text: 5. FUTURE MISSIONS: AFTA AND ATLAST clei that could be probed with AFTA. A 2.4 m space∼- based telescope used at optical and UV wavelengths ( NRM adds signiﬁcant science impact to any pro- 400nm) with an NRM, would probe diﬀerent structures posed space telescope, for minor technical considerations than JWST’s NIRISS AMI, particularly outer disk struc- (i.e. requiring detectors to be sampled at the resolu- ture in nearby massive Seyfert AGNs. It could also inves- tion achievable with the mask, or pixellation of at least tigate the connection between AGNs and star formation. λ/4D). Future missions with AMI mode capabilities For very large aperture future missions (e.g. a 16 m could probe regions deep in the potential well of SMBHs. optical ATLAST), the additional collecting area would For example, an NRM on ATLAST could image close to allow routine use of spectrometers in AMI mode. Simul- the event horizon of Sgr A* and could image to within a few hundred gravitational radii (r = GM/c2) of nearby taneous spectral information would help constrain the ∼ g physical conditions and dynamics of imaged structures 109M(cid:12) Seyfert AGNs. ATLAST AMI could there- in AGNs–important for answers to many of the big ques- fore be used to image whole AGN disks in the nearby § tions outlined in . Suﬃciently large collecting areas on Universe, allowing us to measure disk warps, gaps and future space telescopes would also permit us to image cavities directly. From Table 1 and Fig. 16, we see that structures in polarized light from AGNs (Sivaramakrish- a 16 m ATLAST is signiﬁcantly preferable to an 8m ver- nan et al. 2009b). Finally, while NIRISS AMI can expect sion, since it gives us most of the torus region in nearby ∼ ∼ to observe tens of AGNs, future missions could target Seyferts (IWA 10 vs 20 pc) and better access to the Draft version May 26, 2020 hundreds. outer disk regions of some of the nearest SMBHs. Preprint typeset using LATEX style emulateapj v. 12/16/11 Figure 16 also shows the regions of several galactic nu- AGN AND QUASAR SCIENCE WITH APERTURE MASKING INTERFEROMETRY ON THE JAMES WEBB SPACE TELESCOPE K.E. Saavik Ford1,2, Barry McKernan1,2 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Anand Sivaramakrishnan1,3, Andre´ R. Martel, Anton Koekemoer, 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218 David Lafrenie`re 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Universit´e de Montr´eal, D´epartement de Physique C.P. 6128 Succ. Centre-ville, Qc, H3C 3J7, Canada and Se´bastien Parmentier, 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794 Subject headings: galaxies: active – galaxies: Seyfert – quasars: general – instrumentation: inter- ferometers – techniques: high angular resolution – space vehicles: instruments – methods: data analysis – techniques: image processing – accretion – accretion disks 6. CONCLUSIONS obtainable with long exposure ground-based O/IR inter- ferometry. JWST-NIRISS AMI will be a unique facil- Space-based aperture masking interferometry (AMI) ity for carrying out moderate to high contrast observa- can image extended structures around AGNs and quasars tions of AGNs at high resolution, allowing us to con- at moderate contrast and high angular resolution. Space- strain models of AGNs binarity, fuelling and structure based imaging does not require a prior model of the tar- at levels beyond present and proposed ground-based ob- get (we obtain fringe phase) unlike ground-based O/IR servatories. NRMs should be considered for addition to interferometry. O/IR interferometry also cannot obtain the ﬁlter wheel of most future space telescope missions, closure amplitudes from the ground, due to atmospheric allowing high resolution and moderate to high contrast itnrasDtsratabofiflti1tvy0e.−r4sWioshneoMudaledymb2o6en,sa2tc0rha2i0teevetdhaint sahoprixteelx-tpoo-spuixreesl wcoitnh- images of extended emission around bright sources in the optical/IR/UV bands. JWST-NIRISS AMI, an order of magnitude better than 1 Department of Astrophysics, American Museum of Natural History, New York, NY 10024 2 CUNY Graduate Center, 365 5th Avenue, New York, NY 10016 3 Department of Physics & Astronomy, Stony Brook Univer- sity, Stony Brook, NY 11794

text: FUTURE

text: MISSIONS:

text: AFTA

text: AND

text: ATLAST

text: clei

text: that

text: could

text: be

text: probed

text: with

text: AFTA.

text: A

text: m

text: space∼-

text: based

text: telescope

text: used

text: at

text: optical

text: and

text: UV

text: wavelengths

text: NRM

text: adds

text: signiﬁcant

text: science

text: impact

text: to

text: any

text: pro-

text: with

text: an

text: NRM,

text: would

text: probe

text: diﬀerent

text: structures

text: posed

text: space

text: telescope,

text: for

text: minor

text: technical

text: considerations

text: than

text: JWST’s

text: NIRISS

text: AMI,

text: particularly

text: outer

text: disk

text: struc-

text: (i.e.

text: requiring

text: detectors

text: to

text: be

text: sampled

text: at

text: the

text: resolu-

text: ture

text: in

text: nearby

text: massive

text: Seyfert

text: AGNs.

text: It

text: could

text: also

text: inves-

text: tion

text: achievable

text: with

text: the

text: mask,

text: or

text: pixellation

text: of

text: at

text: least

text: tigate

text: the

text: connection

text: between

text: AGNs

text: and

text: star

text: formation.

text: Future

text: missions

text: with

text: AMI

text: mode

text: capabilities

text: For

text: very

text: large

text: aperture

text: future

text: missions

text: (e.g.

text: a

text: m

text: could

text: probe

text: regions

text: deep

text: in

text: the

text: potential

text: well

text: of

text: SMBHs.

text: optical

text: ATLAST),

text: the

text: additional

text: collecting

text: area

text: would

text: For

text: example,

text: an

text: NRM

text: on

text: ATLAST

text: could

text: image

text: close

text: to

text: allow

text: routine

text: use

text: of

text: spectrometers

text: in

text: AMI

text: mode.

text: Simul-

text: the

text: event

text: horizon

text: of

text: Sgr

text: A*

text: and

text: could

text: image

text: to

text: within

text: a

text: few

text: hundred

text: gravitational

text: radii

text: (r

text: of

text: nearby

text: taneous

text: spectral

text: information

text: would

text: help

text: constrain

text: the

text: g

text: physical

text: conditions

text: and

text: dynamics

text: of

text: imaged

text: structures

text: Seyfert

text: AGNs.

