---
abstract: 'We present the first interferometric detections of Fast Radio Bursts (FRBs), an enigmatic new class of astrophysical transient. In a 180-day survey of the Southern sky we discovered 3 FRBs at 843 MHz with the UTMOST array, as part of commissioning science during a major ongoing upgrade. The wide field of view of UTMOST ($\approx 9$ deg$^{2}$) is well suited to FRB searches. The primary beam is covered by 352 partially overlapping fan-beams, each of which is searched for FRBs in real time with pulse widths in the range 0.655 to 42 ms, and dispersion measures $\leq$2000 pc cm$^{-3}$. Detections of FRBs with the UTMOST array places a lower limit on their distances of $\approx 10^4$ km (limit of the telescope near-field) supporting the case for an astronomical origin. Repeating FRBs at UTMOST or an FRB detected simultaneously with the Parkes radio telescope and UTMOST, would allow a few arcsec localisation, thereby providing an excellent means of identifying FRB host galaxies, if present. Up to 100 hours of follow-up for each FRB has been carried out with the UTMOST, with no repeating bursts seen. From the detected position, we present 3$\sigma$ error ellipses of 15$\arcsec \times 8.4^{\circ}$ on the sky for the point of origin for the FRBs. We estimate an all-sky FRB rate at 843 MHz above a fluence $\cal F_\mathrm{lim}$ of 11 Jy ms of $\sim 78$ events sky$^{-1}$ d$^{-1}$ at the 95 percent confidence level. The measured rate of FRBs at 843 MHz is of order two times higher than we had expected, scaling from the FRB rate at the Parkes radio telescope, assuming that FRBs have a flat spectral index and a uniform distribution in Euclidean space [@Caleb]. We examine how this can be explained by FRBs having a steeper spectral index and/or a flatter log$N$-log$\mathcal{F}$ distribution than expected for a Euclidean Universe.'
author:
- |
    M. Caleb$^{1,2,3}$[^1], C. Flynn$^{2,3}$, M. Bailes$^{2,3}$, E.D. Barr$^{2,3,4}$, T. Bateman$^{6}$, S. Bhandari$^{2,3}$, D. Campbell-Wilson$^{6,3}$ W. Farah$^{2}$, A.J. Green$^{6,3}$, R.W. Hunstead$^{6}$ A. Jameson$^{2,3}$, F. Jankowski$^{2,3}$, E.F. Keane$^{5}$, A. Parthasarathy$^{2,3}$, V. Ravi$^{2,3,7}$, P.A. Rosado$^{2,8}$, W. van Straten$^{2,9}$, V. Venkatraman Krishnan$^{2,3}$\
    \
    $^{1}$ Research School of Astronomy and Astrophysics, Australian National University, ACT, 2611, Australia\
    $^{2}$ Centre for Astrophysics and Supercomputing, Swinburne University of Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia\
    $^{3}$ ARC Centre of Excellence for All-sky Astrophysics\
    $^{4}$ Max-Planck-Institut f[ü]{}r Radioastronomie, Auf dem H[ü]{}gel 69, D-53121 Bonn, Germany\
    $^{5}$ SKA Organisation, Jodrell Bank Observatory, Cheshire, SK11 9DL, UK\
    $^{6}$ Sydney Institute for Astronomy (SIfA), School of Physics, The University of Sydney, NSW 2006, Australia\
    $^{7}$ Cahill Center for Astronomy and Astrophysics, MC249-17, California Institute of Technology, Pasadena, CA 91125, USA\
    $^{8}$ Monash Centre for Astrophysics, School of Physics and Astronomy, Monash University, VIC 3800, Australia\
    $^{9}$ Institute for Radio Astronomy & Space Research, Auckland University of Technology, Private Bag 92006, Auckland 1142, New Zealand
bibliography:
- 'UTMOST\_FRBs.bib'
title: The first interferometric detections of Fast Radio Bursts
---

instrumentation: interferometers – intergalactic medium – surveys – radio continuum – methods: data analysis

Introduction
============

Fast radio bursts (FRBs) are a relatively new class of radio transient that are short, bright and highly dispersed. The pulses are typically of durations of a few milliseconds, and exhibit dispersion sweeps characteristic of propagation through a cold diffuse plasma [@Lorimer; @Thornton]. The dispersion measures (DMs) of these pulses are significantly higher than the contribution from the line-of-sight through the Galactic Interstellar Medium (ISM), suggestive of a cosmological origin in which the large DMs are due to passage through the Intergalactic Medium (IGM). If they are at cosmological distances, their inferred intrinsic energies ($> 10^{31}$ J) and brightness temperatures (${T}_\mathrm{b} >10^{33}$ K) necessitate a coherent emission mechanism, while the short durations of the pulses suggest a very compact source of origin [@Luan; @Dennison]. The 18 FRBs published to date (refer to the FRBCAT repository$^{\ref{frbcat}}$ for the complete list) have been discovered in either post-processing of archival surveys or, in real-time, using the Parkes radio telescope with the exception of two, detected at the Arecibo [@Spitler] and Green Bank telescopes All but one of the bursts have been found at 1.4 GHz, with the exception being the GBT burst, which was seen at 800 MHz.

The observed FRB all-sky rate is very high. [@Champion] derive a rate of $7^{+5}_{-3} \times 10^{3}$ events sky$^{-1}$ d$^{-1}$ at 1.4 GHz for bursts between 0.13 and 1.5 Jy ms in fluence and widths in the range 0.128 ms to 16 ms. The high FRB rate is a major constraint on theories for their origin. Until recently, such theories have generally assumed they are cataclysmic events, in which the progenitor is obliterated. However, one FRB is now known to repeat in a non-periodic manner [FRB 121102, @nat_spitler], opening up possibilities for other progenitor models. Following the discoveries reported in this paper, [@Chatterjee] have achieved sub-arsecond localisation of the FRB 121102 using radio interferometric observations from the Very Large Array. The source has been localised to a $m_{r^\prime} = 25.1$ AB mag low-metallicity, star-forming dwarf galaxy at $z = 0.19273(8)$ [@SriHarsh]. The precise localisation shows that the source is either co-located with a 180 $\upmu$Jy active galactic nucleus or an unresolved type of extragalactic source. However, the exact nature of the FRB progenitor is still unknown.

