Time-averaged Spectrum Research Articles

Synchrotron Polarization of a Hybrid Distribution of Relativistic Thermal and Nonthermal Electrons in Gamma-Ray Burst Prompt Emission

Synchrotron polarization of relativistic nonthermal electrons in gamma-ray bursts (GRBs) has been widely studied. However, recent numerical simulations of relativistic shocks and magnetic reconnection have found that a more realistic electron distribution consists of a power-law component plus a thermal component, which requires observational validation. In this paper, we investigate synchrotron polarization using a hybrid energy distribution of relativistic thermal and nonthermal electrons within a globally toroidal magnetic field in GRB prompt emission. Our results show that compared to the case of solely nonthermal electrons, the synchrotron polarization degrees (PDs) in these hybrid electrons can vary widely, depending on different parameters, and that the PD decreases progressively with frequency in the gamma-ray, X-ray, and optical bands. The time-averaged PD spectrum displays a significant bump in the gamma-ray and X-ray bands, with the PDs higher than ∼60% if the thermal peak energy of the electrons is much smaller than the conjunctive energy of the electrons between the thermal and nonthermal distributions. The high-synchrotron PDs (≳60%) in the gamma-ray and X-ray bands, which generally cannot be produced by solely nonthermal electrons with typical power-law slopes, can be achieved by hybrid electrons and primarily originates from the exponential decay part of the thermal component. Moreover, this model can roughly explain the PDs and spectral properties of some GRBs, where GRB 110301A with a high PD ( 70−22+22% ) may be potential evidence for the existence of relativistic thermal electrons.

Comprehensive investigations on spectral and temporal features of GX 5−1 using AstroSat observations

ABSTRACT Comprehensive spectrotemporal analyses of the Z-type neutron star low-mass X-ray binary GX 5−1 were performed using 10 broad-band observations from AstroSat/Soft X-ray Telescope and Large Area X-ray Proportional Counter (LAXPC) instruments. The LAXPC-20 hardness–intensity diagram showed horizontal and normal branches (HBs and NBs) of the Z track which exhibited secular motion. The time-averaged spectra in the energy range 0.7–25.0 keV could be fitted with the model combination – $\tt {constant}\, \times \, \tt {tbabs}\, \times \, \tt {edge}\, \times \, \tt {edge}\, \times \, \tt {thcomp}\, \times \, \tt {diskbb}$. This yielded $\Gamma \, \sim$ 2, $kT_{\mathrm{ e}}\, \sim$ 3.3 keV, and $F_{\mathrm{ disc}}$/$F_{\mathrm{ total}}\, \sim$ 0.8 indicating the soft/intermediate spectral state of the source during the observations. Flux-resolved spectral analysis revealed a positive correlation between $kT_{\mathrm{ in}}$ and $F_{\mathrm{ bol}}$. However, a negative correlation was observed between them in one of the NBs. Time-averaged temporal analysis revealed multiple HB oscillations (HBOs) and NB oscillations (NBOs), and peaked noise components in the $\sim$5–50 Hz range. Furthermore, flux-resolved temporal analysis showed that the frequency of the HBOs correlates positively whereas the strength of HBOs correlates negatively with $F_{\mathrm{ bol}}$, indicating their probable origin from the accretion disc. In contrast, the frequency and strength of NBOs remain fairly constant with $F_{\mathrm{ bol}}$, suggesting that they originate from a different region in the system. Using the relativistic precession model along with highest frequency of the HBO, the upper limits of the magnetic dipole moment ($\mu$) and field strength (B) at the poles of the neutron star in the system were found to be 25.60$\times \, 10^{25}$ G cm3 and 3.64$\times \, 10^{8}$ G, respectively, for $k_{\mathrm{ A}}$ = 1.

A Multiwavelength Study to Decipher the 2017 Flare of the Blazar OJ 287

In 2017 February, the blazar OJ 287 underwent a period of intense multiwavelength activity. It reached a new historic peak in the soft X-ray (0.3–10 keV) band, as measured by the Swift X-ray Telescope. This event coincides with a very-high-energy (VHE) γ-ray outburst that led VERITAS to detect emission above 100 GeV, with a detection significance of 10σ (from 2016 December 9 to 2017 March 31). The time-averaged VHE γ-ray spectrum was consistent with a soft power law (Γ = −3.81 ± 0.26) and an integral flux corresponding to ∼2.4% that of the Crab Nebula above the same energy. Contemporaneous data from multiple instruments across the electromagnetic spectrum reveal a complex flaring behavior, primarily in the soft X-ray and VHE bands. To investigate the possible origin of such an event, our study focuses on three distinct activity states: before, during, and after the 2017 February peak. The spectral energy distributions during these periods suggest the presence of at least two nonthermal emission zones, with the more compact one responsible for the observed flare. Broadband modeling results and observations of a new radio knot in the jet of OJ 287 in 2017 are consistent with a flare originating from a strong recollimation shock outside the radio core.