text: ATLAST

text: AMI

text: could

text: there-

text: in

text: AGNs–important

text: for

text: answers

text: to

text: many

text: of

text: the

text: big

text: ques-

text: fore

text: be

text: used

text: to

text: image

text: whole

text: AGN

text: disks

text: in

text: the

text: nearby

text: tions

text: outlined

text: in

text: Suﬃciently

text: large

text: collecting

text: areas

text: on

text: Universe,

text: allowing

text: us

text: to

text: measure

text: disk

text: warps,

text: gaps

text: and

text: future

text: space

text: telescopes

text: would

text: also

text: permit

text: us

text: to

text: image

text: cavities

text: directly.

text: From

text: Table

text: and

text: Fig.

text: we

text: see

text: that

text: structures

text: in

text: polarized

text: light

text: from

text: AGNs

text: (Sivaramakrish-

text: a

text: m

text: ATLAST

text: is

text: signiﬁcantly

text: preferable

text: to

text: an

text: ver-

text: et

text: al.

text: Finally,

text: while

text: NIRISS

text: AMI

text: can

text: expect

text: sion,

text: since

text: it

text: gives

text: us

text: most

text: of

text: the

text: torus

text: region

text: in

text: nearby

text: to

text: observe

text: tens

text: of

text: AGNs,

text: future

text: missions

text: could

text: target

text: Seyferts

text: (IWA

text: vs

text: pc)

text: and

text: better

text: access

text: to

text: the

text: Draft

text: version

text: May

text: hundreds.

text: outer

text: disk

text: regions

text: of

text: some

text: of

text: the

text: nearest

text: SMBHs.

text: Preprint

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: Figure

text: also

text: shows

text: the

text: regions

text: of

text: several

text: galactic

text: nu-

text: AGN

text: AND

text: QUASAR

text: SCIENCE

text: WITH

text: APERTURE

text: MASKING

text: INTERFEROMETRY

text: ON

text: THE

text: JAMES

text: WEBB

text: SPACE

text: TELESCOPE

text: K.E.

text: Saavik

text: Barry

text: Department

text: of

text: Science,

text: Borough

text: of

text: Manhattan

text: Community

text: College,

text: City

text: University

text: of

text: New

text: York,

text: New

text: York,

text: NY

text: Anand

text: Andre´

text: R.

text: Martel,

text: Anton

text: Koekemoer,

text: Department

text: of

text: Science,

text: Borough

text: of

text: Manhattan

text: Community

text: College,

text: City

text: University

text: of

text: New

text: York,

text: New

text: York,

text: NY

text: Space

text: Telescope

text: Science

text: Institute,

text: San

text: Martin

text: Drive,

text: Baltimore,

text: MD

text: David

text: Lafrenie`re

text: Department

text: of

text: Science,

text: Borough

text: of

text: Manhattan

text: Community

text: College,

text: City

text: University

text: of

text: New

text: York,

text: New

text: York,

text: NY

text: Universit´e

text: de

text: Montr´eal,

text: D´epartement

text: de

text: Physique

text: C.P.

text: Succ.

text: Centre-ville,

text: Qc,

text: Canada

text: and

text: Se´bastien

text: Parmentier,

text: Department

text: of

text: Science,

text: Borough

text: of

text: Manhattan

text: Community

text: College,

text: City

text: University

text: of

text: New

text: York,

text: New

text: York,

text: NY

text: Department

text: of

text: Physics

text: and

text: Astronomy,

text: Stony

text: Brook

text: University,

text: Stony

text: Brook,

text: NY

text: Subject

text: headings:

text: galaxies:

text: active

text: galaxies:

text: Seyfert

text: quasars:

text: general

text: instrumentation:

text: inter-

text: ferometers

text: techniques:

text: high

text: angular

text: resolution

text: space

text: vehicles:

text: instruments

text: methods:

text: data

text: analysis

text: techniques:

text: image

text: processing

text: accretion

text: accretion

text: disks

text: CONCLUSIONS

text: obtainable

text: with

text: long

text: exposure

text: ground-based

text: O/IR

text: inter-

text: ferometry.

text: JWST-NIRISS

text: AMI

text: will

text: be

text: a

text: unique

text: facil-

text: Space-based

text: aperture

text: masking

text: interferometry

text: (AMI)

text: ity

text: for

text: carrying

text: out

text: moderate

text: to

text: high

text: contrast

text: observa-

text: can

text: image

text: extended

text: structures

text: around

text: AGNs

text: and

text: quasars

text: tions

text: of

text: AGNs

text: at

text: high

text: resolution,

text: allowing

text: us

text: to

text: con-

text: at

text: moderate

text: contrast

text: and

text: high

text: angular

text: resolution.

text: Space-

text: strain

text: models

text: of

text: AGNs

text: binarity,

text: fuelling

text: and

text: structure

text: based

text: imaging

text: does

text: not

text: require

text: a

text: prior

text: model

text: of

text: the

text: tar-

text: at

text: levels

text: beyond

text: present

text: and

text: proposed

text: ground-based

text: ob-

text: get

text: (we

text: obtain

text: fringe

text: phase)

text: unlike

text: ground-based

text: O/IR

text: servatories.

text: NRMs

text: should

text: be

text: considered

text: for

text: addition

text: to

text: interferometry.

text: O/IR

text: interferometry

text: also

text: cannot

text: obtain

text: the

text: ﬁlter

text: wheel

text: of

text: most

text: future

text: space

text: telescope

text: missions,

text: closure

text: amplitudes

text: from

text: the

text: ground,

text: due

text: to

text: atmospheric

text: allowing

text: high

text: resolution

text: and

text: moderate

text: to

text: high

text: contrast

text: sahoprixteelx-tpoo-spuixreesl

text: wcoitnh-

text: images

text: of

text: extended

text: emission

text: around

text: bright

text: sources

text: in

text: the

text: optical/IR/UV

text: bands.

