arXiv subjects

Harold Weaver

Publications and source records attributed to Harold Weaver.

At least 19 recordsLinked to original sources

arXiv2022

Sublimation Origin of Active Asteroid P/2018 P3

Active asteroids show (typically transient) cometary activity, driven by a range of processes. A sub-set, sometimes called main-belt comets, may be driven by sublimation and so could be useful for tracing the present-day distribution of asteroid ice. Object P/2018 P3 has a Tisserand parameter 3.096 but a high eccentricity 0.415, placing it within the dynamical boundary between asteroids and comets. We aim to determine the cause of activity (sublimation or something else) and assess the dynamical stability of P3, in order to better constrain the intrinsic ice content in the main belt. We obtained Hubble Space Telescope images of P3 at the highest angular resolution. We compared the observations with a Monte Carlo model of dust dynamics. We identified and analyzed archival CFHT (2013) and NEOWISE (2018) data. In addition, we numerically integrated the orbits of P3 clones for 100 Myr. P3 has been recurrently active near two successive perihelia (at 1.76 AU), indicative of a sublimation origin. The absence of 4.6 um band excess indicates zero or negligible CO or CO2 gas production from P3. The properties of the ejected dust are remarkably consistent with those found in other main-belt comets (continuous emission of ~0.05-5 mm particles at 0.3-3 m/s speeds), with mass-loss rates of >~2 kg/s. The orbit of P3 is unstable on timescales ~10 Myr. We speculate that P3 has recently arrived from a more stable source (either the Kuiper Belt or elsewhere in the main belt) and has been physically aged at its current location, finally becoming indistinguishable from a weakly sublimating asteroid in terms of its dust properties. Whatever the source of P3, given the dynamical instability of its current orbit, P3 should not be used to trace the native distribution of asteroid ice.

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arXiv2022

Hubble Space Telescope Observations of Active Asteroid P/2020 O1 (Lemmon-PANSTARRS)

We present Hubble Space Telescope observations of active asteroid P/2020 O1 taken to examine its development for a year after perihelion. We find that the mass loss peaks <~1 kg/s in 2020 August and then declines to nearly zero over four months. Long-duration mass loss (~180 days) is consistent with a sublimation origin, indicating that this object is likely an ice-bearing main-belt comet. Equilibrium sublimation of water ice from an area as small as 1580 m^2 can supply the observed mass loss. Time-series photometry shows tentative evidence for extremely rapid rotation (double-peaked period < 2 hr) of the small nucleus (effective radius ~420 m). Ejection velocities of 0.1 mm particles are comparable to the 0.3 m/s gravitational escape speed from the nucleus, while larger particles are ejected at speeds less than the escape velocity. These properties are consistent with the sublimation of near-surface ice aided by centripetal acceleration. If water ice sublimation is confirmed, P/2020 O1 would be the icy asteroid with the smallest semimajor axis (highest temperature), setting new bounds on the distribution of ice in the asteroid belt.

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arXiv2021

Lucy Mission to the Trojan Asteroids: Instrumentation and Encounter Concept of Operations

The Lucy Mission accomplishes its science during a series of five flyby encounters with seven Trojan asteroid targets. This mission architecture drives a concept of operations design that maximizes science return, provides redundancy in observations where possible, features autonomous fault protection and utilizes onboard target tracking near closest approach. These design considerations reduce risk during the relatively short time-critical periods when science data is collected. The payload suite consists of a color camera and infrared imaging spectrometer, a high-resolution panchromatic imager, and a thermal infrared spectrometer. The mission design allows for concurrent observations of all instruments. Additionally, two spacecraft subsystems will also contribute to the science investigations: the Terminal Tracking Cameras will obtain wide field-of-view imaging near closest approach to determine the shape of each of the Trojan targets and the telecommunication subsystem will carry out Doppler tracking of the spacecraft to determine the mass of each of the Trojan targets.