Despite concerted follow-up efforts for almost all FRBs, this remains the only FRB seen to repeat. These efforts have been quite substantial. For instance, $\approx80$ hrs of followup for the Lorimer burst [@Lorimer], $\approx80$ hrs for FRB 131104 [@Ravi] and $\approx110$ hrs of selected FRB positions [@Petroff_followup] at the Parkes radio telescope yielded no repeats. This suggests the possibility of there being two independent classes of FRBs – repeating and non-repeating – with two classes of possible progenitors [@nat_keane]. Progenitor theories include flaring magnetars [@Lyubarsky], giant pulses from pulsars [@Wasserman; @Connor], binary white dwarf mergers [@Kashiyama], neutron star mergers [@Totani] and collapsing supramassive neutron stars [@Falcke]. It is possible that the lack of repetition of pulses for the FRB discoveries at the Parkes radio telescope is merely due to limited sensitivity and follow-up time, and that all FRBs have a common origin [@Scholz]. FRB 010724 is an exception to this however: its extreme brightness ($\sim 30$ Jy) far outweighs the lower gain of Parkes relative to Arecibo, so that one cannot infer its lack of repeat bursts is due to limited sensitivity. Recently, [@RaviSci] have reported the detection of FRB 150708, which is of comparable brightness ($\sim 12$ Jy) to FRB 010724, and exhibits 100 percent polarisation and suggests weak turbulence in the ionised IGM. [@DeLaunay] have associated a $\gamma$-ray transient with the FRB 131104 discovered by [@Ravi]. However [@Ravi2016] in contrast, report on the discovery of a variable source (consistent with an AGN) temporally and spatially coincident with the FRB 131104 but not spatially coincident with the $\gamma$-ray burst, and rule out the association of the $\gamma$-ray burst with the FRB using probabilistic reasoning.

![Frequency vs time behaviour of FRBs 160317, 160410 and 160608 detected at UTMOST at the centre frequency of 834.765 MHz. The top panel in each case shows the frequency-averaged pulse profile. The bottom panel shows that narrow-band RFI has been excised and the effects of inter-channel dispersion have been removed assuming DMs of $1165 \pm11$, $278\pm3$ and $682\pm7$ pc cm$^{-3}$ respectively. The data are uncalibrated as the bandpass of the system varies as a function of meridian angle, and the flux densities are in arbitrary units. Note the different time range on the abscissa for FRB 160410.[]{data-label="fig:frbs1"}](Fig1a){width="50.00000%"}

![Frequency vs time behaviour of FRBs 160317, 160410 and 160608 detected at UTMOST at the centre frequency of 834.765 MHz. The top panel in each case shows the frequency-averaged pulse profile. The bottom panel shows that narrow-band RFI has been excised and the effects of inter-channel dispersion have been removed assuming DMs of $1165 \pm11$, $278\pm3$ and $682\pm7$ pc cm$^{-3}$ respectively. The data are uncalibrated as the bandpass of the system varies as a function of meridian angle, and the flux densities are in arbitrary units. Note the different time range on the abscissa for FRB 160410.[]{data-label="fig:frbs1"}](Fig1b){width="50.00000%"}

![Frequency vs time behaviour of FRBs 160317, 160410 and 160608 detected at UTMOST at the centre frequency of 834.765 MHz. The top panel in each case shows the frequency-averaged pulse profile. The bottom panel shows that narrow-band RFI has been excised and the effects of inter-channel dispersion have been removed assuming DMs of $1165 \pm11$, $278\pm3$ and $682\pm7$ pc cm$^{-3}$ respectively. The data are uncalibrated as the bandpass of the system varies as a function of meridian angle, and the flux densities are in arbitrary units. Note the different time range on the abscissa for FRB 160410.[]{data-label="fig:frbs1"}](Fig1c){width="50.00000%"}

Most published FRBs have been detected with single dish antennas, with relatively poor angular resolution, and we are unable to indisputably rule out a near-field or atmospheric origin for the one-off events until now. The FRB detections made with the multi-beam receiver at the Parkes radio telescope however, are likely to originate at $\gtrsim 20$ km [@Vedantham]. Also FRB 150418 has been proposed to be associated with a galaxy at $z \sim 0.5$. However this association has been called into question by [@Williams] and [@Vedantham], and other models like giant pulses from extragalactic pulsars which could account for the excess DM in the local environment, have been proposed [@Connor]. Better localisation during discovery in the radio requires an interferometric detection.

![The three panels display the total power pulse profiles for one polarisation in three adjacent fan-beams. FRBs 160317 and 160410 were also detected as sub-threshold events in neighbouring fan-beams (in addition to the high S/Ns in the primary detection fan-beams), indicating that they did not occur near the centres of the primary fan-beam. On the contrary, FRB 160608 was only detected in one fan-beam suggesting that it occurred close to the the centre of beam 208 (see bottom panel).[]{data-label="fig:frbs2"}](Fig2a.pdf){width="50.00000%"}

![The three panels display the total power pulse profiles for one polarisation in three adjacent fan-beams. FRBs 160317 and 160410 were also detected as sub-threshold events in neighbouring fan-beams (in addition to the high S/Ns in the primary detection fan-beams), indicating that they did not occur near the centres of the primary fan-beam. On the contrary, FRB 160608 was only detected in one fan-beam suggesting that it occurred close to the the centre of beam 208 (see bottom panel).[]{data-label="fig:frbs2"}](Fig2b.pdf){width="50.00000%"}

![The three panels display the total power pulse profiles for one polarisation in three adjacent fan-beams. FRBs 160317 and 160410 were also detected as sub-threshold events in neighbouring fan-beams (in addition to the high S/Ns in the primary detection fan-beams), indicating that they did not occur near the centres of the primary fan-beam. On the contrary, FRB 160608 was only detected in one fan-beam suggesting that it occurred close to the the centre of beam 208 (see bottom panel).[]{data-label="fig:frbs2"}](Fig2c.pdf){width="50.00000%"}

In a companion paper, we describe how the Molonglo Observatory Synthesis Telescope (sited near Canberra in Australia) is currently undergoing a major upgrade, with the addition of a state-of-the-art correlator to transform it into an FRB finding machine - the UTMOST (Bailes et al., submitted). Two FRB searches were performed with UTMOST in 2015 during the upgrade, when the system was operating at a small fraction of the final expected sensitivity, and only yielded an upper limit of the FRB rate [@Caleb].

We have now undertaken a third FRB survey at UTMOST and discovered 3 FRBs. These are the first FRBs observed with an interferometer, further strengthening the case for an astronomical origin in addition to the detections at other telescopes and in the expected number of beams at Parkes for far-field events, as detection with UTMOST implies the events are in the far-field region $\gtrsim 10^4$ km. Section \[sec:OA\] of this paper briefly outlines the telescope specifications, survey properties and the transient detection pipeline. We present the bursts’ properties and their follow-up observations and localisation areas in Section \[sec:results\]. The event rate estimates of the FRBs at 843 MHz based on the detections of the 3 FRBs and constraints on their spectral index are detailed in Section \[sec:disc\] followed by our conclusions in Section \[sec:conc\].