text: JWST-NIRISS

text: AMI,

text: an

text: order

text: of

text: magnitude

text: better

text: than

text: Department

text: of

text: Astrophysics,

text: American

text: Museum

text: of

text: Natural

text: History,

text: New

text: York,

text: NY

text: CUNY

text: Graduate

text: Center,

text: Avenue,

text: New

text: York,

text: NY

text: Department

text: of

text: Physics

text: Astronomy,

text: Stony

text: Brook

text: Univer-

text: sity,

text: Stony

text: Brook,

text: NY


text: Interferometric imaging of AGN with JWST 19 Preprint typeset using LATEX style emulateapj v. 12/16/11 AGN AND QUASAR SCIENCE WITH APERTURE MASKING INTERFEROMETRY ON THE JAMES WEBB SPACE TELESCOPE K.E. Saavik Ford1,2, Barry McKernan1,2 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Anand Sivaramakrishnan1,3, Andre´ R. Martel, Anton Koekemoer, 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218 David Lafrenie`re 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Universit´e de Montr´eal, D´epartement de Physique C.P. 6128 Succ. Centre-ville, Qc, H3C 3J7, Canada and Se´bastien Parmentier, 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794 Subject headings: galaxies: active – galaxies: Seyfert – quasars: general – instrumentation: inter- ferometers – techniques: high angular resolution – space vehicles: instruments – methods: data analysis – techniques: image processing – accretion – accretion disks ACKNOWLEDGEMENTS tor’s Discretionary Research Fund. KESF acknowledges the support of a BMCC Faculty Development Grant. We The authors thank the referee for insightful and schol- acknowledge many useful discussions with Ron Allen, arly comments, as well as helpful suggestions that im- Julie Comerford, Laura Ferrarese, John Hutchings, Jin proved the paper’s clarity, completeness, and organiza- Koda, John Monnier, and Peter Teuben. KESF would tion. This work is supported in part by the National like to dedicate this paper to her father, who passed away Science Foundation grant AST 08-04417, PHY 11-25915, in June 2013, after doing so much to help his daughter the NASA grant APRA 08-0117, and the STScI Direc- build a career in astrophysics. APPENDIX NON-REDUNDANT MASKING OVERVIEW We brieﬂy review some relevant details of non-redundant masking (NRM) for observers unfamiliar with the technique. More complete descriptions of the technique can be found in Monnier (2003); Monnier & Allen (2013); Ireland (2013). We assume the Fraunhofer approximation of Fourier optics, viz., the image plane complex amplitude a(k) is the Fourier transform of the pupil plane complex amplitude A(x). In the monochromatic case, at wavelength λ, expressing pupil ≡ ≡ plane coordinates x (x, y) in units of the wavelength yields image plane coordinates k (k , k ) in radians. We x y refer the reader to Bracewell (e.g., 2000) for details on Fourier theoretical results utilized in this section. While our simulations have focussed on JWST-NIRISS’ 7-hole mask (Fig. 1), we describe NRM in more general − terms here. N identical holes in a pupil mask generate N (N 1)/2 sinusoidal fringes in the complex amplitude of h h h the image plane. Each fringe is multiplied by an envelope function, the primary beam, which is the Fourier transform of the individual hole transmission function. The vector between a pair of holes in the mask is referred to as a baseline. The image intensity I(k) for the JWST-NIRISS 7-hole mask is shown in Fig. 3 (b and c). I(k) only contains spatial V frequencies of the sky brightness distribution that are transmitted by the pupil mask. The complex visibility, (u), is the Fourier transform of the image intensity I(k). The domain of the complex visibility function is a pupil-like ≡ V ‘spatial frequency’ space, with coordinates u (u, v). (u) is the Fourier component of the image brightness with V spatial frequency u. An inspection of the absolute value of (u) shows that only certain spatial frequencies pass through the mask (see Fig. 6). For a point source target, the quantity plotted in this ﬁgure is commonly known as the modulation transfer function (or MTF) of the optical system. Its support is obviously the region where the autocorrelation function of the pupil mask is non-zero. The isolated areas of signal in this plane, splodges (Lloyd et al. 2006), are twice as wide as the hole size. The centers of splodges are at the vector separations of the hole centers, i.e., V the baselines. Since I(k) is real, (u) is Hermitian, so only one half of this space provides independent information. 1 Department of Astrophysics, American Museum of Natural History, New York, NY 10024 2 CUNY Graduate Center, 365 5th Avenue, New York, NY 10016 3 Department of Physics & Astronomy, Stony Brook Univer- sity, Stony Brook, NY 11794

text: Interferometric

label: paragraph

text: imaging

text: of

text: AGN

text: with

text: JWST

text: Preprint

label: section

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: AGN

text: AND

text: QUASAR

text: SCIENCE

text: WITH

text: APERTURE

text: MASKING

text: INTERFEROMETRY

text: ON

text: THE

text: JAMES

text: WEBB

text: SPACE

text: TELESCOPE

text: K.E.

label: author

text: Saavik

text: Barry

text: Department

text: of

text: Science,

text: Borough

text: of

text: Manhattan

text: Community

text: College,

text: City

text: University

text: of

text: New

text: York,

text: New

text: York,

text: NY

text: Anand

text: Andre´

text: R.

text: Martel,

text: Anton

text: Koekemoer,

text: Department

text: of

text: Science,

text: Borough

text: of

text: Manhattan

text: Community

text: College,

text: City

text: University

text: of

text: New

text: York,

text: New

text: York,

text: NY

text: Space

text: Telescope

text: Science

text: Institute,

text: San

text: Martin

text: Drive,

text: Baltimore,

text: MD

text: David

text: Lafrenie`re

text: Department

text: of

text: Science,

text: Borough

text: of

text: Manhattan

text: Community

text: College,

text: City

text: University

text: of

text: New

text: York,

text: New

text: York,

text: NY

text: Universit´e

text: de

text: Montr´eal,

text: D´epartement

text: de

text: Physique

text: C.P.

text: Succ.

text: Centre-ville,

text: Qc,

text: Canada

text: and

text: Se´bastien

text: Parmentier,

text: Department

text: of

text: Science,

text: Borough

text: of

text: Manhattan

text: Community

text: College,

text: City

text: University

text: of

text: New

text: York,

text: New

text: York,

text: NY

text: Department

text: of

text: Physics

text: and

text: Astronomy,

text: Stony

text: Brook

text: University,

text: Stony

text: Brook,

text: NY

text: Subject

text: headings:

text: galaxies:

text: active

text: galaxies:

text: Seyfert

text: quasars:

text: general

text: instrumentation:

text: inter-

text: ferometers

text: techniques:

text: high

text: angular

text: resolution

text: space

text: vehicles:

text: instruments

text: methods:

text: data

text: analysis

text: techniques:

text: image

text: processing

text: accretion

text: accretion

text: disks

text: ACKNOWLEDGEMENTS

text: tor’s

text: Discretionary

text: Research

text: Fund.