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arXiv2021

Cometary Activity Begins at Kuiper Belt Distances: Evidence from C/2017 K2

We study the development of activity in the incoming long-period comet C/2017 K2 over the heliocentric distance range 9 < r_H < 16 AU. The comet continues to be characterized by a coma of sub-millimeter and larger particles ejected at low velocity. In a fixed co-moving volume around the nucleus we find that the scattering cross-section of the coma is related to the heliocentric distance by a power law with heliocentric index $s = 1.14\pm0.05$. This dependence is significantly weaker than the inverse square variation of the insolation as a result of two effects. These are, first, the heliocentric dependence of the dust velocity and, second, a lag effect due to very slow-moving particles ejected long before the observations were taken. A Monte Carlo model of the photometry shows that dust production beginning at r_H ~ 35 AU is needed to match the measured heliocentric index, with only a slight dependence on the particle size distribution. Mass loss rates in dust at 10 AU are of order 1000 kg/s, while loss rates in gas may be much smaller, depending on the unknown dust to gas ratio. Consequently, the ratio of the non-gravitational acceleration to the local solar gravity may, depending on the nucleus size, attain values comparable to values found in short-period comets at much smaller distances. Non-gravitational acceleration in C/2017 K2 and similarly distant comets, while presently unmeasured, may limit the accuracy with which we can infer the properties of the Oort cloud from the orbits of long-period comets.

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arXiv2020

Component properties and mutual orbit of binary main-belt comet 288P/(300163) 2006 VW139

The binary asteroid 288P/(300163) is unusual both for its combination of wide-separation and high mass ratio and for its comet-like activity. It is not currently known whether there is a causal connection between the activity and the unusual orbit or if instead the activity helped to overcome a strong detection bias against such sub-arcsecond systems. We aim to find observational constraints discriminating between possible formation scenarios and to characterise the physical properties of the system components. We measured the component separation and brightness using point spread function fitting to high-resolution Hubble Space Telescope/Wide Field Camera 3 images from 25 epochs between 2011 and 2020. We constrained component sizes and shapes from the photometry, and we fitted a Keplerian orbit to the separation as a function of time. Approximating the components A and B as prolate spheroids with semi-axis lengths a$<$b and assuming a geometric albedo of 0.07, we find $a_A \leq$ 0.6 km, $b_A \geq$ 1.4 km, $a_B \leq$ 0.5 km, and $b_B \geq$ 0.8 km. We find indications that the dust production may have concentrated around B and that the mutual orbital period may have changed by 1-2 days during the 2016 perihelion passage. Orbit solutions have semi-major axes in the range of (105-109) km, eccentricities between 0.41 and 0.51, and periods of (117.3-117.5) days pre-perihelion and (118.5-119.5) days post-perihelion, corresponding to system masses in the range of (6.67-7.23) $\times$ 10$^{12}$ kg. The mutual and heliocentric orbit planes are roughly aligned. Based on the orbit alignment, we infer that spin-up of the precursor by the YORP effect led to the formation of the binary system. We disfavour (but cannot exclude) a scenario of very recent formation where activity was directly triggered by the break-up, because our data support a scenario with a single active component.

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arXiv2020

Outburst and Splitting of Interstellar Comet 2I/Borisov

We present Hubble Space Telescope observations of a photometric outburst and splitting event in interstellar comet 2I/Borisov. The outburst, first reported with the comet outbound at 2.8 AU (Drahus et al.~2020), was caused by the expulsion of solid particles having a combined cross-section about 100 sq. km and a mass in 0.1 mm sized particles about 2e7 kg. The latter corresponds to 1e-4 of the mass of the nucleus, taken as a sphere of radius 500 m. A transient ``double nucleus'' was observed on UT 2020 March 30 (about three weeks after the outburst), having a cross-section about 0.6 sq. km and corresponding dust mass 1e5 kg. The secondary was absent in images taken on and before March 28, and in images taken on and after April 03. The unexpectedly delayed appearance and rapid disappearance of the secondary are consistent with an origin through rotational bursting of one or more large (meter-sized) boulders under the action of outgassing torques, following their ejection from the main nucleus. Overall, our observations reveal that the outburst and splitting of the nucleus are minor events involving a negligible fraction of the total mass: 2I/Borisov will survive its passage through the planetary region largely unscathed.

