Time-averaged Spectrum Research Articles

Multiwavelength study of high-redshift blazars

ABSTRACT High-redshift blazars are among the most powerful objects in the Universe. The spectral and temporal properties of 33 distant blazars (z > 2.5) detected in the high-energy γ-ray band are investigated by analysing the Fermi-LAT and Swift Ultraviolet and Optical Telescope/X-ray Telescope (UVOT/XRT) data. The considered sources have soft time-averaged γ-ray spectra (Γγ ≥ 2.2) whereas those that have been observed in the X-ray band have hard X-ray spectra (ΓX = 1.01−1.86). The γ-ray flux of high-redshift blazars ranges from 4.84 × 10−10 to 1.50 × 10−7 photon cm−2 s−1 and the luminosity is within (0.10−5.54) × 1048 erg s−1 which during the γ-ray flares increases up to (0.1−1) × 1050 erg s−1. In the X-ray band, only the emission of PKS 0438−43, B2 0743+25, and TXS 0222+185 is found to vary in different Swift XRT observations whereas in the γ-ray band, the emission is variable for fourteen sources: the flux of B3 1343+451 and PKS 0537−286 changes in sub-day scales, that of PKS 0347−211 and PKS 0451−28 in day scales, while the γ-ray variability of the others is in week or month scales. The properties of distant blazar jets are derived by modelling the multiwavelength spectral energy distributions within a one-zone leptonic scenario assuming that the X-ray and γ-ray emissions are produced from inverse Compton scattering of synchrotron and dusty torus photons. From the fitting, the emission region size is found to be ≤0.05 pc and the magnetic field and the Doppler factor are correspondingly within 0.10−1.74 G and 10.0−27.4. By modelling the optical–UV excess, we found that the central black hole masses and accretion disc luminosities are within Ld ≃ (1.09−10.94) × 1046 erg s−1 and (1.69−5.35) × 109 M⊙, respectively.

Spectral Variability of VHS J1256–1257b from 1 to 5 μm

Abstract Multiwavelength time-resolved observations of rotationally modulated variability from brown dwarfs and giant exoplanets are the most effective method for constraining their heterogeneous atmospheric structures. In a companion paper, we reported the discovery of strong near-infrared variability in HST/WFC3/G141 light curves of the very red L-dwarf companion VHS J1256–1257b. In this paper, we present a follow-up 36 hr Spitzer/IRAC Channel 2 light curve together with an in-depth analysis of the Hubble space telescope (HST) and the Spitzer data. The combined data set provides time-resolved light curves of VHS1256b sampling 1.1 to 4.5 μm. The Spitzer light curve is best fit with a single sine wave with a period of 22.04 ± 0.05 hr and a peak-to-peak amplitude of 5.76 ± 0.04%. Combining the period with a previously measured projected rotational velocity ( ), we find that VHS1256b is most consistent with equatorial viewing geometry. The HST/G141+Spitzer spectral energy distribution favors a model with a of 1000 K and low surface gravity with disequilibrium chemistry. The spectral variability of VHS1256b is consistent with predictions from partly cloudy models, suggesting that heterogeneous clouds are the dominant source of the observed modulations. We find evidence at the 3.3σ level for amplitude variations within the 1.67 μm CH4 band, which is the first such detection for a variable L-dwarf. We compare the HST/G141 time-resolved spectra of three red L-dwarfs with high-amplitude near-infrared rotational modulations and find that although their time-averaged spectra are similar, their spectroscopic variabilities exhibit notable differences. This diversity reinforces the advantage of time-resolved spectroscopic observations for understanding the atmospheres of brown dwarfs and directly imaged exoplanets.

Random walks of trains of dissipative solitons

The propagation of light pulses in dual-core nonlinear optical fibers is studied using a model proposed by Sakaguchi and Malomed. The system consists of a supercritical complex Ginzburg-Landau equation coupled to a linear equation. Our analysis includes single standing and walking solitons as well as walking trains of 3, 5, 6, and 12 solitons. For the characterization of the different scenarios, we used ensemble-averaged square displacement of the soliton trajectories and time-averaged power spectrum of the background waves. Power law spectra, indicative of turbulence, were found to be associated with random walks. The number of solitons (or their separations) can trigger anomalous random walks or totally suppress the background waves.

