text: Mon. Not. R. Astron. Soc. 000, 1–8 (2015) Printed 27 May 2020 (MN LATEX style ﬁle v2.2) The rms-ﬂux relation in accreting white dwarfs: another nova-like variable and the ﬁrst dwarf nova M. Van de Sande1(cid:63), S. Scaringi 2,1, C. Knigge3 1 Instituut voor Sterrenkunde, KU Leuven, Celestijnenlaan 200D, B-3001 Heverlee, Belgium 2 Max-Planck-Institute fu¨r Extraterrestrische Physik, Postfach 1312, D-85748 Garching, Germany 3 Department of Physics and Astronomy, University of Southampton, Highﬁeld, Southampton, SO17 1BJ, UK Accepted 2015 January 20. Received 2015 January 20; in original form 2014 December 5 ABSTRACT We report on the detection of the linear rms-ﬂux relation in two accreting white dwarf binary systems: V1504 Cyg and KIC 8751494. The rms-ﬂux relation relates the abso- lute root-mean-square (rms) variability of the light curve to its mean ﬂux. The light curves analysed were obtained with the Kepler satellite at a 58.8 s cadence. The rms- ﬂux relation was previously detected in only one other cataclysmic variable, MV Lyr. This result reenforces the ubiquity of the linear rms-ﬂux relation as a characteristic property of accretion-induced variability, since it has been observed in several black hole binaries, neutron star binaries and active galactic nuclei. Moreover, its detection in V1504 Cyg is the ﬁrst time the rms-ﬂux relation has been detected in a dwarf nova- type CV during quiescence. This result, together with previous studies, hence points towards a common physical origin of accretion-induced variability, independent of the size, mass, or type of the central accreting compact object. Key words: accretion discs, binaries: close, stars: novae, cataclysmic variables, stars: dwarf novae, stars: individual: V344 Lyr, V363 Lyr, V447 Lyr, V516 Lyr, V585 Lyr, V1504 Cyg, KIC 8751494 1 INTRODUCTION the optical and UV domain over time-scales ranging from seconds to hours (Scaringi et al. 2012; Scaringi et al. 2013). Compact interacting binaries (CBs) are binary systems The diﬀerences in energy and time-scale are due to the dif- consisting of a low-mass red dwarf secondary and a compact ference in size of their central compact objects and of the object, such as a white dwarf (WD), neutron star (NS) or inner discs of their accretion discs. In XRBs, the inner rim black hole (BH). An accretion disc surrounding the compact of the disc is located typically at a few km from the cen- object is formed due to Roche-lobe overﬂow of the secondary. tral object. Because of the small size of the inner disc, the As angular momentum is transported outwards within the aperiodic variability emitted by the disc has a larger high- disc, material spirals towards the compact object. The ac- frequency component. Material is also able to reach down cretion disc gives rise to aperiodic broad-band variability, deeper into the gravitational potential of the central object, also known as ﬂickering, in all types of CBs (van der Klis causing it to emit mainly in the X-ray domain. In AGN, the 1995; Warner 2003). The time-scale and energy range at location of the inner rim is dependent on the event horizon, which aperiodic variability can be observed depends on the typically a few AU from the central object. The characteris- central compact object. In X-ray binaries (XRBs, accreting tic time-scales of the emitted aperiodic variability are hence NS or BH) the aperiodic variability is mainly visible in the much longer. In CVs the central object is larger than that of X-ray domain over time-scales ranging from milliseconds to XRBs, such that the inner rim of the disc is located at a few minutes (van der Klis 1995). In active galactic nuclei (AGN, 1000 km. The characteristic time-scales of the emitted ape- accreting supermassive BH) it is mainly visible in the optical riodic variability are therefore somewhat smaller than those and UV domain over time-scales ranging from days to years of AGN, but larger than those of XRBs, while the disc emits (Uttley & McHardy 2001; Uttley et al. 2005). Part of the mainly in the optical and UV domain. energy is also emitted in the X-ray domain, where aperiodic variability has been previously studied in these systems. In Aperiodic variability is characterised by several proper- cataclysmic variables (CVs, accreting WD) it is visible in ties, among which the linear rms-ﬂux relation. This relation relates the absolute root-mean-square (rms) variability of the light curve to its mean ﬂux: the stronger the variabil- (cid:63) E-mail: marie.vandesande@ster.kuleuven.be ity, the brighter the system. The rms-ﬂux relation implies (cid:13) c 2015 RAS

text: Mon.

text: Not.

text: R.

text: Astron.

text: Soc.

text: Printed

text: May

text: (MN

text: LATEX

text: style

text: ﬁle

text: The

label: title

text: rms-ﬂux

text: relation

text: in

text: accreting

text: white

text: dwarfs:

text: another

text: nova-like

text: variable

text: and

text: the

text: ﬁrst

text: dwarf

text: nova

text: M.

text: Van

text: de

text: S.

text: Scaringi

text: C.

text: Instituut

text: voor

text: Sterrenkunde,

text: KU

text: Leuven,

text: Celestijnenlaan

text: Heverlee,

text: Belgium

text: Max-Planck-Institute

text: fu¨r

text: Extraterrestrische

text: Physik,

text: Postfach

text: Garching,

text: Germany

text: Department

text: of

text: Physics

text: and

text: Astronomy,

text: University

text: of

text: Southampton,

text: Highﬁeld,

text: Southampton,

text: UK

text: Accepted

label: date

text: January

text: Received

text: January

text: in

text: original

text: form

text: December

text: ABSTRACT

text: We

label: abstract

text: report

text: on

text: the

text: detection

text: of

text: the

text: linear

text: rms-ﬂux

text: relation

text: in

text: two

text: accreting

text: white

text: dwarf

text: binary

text: systems:

text: Cyg

text: and

text: KIC

text: The

text: rms-ﬂux

text: relation

text: relates

text: the

text: abso-

text: lute

text: root-mean-square

text: (rms)

text: variability

text: of

text: the

text: light

text: curve

text: to

text: its

text: mean

text: ﬂux.

text: The

text: light

text: curves

text: analysed

text: were

text: obtained

text: with

text: the

text: Kepler

text: satellite

text: at

text: a

text: s

text: cadence.

text: The

text: rms-

text: ﬂux

text: relation

text: was

text: previously

text: detected

text: in

text: only

text: one

text: other

text: cataclysmic

text: variable,

text: MV

text: Lyr.

text: This

text: result

text: reenforces

text: the

text: ubiquity

text: of

text: the

text: linear

text: rms-ﬂux

text: relation

text: as

text: a

text: characteristic

text: property

text: of

text: accretion-induced

text: variability,

text: since

text: it

text: has

text: been

text: observed

text: in

text: several

text: black

text: hole

text: binaries,

text: neutron

text: star

text: binaries

text: and

text: active

text: galactic

text: nuclei.

text: Moreover,

text: its

text: detection

text: in

text: Cyg

text: is

text: the

text: ﬁrst

text: time

text: the

text: rms-ﬂux

text: relation

text: has

text: been

text: detected

text: in

text: a

text: dwarf

text: nova-

text: type

text: CV

text: during

text: quiescence.

text: This

text: result,

text: together

text: with

text: previous

text: studies,

text: hence

text: points

text: towards

text: a

text: common

text: physical

text: origin

text: of

text: accretion-induced

text: variability,

text: independent

text: of

text: the

text: size,

text: mass,

text: or

text: type

text: of

text: the

text: central

text: accreting

text: compact

text: object.

text: Key

text: words:

text: accretion

text: discs,

text: binaries:

text: close,

text: stars:

text: novae,

text: cataclysmic

text: variables,

text: stars:

text: dwarf

text: novae,

text: stars:

text: individual:

text: Lyr,

text: Lyr,

text: Lyr,

text: Lyr,

text: Lyr,

text: Cyg,

text: KIC

text: INTRODUCTION

text: the

text: optical

text: and

text: UV

text: domain

text: over

text: time-scales

text: ranging

text: from

text: seconds

text: to

text: hours

text: (Scaringi

text: et

text: al.

text: Scaringi

text: et

text: al.

text: Compact

text: interacting

text: binaries

text: (CBs)

text: are

text: binary

text: systems

text: The

text: diﬀerences

text: in

text: energy

text: and

text: time-scale

text: are

text: due

text: to

text: the

text: dif-

text: consisting

text: of

text: a

text: low-mass

text: red

text: dwarf

text: secondary

text: and

text: a

text: compact

text: ference

text: in

text: size

text: of

text: their

text: central

text: compact

text: objects

text: and

text: of

text: the

text: object,

text: such

text: as

text: a

text: white

text: dwarf

text: (WD),

text: neutron

text: star

text: (NS)

text: or

text: inner

text: discs

text: of

text: their

text: accretion

text: discs.

text: In

text: XRBs,

text: the

text: inner

text: rim

text: black

text: hole

text: (BH).

text: An

text: accretion

text: disc

text: surrounding

text: the

text: compact

text: of

text: the

text: disc

text: is

text: located

text: typically

text: at

text: a

text: few

text: km

text: from

text: the

text: cen-

text: object

text: is

text: formed

text: due

text: to

text: Roche-lobe

text: overﬂow

text: of

text: the

text: secondary.

text: tral

text: object.

text: Because

text: of

text: the

text: small

text: size

text: of

text: the

text: inner

text: disc,

text: the

text: As

text: angular

text: momentum

text: is

text: transported

text: outwards

text: within

text: the

text: aperiodic

text: variability

text: emitted

text: by

text: the

text: disc

text: has

text: a

text: larger

text: high-

text: disc,

text: material

text: spirals

text: towards

text: the

text: compact

text: object.

text: The

text: ac-

text: frequency

text: component.

text: Material

text: is

text: also

text: able

text: to

text: reach

text: down

text: cretion

text: disc

text: gives

text: rise

text: to

text: aperiodic

text: broad-band

text: variability,

text: deeper

text: into

text: the

text: gravitational

text: potential

text: of

text: the

text: central

text: object,

text: also

text: known

text: as

text: ﬂickering,

text: in

text: all

text: types

text: of

text: CBs

text: (van

text: der

text: Klis

text: causing

text: it

text: to

text: emit

text: mainly

text: in

text: the

text: X-ray

text: domain.

text: In

text: AGN,

text: the

text: Warner

text: The

text: time-scale

text: and

text: energy

text: range

text: at

text: location

text: of

text: the

text: inner

text: rim

text: is

text: dependent

text: on

text: the

text: event

text: horizon,

text: which

text: aperiodic

text: variability

text: can

text: be

text: observed

text: depends

text: on

text: the

text: typically

text: a

text: few

text: AU

text: from

text: the

text: central

text: object.

text: The

text: characteris-

text: central

text: compact

text: object.

text: In

text: X-ray

text: binaries

text: (XRBs,

text: accreting

text: tic

text: time-scales

text: of

text: the

text: emitted

text: aperiodic

text: variability

text: are

text: hence

text: NS

text: or

text: BH)

text: the

text: aperiodic

text: variability

text: is

text: mainly

text: visible

text: in

text: the

text: much

text: longer.

text: In

text: CVs

text: the

text: central

text: object

text: is

text: larger

text: than

text: that

text: of

text: X-ray

text: domain

text: over

text: time-scales

text: ranging

text: from

text: milliseconds

text: to

text: XRBs,

text: such

text: that

text: the

text: inner

text: rim

text: of

text: the

text: disc

text: is

text: located

text: at

text: a

text: few

text: minutes

text: (van

text: der

text: Klis

text: In

text: active

text: galactic

text: nuclei

text: (AGN,

text: km.

text: The

text: characteristic

text: time-scales

text: of

text: the

text: emitted

text: ape-

text: accreting

text: supermassive

text: BH)

text: it

text: is

text: mainly

text: visible

text: in

text: the

text: optical

text: riodic

text: variability

text: are

text: therefore

text: somewhat

text: smaller

text: than

text: those

text: and

text: UV

text: domain

text: over

text: time-scales

text: ranging

text: from

text: days

text: to

text: years

text: of

text: AGN,

text: but

text: larger

text: than

text: those

text: of

text: XRBs,

text: while

text: the

text: disc

text: emits

text: (Uttley

text: McHardy

text: Uttley

text: et

text: al.

text: Part

text: of

text: the

text: mainly

text: in

text: the

text: optical

text: and

text: UV

text: domain.

text: energy

text: is

text: also

text: emitted

text: in

text: the

text: X-ray

text: domain,

text: where

text: aperiodic

text: variability

text: has

text: been

text: previously

text: studied

text: in

text: these

text: systems.

text: In

text: Aperiodic

text: variability

text: is

text: characterised

text: by

text: several

text: proper-

text: cataclysmic

text: variables

text: (CVs,

text: accreting

text: WD)

text: it

text: is

text: visible

text: in

text: ties,

text: among

text: which

text: the

text: linear

text: rms-ﬂux

text: relation.

text: This

text: relation

text: relates

text: the

text: absolute

text: root-mean-square

text: (rms)

text: variability

text: of

text: the

text: light

text: curve

text: to

text: its

text: mean

text: ﬂux:

text: the

text: stronger

text: the

text: variabil-

text: E-mail:

text: marie.vandesande@ster.kuleuven.be

text: ity,

text: the

text: brighter

text: the

text: system.

text: The

text: rms-ﬂux

text: relation

text: implies

text: c

text: RAS


text: Rms-ﬂux relation in CVs 3 (a) Kepler Q5 observations of V1504 Cyg (b) Kepler Q5 observations of KIC 8751494 Figure 1. Illustration of the Kepler light curves of V1504 Cyg and KIC 8751494, the measurements during Kepler Q5 are shown in the left panels. The selection of quiescent intervals in DNe is illustrated for V1504 Cyg: in the left panel of ﬁgure 1a, a steady interval has been selected in gray. This is shown in more detail in its right panel. For KIC 8751494, a four day-segment is shown in more detail in the right panel of ﬁgure 1b, where the orbital period can also be seen. Aperiodic variability is clearly visible in both objects. Quiescent states of DNe and observations of NLs will both causes the inter-quarter ﬂux oﬀsets (Jenkins et al. 2010). be called ‘steady’ states or quarters. Obvious outliers, which We have made no attempt to correct for these inter-quarter can be caused by systematic error sources from the telescope ﬂux oﬀsets as we only analyse quarters for NLs and single and spacecraft (Fraquelli & Thompson 2012), are removed steady intervals or steady intervals combined according to from all steady data sets. This has been done by visually their quarter of observation for DNe. It would be interesting analysing the ﬂux distribution of each light curve, plotted to do so in the future. on a semi-logarithmic scale. Using this scaling, any discrep- 2.3 Data analysis methods ancies to the ﬂux distribution can be spotted and removed from the light curve. In the left panels of ﬁgure 1, we show The data analysis method is applied to every steady the light curves observed for V1504 Cyg and KIC 8751494 data set of each object. The aim is to determine whether during Kepler Q5. Various DN outbursts and one superout- the rms-ﬂux relation is present in the steady data sets, and burst are clearly visible in the light curve of the DN- SU if so, to ﬁt a linear model to it. In order to reduce the error UMa V1504 Cyg (ﬁgure 1a). The selection of a quiescent on the intercept of the ﬁtted linear relation, we ﬁrst subtract interval is shown as well in gray, which is shown in detail its the mean ﬂux level from the steady data set. This does not right panel. As is clear from ﬁgure 1b, the NL KIC 8751494 aﬀect the detection of the rms-ﬂux relation, as we are sim- does not show any outbursts. Aperiodic variability is clearly ply subtracting a constant from the entire data set under visible in both objects (right panels of ﬁgure 1). consideration. The data analysis method can be split into Temporal gaps in the light curves are created by re- two main parts: an appropriate binning of the data set fol- moving outbursts or are due to gaps in the Kepler obser- lowed by ﬁtting a linear model to the binned results. We ﬁrst vations itself. As our data analysis method (see section ) is bin the light curve over time. This is done by dividing the insensitive to gaps in the data, we did not join diﬀerent seg- light curve into several segments, each containing the same ments separated by a gap. The steady intervals of DNe can number of measurements. As every measurement is taken at be combined according to their quarter of observation. This a 58.8 s cadence, the number of measurements in each bin also creates temporal gaps in the light curve, which we again corresponds to the time-scale of aperiodic variability under keep as part of the data set. Small oﬀsets due to long terms investigation. The number of bins after the ﬁrst binning is drifts in the light curve were noted, but were small enough hence dependent on the chosen time scale. The mean ﬂux to have a negligible impact on our method. However, we can- value of the measurements and their variance is calculated not combine all their steady intervals over several quarters for each bin. Within each bin, the expected Poisson noise or combine all quarters of NLs, since the mean ﬂux level is level is subtracted from the measured variance in order to shifted when entering a new quarter. These shifts are due to obtain an indicative measure of the intrinsic variance of the ◦ the quarterly 90 rolls the satellite had to perform in order to data. The expected Poisson noise level has been calculated have its solar panels face the Sun. After every roll the object for every sampled time scale and hence bin size. It is known is placed on a diﬀerent CCD camera, which causes the op- however that Kepler light curves suﬀer from additional noise timal aperture to change. It is the change in aperture that components, and thus the Poisson noise estimate used is a (cid:13) c 2015 RAS, MNRAS 000, 1–8 ##LTFigure## ##LTFigure##

text: Rms-ﬂux

label: paragraph

text: relation

text: in

text: CVs

text: (a)

text: Kepler

label: caption

text: observations

text: of

text: Cyg

text: (b)

text: Kepler

text: observations

text: of

text: KIC

text: Figure

text: Illustration

text: of

text: the

text: Kepler

text: light

text: curves

text: of

text: Cyg

text: and

text: KIC

text: the

text: measurements

text: during

text: Kepler

text: are

text: shown

text: in

text: the

text: left

text: panels.

text: The

text: selection

text: of

text: quiescent

text: intervals

text: in

text: DNe

text: is

text: illustrated

text: for

text: Cyg:

text: in

text: the

text: left

text: panel

text: of

text: ﬁgure

text: a

text: steady

text: interval

text: has

text: been

text: selected

text: in

text: gray.

text: This

text: is

text: shown

text: in

text: more

text: detail

text: in

text: its

text: right

text: panel.

text: For

text: KIC

text: a

text: four

text: day-segment

text: is

text: shown

text: in

text: more

text: detail

text: in

text: the

text: right

text: panel

text: of

text: ﬁgure

text: where

text: the

text: orbital

text: period

text: can

text: also

text: be

text: seen.

text: Aperiodic

text: variability

text: is

text: clearly

text: visible

text: in

text: both

text: objects.

text: Quiescent

text: states

text: of

text: DNe

text: and

text: observations

text: of

text: NLs

text: will

text: both

text: causes

text: the

text: inter-quarter

text: ﬂux

text: oﬀsets

text: (Jenkins

text: et

text: al.

text: be

text: called

text: ‘steady’

text: states

text: or

text: quarters.

text: Obvious

text: outliers,

text: which

text: We

text: have

text: made

text: no

text: attempt

text: to

text: correct

text: for

text: these

text: inter-quarter

text: can

text: be

text: caused

text: by

text: systematic

text: error

text: sources

text: from

text: the

text: telescope

text: ﬂux

text: oﬀsets

text: as

text: we

text: only

text: analyse

text: quarters

text: for

text: NLs

text: and

text: single

text: and

text: spacecraft

text: (Fraquelli

text: Thompson

text: are

text: removed

text: steady

text: intervals

text: or

text: steady

text: intervals

text: combined

text: according

text: to

text: from

text: all

text: steady

text: data

text: sets.

text: This

text: has

text: been

text: done

text: by

text: visually

text: their

text: quarter

text: of

text: observation

text: for

text: DNe.

text: It

text: would

text: be

text: interesting

text: analysing

text: the

text: ﬂux

text: distribution

text: of

text: each

text: light

text: curve,

text: plotted

text: to

text: do

text: so

text: in

text: the

text: future.

text: on

text: a

text: semi-logarithmic

text: scale.

text: Using

text: this

text: scaling,

text: any

text: discrep-

text: Data

text: analysis

text: methods

text: ancies

text: to

text: the

text: ﬂux

text: distribution

text: can

text: be

text: spotted

text: and

text: removed

text: from

text: the

text: light

text: curve.

text: In

text: the

text: left

text: panels

text: of

text: ﬁgure

text: we

text: show

text: The

text: data

text: analysis

text: method

text: is

text: applied

text: to

text: every

text: steady

text: the

text: light

text: curves

text: observed

text: for

text: Cyg

text: and

text: KIC

text: data

text: set

text: of

text: each

text: object.

text: The

text: aim

text: is

text: to

text: determine

text: whether

text: during

text: Kepler

text: Various

text: DN

text: outbursts

text: and

text: one

text: superout-

text: the

text: rms-ﬂux

text: relation

text: is

text: present

text: in

text: the

text: steady

text: data

text: sets,

text: and

text: burst

text: are

text: clearly

text: visible

text: in

text: the

text: light

text: curve

text: of

text: the

text: DN-

text: SU

text: if

text: so,

text: to

text: ﬁt

text: a

text: linear

text: model

text: to

text: it.

text: In

text: order

text: to

text: reduce

text: the

text: error

text: UMa

text: Cyg

text: (ﬁgure

text: The

text: selection

text: of

text: a

text: quiescent

text: on

text: the

text: intercept

text: of

text: the

text: ﬁtted

text: linear

text: relation,

text: we

text: ﬁrst

text: subtract

text: interval

text: is

text: shown

text: as

text: well

text: in

text: gray,

text: which

text: is

text: shown

text: in

text: detail

text: its

text: the

text: mean

text: ﬂux

text: level

text: from

text: the

text: steady

text: data

text: set.

text: This

text: does

text: not

text: right

text: panel.

text: As

text: is

text: clear

text: from

text: ﬁgure

text: the

text: NL

text: KIC

text: aﬀect

text: the

text: detection

text: of

text: the

text: rms-ﬂux

text: relation,

text: as

text: we

text: are

text: sim-

text: does

text: not

text: show

text: any

text: outbursts.

text: Aperiodic

text: variability

text: is

text: clearly

text: ply

text: subtracting

text: a

text: constant

text: from

text: the

text: entire

text: data

text: set

text: under

text: visible

text: in

text: both

text: objects

text: (right

text: panels

text: of

text: ﬁgure

text: consideration.

text: The

text: data

text: analysis

text: method

text: can

text: be

text: split

text: into

text: Temporal

text: gaps

text: in

text: the

text: light

text: curves

text: are

text: created

text: by

text: re-

text: two

text: main

text: parts:

text: an

text: appropriate

text: binning

text: of

text: the

text: data

text: set

text: fol-

text: moving

text: outbursts

text: or

text: are

text: due

text: to

text: gaps

text: in

text: the

text: Kepler

text: obser-

text: lowed

text: by

text: ﬁtting

text: a

text: linear

text: model

text: to

text: the

text: binned

text: results.

text: We

text: ﬁrst

text: vations

text: itself.

text: As

text: our

text: data

text: analysis

text: method

text: (see

text: section

text: is

text: bin

text: the

text: light

text: curve

text: over

text: time.

text: This

text: is

text: done

text: by

text: dividing

text: the

text: insensitive

text: to

text: gaps

text: in

text: the

text: data,

text: we

text: did

text: not

text: join

text: diﬀerent

text: seg-

text: light

text: curve

text: into

text: several

text: segments,

text: each

text: containing

text: the

text: same

text: ments

text: separated

text: by

text: a

text: gap.

text: The

text: steady

text: intervals

text: of

text: DNe

text: can

text: number

text: of

text: measurements.

text: As

text: every

text: measurement

text: is

text: taken

text: at

text: be

text: combined

text: according

text: to

text: their

text: quarter

text: of

text: observation.

text: This

text: a

text: s

text: cadence,

text: the

text: number

text: of

text: measurements

text: in

text: each

text: bin

text: also

text: creates

text: temporal

text: gaps

text: in

text: the

text: light

text: curve,

text: which

text: we

text: again

text: corresponds

text: to

text: the

text: time-scale

text: of

text: aperiodic

text: variability

text: under

text: keep

text: as

text: part

text: of

text: the

text: data

text: set.

text: Small

text: oﬀsets

text: due

text: to

text: long

text: terms

text: investigation.

text: The

text: number

text: of

text: bins

text: after

text: the

text: ﬁrst

text: binning

text: is

text: drifts

text: in

text: the

text: light

text: curve

text: were

text: noted,

text: but

text: were

text: small

text: enough

text: hence

text: dependent

text: on

text: the

text: chosen

text: time

text: scale.

text: The

text: mean

text: ﬂux

text: to

text: have

text: a

text: negligible

text: impact

text: on

text: our

text: method.

text: However,

text: we

text: can-

text: value

text: of

text: the

text: measurements

text: and

text: their

text: variance

text: is

text: calculated

text: not

text: combine

text: all

text: their

text: steady

text: intervals

text: over

text: several

text: quarters

text: for

text: each

text: bin.

text: Within

text: each

text: bin,

text: the

text: expected

text: Poisson

text: noise

text: or

text: combine

text: all

text: quarters

text: of

text: NLs,

text: since

text: the

text: mean

text: ﬂux

text: level

text: is

text: level

text: is

text: subtracted

text: from

text: the

text: measured

text: variance

text: in

text: order

text: to

text: shifted

text: when

text: entering

text: a

text: new

text: quarter.

text: These

text: shifts

text: are

text: due

text: to

text: obtain

text: an

text: indicative

text: measure

text: of

text: the

text: intrinsic

text: variance

text: of

text: the

text: the

text: quarterly

text: rolls

text: the

text: satellite

text: had

text: to

text: perform

text: in

text: order

text: to

text: data.

text: The

text: expected

text: Poisson

text: noise

text: level

text: has

text: been

text: calculated

text: have

text: its

text: solar

text: panels

text: face

text: the

text: Sun.

text: After

text: every

text: roll

text: the

text: object

text: for

text: every

text: sampled

text: time

text: scale

text: and

text: hence

text: bin

text: size.

text: It

text: is

text: known

text: is

text: placed

text: on

text: a

text: diﬀerent

text: CCD

text: camera,

text: which

text: causes

text: the

text: op-

text: however

text: that

text: Kepler

text: light

text: curves

text: suﬀer

text: from

text: additional

text: noise

text: timal

text: aperture

text: to

text: change.

text: It

text: is

text: the

text: change

text: in

text: aperture

text: that

text: components,

text: and

text: thus

text: the

text: Poisson

text: noise

text: estimate

text: used

text: is

text: a

text: c

text: RAS,

text: MNRAS

text: ##LTFigure##

text: ##LTFigure##


text: 4 M. Van de Sande, S. Scaringi, C. Knigge conservative one. The mean ﬂux - intrinsic variance results are averaged over by rebinning them into ten bins. All bins are equally weighted. We have checked that the number of bins adopted does not signiﬁcantly aﬀect the ﬁnal results, and have chosen for ten bins in accordance with Scaringi et al. (2012). A linear relation can still be found with fewer or more bins; gradient and intercept of the found rms-ﬂux relation are not aﬀected by the number of bins. The linear relation between mean ﬂux and rms variabil- ity is statistically quantiﬁed by the Spearman’s rank correla- tion coeﬃcient. Whenever this result was close to 1, implying a positive correlation between rms and ﬂux, a linear model was ﬁtted to the data by using a linear regression method. The goodness-of-ﬁt of the linear model is quantiﬁed by its reduced chi-squared value. The errors on the gradient and intercept of the ﬁtted linear model are obtained by boot- (a) V1504 Cyg strapping the original light curve. Bootstrapping is a non- parametric resampling method used to estimate the sam- pling distribution underlying a statistic of interest. In this study, the statistics of interest are the gradient and intercept of the ﬁtted linear rms-ﬂux relation. A bootstrapped light curve is produced by resampling the original light curve with replacement, to which we apply the previously described data analysis method. This process is repeated 500 times. The standard deviations of the obtained sampling distri- butions are then adopted as the errors on their respective statistic of interest. Note that because we have subtracted the mean ﬂux value from the light curve, the correlation be- tween gradient and intercept is removed. This is done as to produce circular error contours. 3 RESULTS We have analysed all CVs listed in table 1, except for (b) KIC 8751494 the previously studied MV Lyr. The rms-ﬂux relation has Figure 2. Rms-ﬂux relations found in the combined steady inter- been detected in all steady data of the DN - SU UMa star vals of Q4 of V1504 Cyg and in the quarter Q2 of KIC 8751494. V1404 Cyg and the NL KIC 8751494. These results will be The time-scale sampled is approximately 10 minutes. (cid:112) discussed in section . No rms-ﬂux relation was detected in the other six CVs, but this can be accounted for by the low ± signal-to-noise (SNR) level in their light curves (see section For most other data sets, we ﬁnd that χ2 = 1 2/K = ± red below). 1 0.5 with K = 8 the degrees of freedom of our model. (cid:29) 3.1 Detection of the rms-ﬂux relation in V1504 For some data sets in V1504 Cyg, we found χ2 1. This red Cyg and KIC 8751494 is because the orbital period of the system is clearly visi- ble during those intervals, which aﬀects the high frequency The rms-ﬂux relation has been detected in all steady aperiodic variability power. intervals of V1504 Cyg, both separately and combined ac- 3.2 Conditions for the detection of the rms-ﬂux cording to their quarter of observation, and all quarters of relation KIC 8751494. The total amount of steady data analysed is listed in table 1. Moreover, the data sets have been analysed Out of the eight CVs we have analysed, we have only de- at multiple time-scales and the rms-ﬂux relation is found tected the rms-ﬂux relation in V1504 Cyg and KIC 8751494. at every sampled time-scale. Figure 2 shows the rms-ﬂux If the rms-ﬂux relation were truly universal, we would ex- relations found in the combined steady intervals of Q4 of pect to detect it in all CVs, provided the data quality is V1504 Cyg and in quarter Q2 of KIC 8751494. The time- suﬃcient. Here we veriﬁed whether the six systems with no scale sampled is equal to ten times the sampling cadence, detection are aﬀected by larger noise levels, thus diminishing approximately ten minutes. The fractional rms level of the the SNR. Aperiodic variability cannot be detected when the DN-SU UMa V1504 Cyg and the NL KIC 8751494 is approx- instrumental noise level of the data set is comparable to or imately equal to 0.15 and 0.05 respectively, which is calcu- larger than its intrinsic rms variability, as the instrumental lated by dividing the rms variability of an entire steady data errors then ‘overpower’ the intrinsic variability emitted by set by its original mean ﬂux. The rms-ﬂux relation does not the system. show any temporal evolution in both objects, unlike MV Lyr It has already been shown by Edelson et al. (2014) that (Scaringi et al. 2012). the instrumental noise levels within Kepler light curves can In all data sets, we ﬁnd that the Spearman’s rank coef- be substantially larger than pure Poisson noise. This can ﬁcient is never below 0.9. The reduced chi-squared values of possibly be attributed to Moire pattern drift (MPD), which linear ﬁts shown in ﬁgure 2 are 1.02 and 1.14 respectively. arises from crosstalk between the four guiding sensors and (cid:13) c 2015 RAS, MNRAS 000, 1–8 ##LTFigure## ##LTFigure## ##LTLine##

text: M.

text: Van

text: de

text: Sande,

text: S.

text: Scaringi,

text: C.

text: Knigge

text: conservative

text: one.

text: The

text: mean

text: ﬂux

text: intrinsic

text: variance

text: results

text: are

text: averaged

text: over

text: by

text: rebinning

text: them

text: into

text: ten

text: bins.

text: All

text: bins

text: are

text: equally

text: weighted.

text: We

text: have

text: checked

text: that

text: the

text: number

text: of

text: bins

text: adopted

text: does

text: not

text: signiﬁcantly

text: aﬀect

text: the

text: ﬁnal

text: results,

text: and

text: have

text: chosen

text: for

text: ten

text: bins

text: in

text: accordance

text: with

text: Scaringi

text: et

text: al.

text: A

text: linear

text: relation

text: can

text: still

text: be

text: found

text: with

text: fewer

text: or

text: more

text: bins;

text: gradient

text: and

text: intercept

text: of

text: the

text: found

text: rms-ﬂux

text: relation

text: are

text: not

text: aﬀected

text: by

text: the

text: number

text: of

text: bins.

text: The

text: linear

text: relation

text: between

text: mean

text: ﬂux

text: and

text: rms

text: variabil-

text: ity

text: is

text: statistically

text: quantiﬁed

text: by

text: the

text: Spearman’s

text: rank

text: correla-

text: tion

text: coeﬃcient.

text: Whenever

text: this

text: result

text: was

text: close

text: to

text: implying

text: a

text: positive

text: correlation

text: between

text: rms

text: and

text: ﬂux,

text: a

text: linear

text: model

text: was

text: ﬁtted

text: to

text: the

text: data

text: by

text: using

text: a

text: linear

text: regression

text: method.

text: The

text: goodness-of-ﬁt

text: of

text: the

text: linear

text: model

text: is

text: quantiﬁed

text: by

text: its

text: reduced

text: chi-squared

text: value.

text: The

text: errors

text: on

text: the

text: gradient

text: and

text: intercept

text: of

text: the

text: ﬁtted

text: linear

text: model

text: are

text: obtained

text: by

text: boot-

text: (a)

text: Cyg

text: strapping

text: the

text: original

text: light

text: curve.

text: Bootstrapping

text: is

text: a

text: non-

text: parametric

text: resampling

text: method

text: used

text: to

text: estimate

text: the

text: sam-

text: pling

text: distribution

text: underlying

text: a

text: statistic

text: of

text: interest.

text: In

text: this

text: study,

text: the

text: statistics

text: of

text: interest

text: are

text: the

text: gradient

text: and

text: intercept

text: of

text: the

text: ﬁtted

text: linear

text: rms-ﬂux

text: relation.

text: A

text: bootstrapped

text: light

text: curve

text: is

text: produced

text: by

text: resampling

text: the

text: original

text: light

text: curve

text: with

text: replacement,

text: to

text: which

text: we

text: apply

text: the

text: previously

text: described

text: data

text: analysis

text: method.

text: This

text: process

text: is

text: repeated

text: times.

text: The

text: standard

text: deviations

text: of

text: the

text: obtained

text: sampling

text: distri-

text: butions

text: are

text: then

text: adopted

text: as

text: the

text: errors

text: on

text: their

text: respective

text: statistic

text: of

text: interest.

text: Note

text: that

text: because

text: we

text: have

text: subtracted

text: the

text: mean

text: ﬂux

text: value

text: from

text: the

text: light

text: curve,

text: the

text: correlation

text: be-

text: tween

text: gradient

text: and

text: intercept

text: is

text: removed.

text: This

text: is

text: done

text: as

text: to

text: produce

text: circular

text: error

text: contours.

text: RESULTS

text: We

text: have

text: analysed

text: all

text: CVs

text: listed

text: in

text: table

text: except

text: for

text: (b)

text: KIC

text: the

text: previously

text: studied

text: MV

text: Lyr.

text: The

text: rms-ﬂux

text: relation

text: has

text: Figure

text: Rms-ﬂux

text: relations

text: found

text: in

text: the

text: combined

text: steady

text: inter-

text: been

text: detected

text: in

text: all

text: steady

text: data

text: of

text: the

text: DN

text: SU

text: UMa

text: star

text: vals

text: of

text: of

text: Cyg

text: and

text: in

text: the

text: quarter

text: of

text: KIC

text: Cyg

text: and

text: the

text: NL

text: KIC

text: These

text: results

text: will

text: be

text: The

text: time-scale

text: sampled

text: is

text: approximately

text: minutes.

text: discussed

text: in

text: section

text: No

text: rms-ﬂux

text: relation

text: was

text: detected

text: in

text: the

text: other

text: six

text: CVs,

text: but

text: this

text: can

text: be

text: accounted

text: for

text: by

text: the

text: low

text: signal-to-noise

text: (SNR)

text: level

text: in

text: their

text: light

text: curves

text: (see

text: section

text: For

text: most

text: other

text: data

text: sets,

text: we

text: ﬁnd

text: that

text: χ2

text: red

text: below).

text: with

text: K

text: the

text: degrees

text: of

text: freedom

text: of

text: our

text: model.

text: Detection

text: of

text: the

text: rms-ﬂux

text: relation

text: in

text: For

text: some

text: data

text: sets

text: in

text: Cyg,

text: we

text: found

text: χ2

text: This

text: red

text: Cyg

text: and

text: KIC

text: is

text: because

text: the

text: orbital

text: period

text: of

text: the

text: system

text: is

text: clearly

text: visi-

text: ble

text: during

text: those

text: intervals,

text: which

text: aﬀects

text: the

text: high

text: frequency

text: The

text: rms-ﬂux

text: relation

text: has

text: been

text: detected

text: in

text: all

text: steady

text: aperiodic

text: variability

text: power.

text: intervals

text: of

text: Cyg,

text: both

text: separately

text: and

text: combined

text: ac-

text: Conditions

text: for

text: the

text: detection

text: of

text: the

text: rms-ﬂux

text: cording

text: to

text: their

text: quarter

text: of

text: observation,

text: and

text: all

text: quarters

text: of

text: relation

text: KIC

text: The

text: total

text: amount

text: of

text: steady

text: data

text: analysed

text: is

text: listed

text: in

text: table

text: Moreover,

text: the

text: data

text: sets

text: have

text: been

text: analysed

text: Out

text: of

text: the

text: eight

text: CVs

text: we

text: have

text: analysed,

text: we

text: have

text: only

text: de-

text: at

text: multiple

text: time-scales

text: and

text: the

text: rms-ﬂux

text: relation

text: is

text: found

text: tected

text: the

text: rms-ﬂux

text: relation

text: in

text: Cyg

text: and

text: KIC

text: at

text: every

text: sampled

text: time-scale.

text: Figure

text: shows

text: the

text: rms-ﬂux

text: If

text: the

text: rms-ﬂux

text: relation

text: were

text: truly

text: universal,

text: we

text: would

text: ex-

text: relations

text: found

text: in

text: the

text: combined

text: steady

text: intervals

text: of

text: of

text: pect

text: to

text: detect

text: it

text: in

text: all

text: CVs,

text: provided

text: the

text: data

text: quality

text: is

text: Cyg

text: and

text: in

text: quarter

text: of

text: KIC

text: The

text: time-

text: suﬃcient.

text: Here

text: we

text: veriﬁed

text: whether

text: the

text: six

text: systems

text: with

text: no

text: scale

text: sampled

text: is

text: equal

text: to

text: ten

text: times

text: the

text: sampling

text: cadence,

text: detection

text: are

text: aﬀected

text: by

text: larger

text: noise

text: levels,

text: thus

text: diminishing

text: approximately

text: ten

text: minutes.

text: The

text: fractional

text: rms

text: level

text: of

text: the

text: the

text: SNR.

text: Aperiodic

text: variability

text: cannot

text: be

text: detected

text: when

text: the

text: DN-SU

text: UMa

text: Cyg

text: and

text: the

text: NL

text: KIC

text: is

text: approx-

text: instrumental

text: noise

text: level

text: of

text: the

text: data

text: set

text: is

text: comparable

text: to

text: or

text: imately

text: equal

text: to

text: and

text: respectively,

text: which

text: is

text: calcu-

text: larger

text: than

text: its

text: intrinsic

text: rms

text: variability,

text: as

text: the

text: instrumental

text: lated

text: by

text: dividing

text: the

text: rms

text: variability

text: of

text: an

text: entire

text: steady

text: data

text: errors

text: then

text: ‘overpower’

text: the

text: intrinsic

text: variability

text: emitted

text: by

text: set

text: by

text: its

text: original

text: mean

text: ﬂux.

text: The

text: rms-ﬂux

text: relation

text: does

text: not

text: the

text: system.

text: show

text: any

text: temporal

text: evolution

text: in

text: both

text: objects,

text: unlike

text: MV

text: Lyr

text: It

text: has

text: already

text: been

text: shown

text: by

text: Edelson

text: et

text: al.

text: that

text: (Scaringi

text: et

text: al.

text: the

text: instrumental

text: noise

text: levels

text: within

text: Kepler

text: light

text: curves

text: can

text: In

text: all

text: data

text: sets,

text: we

text: ﬁnd

text: that

text: the

text: Spearman’s

text: rank

text: coef-

text: be

text: substantially

text: larger

text: than

text: pure

text: Poisson

text: noise.

text: This

text: can

text: ﬁcient

text: is

text: never

text: below

text: The

text: reduced

text: chi-squared

text: values

text: of

text: possibly

text: be

text: attributed

text: to

text: Moire

text: pattern

text: drift

text: (MPD),

text: which

text: linear

text: ﬁts

text: shown

text: in

text: ﬁgure

text: are

text: and

text: respectively.

text: arises

text: from

text: crosstalk

text: between

text: the

text: four

text: guiding

text: sensors

text: and

text: c

text: RAS,

text: MNRAS

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##


text: Rms-ﬂux relation in CVs 5 (a) V1504 Cyg (b) KIC 8751494 (c) BOKS 45906 Figure 3. Representative PSDs of V1504 Cyg, KIC 8751494, and BOKS 45906 in grey. Binned PSDs are shown in black. The lowest frequency sampled is marked by a dashed line, which corresponds to a time-scale of approximately 140 minutes. High-frequency variability is clearly visible in the PSDs of V1504 Cyg and KIC 8751494, while Poisson noise dominates the PSD of BOKS 45906, especially at high frequencies. The PSDs of the other ﬁve CVs that do not show the rms-ﬂux relation are similar to ﬁgure 3c. Table 2. Mean ﬂux level of a representative 10 minute segment of a steady interval for all objects, together with its rms variability and fractional rms level. V1504 Cyg and KIC 8751494 are the only two systems analysed in this paper with detected rms-ﬂux relations, and are also the only two systems where intrinsic variability dominates over instrumental variability. For these systems the fractional rms column shows the intrinsic value for these systems. The remaining objects are dominated by instrumental noise, and the fractional rms column is indicative on the upper limit for intrinsic variability. For completeness, MV Lyr has been included as well. The Kepler data of MV Lyr displayed a greater range of variability amplitudes than the objects studied in the present paper, so the values obtained for this source are expressed as a range of values (Scaringi et al. 2013). In this object intrinsic variability also dominates over instrumental variability. Object Mean ﬂux Rms Fractional Rms dominated by (electrons) (electrons) rms intrinsic variability? BOKS 45906 4,643 645 14% No V344 Lyr 26,762 871 3% No V363 Lyr 23,750 876 4% No V447 Lyr 2,374 529 22% No V516 Lyr 1,038 331 32% No V585 Lyr 13,238 605 5% No V1504 Cyg 116,602 5,078 4% Yes KIC 8751494 242,974 3,390 1% Yes × × MV Lyr 3-13 104 0.1-0.4 104 2-5% Yes the 84 science channel readouts (Kolodziejczak et al. 2010). method for V1504 Cyg, KIC 8751494, and BOKS 45906 It has been also pointed out (Wehrle et al. 2013; Revalski are shown in grey. Overplotted in black are the smoothed et al. 2014) that MPD noise can be a serious problem for PSDs, calculated by binning the PSD and averaging over aperiodic variability studies, especially at high frequencies the bins. We have opted for a Lomb-Scargle method as the for low luminosity sources. As there is no known procedure data sets are unevenly spaced due to the presence of tem- to accurately estimate this noise component we are unable to poral gaps. The lowest frequency we have sampled when place robust constraints on the level of instrumental noise searching for the rms-ﬂux relation is marked by a vertical for the observed systems. We have checked which objects dashed line. From this ﬁgure it can be seen that the PSD were more heavily aﬀected by MPD and found that only of BOKS 45906 displays a white noise dominated PSD at − KIC 8751494 and V1504 Cyg were aﬀected during Q5 and frequencies above 10 5Hz (with positive power law index in Q2, 6, 10, and 14 respectively (Van Cleve & Caldwell 2009). power x frequency). Thus in this system any intrinsic vari- The rms-ﬂux relation is detected in all non MPD-aﬀected, ability cannot be measured above that frequency. The PSD as well as MPD-aﬀected quarters. However, we can inspect shape of BOKS 45906 is representative of the PSDs of other the PSDs of all objects to determine at which frequencies in- objects which do not show the rms-ﬂux relation in our sam- strumental noise begins to dominate the intrinsic aperiodic ple, as they are also dominated by instrumental white noise. variability. This is because instrumental noise will appear as We therefore conclude that the rms-ﬂux relation is detected white noise in the PSDs (constant power across all frequen- within all objects for which the data quality is suﬃcient for cies), whilst accreting systems are known to display strictly its detection. negative power-law indices (with breaks at speciﬁc charac- We can however try and place upper limits on the level teristic frequencies). of intrinsic rms for systems where the rms-ﬂux relation is In ﬁgure 3, PSDs calculated with a Lomb-Scargle not detected. Table 2 shows the mean ﬂux and rms levels (cid:13) c 2015 RAS, MNRAS 000, 1–8 ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: Rms-ﬂux

text: relation

text: in

text: CVs

text: (a)

text: Cyg

text: (b)

text: KIC

text: (c)

text: BOKS

text: Figure

text: Representative

text: PSDs

text: of

text: Cyg,

text: KIC

text: and

text: BOKS

text: in

text: grey.

text: Binned

text: PSDs

text: are

text: shown

text: in

text: black.

text: The

text: lowest

text: frequency

text: sampled

text: is

text: marked

text: by

text: a

text: dashed

text: line,

text: which

text: corresponds

text: to

text: a

text: time-scale

text: of

text: approximately

text: minutes.

text: High-frequency

text: variability

text: is

text: clearly

text: visible

text: in

text: the

text: PSDs

text: of

text: Cyg

text: and

text: KIC

text: while

text: Poisson

text: noise

text: dominates

text: the

text: PSD

text: of

text: BOKS

text: especially

text: at

text: high

text: frequencies.

text: The

text: PSDs

text: of

text: the

text: other

text: ﬁve

text: CVs

text: that

text: do

text: not

text: show

text: the

text: rms-ﬂux

text: relation

text: are

text: similar

text: to

text: ﬁgure

text: Table

text: Mean

text: ﬂux

text: level

text: of

text: a

text: representative

text: minute

text: segment

text: of

text: a

text: steady

text: interval

text: for

text: all

text: objects,

text: together

text: with

text: its

text: rms

text: variability

text: and

text: fractional

text: rms

text: level.

text: Cyg

text: and

text: KIC

text: are

text: the

text: only

text: two

text: systems

text: analysed

text: in

text: this

text: paper

text: with

text: detected

text: rms-ﬂux

text: relations,

text: and

text: are

text: also

text: the

text: only

text: two

text: systems

text: where

text: intrinsic

text: variability

text: dominates

text: over

text: instrumental

text: variability.

text: For

text: these

text: systems

text: the

text: fractional

text: rms

text: column

text: shows

text: the

text: intrinsic

text: value

text: for

text: these

text: systems.

text: The

text: remaining

text: objects

text: are

text: dominated

text: by

text: instrumental

text: noise,

text: and

text: the

text: fractional

text: rms

text: column

text: is

text: indicative

text: on

text: the

text: upper

text: limit

text: for

text: intrinsic

text: variability.

text: For

text: completeness,

text: MV

text: Lyr

text: has

text: been

text: included

text: as

text: well.

text: The

text: Kepler

text: data

text: of

text: MV

text: Lyr

text: displayed

text: a

text: greater

text: range

text: of

text: variability

text: amplitudes

text: than

text: the

text: objects

text: studied

text: in

text: the

text: present

text: paper,

text: so

text: the

text: values

text: obtained

text: for

text: this

text: source

text: are

text: expressed

text: as

text: a

text: range

text: of

text: values

text: (Scaringi

text: et

text: al.

text: In

text: this

text: object

text: intrinsic

text: variability

text: also

text: dominates

text: over

text: instrumental

text: variability.

text: Object

text: Mean

text: ﬂux

text: Rms

text: Fractional

text: Rms

text: dominated

text: by

text: (electrons)

text: (electrons)

text: rms

text: intrinsic

text: variability?

text: BOKS

text: No

text: Lyr

text: No

text: Lyr

text: No

text: Lyr

text: No

text: Lyr

text: No

text: Lyr

text: No

text: Cyg

text: Yes

text: KIC

text: Yes

text: MV

text: Lyr

text: Yes

text: the

text: science

text: channel

text: readouts

text: (Kolodziejczak

text: et

text: al.

text: method

text: for

text: Cyg,

text: KIC

text: and

text: BOKS

text: It

text: has

text: been

text: also

text: pointed

text: out

text: (Wehrle

text: et

text: al.

text: Revalski

text: are

text: shown

text: in

text: grey.

text: Overplotted

text: in

text: black

text: are

text: the

text: smoothed

text: et

text: al.

text: that

text: MPD

text: noise

text: can

text: be

text: a

text: serious

text: problem

text: for

text: PSDs,

text: calculated

text: by

text: binning

text: the

text: PSD

text: and

text: averaging

text: over

text: aperiodic

text: variability

text: studies,

text: especially

text: at

text: high

text: frequencies

text: the

text: bins.

text: We

text: have

text: opted

text: for

text: a

text: Lomb-Scargle

text: method

text: as

text: the

text: for

text: low

text: luminosity

text: sources.

text: As

text: there

text: is

text: no

text: known

text: procedure

text: data

text: sets

text: are

text: unevenly

text: spaced

text: due

text: to

text: the

text: presence

text: of

text: tem-

text: to

text: accurately

text: estimate

text: this

text: noise

text: component

text: we

text: are

text: unable

text: to

text: poral

text: gaps.

text: The

text: lowest

text: frequency

text: we

text: have

text: sampled

text: when

text: place

text: robust

text: constraints

text: on

text: the

text: level

text: of

text: instrumental

text: noise

text: searching

text: for

text: the

text: rms-ﬂux

text: relation

text: is

text: marked

text: by

text: a

text: vertical

text: for

text: the

text: observed

text: systems.

text: We

text: have

text: checked

text: which

text: objects

text: dashed

text: line.

text: From

text: this

text: ﬁgure

text: it

text: can

text: be

text: seen

text: that

text: the

text: PSD

text: were

text: more

text: heavily

text: aﬀected

text: by

text: MPD

text: and

text: found

text: that

text: only

text: of

text: BOKS

text: displays

text: a

text: white

text: noise

text: dominated

text: PSD

text: at

text: KIC

text: and

text: Cyg

text: were

text: aﬀected

text: during

text: and

text: frequencies

text: above

text: (with

text: positive

text: power

text: law

text: index

text: in

text: and

text: respectively

text: (Van

text: Cleve

text: Caldwell

text: power

text: x

text: frequency).

text: Thus

text: in

text: this

text: system

text: any

text: intrinsic

text: vari-

text: The

text: rms-ﬂux

text: relation

text: is

text: detected

text: in

text: all

text: non

text: MPD-aﬀected,

text: ability

text: cannot

text: be

text: measured

text: above

text: that

text: frequency.

text: The

text: PSD

text: as

text: well

text: as

text: MPD-aﬀected

text: quarters.

text: However,

text: we

text: can

text: inspect

text: shape

text: of

text: BOKS

text: is

text: representative

text: of

text: the

text: PSDs

text: of

text: other

text: the

text: PSDs

text: of

text: all

text: objects

text: to

text: determine

text: at

text: which

text: frequencies

text: in-

text: objects

text: which

text: do

text: not

text: show

text: the

text: rms-ﬂux

text: relation

text: in

text: our

text: sam-

text: strumental

text: noise

text: begins

text: to

text: dominate

text: the

text: intrinsic

text: aperiodic

text: ple,

text: as

text: they

text: are

text: also

text: dominated

text: by

text: instrumental

text: white

text: noise.

text: variability.

text: This

text: is

text: because

text: instrumental

text: noise

text: will

text: appear

text: as

text: We

text: therefore

text: conclude

text: that

text: the

text: rms-ﬂux

text: relation

text: is

text: detected

text: white

text: noise

text: in

text: the

text: PSDs

text: (constant

text: power

text: across

text: all

text: frequen-

text: within

text: all

text: objects

text: for

text: which

text: the

text: data

text: quality

text: is

text: suﬃcient

text: for

text: cies),

text: whilst

text: accreting

text: systems

text: are

text: known

text: to

text: display

text: strictly

text: its

text: detection.

text: negative

text: power-law

text: indices

text: (with

text: breaks

text: at

text: speciﬁc

text: charac-

text: We

text: can

text: however

text: try

text: and

text: place

text: upper

text: limits

text: on

text: the

text: level

text: teristic

text: frequencies).

text: of

text: intrinsic

text: rms

text: for

text: systems

text: where

text: the

text: rms-ﬂux

text: relation

text: is

text: In

text: ﬁgure

text: PSDs

text: calculated

text: with

text: a

text: Lomb-Scargle

text: not

text: detected.

text: Table

text: shows

text: the

text: mean

text: ﬂux

text: and

text: rms

text: levels

text: c

text: RAS,

text: MNRAS

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Mon. Not. R. Astron. Soc. 000, 1–11 (2015) Printed 27 May 2020 (MN LATEX style ﬁle v2.2) Bayesian Inference for Radio Observations Michelle Lochner1,2,3,4, Iniyan Natarajan5, Jonathan T. L. Zwart5,6, Oleg Smirnov7,8, Bruce A. Bassett1,2,3, Nadeem Oozeer1,8,9 and Martin Kunz1,10 1African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg, 7945, South Africa 2Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, Cape Town, 7700, South Africa 3South African Astronomical Observatory, Observatory Road, Observatory, Cape Town, 7935, South Africa 4Department of Physics & Astronomy, University College London, London, WC1E 6BT, U.K. 5Astrophysics, Cosmology and Gravity Centre (ACGC), Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa 6Department of Physics & Astronomy, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa 7Department of Physics and Electronics, Rhodes University, PO Box 94, Grahamstown, 6140, South Africa 8SKA South Africa, 3rd Floor, The Park, Park Road, Pinelands, 7405, South Africa 9Centre for Space Research, North-West University, Potchefstroom 2520, South Africa 10D´epartement de Physique Th´eorique and Center for Astroparticle Physics, Universit´e de Gen`eve, Quai E. Ansermet 24, CH-1211 Gen`eve 4, Switzerland ABSTRACT New telescopes like the Square Kilometre Array (SKA) will push into a new sensitivity regime and expose systematics, such as direction-dependent eﬀects, that could previ- ously be ignored. Current methods for handling such systematics rely on alternating best estimates of instrumental calibration and models of the underlying sky, which can lead to inadequate uncertainty estimates and biased results because any correlations between parameters are ignored. These deconvolution algorithms produce a single im- age that is assumed to be a true representation of the sky, when in fact it is just one realisation of an inﬁnite ensemble of images compatible with the noise in the data. In contrast, here we report a Bayesian formalism that simultaneously infers both system- atics and science. Our technique, Bayesian Inference for Radio Observations (BIRO), determines all parameters directly from the raw data, bypassing image-making en- tirely, by sampling from the joint posterior probability distribution. This enables it to derive both correlations and accurate uncertainties, making use of the ﬂexible software MeqTrees to model the sky and telescope simultaneously. We demonstrate BIRO with two simulated sets of Westerbork Synthesis Radio Telescope datasets. In the ﬁrst, we perform joint estimates of 103 scientiﬁc (ﬂux densities of sources) and instrumental (pointing errors, beam width and noise) parameters. In the second example, we perform source separation with BIRO. Using the Bayesian evidence, we can accurately select between a single point source, two point sources and an extended Gaussian source, allowing for ‘super-resolution’ on scales much smaller than the synthesised beam. Key words: methods: data analysis – methods: statistical – techniques: intefero- metric. 1 INTRODUCTION CLEAN (Ho¨gbom 1974), alternating with sky and instru- mental modelling to determine the best-ﬁtting, calibrated The high sensitivity of the SKA (up to 50 times image (Pearson & Readhead 1984; Kazemi et al. 2011; more sensitive than current instruments (Carilli & Rawl- Kazemi & Yatawatta 2013; Bhatnagar et al. 2008). This pro- ings 2004)) combined with a relatively cheap antenna de- vides only a point estimate of the model parameters which sign means a far more careful and detailed treatment of will in general diﬀer from the true parameters due to random systematics will be required to fully exploit this telescope noise (Enßlin et al. 2014). (Noordam 2000). The current approach to this calibration problem iteratively applies deconvolution methods such as A more rigorous approach is to infer the science and (cid:13) c 2015 RAS

text: Mon.

text: Not.

text: R.

text: Astron.

text: Soc.

text: Printed

text: May

text: (MN

text: LATEX

text: style

text: ﬁle

text: Bayesian

text: Inference

text: for

text: Radio

text: Observations

text: Michelle

label: author

text: Iniyan

text: Jonathan

text: T.

text: L.

text: Oleg

text: Bruce

text: A.

text: Nadeem

text: and

text: Martin

text: Institute

text: for

text: Mathematical

text: Sciences,

text: Melrose

text: Road,

text: Muizenberg,

text: South

text: Africa

text: of

text: Mathematics

text: and

text: Applied

text: Mathematics,

text: University

text: of

text: Cape

text: Town,

text: Rondebosch,

text: Cape

text: Town,

text: South

text: Africa

text: African

text: Astronomical

text: Observatory,

text: Observatory

text: Road,

text: Observatory,

text: Cape

text: Town,

text: South

text: Africa

text: of

text: Physics

text: Astronomy,

text: University

text: College

text: London,

text: London,

text: U.K.

text: Cosmology

text: and

text: Gravity

text: Centre

text: (ACGC),

text: Department

text: of

text: Astronomy,

text: University

text: of

text: Cape

text: Town,

text: Private

text: Bag

text: Rondebosch

text: South

text: Africa

text: of

text: Physics

text: Astronomy,

text: University

text: of

text: the

text: Western

text: Cape,

text: Private

text: Bag

text: Bellville

text: South

text: Africa

text: of

text: Physics

text: and

text: Electronics,

text: Rhodes

text: University,

text: PO

text: Box

text: Grahamstown,

text: South

text: Africa

text: South

text: Africa,

text: Floor,

text: The

text: Park,

text: Park

text: Road,

text: Pinelands,

text: South

text: Africa

text: for

text: Space

text: Research,

text: North-West

text: University,

text: Potchefstroom

text: South

text: Africa

text: de

text: Physique

text: Th´eorique

text: and

text: Center

text: for

text: Astroparticle

text: Physics,

text: Universit´e

text: de

text: Gen`eve,

text: Quai

text: E.

text: Ansermet

text: Gen`eve

text: Switzerland

text: ABSTRACT

text: New

text: telescopes

text: like

text: the

text: Square

text: Kilometre

text: Array

text: (SKA)

text: will

text: push

text: into

text: a

text: new

text: sensitivity

text: regime

text: and

text: expose

text: systematics,

text: such

text: as

text: direction-dependent

text: eﬀects,

text: that

text: could

text: previ-

text: ously

text: be

text: ignored.

text: Current

text: methods

text: for

text: handling

text: such

text: systematics

text: rely

text: on

text: alternating

text: best

text: estimates

text: of

text: instrumental

text: calibration

text: and

text: models

text: of

text: the

text: underlying

text: sky,

text: which

text: can

text: lead

text: to

text: inadequate

text: uncertainty

text: estimates

text: and

text: biased

text: results

text: because

text: any

text: correlations

text: between

text: parameters

text: are

text: ignored.

text: These

text: deconvolution

text: algorithms

text: produce

text: a

text: single

text: im-

text: age

text: that

text: is

text: assumed

text: to

text: be

text: a

text: true

text: representation

text: of

text: the

text: sky,

text: when

text: in

text: fact

text: it

text: is

text: just

text: one

text: realisation

text: of

text: an

text: inﬁnite

text: ensemble

text: of

text: images

text: compatible

text: with

text: the

text: noise

text: in

text: the

text: data.

text: In

text: contrast,

text: here

text: we

text: report

text: a

text: Bayesian

text: formalism

text: that

text: simultaneously

text: infers

text: both

text: system-

text: atics

text: and

text: science.

text: Our

text: technique,

text: Bayesian

text: Inference

text: for

text: Radio

text: Observations

text: (BIRO),

text: determines

text: all

text: parameters

text: directly

text: from

text: the

text: raw

text: data,

text: bypassing

text: image-making

text: en-

text: tirely,

text: by

text: sampling

text: from

text: the

text: joint

text: posterior

text: probability

text: distribution.

text: This

text: enables

text: it

text: to

text: derive

text: both

text: correlations

text: and

text: accurate

text: uncertainties,

text: making

text: use

text: of

text: the

text: ﬂexible

text: software

text: MeqTrees

text: to

text: model

text: the

text: sky

text: and

text: telescope

text: simultaneously.

text: We

text: demonstrate

text: BIRO

text: with

text: two

text: simulated

text: sets

text: of

text: Westerbork

text: Synthesis

text: Radio

text: Telescope

text: datasets.

text: In

text: the

text: ﬁrst,

text: we

text: perform

text: joint

text: estimates

text: of

text: scientiﬁc

text: (ﬂux

text: densities

text: of

text: sources)

text: and

text: instrumental

text: (pointing

text: errors,

text: beam

text: width

text: and

text: noise)

text: parameters.

text: In

text: the

text: second

text: example,

text: we

text: perform

text: source

text: separation

text: with

text: BIRO.

text: Using

text: the

text: Bayesian

text: evidence,

text: we

text: can

text: accurately

text: select

text: between

text: a

text: single

text: point

text: source,

text: two

text: point

text: sources

text: and

text: an

text: extended

text: Gaussian

text: source,

text: allowing

text: for

text: ‘super-resolution’

text: on

text: scales

text: much

text: smaller

text: than

text: the

text: synthesised

text: beam.

text: Key

text: words:

text: methods:

text: data

text: analysis

text: methods:

text: statistical

text: techniques:

text: intefero-

text: metric.

text: INTRODUCTION

text: CLEAN

text: (Ho¨gbom

text: alternating

text: with

text: sky

text: and

text: instru-

text: mental

text: modelling

text: to

text: determine

text: the

text: best-ﬁtting,

text: calibrated

text: The

text: high

text: sensitivity

text: of

text: the

text: SKA

text: (up

text: to

text: times

text: image

text: (Pearson

text: Readhead

text: Kazemi

text: et

text: al.

text: more

text: sensitive

text: than

text: current

text: instruments

text: (Carilli

text: Rawl-

text: Kazemi

text: Yatawatta

text: Bhatnagar

text: et

text: al.

text: This

text: pro-

text: ings

text: combined

text: with

text: a

text: relatively

text: cheap

text: antenna

text: de-

text: vides

text: only

text: a

text: point

text: estimate

text: of

text: the

text: model

text: parameters

text: which

text: sign

text: means

text: a

text: far

text: more

text: careful

text: and

text: detailed

text: treatment

text: of

text: will

text: in

text: general

text: diﬀer

text: from

text: the

text: true

text: parameters

text: due

text: to

text: random

text: systematics

text: will

text: be

text: required

text: to

text: fully

text: exploit

text: this

text: telescope

text: noise

text: (Enßlin

text: et

text: al.

text: (Noordam

text: The

text: current

text: approach

text: to

text: this

text: calibration

text: problem

text: iteratively

text: applies

text: deconvolution

text: methods

text: such

text: as

text: A

text: more

text: rigorous

text: approach

text: is

text: to

text: infer

text: the

text: science

text: and

text: c

text: RAS


text: Bayesian Inference for Radio Observations 9 Figure 10: Left column: The true sky for the extended Gaus- Figure 11: Relative natural log-evidence (i.e. the natural log- sian, single point source and two point sources models (from arithm of the ratio of the Bayesian evidence for the true top to bottom). Middle column: The CLEANed image for model to that of a single point source) as a function of Gaus- the three models. Right column: The maximum posterior sian source size, for the extended source input model, show- BIRO image for the three models. The purple contour in ing the evidence-crossover points for diﬀerent source sizes each image indicates the size of the synthesised beam, as and signal-to-noise ratios (peak ﬂux to background noise). returned by CLEAN (note that the sources are all much The horizontal axis gives the size of the circular Gaussian smaller than the synthesised beam). BIRO recovers the cor- source in the input model (the reader is reminded that the rect input model each time while CLEAN is unable to dis- FWHM of the synthesised beam is around 13 arcseconds). tinguish between the models at the same SNR (in this case The vertical axis gives the odds in favour of the Gaussian the SNR was 1000). source model when model comparison is performed for the Gaussian model against a point-source model. The more pos- itive the relative log-evidence is, the more strongly is the Hypothesis Gaussian model favoured. Each curve on the graph is for a diﬀerent noise level with the approximate (map) SNRs Simulation A B C shown in the legend. A 1 : 1 10593 : 1 107200 : 1 B 10993 : 1 1 : 1 105079 : 1 tions are complicated and non-negligible. BIRO eﬀortlessly C 62 : 1 857 : 1 1 : 1 incorporates the eﬀects of the correlations in the estimates of the marginalised uncertainties on the individual parame- ters, as well as providing a way to study these correlations in Table 2: Relative evidences for each model in each simu- the form of the covariance matrix. We compared our results lated dataset. A is the two-source model, B is the extended to a standard CLEAN algorithm, without calibration (since source model and C is the one-source model. The evidences our simulated data contains only direction-dependent eﬀects are relative to the model used to generate the dataset (so, and publicly available calibration algorithms only deal with for example, for the two-point-source dataset, the evidence direction-independent eﬀects). Because of the time-varying for each model is compared to the two point source model). pointing errors we introduce to the dataset, CLEAN is only The maximum error in log-evidence is 1.5. High odds indi- able to ﬁnd 5 out of the 17 sources and returns biased ﬂux cate the input model is favoured (as it is in all three cases), densities for them, while BIRO returns unbiased ﬂux densi- showing that nested sampling selects the correct model at ties for all sources. BIRO is also able to correctly determine high signiﬁcance (at a SNR of 1000). the coeﬃcients of the time-varying pointing errors, the pri- mary beam width and the noise on the visibilities. In the second example problem, we addressed the issue (SNRs). Any evidence lower than ‘strong’ is not usually con- of how to determine the best sky model for the data. We sidered high enough to say either way which model is cor- worked with three models: a single point source, a Gaussian rect. Perhaps obviously, at high SNR extremely small sources extended source and two point sources. We simulated data can be detected (around 1.0 arcseconds) and sources become for each of the three models and then, for each dataset, ran more diﬃcult to distinguish as the SNR is reduced. MultiNest to ﬁt each of the three models and determine Video 1 in the online-only content shows visually how the Bayesian evidence. The evidence then determines the MultiNest converges to the correct model, exploring the pos- selection of the correct model. All of the sources detected terior as it goes, for the extended source model. Each frame were several times smaller than the synthesised beam, hence is an image generated using the parameters from every 40th we successfully achieved super-resolution as well as source- step of the chain. separation. 6 DISCUSSION AND CONCLUSIONS This paper constitutes a proof of concept but more work We have introduced the technique Bayesian Inference is required before the technique can be easily applied to in- for Radio Observations (BIRO), a Bayesian approach to the terferometric dataset: deconvolution problem of radio interferometry. Instead of (i) Firstly, while using a WSRT simulation has relevance making an image and then performing source extraction, to the SKA due to the similar instrumental setup, the SKA BIRO uses MCMC or nested sampling to ﬁt models directly will have many more antennas (on the order of a thou- to the visibility data and obtain the posterior for the param- sand) which will of course result in many more instrumental eters of interest, as well as nuisance parameters. parameters (and indirectly more science parameters as the In the ﬁrst example problem, we focused on the rela- source count increases with sensitivity). Fortunately, while tionship between scientiﬁc and instrumental parameters. It the number of instrumental parameters scales as the number was found that all parameter estimates from BIRO were con- of antennas, N, the number of datapoints scales as the num- O sistent within their error bars with the true values. As well ber of baselines, i.e. (N2), meaning it is plausible that one as determining the uncertainties of the parameters, BIRO could simultaneously determine the sky and instrumental also returns the covariance matrix between them, as a by- parameters for large N. While the precedent for sampling an product of the full posterior. Our work shows these correla- extremely large parameter space exists (Jasche & Wandelt (cid:13) c 2015 RAS, MNRAS 000, 1–11 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Bayesian

text: Inference

text: for

text: Radio

text: Observations

text: Figure

text: Left

text: column:

text: The

text: true

text: sky

text: for

text: the

text: extended

text: Gaus-

text: Figure

text: Relative

text: natural

text: log-evidence

text: (i.e.

text: the

text: natural

text: log-

text: sian,

text: single

text: point

text: source

text: and

text: two

text: point

text: sources

text: models

text: (from

text: arithm

text: of

text: the

text: ratio

text: of

text: the

text: Bayesian

text: evidence

text: for

text: the

text: true

text: top

text: to

text: bottom).

text: Middle

text: column:

text: The

text: CLEANed

text: image

text: for

text: model

text: to

text: that

text: of

text: a

text: single

text: point

text: source)

text: as

text: a

text: function

text: of

text: Gaus-

text: the

text: three

text: models.

text: Right

text: column:

text: The

text: maximum

text: posterior

text: sian

text: source

text: size,

text: for

text: the

text: extended

text: source

text: input

text: model,

text: show-

text: BIRO

text: image

text: for

text: the

text: three

text: models.

text: The

text: purple

text: contour

text: in

text: ing

text: the

text: evidence-crossover

text: points

text: for

text: diﬀerent

text: source

text: sizes

text: each

text: image

text: indicates

text: the

text: size

text: of

text: the

text: synthesised

text: beam,

text: as

text: and

text: signal-to-noise

text: ratios

text: (peak

text: ﬂux

text: to

text: background

text: noise).

text: returned

text: by

text: CLEAN

text: (note

text: that

text: the

text: sources

text: are

text: all

text: much

text: The

text: horizontal

text: axis

text: gives

text: the

text: size

text: of

text: the

text: circular

text: Gaussian

text: smaller

text: than

text: the

text: synthesised

text: beam).

text: BIRO

text: recovers

text: the

text: cor-

text: source

text: in

text: the

text: input

text: model

text: (the

text: reader

text: is

text: reminded

text: that

text: the

text: rect

text: input

text: model

text: each

text: time

text: while

text: CLEAN

text: is

text: unable

text: to

text: dis-

text: FWHM

text: of

text: the

text: synthesised

text: beam

text: is

text: around

text: arcseconds).

text: tinguish

text: between

text: the

text: models

text: at

text: the

text: same

text: SNR

text: (in

text: this

text: case

text: The

text: vertical

text: axis

text: gives

text: the

text: odds

text: in

text: favour

text: of

text: the

text: Gaussian

text: the

text: SNR

text: was

text: source

text: model

text: when

text: model

text: comparison

text: is

text: performed

text: for

text: the

text: Gaussian

text: model

text: against

text: a

text: point-source

text: model.

text: The

text: more

text: pos-

text: itive

text: the

text: relative

text: log-evidence

text: is,

text: the

text: more

text: strongly

text: is

text: the

text: Hypothesis

text: Gaussian

text: model

text: favoured.

text: Each

text: curve

text: on

text: the

text: graph

text: is

text: for

text: a

text: diﬀerent

text: noise

text: level

text: with

text: the

text: approximate

text: (map)

text: SNRs

text: Simulation

text: A

text: B

text: C

text: shown

text: in

text: the

text: legend.

text: A

text: B

text: tions

text: are

text: complicated

text: and

text: non-negligible.

text: BIRO

text: eﬀortlessly

text: C

text: incorporates

text: the

text: eﬀects

text: of

text: the

text: correlations

text: in

text: the

text: estimates

text: of

text: the

text: marginalised

text: uncertainties

text: on

text: the

text: individual

text: parame-

text: ters,

text: as

text: well

text: as

text: providing

text: a

text: way

text: to

text: study

text: these

text: correlations

text: in

text: Table

text: Relative

text: evidences

text: for

text: each

text: model

text: in

text: each

text: simu-

text: the

text: form

text: of

text: the

text: covariance

text: matrix.

text: We

text: compared

text: our

text: results

text: lated

text: dataset.

text: A

text: is

text: the

text: two-source

text: model,

text: B

text: is

text: the

text: extended

text: to

text: a

text: standard

text: CLEAN

text: algorithm,

text: without

text: calibration

text: (since

text: source

text: model

text: and

text: C

text: is

text: the

text: one-source

text: model.

text: The

text: evidences

text: our

text: simulated

text: data

text: contains

text: only

text: direction-dependent

text: eﬀects

text: are

text: relative

text: to

text: the

text: model

text: used

text: to

text: generate

text: the

text: dataset

text: (so,

text: and

text: publicly

text: available

text: calibration

text: algorithms

text: only

text: deal

text: with

text: for

text: example,

text: for

text: the

text: two-point-source

text: dataset,

text: the

text: evidence

text: direction-independent

text: eﬀects).

text: Because

text: of

text: the

text: time-varying

text: for

text: each

text: model

text: is

text: compared

text: to

text: the

text: two

text: point

text: source

text: model).

text: pointing

text: errors

text: we

text: introduce

text: to

text: the

text: dataset,

text: CLEAN

text: is

text: only

text: The

text: maximum

text: error

text: in

text: log-evidence

text: is

text: High

text: odds

text: indi-

text: able

text: to

text: ﬁnd

text: out

text: of

text: the

text: sources

text: and

text: returns

text: biased

text: ﬂux

text: cate

text: the

text: input

text: model

text: is

text: favoured

text: (as

text: it

text: is

text: in

text: all

text: three

text: cases),

text: densities

text: for

text: them,

text: while

text: BIRO

text: returns

text: unbiased

text: ﬂux

text: densi-

text: showing

text: that

text: nested

text: sampling

text: selects

text: the

text: correct

text: model

text: at

text: ties

text: for

text: all

text: sources.

text: BIRO

text: is

text: also

text: able

text: to

text: correctly

text: determine

text: high

text: signiﬁcance

text: (at

text: a

text: SNR

text: of

text: the

text: coeﬃcients

text: of

text: the

text: time-varying

text: pointing

text: errors,

text: the

text: pri-

text: mary

text: beam

text: width

text: and

text: the

text: noise

text: on

text: the

text: visibilities.

text: In

text: the

text: second

text: example

text: problem,

text: we

text: addressed

text: the

text: issue

text: (SNRs).

text: Any

text: evidence

text: lower

text: than

text: ‘strong’

text: is

text: not

text: usually

text: con-

text: of

text: how

text: to

text: determine

text: the

text: best

text: sky

text: model

text: for

text: the

text: data.

text: We

text: sidered

text: high

text: enough

text: to

text: say

text: either

text: way

text: which

text: model

text: is

text: cor-

text: worked

text: with

text: three

text: models:

text: a

text: single

text: point

text: source,

text: a

text: Gaussian

text: rect.

text: Perhaps

text: obviously,

text: at

text: high

text: SNR

text: extremely

text: small

text: sources

text: extended

text: source

text: and

text: two

text: point

text: sources.

text: We

text: simulated

text: data

text: can

text: be

text: detected

text: (around

text: arcseconds)

text: and

text: sources

text: become

text: for

text: each

text: of

text: the

text: three

text: models

text: and

text: then,

text: for

text: each

text: dataset,

text: ran

text: more

text: diﬃcult

text: to

text: distinguish

text: as

text: the

text: SNR

text: is

text: reduced.

text: MultiNest

text: to

text: ﬁt

text: each

text: of

text: the

text: three

text: models

text: and

text: determine

text: Video

text: in

text: the

text: online-only

text: content

text: shows

text: visually

text: how

text: the

text: Bayesian

text: evidence.

text: The

text: evidence

text: then

text: determines

text: the

text: MultiNest

text: converges

text: to

text: the

text: correct

text: model,

text: exploring

text: the

text: pos-

text: selection

text: of

text: the

text: correct

text: model.

text: All

text: of

text: the

text: sources

text: detected

text: terior

text: as

text: it

text: goes,

text: for

text: the

text: extended

text: source

text: model.

text: Each

text: frame

text: were

text: several

text: times

text: smaller

text: than

text: the

text: synthesised

text: beam,

text: hence

text: is

text: an

text: image

text: generated

text: using

text: the

text: parameters

text: from

text: every

text: we

text: successfully

text: achieved

text: super-resolution

text: as

text: well

text: as

text: source-

text: step

text: of

text: the

text: chain.

text: separation.

text: DISCUSSION

text: AND

text: CONCLUSIONS

text: This

text: paper

text: constitutes

text: a

text: proof

text: of

text: concept

text: but

text: more

text: work

text: We

text: have

text: introduced

text: the

text: technique

text: Bayesian

text: Inference

text: is

text: required

text: before

text: the

text: technique

text: can

text: be

text: easily

text: applied

text: to

text: in-

text: for

text: Radio

text: Observations

text: (BIRO),

text: a

text: Bayesian

text: approach

text: to

text: the

text: terferometric

text: dataset:

text: deconvolution

text: problem

text: of

text: radio

text: interferometry.

text: Instead

text: of

text: (i)

text: Firstly,

text: while

text: using

text: a

text: WSRT

text: simulation

text: has

text: relevance

text: making

text: an

text: image

text: and

text: then

text: performing

text: source

text: extraction,

text: to

text: the

text: SKA

text: due

text: to

text: the

text: similar

text: instrumental

text: setup,

text: the

text: SKA

text: BIRO

text: uses

text: MCMC

text: or

text: nested

text: sampling

text: to

text: ﬁt

text: models

text: directly

text: will

text: have

text: many

text: more

text: antennas

text: (on

text: the

text: order

text: of

text: a

text: thou-

text: to

text: the

text: visibility

text: data

text: and

text: obtain

text: the

text: posterior

text: for

text: the

text: param-

text: sand)

text: which

text: will

text: of

text: course

text: result

text: in

text: many

text: more

text: instrumental

text: eters

text: of

text: interest,

text: as

text: well

text: as

text: nuisance

text: parameters.

text: parameters

text: (and

text: indirectly

text: more

text: science

text: parameters

text: as

text: the

text: In

text: the

text: ﬁrst

text: example

text: problem,

text: we

text: focused

text: on

text: the

text: rela-

text: source

text: count

text: increases

text: with

text: sensitivity).

text: Fortunately,

text: while

text: tionship

text: between

text: scientiﬁc

text: and

text: instrumental

text: parameters.

text: It

text: the

text: number

text: of

text: instrumental

text: parameters

text: scales

text: as

text: the

text: number

text: was

text: found

text: that

text: all

text: parameter

text: estimates

text: from

text: BIRO

text: were

text: con-

text: of

text: antennas,

text: N,

text: the

text: number

text: of

text: datapoints

text: scales

text: as

text: the

text: num-

text: O

text: sistent

text: within

text: their

text: error

text: bars

text: with

text: the

text: true

text: values.

text: As

text: well

text: ber

text: of

text: baselines,

text: i.e.

text: meaning

text: it

text: is

text: plausible

text: that

text: one

text: as

text: determining

text: the

text: uncertainties

text: of

text: the

text: parameters,

text: BIRO

text: could

text: simultaneously

text: determine

text: the

text: sky

text: and

text: instrumental

text: also

text: returns

text: the

text: covariance

text: matrix

text: between

text: them,

text: as

text: a

text: by-

text: parameters

text: for

text: large

text: N.

text: While

text: the

text: precedent

text: for

text: sampling

text: an

text: product

text: of

text: the

text: full

text: posterior.

text: Our

text: work

text: shows

text: these

text: correla-

text: extremely

text: large

text: parameter

text: space

text: exists

text: (Jasche

text: Wandelt

text: c

text: RAS,

text: MNRAS

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##