arXiv subjects

A. R. Offringa

Publications and source records attributed to A. R. Offringa.

At least 37 records · Page 2Linked to original sources

arXiv2023

A novel radio imaging method for physical spectral index modelling

We present a new method, called "forced-spectrum fitting", for physically-based spectral modelling of radio sources during deconvolution. This improves upon current common deconvolution fitting methods, which often produce inaccurate spectra. Our method uses any pre-existing spectral index map to assign spectral indices to each model component cleaned during the multi-frequency deconvolution of WSClean, where the pre-determined spectrum is fitted. The component magnitude is evaluated by performing a modified weighted linear least-squares fit. We test this method on a simulated LOFAR-HBA observation of the 3C196 QSO and a real LOFAR-HBA observation of the 4C+55.16 FRI galaxy. We compare the results from the forced-spectrum fitting with traditional joined-channel deconvolution using polynomial fitting. Because no prior spectral information was available for 4C+55.16, we demonstrate a method for extracting spectral indices in the observed frequency band using "clustering". The models generated by the forced-spectrum fitting are used to improve the calibration of the datasets. The final residuals are comparable to existing multi-frequency deconvolution methods, but the output model agrees with the provided spectral index map, embedding correct spectral information. While forced-spectrum fitting does not solve the determination of the spectral information itself, it enables the construction of accurate multi-frequency models that can be used for wide-band calibration and subtraction.

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arXiv2023

Measurement of the Anisotropy Power Spectrum of the Radio Synchrotron Background

We present the first targeted measurement of the power spectrum of anisotropies of the radio synchrotron background, at 140 MHz where it is the overwhelmingly dominant photon background. This measurement is important for understanding the background level of radio sky brightness, which is dominated by steep-spectrum synchrotron radiation at frequencies below 0.5 GHz and has been measured to be significantly higher than that which can be produced by known classes of extragalactic sources and most models of Galactic halo emission. We determine the anisotropy power spectrum on scales ranging from 2 degrees to 0.2 arcminutes with LOFAR observations of two 18 square degree fields -- one centered on the Northern hemisphere coldest patch of radio sky where the Galactic contribution is smallest and one offset from that location by 15 degrees. We find that the anisotropy power is higher than that attributable to the distribution of point sources above 100 micro-Jy in flux. This level of radio anisotropy power indicates that if it results from point sources, those sources are likely at low fluxes and incredibly numerous, and likely clustered in a specific manner.

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arXiv2023

Apertif 1.4 GHz continuum observations of the Boötes field and their combined view with LOFAR

We present a new image of a 26.5 square degree region in the Boötes constellation obtained at 1.4 GHz using the Aperture Tile in Focus (Apertif) system on the Westerbork Synthesis Radio Telescope. We use a newly developed processing pipeline which includes direction-dependent self-calibration which provides a significant improvement of the quality of the images compared to those released as part of the Apertif first data release. For the Boötes region, we mosaic 187 Apertif images and extract a source catalog. The mosaic image has an angular resolution of 27${\times}$11.5 arcseconds and a median background noise of 40 $μ$Jy/beam. The catalog has 8994 sources and is complete down to the 0.3 mJy level. We combine the Apertif image with LOFAR images of the Boötes field at 54 and 150 MHz to study spectral properties of the sources. We find a spectral flattening towards low flux density sources. Using the spectral index limits from Apertif non-detections we derive that up to 9 percent of the sources have ultra-steep spectra with a slope steeper than -1.2. Steepening of the spectral index with increasing redshift is also seen in the data showing a different dependency for the low-frequency spectral index and the high frequency one. This can be explained by a population of sources having concave radio spectra with a turnover frequency around the LOFAR band. Additionally, we discuss cases of individual extended sources with an interesting resolved spectral structure. With the improved pipeline, we aim to continue processing data from the Apertif wide-area surveys and release the improved 1.4 GHz images of several famous fields.