UTMOST specifications and survey properties {#sec:OA}
===========================================

The UTMOST consists of an East-West (E-W) aligned cylindrical paraboloid divided into two ‘arms’ (separated by a 15-m gap), each 11.6-m wide and 778-m long, with 7744 right circularly polarised ring antennas operating at 843 MHz on a line feed system at its focus. Groups of 22 consecutive ring antennas (these groups are termed ‘modules’) are phased to the physical centre of the module, forming 352 unique inputs (each with a beam $4.0^{\circ} \times 2.8^{\circ}$ FWHP) which are then beamformed (Bailes at al., submitted). We operate the telescope by tilting the arms North-South and steering the ring antennae East-West by differential rotation. UTMOST can access the sky South of $\delta=+18^{\circ}$ with the East-West steering limited to $\pm60^{\circ}$. The telescope’s field of view, sensitivity and high duty cycle make it a near ideal survey instrument for finding FRBs and other radio transients. Since late 2015, we have been using UTMOST to search for fast radio transients for an average of 18 hours a day, while simultaneously timing more than 300 pulsars weekly (Bailes et al., submitted, Jankowski et al., in prep).

In FRB search mode, the 4.0 degrees FWHP of the primary beam is tiled in the E-W direction by 352 elliptical, coherent, tied-array beams (called ‘fan-beams’, each 46$\arcsec$ wide), spaced 41$\arcsec$ apart and overlapping at very close to their half power points at 843 MHz. In the N-S direction the resolution of the fan-beams is the same as that of the primary beam ($\approx2.8$ degrees). The fan-beams are numbered from 1 to 352 running from East to West across the primary beam, with fan-beam 177 directly centred on boresight. The sensitivity of the telescope to bursts can be estimated using the radiometer equation:

$$S_\mathrm{min} = \beta \, \frac {(\mathrm{S/N_\mathrm{min}}) \, T_\mathrm{sys}} {G \, {\sqrt{\Delta\nu \, W \, N_\mathrm{p}}}}$$

![image](Fig3.pdf){width="5.0"}

where ${S_\mathrm{min}}$ is the minimum detectable flux for a threshold signal-to-noise $\mathrm{S/N_{min}}$, $\beta$ is the digitisation factor, $\Delta\nu$ is the bandwidth in Hz, ${N_\mathrm{p}}$ is the number of polarisations (${N_\mathrm{p}}=1$ for UTMOST as it is right circularly polarised only), $W$ is the pulse width in ms, ${T_\mathrm{sys}}$ is the system temperatures in K respectively, and $G$ is the system gain in K Jy$^{-1}$. We define S/N as the ratio of the sum of the on-pulse flux to the product of the rms of the off-pulse flux and square root of number of on-pulse bins ($\mathrm{S/N} = \frac{\mathrm{I_{on}}}{\sqrt{\mathrm{nbin}}\,\mathrm{I_{off}}}$). For the fully upgraded instrument, we expect $S_\mathrm{min} = 1.6$ Jyms for a 10$\sigma$ 1-ms wide pulse, 3.5 K Jy$^{-1}$ gain, 100 K system temperature and 31.25 MHz bandwidth. The system bandwidth is however only about half of the initially anticipated 31.25 MHz bandwidth, as the ring antennas have a significant roll-off in sensitivity away from 843 MHz. This has been measured using integrated pulses from the pulsar J1644$-$4559. We find that on average $\sim 86$ percent of the total S/N is concentrated in the upper half of the band ($\sim836 - 850$) as the antennas are tuned to maximum sensitivity at 843 MHz. We adopt a bandwidth of 16 MHz for the sensitivity calculations in the paper, to be conservative.

During the upgrade, we characterise the system sensitivity by a fraction of the final expected gain $\upepsilon$. This factor encompasses systemic losses due to (1) pointing errors (from physical misalignment in the modules N-S, and phasing errors in the antenna system E-W), (2) self-generated radio frequency interference (RFI) mainly due to improperly shielded electronics in the receiver boxes near the telescope, (3) coherent noise in the receiver boxes, which affects some sets of adjacent modules, and other inefficiencies in the system performance that we are still characterising, such as systematic errors in the phase/delay solutions across the interferometer (Bailes et al. in submitted).

At present (October 2016), we estimate $\upepsilon \approx 0.14$, based on observations of strong calibrators of known flux densities and a number of high DM pulsars with relatively stable flux densities. This implies an effective $T_\mathrm{sys}$ of $400 \pm 100$ K. This is significantly higher than the system temperature seen on the best performing modules, which can be as low as 100 K. We note that $\upepsilon$ can vary from day to day as modules are either serviced in the field or have electronics maintenance in the workshops, and typically lie in the range $0.15 < \upepsilon <
0.20$. Occasionally, if only one arm is operational, we have the option to continue surveys at half sensitivity (i.e. $0.07 < \upepsilon < 0.10$). The telescope can access the Southern sky for $\delta < +18^{\circ}$, and for most parts of the sky we tend to observe reasonably close to the meridian, in order to maximise sensitivity. The sensitivity is reduced by projection effects away from the meridian.

![image](Fig4a.pdf){width="47.00000%"}![image](Fig4b.pdf){width="47.00000%"} ![image](Fig4c.pdf){width="47.00000%"}

\[sec:pipeline\]

In November 2015, we commenced our third FRB survey “V3.0”. It ran for a total of 159.0 days on sky (between 01-11-2015 and 30-11-2016), at $\upepsilon \approx 0.14$ of the final target telescope sensitivity. Our fluence limit of the survey, that is the fluence of the narrowest detectable pulse $\cal{F}_\mathrm{lim}$ can be parametrized as,

$$\label{eq:lim_flue}
{\cal{F}_\mathrm{lim}} \approx 11 \Bigg(\frac{W}{\mathrm{ms}}\Bigg)^{1/2}\,  \mathrm{Jy \, ms}$$

where, 11 Jy is the UTMOST flux limit for $\mathrm{S/N}=10$, $G=3.0$ K Jy$^{-1}$, $\Delta\nu=16$ MHz, $W=1$ ms, $N_\mathrm{p}=1$ and $T_\mathrm{sys}=400$ K. It should be noted that this is not the same as the fluence completeness limit $\cal{F}_\mathrm{complete}$. Between $\cal{F}_\mathrm{lim}$ and $\cal{F}_\mathrm{complete}$ we are incomplete and not all FRBs with fluences in this range are detectable. This incompleteness region corresponds to the pink shaded region in Figure \[fig:fluence\]. The two previous surveys (V1.0 and V2.0) reported in [@Caleb] yielded no FRB events. Relative to V3.0, V1.0 ran for 19.5 days at lower sensitivity ($\upepsilon = 0.07$), while V2.0 operated for 9.4 days at the same sensitivity ($\upepsilon =
0.14$). FRB survey V3.0 consists primarily of pointings taken commensally during pulsar timing observations. In this mode, the time series data from 352 fan-beams are searched for dispersed single pulses in real time, using a custom version of the heimdall software on 8 Nvidia GeForce GTX TITAN X (Maxwell) GPUs with a latency of 8-s. The resulting candidates were then processed offline, typically the following morning for overnight pulsar timing (RFI is much reduced at night, and the telescope is made available for maintenance on week days). On weekends, the telescope is usually operated continuously. The candidate processing pipeline used is described in detail in [@Caleb]. The process followed is:

1.  obtain 352 data streams (8-bits/sample), one for each fan-beam, at 655.36-$\upmu$s sampling

2.  search time series for single pulses with width, $0.65536 < W <
      41.943$ ms ($W = 2^\mathrm{N} \times 0.65536$ ms, where N = 0,1,2...) and DMs in the range $100 < \mathrm{DM} < 2000$ pc cm$^{-3}$,

3.  remove events occurring simultaneously in more than 3 fan-beams at a given instant in time,

4.  classify only events with S/N $\geq10$, DM $\geq100$ pc cm$^{-3}$ and $W\leq41.943$ ms as potential FRB candidates. These then require human scrutiny of the diagnostic plots, to remove candidates that were RFI, almost always due to narrow-band mobile handset emissions in our operating passband and single pulses from known pulsars.