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arXiv2020

Coma Anisotropy and the Rotation Pole of Interstellar Comet 2I/Borisov

Hubble Space Telescope observations of interstellar comet 2I/Borisov near perihelion show the ejection of large (>~100 um) particles at <~9 m/s speeds, with estimated mass-loss rates of ~35 kg/s. The total mass loss from comet Borisov corresponds to loss of a surface shell on the nucleus only ~0.4 m thick. This shell is thin enough to be susceptible to past chemical processing in the interstellar medium by cosmic rays, meaning that the ejected materials cannot necessarily be considered as pristine. Our high-resolution images reveal persistent asymmetry in the dust coma, best explained by a thermal lag on the rotating nucleus causing peak mass loss to occur in the comet nucleus afternoon. In this interpretation, the nucleus rotates with an obliquity of 30 deg (pole direction RA = 205 deg and Dec. = 52 deg). The subsolar latitude varied from -35 deg (southern solstice) at the time of discovery to 0 deg (equinox) in 2020 January, suggesting the importance of seasonal effects. Subsequent activity likely results from regions freshly activated as the northern hemisphere is illuminated for the first time.

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arXiv2020

Hubble Space Telescope Search for Activity in High Perihelion Objects

Solar system objects with perihelia beyond the orbit of Jupiter ($q >$ 5 AU) are too cold for water ice to generate an appreciable coma via sublimation. Despite this, numerous high perihelion objects (HPOs) including many comets and recently escaped Kuiper belt objects (``Centaurs'') are observed to be active out at least to the orbit of Saturn ($q \sim$ 10 AU). Peak equilibrium temperatures at 10 AU ($\sim$125 K), while far too low to sublimate water ice, are sufficient to sublimate super-volatiles such as CO and CO$_2$ ice. Temperatures at 10 AU are also high enough to trigger the rapid crystallization of exposed amorphous ice, thus constituting another possible driver of distant activity. While supervolatile ices can sublimate strongly (as $r_H^{-2}$) to at least Kuiper belt (30 AU) distances, crystallization is an exponential function of temperature that cannot be sustained much beyond $\sim$10 AU. The heliocentric dependence of the activity thus suggests an observational test. If activity in high perihelion objects is triggered by crystallization, then no examples of activity should be found with perihelia $q >>$ 10 AU. If, on the other hand, activity is due to free sublimation of exposed supervolatile ices, or another cause, then distant activity might be detected. We obtained sensitive, high resolution Hubble Space Telescope observations of HPOs to search for activity beyond the crystallization zone. No examples of activity were detected in 53 objects with $q >$ 15 AU, consistent with the crystallization trigger hypothesis. However, sensitivity limits are such that we cannot reject the alternative hypothesis that mass loss is driven by the sublimation of supervolatile ices. We also searched for binary companions in our sample, finding none and setting an empirical 3$σ$ limit to the binary fraction of $<8$\%.

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arXiv2020

The Nucleus of Interstellar Comet 2I/Borisov

We present high resolution imaging observations of interstellar comet 2I/Borisov (formerly C/2019 Q4) obtained using the Hubble Space Telescope. Scattering from the comet is dominated by a coma of large particles (characteristic size 0.1 mm) ejected anisotropically. Convolution modeling of the coma surface brightness profile sets a robust limit to the spherical-equivalent nucleus radius r_n < 0.5 km (geometric albedo 0.04 assumed). We obtain an independent constraint based on the non-gravitational acceleration of the nucleus, finding r_n > 0.2 km (nucleus density 500 kg/m3 assumed). The profile and the non-gravitational constraints cannot be simultaneously satisfied if density < 25 kg/m3; the nucleus of comet Borisov cannot be a low density fractal assemblage of the type proposed elsewhere for the nucleus of 1I/'Oumuamua. We show that the spin-up timescale to outgassing torques, even at the measured low production rates, is comparable to or shorter than the residence time in the Sun's water sublimation zone. The spin angular momentum of the nucleus should be changed significantly during the current solar fly-by. Lastly, we find that the differential interstellar size distribution in the 0.5 mm to 100 m size range can be represented by power laws with indices < 4 and that interstellar bodies of 100 m size scale strike Earth every one to two hundred million years.