The obscured X-ray binaries V404 Cyg, Cyg X–3, V4641 Sgr, and GRS 1915+105

Aims. V404 Cyg, Cyg X–3, V4641 Sgr, and GRS 1915+105 are among the brightest X-ray binaries and display complex behavior in their multiwavelength emission. Except for Cyg X–3, the other three sources have large accretion disks, and there is evidence of a high orbital inclination. Therefore, any large-scale geometrical change in the accretion disk can cause local obscuration events. On the other hand, Cyg X–3 orbits its Wolf-Rayet companion star inside the heavy stellar wind obscuring the X-ray source. We study here whether the peculiar X-ray spectra observed from all four sources can be explained by local obscuration events. Methods. We used spectra obtained with the Nuclear Spectroscopic Telescope Array and Rossi X-ray Timing Explorer to study the spectral evolution of the four luminous hard X-ray sources. We fit the time-averaged spectra, and also time-resolved spectra in case of V404 Cyg, with two physically motivated models describing either a scenario where all the intrinsic emission is reprocessed in the surrounding matter or where the emitter is surrounded by a thick torus with variable opening angle. Results. We show that the X-ray spectra during specific times are very similar in all four sources, likely arising from the high-density environments where they are embedded. The fitted models suggest that a low-luminosity phase preceding an intense flaring episode in the 2015 outburst of V404 Cyg is heavily obscured, but intrinsically very bright (super-Eddington) accretion state. Similar spectral evolution to that of V404 Cyg is observed from the recent X-ray state of GRS 1915+105 that presented unusually low luminosity. The modeling results point to a geometry change in the (outflowing) obscuring matter in V404 Cyg and GRS 1915+105, which is also linked to the radio (jet) evolution. Within the framework of the models, all sources display obscured X-ray emission, but with different intrinsic luminosities ranging from lower than 1% of the Eddington luminosity up to the Eddington limit. This indicates that different factors cause the obscuration. This work highlights the importance of taking the reprocessing of the X-ray emission in the surrounding medium into account in modeling the X-ray spectra. This may well take place in other sources as well.

An experimental study of trailing edge noise from a heaving airfoil.

In this study, the far-field noise and near-field flow properties from a heaving NACA 0012 airfoil at the Reynolds number of 6.6×104 were investigated experimentally in a 0.4 m2 anechoic wind tunnel. The airfoil had an incident angle of 0° and followed a sinusoidal heaving motion. The Strouhal number, controlled by changing the heaving frequency and amplitude, varied from 0.0024 to 0.008. The acoustic properties were measured by a free-field microphone placed at a distance of 1.2 m away from the tunnel central line, and the flow structures near the trailing edge were acquired using the particle image velocimetry. It was found that the heaving motion could reduce the sound pressure level (SPL) of the primary peak in the time-averaged spectra. The spectrograms obtained by the short-time Fourier transform revealed that the discrete tones were produced when the airfoil passed through the maximum heaving position. During the corresponding period, a sequence of large-scaled vortices convected on the airfoil surface was observed, and then was shed from the trailing edge to the wake region at the same frequency as the primary tone of the induced sound. With the increase of Strouhal number, the sound signals tended to be broadband, and the overall SPL was increased in the far field.

NuSTAR observation of the supergiant fast X-ray transient IGR J11215−5952 during its 2017 outburst

We report on the results of a NuSTAR observation of the supergiant fast X-ray transient pulsar IGR J11215−5952 during the peak of its outburst in June 2017. IGR J11215−5952 is the only SFXT undergoing strictly periodic outbursts (every 165 days). NuSTAR caught several X-ray flares, spanning a dynamic range of 100, and detected X-ray pulsations at 187.0 s, which is consistent with previous measurements. The spectrum from the whole observation is well described by an absorbed power law (with a photon index of 1.4), which is modified, above ∼7 keV, by a cutoff with an e-folding energy of ∼24 keV. A weak emission line is present at 6.4 keV, consistent with Kα emission from cold iron in the supergiant wind. The time-averaged flux is ∼1.5 × 10−10 erg cm−2 s−1 (3−78 keV, corrected for the absorption), translating into an average luminosity of about 9 × 1035 erg s−1 (1–100 keV, assuming a distance of 6.5 kpc). The NuSTAR observation allowed us to perform the most sensitive search for cyclotron resonant scattering features in the hard X-ray spectrum, resulting in no significant detection in any of the different spectral extractions adopted (time-averaged, temporally selected, spin-phase-resolved and intensity-selected spectra). The pulse profile showed an evolution with both the energy (3−12 keV energy range compared with 12−78 keV band) and the X-ray flux: a double-peaked profile was evident at higher fluxes (and in both energy bands), while a single-peaked, sinusoidal profile was present at the lowest intensity state achieved within the NuSTAR observations (in both energy bands). The intensity-selected analysis allowed us to observe an anti-correlation of the pulsed fraction with the X-ray luminosity. The pulse profile evolution can be explained by X-ray photon scattering in the accreting matter above magnetic poles of a neutron star at the quasi-spherical settling accretion stage.