A Possible X-ray Quasi-periodic Oscillation in the Narrow-line Seyfert 1 Galaxy Mrk 142

A possible quasi-periodic oscillation (QPO) at frequency 7.045 × 10−5 Hz is found in the narrow-line Seyfert 1 galaxy Mrk 142 in the data of XMM-Newton collected on 2020 April 11. We find that the QPO signal is statistically significantly larger than the 5σ level and highly coherent with quality factor Q > 5 at the 0.3–10 keV band by using the method of the Lomb–Scargle Periodogram, the Weighted Wavelet Z-transform and the REDFIT. We analyze the data in 0.3–0.6 keV, 0.6–1 keV, 1–3 keV and 3–10 keV energy bands, and find obvious QPO signals at 0.3–0.6 keV and 1–3 keV bands. We then analyze the time-average spectra and time variability at the QPO frequency of 7.045 × 10−5 Hz, and use a model to fit them. We find that the QPO signal mainly comes from the X-ray hot corona.

The dispersion of the Ep, i–Liso correlation of long gamma-ray bursts is partially due to assembling different sources

Context. Long gamma-ray burst (GRB) prompt emission shows a correlation between the intrinsic peak energy, Ep, i, of the time-average νFν spectrum and the isotropic-equivalent peak gamma-ray luminosity, Lp, iso, as well as the total released energy, Eiso. The same correlation is found within individual bursts, when time-resolved Ep, i and Liso are considered. These correlations are characterised by an intrinsic dispersion, whose origin is still unknown. Discovering the origin of the correlation and of its dispersion would shed light on the still poorly understood prompt emission and would propel GRBs to powerful standard candles. Aims. We studied the dispersion of both isotropic-equivalent and collimation-corrected time-resolved correlations. We also investigated whether the intrinsic dispersion computed within individual GRBs is different from that obtained including different bursts into a unique sample. We then searched for correlations between key features, such as the Lorentz factor and jet opening angle, and intrinsic dispersion, when the latter is treated as one of the characterising properties. Methods. We performed a time-resolved spectral analysis of 20 long type-II or collapsar-candidate GRBs detected by the Fermi Gamma-ray Burst Monitor with a known redshift and estimates of the jet opening angle and/or the Lorentz factor. Time intervals were determined using Bayesian blocks. Then we carried out a statistical analysis starting from distributions of simulated values of the intrinsic dispersion of each burst in the sample. Results. The collimation-corrected correlation appears to be no less dispersed than the isotropic-equivalent one. Also, individual GRBs are significantly less dispersed than the whole sample. We excluded (at a 4.2σ confidence level) the difference in samples’ sizes as the possible reason, thus confirming that individual GRBs are intrinsically less dispersed than the whole sample. No correlation was found between intrinsic dispersion and other key properties for the few GRBs with available information. Conclusions. The contribution to the dispersion by the jet opening angle is not relevant. Moreover, our results prove that the intrinsic dispersion that affects the Ep, i − Liso correlation is partially, though not entirely, due to assembling different GRBs. We therefore conclude that the presence of different GRBs significantly contributes to the observed dispersion of both time-average Ep, i − Lp, iso and Ep, i − Eiso correlations.

Potential of K-means clustering for preliminary labeling of acoustic data samples

Labeling data for use in supervised learning algorithms can be a long and arduous process, especially for large datasets. This work explores if an unsupervised k-means clustering algorithm can act as a tool to efficiently obtain preliminary labels for large acoustic datasets. This approach is tested on vector sensor data from Monterey Accelerated Research System (MARS). To determine the optimal number of clusters in the k-means algorithm, the silhouette method is used. The clustering of different types of input data will be compared. Specifically, time average spectra from the pressure sensor, the acoustic particle velocity, and intensity will be considered. In addition, the results of clustering data from different times of day and months over a two-year period will be compared. Results of the clustering will be presented along with an assessment of the potential for using a k-means clustering as a preliminary labeling tool for large acoustic sets. [Undergraduate research funded by the College of Physical and Mathematical Sciences as Brigham Young University.]