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arXiv2019

Distribution and Energy Balance of Pluto's Nitrogen Ice, as seen by New Horizons in 2015

Pluto's surface is geologically complex because of volatile ices that are mobile on seasonal and longer time scales. Here we analyzed New Horizons LEISA spectral data to globally map the nitrogen ice, including nitrogen with methane diluted in it. Our goal was to learn about the seasonal processes influencing ice redistribution, to calculate the globally averaged energy balance, and to place a lower limit on Pluto's N2 inventory. We present the average latitudinal distribution of nitrogen and investigate the relationship between its distribution and topography on Pluto by using maps that include the shifted bands of methane in solid solution with nitrogen to more completely map the distribution of the nitrogen ice. We find that the global average bolometric albedo is 0.83 +\- 0.11, similar to that inferred for Triton, and that a significant fraction of Pluto's N2 is stored in Sputnik Planitia. Under the assumption that Pluto's nitrogen-dominated 11.5 microbar atmosphere is in vapor pressure equilibrium with the nitrogen ice, the ice temperature is 36.93 +/- 0.10 K, as measured by New Horizons. Combined with our global energy balance calculation, this implies that the average bolometric emissivity of Pluto's nitrogen ice is probably in the range 0.47 - 0.72. This is consistent with the low emissivities estimated for Triton based on Voyager, and may have implications for Pluto's atmospheric seasonal variations, as discussed below. The global pattern of volatile transport at the time of the encounter was from north to south, and the transition between condensation and sublimation within Sputnik Planitia is correlated with changes in the grain size and CH4 concentration derived from the spectral maps. The low emissivity of Pluto's N2 ice suggests that Pluto's atmosphere may undergo an extended period of constant pressure even as Pluto recedes from the Sun in its orbit.

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arXiv2019

The Color and Binarity of (486958) 2014 MU69 and Other Long-Range New Horizons Kuiper Belt Targets

The Hubble Space Telescope (HST) measured the colors of eight Kuiper Belt Objects (KBOs) that will be observed by the New Horizons spacecraft including its 2019 close fly-by target the Cold Classical KBO (486958) 2014 MU69. We find that the photometric colors of all eight objects are red, typical of the Cold Classical dynamical population within which most reside. Because 2014 MU69 has a similar color to that of other KBOs in the Cold Classical region of the Kuiper Belt, it may be possible to use the upcoming high-resolution New Horizons observations of 2014 MU69 to draw conclusions about the greater Cold Classical population. Additionally, HST found none of these KBOs to be binary within separations of ~0.06 arcsec (~2000 km at 44 AU range) and Δm less than or equal to 0.5. This conclusion is consistent with the lower fraction of binaries found at relatively wide separations. A few objects appear to have significant photometric variability, but our observations are not of sufficient signal-to-noise or time duration for further interpretation.

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arXiv2018

Active Asteroid P/2017 S5 (ATLAS)

Observations of active asteroid P/2017 S5 when near perihelion reveal the ejection of large (0.1 to 10 mm) particles at 0.2 to 2 m/s speeds, with estimated mass-loss rates of a few kg/s. The protracted nature of the mass loss (continuous over 150 days) is compatible with a sublimation origin, meaning that this object is likely an ice-bearing main-belt comet. Equilibrium sublimation of exposed water ice covering as little as 0.1 sq. km can match the data. Observations a year after perihelion show the object in an inactive state from which we deduce a nucleus effective radius 450(+100/-60) m (albedo 0.06+/-0.02 assumed). The gravitational escape speed from a body of this size is just 0.3 m/s, comparable to the inferred ejection speed of the dust. Time-series photometry provides tentative evidence for rapid rotation (lightcurve period 1.4 hour) that may also play a role in the loss of mass and which, if real, is a likely consequence of spin-up by sublimation torques. P/2017 S5 shares both physical and orbital similarities with the split active asteroid pair P/2016 J1-A and J1-B, and all three objects are likely members of the 7 Myr old, collisionally produced, Theobalda family.