text: KESF

text: acknowledges

text: the

text: support

text: of

text: a

text: BMCC

text: Faculty

text: Development

text: Grant.

text: We

text: The

text: authors

text: thank

text: the

text: referee

text: for

text: insightful

text: and

text: schol-

text: acknowledge

text: many

text: useful

text: discussions

text: with

text: Ron

text: Allen,

text: arly

text: comments,

text: as

text: well

text: as

text: helpful

text: suggestions

text: that

text: im-

text: Julie

text: Comerford,

text: Laura

text: Ferrarese,

text: John

text: Hutchings,

text: Jin

text: proved

text: the

text: paper’s

text: clarity,

text: completeness,

text: and

text: organiza-

text: Koda,

text: John

text: Monnier,

text: and

text: Peter

text: Teuben.

text: KESF

text: would

text: tion.

text: This

text: work

text: is

text: supported

text: in

text: part

text: by

text: the

text: National

text: like

text: to

text: dedicate

text: this

text: paper

text: to

text: her

text: father,

text: who

text: passed

text: away

text: Science

text: Foundation

text: grant

text: AST

text: PHY

text: in

text: June

text: after

text: doing

text: so

text: much

text: to

text: help

text: his

text: daughter

text: the

text: NASA

text: grant

text: APRA

text: and

text: the

text: STScI

text: Direc-

text: build

text: a

text: career

text: in

text: astrophysics.

text: APPENDIX

text: NON-REDUNDANT

text: MASKING

text: OVERVIEW

text: We

text: brieﬂy

text: review

text: some

text: relevant

text: details

text: of

text: non-redundant

text: masking

text: (NRM)

text: for

text: observers

text: unfamiliar

text: with

text: the

text: technique.

text: More

text: complete

text: descriptions

text: of

text: the

text: technique

text: can

text: be

text: found

text: in

text: Monnier

text: Monnier

text: Allen

text: Ireland

text: We

text: assume

text: the

text: Fraunhofer

text: approximation

text: of

text: Fourier

text: optics,

text: viz.,

text: the

text: image

text: plane

text: complex

text: amplitude

text: a(k)

text: is

text: the

text: Fourier

text: transform

text: of

text: the

text: pupil

text: plane

text: complex

text: amplitude

text: A(x).

text: In

text: the

text: monochromatic

text: case,

text: at

text: wavelength

text: λ,

text: expressing

text: pupil

text: plane

text: coordinates

text: x

text: (x,

text: y)

text: in

text: units

text: of

text: the

text: wavelength

text: yields

text: image

text: plane

text: coordinates

text: k

text: (k

text: k

text: in

text: radians.

text: We

text: x

text: y

text: refer

text: the

text: reader

text: to

text: Bracewell

text: (e.g.,

text: for

text: details

text: on

text: Fourier

text: theoretical

text: results

text: utilized

text: in

text: this

text: section.

text: While

text: our

text: simulations

text: have

text: focussed

text: on

text: JWST-NIRISS’

text: mask

text: (Fig.

text: we

text: describe

text: NRM

text: in

text: more

text: general

text: terms

text: here.

text: N

text: identical

text: holes

text: in

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text: stable when Q < 1. Since κc drops more rapidly than Σ the tests outlined above, but may be possible with future s in the outskirts of realistic gas disk models (e.g. Sirko & telescopes. Goodman 2003), we expect outer AGN disks to be vul- 2.6. Testing the standard model of AGNs nerable to clumping and rapid collapse into star forming The standard model of AGNs assumes that orienta- regions (e.g. Shlosman & Begelman 1987; Goodman & tion is a key determinant of observed properties (An- Tan 2004). The AGN outer disk may therefore consist tonucci 1993; Urry & Padovani 1995). However, random of many unstable, colliding clouds or density ﬂuctuations accretion events onto SMBHs should result in misalign- (e.g. Nenkova et al. 2008; Mor et al. 2009). By imaging ment between axisymmetric galactic disks, AGN accre- the outskirts of AGNs we can put limits on the number, tion disks and the spin of the SMBH (Volonteri et al. size, extent, supply and luminosity of warm clouds in the 2007). A rotating black hole will torque the accretion outer AGN disk, directly testing the link between star disk into its equatorial plane, leading to a warp in a mis- formation and AGNs. NIRISS in AMI mode should im- aligned accretion disk (Bardeen & Petterson 1975; King age the outer accretion disk (torus) of about 20 nearby & Pringle 2006). Thus, obscuration in many AGNs could AGNs; future missions will be required to assemble a ∼ actually be due to misaligned or warped disks (Lawrence larger ( 100’s) sample. & Elvis 2010). Images of the outskirts of AGN disks will 2.5. Searching for gaps and cavities in disks reveal warps and misalignments between the disk and Analogous to protoplanetary accretion disks, gaps and the plane of the host galaxy and test models of stochas- cavities can appear in AGN disks because of the pres- tic accretion. Alignment checks between host galaxy and ence of intermediate mass or SMBHs in the AGN disk fuelling region are achievable with NIRISS in AMI mode (Artymowicz et al. 1993; Syer & Clarke 1995; Ivanov for tens of nearby AGNs; warping in the outer accretion et al. 1999; Levin 2007; Kocsis et al. 2012; McKernan disk is also observable for a smaller sample. Observing et al. 2014). The critical mass ratio (q = q ) of sec- warps in the inner disk awaits instruments with smaller crit ondary black hole to primary SMBH, above which a gap inner working angles. is opened in a disk i(cid:18)s (Lin(cid:19)& Pa(cid:18)palo(cid:19)izou 1986; McKernan Another important source of AGN obscuration is a et al. 2014) large covering of orbiting clouds, independent of the AGN disk orientation (e.g. Krolik 1999; Weedman & ≈ 27π 1/2 H 5/2 Houck 2009; McKernan et al. 2010a). The orbiting qcrit 8 r α1/2 (2) clouds may originate among the unstable clumps be- lieved to exist in the outskirts of the disk. Images of the outskirts of AGN disks (well within range of NIRISS in where H/r is the disk thickness and α is the viscos- AMI mode) will constrain the geometry and ratio of or- ity parameter (Shakura & Sunyaev 1973). Images of bital cloud mass to torus cloud mass and will allow us to gaps and cavities in AGN disks will directly constrain test models of the origin of the orbiting clouds/stars (e.g. models of disk thickness and viscosity as well as models McKernan & Yaqoob 1998; Risaliti et al. 2002; Turner ofDinratefrtmveedrisaiotne Mmaayss26b,l2a0c2k0 hole formation (McKernan & Miller 2009, and references therein). etParle.pr2in0t1t2y)p.esIemt uasginigngLATgEaXpsstyoler ecmauvlaittieeaspjinv. A12G/1N6/1d1isks re- quires much higher angular resolution than required for AGN AND QUASAR SCIENCE WITH APERTURE MASKING INTERFEROMETRY ON THE JAMES WEBB SPACE TELESCOPE K.E. Saavik Ford1,2, Barry McKernan1,2 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Anand Sivaramakrishnan1,3, Andre´ R. Martel, Anton Koekemoer, 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218 David Lafrenie`re 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Universit´e de Montr´eal, D´epartement de Physique C.P. 6128 Succ. Centre-ville, Qc, H3C 3J7, Canada and Se´bastien Parmentier, 0 Department of Science, Borough of Manhattan Community College, City University of New York, New York, NY 10007 Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794 Subject headings: galaxies: active – galaxies: Seyfert – quasars: general – instrumentation: inter- ferometers – techniques: high angular resolution – space vehicles: instruments – methods: data analysis – techniques: image processing – accretion – accretion disks 1 Department of Astrophysics, American Museum of Natural ##LTLine## ##LTLine##