Deep neural networks for estimation of gamma-ray burst redshifts

Abstract While the available set of Gamma-ray Burst (GRB) data with known redshift is currently limited, a much larger set of GRB data without redshift is available from different instruments. This data includes well-measured prompt gamma-ray flux and spectral information. We estimate the redshift of a selection of these GRBs detected by Fermi-GBM and Konus-Wind using Machine Learning techniques that are based on spectral parameters. We find that Deep Neural Networks with Random Forest models employing non-linear relations among input parameters can reasonably reproduce the pseudo-redshift distribution of GRBs, mimicking the distribution of GRBs with spectroscopic redshift. Furthermore, we find that the pseudo-redshift samples of GRBs satisfy (i) Amati relation between the peak photon energy of the time-averaged energy spectrum in the cosmological rest frame of the GRB Ei, p and the isotropic-equivalent radiated energy Eiso during the prompt phase; and (ii) Yonetoku relation between Ei, p and isotropic-equivalent luminosity Liso, both measured during the peak flux interval.

Effects of liquid properties on the development of nanosecond-pulsed plasma inside of liquid: comparison of water and liquid nitrogen

In this manuscript, we report on observations of the development of nanosecond-pulsed plasma in liquids and examine liquids with two drastically different properties: water and liquid nitrogen. Here, we compare the discharge appearance using high-speed imaging, examine bubble formation using shadow imaging, and measure the time-averaged optical emission spectra of these plasmas. Because the liquid nitrogen plasma is ignited in a liquid that is at boiling temperature, we also study the water discharge at various temperatures, up to boiling. We demonstrate that the discharge development appears not to be affected by this type of liquid. Optical emission, however, is strikingly different: in water, we observe continuum emission in the UV region only and no black-body continuum or atomic lines, whereas the liquid nitrogen spectrum is populated by molecular and longer wavelength broadband emissions.

Time-averaged atomic volume spectrum: locating and identifying vacancies.

Vacancies, as well as their derivatives, usually play a crucial role in many essential properties of materials. However, they always behave erratically, especially under universal thermal vibration, and are therefore difficult to accurately locate. Until now, the lack of an accurate and flexible method for locating and identifying vacancies has hampered the development of relative fields. In this paper, we present a new method to solve this problem. The strategy is to target the atomic cage enwrapping vacancies instead of the vacancies themselves. The core of the method is a time-averaged atomic volume spectrum (TAVS). The key to this method is to identify atoms using time-averaged rather than transient atomic volume, thereby simultaneously denoising intrinsic thermal vibration and avoiding vacancy migration jump. Using this method, we have succeeded for the first time in obtaining the panoramic maps of spontaneously trapped defects in quenched and annealed face-centered cubic aluminum and even the instantaneous images of a steady trapping process. All characteristics of each trapped vacancy, including location, dimension, volume and morphology, as well as aggregate statistical data such as vacancy amount and concentration, can be completely and accurately obtained. Furthermore, these first maps of defects (vacancies) revealed some surprising and interesting phenomena for future exploration. In conclusion, this method provides not only a means of locating and catching vacancies, but also a strategy for identifying and characterizing vacancies. On the basis of its successful application in FCC Al, the TAVS method can be easily extended to other systems as well.

Small-scale dynamo in cool stars

Context. Some of the quiet solar magnetic flux could be attributed to a small-scale dynamo (SSD) operating in the convection zone. An SSD operating in cool main-sequence stars is expected to affect the atmospheric structure, in particular, the convection, and should have observational signatures. Aims. We investigate the distribution of SSD magnetic fields and their effect on bolometric intensity characteristics, vertical velocity, and spatial distribution of the kinetic energy (KE) and magnetic energy (ME) in the lower photosphere of different spectral types. Methods. We analyzed the SSD and purely hydrodynamic simulations of the near surface layers of F3V, G2V, K0V, and M0V stars. We compared the time-averaged distributions and power spectra in SSD setups relative to the hydrodynamic setup. The properties of the individual magnetic fields are also considered. Results. The probability density functions with a field strength at the τ = 1 surface are quite similar for all cases. The M0V star displays the strongest fields, but relative to the gas pressure, the fields on the F3V star reach the highest values. In all stars, the horizontal field is stronger than the vertical field in the middle photosphere, and this excess becomes increasingly prominent toward later spectral types. These fields result in a decrease in the upflow velocities and a slight decrease in granule size, and also lead to formation of bright points in intergranular lanes. The spatial distribution of the KE and ME is also similar for all cases, implying that important scales are proportional to the pressure scale height. Conclusions. The SSD fields have rather similar effects on the photospheres of cool main-sequence stars: a significant reduction in convective velocities, as well as a slight reduction in granule size and a concentration of the field to kilogauss levels in intergranular lanes that is associated with the formation of bright points. The distribution of the field strengths and energies is also rather similar.