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arXiv2018

Distant Comet C/2017 K2 and the Cohesion Bottleneck

Distant long-period comet C/2017 K2 has been outside the planetary region of the solar system for 3 Myr, negating the possibility that heat retained from the previous perihelion could be responsible for its activity. This inbound comet is also too cold for water ice to sublimate and too cold for amorphous water ice, if present, to crystallize. C/2017 K2 thus presents an ideal target in which to investigate the mechanisms responsible for activity in distant comets. We have used Hubble Space Telescope to study the comet in the pre-perihelion distance range 13.8 to 15.9 AU. The coma maintains a logarithmic surface brightness gradient $m = -1.010\pm$0.004, consistent with steady-state mass loss. The absence of a radiation pressure swept tail indicates that the effective particle size is large (0.1 mm) and the mass loss rate is $\sim$200 kg s$^{-1}$, remarkable for a comet still beyond the orbit of Saturn. Extrapolation of the photometry indicates that activity began in 2012.1, at 25.9$\pm$0.9 AU, where the blackbody temperature is only 55 K. This large distance and low temperature suggest that cometary activity is driven by the sublimation of a super-volatile ice (e.g.~CO), presumably preserved by K2's long-term residence in the Oort cloud. The mass loss rate can be sustained by CO sublimation from an area $\lesssim 2$ km$^2$, if located near the hot sub-solar point on the nucleus. However, while the drag force from sublimated CO is sufficient to lift millimeter sized particles against the gravity of the cometary nucleus, it is 10$^2$ to 10$^3$ times too small to eject these particles against inter-particle cohesion. Our observations thus require either a new understanding of the physics of inter-particle cohesion or the introduction of another mechanism to drive distant cometary mass loss. We suggest thermal fracture and electrostatic supercharging in this context.

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arXiv2018

The Nucleus of Active Asteroid 311P/(2013 P5) PANSTARRS

The unique inner-belt asteroid 311P/PANSTARRS (formerly P/2013 P5) is notable for its sporadic, comet-like ejection of dust in nine distinct epochs spread over $\sim$250 days in 2013. This curious behavior has been interpreted as the product of localized, equator-ward landsliding from the surface of an asteroid rotating at the brink of instability. We obtained new Hubble Space Telescope observations to directly measure the nucleus and to search for evidence of its rapid rotation. However, instead of providing photometric evidence for rapid nucleus rotation, our data set a lower limit to the lightcurve period, $P \ge$ 5.4 hour. The dominant feature of the lightcurve is a V-shaped minimum, $\sim$0.3 magnitudes deep, that is suggestive of an eclipsing binary. Under this interpretation, the time-series data are consistent with a secondary/primary mass ratio, $m_s/m_p \sim$ 1:6, a ratio of separation/primary radius, $r/r_p \sim$ 4 and an orbit period $\sim$0.8 days. These properties lie within the range of other asteroid binaries that are thought to be formed by rotational breakup. While the lightcurve period is long, centripetal dust ejection is still possible if one or both components rotates rapidly ($\lesssim$ 2 hour) and has a small lightcurve variation because of azimuthal symmetry. Indeed, radar observations of asteroids in critical rotation reveal "muffin-shaped" morphologies which are closely azimuthally symmetric and which show minimal lightcurves. Our data are consistent with 311P being a close binary in which one or both components rotates near the centripetal limit. The mass loss in 2013 suggests that breakup occurred recently and could even be on-going. A search for fragments that might have been recently ejected beyond the Hill sphere reveals none larger than effective radius $r_e \sim$ 10 m.

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arXiv2018

Evolution of H$_2$O Production in Comet C/2012 S1 (ISON) as Inferred from Forbidden Oxygen and OH Emission

We present H$_2$O production rates for comet C/2012 S1 (ISON) derived from observations of [OI] and OH emission during its inbound leg, covering a heliocentric distance range of 1.8-0.44 AU. Our production rates are in agreement with previous measurements using a variety of instruments and techniques and with data from the various observatories greatly differing in their projected fields of view. The consistent results across all data suggest the absence of an extended source of H$_2$O production, for example sublimation of icy grains in the coma, or a source with spatial extent confined to the dimensions of the smallest projected field of view (in this case $<$ 1,000 km). We find that ISON had an active area of around 10 km$^2$ for heliocentric distances R$_h$ > 1.2 AU, which then decreased to about half this value from R$_h$=1.2-0.9 AU. This was followed by a rapid increase in active area at about R$_h$=0.6 AU, corresponding to the first of three major outbursts ISON experienced inside of 1 AU. The combination of a detected outburst in the light curve and rapid increase in active area likely indicates a major nucleus fragmentation event. The 5-10 km$^2$ active area observed outside of R$_h$=0.6 AU is consistent with a 50-100% active fraction for the nucleus, larger than typically observed for cometary nuclei. Although the absolute value of the active area is somewhat dependent on the thermal model employed, the changes in active area observed are consistent among models. The conclusion of a 50-100+% active fraction is robust for realistic thermal models of the nucleus. However the possibility of a contribution of a spatially unresolved distribution of icy grains cannot be discounted.

