Part IV

X-ray Binaries

X-ray binaries were the first observational evidence for stellar-mass black holes — Cyg X-1 dynamically confirmed in 1972 — and remain the richest laboratory for accretion physics. This module covers mass measurements, state transitions (hardness-intensity diagrams), and relativistic jet launching.

Dynamical Mass Measurement

In a BH-companion binary, the companion’s Doppler velocity K1 and orbital period P determine the mass function:

\[ f(M_2) \;=\; \frac{P\,K_1^3}{2\pi G} \;=\; \frac{M_2^3 \sin^3 i}{(M_1 + M_2)^2} \]

A measured f(M₂) > 3 M_☉ is essentially diagnostic of a black hole, because the companion must have mass at least f/sin³i ≥ TOV limit. Cygnus X-1 (BH ~21 M_☉), LMC X-3, GRO J1655-40, V404 Cyg, GX 339-4 are the canonical dynamically-confirmed stellar BHs.

Spectral States & HID

Transient XRBs cycle through distinct accretion states during outbursts: low/hard (dominated by a hot corona + jet), high/soft (disk-dominated, no jet), and intermediate states. The hardness-intensity diagram traces a characteristic “q” shape. Fender 2004 mapped the unified model linking state transitions to jet launching.

Simulation: Hardness-Intensity Diagram

Python

script.py36 lines

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

# Hardness-Intensity Diagram for canonical transient XRB (GX 339-4 style)
# outburst: hard -> soft -> hard
t = np.linspace(0, 180, 200)
hardness = np.where(t < 20, 0.8,
             np.where(t < 40, 0.8 - 0.03*(t-20),
               np.where(t < 140, 0.15 + 0.01*np.sin(t/10),
                 np.where(t < 160, 0.15 + 0.035*(t-140), 0.85))))
intensity = np.where(t < 40, 0.3 + 0.03*t,
              np.where(t < 140, 1.5 - 0.005*(t-40)*np.sin(t/20),
                np.where(t < 160, 1.0 - 0.04*(t-140), 0.2)))

fig, ax = plt.subplots(figsize=(10, 7), facecolor='#0a0a1a')
ax.set_facecolor('#111827'); ax.tick_params(colors='#cbd5e1')
for s in ax.spines.values(): s.set_color('#334155')
# HID path
ax.plot(hardness, intensity, color='#fbbf24', lw=2, alpha=0.4)
sc = ax.scatter(hardness, intensity, c=t, cmap='plasma', s=30)
# State labels
ax.annotate('Quiescent', (0.85, 0.35), color='#cbd5e1', fontweight='bold')
ax.annotate('Low/hard', (0.75, 1.0), color='#cbd5e1', fontweight='bold')
ax.annotate('Intermediate', (0.4, 1.4), color='#cbd5e1', fontweight='bold')
ax.annotate('High/soft', (0.15, 1.2), color='#cbd5e1', fontweight='bold')
ax.invert_xaxis()
ax.set_xlabel('Hardness (hard/soft X-ray)', color='#cbd5e1')
ax.set_ylabel('Intensity (L/L_Edd)', color='#cbd5e1')
ax.set_title('Canonical XRB outburst HID: q-shape',
             color='#fde68a', fontweight='bold')
plt.colorbar(sc, ax=ax, label='Time (days)')
plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')
print('XRBs cycle through low/hard -> high/soft -> low/hard states via HID q-track')
print('Corona-disk dominance alternates; relativistic jet launched in hard state')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Spin Measurement

Two techniques constrain BH spin: (a) continuum fitting of the high/soft-state disk spectrum (requires known distance, inclination, mass) identifies the innermost stable circular orbit (ISCO); (b) iron-line reflection spectroscopy fits a relativistically-broadened Fe Kα 6.4 keV line from reflected emission. Observed spins span 0 to >0.98 (GRS 1915+105 near-extremal); distributions reveal formation-channel preferences.

Key References

• Webster, B. L. & Murdin, P. (1972). “Cygnus X-1: a spectroscopic binary with a heavy companion?” Nature, 235, 37–38.

• Fender, R. P., Belloni, T. M. & Gallo, E. (2004). “Towards a unified model for black hole X-ray binary jets.” Mon. Not. R. Astron. Soc., 355, 1105–1118.

• McClintock, J. E. & Remillard, R. A. (2006). “Black hole binaries.” In Compact Stellar X-Ray Sources, Cambridge UP.

• Reynolds, C. S. (2021). “Observational constraints on black hole spin.” Annu. Rev. Astron. Astrophys., 59, 117–154.

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