Vibrational dynamics and spectroscopy of water at porous g-C3N4 and C2N surfaces.

Porous graphitic materials containing nitrogen are promising catalysts for photo(electro)chemical reactions, notably water splitting, but can also serve as "molecular sieves". Nitrogen increases the hydrophilicity of the graphite parent material, among other effects. A deeper understanding of how water interacts with C- and N-containing layered materials, if and which differences exist between materials with different N content and pore size, and what the role of water dynamics is - a prerequsite for catalysis and sieving - is largely absent, however. Vibrational spectroscopy can answer some of these questions. In this work, the vibrational dynamics and spectroscopy of deuterated water molecules (D2O) mimicking dense water layers at room temperature on the surfaces of two different C/N-based materials with different N content and pore size, namely graphitic C3N4 (g-C3N4) and C2N, are studied using ab initio molecular dynamics (AIMD). In particular, time-dependent vibrational sum frequency generation (TD-vSFG) spectra of the OD modes and also time-averaged vSFG spectra and OD frequency distributions are computed. This allows us to distinguish "free" (dangling) OD bonds from OD bonds that are bound in a H-bonded water network or at the surface - with subtle differences between the two surfaces and also to a pure water/air interface. It is found that the temporal decay of OD modes is very similar on both surfaces with a correlation time near 4 ps. In contrast, TD-vSFG spectra reveal that the interconversion time from "bonded" to "free" OD bonds is about 8 ps for water on C2N and thus twice as long as for g-C3N4, demonstrating a propensity of the former material to stabilize bonded OD bonds.

High-energy Neutrinos from Gamma-Ray-faint Accretion-powered Hypernebulae

Hypernebulae are inflated by accretion-powered winds accompanying hyper-Eddington mass transfer from an evolved post-main-sequence star onto a black hole or neutron star companion. The ions accelerated at the termination shock—where the collimated fast disk winds and/or jet collide with the slower, wide-angled wind-fed shell—can generate high-energy neutrinos via hadronic proton–proton reactions, and photohadronic (p γ) interactions with the disk thermal and Comptonized nonthermal background photons. It has been suggested that some fast radio bursts (FRBs) may be powered by such short-lived jetted hyper-accreting engines. Although neutrino emission associated with the millisecond duration bursts themselves is challenging to detect, the persistent radio counterparts of some FRB sources—if associated with hypernebulae—could contribute to the high-energy neutrino diffuse background flux. If the hypernebula birth rate follows that of stellar-merger transients and common envelope events, we find that their volume-integrated neutrino emission—depending on the population-averaged mass-transfer rates—could explain up to ∼25% of the high-energy diffuse neutrino flux observed by the IceCube Observatory and the Baikal Gigaton Volume Detector Telescope. The time-averaged neutrino spectrum from hypernebula—depending on the population parameters—can also reproduce the observed diffuse neutrino spectrum. The neutrino emission could in some cases furthermore extend to >100 PeV, detectable by future ultra-high-energy neutrino observatories. The large optical depth through the nebula to Breit–Wheeler (γ γ) interaction attenuates the escape of GeV–PeV gamma rays coproduced with the neutrinos, rendering these gamma-ray-faint neutrino sources, consistent with the Fermi observations of the isotropic gamma-ray background.

Temporal and Spectral Properties of the Persistent Radio Source Associated with FRB 20190520B with the VLA