Results {#sec:results}
=======

The false positive rate at UTMOST is high due to RFI caused by mobile phone handsets, which produce narrow band (5-MHz) emission in our band, typically in $\approx 20$ ms pulses. These can be eliminated because celestial pulses are expected to be broadband, modulated by a frequency dependent response across the 31.25 MHz bandwidth. This process has been validated using individual pulses from about 20 bright pulsars seen to date. We are presently automating this process using machine learning algorithms, so that pulses can trigger a full voltage dump of the raw data while they are still in the $\approx30$ seconds of ring buffer storage, with alerts issued in near real-time. RFI occurs predominantly at low DM, but the rate is high enough to produce a few hundred spurious candidates above our DM limit of 100 pccm$^{-3}$ daily. Candidates were typically vetted each morning after data taking.

In 2016 March, April and June we made the first interferometric detections of FRBs at 843 MHz: FRB 160317, FRB 160410 and FRB 160608, as shown in Figure \[fig:frbs1\].

FRB 160317
----------

This was detected on 2016 March 17 at 09:00:36.530 UTC while observing an X-ray magnetar SGR 0755$-$23, in response to an Astronomers Telegram [@Barthelmy]. The burst occurred about 0.4 degrees East of the magnetar, and was detected $\sim$1$^{\circ}$ off the Galactic plane with a DM of 1165(11) pc cm$^{-3}$. The DM due to the ISM at this sight-line is $\sim320$ pc cm$^{-3}$ from the NE2001 model by [@Cordes] and $\sim395$ pc cm$^{-3}$ from the YMW16 model [@Yao]. The burst with S/N $ \sim 13$, occurred East of the centre of the primary fan-beam of detection (Beam 212) since it appeared weakly in the adjacent fan-beam with S/N $\sim 5$ (Beam 213) as shown in Figure \[fig:frbs2\].

FRB 160410
----------

Similarly to FRB 160317, this FRB was also detected in two adjacent fan-beams (Beam 085 with S/N $\sim 13$ and Beam 084 with S/N $\sim 4$) as seen in Figure \[fig:frbs2\]. A single dispersed pulse was discovered on 2016 April 04 at 08:33:39.680 UTC, in an observation of the pulsar J0837$+$0410 at the telescope’s boresight. This pulsar is so bright that individual pulses were seen from it as the FRB occurred, meaning the flux density scale and bandpass response of the observation were well understood. The FRB was seen $\sim 1^{\circ}$ away from boresight. This pulse was detected at Galactic latitude, $\sim27^{\circ}$ with the line-of-sight DM accounting for only $\sim58$ pc cm$^{-3}$ of the total observed DM from the NE2001 model. The YMW16 model estimates $\sim 63$ pc cm$^{-3}$. FRB 160410 has the lowest DM excess $\sim220$ pc cm$^{-3}$ of any published FRB, potentially making it the closest FRB discovered to date and an excellent candidate to search for repeat pulses.

FRB 160608
----------

The burst occurred in an observation of the pulsar J0738$-$4042 at $l = 254.11$ deg and $b=-9.54$ deg on 2016 June 06 at 03:53:01.088 UT with a total DM of $\sim682$ pc cm$^{-3}$ and $\sim238$ pc cm$^{-3}$ contribution from the Milky Way (NE2001). The YMW16 model’s estimate however is $\sim 310$ pc cm$^{-3}$. It was seen $\sim0.5^{\circ}$ from the boresight position. FRB 160608 was detected with S/N $\sim 12$, just above the detection threshold of 10 and it occurred towards the centre of the primary detection fan-beam (Beam 208). No pulse was detected in the adjacent fan-beams (see Figure \[fig:frbs2\]). This was initially of concern, but tests with the Vela pulsar placed sufficiently far South of the telescope boresight, to produce an individual pulse with the same S/N showed that detection in a single fan-beam occurred $\approx20\%$ of the time. The localisation of this FRB is thus slightly poorer ($21\arcsec \times 8.4^{\circ}$) than for the other two FRBs, for which 2 fan-beam detections allow more accurate positions.\

\[tab:specs\]

  ------------------------------------------------------------- ------------------------- ------------------------- -------------------------
                            Parameter                                  FRB 160317                FRB 160410                FRB 160608
                       \[0.5ex\] UTC start                         2016-03-17-08:30:58       2016-04-10-08:16:54       2016-06-08-03:52:24
                            UTC event                            2016-03-17-09:00:36.530   2016-04-10-08:33:39.680   2016-06-08-03:53:01.088
                       RA J2000 (hh:mm:ss)                              07:53:47                  08:41:25                  07:36:42
                      DEC J2000 (dd:mm:ss)                             $-$29:36:31                +06:05:05                $-$40:47:52
                            $l$ (deg)                                    246.05                    220.36                    254.11
                            $b$ (deg)                                    $-$0.99                    27.19                    $-$9.54
                          Detection S/N                                    13                        13                        12
                   $S_\mathrm{peak, obs}$ (Jy)                           $>3.0$                    $>7.0$                    $>4.3$
                    Boresight Fluence (Jyms)                              $>69$                     $>34$                     $>37$
                  Isotropic energy, $E_{0}$ (J)                      $\sim 10^{34}$            $\sim 10^{32}$            $\sim 10^{33}$
                    Observed width, $W$ (ms)                              21(7)                     4(1)                      9(6)
         DM smearing, $\uptau_{843 \,\mathrm{MHz}}$ (ms)                  12.6                       3.0                       7.4
            $\mathrm{DM}_\mathrm{FRB}$ (pc cm$^{-3}$)                   1165(11)                   278(3)                    682(7)
   $\mathrm{DM}_{\mathrm{Gal},\mathrm{NE2001}}$ (pc cm$^{-3}$)            319.6                     57.7                      238.3
   $\mathrm{DM}_{\mathrm{Gal},\mathrm{YMW16}}$ (pc cm$^{-3}$)             394.6                     62.5                      310.3
                     Inferred redshift, $z$                                0.7                       0.2                       0.4
                    Luminosity Distance (Gpc)                             4.30                      0.89                      1.97
                    Co-moving Distance (Gpc)                              2.52                      0.75                      1.44
                        Boresight source                             SGR 0755$-$2933            J0837$+$0610              J0738$-$4042
                             \[1ex\]                                                                                
  ------------------------------------------------------------- ------------------------- ------------------------- -------------------------

