arXiv subjects

Farhad Yusef-Zadeh

Publications and source records attributed to Farhad Yusef-Zadeh.

At least 19 recordsLinked to original sources

arXiv2026

The Nearest Galactic Nucleus: Studying the Galactic Centre with SKA-Mid

The Galactic Centre is the nearest nucleus of a galaxy and the most extreme environment that we can observe down to physical scales of a few hundred astronomical units. There is no other region in the Milky Way that can match its unique characteristics, such as its stellar density, turbulence and temperature of the interstellar medium, strong large scale magnetic field, concentration of stellar remnants, or mean star formation rate. The Galactic Centre is a unique target to understand the physics of galactic nuclei and study a large number of rare objects, such as extremely massive stars and stellar remnants, at a well-defined distance. The Galactic Centre has been and is being studied intensively with the most advanced facilities. In this chapter, we advocate for a large-area, multi-wavelength continuum survey with the Square Kilometre Array of an area of about 2.0deg x 0.4deg (~290pc x 60pc), centred on the massive black hole Sagittarius A* and for repeated deep observations of the nuclear star cluster over a decade, which will allow the community to address multiple science problems with single dataset.

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arXiv2025

The SKA Galactic Centre Survey -- A White Paper

With its extreme density of stars and stellar remnants, dense young massive clusters, high specific star formation rate, intense radiation field, high magnetic field strength, and properties of the interstellar medium that resemble those in high redshift galaxies and starbursts, the Galactic Centre is the most extreme environment that we can observe in detail. It is also the only nucleus of a galaxy that we can observe with a resolution of just a few milli parsecs. This makes it a crucial target to understand the physics of galactic nuclei and star formation, as well as the connection between them. It enables studies of a large number of otherwise rare objects, such as extremely massive stars and stellar remnants, at a well-defined distance, thus facilitating the interpretation of their properties. The Galactic Centre has been and is being studied intensively with the most advanced facilities. In this White Paper, we advocate for a large-area, multi-wavelength survey with the Square Kilometre Array of an area of about 1.25x0.3 deg**2 (180x40 pc**2), centered on the massive black hole Sagittarius A* and for repeated deep observations of the nuclear star cluster over a decade, which will allow the community to address multiple science problems with a single data set.

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arXiv2024

Multiwavelength Observations of Sgr A*. II. 2019 July 21 and 26

We report on the final two days of a multiwavelength campaign of Sgr A* observing in the radio, submillimeter, infrared, and X-ray bands in July 2019. Sgr A* was remarkably active, showing multiple flaring events across the electromagnetic spectrum. We detect a transient $\sim35$-minute periodicity feature in Spitzer Space Telescope light curves on 21 July 2019. Time-delayed emission was detected in ALMA light curves, suggesting a hotspot within the accretion flow on a stable orbit. On the same night, we observe a decreased flux in the submillimeter light curve following an X-ray flare detected by the Chandra X-ray Observatory and model the feature with an adiabatically expanding synchrotron hotspot occulting the accretion flow. The event is produced by a plasma $0.55~R_{\text{S}}$ in radius with an electron spectrum $p=2.84$. It is threaded by a $\sim130$ Gauss magnetic field and expands at $0.6\%$ the speed of light. Finally, we reveal an unambiguous flare in the infrared, submillimeter, and radio, demonstrating that the variable emission is intrinsically linked. We jointly fit the radio and submillimeter light curves using an adiabatically expanding synchrotron hotspot and find it is produced by a plasma with an electron spectrum $p=0.59$, $187$ Gauss magnetic field, and radius $0.47~R_{\text{S}}$ that expands at $0.029c$. In both cases, the uncertainty in the appropriate lower and upper electron energy bounds may inflate the derived equipartition field strengths by a factor of 2 or more. Our results confirm that both synchrotron- and adiabatic-cooling processes are involved in the variable emission's evolution at submillimeter and infrared wavelengths.