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arXiv2017

A binary main belt comet

The asteroids are primitive solar system bodies which evolve both collisionally and through disruptions due to rapid rotation [1]. These processes can lead to the formation of binary asteroids [2-4] and to the release of dust [5], both directly and, in some cases, through uncovering frozen volatiles. In a sub-set of the asteroids called main-belt comets (MBCs), the sublimation of excavated volatiles causes transient comet-like activity [6-8]. Torques exerted by sublimation measurably influence the spin rates of active comets [9] and might lead to the splitting of bilobate comet nuclei [10]. The kilometer-sized main-belt asteroid 288P (300163) showed activity for several months around its perihelion 2011 [11], suspected to be sustained by the sublimation of water ice [12] and supported by rapid rotation [13], while at least one component rotates slowly with a period of 16 hours [14]. 288P is part of a young family of at least 11 asteroids that formed from a ~10km diameter precursor during a shattering collision 7.5 million years ago [15]. Here we report that 288P is a binary main-belt comet. It is different from the known asteroid binaries for its combination of wide separation, near-equal component size, high eccentricity, and comet-like activity. The observations also provide strong support for sublimation as the driver of activity in 288P and show that sublimation torques may play a significant role in binary orbit evolution.

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arXiv2017

A Comet Active Beyond the Crystallization Zone

We present observations showing in-bound long-period comet C/2017 K2 (PANSTARRS) to be active at record heliocentric distance. Nucleus temperatures are too low (60 K to 70 K) either for water ice to sublimate or for amorphous ice to crystallize, requiring another source for the observed activity. Using the Hubble Space Telescope we find a sharply-bounded, circularly symmetric dust coma 10$^5$ km in radius, with a total scattering cross section of $\sim$10$^5$ km$^2$. The coma has a logarithmic surface brightness gradient -1 over much of its surface, indicating sustained, steady-state dust production. A lack of clear evidence for the action of solar radiation pressure suggests that the dust particles are large, with a mean size $\gtrsim$ 0.1 mm. Using a coma convolution model, we find a limit to the apparent magnitude of the nucleus $V >$ 25.2 (absolute magnitude $H >$ 12.9). With assumed geometric albedo $p_V$ = 0.04, the limit to the nucleus circular equivalent radius is $<$ 9 km. Pre-discovery observations from 2013 show that the comet was also active at 23.7 AU heliocentric distance. While neither water ice sublimation nor exothermic crystallization can account for the observed distant activity, the measured properties are consistent with activity driven by sublimating supervolatile ices such as CO$_2$, CO, O$_2$ and N$_2$. Survival of supervolatiles at the nucleus surface is likely a result of the comet's recent arrival from the frigid Oort cloud.

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arXiv2017

Anatomy of an Asteroid Break-Up: The Case of P/2013 R3

We present an analysis of new and published data on P/2013 R3, the first asteroid detected while disintegrating. Thirteen discrete components are measured in the interval between UT 2013 October 01 and 2014 February 13. We determine a mean, pair-wise velocity dispersion amongst these components of $Δv = 0.33\pm0.03$ m s$^{-1}$ and find that their separation times are staggered over an interval of $\sim$5 months. Dust enveloping the system has, in the first observations, a cross-section $\sim$30 km$^2$ but fades monotonically at a rate consistent with the action of radiation pressure sweeping. The individual components exhibit comet-like morphologies and also fade except where secondary fragmentation is accompanied by the release of additional dust. We find only upper limits to the radii of any embedded solid nuclei, typically $\sim$100 to 200 m (geometric albedo 0.05 assumed). Combined, the components of P/2013 R3 would form a single spherical body with radius $\lesssim$400 m, which is our best estimate of the size of the precursor object. The observations are consistent with rotational disruption of a weak (cohesive strength $\sim$50 to 100 N m$^{-2}$) parent body, $\sim$400 m in radius. Estimated radiation (YORP) spin-up times of this parent are $\lesssim$1 Myr, shorter than the collisional lifetime. If present, water ice sublimating at as little as 10$^{-3}$ kg s$^{-1}$ could generate a torque on the parent body rivaling the YORP torque. Under conservative assumptions about the frequency of similar disruptions, the inferred asteroid debris production rate is $\gtrsim$10$^3$ kg s$^{-1}$, which is at least 4% of the rate needed to maintain the Zodiacal Cloud.

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