The primary advantage of the array is that a pulse from a far-field point source is detected in a maximum of 3 adjacent fan-beams at any given time, confirmed by extensive pulsar observations. RFI is typically near-field, and predominantly appears in more than 3 adjacent fan-beams, meaning that it can be reliably excised to reduce false positive rates when searching for transients. Using the adjacent fan-beam detections of FRB 160317, we have modeled the point of separation between the near-field or Fresnel region and the far-field or Fraunhofer region of the telescope. Assuming a point source at $10^{6}$ km, we compute the S/N for a tied-array beam (e.g. fan-beam 212) phased at an offset of 0.3 from the centre of the beam to ensure a two fan-beam detection. We compute the path length to each module, the phase of the signal along the array and perpendicular to the array, and add all these as a vector sum weighted by the module performance, to get the “boresight” S/N. We see that in Figure \[fig:fresnel\] at a distance of $\gtrsim 10^4$ km, we achieve a two fan-beam detection with S/N $\sim 13$ in the primary detection beam and S/N $\sim 5$ in the secondary detection beam, similar to the FRB being modeled. Detections of FRBs in 1 or 2 fan-beams only, thus allows us to identify them as sources more distant than this, placing them well away from the Earth and hence effectively rule out sources of local origin.

The discovery observations containing the FRBs were carefully inspected to check for similar events at the same time and with the same DM as the FRB, in other fan-beams. No other broadband pulses were detected in any other fan-beams within approximately 60 seconds of the bursts. Moreover, in addition to all the tied array fan-beams, we form a single special fan-beam as the incoherent sum of all the other fan-beams. This “total power” fan-beam was also searched for events near the UTC of the 3 bursts. For the 3 FRBs, this fan-beam contained no unusual sources of RFI. Only twice during the 3 surveys did we find FRB-like candidates (i.e. appearing across the band) which were identified as RFI upon closer analysis. In each case, similar events could be found in dozens to hundreds of fan beams, and were thus obvious near-field RFI. These false candidates also had ‘patchy’ power across the observing band, indicative of RFI generated from different carrier handsets operating at the same time in our band.

![Simulations of the detections of FRB 160317 in adjacent fan-beams (FB) to determine the Fresnel limit of the telescope. At a distance of $\gtrsim 10^{4}$ km, the S/Ns of the modeled pulse in FB 212 and FB 213, match that of the observations with non-detections in the other fan-beams.[]{data-label="fig:fresnel"}](Fig5.pdf){width="3.5"}

![image](Fig6){width="6.0"}

Two of the three FRBs have been discovered relatively close to the Galactic plane, with the locations marked as stars in Figure \[fig:skydist\]. All three have DMs significantly in excess of the Galactic contribution, suggesting an extragalactic or cosmological origin. Under this assumption, the contribution from the IGM to the DM can be used to infer a redshift, using the scaling relation in [@Ioka] and [@Inoue]. This places FRBs 160317, 160410 and 160608 at a redshift upper limit of 0.7, 0.2 and 0.4 respectively, assuming zero contribution from any potential host galaxy. Any contribution from a host galaxy or the immediate vicinity of an associated source, could be a significant fraction of the total DM depending on its orientation and location. The average DM for elliptical galaxies is 37 pc cm$^{-3}$ and for spiral galaxies is 45 pc cm$^{-3}$ based on the probability distribution of DMs computed for a range of host galaxies [@Xu]. For spirals, the weighted average over a range of inclination angles is estimated to be 142 pc cm$^{-3}$. However the host contribution to the DM from high redshift galaxies can be small due to cosmological time dilation and the corresponding redshifting of frequency [@Zhou]. It also does not account for any bias in FRB locations within galaxies.

The S/Ns, DMs and widths of all three FRBs have been computed using the <span style="font-variant:small-caps;">destroy</span> single pulse search software, <span style="font-variant:small-caps;">psrchive</span> with scripts made publicly available through the FRBCAT repository. The observed widths of all three FRBs are dominated by dispersion smearing as shown in Table \[tab:specs\]. This is due to our small bandwidth and limited number of channels (40 channels). We have now implemented a fine channel mode (320 channels) which will potentially increase our sensitivity and the FRB detection rate by a factor of $\sqrt{8}$. Our total bandwidth of only 31.25 MHz is too narrow to permit a measurement of dispersion index. Single pulses from the Vela pulsar were used to test our sensitivity to the DM index. The DM and the DM index $\delta$ where the dispersive delay is given by, $$\Delta t \propto \Delta\nu^{-\delta}$$ are found to be highly correlated, so that we can place no practical limit on $\delta$. We therefore set the DM index to $\delta = -2$.

![image](Fig7.pdf){width="5.0"}

![image](Fig8.pdf){width="5.0"}

Localisation
------------

All 3 FRBs were discovered while following up known sources, which were at “bore-sight”, and thus centred on Beam 177. We localise each FRB’s position on the sky, using the angular separation between the FRB’s position in fan-beam space from the bore-sight fan-beam. The 1$\sigma$ uncertainty in the direction of the semi-major axis is defined by the primary beam ($\sim 2.8^{\circ}$) while the uncertainty in the semi-minor axis is controlled by the fractional S/N measured from the adjacent fan-beam detections. For FRBs detected in adjacent fan-beams, we perform a simple linear interpolation based on the S/N to localise the event in the fan-beam grid. For the FRB with a single fan-beam detection only, we assume the centre of the fan-beam for the localisation. This allows us to construct a trace on the sky relative to the bore-sight (RAJ, DECJ), taking into account the meridian angle of the observation at the time of the FRB, and the known, slight skew and tilt of the East-West arms relative to true East-West and horizontal. The trace is a strong function of the hour angle of the observation, as one would expect in an East-West array. The uncertainty in the direction of the semi-minor axis was confirmed observationally using single pulse detections from bright pulsars. Single pulses that were detected in two adjacent fan-beams with S/N similar to our FRBs were chosen to estimate our localisation accuracy on the sky. These fan-beam localisations were then compared to the true position of the pulsar. The 1$\sigma$ scatter of the calculated position of the pulsar from individual pulses, compared to the known position of the pulsar, is $\sim 0.1$ fan-beams or $\sim 5\arcsec$, in the direction perpendicular to the fan-beam. For single fan-beam detections at low S/N, a similar analysis yielded a slightly poorer localisation precision of 7$\arcsec$. Two of the FRBs found with UTMOST have FWHP error ellipses of 5$\arcsec \times 2.8^{\circ}$ ($\sim 11$ arcmin$^2$) on the sky as seen, in Figure \[fig:frbspos\]. For comparison, single beam FRB detections (with FWHP beamwidth of 14.0$\arcsec$) at Parkes, are localised to $\sim160$ arcmin$^2$. The probability density of the localisation is shown in right ascension (RA) and declination (DEC), with the cross marking the most likely position of the burst for each FRB.