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arXiv2023

Polarized signatures of adiabatically expanding hotspots in Sgr A*'s accretion flow

We report 235 GHz linear and circular polarization (LP and CP) detections of Sgr A* at levels of $\sim10\%$ and $\sim-1\%$, respectively, using ALMA. We describe the first full-Stokes modeling of an observed submillimeter flare with an adiabatically-expanding synchrotron hotspot using a polarized radiative transfer prescription. Augmented with a simple full-Stokes model for the quiescent emission, we jointly characterize properties of both the quiescent and variable components by simultaneously fitting all four Stokes parameter light curves. The hotspot has magnetic field strength $71$ G, radius $0.75$ Schwarzschild radii, and expands at speed $0.013$c assuming magnetic equipartition. The magnetic field's position angle projected in the plane-of-sky is $\approx55^\circ$ East of North, which previous analyses reveal as the accretion flow's angular momentum axis and further supports Sgr A* hosting a magnetically-arrested disk. The magnetic field is oriented approximately perpendicular to the line of sight, which suggests repolarization as the cause of the high circular-to-linear polarization ratio observed at radio frequencies. We additionally recover several properties of the quiescent emission, consistent with previous analyses of the accretion flow, such as a rotation measure $\approx-4.22\times10^{5}$ rad m$^{-2}$. Our findings provide critical constraints for interpreting and mitigating the polarized variable emission in future Event Horizon Telescope images of Sgr A*.

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arXiv2021

Multi-wavelength Observations of Sgr A*. I. 2019 July 18

We present and analyze ALMA submillimeter observations from a multi-wavelength campaign of Sgr A* during 18 July 2019. In addition to the submillimeter, we utilize concurrent mid-IR (Spitzer) and X-ray (Chandra) observations. The submillimeter emission lags less than $δt\approx30$ minutes behind the mid-IR data. However, the entire submillimeter flare was not observed, raising the possibility that the time delay is a consequence of incomplete sampling of the light curve. The decay of the submillimeter emission is not consistent with synchrotron cooling. Therefore, we analyze these data adopting an adiabatically expanding synchrotron source that is initially optically thick or thin in the submillimeter, yielding time-delayed or synchronous flaring with the IR, respectively. The time-delayed model is consistent with a plasma blob of radius $0.8~R_{\text{S}}$ (Schwarzschild radius), electron power-law index $p=3.5$ ($N(E)\propto E^{-p}$), equipartition magnetic field of $B_{\text{eq}}\approx90$ Gauss, and expansion velocity $v_{\text{exp}}\approx0.004c$. The simultaneous emission is fit by a plasma blob of radius $2~R_{\text{S}}$, $p=2.5$, $B_{\text{eq}}\approx27$ Gauss, and $v_{\text{exp}}\approx0.014c$. Since the submillimeter time delay is not completely unambiguous, we cannot definitively conclude which model better represents the data. This observation presents the best evidence for a unified flaring mechanism between submillimeter and X-ray wavelengths and places significant constraints on the source size and magnetic field strength. We show that concurrent observations at lower frequencies would be able to determine if the flaring emission is initially optically thick or thin in the submillimeter.

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arXiv2021

Detection of a 20 minute time lag observed from Sgr A* between 8 and 10 GHz with the VLA

We report the detection and analysis of a radio flare observed on 17 April 2014 from Sgr A* at $9$ GHz using the VLA in its A-array configuration. This is the first reported simultaneous radio observation of Sgr A* across $16$ frequency windows between $8$ and $10$ GHz. We cross correlate the lowest and highest spectral windows centered at $8.0$ and $9.9$ GHz, respectively, and find the $8.0$ GHz light curve lagging $18.37^{+2.17}_{-2.18}$ minutes behind the $9.9$ GHz light curve. This is the first time lag found in Sgr A*'s light curve across a narrow radio frequency bandwidth. We separate the quiescent and flaring components of Sgr A* via flux offsets at each spectral window. The emission is consistent with an adiabatically-expanding synchrotron plasma, which we fit to the light curves to characterize the two components. The flaring emission has an equipartition magnetic field strength of $2.2$ Gauss, size of $14$ Schwarzschild radii, average speed of $12000$ km s$^{-1}$, and electron energy spectrum index ($N(E)\propto E^{-p}$), $p = 0.18$. The peak flare flux at $10$ GHz is approximately $25$% of the quiescent emission. This flare is abnormal as the inferred magnetic field strength and size are typically about $10$ Gauss and few Schwarzschild radii. The properties of this flare are consistent with a transient warm spot in the accretion flow at a distance of $10$-$100$ Schwarzschild radii from Sgr A*. Our analysis allows for independent characterization of the variable and quiescent components, which is significant for studying temporal variations in these components.