Among more than 800 known fast radio bursts (FRBs), only two, namely FRB 20121102A and FRB 20190520B, are confirmed to be associated with persistent radio sources (PRSs). Here, we report evidence of apparent temporal variability in the PRS associated with the bursting FRB 20190520B based on Karl G. Jansky Very Large Array observations taken in 2020 and 2021. Based on the analysis of the epoch-to-epoch variability of the PRS at the L, S, C, and X bands (1–12 GHz), we detected not only overall marginal variability, but also a likely radio flux decrease (σ ∼ 3.2) between the observations taken in 2020 and 2021 at 3 GHz. Assuming no spectral variation in the PRS during these observations, we found evidence for an overall broadband radio flux decrease by about 20% between the 2020 and the 2021 observations, suggesting that the PRS probably evolves on a yearly timescale. If we attribute the marginal variability at 3 GHz as intrinsic or due to scintillation, the size of the potential variable component of the PRS is constrained to be subparsec. On the other hand, the size of the PRS can be also constrained to ≳0.22 pc from the time-averaged radio spectrum and the integrated radio luminosity in the 1–12 GHz band, based on equipartition and self-absorption arguments. We discuss the potential origins of the PRS and suggest that an accreting compact object origin might be able to explain the PRS’s temporal and spectral properties. Confirmation of the variability or flux decline of the PRS will be critical for our understanding of the PRS and its relation to the bursting source.

Variability as a Predictor for the Hard-to-soft State Transition in GX 339−4

During the outbursts of black hole X-ray binaries (BHXRBs), their accretion flows transition through several states. The source luminosity rises in the hard state, dominated by nonthermal emission, before transitioning to the blackbody-dominated soft state. As the luminosity decreases, the source transitions back into the hard state and fades to quiescence. This picture does not always hold, as ≈40% of the outbursts never leave the hard state. Identifying the physics that govern state transitions remains one of the outstanding open questions in black hole astrophysics. In this paper we present an analysis of archival RXTE data of multiple outbursts of GX 339−4. We compare the properties of the X-ray variability and time-averaged energy spectrum and demonstrate that the variability (quantified by the power spectral hue) systematically evolves ≈10–40 days ahead of the canonical state transition (quantified by a change in spectral hardness); no such evolution is found in hard-state-only outbursts. This indicates that the X-ray variability can be used to predict if and when the hard-to-soft state transition will occur. Finally, we find a similar behavior in 10 outbursts of four additional BHXRBs with more sparse observational coverage. Based on these findings, we suggest that state transitions in BHXRBs might be driven by a change in the turbulence in the outer regions of the disk, leading to a dramatic change in variability. This change is only seen in the spectrum days to weeks later, as the fluctuations propagate inwards toward the corona.

Type-A quasi-periodic oscillation in the black hole transient MAXI J1348−630

ABSTRACT We present a detailed analysis of the spectral and timing characteristics of a 7-Hz type-A quasi-periodic oscillation (QPO) detected in NICER observations of the black hole X-ray binary MAXI J1348−630 during its high-soft state. The QPO is broad and weak, with an integrated fractional rms amplitude of 0.9 per cent in the 0.5–10 keV band. Thanks to the large effective area of NICER, combined with the high flux of the source and a relatively long accumulative exposure time, we construct the first rms and phase-lag spectra for a type-A QPO. Our analysis reveals that the fractional rms amplitude of the QPO increases with energy from below 1 per cent at 1 keV to ∼3 per cent at 6 keV. The shape of the QPO spectrum is similar to that of the Comptonized component, suggesting that the Comptonized region is driving the variability. The phase lags at the QPO frequency are always soft taking the lowest energy as reference. By jointly fitting the time-averaged spectrum of the source and the rms and phase-lag spectra of the QPO with the time-dependent Comptonization model vkompthdk, we find that the radiative properties of the type-A QPO can be explained by a vertically extended Comptonized region with a size of ∼2300 km.

A rapid dynamic headspace method for authentication of whiskies using artificial neural networks

A rapid headspace analysis method for the authenticity testing of whiskies of different brands and years was developed for a low cost, deployable atmospheric pressure ionisation mass spectrometer, which required minimal sample preparation. Principal component analysis was applied to the time-averaged mass spectra, the classification results for which were compared against artificial neural network methods. The artificial neural network was found to outperform PCA, achieving ≥95% accuracy for all sampling conditions, with only two misclassifications under the ideal conditions, while requiring less development time.

A variable corona during the transition from type-C to type-B quasi-periodic oscillations in the black hole X-ray binary MAXI J1820+070