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arXiv2023

Diffuse Sources, Clustering and the Excess Anisotropy of the Radio Synchrotron Background

We present the largest low frequency (120~MHz) arcminute resolution image of the radio synchrotron background (RSB) to date, and its corresponding angular power spectrum of anisotropies (APS) with angular scales ranging from $3^\circ$ to $0.3^\prime$. We show that the RSB around the North Celestial Pole has a significant excess anisotropy power at all scales over a model of unclustered point sources based on source counts of known source classes. This anisotropy excess, which does not seem attributable to the diffuse Galactic emission, could be linked to the surface brightness excess of the RSB. To better understand the information contained within the measured APS, we model the RSB varying the brightness distribution, size, and angular clustering of potential sources. We show that the observed APS could be produced by a population of faint clustered point sources only if the clustering is extreme and the size of the Gaussian clusters is $\lesssim 1'$. We also show that the observed APS could be produced by a population of faint diffuse sources with sizes $\lesssim 1'$, and this is supported by features present in our image. Both of these cases would also cause an associated surface brightness excess. These classes of sources are in a parameter space not well probed by even the deepest radio surveys to date.

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arXiv2023

An interference detection strategy for Apertif based on AOFlagger 3

Context. Apertif is a multi-beam receiver system for the Westerbork Synthesis Radio Telescope that operates at 1.1-1.5 GHz, which overlaps with various radio services, resulting in contamination of astronomical signals with radio-frequency interference (RFI). Aims. We analyze approaches to mitigate Apertif interference and design an automated detection procedure for its imaging mode. Using this approach, we present long-term RFI detection results of over 300 Apertif observations. Methods. Our approach is based on the AOFlagger detection approach. We introduce several new features, including ways to deal with ranges of invalid data (e.g. caused by shadowing) in both the SumThreshold and scale-invariant rank operator steps; pre-calibration bandpass calibration; auto-correlation flagging; and HI flagging avoidance. These methods are implemented in a new framework that uses the Lua language for scripting, which is new in AOFlagger version 3. Results. Our approach removes RFI fully automatically, and is robust and effective enough for further calibration and (continuum) imaging of these data. Analysis of 304 observations show an average of 11.1% of lost data due to RFI with a large spread. We observe 14.6% RFI in auto-correlations. Computationally, AOFlagger achieves a throughput of 370 MB/s on a single computing node. Compared to published machine learning results, the method is one to two orders of magnitude faster.

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arXiv2022

First release of Apertif imaging survey data

(Abridged) Apertif is a phased-array feed system for WSRT, providing forty instantaneous beams over 300 MHz of bandwidth. A dedicated survey program started on 1 July 2019, with the last observations taken on 28 February 2022. We describe the release of data products from the first year of survey operations, through 30 June 2020. We focus on defining quality control metrics for the processed data products. The Apertif imaging pipeline, Apercal, automatically produces non-primary beam corrected continuum images, polarization images and cubes, and uncleaned spectral line and dirty beam cubes for each beam of an Apertif imaging observation. For this release, processed data products are considered on a beam-by-beam basis within an observation. We validate the continuum images by using metrics that identify deviations from Gaussian noise in the residual images. If the continuum image passes validation, we release all processed data products for a given beam. We apply further validation to the polarization and line data products. We release all raw observational data from the first year of survey observations, for a total of 221 observations of 160 independent target fields, covering approximately one thousand square degrees of sky. Images and cubes are released on a per beam basis, and 3374 beams are released. The median noise in the continuum images is 41.4 uJy/bm, with a slightly lower median noise of 36.9 uJy/bm in the Stokes V polarization image. The median angular resolution is 11.6"/sin(Dec). The median noise for all line cubes, with a spectral resolution of 36.6 kHz, is 1.6 mJy/bm, corresponding to a 3-sigma HI column density sensitivity of 1.8 x 10^20 atoms cm^-2 over 20 km/s (for a median angular resolution of 24" x 15"). We also provide primary beam images for each individual Apertif compound beam. The data are made accessible using a Virtual Observatory interface.

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arXiv2022

Continuum source catalog for the first APERTIF data release

The first data release of Apertif survey contains 3074 radio continuum images covering a thousand square degrees of the sky. The observations were performed during August 2019 to July 2020. The continuum images were produced at a central frequency 1355 MHz with the bandwidth of $\sim$150 MHz and angular resolution reaching 10". In this work we introduce and apply a new method to obtain a primary beam model using a machine learning approach, Gaussian process regression. The primary beam models obtained with this method are published along with the data products for the first Apertif data release. We apply the method to the continuum images, mosaic them and extract the source catalog. The catalog contains 249672 radio sources many of which are detected for the first time at these frequencies. We cross-match the coordinates with the NVSS, LOFAR/DR1/value-added and LOFAR/DR2 catalogs resulting in 44523, 22825 and 152824 common sources respectively. The first sample provides a unique opportunity to detect long term transient sources which have significantly changed their flux density for the last 25 years. The second and the third ones combined together provide information about spectral properties of the sources as well as the redshift estimates.