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arXiv2021

Revealing the Physical Conditions around Sgr A* using Bayesian Inference -- I. Observations and Radiative Transfer

We report sub-arcsecond ALMA observations between 272 - 375 GHz towards Sgr A*'s Circumnuclear disk (CND). Our data comprises 8 individual pointings, with significant SiO (8(7) - 7(6)) and SO (7 - 6) emission detected towards 98 positions within these pointings. Additionally, we identify H2CS (9(1,9) - 8(1,8)), OCS (25 - 24) and CH3OH (2(1,1) - 2(0,2)) towards a smaller subset of positions. By using the observed peak line flux density together with a Bayesian Inference technique informed by radiative transfer models, we systematically recover the physical gas conditions towards each of these positions. We estimate that the bulk of the surveyed gas has temperature T < 500 K and density n $\lessapprox 10^{6}$ cm$^{-3}$, consistent with previous studies of similar positions as traced by HCN clumps. However, we identify an uncharacteristically hot (T $\approx 600$ K) and dense (n $\approx 10^{6}$ cm$^{-3}$) source in the Northeastern Arm. This position is found to be approximately consistent with a gravitationally bound region dominated by turbulence. We also identify a nearby cold (T $\approx 60$ K) and extremely dense (n $\approx 10^{7}$ cm$^{-3}$) position that is again potentially bound and dominated by turbulence. We also determine that the total gas mass contained within the CND is M $\approx 4 \times 10^{4}$ $M_{\odot}$. Furthermore, we qualitatively note that the observed chemical enrichment across large scales within the CND is consistent with bulk grain processing, though multiple desorption mechanisms are plausibly responsible. Further chemical modelling is required to identify the physical origin of the grain-processing, as well as the localised H2CS and OCS emission.

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arXiv2019

Chandra Spectral and Timing Analysis of Sgr A*'s Brightest X-ray Flares

We analyze the two brightest Chandra X-ray flares detected from Sagittarius A*, with peak luminosities more than 600 x and 245 x greater than the quiescent X-ray emission. The brightest flare has a distinctive double-peaked morphology --- it lasts 5.7 ksec ($\sim 2$ hours), with a rapid rise time of 1500 sec and a decay time of 2500 sec. The second flare lasts 3.4 ksec, with rise and decay times of 1700 sec and 1400 sec. These luminous flares are significantly harder than quiescence: the first has a power law spectral index $Γ= 2.06\pm 0.14$ and the second has $Γ= 2.03\pm 0.27$, compared to $Γ= 3.0\pm0.2$ for the quiescent accretion flow. These spectral indices (as well as the flare hardness ratios) are consistent with previously-detected Sgr A* flares, suggesting that bright and faint flares arise from similar physical processes. Leveraging the brightest flare's long duration and high signal-to-noise, we search for intraflare variability and detect excess X-ray power at a frequency of $ν\approx 3$ mHz, but show that it is an instrumental artifact and not of astrophysical origin. We find no other evidence (at the 95% confidence level) for periodic or quasi-periodic variability in either flares' time series. We also search for non-periodic excess power but do not find compelling evidence in the power spectrum. Bright flares like these remain our most promising avenue for identifying Sgr A*'s short timescale variability in the X-ray, which may probe the characteristic size scale for the X-ray emission region.

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arXiv2019

2 mm GISMO Observations of the Galactic Center. II. A Nonthermal Filament in the Radio Arc and Compact Sources

We have used the Goddard IRAM 2-Millimeter Observer (GISMO) with the 30 m IRAM telescope to carry out a 2 mm survey of the Galaxy's central molecular zone (CMZ). These observations detect thermal emission from cold ISM dust, thermal free-free emission from ionized gas, and nonthermal synchrotron emission from relatively flat-spectrum sources. Archival data sets spanning $3.6 μ$m to 90 cm are used to distinguish different emission mechanisms. After the thermal emission of dust is modeled and subtracted, the remaining 2 mm emission is dominated by free-free emission, with the exception of the brightest nonthermal filament (NTF) that runs though the middle of the bundle of filaments known as the Radio Arc. This is the shortest wavelength at which any NTF has been detected. The GISMO observations clearly trace this NTF over a length of ~0.2$^\circ$, with a mean 2 mm spectral index which is steeper than at longer wavelengths. The 2 mm to 6 cm (or 20 cm) spectral index steepens from $α\approx -0.2$ to $-0.7$ as a function distance from the Sickle H II region, suggesting that this region is directly related to the NTF. A number of unresolved (at $21''$) 2 mm sources are found nearby. One appears to be thermal dust emission from a molecular cloud that is associated with an enigmatic radio point source whose connection to the Radio Arc is still debated. The morphology and colors at shorter IR wavelengths indicate other 2 mm unresolved sources are likely to be compact H II regions.