text: stable

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text: Sirko

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text: SMBHs

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text: Nenkova

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text: directly

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text: link

text: between

text: star

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text: its

text: equatorial

text: plane,

text: leading

text: to

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text: warp

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text: mis-

text: formation

text: and

text: AGNs.

text: NIRISS

text: in

text: AMI

text: mode

text: should

text: im-

text: aligned

text: accretion

text: disk

text: (Bardeen

text: Petterson

text: King

text: age

text: the

text: outer

text: accretion

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text: (torus)

text: of

text: about

text: nearby

text: Pringle

text: Thus,

text: obscuration

text: in

text: many

text: AGNs

text: could

text: AGNs;

text: future

text: missions

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text: be

text: required

text: to

text: assemble

text: a

text: actually

text: be

text: due

text: to

text: misaligned

text: or

text: warped

text: disks

text: (Lawrence

text: larger

text: sample.

text: Elvis

text: Images

text: of

text: the

text: outskirts

text: of

text: AGN

text: disks

text: will

text: Searching

text: for

text: gaps

text: and

text: cavities

text: in

text: disks

text: reveal

text: warps

text: and

text: misalignments

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text: the

text: disk

text: and

text: Analogous

text: to

text: protoplanetary

text: accretion

text: disks,

text: gaps

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text: plane

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text: the

text: host

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text: test

text: models

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text: cavities

text: can

text: appear

text: in

text: AGN

text: disks

text: because

text: of

text: the

text: pres-

text: tic

text: accretion.

text: Alignment

text: checks

text: between

text: host

text: galaxy

text: and

text: ence

text: of

text: intermediate

text: mass

text: or

text: SMBHs

text: in

text: the

text: AGN

text: disk

text: fuelling

text: region

text: are

text: achievable

text: with

text: NIRISS

text: in

text: AMI

text: mode

text: (Artymowicz

text: et

text: al.

text: Syer

text: Clarke

text: Ivanov

text: for

text: tens

text: of

text: nearby

text: AGNs;

text: warping

text: in

text: the

text: outer

text: accretion

text: et

text: al.

text: Levin

text: Kocsis

text: et

text: al.

text: McKernan

text: disk

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text: al.

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text: critical

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text: warps

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text: above

text: which

text: a

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text: working

text: angles.

text: is

text: opened

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text: disk

text: McKernan

text: Another

text: important

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text: a

text: et

text: al.

text: large

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text: orbiting

text: clouds,

text: independent

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text: the

text: AGN

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text: (e.g.

text: Krolik

text: Weedman

text: H

text: Houck

text: McKernan

text: et

text: al.

text: The

text: orbiting

text: qcrit

text: r

text: clouds

text: may

text: originate

text: among

text: the

text: unstable

text: clumps

text: be-

text: lieved

text: to

text: exist

text: in

text: the

text: outskirts

text: of

text: the

text: disk.

text: Images

text: of

text: the

text: outskirts

text: of

text: AGN

text: disks

text: (well

text: within

text: range

text: of

text: NIRISS

text: in

text: where

text: H/r

text: is

text: the

text: disk

text: thickness

text: and

text: α

text: is

text: the

text: viscos-

text: AMI

text: mode)

text: will

text: constrain

text: the

text: geometry

text: and

text: ratio

text: of

text: or-

text: ity

text: parameter

text: (Shakura

text: Sunyaev

text: Images

text: of

text: bital

text: cloud

text: mass

text: to

text: torus

text: cloud

text: mass

text: and

text: will

text: allow

text: us

text: to

text: gaps

text: and

text: cavities

text: in

text: AGN

text: disks

text: will

text: directly

text: constrain

text: test

text: models

text: of

text: the

text: origin

text: of

text: the

text: orbiting

text: clouds/stars

text: (e.g.

text: models

text: of

text: disk

text: thickness

text: and

text: viscosity

text: as

text: well

text: as

text: models

text: McKernan

text: Yaqoob

text: Risaliti

text: et

text: al.

text: Turner

text: ofDinratefrtmveedrisaiotne

text: hole

text: formation

text: (McKernan

text: Miller

text: and

text: references

text: therein).

text: uasginigngLATgEaXpsstyoler

text: ecmauvlaittieeaspjinv.

text: re-

text: quires

text: much

text: higher

text: angular

text: resolution

text: than

text: required

text: for

text: AGN

text: AND

text: QUASAR

text: SCIENCE

text: WITH

text: APERTURE

text: MASKING

text: INTERFEROMETRY

text: ON

text: THE

text: JAMES

text: WEBB

text: SPACE

text: TELESCOPE

text: K.E.

text: Saavik

text: Barry

text: Department

text: of

text: Science,

text: Borough

text: of

text: Manhattan

text: Community

text: College,

text: City

text: University

text: of

text: New

text: York,

text: New

text: York,

text: NY

text: Anand

text: Andre´

text: R.