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arXiv2022

Degree-Scale Galactic Radio Emission at 122 MHz around the North Celestial Pole with LOFAR-AARTFAAC

Aims: Contamination from bright diffuse Galactic thermal and non-thermal radio emission poses crucial challenges in experiments aiming to measure the 21-cm signal of neutral hydrogen from the Cosmic Dawn and Epoch of Reionization. If not included in calibration, this diffuse emission can severely impact the analysis and signal extraction in 21-cm experiments. We examine large-scale diffuse Galactic emission at 122~MHz, around the North Celestial Pole, using the AARTFAAC-HBA system. Methods: In this pilot project, we present the first-ever wide-field image produced with a single sub-band of the data recorded with the AARTFAAC-HBA. We demonstrate two methods: multiscale CLEAN and shapelet decomposition, to model the diffuse emission revealed in the image. We use angular power spectrum metrics to quantify different components of the emission and compare the performance of the two diffuse structure modelling approaches. Results: We observe that the point sources dominate the angular power spectrum ($\ell(\ell+1)C_{\ell}/2π\equiv Δ^2(\ell)$) of the emission in the field on scales $\ell\gtrsim 60$ ($\lesssim 3$~degree). The angular power spectrum after subtraction of compact sources is flat within $20\lesssim \ell \lesssim200$ range, suggesting that the residual power is dominated by the diffuse emission on scales $\ell\lesssim200$. The residual diffuse emission has a brightness temperature variance of $Δ^2_{\ell=180} = (145.64 \pm 13.61)~{\rm K}^2$ at 122~MHz on angular scales of 1~degree, and is consistent with a power-law following $C_{\ell}\propto \ell^{-2.0}$ in $20\lesssim \ell \lesssim200$ range. We also find that, in the current setup, the multiscale CLEAN is suitable to model the compact and diffuse structures on a wide range of angular scales, whereas the shapelet decomposition method better models the large scales, which are of the order of a few degrees and wider.

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arXiv2022

Statistical analysis of the causes of excess variance in the 21 cm signal power spectra obtained with the Low-Frequency Array

The detection of the 21 cm signal of neutral hydrogen from the Epoch of Reionization (EoR) is challenging due to bright foreground sources, radio frequency interference (RFI), the ionosphere, and instrumental effects. Even after correcting for these effects in the calibration step and applying foreground removal techniques, the remaining residuals in the observed 21 cm power spectra are still above the thermal noise, which is referred to as the "excess variance." We study potential causes of this excess variance based on 13 nights of data obtained with the Low-Frequency Array (LOFAR). We focused on the impact of gain errors, the sky model, and ionospheric effects on the excess variance by correlating the relevant parameters such as the gain variance over time or frequency, local sidereal time (LST), diffractive scale, and phase structure-function slope with the level of excess variance. Our analysis shows that excess variance has an LST dependence, which is related to the power from the sky. And the simulated Stokes I power spectra from bright sources and the excess variance show a similar progression over LST with the minimum power appearing at LST bin 6h to 9h. This LST dependence is also present in sky images of the residual Stokes I of the observations. In very-wide sky images, we demonstrate that the extra power comes exactly from the direction of bright and distant sources Cassiopeia A and Cygnus A with the array beam patterns. These results suggest that the level of excess variance in the 21 cm signal power spectra is related to sky effects and, hence, it depends on LST. In particular, very bright and distant sources such as Cassiopeia A and Cygnus A can dominate the effect. This is in line with earlier studies and offers a path forward toward a solution since the correlation between the sky-related effects and the excess variance is non-negligible.

