Radiation, Spectra, and Variability

Radiation is how an accreting black hole becomes observable, but the detector never receives a direct photograph of the flow. Local dissipation creates a multitemperature thermal continuum; hot electrons redistribute seed photons; that continuum illuminates matter and produces reflection; propagation and light travel imprint frequency-dependent variability. Instrument response and sampling then mix the result. This chapter builds that physical and inferential chain from spectra to cross-spectra. It covers thermal continua, Comptonization, reflection, reverberation, oscillations, and response-aware inference; it excludes a unique mass or spin claim unless the model and systematic alternatives are carried through.

Helpful background. Black-Hole Geometry and Mechanics supplies orbital, redshift, and light-travel scales; Relativistic Matter and Compact Objects supplies fluid and radiative-matter physics; Black-Hole Astrophysics supplies accretion-state and observational context; Waves, Green Functions, and Geometric Optics supplies transfer functions and Fourier propagation; and Computational Relativity supplies ray transfer, forward simulation, and recovery tests.

Enter the radiation chain

A compact schematic for the expected count rate in detector channel $i$ is

$$\mu_i(t)=\int R_i(E,t)A(E,t) \left[F_E^{\rm disk}(t)+F_E^{\rm C}(t)+F_E^{\rm refl}(t)\right]dE+B_i(t).$$

Here each $F_E$ is a photon number flux density, $R_i$ is a redistribution probability, $A$ is effective area, and $B_i$ is background count rate. An energy flux density must first be divided by photon energy $E$ in consistent units; expected counts follow by integrating $\mu_i(t)$ over an exposure. The disk term depends on dissipation, inner boundary, atmosphere, inclination, and relativistic transfer. The Comptonized term depends on electron temperature, optical depth, seed photons, geometry, and pairs or nonthermal electrons. Reflection depends on illumination, density, ionization, abundance, energy shifts, and delays. Spectral fitting and timing analysis are therefore two projections of one forward problem.

The mean spectrum constrains energy redistribution. A power spectrum constrains variance over timescale. A cross-spectrum

$$C_{ij}(f)=\left\langle X_i(f)X_j^*(f)\right\rangle$$

adds energy-dependent phase and coherent amplitude. With the positive analysis exponent and negative synthesis exponent defined on Reverberation Mapping, a delayed comparison band $y(t)=x(t-\tau)$ has positive phase in $C_{yx}=\langle YX^*\rangle$ before wrapping. A viable physical model should predict all products jointly under one declared Fourier convention, response, energy selection, and source state. Ingram gives a unified account in Ingram 2023, §§ 2–3.

Six routes through the chapter

Read the pages in order when building the full inference chain, or enter at the observable you need:

1. Thermal Disk Continua derives the thin-disk flux and color-corrected continuum, then separates a stable apparent radius from the additional assumptions needed to identify an inner edge.
2. Comptonizing Coronae connects temperature, optical depth, the Compton parameter, compactness, seed feedback, and the high-energy rollover without assigning a unique coronal shape.
3. Reflection Spectra and Fluorescent Lines builds ionized reflection and relativistic transfer, emphasizing continuum, density, emissivity, abundance, and calibration degeneracies.
4. Reverberation Mapping defines a signed cross-spectrum and shows how propagation, dilution, phase wrapping, and extended responses complicate the conversion from lag to distance.
5. Quasi-Periodic Oscillations and Timing Features compares orbital, precession, diskoseismic, instability, propagating, and stochastic explanations using widths, harmonics, energy dependence, phase, trials, and state evolution.
6. Spectral–Timing Systematics closes the loop with response-aware likelihoods, mismatch models, simulations, and interval-coverage tests.

The first three pages move from local radiation physics to time-averaged detector spectra. The next two use variability to constrain transfer and dynamics. The last asks whether the complete procedure would recover a known truth. That order prevents a common failure: interpreting a fitted geometric parameter before testing the radiation and detector assumptions that created it.

Observation anchors and what they do not prove

The examples are tied to named archival releases rather than generic “representative” spectra. Their role is to make selections and calibration boundaries concrete, not to claim new fits. The source state, event release, response, energy band, screening, and statistical procedure belong to the result.