text: Martel,

text: Anton

text: Koekemoer,

text: Department

text: of

text: Science,

text: Borough

text: of

text: Manhattan

text: Community

text: College,

text: City

text: University

text: of

text: New

text: York,

text: New

text: York,

text: NY

text: Space

text: Telescope

text: Science

text: Institute,

text: San

text: Martin

text: Drive,

text: Baltimore,

text: MD

text: David

text: Lafrenie`re

text: Department

text: of

text: Science,

text: Borough

text: of

text: Manhattan

text: Community

text: College,

text: City

text: University

text: of

text: New

text: York,

text: New

text: York,

text: NY

text: Universit´e

text: de

text: Montr´eal,

text: D´epartement

text: de

text: Physique

text: C.P.

text: Succ.

text: Centre-ville,

text: Qc,

text: Canada

text: and

text: Se´bastien

text: Parmentier,

text: Department

text: of

text: Science,

text: Borough

text: of

text: Manhattan

text: Community

text: College,

text: City

text: University

text: of

text: New

text: York,

text: New

text: York,

text: NY

text: Department

text: of

text: Physics

text: and

text: Astronomy,

text: Stony

text: Brook

text: University,

text: Stony

text: Brook,

text: NY

text: Subject

text: headings:

text: galaxies:

text: active

text: galaxies:

text: Seyfert

text: quasars:

text: general

text: instrumentation:

text: inter-

text: ferometers

text: techniques:

text: high

text: angular

text: resolution

text: space

text: vehicles:

text: instruments

text: methods:

text: data

text: analysis

text: techniques:

text: image

text: processing

text: accretion

text: accretion

text: disks

text: Department

text: of

text: Astrophysics,

text: American

text: Museum

text: of

text: Natural

text: ##LTLine##

text: ##LTLine##


text: 6 Ford et al. 3. IMAGING EXTENDED STRUCTURES USING JWST-NIRISS AMI Here we describe the JWST-NIRISS AMI mode, our basic observational strategy and image reconstruction technique. To make our exposition accessible to those un- familiar with imaging interferometry, we include 3 didac- tic appendices on basic topics. We begin with an expla- nation of the JWST-NIRISS AMI system; then discuss our simulated observations including sources of noise; we motivate and describe a move to the Fourier plane; we then explain our standard deconvolution strategy, show- ing results for several models. We then demonstrate some extensions to our standard method, and end with a comparison to ground-based capabilities. We highlight the importance of model-free image reconstruction with NIRISS in AMI mode, since a priori models of AGNs environs are extremely diverse. 3.1. JWST-NIRISS AMI implementation NIRISS’s non-redundant mask consists of a seven- hole, ∼ 15% throughput, titanium mask in a slot in the Fig. 1.— A prototype of the NIRISS 7-hole non-redundant pupil mask (AMI mode) with engraved outline of primary mirror seg- pupil wheel of NIRISS (Doyon et al. 2012). A full-scale ments and secondary supports. Engraving is on the reverse side prototype is shown in ﬁg. 1. The NRM can be used of the mask, so this image is looking through the back of it (ﬂip with NIRISS’s F277W, F380M, F430M or F480M ﬁl- and rotate to compare to ﬁgure 2). The mask has undersized holes compared to the segment size to mitigate against pupil shear up ters, which are centered at 2.77, 3.80, 4.30 and 4.80µm, to 3.8%. The projection of the JWST pupil is nominally circum- with bandwidths of 25, 5, 5 and 8%, respectively. Fig. 2 scribed by a circle 39 mm in diameter. The part diameter is 50 mm. demonstrates the eﬀect of increasingly complex apertures on PSF shape, ending with AMI mode’s theoretical PSF. ﬁeld error of 0.1%; variable intra-pixel response with a Clearly, it is more complicated than the unmasked JWST ± Gaussian proﬁle (1 at center, 0.8 0.05(rms) at the cor- PSF. However, the core of the NRM PSF is noticeably ner); read noise; dark current; background and photon smaller, and it is surrounded by a deep null. The the- noise; a pointing error of 5 mas rms per axis assumed oretical angular resolution is thus 0.5λ/D (65 mas at between each exposure. We expect spectral smearing 4µm), which is more than a factor of 2 smaller than the ∼ ( λ/∆λ) to be negligible when working within 2.5λ/D. 1.22λ/D Rayleigh limit. NIRISS’s 65 mas detector pixels Observations were simulated on a grid 11x ﬁner than are Nyquist-spaced at a wavelength of 4 µm. Fig. 3 shows NIRISS pixels. To simulate a PSF calibrator observation, both oversampled and detector pixel-sampled PSFs (all the noisy point source exposures were binned up by a fac- simulated sources of noise are included in the last panel). tor of 11, as shown in the last panel of Fig. 3. We gen- The NRM image is most useful in a region between an erated a model of extended structure around the point inner and outer working angle (IWA and OWA). For source AGN on the same oversampled grid; convolved NIRISS, the IWA is 0.5λ/D, where λ is the wavelength the model image with a noiseless oversampled PSF. We and D the primary mirror diameter or longest baseline. then shifted the oversampled, convolved model image to The OWA is 4–5 λ/D — at wider angular scales direct match the pointing error of the N th exposure of the imaging is more eﬃcient than NRM. Thus NIRISS’ AMI exp simulated noisy point source sequence. We then binned mode is most interesting at separations between about up the noiseless, shifted, convolved extended structure by 65 mas and 600 mas. Further work may show that in a factor of 11 and added it to the noisy point source ob- the shortest wavelength ﬁlter, F277W, AMI observations servation sequence (with matching pointing error). Our could be dithered to provide slightly smaller IWA than approach is simpler than simulating the point source to- 65 mas (Koekemor & Lindsay 2005; Greenbaum et al. gether with extended structure through the whole optical 2013). path including noise, since noise sources associated with 3.2. Simulated observations and noise sources the point source will dominate. Fig. 4 shows our ﬁducial We simulated observations of a 7.5 magnitude (mag) extended structure model and the resulting NIRISS AMI point source with NIRISS in AMI mode, conservatively mode interference fringes. Our model is a horizontal bar, assuming a detector with a 1% bandpass centered on 9 NIRISS pixels long (585 mas), with integrated ﬂux of 4.5µm. Exposure times to actual NIRISS ﬁlters cen- 8.5 mag, around a 7.5 mag point source AGN. We also tered on 3.80, 4.30, and 4.80µm are extrapolated from considered models non-aligned with our detector grid, our initial simulations; what we simulated as a 1 second which showed similar behavior. exposure would actually take 0.25 seconds with NIRISS We used the ﬁrst 28 noisy exposures of the point source F380M or F430M and 0.16 seconds using NIRISS F480M, as our PSF calibrator, so exposure times were 18.48 given their 5, 5 and 8 % bandwidths, respectively. The (11.76) sec at one orientation for the F380M, F430M observation consisted of 10 sequences of N = 28 ex- and (F480M) ﬁlters. The remaining 9 sequences of 28 exp posures, each exposure equivalent to 0.66 (0.42) seconds exposures (with added extended structure and diﬀerent for the F380M, F430M and (F480M) ﬁlters, respectively. noise realizations) were used to simulate science tar- Simulated noise sources included: a polychromatic PSF get exposures, so science exposure times were 166.32 according to a speciﬁed transmission proﬁle; pixel ﬂat- (105.84) sec at one orientation for the F380M, F430M ##LTFigure## ##LTFigure##