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arXiv2022

Deep sub-arcsecond widefield imaging of the Lockman Hole field at 144 MHz

High quality low-frequency radio surveys have the promise of advancing our understanding of many important topics in astrophysics, including the life cycle of active galactic nuclei (AGN), particle acceleration processes in jets, the history of star formation, and exoplanet magnetospheres. Currently leading low-frequency surveys reach an angular resolution of a few arcseconds. However, this resolution is not yet sufficient to study the more compact and distant sources in detail. Sub-arcsecond resolution is therefore the next milestone in advancing these fields. The biggest challenge at low radio frequencies is the ionosphere. If not adequately corrected for, ionospheric seeing blurs the images to arcsecond or even arcminute scales. Additionally, the required image size to map the degree-scale field of view of low-frequency radio telescopes at this resolution is far greater than what typical soft- and hardware is currently capable of handling. Here we present for the first time (to the best of our knowledge) widefield sub-arcsecond imaging at low radio frequencies. We derive ionospheric corrections in a few dozen individual directions and apply those during imaging efficiently using a recently developed imaging algorithm (arXiv:1407.1943, arXiv:1909.07226). We demonstrate our method by applying it to an eight hour observation of the International LOw Frequency ARray (LOFAR) Telescope (ILT) (arXiv:1305.3550). Doing so we have made a sensitive $7.4\ \mathrm{deg}^2$ $144\ \mathrm{MHz}$ map at a resolution of $0.3''$ reaching $25\ μ\mathrm{Jy\ beam}^{-1}$ near the phase centre. The estimated $250,000$ core hours used to produce this image, fit comfortably in the budget of available computing facilities. This result will enable future mapping of the entire northern low-frequency sky at sub-arcsecond resolution.

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arXiv2021

Apercal -- The Apertif Calibration Pipeline

Apertif (APERture Tile In Focus) is one of the Square Kilometre Array (SKA) pathfinder facilities. The Apertif project is an upgrade to the 50-year-old Westerbork Synthesis Radio Telescope (WSRT) using phased-array feed technology. The new receivers create 40 individual beams on the sky, achieving an instantaneous sky coverage of 6.5 square degrees. The primary goal of the Apertif Imaging Survey is to perform a wide survey of 3500 square degrees (AWES) and a medium deep survey of 350 square degrees (AMES) of neutral atomic hydrogen (up to a redshift of 0.26), radio continuum emission and polarisation. Each survey pointing yields 4.6 TB of correlated data. The goal of Apercal is to process this data and fully automatically generate science ready data products for the astronomical community while keeping up with the survey observations. We make use of common astronomical software packages in combination with Python based routines and parallelisation. We use an object oriented module-based approach to ensure easy adaptation of the pipeline. A Jupyter notebook based framework allows user interaction and execution of individual modules as well as a full automatic processing of a complete survey observation. If nothing interrupts processing, we are able to reduce a single pointing survey observation on our five node cluster with 24 physical cores and 256 GB of memory each within 24h keeping up with the speed of the surveys. The quality of the generated images is sufficient for scientific usage for 44 % of the recorded data products with single images reaching dynamic ranges of several thousands. Future improvements will increase this percentage to over 80 %. Our design allowed development of the pipeline in parallel to the commissioning of the Apertif system.

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arXiv2021

Multi-scale feedback and feeding in the closest radio galaxy Centaurus A

Supermassive black holes and supernovae explosions at the centres of active galaxies power cycles of outflowing and inflowing gas that affect galactic evolution and the overall structure of the Universe. While simulations and observations show that this must be the case, the range of physical scales (over ten orders of magnitude) and paucity of available tracers, make both the simulation and observation of these effects difficult. By serendipity, there lies an active galaxy, Centaurus A (NGC 5128), at such a close proximity as to allow its observation over this entire range of scales and across the entire electromagnetic spectrum. In the radio band, however, details on scales of 10-100 kpc from the supermassive black hole have so far been obscured by instrumental limitations. Here we report low-frequency radio observations that overcome these limitations and show evidence for a broad, bipolar outflow with velocity 1100 km per s and mass outflow rate of 2.9 solar masses per year on these scales. We combine our data with the plethora of multi-scale, multi-wavelength historical observations of Centaurus A to probe a unified view of feeding and feedback, which we show to be consistent with the Chaotic Cold Accretion self-regulation scenario.