Ion energy distributions at substrate in bipolar HiPIMS: effect of positive pulse delay, length and amplitude

A bipolar HiPIMS discharge with a rectangular positive voltage pulse (with controllable amplitude, delay after the main negative pulse and positive pulse length) was systematically investigated by mass spectroscopy. The time-averaged spectra of ions measured at the substrate position exhibit a prominent high-energy peak. It is shown that the position of the peak can be varied by the positive pulse amplitude, its magnitude scales with the pulse length and its width can be slightly influenced by the length of the delay interval. Measurements of the plasma potential at the mass spectrometer position and time-resolved mass spectroscopy clearly show that the high-energy peak is formed by ions accelerated by the elevated plasma potential during the positive pulse. In addition, fine details of the ion energy distribution functions related to the plasma potential transients at the start of the positive pulse are identified. The presented results are beneficial for the optimisation of the parameters of the positive pulse in experiments implementing the bipolar HiPIMS technology.

Testing The Lamp-Post and Wind Reverberation Models with XMM-Newton Observations of NGC 5506.

The lamp-post geometry is often used to model X-ray data of accreting black holes. Despite its simple assumptions, it has proven to be powerful in inferring fundamental black hole properties such as the spin. Early results of X-ray reverberations showed support for such a simple picture, though wind-reverberation models have also been shown to explain the observed delays. Here, we analyze new and old XMM-Newton observations of the variable Seyfert-1 galaxy NGC 5506 to test these models. The source shows an emission line feature around 6.7 keV that is delayed relative to harder and softer energy bands. The spectral feature can be modeled with either a weakly relativistic disk line or by scattering in distant material. By modeling both the spectral and timing signatures, we find that the reflection fraction needed to explain the lags is larger than observed in the time-averaged spectrum, ruling out both a static lamp-post and simple wind reverberation models.

Venturing beyond the ISCO: detecting X-ray emission from the plunging regions around black holes

ABSTRACT We explore how X-ray reverberation around black holes may reveal the presence of the innermost stable circular orbit (ISCO), predicted by general relativity, and probe the dynamics of the plunging region between the ISCO and the event horizon. Being able to directly detect the presence of the ISCO and probe the dynamics of material plunging through the event horizon represents a unique test of general relativity in the strong field regime. X-ray reverberation off of the accretion disc and material in the plunging region is modelled using general relativistic ray tracing simulations. X-ray reverberation from the plunging region has a minimal effect on the time-averaged X-ray spectrum and the overall lag-energy spectrum, but is manifested in the lag in the highest frequency Fourier components, above $0.01\, c^{3}\, (GM)^{-1}$ (scaled for the mass of the black hole) in the 2–4 keV energy band for a non-spinning black hole or the 1–2 keV energy band for a maximally spinning black hole. The plunging region is distinguished from disc emission not just by the energy shifts characteristic of plunging orbits, but by the rapid increase in ionization of material through the plunging region. Detection requires measurement of time lags to an accuracy of 20 per cent at these frequencies. Improving accuracy to 12 per cent will enable constraints to be placed on the dynamics of material in the plunging region and distinguish plunging orbits from material remaining on stable circular orbits, confirming the existence of the ISCO, a prime discovery space for future X-ray missions.

Modeling the Upper kHz QPOs of 4U 1728-34 with X-Ray Reverberation

Abstract While kilohertz quasi-periodic oscillations (kHz QPOs) have been well studied for decades since their initial discovery, the cause of these signals remains unknown, as no model has been able to accurately predict all of their spectral and timing properties. Separately, X-ray reverberation lags have been detected in active galactic nuclei and stellar-mass black hole binaries, and reverberation may be expected to occur in neutron star systems as well, producing lags of the same amplitude as the lags measured of the kHz QPOs. Furthermore, the detection of a relativistically reflected Fe K line in the time-averaged spectra of many neutron star systems provides an additional motivation for testing reverberation. While it has been shown that the lag-energy properties of the lower kHz QPOs are unlikely to be produced by X-ray reverberation, the upper kHz QPOs have not yet been explored. We therefore model the upper kHz QPO lag-energy spectra using relativistic ray-tracing functions and apply them to archival Rossi X-Ray Timing Explorer data on 4U 1728–34 where upper kHz QPOs have been detected. By modeling the time-averaged spectra in which upper kHz QPOs had been significantly detected, we determine the reflected flux fraction across all energies and produce a model for the lag-energy spectra from X-ray reverberation. We explore the dependence of the modeled lag properties on several different types of reflection models but are unable to successfully reproduce the measured lags of 4U 1728–34. We conclude that reverberation alone does not explain the measured time lags detected in upper kHz QPOs.