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arXiv2019

2 mm GISMO Observations of the Galactic Center. I. Dust Emission

The Central Molecular Zone (CMZ), covering the inner ~1$^\circ$ of the Galactic plane has been mapped at 2 mm using the GISMO bolometric camera on the 30 m IRAM telescope. The $21''$ resolution maps show abundant emission from cold molecular clouds, from star forming regions, and from one of the Galactic center nonthermal filaments. In this work we use the Herschel Hi-GAL data to model the dust emission across the Galactic center. We find that a single-temperature fit can describe the 160 -- 500 $μ$m emission for most lines of sight, if the long-wavelength dust emissivity scales as $λ^{-β}$ with $β\approx 2.25$. This dust model is extrapolated to predict the 2 mm dust emission. Subtraction of the model from the GISMO data provides a clearer look at the 2 mm emission of star-forming regions and the brightest nonthermal filament.

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arXiv2019

ALMA Detection of Extreme Blue-Shifted Ionized Gas Within 0.2 pc of Sgr A* from -480 to -300 km/s

We have used the capabilities of ALMA to probe the ionized gas towards the Galactic center with the H30alpha mm hydrogen recombination line within 30'' of the Galaxy's dynamical center. The observations are made with spatial and spectral resolutions of 0.46'' X 0.35'' and 3 km/s, respectively. Multiple compact and extended sources are detected in the mini-cavity region with extreme negative radial velocities ranging from -480 to -300 km/s, 2-3'' (0.08-0.12 pc) from Sgr A*. This is the highest radial velocity of ionized gas detected beyond the inner 1'' of Sgr A*. This new component is kinematically isolated from the orbiting ionized gas. The detected gas has a velocity gradient ranging from -50 to -200 km/s per arcsecond located to the southwest of Sgr A* at a position angle of roughly -160 degrees. Previous proper motion measurements in the immediate vicinity of these extreme high-velocity ionized components have been measured and show transverse velocities that range from 313 - 865 km/s. If we assume that they are associated with each other, then the total velocity implies these components are gravitationally unbound. In addition, we report the kinematics of cometary radio and infrared sources. These sources are diagonally distributed with a position angle of roughly 50 degrees within 14'' of Sgr A*. We interpret the highly blue-shifted features to the SW where the mini-cavity is located and red-shifted cometary sources to the NE in terms of the interaction of a collimated outflow with an opening angle of roughly 30 degrees. An expected mass outflow rate of 2E-7 or 4E-5 solar mass per yr is estimated by a relativistic jet-driven outflow or collimated stellar winds, respectively.

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arXiv2018

SWAG: Distribution and Kinematics of an Obscured AGB Population toward the Galactic Center

Outflows from AGB stars enrich the Galactic environment with metals and inject mechanical energy into the ISM. Radio spectroscopy can recover both properties through observations of molecular lines. We present results from SWAG: "Survey of Water and Ammonia in the Galactic Center". The survey covers the entire Central Molecular Zone (CMZ), the inner 3.35deg x 0.9deg (~480 x 130pc) of the Milky Way that contains ~5x10^7 Mo of molecular gas. Although our survey primarily targets the CMZ, we observe across the entire sightline through the Milky Way. AGB stars are revealed by their signature of double peaked 22 GHz water maser lines. They are distinguished by their spectral signatures and their luminosities, which reach up to 10^-7 Lo. Higher luminosities are usually associated with Young Stellar Objects located in CMZ star forming regions. We detect a population of ~600 new water masers that can likely be associated with AGB outflows.

