Reflection Spectra and Fluorescent Lines

X-ray reflection begins when a hard continuum irradiates optically thick matter. Photoelectric absorption, fluorescence, recombination, and Compton scattering produce a rest-frame spectrum containing lines, edges, a soft forest, and a broad Compton hump. Relativistic transport then shifts and blends those features. The inference problem is layered: atomic reflection, ionization and density, illumination, absorption, distant reprocessing, disk geometry, and detector response must be separated before an inner radius can be interpreted. A unique spin claim without these systematics is outside this chapter.

Required background. Waves, Green Functions, and Geometric Optics supplies scattering, frequency transport, and response convolution used to connect local reflection to observed counts.

Helpful background. Black-Hole Geometry and Mechanics supplies Kerr redshift and orbital scales; Relativistic Matter and Compact Objects supplies ionized-matter and opacity concepts; Black-Hole Astrophysics supplies accretion-state context; and Computational Relativity supplies ray-transfer and synthetic-recovery checks.

Atomic reflection before relativistic smearing

The local reflected spectrum depends on the illuminating continuum, composition, density, column, and ionization state. A standard control parameter is

$$\xi=\frac{4\pi F_{\rm X}}{n_e},$$

where $F_{\rm X}$ is incident ionizing energy flux and $n_e$ is electron density. This is the convention of García et al. 2013, § 2, equation (10), using 1–1000 Rydbergs (13.6 eV–13.6 keV); $\xi$ has units $\mathrm{erg\,cm\,s^{-1}}$. Wind photoionization models often use hydrogen density $n_H$ instead, so $\xi_H=(n_e/n_H)\xi_e$ for the same flux. Translate the density as well as the band before comparing them. Low-ionization iron can fluoresce near 6.4 keV, while He-like and H-like species produce features at higher rest energies. Photoelectric opacity shapes edges and soft emission; Compton scattering produces broad curvature at tens of keV.

One value of $\xi$ does not describe a vertically stratified atmosphere, and a real disk can have radial ionization and density gradients. Atomic rates, elemental abundances, incident cutoff, angle dependence, and high-density physics alter both line and continuum. García and collaborators construct a self-consistent ionized-reflection grid and document these dependencies in García et al. 2013, §§ 2.1–2.2 and 3.1–3.6.

Illumination and relativistic transfer

Let the disk-frame specific intensity be $I_E^{\rm ref}(r,\mu_e)$. Transport to a distant observer again uses invariance of $I_\nu/\nu^3$:

$$F_{E_{\rm obs}}=\int_{\rm image}g^3 I^{\rm ref}_{E_{\rm obs}/g}(r,\mu_e)\,d\Omega_{\rm obs}.$$

The distribution of $g$ over the image creates the familiar asymmetric broad line: Doppler shifts split approaching and receding contributions, gravitational redshift extends a low-energy wing, and lensing changes their weights. The result also depends on inclination, inner and outer radii, disk thickness, and angular emissivity.

The illumination profile is often approximated as $\epsilon(r)\propto r^{-q}$ or computed from a compact source at a height. A broken power law can fit many profiles, but added flexibility can trade against inner radius. Returning radiation and disk thickness change which patches illuminate others. Reynolds reviews the physical and inference layers behind relativistic reflection in Reynolds 2021, §§ 3.2.1–3.2.4.

Reflection fraction is definition dependent

“Reflection fraction” may mean a ratio of incident intensities, emitted luminosities, or observed fluxes in a chosen band. These differ when light bending directs photons toward the disk or horizon rather than the observer. A self-consistent geometry should connect its continuum and reflected normalization, whereas a phenomenological fit may allow them to vary independently.

Dauser and collaborators define a geometry-aware reflection fraction for the relxill family and distinguish it from a band-limited observed reflection strength in Dauser et al. 2016, §§ 2–4. The distinction is an adversarial check: an inferred compact source height should predict a compatible illumination and reflection fraction. Failure may indicate an extended corona, anisotropy, disk truncation, time variability, or a mismatched continuum.

Separating broad reflection from impostors

Continuum curvature can imitate the red wing of a line, especially when the cutoff is weakly constrained. Ionized absorption produces narrow or blended structure; a stellar wind can imprint lines that cross the iron band; distant cold matter adds a narrow core and its own Compton hump. Gain errors or pile-up can distort the line profile. A credible analysis therefore fits a broadband continuum and absorption jointly, rather than subtracting an arbitrary power law and fitting residuals alone.

