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At least 307 records · Page 17

Spectroscopy of Chlorosyl Fluoride, FClO

FClO has been proposed as an intermediate in reactions involving ClF, Cl2O, and ClF3O, and it has been suggested as a molecule of atmospheric interest. It has been prepared in situ by the hydrolysis of ClF3. The pure rotational spectrum of FClO has been studied by conventional millimeter wave techniques and by microwave Fourier transform spectroscopy. Selected transitions were searched for using predictions based on an analysis of the nu(sub 1) band. FClO is an asymmetric prolate top, kappa = -0.8950 for F(35)ClO, with a rather small dipole component of 0.093 (4) D along the a-axis and a larger one of 1.93 (5) D along the b-axis. Transitions with 0 <= J <= 54 and 0 <= K(sub a) <= 18 were observed. Cl hyperfine splitting was generally observable throughout the spectrum with F-19 spin-rotation splitting observable as well in the microwave region. Structural parameters, harmonic force constants, and nuclear magnetic shielding parameters were derived and will be compared with data of related molecules, such as ClF3, ClF, FClO2, and FClO3. High resolution infrared spectra were taken in the regions of the FCl stretching mode and bending mode around 600 and 310/cm, respectively. A preliminary analysis indicates that the FCl stretch, near 596.86/cm for F(35)ClO, is in resonance with the dark overtone of delta near 617/cm. A brief progress report will be given.

Mueller, Holger S. P.↗

Spin-lattice relaxation and the calculation of gain, pump power, and noise temperature in ruby

The use of a quantitative analysis of the dominant source of relaxation in ruby spin systems to make predictions of key maser amplifier parameters is described. The spin-lattice Hamiltonian which describes the interaction of the electron spins with the thermal vibrations of the surrounding lattice is obtained from the literature. Taking into account the vibrational anisotropy of ruby, Fermi's rule is used to calculate the spin transition rates between the maser energy levels. The spin population rate equations are solved for the spin transition relaxation times, and a comparison with previous calculations is made. Predictions of ruby gain, inversion ratio, and noise temperature as a function of physical temperature are made for 8.4-GHz and 32-GHz maser pumping schemes. The theory predicts that ruby oriented at 90 deg will have approximately 50 percent higher gain in dB and slightly lower noise temperature than a 54.7-deg ruby at 32 GHz (assuming pump saturation). A specific calculation relating pump power to inversion ratio is given for a single channel of the 32-GHz reflected wave maser.

Lyons, J. R.↗

Lessons from the GP-B Experience for Future Fundamental Physics Missions in Space

Gravity Probe B launched in April 2004 and completed its science data collection in September 2005, with the objective of sub-milliarcsec measurement of two General Relativistic effects on the spin axis orientation of orbiting gyroscopes. Much of the technology required by GP-B has potential application in future missions intended to make precision measurements. The philosophical approach and experiment design principles developed for GP-B are equally adaptable to these mission concepts. This talk will discuss GP-B's experimental approach and the technological and philosophical lessons learned that apply to future experiments in fundamental physics. Measurement of fundamental constants to high precision, probes of short-range forces, searches for equivalence principle violations, and detection of gravitational waves are examples of concepts and missions that will benefit kern GP-B's experience.

Kolodziejczak, Jeffery↗

Manufacturing Precise, Lightweight Paraboloidal Mirrors

A process for fabricating a precise, diffraction- limited, ultra-lightweight, composite- material (matrix/fiber) paraboloidal telescope mirror has been devised. Unlike the traditional process of fabrication of heavier glass-based mirrors, this process involves a minimum of manual steps and subjective judgment. Instead, this process involves objectively controllable, repeatable steps; hence, this process is better suited for mass production. Other processes that have been investigated for fabrication of precise composite-material lightweight mirrors have resulted in print-through of fiber patterns onto reflecting surfaces, and have not provided adequate structural support for maintenance of stable, diffraction-limited surface figures. In contrast, this process does not result in print-through of the fiber pattern onto the reflecting surface and does provide a lightweight, rigid structure capable of maintaining a diffraction-limited surface figure in the face of changing temperature, humidity, and air pressure. The process consists mainly of the following steps: 1. A precise glass mandrel is fabricated by conventional optical grinding and polishing. 2. The mandrel is coated with a release agent and covered with layers of a carbon- fiber composite material. 3. The outer surface of the outer layer of the carbon-fiber composite material is coated with a surfactant chosen to provide for the proper flow of an epoxy resin to be applied subsequently. 4. The mandrel as thus covered is mounted on a temperature-controlled spin table. 5. The table is heated to a suitable temperature and spun at a suitable speed as the epoxy resin is poured onto the coated carbon-fiber composite material. 6. The surface figure of the optic is monitored and adjusted by use of traditional Ronchi, Focault, and interferometric optical measurement techniques while the speed of rotation and the temperature are adjusted to obtain the desired figure. The proper selection of surfactant, speed or rotation, viscosity of the epoxy, and temperature make it possible to obtain the desired diffraction-limited, smooth (1/50th wave) parabolic outer surface, suitable for reflective coating. 7. A reflective coat is applied by use of conventional coating techniques. 8. Once the final figure is set, a lightweight structural foam is applied to the rear of the optic to ensure stability of the figure.

Hermann, Frederick Thomas↗

Active control of fan-generated plane wave noise

Subsonic propulsion systems for future aircraft may incorporate ultra-high bypass ratio ducted fan engines whose dominant noise source is the fan with blade passage frequency less than 1000 Hz. This low frequency combines with the requirement of a short nacelle to diminish the effectiveness of passive duct liners. Active noise control is seen as a viable method to augment the conventional passive treatments. An experiment to control ducted fan noise using a time domain active adaptive system is reported. The control sound source consists of loudspeakers arrayed around the fan duct. The error sensor location is in the fan duct. The purpose of this experiment is to demonstrate that the in-duct error sensor reduces the mode spillover in the far field, thereby increasing the efficiency of the control system. In this first series of tests, the fan is configured so that predominantly zero order circumferential waves are generated. The control system is found to reduce the blade passage frequency tone significantly in the acoustic far field when the mode orders of the noise source and of the control source are the same. The noise reduction is not as great when the mode orders are not the same even though the noise source modes are evanescent, but the control system converges stably and global noise reduction is demonstrated in the far field. Further experimentation is planned in which the performance of the system will be evaluated when higher order radial and spinning modes are generated.

Gerhold, Carl H.↗

Long-term Coherent Timing of the Accreting Millisecond Pulsar IGR J17062–6143

We report on a coherent timing analysis of the 163 Hz accreting millisecond X-ray pulsar IGR J17062–6143. Using data collected with the Neutron Star Interior Composition Explorer and XMM-Newton, we investigated the pulsar evolution over a time span of four years. We obtained a unique phase-coherent timing solution for the stellar spin, finding the source to be spinning up at a rate of (3.77 ± 0.09) × 10−15 Hz s−1. We further find that the 0.4–6 keV pulse fraction varies gradually between 0.5% and 2.5% following a sinusoidal oscillation with a 1210 ± 40 day period. Finally, we supplemented this analysis with an archival Rossi X-ray Timing Explorer observation and obtained a phase-coherent model for the binary orbit spanning 12 yr, yielding an orbital period-derivative measurement of (8.4 ± 2.0) × 10−12 s s−1. This large orbital period derivative is inconsistent with a binary evolution that is dominated by gravitational wave emission and is suggestive of highly non-conservative mass transfer in the binary system.

stars: neutron↗

Population Properties of Compact Objects from the Second LIGO-Virgo Gravitational-Wave Transient Catalog

We report on the population of 47 compact binary mergers detected with a false-alarm rate of <1yr^(−1) in the second LIGO–Virgo Gravitational-Wave Transient Catalog. We observe several characteristics of the merging binary black hole (BBH) population not discernible until now. First, the primary mass spectrum contains structure beyond a power law with a sharp high-mass cutoff; it is more consistent with a broken power law with a break at 39.7^(+20.3)(-9.1)M or a power law with a Gaussian feature peaking at 33.1^(+4.0)(-5.6)M (90% credible interval). While the primary mass distribution must extend to ~65M or beyond, only 2.9^(+3.5)(-1.7)M% of systems have primary masses greater than 45M. Second, we find that a fraction of BBH systems have component spins misaligned with the orbital angular momentum, giving rise to precession of the orbital plane. Moreover, 12% to 44% of BBH systems have spins tilted by more than 90°, giving rise to a negative effective inspiral spin parameter, χeff. Under the assumption that such systems can only be formed by dynamical interactions, we infer that between 25% and 93% of BBHs with nonvanishing |χeff| > 0.01 are dynamically assembled. Third, we estimate merger rates, finding RBBH = 23.9^(+14.3)(-8.6) Gpc^(-3) yr^(-1) for BBHs and RBNS = 320^(+490)(-240) Gpc^(-3) yr^(-1) for binary neutron stars. We find that the BBH rate likely increases with redshift (85% credibility) but not faster than the star formation rate (86% credibility). Additionally, we examine recent exceptional events in the context of our population models, finding that the asymmetric masses of GW190412 and the high component masses of GW190521 are consistent with our models, but the low secondary mass of GW190814 makes it an outlier.

R Abbott↗

Spin-Up Instability of a Levitated Molten Drop in MHD-Flow Transition to Turbulence

When an alternating magnetic field interacts with induced eddy currents in a conducting body, there will be a repulsive force between the body and the driving coil system generating the field. This repulsive force is the basis of electromagnetic levitation, which allows containerless processing of different materials. The eddy currents in the conducting body also generate Joule heating. Axial rotation of electromagnetically levitated objects is a common observation in levitation systems and often an undesirable side effect of such experiments on 1-g and -g. There have been recent efforts to use magnetic damping and suppress this tendency of body rotation. The first report of rotation in EML drops was attributed to a slight asymmetry of the shape and location of the levitation coils could change the axis and speed of rotation. Other theories of sample rotation include a frequency difference in the traveling electromagnetic waves and a phase difference in two different applied fields of the same frequency. All of these different mechanisms share the following characteristics: the torque is small, constant for constant field strength, and very weakly dependent on the sample's temperature and phase (solid or liquid). During experiments on the MSL-1 (First Microgravity Science Laboratory) mission of the Space Shuttle (STS-83 and STS-94, April and July 1997), a droplet of palladium-silicon alloy was electromagnetically levitated for viscosity measurements. For the non-deforming droplet, the resultant MHD flow inside the drop is inferred from motion of impurities on the surface. These observations indicate formation of a pair of co-rotating toroidal flow structures inside the spheroidal levitated drop that undergo secondary flow instabilities. As rise in the fluid temperature rises, the viscosity falls and the internal flow accelerates and becomes oscillatory; and beyond a point in the experiments, the surface impurities exhibit non-coherent chaotic motion signifying emergence of turbulence inside the drop. In this work, a background of these set of observations will be given followed by a presentation of our results on the digital particle tracking analysis that has been performed on a number of available videos. The analysis indicates that the levitated drop attains a constant rotational speed during the melting phase and formation of the co-rotating axi-symmetric laminar toroidal structures. However, the rate of axial rotation increases dramatically during the deformation of the toroidal structures anti their breakup into chaotic entities. This new data suggests an interaction between the flow inside the levitated molten drop and the driving coils in the experiments. Possible mechanisms for this interaction will be reviewed. The data will also be used to make an assessment of existing theories on droplet rotation.

Abedian, B.↗

Feasibility Study of Two Candidate Reaction Wheel/thruster Hybrid Control Architecture Designs for the Cassini Spacecraft

As the first spacecraft to achieve orbit at Saturn in 2004, Cassini has collected science data throughout its four-year prime mission (2004-08), and has since been approved for a first and second extended mission through 2017. Cassini carries a set of three "fixed" reaction wheels and a backup reaction wheel (reaction wheel #4) is mounted on top of an articulable platform. If necessary, this platform could be articulated to orient the backup reaction wheel with the degraded wheel. The reaction wheels are used primarily for attitude control when precise and stable pointing of a science instrument such as the narrow angle camera is required. In 2001-02, reaction wheel #3 exhibited signs of bearing cage instability. As a result, reaction wheel #4 was articulated to align with reaction wheel #3. Beginning in July 2003, Cassini was controlled using wheel #1, #2, and #4. From their first use in the spring of 2000 until today, reaction wheels #1 and #2 have accumulated more than3.5 billions revolutions each. As such, in spite of very carefully management of the wheel spin rates by the mission operation team, there are some observed increases in the drag torque of the wheels' bearings. Hence, the mission operations team must prepare for the contingency scenario in which the reaction wheel #1 (in addition to wheel #3) had degraded. In this hypothetical fault scenario, the two remaining reaction wheels (#2 and #4) will not be able to provide precise and stable three-axis control of the spacecraft. In this study, we evaluate the feasibility of controlling Cassini using the two remaining reaction wheels and four thrusters to meet the science pointing requirements for two key science operational modes: the Optical Remote Sensing and Downlink, Fields, Particles, & Waves operation modes. The performance (e.g., pointing control error, pointing stability, hydrazine consumption rate, etc.) of the hybrid controllers in both operations scenarios will be compared with those achieved using an all-thruster controller design. Strength and weaknesses of the hybrid control architecture are assessed quantitatively.

Cassini↗

Fine structure of low-energy H(+) in the nightside auroral region

Low-energy H(+) data with 6-s resolution from the retarding ion mass spectrometer instrument on Dynamics Explorer (DE) 1 have been analyzed to reveal the fine structure at middle altitudes of the nightside auroral region. A new method for deconvolving the energy-integrated count rate in the spin plane of the satellite has been used to derive the two-dimensional phase space density. A detailed analysis reveals an alternating conic-beam-conic pattern with the observed conics correlated with large earthward currents in the auroral region. The strong downward current (larger than 1 microamperes per sq m (equivalent value at ionosphere)) provides a free energy source for the perpendicular ion heating, that generates the ion conics with energies from several eV to tens of eV. The bowl shape distribution of the low-energy H(+) is caused by the extended perpendicular heating. The strong correlation between conics and large downward currents suggests that the current-driven electrostatic ion cyclotron wave is an appropriate candidate for the transverse heating mechanism.

Liu, Chao↗

Pattern Formation in Diffusion Flames Embedded in von Karman Swirling Flows

Pattern formation is observed in nature in many so-called excitable systems that can support wave propagation. It is well-known in the field of combustion that premixed flames can exhibit patterns through differential diffusion mechanism between heat and mass. However, in the case of diffusion flames where fuel and oxidizer are separated initially there have been only a few observations of pattern formation. It is generally perceived that since diffusion flames do not possess an inherent propagation speed they are static and do not form patterns. But in diffusion flames close to their extinction local quenching can occur and produce flame edges which can propagate along stoichiometric surfaces. Recently, we reported experimental observations of rotating spiral flame edges during near-limit combustion of a downward-facing polymethylmethacrylate disk spinning in quiescent air. These spiral flames, though short-lived, exhibited many similarities to patterns commonly found in quiescent excitable media including compound tip meandering motion. Flame disks that grow or shrink with time depending on the rotational speed and in-depth heat loss history of the fuel disk have also been reported. One of the limitations of studying flame patterns with solid fuels is that steady-state conditions cannot be achieved in air at normal atmospheric pressure for experimentally reasonable fuel thickness. As a means to reproduce the flame patterns observed earlier with solid fuels, but under steady-state conditions, we have designed and built a rotating, porous-disk burner through which gaseous fuels can be injected and burned as diffusion flames. The rotating porous disk generates a flow of air toward the disk by a viscous pumping action, generating what is called the von K rm n boundary layer which is of constant thickness over the entire burner disk. In this note we present a map of the various dynamic flame patterns observed during the combustion of methane in air as a function of fuel flow rate and the burner rotational speed.

Nayagam, Vedha↗

Edge-Flames in Von Karman Swirling Flows

Classical understanding of diffusion flames dictates that they, unlike the premixed flames, do not possess a characteristic propagation velocity and are constrained by stoichiometric requirements at the flame surface. However, it has been commonly observed that when local extinction occurs within a diffusion flame sheet, the edges that are formed propagate with distinct speeds. In general, the propagation speed of these edges depend on their geometrical shape (concave, convex, or straight) among other factors. Recently, Buckmaster investigated the dynamics of straight diffusion flame edges separating burning and quenched regions using simplified one-dimensional models. He showed that these flame edges can have positive, negative, or zero velocity depending on the Damkoehler number of the equilibrium diffusion flame that support them. It was also shown that this unsteady flame-edge behavior is intrinsically linked to S-curve behavior of the diffusion flame with varying Damkoehler number. When the system Damkoehler number lies between the extinction and ignition limits, flame edges can propagate as an "ignition wave" or as a "failure wave," and for a critical Damkoehler number remain as a stationary flame-edge. We have extend Buckmaster's 1-d model to more general edge-flame configurations where the edges appear as "flame holes" or as "flame disks". These two configurations along with the straight-edge case cover the entire range of possible edge-flame geometry observable in planar diffusion-flame sheets. A generalized map of edge-flame propagation velocities as a function of the system Damkoehler number and the edge-flame radius is presented. Experimentally we show that edge flames can be created using diffusion flames embedded in von Karman boundary layers. In a von Karman boundary layer, the flow is generated by spinning a solid (fuel) disk in a quiescent ambient gas. Under normal gravity we were able to produce "flame disks" over a range of fuel-disk rotational velocities varying from 0 to 20 revolutions per second, by orienting the burning surface of the fuel disk facing downward.