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arXiv2017

SWAG Water Masers in the Galactic Center

The Galactic Center contains large amounts of molecular and ionized gas as well as a plethora of energetic objects. Water masers are an extinction-insensitive probe for star formation and thus ideal for studies of star formation stages in this highly obscured region. With the Australia Telescope Compact Array, we observed 22 GHz water masers in the entire Central Molecular Zone with sub-parsec resolution as part of the large SWAG survey: ``Survey of Water and Ammonia in the Galactic Center''. We detect of order 600 22 GHz masers with isotropic luminosities down to ~10^-7 Lo. Masers with luminosities of >~10^-6 Lo are likely associated with young stellar objects. They appear to be close to molecular gas streamers and may be due to star formation events that are triggered at pericenter passages near Sgr A*. Weaker masers are more widely distributed and frequently show double line features, a tell-tale sign for an origin in evolved star envelopes.

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arXiv2017

The Survey of Water and Ammonia in the Galactic Center (SWAG): Molecular Cloud Evolution in the Central Molecular Zone

The Survey of Water and Ammonia in the Galactic Center (SWAG) covers the Central Molecular Zone (CMZ) of the Milky Way at frequencies between 21.2 and 25.4 GHz obtained at the Australia Telescope Compact Array at $\sim 0.9$ pc spatial and $\sim 2.0$ km s$^{-1}$ spectral resolution. In this paper, we present data on the inner $\sim 250$ pc ($1.4^\circ$) between Sgr C and Sgr B2. We focus on the hyperfine structure of the metastable ammonia inversion lines (J,K) = (1,1) - (6,6) to derive column density, kinematics, opacity and kinetic gas temperature. In the CMZ molecular clouds, we find typical line widths of $8-16$ km s$^{-1}$ and extended regions of optically thick ($τ> 1$) emission. Two components in kinetic temperature are detected at $25-50$ K and $60-100$ K, both being significantly hotter than dust temperatures throughout the CMZ. We discuss the physical state of the CMZ gas as traced by ammonia in the context of the orbital model by Kruijssen et al. (2015) that interprets the observed distribution as a stream of molecular clouds following an open eccentric orbit. This allows us to statistically investigate the time dependencies of gas temperature, column density and line width. We find heating rates between $\sim 50$ and $\sim 100$ K Myr$^{-1}$ along the stream orbit. No strong signs of time dependence are found for column density or line width. These quantities are likely dominated by cloud-to-cloud variations. Our results qualitatively match the predictions of the current model of tidal triggering of cloud collapse, orbital kinematics and the observation of an evolutionary sequence of increasing star formation activity with orbital phase.

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arXiv2017

The CARMA 3mm Survey of the Inner $0.7^\circ\times0.4^\circ$ of the Central Molecular Zone

The Central Molecular Zone (CMZ) of the Galactic Center has to date only been fully mapped at mm wavelengths with singledish telescopes, with resolution about 30$^{\prime\prime}$ (1.2 pc). Using CARMA, we mapped the innermost 0.25 square degrees of the CMZ over the region between -0$.\kern-.25em ^{^\circ}$2$ \leq l \leq $0$.\kern-.25em ^{^\circ}$5 and -0$.\kern-.25em ^{^\circ}$2$ \leq b \leq $0$.\kern-.25em ^{^\circ}$2 (90$\times$50 pc) with spatial and spectral resolution of $\sim$10$^{\prime\prime}$ (0.4 pc) and $\sim$2.5 km/s, respectively. We provide a catalog of 3mm continuum sources as well as spectral line images of SiO(J=2-1), HCO$^{+}$(J=1-0), HCN(J=1-0), N$_{2}$H$^{+}$(J=1-0), and CS(J=2-1), with velocity coverage VLSR= -200 to 200 km/s. To recover the large scale structure resolved out by the interferometer, the continuum-subtracted spectral line images were combined with data from the Mopra 22-m telescope survey, thus providing maps containing all spatial frequencies down to the resolution limit. We find that integrated intensity ratio of I(HCN)/I(HCO$^{+}$) is anti-correlated with the intensity of the 6.4 keV Fe K$α$, which is excited either by high energy photons or low energy cosmic rays, and the gas velocity dispersion as traced by HCO$^{+}$ is correlated with Fe K$α$ intensity. The intensity ratio and velocity dispersion patterns are consistent with variation expected from the interaction of low energy cosmic rays with molecular gas.