Useful cross-checks include resolving narrow absorption with a grating when possible, varying the continuum kernel, fitting the iron band and the full reflection spectrum, allowing density and ionization gradients, and checking whether time lags share the same energy structure. Posterior-predictive residuals should be examined in count space.

Observation and calibration boundary through 2026-08-18

The named dataset version is the simultaneous hard-state Cygnus X-1 release used by Parker and collaborators: NuSTAR ObsID 30001011007 and Suzaku ObsID 409049010 Parker et al. 2015, § 2 and Table 1. It combines NuSTAR FPMA/FPMB with simultaneous Suzaku XIS1 and broader PIN/GSO coverage. This page performs no new fit.

Inference layer	Declared historical choice	Limitation and required check
Response and calibration	NuSTARDAS 1.4.1 with CALDB 20140414; HEASoft 6.16 with Suzaku 2014-07-01 calibration	Preserve gain, redistribution, effective area, background, pile-up excision, dead time, and cross-normalization
Selection	NuSTAR 3–79 keV; selected XIS1 bands around excluded calibration structure; hard-detector ranges and background files stated in the paper	Results depend on energy cuts, grouping statistic, source stationarity, and the treatment of the companion wind
Physical model	Incident Comptonized continuum, ionized disk reflection, relativistic transfer, narrow distant line, absorption, and wind component	Density, abundances, cutoff, ionization gradient, emissivity, thickness, truncation, and returning radiation are degenerate
Statistics	Count-space likelihood with priors on inclination, inner radius, emissivity or source geometry, density, and cross-calibration	A credible interval is conditional on the continuum and atomic model family; fit statistic alone does not test misspecification

The historical analysis used XSPEC 12.8.2a and compared iron-band and full broadband fits Parker et al. 2015, §§ 3.2–3.3 and Tables 3–4. The software path for a current reproduction must archive the event files, response and background products, calibration release, exact atomic/reflection and relativistic-transfer versions, abundance table, absorption grid, priors, and scripts. It should refit alternative continua, density families, and emissivity geometries, then report posterior-predictive residuals and the shift in inner radius. The published disagreement among fitted iron abundances under different spectral windows is itself a warning that model systematics can dominate a formally precise parameter.

What can be inferred

A broad iron feature plus Compton hump can establish reprocessing in fast-moving, strongly shifted material more robustly than either feature alone. They do not by themselves specify a unique corona or prove that the fitted inner edge equals the ISCO. A stable conclusion must survive absorption, continuum, density, illumination, and calibration alternatives and agree with independent timing or continuum constraints.

Common pitfalls

Treating every broad residual as a line. Continuum and absorption errors produce structured residuals. Reflection must be tested as a broadband atomic spectrum.

Equating a fit radius with spin. Spin follows only after the disk edge, geometry, and Kerr assumptions are declared and their uncertainties propagated.

Exercise

A disk patch has ionizing flux $F_{\rm X}=10^{18}\ {\rm erg\,cm^{-2}\,s^{-1}}$ and $n_e=10^{15}\ {\rm cm^{-3}}$. Calculate $\xi$. If the true density is ten times larger at the same incident flux, what happens, and why can a fixed-density reflection grid bias an inner-radius inference?

Solution

The first value is

$$\xi=\frac{4\pi(10^{18})}{10^{15}} =1.26\times10^4\ {\rm erg\,cm\,s^{-1}}.$$

Increasing the density by ten lowers $\xi$ to $1.26\times10^3\ {\rm erg\,cm\,s^{-1}}$. The line ratios, soft emission, opacity, and continuum curvature therefore change even though the geometry has not. If a fixed-density grid cannot represent that spectrum, the fit may compensate with abundance, emissivity, inclination, or relativistic broadening, shifting the apparent inner radius.

References

• Dauser, T., García, J., Walton, D. J., Eikmann, W., Kallman, T., McClintock, J., and Wilms, J. (2016). “Normalizing a Relativistic Model of X-ray Reflection: Definition of the Reflection Fraction and Its Implementation in relxill.” Astronomy & Astrophysics 590, A76. DOI; Open PDF.
• García, J., Dauser, T., Reynolds, C. S., Kallman, T. R., McClintock, J. E., Wilms, J., and Eikmann, W. (2013). “X-Ray Reflected Spectra from Accretion Disk Models. III. A Complete Grid of Ionized Reflection Calculations.” The Astrophysical Journal 768, 146. 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.