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arXiv2021

A numerical study of 21-cm signal suppression and noise increase in direction-dependent calibration of LOFAR data

We investigate systematic effects in direction dependent gain calibration in the context of the Low-Frequency Array (LOFAR) 21-cm Epoch of Reionization (EoR) experiment. The LOFAR EoR Key Science Project aims to detect the 21-cm signal of neutral hydrogen on interferometric baselines of $50-250 λ$. We show that suppression of faint signals can effectively be avoided by calibrating these short baselines using only the longer baselines. However, this approach causes an excess variance on the short baselines due to small gain errors induced by overfitting during calibration. We apply a regularised expectation-maximisation algorithm with consensus optimisation (sagecal-co) to real data with simulated signals to show that overfitting can be largely mitigated by penalising spectrally non-smooth gain solutions during calibration. This reduces the excess power with about a factor 4 in the simulations. Our results agree with earlier theoretical analysis of this bias-variance trade off and support the gain-calibration approach to the LOFAR 21-cm signal data.

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arXiv2021

Apertif, Phased Array Feeds for the Westerbork Synthesis Radio Telescope

We describe the APERture Tile In Focus (Apertif) system, a phased array feed (PAF) upgrade of the Westerbork Synthesis Radio Telescope which has transformed this telescope into a high-sensitivity, wide field-of-view L-band imaging and transient survey instrument. Using novel PAF technology, up to 40 partially overlapping beams can be formed on the sky simultaneously, significantly increasing the survey speed of the telescope. With this upgraded instrument, an imaging survey covering an area of 2300 deg2 is being performed which will deliver both continuum and spectral line data sets, of which the first data has been publicly released. In addition, a time domain transient and pulsar survey covering 15,000 deg2 is in progress. An overview of the Apertif science drivers, hardware and software of the upgraded telescope is presented, along with its key performance characteristics.

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arXiv2021

The GLEAM 200 MHz Local Radio Luminosity Function for AGN and Star-forming Galaxies

The GaLactic and Extragalactic All-sky Murchison Widefield Array (GLEAM) is a radio continuum survey at 76-227 MHz of the entire southern sky (Declination $<+30°$) with an angular resolution of $\approx 2$ arcmin. In this paper, we combine GLEAM data with optical spectroscopy from the 6dF Galaxy Survey to construct a sample of 1,590 local (median $z \approx 0.064$) radio sources with $S_{200\,\mathrm{MHz}} > 55$ mJy across an area of $\approx 16,700~\mathrm{deg}^{2}$. From the optical spectra, we identify the dominant physical process responsible for the radio emission from each galaxy: 73 per cent are fuelled by an active galactic nucleus (AGN) and 27 per cent by star formation. We present the local radio luminosity function for AGN and star-forming galaxies at 200 MHz and characterise the typical radio spectra of these two populations between 76 MHz and $\sim 1$ GHz. For the AGN, the median spectral index between 200 MHz and $\sim 1$ GHz, $α_{\mathrm{high}}$, is $-0.600 \pm 0.010$ (where $S \propto ν^α$) and the median spectral index within the GLEAM band, $α_{\mathrm{low}}$, is $-0.704 \pm 0.011$. For the star-forming galaxies, the median value of $α_{\mathrm{high}}$ is $-0.650 \pm 0.010$ and the median value of $α_{\mathrm{low}}$ is $-0.596 \pm 0.015$. Among the AGN population, flat-spectrum sources are more common at lower radio luminosity, suggesting the existence of a significant population of weak radio AGN that remain core-dominated even at low frequencies. However, around 4 per cent of local radio AGN have ultra-steep radio spectra at low frequencies ($α_{\mathrm{low}} < -1.2$). These ultra-steep-spectrum sources span a wide range in radio luminosity, and further work is needed to clarify their nature.

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arXiv2021

Diffuse galaxy cluster emission at 168 MHz within the Murchison Widefield Array Epoch of Reionization 0-hour field