NuSTAR/XMM–Newton monitoring of the Seyfert 1 galaxy HE 1143-1810

Aims. We test the two-corona accretion scenario for active galactic nuclei in the case of the “bare” Seyfert 1 galaxy HE 1143-1810. Methods. We perform a detailed study of the broad-band UV–X-ray spectral properties and of the short-term variability of HE 1143-1810. We present results of a joint XMM–Newton and NuSTAR monitoring of the source, consisting of 5 × 20 ks observations, each separated by 2 days, performed in December 2017. Results. The source is variable in flux among the different observations, and a correlation is observed between the UV and X-ray emission. Moderate spectral variability is observed in the soft band. The time-averaged X-ray spectrum exhibits a cut-off at ∼100 keV consistent with thermal Comptonization. We detect an iron Kα line consistent with being constant during the campaign and originating from a mildly ionized medium. The line is accompanied by a moderate, ionized reflection component. A soft excess is clearly present below 2 keV and is well described by thermal Comptonization in a “warm” corona with a temperature of ∼0.5 keV and a Thomson optical depth of ∼17 − 18. For the hot hard X-ray emitting corona, we obtain a temperature of ∼20 keV and an optical depth of ∼4 assuming a spherical geometry. A fit assuming a jet-emitting disc (JED) for the hot corona also provides a nice description of the broad-band spectrum. In this case, the data are consistent with an accretion rate varying between ∼0.7 and ∼0.9 in Eddington units and a transition between the outer standard disc and the inner JED at ∼20 gravitational radii. Conclusions. The broad-band high-energy data agree with an accretion flow model consisting of two phases: an outer standard accretion disc with a warm upper layer, responsible for the optical–UV emission and the soft X-ray excess, and an inner slim JED playing the role of a hard X-ray emitting hot corona.

NuSTAR observation of Ark 564 reveals the variation of coronal temperature with flux

ABSTRACT The hard X-ray spectral index of some active galactic nuclei (AGN) has been observed to steepen with the source flux. This has been interpreted in a Comptonization scenario, where an increase in the soft flux decreases the temperature of the corona, leading to steepening of the photon index. However, the variation of the coronal temperature with flux has been difficult to measure due to the presence of complex reflection component in the hard X-rays and the lack of high-quality data at that energy band. Recently, a 200 ks Nuclear Spectroscopic Telescope Array(NuSTAR) observation of Ark 564 in 3–50 keV band revealed the presence of one of the coolest coronae with temperature kTe ∼ 15 keV in the time-averaged spectrum. Here, we reanalyse the data and examined the spectra in four flux levels. Our analysis shows that the coronal temperature decreased from ∼17 to ∼14 keV as the flux increased. The high energy photon index Γ ∼ 2.3 varied by less than 0.1, implying that the optical depth of the corona increased by about 10 per cent as the flux increased. This first reporting of coronal temperature variation with flux shows that further long observation by NuSTAR of this and other sources would shed light on the geometry and dynamics of the inner regions of the accretion flow.