One of the advantages of UTMOST is that we can localise pulses to a few arcsec radius if the source is observed at different hour angles. The tilt of the error ellipses seen in Figure \[fig:frbspos\] demonstrates this and is a result of the geometry of the telescope. The most likely position of the FRB is marked by the cross. The fan-beams rotate in position according to the hour angle pointing of the telescope. A repeat FRB pulse, at a significantly different hour angle, allows us to localise the bursts to a few arcsec, depending on the S/N of the event. We have tested this scenario using individual pulses from four different pulsars, with results for four different hour angles (and offsets from boresight) for the Vela pulsar shown in Figure \[fig:vela\]. Localisations to about 5${\arcsec}$ accuracy, or 0.1 fan-beam widths, should be possible for a bright, repeating FRB. UTMOST is part of the shadowing campaign of the Survey for Pulsars and Extragalactic Radio Bursts (SUPERB) at the Parkes radio telescope (Keane et al. in prep). An FRB detected simultaneously with Parkes and UTMOST will yield a good localisation even if the burst does not repeat. The sky positions of the 3 FRBs were also re-observed at different HAs, to look for additional bursts. We spent 105 hours following FRB 160317, 43 hours on FRB 160410 and 35 hours on FRB 160608. The data were searched offline for pulses with S/N $\geq10$ and with $\pm20\%$ of the DM of the FRB, using the pipeline described in Section \[sec:pipeline\]. No repeat pulses were found from any of the FRB positions.

FRB event rate at UTMOST {#sec:disc}
========================

Event rate analysis {#sec:rate}
-------------------

[@Liam] present detailed analyses constraining FRB rates at various telescopes, scaling from a single FRB discovered at 800 MHz at the GBT (FRB 110523) [@Masui]. They estimate a rate of $4.2^{+19.6}_{-3.2} \times 10^{-1}$ d$^{-1}$ at UTMOST operating at its design sensitivity, based on comparisons between the sensitivities and areas surveyed by the two telescopes. This estimate is consistent with [@Caleb]. Using the same method as outlined in [@Liam] we calculate a rate of $0.014^{+0.05}_{-0.013}$ d$^{-1}$, which agrees with the rate we measure at UTMOST. We have spent a total of 180 days on sky and discovered 3 FRBs with a FoV of 8.8 deg$^{2}$. Based on this, we measure a detectable event rate of ($\mathcal{R}$),

$$\label{eq:rate}
\mathcal{R}\, (\mathcal{F} \gtrsim 11 \, \mathrm{Jy\, ms}) \gtrsim 0.78^{+1.24}_{-0.57} \, \times 10^{2} \, \mathrm{events \,\, sky^{-1} \, d^{-1}}$$

at the 95 percent confidence level [@Gehrels], above a full power boresight fluence of 11 Jy ms as parametrised by Equation \[eq:lim\_flue\], at the half-power FoV. The rate is given as a lower limit since all searches are incomplete in the fluence-width plane. Following [@KeanePetroff], our fluence complete rate is $$\mathcal{R}\, (\mathcal{F} \gtrsim 69  \, \mathrm{Jy\, ms}) \sim 5.0^{+18.7}_{-4.7}  \, \mathrm{events \,\, sky^{-1} \, d^{-1}},
%{\color{red}\mathcal{R}\, (\mathcal{F} \gtrsim 69  \, \mathrm{Jy\, ms}) \sim 0.049^{+0.187}_{-0.047}  \times 10^{2} \, \mathrm{events \,\, sky^{-1} \, d^{-1}}.}$$

as shown in Figure \[fig:fluence\].

In [@Caleb_sim; @Caleb] we have made estimates of the event rate expected at UTMOST, scaling from the event rate at Parkes, under assumptions about the integral source count distribution (log$N$-log$\mathcal{F}$ relation) and the spectral index of FRBs. To do this we assume that the spectral energy distribution is flat between the Parkes 1.4 GHz and UTMOST’s 843 MHz operating frequencies, and that the source count distribution scales as,

$$N(>\mathcal{F}_\mathrm{lim}) = \mathcal{F}_\mathrm{lim}^{\alpha}$$

where $\alpha=-3/2$ for events populated in a Euclidean Universe. Under these assumptions, [@Caleb] predict a rate of $\gtrsim0.008(0.004)$ events d$^{-1}$ for a 10$\sigma$, 1-ms wide FRB to a minimum detectable fluence of $\mathcal{F}_\mathrm{lim} = 11$ Jy ms at boresight (see Equation \[eq:lim\_flue\]). If we correct this rate for the measured primary beam dimensions being 10% larger (Bailes et al., submitted) than adopted in [@Caleb] we get a rate of $\gtrsim0.007(0.004)$ events d$^{-1}$ which translates to being able to detect 1.3 events in 180 days on sky. This is in mild tension with our discovery of 3 events in the survey. We quantify this tension by calculating the probability of observing 3 or more events to be 14.3%, assuming Poisson statistics with a mean of 1.3. Note that the predicted rate at UTMOST takes into account pulse-width broadening in the current implementation of the backend (channel widths $\sim780$ kHz) but does not account for possibly highly atypical scintillation properties along specific lines of sight to FRB events. The difference in the estimated and measured rates could be due to FRBs being brighter than expected at 843 MHz, and/or the slope of the source count distribution $\alpha$, being shallower than the assumed value. Simultaneous broad-band detection of an FRB (e.g. Parkes + UTMOST) would help constrain the spectral index and resolve the question. In [@Caleb_sim], we measured $\alpha \approx -0.9 \pm 0.3$, from 9 FRBs discovered in the high latitude sub-survey of the HTRU survey at Parkes. This is consistent with the events occurring at cosmological distances in a $\Lambda$CDM Universe, in which the Euclidean value for $\alpha$ does not hold. Assuming a flat spectral index for FRBs ($\gamma=0$), if we scale the rate at UTMOST from Equation \[eq:rate\], assuming $\alpha=-1.0$ for the slope of the log$N$-log$\cal{F}$ relation based on the best-fit from the bottom panel in Figure \[fig:spectral\_index\], we obtain a rate of $\gtrsim 2.1 \times 10^{3}$ events sky$^{-1}$ d$^{-1}$ at 1.4 GHz, consistent with the observed Parkes rate at the 2$\sigma$ level [@Champion].

Constraints on spectral and source count distribution indices
-------------------------------------------------------------

As discussed in the previous section, the observed FRB rate at UTMOST at 843 MHz can be brought to consistency with the rate found at 1.4 GHz at Parkes if we assume FRBs are flat spectrum sources on average, and that the log$N$-log$\mathcal{F}$ relation has slope $\alpha = -1.0$ (best-fit for $\alpha$ assuming $\gamma = 0$, in bottom panel of Figure \[fig:spectral\_index\]). Alternatively, we can relax the flat spectrum assumption, and ascribe the higher than expected rate to FRBs being brighter at 843 MHz than at 1.4 GHz. Assuming the log$N$-log$\mathcal{F}$ relation has slope $\alpha$ and that the FRBs have a power law spectral energy distribution with index $\gamma$ such that $S \propto \nu^{\gamma}$, we examine the following scenarios:

1.  *$\alpha=-3/2$*: Based on the detection rates at the Parkes, GBT and UTMOST telescopes, we can constrain a spectral index for FRBs as shown in Figure \[fig:spectral\_index\]. The rate of $\sim 7000$ events sky$^{-1}$ d$^{-1}$ at 1.4 GHz at Parkes, above a fluence limit of 0.4 Jy ms [@Champion] scales to $\sim56$ events sky$^{-1}$ d$^{-1}$ above UTMOST’s fluence limit of 11 Jy ms, and assuming the spectral index to be flat ($\gamma=0$) (see Section \[sec:rate\]). Similarly, the rate of $\sim 2.7 \times 10^{4}$ events sky$^{-1}$ d$^{-1}$ at 800 MHz at the GBT, above a fluence threshold of 0.3 Jy ms [@Liam] and scaled to UTMOST’s fluence threshold of 11 Jy ms is $\sim116$ events sky$^{-1}$ d$^{-1}$. Using these values we fit for the FRB spectra to be $\gamma = -1.1(1.2)$ (top panel of Figure \[fig:spectral\_index\]). This is found to be consistent with the observed constraint in [@nat_keane] albeit for just one FRB.

2.  *$\gamma=0$*: A similar analysis can be done to constrain the index $\alpha$ of the integral source count distribution assuming a flat spectral index. We constrain a value of $\alpha=-1.0(1.1)$ for $\gamma=0$ (bottom panel of Figure \[fig:spectral\_index\]). This value of $\alpha$ gives scaled rates of $\sim 270$ events sky$^{-1}$ d$^{-1}$ at 1.4 GHz at Parkes and $\sim 690$ events sky$^{-1}$ d$^{-1}$ at 800 MHz at GBT.

Another possible scenario is that FRBs are giant pulses from pulsars [@Wasserman]. The average observed spectral index for pulsars is $\gamma = -1.6$ (Jankowski et al., in submitted). If we assume this to be typical of FRBs, we fit a slope of $\alpha = -1.76$ for their log$N$-log$\cal{F}$ distribution. We note that the repeat FRB pulses from the Arecibo FRB 121102 exhibit a wide range of spectral indices [$\gamma \sim \, -10 \, \mathrm{to} \,+14$ ; @nat_spitler], similar to giant pulses from the Crab pulsar. For example giant pulses from the Crab pulsar exhibit spectral volatility in their broad range of spectral indices [$\gamma \sim  \, -15 \, \mathrm{to} \, +10$ ; @Karuppusamy], therefore it will be difficult to estimate the mean of the spectral indices until the numbers are sufficiently high. From simultaneous observations of FRB 150418 with Parkes at 1.4 GHz and the Murchison Widefield Array (MWA) at 150 MHz [@nat_keane], the non-detection at the MWA places a limit of $\gamma > -3.0$. [@Vfastr] estimate a weak constraint of $-7.6 < \gamma < 5.8$ based on the detection sensitivity of [@Champion] which is consistent with our estimated values. These constraints are only valid if the spectral energy distribution (SED) is an unbroken broadband power law and insignificantly affected by scintillation. This remains to be observationally proven. Future broadband instruments like CHIME [@Bandura] should have high FRB discovery rates and spectral coverage to definitively test this. Using the method in [@Caleb] we scale the observed rate at UTMOST for a boresight fluence of 11 Jy ms, to estimate the rates at CHIME and HIRAX [@Newburgh] under a Euclidean Universe assumption. We expect CHIME to detect $\sim 70$ events beam$^{-1}$ d$^{-1}$ for $T_\mathrm{sys} = 50$ K, S/N $= 10$, $G = 1.38$ K Jy$^{-1}$, $N_\mathrm{p} = 2$ and FoV = 250 deg$^{2}$ [@Connor; @Ng]. Similarly, we expect 350 events beam$^{-1}$ d$^{-1}$ at HIRAX for $T_\mathrm{sys} = 50$ K, S/N $= 10$, $G = 10.5$ K Jy$^{-1}$, $N_\mathrm{p} = 2$ and FoV = 56 deg$^{2}$ [@Newburgh].

Conclusions {#sec:conc}
===========

In this paper we present the first interferometric detections of FRBs, found during 180 days on sky at UTMOST. The events are beyond the $\approx 10^4$ km near-field limit of the telescope, ruling out local (terrestrial) sources of interference as a possible origin. We demonstrate with pulsars that a repeating FRB seen at UTMOST has the potential to be localised to $\approx 15\arcsec$ diameter error circle, an exciting prospect for identifying the host.

An all-sky rate of $\mathcal{R}(\gtrsim 11$ Jyms) $\gtrsim
0.78^{+1.24}_{-0.57} \times 10^{2}\, \mathrm{events \,\, sky^{-1} \,
d^{-1}}$ at 843 MHz is calculated from our 3 events, at the boresight fluence out to the half-power FoV. Based on the time spent on sky and the number of detections made, we measure a rate of $0.017^{+0.03}_{-0.01}$ events beam$^{-1}$ d$^{-1}$ at UTMOST, for the sensitivity achieved during the upgrade. The rates estimated by [@Caleb] for the present sensitivity, is only 14.3% unlikely assuming Poisson statistics with a mean of 1.3. One possibility could be due to the log$N$-log$\cal F$ relation for events being flatter than for a Euclidean Universe, having a slope $\alpha \approx -1.0$, rather than $\alpha = -3/2$. In this case, searching for FRBs with a less sensitive, but wider field of view instrument, appears to be a competitive strategy, [e.g. @Vedantham]. Alternatively, FRBs may simply be brighter at 843 MHz on average than at 1.4 GHz, implying a steeper spectral index for FRBs. Assuming a Euclidean Universe scaling, we find a best fit spectral index of $\gamma = -1.1(1.2)$.

Our ongoing work, and the work of others at many other facilities, will settle these questions once sufficient numbers of FRBs are detected over a broad frequency range. Understanding the spectra and log$N$-log$\mathcal{F}$ distributions are vital in the quest to understand this enigmatic population.

