The gravitational redshift of an atomic line from the surface layer of a neutron star leads to a unique determination of the relation between its mass and radius. The detection of a rotationally-broadened atomic line from a rapidly spinning neutron star offers the additional possibility of measuring directly the stellar radius [115, 32] and, therefore, of determining its mass, as well. The profile of a rotationally-broadened atomic line can be used to study frame-dragging effects in the strong-field regime . Moreover, detecting a gravitationally-redshifted and rotationally-broadened atomic line can lead to a measurement of the oblateness of the spinning star , which is determined by the strong-field coupling of matter with the gravitationally field. Unfortunately, this is one of the very few astrophysical settings discussed in this review in which observations significantly trail behind theoretical investigations.
Despite many optimistic expectations and early claims (see, e.g., ), the observed spectra of almost all weakly-magnetic neutron stars are remarkably featureless. The best studied case is that of the nearby isolated neutron star RX J1856–3754, which was observed for 450 ks with the Chandra X-ray Observatory and showed no evidence for any atomic lines from heavy elements . This is, in fact, not surprising, given that heavy elements drift into the photosphere in timescales of minutes  and it takes only of light elements to blanket a heavy element surface.
There are two types of neutron stars, however, in the atmospheres of which heavy elements may abound: young cooling neutron stars and accreting X-ray bursters . On the one hand, the escaping latent heat of the supernova explosion makes young neutron stars relatively bright sources of X-rays. Their strong magnetic fields can inhibit the accretion of light elements either from the supernova fallback or from the interstellar medium, leaving the surface heavy elements exposed. On the other hand, in the atmospheres of accreting, weakly-magnetic neutron stars, heavy elements are continuously replenished. Moreover, large radiation fluxes pass through their atmospheres during thermonuclear bursts  making them very bright and easily detectable.
The most promising detection to date of gravitationally-redshifted lines from the surface of a neutron star came from an observation with XMM-Newton of the source EXO 0748–676, which showed redshifted atomic lines during thermonuclear flashes . This is a slowly spinning neutron star (47 Hz ) and hence its external spacetime can be accurately described by the Schwarzschild metric. In this case, the measurement of a gravitational redshift of z = 0.35 leads to a unique determination of the relation between the mass and the radius of the neutron star, i.e., . The combination of this result with the spectral properties of thermonuclear bursts during periods of photospheric radius expansion and in the cooling tails also allowed for an independent determination of the mass and radius of the neutron star .
Future observations of bursting or young neutron stars with upcoming X-ray missions such as IXO  and XEUS  have the potential to detect many gravitationally-redshifted atomic lines and, hence, to probe the coupling of matter to the strong gravitational fields found in the interiors of neutron stars.
Astrophysical black holes in active galactic nuclei accreting at moderate rates offer another possibility for probing strong gravitational fields using atomic spectroscopy (for an extensive review on the subject see ; see also  for a review of iron line observations from stellar-mass black holes). The relatively cool accretion disks in these systems act as large mirrors, reflecting the high-energy radiation that is believed to be produced in the disk coronae by magnetic flaring . The spectrum of reflected radiation in hard X-rays is determined by electron scattering, whereas the spectrum in the soft X-rays is characterized by a large number of fluorescent lines caused by bound-bound transitions of the partially ionized material. The combination of the high yield and relatively high abundance of iron atoms in the accreting material make the iron K line, with a rest energy of 6.4 keV for a neutral atom, the most prominent feature of the spectrum.
The profile of the fluorescent iron line as observed at infinity is determined mainly by general and special relativistic effects that influence the propagation of photons from the point of reflection to the observer . Dividing an accretion disk into a series of concentric rings orbiting at the local Keplerian frequency, special relativistic effects produce a rotational splitting of the line emerging from each ring, whereas general relativistic effects generate an overall redshift . The combination of these effects integrated over the entire surface of the accretion disk leads to a characteristic profile for the iron reflection line, which is broad with a shallow and extended red wing (Figure 8).
The magnitude of the relativistic effects depends on the specifics of the spacetime of the black hole, the position and orientation of the observer, the position and properties of the source of X-rays above the accretion disk, and the dependence of fluorescence yield on position of the accretion disk through its dependence on the ionization states of the elements . Given a model for the source of X-rays and the accretion disk, fitting the profile of an iron line from an accreting black hole can lead, in principle, to a direct mapping of its spacetime. Unfortunately, the source of X-ray illumination and the physical properties of the accretion flows themselves are poorly understood.
If we make assumptions regarding these astrophysical complications that are largely model independent, a general property of the spacetime, such as the spin of the black hole, can be measured. The accretion disk is typically modeled as a geometrically thin reflecting surface at the rotational equator of the black hole that extends inwards to the radius of the innermost stable circular orbit. Even though the density of the material inside this radius is significant and might reflect the illuminating X-rays, its ionization state changes rapidly, leading to small changes in the resulting iron line profile [132, 23]. The extent of the iron line towards lower energies is a measure of the innermost radius of the accretion disk. By assumption, this radius is set as the radius of the innermost stable circular orbit, which depends on the spin of the black hole. Fitting theoretical models to observations can, therefore, lead to a measurement of the black-hole spin.
The uncertainties in the position of the illuminating source and in the disk structure are often modeled by a single function for the “emissivity” of the iron line, which measures the flux in the iron line that emerges locally from each patch on the accretion disk. This is typically taken to be axisymmetric and to have a power-law dependence on radius, i.e., . Increasing the emissivity index results in iron-line profiles with more extended red wings, which is degenerate with increasing spin of the black hole (see Figure 8 and ). This uncertainty can introduce significant systematic errors in modeling iron-line profiles from slowly-spinning black holes. For rapidly-spinning black holes, however, masking the effect of the black-hole spin by steepening the emissivity function requires an unphysically high value for the emissivity index .
Since the original observation of broadened iron lines from the supermasive black hole MCG-6-15-30 with ASCA , observations of other active galactic nuclei with ASCA , XMM-Newton , and more recently with Suzaku , as well as of stellar-mass black holes , have revealed many more examples of such redshifted atomic lines. The best studied case remains MCG-6-15-30 (see Figure 9), in which the extended red wing of the line has been discussed as evidence for a rapidly-spinning black hole ( ).
Perhaps the most challenging, although most rewarding to understand, property of iron lines is their time variability. Current observations of iron lines from accreting black holes (e.g., the one shown in Figure 9) are integrated over a time that is equal to many hundred times the dynamical timescale in the accretion-disk region, where the lines are formed. As a result, an observed line profile is not the result of reflection from an accretion disk of a single flaring event, but rather the convolution of many such events that occurred over the duration of the observation. Moreover, the continuum spectrum of the black hole, which is presumably reflected off the accretion disk to produce the fluorescent iron line, changes over longer timescales, implying a correlated variability of the line itself.
Observations with current instruments can only investigate the correlated variability of the iron line with the continuum spectrum (see, however, ). They have shown that the flux in the line remains remarkably constant, even though the continuum flux changes by almost an order of magnitude . General relativistic light bending, which leads to focusing of the photon rays towards the innermost regions of the accretion disk, may be responsible for this puzzling effect .
Future observations with upcoming X-ray missions, such as IXO  and XEUS , will resolve the time evolution of the reflected iron line from a single magnetic flare . Because density inhomogeneities in the turbulent accretion flow move, roughly, in test-particle orbits , the time evolution of the redshift of the iron line from a single flare reflected mainly off a localized density inhomogeneity will allow for a direct mapping of the spacetime around the black hole.
This work is licensed under a Creative Commons License.