text: Ford

text: et

text: al.

text: IMAGING

text: EXTENDED

text: STRUCTURES

text: USING

text: JWST-NIRISS

text: AMI

text: Here

text: we

text: describe

text: the

text: JWST-NIRISS

text: AMI

text: mode,

text: our

text: basic

text: observational

text: strategy

text: and

text: image

text: reconstruction

text: technique.

text: To

text: make

text: our

text: exposition

text: accessible

text: to

text: those

text: un-

text: familiar

text: with

text: imaging

text: interferometry,

text: we

text: include

text: didac-

text: tic

text: appendices

text: on

text: basic

text: topics.

text: We

text: begin

text: with

text: an

text: expla-

text: nation

text: of

text: the

text: JWST-NIRISS

text: AMI

text: system;

text: then

text: discuss

text: our

text: simulated

text: observations

text: including

text: sources

text: of

text: noise;

text: we

text: motivate

text: and

text: describe

text: a

text: move

text: to

text: the

text: Fourier

text: plane;

text: we

text: then

text: explain

text: our

text: standard

text: deconvolution

text: strategy,

text: show-

text: ing

text: results

text: for

text: several

text: models.

text: We

text: then

text: demonstrate

text: some

text: extensions

text: to

text: our

text: standard

text: method,

text: and

text: end

text: with

text: a

text: comparison

text: to

text: ground-based

text: capabilities.

text: We

text: highlight

text: the

text: importance

text: of

text: model-free

text: image

text: reconstruction

text: with

text: NIRISS

text: in

text: AMI

text: mode,

text: since

text: a

text: priori

text: models

text: of

text: AGNs

text: environs

text: are

text: extremely

text: diverse.

text: JWST-NIRISS

text: AMI

text: implementation

text: NIRISS’s

text: non-redundant

text: mask

text: consists

text: of

text: a

text: seven-

text: hole,

text: throughput,

text: titanium

text: mask

text: in

text: a

text: slot

text: in

text: the

text: Fig.

text: A

text: prototype

text: of

text: the

text: NIRISS

text: non-redundant

text: pupil

text: mask

text: (AMI

text: mode)

text: with

text: engraved

text: outline

text: of

text: primary

text: mirror

text: seg-

text: pupil

text: wheel

text: of

text: NIRISS

text: (Doyon

text: et

text: al.

text: A

text: full-scale

text: ments

text: and

text: secondary

text: supports.

text: Engraving

text: is

text: on

text: the

text: reverse

text: side

text: prototype

text: is

text: shown

text: in

text: ﬁg.

text: The

text: NRM

text: can

text: be

text: used

text: of

text: the

text: mask,

text: so

text: this

text: image

text: is

text: looking

text: through

text: the

text: back

text: of

text: it

text: (ﬂip

text: with

text: NIRISS’s

text: or

text: ﬁl-

text: and

text: rotate

text: to

text: compare

text: to

text: ﬁgure

text: The

text: mask

text: has

text: undersized

text: holes

text: compared

text: to

text: the

text: segment

text: size

text: to

text: mitigate

text: against

text: pupil

text: shear

text: up

text: ters,

text: which

text: are

text: centered

text: at

text: and

text: to

text: The

text: projection

text: of

text: the

text: JWST

text: pupil

text: is

text: nominally

text: circum-

text: with

text: bandwidths

text: of

text: and

text: respectively.

text: Fig.

text: scribed

text: by

text: a

text: circle

text: mm

text: in

text: diameter.

text: The

text: part

text: diameter

text: is

text: mm.

text: demonstrates

text: the

text: eﬀect

text: of

text: increasingly

text: complex

text: apertures

text: on

text: PSF

text: shape,

text: ending

text: with

text: AMI

text: mode’s

text: theoretical

text: PSF.

text: ﬁeld

text: error

text: of

text: variable

text: intra-pixel

text: response

text: with

text: a

text: Clearly,

text: it

text: is

text: more

text: complicated

text: than

text: the

text: unmasked

text: JWST

text: Gaussian

text: proﬁle

text: at

text: center,

text: at

text: the

text: cor-

text: PSF.

text: However,

text: the

text: core

text: of

text: the

text: NRM

text: PSF

text: is

text: noticeably

text: ner);

text: read

text: noise;

text: dark

text: current;

text: background

text: and

text: photon

text: smaller,

text: and

text: it

text: is

text: surrounded

text: by

text: a

text: deep

text: null.

text: The

text: the-

text: noise;

text: a

text: pointing

text: error

text: of

text: mas

text: rms

text: per

text: axis

text: assumed

text: oretical

text: angular

text: resolution

text: is

text: thus

text: mas

text: at

text: between

text: each

text: exposure.

text: We

text: expect

text: spectral

text: smearing

text: which

text: is

text: more

text: than

text: a

text: factor

text: of

text: smaller

text: than

text: the

text: λ/∆λ)

text: to

text: be

text: negligible

text: when

text: working

text: within

text: Rayleigh

text: limit.

text: NIRISS’s

text: mas

text: detector

text: pixels

text: Observations

text: were

text: simulated

text: on

text: a

text: grid

text: ﬁner

text: than

text: are

text: Nyquist-spaced

text: at

text: a

text: wavelength

text: of

text: µm.