Estimation of Time-Varying Spectral Peaks and Decomposition of EEG Spectrograms

Detection of spectral peaks and estimation of their properties, including frequency and amplitude, are fundamental to many applications of signal processing. Electroencephalography (EEG) of sleep, in particular, displays characteristic oscillations that change continuously throughout the night. Capturing these dynamics is essential to understanding the sleep process and characterizing the heterogeneity observed across individuals. Most sleep EEG analyses rely on either time-averaged spectra or bandpassed amplitude/power. Unfortunately, these approaches obscure the time-variability of peak properties, require specification of a priori criteria, and cannot distinguish power from nearby oscillations. More sophisticated approaches, using various spectral models, have been proposed to better estimate oscillatory properties, but these too have limitations. We present an improved approach to spectrogram decomposition, tracking time-varying parameterized peak functions and dynamically estimating their parameters using a modified form of the iterated extended Kalman filter (IEKF) that incorporates discrete On/Off-switching of peak combinations and a sampling step to draw the initial reference trajectory. We evaluate this approach on two types of simulated examples—one nearly within the model class and one outside. We find excellent performance, in terms of spectral fits and accuracy of estimated states, for both simulation types. We then apply the approach to real EEG data of sleep onset, obtaining quality spectral estimates with estimated peak combinations closely matching the expert-scored sleep stages. This approach offers not only the ability to estimate time-varying parameters of spectral peaks but, moving forward, the potential to estimate the governing dynamics and analyze their variability across nights, subjects, and clinical groups.

Floquet formalism for multiphoton dynamics in superconducting quantum circuits

SynopsisHigh-order nonlinear multiphoton dynamics of closed and open periodically driven quantum systems are studied within the semiclassical Floquet formalism. Multiphoton up/down frequency conversion in a super-conducting circuit is realized and faithfully reproduced with calculations. The absorption spectra are suppressed at sharp avoided crossings of the Floquet states. We include the effect of coupling to cavity by time evolving the master equation. The characteristics of the time-averaged absorption spectra are investigated by probing details near and around the exceptional points where suppression occurs.

Search for spectral features in extragalactic background light with gamma-ray telescopes

Context. Cumulative optical and infrared emission from galaxies accumulated over cosmological time scales, the extragalactic background light (EBL), could be probed by complementary techniques of direct observations and source counting in the visible and infrared as well as via its imprint on the signal of distant active galactic nuclei in γ-rays. Aims. We compare the visible and infrared measurements with the γ-ray constraints and study if the discrepancies of the measurements with different methods could be due to the presence of features in the EBL spectrum that are localised in the micron wavelength range. Methods. We combined data on time-averaged spectra of selected blazars that were obtained by Fermi and ground-based gamma-ray telescopes. We also modelled the effect of absorption on EBL while allowing for the existence of a previously unaccounted spectral feature. Results. We show that a previously reported “excess” in EBL flux in the ∼1 micron wavelength range is consistent with γ-ray measurements, that is, if the excess has the form of a narrow feature of the width δλ < λ and an overall flux of up to 15 nW m−2 sr−1 above the “minimal” EBL, which is estimated from the visible and infrared source counts. Such ’bump-like’ spectral features could originate, for example, from decaying dark-matter particles, or either axions or peculiar astrophysical processes in the course of star-formation history. We discuss the possibilities for the search of spectral features in the EBL with the Cherenkov Telescope Array.

The CARMENES search for exoplanets around M dwarfs

The He I infrared (IR) triplet at 10 830 Å is an important activity indicator for the Sun and in solar-type stars, however, it has rarely been studied in relation to M dwarfs to date. In this study, we use the time-averaged spectra of 319 single stars with spectral types ranging from M0.0 V to M9.0 V obtained with the CARMENES high resolution optical and near-infrared spectrograph at Calar Alto to study the properties of the He I IR triplet lines. In quiescence, we find the triplet in absorption with a decrease of the measured pseudo equivalent width (pEW) towards later sub-types. For stars later than M5.0 V, the He I triplet becomes undetectable in our study. This dependence on effective temperature may be related to a change in chromospheric conditions along the M dwarf sequence. When an emission in the triplet is observed, we attribute it to flaring. The absence of emission during quiescence is consistent with line formation by photo-ionisation and recombination, while flare emission may be caused by collisions within dense material. The He I triplet tends to increase in depth according to increasing activity levels, ultimately becoming filled in; however, we do not find a correlation between the pEW(He IR) and X-ray properties. This behaviour may be attributed to the absence of very inactive stars (LX∕Lbol < −5.5) in our sample or to the complex behaviour with regard to increasing depth and filling in.