Acknowledgments {#acknowledgments .unnumbered}
===============

The authors would like to thank the referee for their insightful suggestions. We would also like to thank Jr-Wei Tsai and Liam Connor for useful discussions. The Molonglo Observatory is owned and operated by the University of Sydney with support from the School of Physics. The upgrade to the observatory has been supported by the University of Sydney through the Federation Fellowship FF0561298 and the Science Leveraging Fund of the New South Wales Department of Trade and Investment. Support for operations and continuing development of the observatory is provided by the University of Sydney, Swinburne University of Technology, the Australian Research Council Centre for All-Sky Astrophysics (CAASTRO), through project number CE110001020, and the Laureate Fellowship FL150100148. The late Professor George Collins allocated strategic funds for the purchase of the supercomputer in use at the facility from Swinburne University and was a passionate advocate for the project.

Positional coordinates of FRBs 160317, 160410 and 160608
========================================================

The coordinates of the FRB localisation ellipses in Figure \[fig:frbspos\] are given in Table \[table:coords\].

[c c c | c c c | c c c ]{} & &\
\[0.5ex\] RA (hrs) & DEC (deg) & Prob. & RA (hrs) & DEC (deg) & Prob. & RA (hrs) & DEC (deg) & Prob.\
\
\[0.5ex\] 7.9356 & $-32.4208$ & 0.0027 & 8.6865 & 3.3129 & 0.0028 & 7.6415 & $-43.5684$ & 0.0031\
7.9332 & $-32.2575$ & 0.0038 & 8.6867 & 3.4761 & 0.0039 & 7.6396 & $-43.4052$ & 0.0043\
7.9308 & $-32.0943$ & 0.0053 & 8.6869 & 3.6394 & 0.0054 & 7.6378 & $-43.2419$ & 0.0058\
7.9284 & $-31.9310$ & 0.0072 & 8.6870 & 3.8027 & 0.0072 & 7.6359 & $-43.0786$ & 0.0077\
7.9260 & $-31.7677$ & 0.0095 & 8.6872 & 3.9659 & 0.0096 & 7.6341 & $-42.9154$ & 0.0101\
7.9237 & $-31.6045$ & 0.0124 & 8.6874 & 4.1292 & 0.0123 & 7.6323 & $-42.7521$ & 0.0129\
7.9213 & $-31.4412$ & 0.0159 & 8.6876 & 4.2924 & 0.0156 & 7.6305 & $-42.5888$ & 0.0163\
7.9190 & $-31.2780$ & 0.0199 & 8.6878 & 4.4557 & 0.0194 & 7.6287 & $-42.4256$ & 0.0200\
7.9167 & $-31.1147$ & 0.0243 & 8.6881 & 4.6190 & 0.0236 & 7.6269 & $-42.2623$ & 0.0242\
7.9145 & $-30.9514$ & 0.0291 & 8.6883 & 4.7822 & 0.0282 & 7.6252 & $-42.0991$ & 0.0287\
7.9122 & $-30.7882$ & 0.0342 & 8.6885 & 4.9455 & 0.0330 & 7.6234 & $-41.9358$ & 0.0334\
7.9100 & $-30.6249$ & 0.0393 & 8.6888 & 5.1088 & 0.0378 & 7.6217 & $-41.7725$ & 0.0381\
7.9078 & $-30.4616$ & 0.0442 & 8.6890 & 5.2720 & 0.0425 & 7.6200 & $-41.6093$ & 0.0426\
7.9056 & $-30.2984$ & 0.0486 & 8.6893 & 5.4353 & 0.0468 & 7.6183 & $-41.4460$ & 0.0467\
7.9034 & $-30.1351$ & 0.0524 & 8.6895 & 5.5986 & 0.0505 & 7.6166 & $-41.2827$ & 0.0502\
7.9012 & $-29.9718$ & 0.0554 & 8.6898 & 5.7618 & 0.0535 & 7.6149 & $-41.1195$ & 0.0530\
7.8991 & $-29.8086$ & 0.0572 & 8.6901 & 5.9251 & 0.0555 & 7.6133 & $-40.9562$ & 0.0548\
7.8970 & $-29.6453$ & 0.0579 & 8.6904 & 6.0884 & 0.0564 & 7.6116 & $-40.7929$ & 0.0556\
7.8949 & $-29.4820$ & 0.0574 & 8.6907 & 6.2516 & 0.0562 & 7.6100 & $-40.6297$ & 0.0553\
7.8928 & $-29.3188$ & 0.0557 & 8.6910 & 6.4149 & 0.0549 & 7.6084 & $-40.4664$ & 0.0540\
7.8907 & $-29.1555$ & 0.0529 & 8.6913 & 6.5782 & 0.0526 & 7.6068 & $-40.3031$ & 0.0517\
7.8887 & $-28.9922$ & 0.0492 & 8.6916 & 6.7414 & 0.0493 & 7.6052 & $-40.1399$ & 0.0485\
7.8866 & $-28.8290$ & 0.0448 & 8.6919 & 6.9047 & 0.0454 & 7.6036 & $-39.9766$ & 0.0447\
7.8846 & $-28.6657$ & 0.0400 & 8.6923 & 7.0680 & 0.0409 & 7.6021 & $-39.8133$ & 0.0404\
7.8826 & $-28.5024$ & 0.0349 & 8.6926 & 7.2312 & 0.0362 & 7.6005 & $-39.6501$ & 0.0357\
7.8806 & $-28.3392$ & 0.0299 & 8.6930 & 7.3945 & 0.0313 & 7.5990 & $-39.4868$ & 0.0310\
7.8787 & $-28.1759$ & 0.0250 & 8.6933 & 7.5578 & 0.0266 & 7.5975 & $-39.3235$ & 0.0265\
7.8767 & $-28.0126$ & 0.0205 & 8.6937 & 7.7210 & 0.0221 & 7.5960 & $-39.1603$ & 0.0221\
7.8748 & $-27.8494$ & 0.0165 & 8.6941 & 7.8843 & 0.0181 & 7.5945 & $-38.9970$ & 0.0181\
7.8729 & $-27.6861$ & 0.0130 & 8.6945 & 8.0476 & 0.0144 & 7.5930 & $-38.8337$ & 0.0146\
7.8710 & $-27.5229$ & 0.0100 & 8.6949 & 8.2108 & 0.0113 & 7.5915 & $-38.6705$ & 0.0115\
7.8691 & $-27.3596$ & 0.0075 & 8.6953 & 8.3741 & 0.0087 & 7.5900 & $-38.5072$ & 0.0089\
7.8672 & $-27.1963$ & 0.0056 & 8.6957 & 8.5373 & 0.0065 & 7.5886 & $-38.3440$ & 0.0067\
7.8654 & $-27.0331$ & 0.0040 & 8.6961 & 8.7006 & 0.0048 & 7.5871 & $-38.1807$ & 0.0050\
7.8635 & $-26.8698$ & 0.0029 & 8.6965 & 8.8639 & 0.0035 & 7.5857 & $-38.0174$ & 0.0037\

[^1]: Email: manisha.caleb@anu.edu.au