ABSTRACT We analyse a Neutron Star Interior Composition Explorer observation of the black hole X-ray binary MAXI J1820+070 during a transition from type-C to type-B quasi-periodic oscillations (QPOs). We find that below ∼2 keV, for the type-B QPOs the rms amplitude is lower and the magnitude of the phase lags is larger than for the type-C QPOs. Above that energy, the rms and phase-lag spectra of the type-B and type-C QPOs are consistent with being the same. We perform a joint fit of the time-averaged spectra of the source, and the rms and phase-lag spectra of the QPOs with the time-dependent Comptonization model vkompth to study the geometry of the corona during the transition. We find that the data can be well fitted with a model consisting of a small and a large corona that are physically connected. The sizes of the small and large coronae increase gradually during the type-C QPO phase whereas they decrease abruptly at the transition to type-B QPO. At the same time, the inner radius of the disc moves inward at the QPO transition. Combined with simultaneous radio observations showing that discrete jet ejections happen around the time of the QPO transition, we propose that a corona that expands horizontally during the type-C QPO phase, from ∼104 km ($\sim 800\, R_{\rm g}$) to 105 km ($\sim 8000\, R_{\rm g}$) overlying the accretion disc, transforms into a vertical jet-like corona extending over ∼104 km ($\sim 800\, R_{\rm g}$) during the type-B QPO phase.

The X-ray corona in the black hole binary GRO J1655−40 from the properties of non-harmonically related quasi-periodic oscillations

ABSTRACT The study of quasi-periodic oscillations (QPOs) plays a vital role in understanding the nature and geometry of the Comptonizing medium around black hole X-ray binaries. The spectral-state dependence of various types of QPOs (namely A, B, and C) suggests that they could have different origins. The simultaneous presence of different types of QPOs would therefore imply the simultaneous occurrence of different mechanisms. In this work, we study the radiative properties of two non-harmonically related QPOs in the black hole binary GRO J1655−40 detected at the peak of the ultraluminous state during the 2005 outburst of the source. The two QPOs have been previously identified as types B and C, respectively. We jointly fit the phase-lag and rms spectra of the QPOs and the time-averaged spectrum of the source with the time-dependent Comptonization model vkompth to infer the geometry of the media producing the QPOs. The time-averaged spectrum required a hot disc of 2.3 keV and a steep power law with index 2.7, revealing that the source was in an ultraluminous state. The corona that drives the variability of the type-B QPO is smaller in size and has a lower feedback fraction than the one that drives the variability of the type-C QPO. This suggests the simultaneous presence of a horizontally extended corona covering the accretion disc and a vertically elongated jet-like corona that are responsible for the type-C and B QPOs, respectively.

Ion Energization and Thermalization in Magnetic Reconnection Exhaust Region in the Solar Wind

Plasma energization and thermalization in magnetic reconnection is an important topic in astrophysical studies. We select two magnetic reconnection exhausts encountered by Solar Orbiter and analyze the associated ion heating in the kinetic regime. Both cases feature asymmetric plasma merging in the exhaust and anisotropic heating. For a quantitative investigation of the associated complex velocity-space structures, we adopt a three-dimensional Hermite representation of the proton velocity distribution function to produce the distribution of Hermite moments. We also derive the enstrophy and Hermite spectra to analyze the free energy conversion and transfer in phase space. We find a depletion of Hermite power at small m (corresponding to large-scale structures in velocity space) inside the reconnection exhaust region, concurrent with enhanced proton temperature and decreased enstrophy. Furthermore, the slopes of the 1D time-averaged parallel Hermite spectra are lower inside the exhaust and consistent with the effect of phase mixing that creates small fluctuations in velocity space. These fluctuations store free energy at higher m and are smoothed by weak collisionality, leading to irreversible thermalization. We also suggest that the perpendicular heating may happen via perpendicular phase mixing resulting from finite Larmor radius effects around the exhaust boundary.

Investigating the Impact of Vertically Extended Coronae on X-Ray Reverberation Mapping

Accreting black holes commonly exhibit hard X-ray emission, originating from a region of hot plasma near the central engine referred to as the corona. The origin and geometry of the corona are poorly understood, and models invoking either inflowing or outflowing material (or both) can successfully explain only parts of the observed phenomenology. In particular, recent works indicate that the time-averaged and variability property might originate in different regions of the corona. In this paper we present a model designed to move beyond the lamppost paradigm, with the goal of accounting for the vertical extent of the corona. In particular, we highlight the impact of including self-consistently a second lamppost, mimicking, for example, an extended jet base. We fully include the effect that the second source has on the time-dependent disk ionization, reflection spectrum, and reverberation lags. We also present an application of this new model to NICER observations of the X-ray binary MAXI J1820+070 near its hard-to-soft state transition. We demonstrate that in these observations, a vertically extended corona can capture both spectral and timing properties, while a single-lamppost model can not. In this scenario, the illumination responsible for the time-averaged spectrum originates close to the black hole, while the variability is likely associated with the ballistic jet.