We detect and characterise extended, diffuse radio emission from galaxy clusters at 168 MHz within the Epoch of Reionization 0-hour field: a $45^\circ \times 45^\circ$ region of the southern sky centred on R.~A.${}= 0^\circ$, decl.${}=-27^\circ$. We detect 29 sources of interest; a newly detected halo in Abell 0141; a newly detected relic in Abell 2751; 4 new halo candidates and a further 4 new relic candidates; and a new phoenix candidate in Abell 2556. Additionally, we find 9 clusters with unclassifiable, diffuse steep-spectrum emission as well as a candidate double relic system associated with RXC J2351.0-1934. We present measured source properties such as their integrated flux densities, spectral indices ($α$, where $S_ν\propto ν^α$), and sizes where possible. We find several of the diffuse sources to have ultra-steep spectra including the halo in Abell 0141, if confirmed, showing $α\leq -2.1 \pm 0.1$ with the present data making it one of the steepest-spectrum haloes known. Finally, we compare our sample of haloes with previously detected haloes and revisit established scaling relations of the radio halo power ($P_{1.4}$) with the cluster X-ray luminosity ($L_{\mathrm{X}}$) and mass ($M_{500}$). We find that the newly detected haloes and candidate haloes are consistent with the $P_{1.4}$-$L_{\mathrm{X}}$ and $P_{1.4}$-$M_{500}$ relations, and see an increase in scatter in the previously found relations with increasing sample size likely caused by inhomogeneous determination of $P_{1.4}$ across the full halo sample. We show that the MWA is capable of detecting haloes and relics within most of the galaxy clusters within the Planck catalogue of Sunyaev-Zel'dovich sources depending on exact halo or relic properties.

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arXiv2020

The AARTFAAC Cosmic Explorer: observations of the 21-cm power spectrum in the EDGES absorption trough

The 21-cm absorption feature reported by the EDGES collaboration is several times stronger than that predicted by traditional astrophysical models. If genuine, a deeper absorption may lead to stronger fluctuations on the 21-cm signal on degree scales (up to 1~Kelvin in rms), allowing these fluctuations to be detectable in nearly 50~times shorter integration times compared to previous predictions. We commenced the "AARTFAAC Cosmic Explorer" (ACE) program, that employs the AARTFAAC wide-field imager, to measure or set limits on the power spectrum of the 21-cm fluctuations in the redshift range $z = 17.9-18.6$ ($Δν= 72.36-75.09$~MHz) corresponding to the deep part of the EDGES absorption feature. Here, we present first results from two LST bins: 23.5-23.75h and 23.5-23.75h, each with 2~h of data, recorded in `semi drift-scan' mode. We demonstrate the application of the new ACE data-processing pipeline (adapted from the LOFAR-EoR pipeline) on the AARTFAAC data. We observe that noise estimates from the channel and time-differenced Stokes~$V$ visibilities agree with each other. After 2~h of integration and subtraction of bright foregrounds, we obtain $2σ$ upper limits on the 21-cm power spectrum of $Δ_{21}^2 < (8139~\textrm{mK})^2$ and $Δ_{21}^2 < (8549~\textrm{mK})^2$ at $k = 0.144~h\,\textrm{cMpc}^{-1}$ for the two LST bins. Incoherently averaging the noise bias-corrected power spectra for the two LST bins yields an upper limit of $Δ_{21}^2 < (7388~\textrm{mK})^2$ at $k = 0.144~h\,\textrm{cMpc}^{-1}$. These are the deepest upper limits thus far at these redshifts.

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arXiv2020

Extreme intra-hour variability of the radio source J1402+5347 discovered with Apertif

The propagation of radio waves from distant compact radio sources through turbulent interstellar plasma in our Galaxy causes these sources to twinkle, a phenomenon called interstellar scintillation. Such scintillations are a unique probe of the micro-arcsecond structure of radio sources as well as of the sub-AU-scale structure of the Galactic interstellar medium. Weak scintillations (i.e. an intensity modulation of a few percent) on timescales of a few days or longer are commonly seen at centimetre wavelengths and are thought to result from the line-of-sight integrated turbulence in the interstellar plasma of the Milky Way. So far, only three sources were known that show more extreme variations, with modulations at the level of some dozen percent on timescales shorter than an hour. This requires propagation through nearby (d <~10 pc) anomalously dense (n_e ~10^2 cm^-3) plasma clouds. Here we report the discovery with Apertif of a source (J1402+5347) showing extreme (~50%) and rapid variations on a timescale of just 6.5 minutes in the decimetre band (1.4 GHz). The spatial scintillation pattern is highly anisotropic, with a semi-minor axis of about 20,000 km. The canonical theory of refractive scintillation constrains the scattering plasma to be within the Oort cloud. The sightline to J1402+5347, however, passes unusually close to the B3 star Alkaid (eta UMa) at a distance of 32 pc. If the scintillations are associated with Alkaid, then the angular size of J1402+5347 along the minor axis of the scintels must be smaller than ~10 micro arcsec yielding an apparent brightness temperature for an isotropic source of >~ 10^ 14K. }

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