A NICER Look at Strong X-Ray Obscuration in the Seyfert-2 Galaxy NGC 4388

Abstract We present an analysis of the time-averaged spectrum of the Seyfert-2 active galaxy NGC 4388, obtained by NICER. The intrinsic strength of the reflection spectrum in NGC 4388, the large collecting area and favorable passband of NICER, and a net exposure of 105.6 ks yielded an exceptionally sensitive spectrum. Using two independent families of models, the intrinsic spectrum from the central engine is found to be highly obscured but not Compton-thick. Enforcing physical self-consistency within each model, the independent treatments give formally consistent results: or . Past measurements made with Suzaku and XMM-Newton are in broad agreement with these column density values. A more recent measurement with NuSTAR (in late 2013) recorded a column density about twice as large; the robustness of this variability is reinforced by the use of consistent models and procedures. The neutral Fe Kα line in the NICER spectrum is nominally resolved and consistent with an origin in the optical broad-line region). The data also require ionized absorption in the Fe K band, similar to the “warm absorbers” detected in Seyfert-1 active galactic nuclei. The low-energy spectrum is consistent with a set of ionized plasma components. We discuss these findings and note that the geometric inferences that derive from this analysis can be tested with XRISM and Athena.

X-ray and optical observations of the black hole candidate MAXI J1828−249

AbstractWe report results from X-ray and optical observations of the Galactic black hole candidate MAXI J1828−249 performed with Suzaku and the Kanata telescope around the X-ray flux peak in the 2013 outburst. The time-averaged X-ray spectrum covering 0.6–168 keV was approximately characterized by a strong multi-color disk blackbody component with an inner disk temperature of ∼0.6 keV, and a power-law tail with a photon index of ∼2.0. We detected an additional structure at 5–10 keV, which can be modeled neither with X-ray reflection on the disk nor relativistic broadening of the disk emission. Instead, it was successfully reproduced with a Comptonization of disk photons by thermal electrons with a relatively low temperature (≲10 keV). We infer that the source was in the intermediate state, considering its long-term trend in the hardness intensity diagram, the strength of the spectral power-law tail, and its variability properties. The low-temperature Comptonization component could be produced in a boundary region between the truncated standard disk and the hot inner flow, or a Comptonizing region that somehow developed above the disk surface. The multi-wavelength spectral energy distribution suggests that the optical and ultraviolet fluxes were dominated by irradiated outer disk emission.

A public relativistic transfer function model for X-ray reverberation mapping of accreting black holes

ABSTRACTWe present the publicly available model reltrans that calculates the light-crossing delays and energy shifts experienced by X-ray photons originally emitted close to the black hole when they reflect from the accretion disc and are scattered into our line of sight, accounting for all general relativistic effects. Our model is fast and flexible enough to be simultaneously fit to the observed energy-dependent cross-spectrum for a large range of Fourier frequencies, as well as to the time-averaged spectrum. This not only enables better geometric constraints than only modelling the relativistically broadened reflection features in the time-averaged spectrum, but additionally enables constraints on the mass of supermassive black holes in active galactic nuclei and stellar-mass black holes in X-ray binaries. We include a self-consistently calculated radial profile of the disc ionization parameter and properly account for the effect that the telescope response has on the predicted time lags. We find that a number of previous spectral analyses have measured artificially low source heights due to not accounting for the former effect and that timing analyses have been affected by the latter. In particular, the magnitude of the soft lags in active galactic nuclei may have been underestimated, and the magnitude of lags attributed to thermal reverberation in X-ray binaries may have been overestimated. We fit reltrans to the lag-energy spectrum of the Seyfert galaxy Mrk 335, resulting in a best-fitting black hole mass that is smaller than previous optical reverberation measurements (∼7 million compared with ∼14–26 million M⊙).

An X-ray reverberation mass measurement of Cygnus X-1

ABSTRACT We present the first X-ray reverberation mass measurement of a stellar-mass black hole. Accreting stellar-mass and supermassive black holes display characteristic spectral features resulting from reprocessing of hard X-rays by the accretion disc, such as an Fe Kα line and a Compton hump. This emission probes the innermost region of the accretion disc through general relativistic distortions to the line profile. However, these spectral distortions are insensitive to black hole mass, since they depend on disc geometry in units of gravitational radii. Measuring the reverberation lag resulting from the difference in path-length between direct and reflected emission calibrates the absolute length of the gravitational radius. We use a relativistic model able to reproduce the behaviour of the lags as a function of energy for a wide range of variability time-scales, addressing both the reverberation lags on short time-scales and the intrinsic hard lags on longer time-scales. We jointly fit the time-averaged spectrum and the real and imaginary parts of the cross-spectrum as a function of energy for a range of Fourier frequencies to Rossi X-ray Timing Exporer data from the X-ray binary Cygnus X-1. We also show that introducing a self-consistently calculated radial ionisation profile in the disc improves the fit, but requires us to impose an upper limit on ionization profile peak to allow a plausible value of the accretion disc density. This limit leads to a mass value more consistent with the existing dynamical measurement.