Part IV

Stellar-Mass BH Formation

Stellar-mass black holes (5–100 M_☉) form at the endpoint of massive-star evolution. Core collapse of stars above ~25 M_☉initial mass produces a black hole directly; slightly less-massive stars produce neutron stars that can later collapse. LIGO has now observed 100+ binary black-hole coalescences, constraining the mass function and rate.

Core-Collapse Threshold

The Tolman-Oppenheimer-Volkoff limit (~2.2–2.5 M_☉) bounds the maximum stable neutron-star mass. Remnants above TOV collapse to form black holes. Progenitor star mass of ~20–25 M_☉ on the main sequence produces ~8–15 M_☉ BH after mass loss. Pair-instability supernovae eliminate the 60–120 M_☉“upper mass gap.”

Natal Kicks

Asymmetric supernova ejection imparts natal velocity kicks up to ~500 km/s to neutron stars; BH natal kicks are controversial but probably smaller (<100 km/s) for direct collapse. Natal kicks are critical for binary BH formation: too-large kick disrupts the binary; too-small produces too many merging systems.

Simulation: BH Mass Function

Python

script.py28 lines

import numpy as np, matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Mass function of stellar-mass black holes from LIGO O1-O3 + EM
masses = [5, 7, 10, 15, 20, 25, 35, 45, 60, 85]
counts = [3, 6, 12, 18, 15, 10, 8, 6, 4, 2]
# Mass gap 2-5 Msun visible
all_masses = np.concatenate([[0.5,1.0,1.4,1.4,1.4,1.4,2.0], masses])
all_counts = np.concatenate([[5,8,10,12,15,10,3], counts])

fig, ax = plt.subplots(figsize=(11, 6), facecolor='#0a0a1a')
ax.set_facecolor('#111827'); ax.tick_params(colors='#cbd5e1')
for s in ax.spines.values(): s.set_color('#334155')
ax.bar(all_masses, all_counts, color='#fbbf24', edgecolor='#fde68a', width=1.5)
ax.axvspan(2.5, 5.0, alpha=0.15, color='#dc2626', label='Mass gap 2.5-5 M_sun')
ax.axvline(1.4, color='#60a5fa', ls='--', label='Neutron star peak')
ax.axvspan(50, 65, alpha=0.15, color='#a78bfa', label='PISN upper mass gap')
ax.set_xlabel('Compact-object mass (M_sun)', color='#cbd5e1')
ax.set_ylabel('Detection count', color='#cbd5e1')
ax.set_title('Stellar BH mass function from LIGO + EM observations',
             color='#fde68a', fontweight='bold')
ax.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')
ax.grid(True, color='#334155', alpha=0.3)
plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')
print('Stellar BH masses 5-80 Msun; neutron stars peak at 1.4 Msun')
print('Pair-instability SN gap ~50-120 Msun; BH mass gap 2.5-5 Msun both active research topics')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Mass Gaps

The lower mass gap (2.5–5 M_☉) was apparent in X-ray-binary samples but has partially filled in with LIGO observations (e.g., GW200105 at 8.9 M_☉primary + 1.9 M_☉ secondary). The upper mass gap (~50–120 M_☉) from pair-instability is consistent with LIGO data but exceptional events like GW190521 (85 + 66 M_☉) suggest hierarchical mergers may populate it.

Key References

• Woosley, S. E. & Weaver, T. A. (1995). “The evolution and explosion of massive stars. II.” Astrophys. J., 440, 717–744.

• Belczynski, K. et al. (2016). “The first gravitational-wave source from the isolated evolution of two stars.” Nature, 534, 512–515.

• The LIGO/Virgo/KAGRA Collaboration (2023). “GWTC-3: compact binary coalescences observed by LIGO and Virgo during the second part of the third observing run.” Phys. Rev. X, 13, 041039.

• Mandel, I. & Farmer, A. (2022). “Merging stellar-mass binary black holes.” Phys. Rep., 955, 1–24.

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