via the 1.602 × 10⁻¹³ J MeV⁻¹ factor and remain additive because the neutrinos escape freely.
At this point the strong-interaction potential energy is no longer negligible, so the APR (Akmal–Pandharipande–Ravenhall) pure-neutron-matter EOS table (see Table V, pg. 36) is interpolated at n = 0.298 fm⁻³ to give E/A_EOS ≈ 40 MeV above the free-neutron rest mass; subtracting the already-computed free-gas kinetic energy per nucleon isolates the attractive potential contribution E_int/A ≈ −11.4 MeV, which is then scaled by N to yield a macroscopic release term of order −2 × 10³⁹ J. Gravitational self-energy is evaluated from the uniform-sphere formula and shown to be four orders of magnitude smaller, hence dropped.
Because x_F lies in the mildly relativistic regime, the Fermi momentum p_F is obtained from the exact relativistic dispersion, after which the total kinetic energy of the degenerate neutron gas is evaluated not via the crude non-relativistic 3/5 N E_F but through the exact integral expression E_k = (m_n c²/π²) (m_n c/ℏ)³ V_f I(x_F), where I(x) is the standard Fermi integral ∫₀ˣ √(1+t²) t² dt; the series expansion I(x) = x³/3 − x⁵/5×(3/2) + … truncated at x¹¹/11×(35/2) suffices because higher-order coefficients remain negligible (<10⁻⁶ relative) at this x_F, delivering a 3.5% downward correction relative to the non-relativistic path and confirming internal consistency.
The nuclear contributions are isolated next because the initial state consists of Fe-56 nuclei (≈ 97% in the inner core), whose binding must be fully overcome before the system can reach the uniform neutron fluid: the dissociation term is therefore N × 8.7906 MeV per nucleon (NNDC value), while the subsequent neutronization p + e⁻ → n + ν_e is costed at the exact Q-value 0.7823 MeV per proton (arising from the n–p–e mass difference), multiplied by the proton fraction Z/A = 26/56; both are converted to SI units via the 1.602 × 10⁻¹³ J MeV⁻¹ factor and remain additive because the neutrinos escape freely.
At this point the strong-interaction potential energy is no longer negligible, so the APR (Akmal–Pandharipande–Ravenhall) pure-neutron-matter EOS table (see Table V, pg. 36) is interpolated at n = 0.298 fm⁻³ to give E/A_EOS ≈ 40 MeV above the free-neutron rest mass; subtracting the already-computed free-gas kinetic energy per nucleon isolates the attractive potential contribution E_int/A ≈ −11.4 MeV, which is then scaled by N to yield a macroscopic release term of order −2 × 10³⁹ J. Gravitational self-energy is evaluated from the uniform-sphere formula and shown to be four orders of magnitude smaller, hence dropped.
It’s explicitly said that Krypton’s destruction was caused by terraforming attempts which changed the density of the planet’s core such that it eventually transformed to a neutron star, replicated again by the Kryptonians in using the World Engine on Earth. Hence the derived astrophys method.
Any input would be appreciated.
Last edited:
