Equatorial Orbit Research Articles

Recent Impact of COSMIC-2 with Improved Radio Occultation Data Assimilation Algorithms

Abstract A Constellation Observing System for Meteorology, Ionosphere and Climate (COSMIC) follow-on constellation, COSMIC-2, was successfully launched into equatorial orbit on 24 June 2019. With an increased signal-to-noise ratio due to improved receivers and digital beam-steering antennas, COSMIC-2 is producing about 5000 high-quality radio occultation (RO) profiles daily over the tropics and subtropics. The initial evaluation of the impact of assimilating COSMIC-2 into NOAA’s Global Forecast System (GFS) showed mixed results, and adjustments to quality control procedures and observation error characteristics had to be made prior to the assimilation of this dataset in the operational configuration in May 2020. Additional changes in the GFS that followed this initial operational implementation resulted in a larger percentage of rejection (∼90%) of all RO observations, including COSMIC-2, in the mid- to lower troposphere. Since then, two software upgrades directly related to the assimilation of RO bending angle observations were developed. These improvements were aimed at optimizing the utilization of COSMIC-2 and other RO observations to improve global weather analyses and forecasts. The first upgrade was implemented operationally in September 2021 and the second one in November 2022. This study describes both RO software upgrades and evaluates the impact of COSMIC-2 with this most recently improved configuration. Specifically, we show that the assimilation of COSMIC-2 observations has a significant impact in improving temperature and winds in the tropics, though benefits also extend to the extratropical latitudes.

Connecting the Equinoctial Elements and Rodrigues Parameters: A New Set of Elements

A geometric interpretation of the equinoctial elements is given with a connection to orthogonal rotations and attitude dynamics in Euclidean 3-space. An identification is made between the equinoctial elements and classic Rodrigues parameters. A new set of equinoctial elements is developed using the modified Rodrigues parameters, thereby removing the coordinate singularity for retrograde equatorial orbits present in previous versions of these elements. A low-thrust trajectory optimization problem is set up using the new elements to numerically verify convergence for the two-point boundary problem, as compared to their predecessors.

A Method for Measuring Gravitational Potential of Satellite’s Orbit Using Frequency Signal Transfer Technique between Satellites

We introduce an approach for the direct measurement of the gravitational potential (GP) along the trajectory of a satellite, with a specific focus on Low-Earth Orbit (LEO) satellites. A LEO satellite communicates with several Geosynchronous Equatorial Orbit (GEO) satellites via frequency signal links. The GP difference can be measured in real-time using the gravitational frequency shift approach by equipping both LEO and GEO satellites with precise atomic clocks. Since the GP at the high orbits of the GEO satellites can be precisely determined by the present gravitational field model EGM2008, the GP along the LEO satellite’s trajectory can be determined. In this study, simulation experiments were conducted, featuring a GRACE-type satellite as the LEO satellite in communication with three equidistant GEO satellites. The results indicated that the accuracy of the GP measurements along the LEO satellite’s trajectory primarily depends on the precision of the onboard atomic clocks. Supposing optical atomic clocks attain an instability level of 1×10−17τ−1/2 (τ in seconds), we determined the GP distribution covered by the LEO satellite’s trajectories with 30-day observations. Then, we determined a gravitational field at the centimeter level based on the GP distribution. The GP data derived from the trajectory of a LEO satellite can be utilized to establish temporal gravitational fields, which have broad applications in different disciplines.

Orbital motion and quasi-periodic oscillations with periastron and Lense–Thirring precession of slowly rotating Einstein–Æther black hole

We study the orbital and oscillatory motion of test particles moving around slowly rotating first and second kinds of Einstein–Æther black holes. In relation to the black hole parameters, we find analytical solutions for the radial profiles of specific energy and specific angular momentum of the equatorial stable circular orbits. The properties of the co-rotating as well as contra-rotating innermost stable circular orbits are analyzed. We examine the radial profiles of the frequencies of latitudinal and radial harmonic oscillations as a function of the black hole mass and dimensionless coupling constants of the theory. The key features of quasi-periodic oscillations of test particles near the stable circular orbits in an equatorial plane of the black hole are discussed. We investigate the positions of resonant radii for high-frequency quasi-periodic oscillations models, namely epicyclic resonance and its variants, relativistic precession and its variants, tidal disruption, as well as warped disc models, considered in the background of slowly rotating first and second kinds of slowly rotating Einstein–Æther black holes. Furthermore, Periastron and Lense–Thirring precessions have been discussed. We demonstrate that the dimensionless coupling parameters of the theory have a strong influence on particle motion around Einstein–Æther black holes.

Distinct topological configurations of equatorial timelike circular orbit for spherically symmetric (hairy) black holes

Topology is a promising approach toward to the light ring in a generic black hole background, and equatorial timelike circular orbit in a stationary black hole background. In this paper, we consider the distinct topological configurations of the timelike circular orbits in static, spherically symmetric, and asymptotic flat black holes. By making use of the equation of motion of the massive particles, we construct a vector with its zero points exactly relating with the timelike circular orbits. Since each zero point of the vector can be endowed with a winding number, the topology of the timelike circular orbits is well established. Stable and unstable timelike circular orbits respectively have winding number +1 and -1. In particular, for given angular momentum, the topological number of the timelike circular orbits also vanishes whether they are rotating or not. Moreover, we apply the study to the Schwarzschild, scalarized Einstein-Maxwell, and dyonic black holes, which have three distinct topological configurations, representations of the radius and angular momentum relationship, with one or two pairs timelike circular orbits at most. It is shown that although the existence of scalar hair and quasi-topological term leads to richer topological configurations of the timelike circular orbits, they have no influence on the total topological number. These results indicate that the topological approach indeed provides us a novel way to understand the timelike circular orbits. Significantly, different topological configurations can share the same topology number, and hence belong to the same topological class. More information is expected to be disclosed when other different topological configurations are present.

Detecting vector charge with extreme mass ratio inspirals onto Kerr black holes

Extreme mass ratio inspirals (EMRIs) are excellent sources for space-based observatories to explore the properties of black holes and test no-hair theorems. We consider EMRIs with a charged compact object inspiralling onto a Kerr black hole in quasi-circular orbits. Using the Teukolsky and generalized Sasaki-Nakamura formalisms for the gravitational and vector perturbations about a Kerr black hole, we numerically calculate the energy fluxes for both gravitational and vector perturbations induced by a charged particle moving in equatorial circular orbits. With one-year observations of EMRIs, we apply the Fisher information matrix method to estimate the charge uncertainty detected by space-based gravitational wave detectors such as the Laser Interferometer Space Antenna, TianQin, and Taiji, and we find that it is possible to detect vector charges as small as q ∼ 0.0049. The results show that EMRIs composed of a Kerr black hole with a higher spin a and lighter mass M, and a secondary charged object with more vector charge give smaller relative error on the charge, thus constrain the charge better. The positive spin of the Kerr black hole can decrease the charge uncertainty by about one or two orders of magnitude.

Comprehensive review of the effects of ionizing radiations on the ZERODUR® glass ceramic

The low thermal expansion glass ceramic ZERODUR® has a long and successful history in spaceborne optical applications. This material is especially used where precise shape invariance is required, i.e., for the mirror substrates dimensional stability when subject to temperature gradients and transients. In space, temperature may not be the exclusive driving force impacting the form stability; influence from ionizing radiations require considerations. The real impact of ionizing radiation on ZERODUR® has become a matter of reconciling, on the one hand, in situ experience, e.g., that the secondary mirror of low Earth orbit (LEO) Hubble Space Telescope crossing the South Atlantic Anomaly or the overall optics of geosynchronous equatorial orbit (GEO) Chandra X-ray Observatory are not reporting any specific problems related to dimensional stability at the optical form level. On the other hand, finite element simulation based on early lab experiments of ZERODUR® compaction are suggesting the opposite. This debate was brought to the forefront with the SILEX mission, where radiative ageing models were significantly overestimating the deformation experimentally observed on the lab replicas and were in even stronger disagreement with the observations collected over the mission. It has been speculated that an erroneous form factor in the physical model used to derive the phenomenological compaction law was responsible for these discrepancies. Following this hypothesis, we readdressed the effect of ionizing radiation induced by γ, electron, and proton fluences on ZERODUR® compaction. For each of these, we present and discuss the irradiation source, the experimental setup, the sample design, and the measurement procedure as well as the observations. Consistent with the feedback gathered over many different space missions, we confirm that the compaction observed is significantly smaller than the estimations available in the prior literature.

Vacuum Arcs on Solar Array Surfaces

Arcs on the solar arrays appear to be the most plausible source for the electromagnetic radiation observed with the Arecibo radio telescope. In the laboratory, electrostatic discharges (arcs) on solar array coupons were studied in a simulated geosynchronous equatorial orbit environment. Arc current pulses were recorded with a high time resolution (up to 0.2 ns). Spectral properties of these pulses demonstrated no monotonic behavior but rather separated peaks within the frequency range of 100–1000 MHz. Electromagnetic radiation from these arcs was registered and recorded with a Yagi antenna 8 m distant from the arc site, and a high-speed oscilloscope. Analysis of the spectra resulted in the discovery of well-separated “lines.” Evaluation of arc current spectra and radio-frequency spectra revealed statistically significant patterns in the frequency domain. Measured radio fluxes and theoretical estimates of the power of electromagnetic radiation from arcs on spacecraft solar arrays strongly argue in favor of the arcing hypothesis for the source of this radiation. These results have enabled considerable progress to be made in understanding the physical nature of the generation of electromagnetic radiation by transient vacuum discharges.

Spatial–Temporal Fusion of 10-Min Aerosol Optical Depth Products with the GEO–LEO Satellite Joint Observations

Geosynchronous equatorial orbit (GEO) satellite-derived AOD possesses huge advantages for monitoring atmospheric aerosol with high frequency; however, the data missing existing in the satellite-derived AOD products dramatically limits this expected advantage due to cloud obscuration and aerosol retrieval algorithm. In recent years, numerous AOD fusion algorithms have been proposed, while these algorithms are mostly developed to blend daily AOD products derived from low Earth orbit (LEO) satellites and generally neglect discrepancies from different categories of products. Therefore, a spatiotemporal fusion framework based on the Bayesian maximum entropy theorem, blending GEO with LEO satellite observations and incorporating data discrepancies (GL-BME), is developed to complementarily recover the Advanced Himawari-8 Imager (AHI) AOD products over East Asia. The results show that GL-BME significantly improves the average spatial completeness of AOD from 20.3% to 67.6% with ensured reliability, and the accuracy of merged AODs nearly maintains that of original AHI AODs. Moreover, a comparison of the monthly aerosol spatial distribution between the merged and original AHI AODs is conducted to evaluate the performance and significance of GL-BME, which indicates that GL-BME could further restore the real atmospheric aerosol situation to a certain extent on the basis of dramatic spatial coverage improvement.

Modeling horizon absorption in spinning binary black holes using effective worldline theory

The mass and spin of black holes (BHs) in binary systems may change due to the infall of gravitational-wave (GW) energy down the horizons. For spinning BHs, this effect enters at 2.5 post-Newtonian (PN) order relative to the leading-order energy flux at infinity. There is currently a discrepancy in the literature in the expressions of these horizon fluxes in the test-body limit at 4PN order (relative 1.5PN order). Here, we model the horizon absorption as tidal heating in an effective worldline theory of a spinning particle equipped with tidally-induced quadrupole and octupole moments. We match the tidal response to analytic solutions of the Teukolsky equation in a scattering scenario, and obtain general formulae for the evolution of mass and spin. We then specialize to the case of aligned-spin--quasi-circular binaries, obtaining the corresponding contributions to the GW phasing through 4PN order. Importantly, we find that the number of GW cycles due to horizon fluxes with masses observed by LIGO-Virgo-KAGRA detectors is about 2-3 orders of magnitude smaller than the other contributions to the phasing at the same PN order. Furthermore, in the test-body limit, we find full agreement with results obtained earlier from BH perturbation theory, with a small mass in an equatorial circular orbit treated as a source perturbing the Kerr metric. Thus, we weigh in on one side of the previous discrepancy.

Sensitivity and Consistency Analysis of BDS-3 GEO Satellite Orbit Maneuver Detection

In this study, 22 maneuvering events of C59 and C60 geostationary equatorial orbit (GEO) satellites of the BeiDou global navigation satellite system (BDS-3) in 2021 were detected and analyzed based on a time-differenced carrier phase velocity measurement algorithm combined with a broadcast ephemeris and station data. We set different empirical thresholds to analyze the effect of threshold selection on the maneuvering detection sensitivity. The sensitivity and consistency of this algorithm for GEO satellite maneuvering detection were analyzed by combining BDS-3 with other Global navigation satellite systems (GNSS) and selecting different geographic stations. The results demonstrate that changing the threshold significantly affects the maneuvering detection of C60, with the optimal maneuvering thresholds of C59 and C60 being 0.045 and 0.02, respectively. A combination of BDS-3 and GPS with equal weights was identified to be most suitable for maneuvering detection of GEO satellites. For any specific maneuver, the difference between the maneuvering time detection of C59 by stations at different geographical locations was no more than 1 min. During the maneuvering detection of C60, the start time of maneuvering detected by African stations was 4.26–8.61 min earlier than that detected by the Chinese and Australian stations, and the end time of maneuvering detected by the African stations was consistent with that detected by the Chinese and Australian stations.

Topology of equatorial timelike circular orbits around stationary black holes

A topological approach has been successfully used to study the properties of the light ring and the null circular orbit, in a generic black hole background. However, for the equatorial timelike circular orbit, quite different from the light ring case, its radius is closely dependent of the energy and angular momentum of a test particle. This fact seems to restrict the extension of the topological treatment to the timelike circular orbit. In this paper, we confirm that the angular momentum does not affect the asymptotic behavior of the constructed vector with its zero points denoting the equatorial timelike circular orbits. As a result, a well-behaved topology to characterize the equatorial timelike circular orbits can be constructed. Our study shows that the total topological number of the timelike circular orbits vanishes for a generic black hole, which is dependent of the energy of the particle. Significantly, it reveals that if there the timelike circular orbits exist, they always come in pairs for fixed angular momentum. Meanwhile, the stable and unstable timelike circular orbits have positive or negative winding number. Of particular interest is that the marginally stable circular orbit corresponds to the bifurcation point of the zero point of the constructed vector. Moreover, we also examine the case when the particle energy acts as the control parameter. It is shown that there will be topological phase transition when the value of the particle energy is one. Below this value, the timelike circular orbits always come in pairs for fixed energy. Otherwise, we will have one more unstable timelike circular orbit. We further apply the treatment to the Kerr black hole. All the results given in a generic black hole background are exactly reproduced. These strongly indicate that our topological approach can be generalized to the equatorial timelike circular orbits.

The orbit schemes to monitor Martian dust storms: Benefits to China’s future Mars missions

<p indent="0mm">Mars is most often the focus of planetary explorations because it promises to have signs of extraterrestrial life. China’s first Mars mission, called Tianwen-1, was launched on 22 May 2021 and was successful in “orbiting, landing and roving” the planet and exploring its environment. Following this success, China planned to carry out another Mars mission, known as Tianwen-3, to collect and return samples. However, dust storms posed a significant threat. Such activity on Mars has an evident pattern of seasonal variation, frequently occurring during spring and summer in the southern hemisphere. Under certain conditions, this dust activity can develop into a global storm, lasting several months, and can strongly perturb the upper atmosphere, which is crucial for the aerocapture of probes. Even regional storms can modify local visibility, temperature, wind, and the density of the lower and middle atmosphere, which impact parachutes, navigational imaging, and other decent-entry-landing and take-off probe processes. Indeed, the tiniest grains of dust pose a threat to human soft tissues and are harmful to the surface probes’ mechanisms and filters. Solar panels, cameras, etc., are also adversely impacted by dust deposition, which peaks during the fallout of a global dust storm. Thus, forecasting dust storms is invaluable for reducing or avoiding damage to these probes. However, the physical complexity of dust storms on Mars is not fully understood, making accurate forecasts of dust storms, and particularly their onset, challenging. To aid future Mars missions such as the Tianwen-3, we propose that China establish a network to monitor Martian dust storms and atmospheric weather that is based on observations of Mars from a spacecraft constellation. This constellation consists of three spacecraft in an equatorial synchronous orbit with an altitude of about <sc>17000 km</sc> and one spacecraft in a large-elliptic polar orbit whose perigee altitude is about <sc>150 km.</sc> The constellation with “four-point integration” can form a monitoring and alert network that provides simultaneous, continuous, and global-scale observations of the Martian atmosphere and surface. As an innovative platform for space-based science, this network has many advantages for Mars exploration, because it can monitor Martian dust storms and geomorphic changes on the surface, as well as the space environment and atmospheric weather. It is also beneficial for spacecraft relaying communication from probes on the Martian surface, allowing continuous data to be uplinked. Finally, the constellation is an ideal platform for on-orbit computing, delivering computer resources to surface assets, and global navigation of Mars.

The Gas Pixel Detectors for the Imaging X-Ray Polarimetry Explorer mission

On December 9th 2021, the Imaging X-ray Polarimetry Explorer (IXPE) was launched aboard a Falcon 9 to its equatorial low Earth orbit, where it began scientific observations on January 11th 2022. Equipped with three identical telescopes – each providing simultaneous polarimetric, spatial, spectroscopic and temporal information – IXPE will measure, for the first time in the soft X-ray band, the polarization of tens of celestial objects of different classes: supernova remnants, pulsars and pulsar wind nebulae, magnetars, active galactic nuclei and accreting black holes.Here I will describe the design and construction of the innovative polarization-sensitive gaseous detectors mounted in the focal plane of the IXPE telescopes, as well as report on the first scientific results of the mission.

Orbits of Particles and Photons around Regular Rotating Black Holes and Solitons

We briefly overview the basic properties and generic behavior of circular equatorial particle orbits and light rings around regular rotating compact objects with dark energy interiors, which are described by regular metrics of the Kerr–Schild class and include rotating black holes and self-gravitating spinning solitons replacing naked singularities. These objects have an internal de Sitter vacuum disk and can have two types of dark interiors, depending on the energy conditions. The first type reduces to the de Sitter disk, the second contains a closed de Sitter surface and an S surface with the de Sitter disk as the bridge and an anisotropic phantom fluid in the regions between the S surface and the disk. In regular geometry, the potentials decrease from V(r)→∞ to their minima, which ensures the existence of the innermost stable photon and particle orbits that are essential for processes of energy extraction occurring within the ergoregions, which for the second type of interiors contain the phantom energy. The innermost orbits provide a diagnostic tool for investigation of dark interiors of de Sitter–Kerr objects. They include light rings which confine these objects and ensure the most informative observational signature for rotating black holes presented by their shadows.

Coherent radiation from a chain of charged particles on a circular orbit around a dielectric ball

We investigate the spectral and angular distribution of the electromagnetic radiation from a chain of relativistic charged particles uniformly rotating along equatorial orbit around a dielectric ball. It is shown that, for weak absorption in the ball material and under relatively mild conditions on the distribution of the particles, the radiation intensity at specific rotation frequencies is essentially stronger than the corresponding radiation for a chain circulating in free space or in a homogeneous transparent medium with the same dielectric constant as that for the ball. We determine the values of parameters of the problem for which the charges in the chain emit coherently and the radiation intensity on a given harmonic increases in proportion to the square of the number of emitting charges. We also show that relative shifts in the particles locations up to 10% do not destroy the coherence properties of the radiation. It is demonstrated that the coherence effects may also dominate in the radiation intensity for chains with non-equidistant distributions of particles. The numerical results obtained for different dielectric balls have revealed the emitted radiation to be in the GHz/THz frequency ranges. The high-power radiation from the chain is confined near the rotation plane within the angular region determined by the Cherenkov angle for the velocity of the chain image on the ball surface. In the special case of an equidistant distribution of charged particles along the orbit the results of the present paper for angle integrated frequency distribution of the radiation are in agreement with those previously obtained by our group. We argue that similar coherence effects will be present in the radiation from a chain of bunches circulating around the ball.

Robust Radiometric Normalization of the Near Equatorial Satellite Images Using Feature Extraction and Remote Sensing Analysis

Relative radiometric normalization (RRN) minimizes radiometric differences among images caused by inconsistencies of acquisition conditions rather than changes in surface. Scale invariant feature transform (SIFT) has the ability to automatically extract control points (CPs) and is commonly used for remote sensing images. However, its results are mostly inaccurate and sometimes contain incorrect matching caused by generating a small number of false CP pairs. These CP pairs have high false alarm matching. This paper presents a modified method to improve the performance of SIFT CPs matching by applying sum of absolute difference (SAD) in a different manner for the new optical satellite generation called near-equatorial orbit satellite and multi-sensor images. The proposed method, which has a significantly high rate of correct matches, improves CP matching. The data in this study were obtained from the RazakSAT satellite a new near equatorial satellite system. The proposed method involves six steps: 1) data reduction, 2) applying the SIFT to automatically extract CPs, 3) refining CPs matching by using SAD algorithm with empirical threshold, and 4) calculation of true CPs intensity values over all image’ bands, 5) preforming a linear regression model between the intensity values of CPs locate in reverence and sensed image’ bands, 6) Relative radiometric normalization conducting using regression transformation functions. Different thresholds have experimentally tested and used in conducting this study (50 and 70), by followed the proposed method, and it removed the false extracted SIFT CPs to be from 775, 1125, 883, 804, 883 and 681 false pairs to 342, 424, 547, 706, 547, and 469 corrected and matched pairs, respectively.

Spin contributions to the gravitational-waveform modes for spin-aligned binaries at the 3.5PN order

We complete the post-Newtonian (PN) prediction at the 3.5PN order for the spin contributions to the gravitational waveforms emitted by inspiraling compact binaries, in the case of quasi-circular, equatorial orbits, where both spins are aligned with the orbital angular momentum. Using results from the multipolar post-Minkowskian wave generation formalism, we extend previous works that derived the dynamics and gravitational-wave energy flux and phasing, by computing the full waveform decomposed in spin-weighted spherical harmonics. This new calculation requires the computation of multipolar moments of higher multipolar order, new quadratic-in-spin contributions to the hereditary tail terms entering at the 3.5PN order, as well as other non-linear interactions between moments. When specialized to the test-mass limit, our results are equivalent to those obtained in the literature for the waveform emitted by a test-mass in equatorial, circular orbits around a Kerr black hole. We also compute the factorized modes for use in effective-one-body waveform models, correcting the 2.5PN nonspinning and 3PN quadratic-in-spin terms in the (2,1) mode used in current models.

Examining the dust of the tailless Oort-cloud comet C/2020 T2

We report our latest analysis of the Oort-cloud comet C/2020 T2 (also named Palomar or T2) observed at 2.06 au from the Sun (phase angle of 28.°5) roughly two weeks before perihelion. It lacks a significant dust tail in scattered light, showing a strong central condensation of the coma throughout the apparition that is reminiscent of so-called Manx comets. Its spectral slope of polarized light increases and decreases in the J (1.25 μm) and H (1.65 μm) bands, respectively, resulting in an overall negative (blue) slope (−0.31 ±0.14% μm−1 ) in contrast to the red polarimetric color of active comets observed at similar geometries. The average polarization degree of T2 is 2.86±0.17% for the J and 2.75±0.16% for the H bands. Given that near-infrared wavelengths are sensitive to the intermediate-scale structure of cometary dust (i.e., dust aggregates), our light-scattering modeling of ballistic aggregates with different porosities and compositions shows that the polarimetric properties of T2 are compatible with low-porosity (~66%), absorbing dust aggregates with negligible ice contents on a scale of 10–100 μm (density of ~652 kg m−3). This is supported by the coma morphology of T2 which has a viable β (the relative importance of solar radiation pressure on dust particles) range of ≲10−4. The secular evolution of r-band activity of T2 from archival data reveals that the increase in its brightness accelerates around 2.4 au pre-perihelion, with its overall dust production rate that is about 100 times smaller than those of active Oort-cloud comets. We also found an apparent concentration of T2 and Manx comets toward ecliptic orbits. This paper underlines the heterogeneous nature of Oort-cloud comets, which may be investigated in the near future with dedicated studies of their dust characteristics.

Internal structures and circular orbits for test particles

We explore how the internal structure of a test particle affects its equatorial stable circular orbits around the Kerr black hole with or without a cosmological constant. To this end, we first explicitly write equations of motion for a test particle in the pole-dipole-quadrupole approximation specifying the quadrupole momentum tensor to a spin-induced model. Then we calculate characteristic quantities – radius, angular momentum, energy, angular velocity, and impact parameter – for the particles on the stable circular orbits. Once the pole-dipole-quadrupole approximation is taken, we find that for a particle on an innermost stable circular orbit, all characteristic quantities, except the angular velocity, become greater relative to the pole-dipole case. In contrast, for a particle on an outermost stable circular orbit, which only exists in the case of the spacetime background being asymptotically de Sitter, it is the radius that becomes smaller while all other quantities become greater.