Nayagam, Vedha↗

Rapid oscillations in cataclysmic variables. IX - BG Canis Minoris (=3A 0729+103)

Results of a 10-yr photometric program on BG Canis Minoris, a novalike variable distinguished by the presence of a strictly coherent 913-s wave in its light curve, are presented. Long-term ephemerides for the orbital clock, which manifests itself through a 3.2-h light modulation, and the 913-s clock, are presented. The light curve exhibits a persistent 0.3 mag modulation at a period consistent with the 3.2-h spectroscopic period. Well-distributed timings of the orbital minima prove that the period is 0.1347486 day. It is inferred from the shape of the waveform and from the existence of X-ray dips with a sharp energy dependence that the modulation is caused by absorption. Power spectra of the nightly light curves show variable secondary features, in addition to the regular 913-s pulse. The 913-s period, presumably the first harmonic of the true white-dwarf spin period, is decreasing by 0.0022 s/yr.

Patterson, Joseph↗

Radiometric observations of atmospheric attenuation at 20.6 and 31.65 GHz: The Wave Propagation Laboratory data base

The National Oceanic and Atmospheric Administration (NOAA) Wave Propagation Laboratory (WPL) presently operates five dual-channel microwave radiometers, one triple-channel microwave radiometer, and one six-channel microwave radiometer. The dual-channel radiometers operate at frequencies of 20.6 or 23.87 GHz and 31.4 or 31.65 GHz. The triple-channel radiometer operates at 20.6, 31.65, and 90.0 GHz. The six-channel radiometer operates at frequencies of 20.6, 31.65, 52.85, 53.85, 55.45, and 58.8 GHz. Recent brightness temperature measurements and attenuation values from some of the above radiometers are presented. These radiometric measurements, taken in different locations throughout the world, have given WPL a diverse set of measurements under a variety of atmospheric conditions. We propose to do a more complete attenuation analysis on these measurements in the future. In addition, a new spinning reflector was installed recently for the dual-channel radiometer at the Platteville, Colorado site. This reflector will extend our measurement capabilities during precipating conditions. Locating the three-channel and portable dual-channel radiometers at or near Greeley, Colorado to support the Advanced Communications Technology Satellite (ACTS) program is discussed.

Jacobson, Mark D.↗

Preferential heating of light ions during an ionospheric Ar(+) injection experiment

The Argon Release for Controlled Studies (ARCS) 4 sounding rocket was launched northward into high altitude from Poker Flat Research Range on February 23, 1990. The vehicle crossed geomagnetic field lines containing discrete auroral activity. An instrumented subpayload released 100-eV and 200-eV Ar(+) ion beams sequentially, in a direction largely perpendicular to both the local geomagnetic field and the subpayload spin axis. The instrumented main payload was separated along field lines from the beam emitting subpayload by a distance which increased at a steady rate of approximately 2.4 m/s. Three dimensional mass spectrometric ion observations of ambient H(+) and O(+) ions, obtained on board the main payload, are presented. Main payload electric field observations in the frequency range 0-16 kHz, are also presented. These observations are presented to demonstrate the operation of transverse ion acceleration, which was differential with respect to ion mass, primarily during 100-eV beam operations. The preferential transverse acceleration of ambient H(+) ions, as compared with ambient O(+) ions, during the second, third, fourth, and fifth 100-eV beam operations, is attributed to a resonance among the injected Ar(+) ions, beam-generated lower hybrid waves, and H(+) ions in the tail of the ambient thermal distribution. This work provides experimental support of processes predicted by previously published theory and simulations.

Pollock, C. J.↗

Search for Intermediate Mass Black Hole Binaries in the First and Second Observing Runs of the Advanced LIGO and VIRGO Network

Gravitational-wave astronomy has been firmly established with the detection of gravitational waves from the merger of ten stellar-mass binary black holes and a neutron star binary. This paper reports on the all-sky search for gravitational waves from intermediate mass black hole binaries in the first and second observing runs of the Advanced LIGO and Virgo network. The search uses three independent algorithms: two based on matched filtering of the data with waveform templates of gravitational-wave signals from compact binaries, and a third, model-independent algorithm that employs no signal model for the incoming signal. No intermediate mass black hole binary event is detected in this search. Consequently, we place upper limits on the merger rate density for a family of intermediate mass black hole binaries. In particular, we choose sources with total masses 𝑀=𝑚1+𝑚2∈[120,800] 𝑀⊙ and mass ratios 𝑞=𝑚2/𝑚1∈[0.1,1.0]. For the first time, this calculation is done using numerical relativity waveforms (which include higher modes) as models of the real emitted signal. We place a most stringent upper limit of 0.20 Gpc−3 yr−1 (in comoving units at the 90% confidence level) for equal-mass binaries with individual masses 𝑚1,2=100 𝑀⊙ and dimensionless spins 𝜒1,2=0.8 aligned with the orbital angular momentum of the binary. This improves by a factor of ∼5 that reported after Advanced LIGO’s first observing run.

B. P. Abbott↗

Preferential heating of light ions during an ionospheric AR(+) injection experiment

The ARCS 4 sounding rocket was launched northward into high altitude from Poker Flat Research Range on February 23, 1990. The vehicle crossed geomagnetic field lines containing discrete auroral activity. An instrumented subpayload released 100 ev and 200 ev Ar(+) ion beams sequentially, in a direction largely perpendicular to both the local geomagnetic field and the subpayload spin axis. The instrumented main payload was separated along field lines from the beam emitting subpayload by a distance which increased at a steady rate of approx. 2.4 m/s. Three-dimensional mass spectrometric ion observations of ambient H(+) and O(+) ions, obtained onboard the main payload, are presented. Main payload electric field observations in the frequency range 0-16 kHz, are also presented. These observations are presented to demonstrate the operation of transverse ion acceleration, that was differential with respect to ion mass, primarily during 100-ev beam operations. The preferential transverse acceleration of ambient H(+) ions, as compared with ambient O(+) ions during the second, third, fourth, and fifth 100-ev beam operations, is attributed to a resonance at the injected the beam drift velocity among the thermal H(+) ions and plasma waves generated by the injected beam and propagating at the beam drift speed. This work provides experimental support of processes predicted by previously published theory and simulations.

Pollock, C. J.↗

Candidate Electromagnetic Counterpart to the Binary Black Hole Merger Gravitational-Wave Event S190521g

We report the first plausible optical electromagnetic counterpart to a (candidate) binary black hole merger. Detected by the Zwicky Transient Facility, the electromagnetic flare is consistent with expectations for a kicked binary black hole merger in the accretion disk of an active galactic nucleus [B. McKernan, K. E. S. Ford, I. Bartoset al., Astrophys. J. Lett.884, L50 (2019)] and is unlikely [<O(0.01%))] due to intrinsic variability of this source. The lack of color evolution implies that it is not a supernova and instead is strongly suggestive of a constant temperature shock. Other false-positive events, such as microlensing ora tidal disruption event, are ruled out or constrained to be <O(0.1%). If the flare is associated withS190521g, we find plausible values of total mass M(BBH) ∼ 100 Mꙩ, kick velocity v(k) ∼ 200 km/s at θ ∼ 60° in a disk with aspect ratio H/a ∼ 0.01(i.e., disk height H at radius a) and gas density ρ ∼ 10^(−10)g/cu.cm. The merger could have occurred at a disk migration trap (a ∼ 700 r(g); r(g) ≡ GM(SMBH)/sq.c, where M(SMBH) is the mass of the active galactic nucleus supermassive black hole). The combination of parameters implies a significant spin for at least one of the black holes in S190521g. The timing of our spectroscopy prevents useful constraints on broad-line asymmetry due to an off-center flare. We predict a repeat flare in this source due to a reencountering with the disk in ∼ 1.6 yr(M(SMBH)/10^(8) Mꙩ)(a/10^(3)r(g))^(3/2).

M. J. Graham↗