Anchor	Use in this chapter	Boundary
LMC X-3 RXTE 1996–2009 thermal sample	Tests stability of selected thin-disk normalizations across many epochs	The published sample is a paper-defined selection; a modern rerun must enumerate products and pin HEASoft, CALDB, background, and responses Steiner et al. 2010, §§ 2–4
Cygnus X-1 NuSTAR 30001011007 plus Suzaku 409049010	Connects coronal curvature and reflection under broad simultaneous coverage	NuSTARDAS 1.4.1/CALDB 20140414 and HEASoft 6.16/2014-07-01 calibration define the historical result Parker et al. 2015, § 2 and Table 1
1H 0707−495 XMM O4 ObsIDs 0511580101–0401	Grounds reverberation and joint spectral–timing systematics in released event lists	SAS 8.0.0 selections are historical; modern clock, calibration, pile-up, response, reference-band, gap, and stationarity choices must be pinned Miller et al. 2010, § 2 and Table 1
GX 339−4 RXTE 92085-01-02-03	Supplies a named 10.59-Hz QPO with measured width and rms	The peak is observed; mapping it to nodal precession at the innermost stable orbit is a model condition Franchini et al. 2017, §§ 2–3, Table 1, and Appendix A

These anchors also illustrate distinct selection effects. A thermal-state cut removes spectra in which disk assumptions are least credible. A broadband spectral observation can still average source evolution. Lag measurements depend on coherence and reference band. A QPO search depends on the noise continuum, frequency range, quality threshold, and trials. Nondetection outside those selections is not a physical null unless the sensitivity to the predicted signal is quantified.

From a fit to a defensible inference

Use three layers. First, describe the measured count-space result: data release, instrument mode, calibration, response, background, selectors, and likelihood. Second, state the radiation or variability model and derive which combination of parameters the observable constrains. Third, test the interpretation against alternative continua, geometries, transfer functions, calibration perturbations, and sampling choices.

Statistical precision belongs to the chosen likelihood and prior. Systematic uncertainty includes both instrument calibration and source-model mismatch. The latter cannot always be represented by adding a fixed percentage: changing disk hardening, coronal geometry, density, absorption, propagation, or illumination can shift the posterior asymmetrically or create another mode. Posterior-predictive residuals reveal disagreement with the fitted data; response-folded injections test bias and interval coverage. Both are needed.

The evidence discussion is current through 2026-08-18. Durable elements include thermal and Comptonized continua, fluorescent reflection, energy-dependent lags, and QPO peaks. Frontier elements include the detailed geometry of the corona, division of lags among propagation and reverberation, physical identity of QPO modes, and the adequacy of joint mismatch models. Reynolds reviews how continuum, reflection, and timing routes inherit different geometric systematics in Reynolds 2021, §§ 3.2–3.4.

Review the chapter

A low-frequency soft-band lag is modeled as a direct component plus a delayed reflection component. Suppose response folding implies that the delayed component supplies a fraction $q=0.25$ of the variable soft-band signal, and the observed low-frequency lag is $30$ s. Estimate the intrinsic delay in the small-phase limit. If an alternative calibration and continuum decomposition gives $q=0.20$, how much does the inferred delay change?

Solution

For direct and delayed components with a small phase, the measured lag is diluted approximately as

$$\tau_{\rm obs}\simeq q\tau_{\rm int}.$$

With $q=0.25$,

$$\tau_{\rm int}=\frac{30\ \mathrm{s}}{0.25}=120\ \mathrm{s}.$$

With $q=0.20$, the same observed lag gives

$$\tau_{\rm int}=\frac{30\ \mathrm{s}}{0.20}=150\ \mathrm{s},$$

a $25$ per cent increase. The timing statistic did not change; the spectral decomposition and response altered its physical conversion. This is why mean spectrum, covariance, response, and lag must be inferred together. Even $150$ s is not automatically a coronal height: an extended transfer function, propagation phase, mass uncertainty, and relativistic path geometry remain to be modeled.

References

• Franchini, A., Motta, S. E., and Lodato, G. (2017). “Constraining Black Hole Spins with Low-Frequency Quasi-Periodic Oscillations in Soft States.” Monthly Notices of the Royal Astronomical Society 467, 145–154. DOI.
• Ingram, A. (2023). “Fourier Spectral-Timing Techniques for the Study of Accreting Black Holes.” Astronomische Nachrichten 344, e20230002. DOI; Open PDF.
• Miller, L., Turner, T. J., Reeves, J. N., and Braito, V. (2010). “X-Ray Reverberation in 1H 0707−495 Revisited.” Monthly Notices of the Royal Astronomical Society 408, 1928–1935. DOI; Open PDF.
• Parker, M. L., Tomsick, J. A., Miller, J. M., et al. (2015). “NuSTAR and Suzaku Observations of the Hard State in Cygnus X-1: Locating the Inner Accretion Disk.” The Astrophysical Journal 808, 9. DOI; Open PDF.
• Reynolds, C. S. (2021). “Observational Constraints on Black Hole Spin.” Annual Review of Astronomy and Astrophysics 59, 117–154. DOI; Open PDF.
• Steiner, J. F., McClintock, J. E., Remillard, R. A., Gou, L., Yamada, S., and Narayan, R. (2010). “The Constant Inner-Disk Radius of LMC X-3: A Basis for Measuring Black Hole Spin.” The Astrophysical Journal Letters 718, L117–L121. DOI; Open PDF.