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arXiv2017

Simultaneous Monitoring of X-ray and Radio Variability in Sagittarius A*

Monitoring of Sagittarius A* from X-ray to radio wavelengths has revealed structured variability --- including X-ray flares --- but it is challenging to establish correlations between them. Most studies have focused on variability in the X-ray and infrared, where variations are often simultaneous, and because long time series at sub-millimeter and radio wavelengths are limited. Previous work on sub-mm and radio variability hints at a lag between X-ray flares and their candidate sub-millimeter or radio counterparts, with the long wavelength data lagging the X-ray. However, there is only one published time lag between an X-ray flare and a possible radio counterpart. Here we report 9 contemporaneous X-ray and radio observations of Sgr A*. We detect significant radio variability peaking $\gtrsim$176 minutes after the brightest X-ray flare ever detected from Sgr A*. We also report other potentially associated X-ray and radio variability, with the radio peaks appearing $\lesssim$80 minutes after these weaker X-ray flares. Taken at face value, these results suggest that stronger X-ray flares lead to longer time lags in the radio. However, we also test the possibility that the variability at X-ray and radio wavelengths is not temporally correlated. We cross-correlate data from mismatched X-ray and radio epochs and obtain comparable correlations to the matched data. Hence, we find no overall statistical evidence that X-ray flares and radio variability are correlated, underscoring a need for more simultaneous, long duration X-ray--radio monitoring of Sgr A*.

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arXiv2014

On the origin of the central 1" hole in the stellar disk of Sgr A* and the Fermi gamma-ray bubbles

The supermassive black hole Sgr A* at the center of the Galaxy is surrounded by two misaligned disks of young, massive stars extending from ~0.04 to 0.4 pc. The stellar surface density increases as ~ r^-2 towards Sgr A* but is truncated within 1" (0.04pc). We explore the origin of this annulus using a model in which star formation occurs in a disk of gas created through the partial capture of a gas cloud as it sweeps through the inner few parsecs of the galaxy and temporarily engulfs Sgr A*. We identify the locations within which star formation and/or accretion onto Sgr A* take place. Within 0.04 pc the disk is magnetically active and the associated heating and enhanced pressure prevents the disk from becoming self gravitating. Instead, it forms a magneto-turbulent disk that drains onto Sgr A* within 3 Myr. Meanwhile, fragmentation of the gas beyond the central 0.04 pc hole creates the observed young stellar disk. The two large scale bubbles of gamma-ray emission extending perpendicular to the Galactic plane may be created by a burst of accretion of ~10^5 Msun of gas lying between 0.01 and 0.03 pc. The observed stellar ages imply that this capture event occurred ~10^6.5 yr ago, thus such events occurring over the life time of the Galaxy could have significantly contributed to the current mass of Sgr A* and to the inner few parsec of the nuclear star cluster. We suggest that these events also occur in extragalactic systems.

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arXiv2014

Infrared H3+ and CO Studies of the Galactic Core: GCIRS 3 and GCIRS 1W

We have obtained improved spectra of key fundamental band lines of H3+, R(1,1)l, R(3,3)l, and R(2,2)l, and ro-vibrational transitions of CO on sightlines toward the luminous infrared sources GCIRS 3 and GCIRS 1W, each located in the Central Cluster of the Galactic center within several arcseconds of Sgr A*. The spectra reveal absorption occurring in three kinds of gaseous environments: (1) cold dense and diffuse gas associated with foreground spiral/lateral arms; (2) warm and diffuse gas absorbing over a wide and mostly negative velocity range, which appears to fill a significant fraction of the Galaxy's Central Molecular Zone (CMZ); and (3) warm, dense and compact clouds with velocities near +50 km s^-1 probably within 1-2 pc of the center. The absorptions by the first two cloud types are nearly identical for all the sources in the Central Cluster, and are similar to those previously observed on sightlines from Sgr A* to 30 pc east of it. Cloud type (3), which has only been observed toward the Central Cluster, shows distinct differences between the sightlines to GCIRS 3 and GCIRS 1W, which are separated on the sky by only 0.33 pc in projection. We identify this material as part of an inward extension of the Circumnuclear Disk previously known from HCN mapping. Lower limits on the products of the hydrogen ionization rate zeta and the path length L are 2.3 x 10^5 cm s^-1 and 1.5 x 10^3 cm s^-1 for the warm and diffuse CMZ gas and for the warm and dense clouds in the core, respectively. The limits indicate that the ionization rates in these regions are well above 10^-15 s^-1.

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