text: Fig.

text: shows

text: NIRISS

text: pixels.

text: To

text: simulate

text: a

text: PSF

text: calibrator

text: observation,

text: both

text: oversampled

text: and

text: detector

text: pixel-sampled

text: PSFs

text: (all

text: the

text: noisy

text: point

text: source

text: exposures

text: were

text: binned

text: up

text: by

text: a

text: fac-

text: simulated

text: sources

text: of

text: noise

text: are

text: included

text: in

text: the

text: last

text: panel).

text: tor

text: of

text: as

text: shown

text: in

text: the

text: last

text: panel

text: of

text: Fig.

text: We

text: gen-

text: The

text: NRM

text: image

text: is

text: most

text: useful

text: in

text: a

text: region

text: between

text: an

text: erated

text: a

text: model

text: of

text: extended

text: structure

text: around

text: the

text: point

text: inner

text: and

text: outer

text: working

text: angle

text: (IWA

text: and

text: OWA).

text: For

text: source

text: AGN

text: on

text: the

text: same

text: oversampled

text: grid;

text: convolved

text: NIRISS,

text: the

text: IWA

text: is

text: where

text: λ

text: is

text: the

text: wavelength

text: the

text: model

text: image

text: with

text: a

text: noiseless

text: oversampled

text: PSF.

text: We

text: and

text: D

text: the

text: primary

text: mirror

text: diameter

text: or

text: longest

text: baseline.

text: then

text: shifted

text: the

text: oversampled,

text: convolved

text: model

text: image

text: to

text: The

text: OWA

text: is

text: λ/D

text: at

text: wider

text: angular

text: scales

text: direct

text: match

text: the

text: pointing

text: error

text: of

text: the

text: N

text: th

text: exposure

text: of

text: the

text: imaging

text: is

text: more

text: eﬃcient

text: than

text: NRM.

text: Thus

text: NIRISS’

text: AMI

text: exp

text: simulated

text: noisy

text: point

text: source

text: sequence.

text: We

text: then

text: binned

text: mode

text: is

text: most

text: interesting

text: at

text: separations

text: between

text: about

text: up

text: the

text: noiseless,

text: shifted,

text: convolved

text: extended

text: structure

text: by

text: mas

text: and

text: mas.

text: Further

text: work

text: may

text: show

text: that

text: in

text: a

text: factor

text: of

text: and

text: added

text: it

text: to

text: the

text: noisy

text: point

text: source

text: ob-

text: the

text: shortest

text: wavelength

text: ﬁlter,

text: AMI

text: observations

text: servation

text: sequence

text: (with

text: matching

text: pointing

text: error).

text: Our

text: could

text: be

text: dithered

text: to

text: provide

text: slightly

text: smaller

text: IWA

text: than

text: approach

text: is

text: simpler

text: than

text: simulating

text: the

text: point

text: source

text: to-

text: mas

text: (Koekemor

text: Lindsay

text: Greenbaum

text: et

text: al.

text: gether

text: with

text: extended

text: structure

text: through

text: the

text: whole

text: optical

text: path

text: including

text: noise,

text: since

text: noise

text: sources

text: associated

text: with

text: Simulated

text: observations

text: and

text: noise

text: sources

text: the

text: point

text: source

text: will

text: dominate.

text: Fig.

text: shows

text: our

text: ﬁducial

text: We

text: simulated

text: observations

text: of

text: a

text: magnitude

text: (mag)

text: extended

text: structure

text: model

text: and

text: the

text: resulting

text: NIRISS

text: AMI

text: point

text: source

text: with

text: NIRISS

text: in

text: AMI

text: mode,

text: conservatively

text: mode

text: interference

text: fringes.

text: Our

text: model

text: is

text: a

text: horizontal

text: bar,

text: assuming

text: a

text: detector

text: with

text: a

text: bandpass

text: centered

text: on

text: NIRISS

text: pixels

text: long

text: mas),

text: with

text: integrated

text: ﬂux

text: of

text: Exposure

text: times

text: to

text: actual

text: NIRISS

text: ﬁlters

text: cen-

text: mag,

text: around

text: a

text: mag

text: point

text: source

text: AGN.

text: We

text: also

text: tered

text: on

text: and

text: are

text: extrapolated

text: from

text: considered

text: models

text: non-aligned

text: with

text: our

text: detector

text: grid,

text: our

text: initial

text: simulations;

text: what

text: we

text: simulated

text: as

text: a

text: second

text: which

text: showed

text: similar

text: behavior.

text: exposure

text: would

text: actually

text: take

text: seconds

text: with

text: NIRISS

text: We

text: used

text: the

text: ﬁrst

text: noisy

text: exposures

text: of

text: the

text: point

text: source

text: or

text: and

text: seconds

text: using

text: NIRISS

text: as

text: our

text: PSF

text: calibrator,

text: so

text: exposure

text: times

text: were

text: given

text: their

text: and

text: bandwidths,

text: respectively.

text: The

text: sec

text: at

text: one

text: orientation

text: for

text: the

text: observation

text: consisted

text: of

text: sequences

text: of

text: N

text: ex-

text: and

text: ﬁlters.

text: The

text: remaining

text: sequences

text: of

text: exp

text: posures,

text: each

text: exposure

text: equivalent

text: to

text: seconds

text: exposures

text: (with

text: added

text: extended

text: structure

text: and

text: diﬀerent

text: for

text: the

text: and

text: ﬁlters,

text: respectively.

text: noise

text: realizations)

text: were

text: used

text: to

text: simulate

text: science

text: tar-

text: Simulated

text: noise

text: sources

text: included:

text: a

text: polychromatic

text: PSF

text: get

text: exposures,

text: so

text: science

text: exposure

text: times

text: were

text: according

text: to

text: a

text: speciﬁed

text: transmission

text: proﬁle;

text: pixel

text: ﬂat-

text: sec

text: at

text: one

text: orientation

text: for

text: the

text: ##LTFigure##

text: ##LTFigure##


text: Interferometric imaging of AGN with JWST 9 1995, and appendix ). While this trade also exists for full- aperture imaging, the inherently smaller PSF core and consequent improvement in theoretical angular resolu- tion of an NRM system means that any result achievable with a full-aperture deconvolution can be improved upon in NRM imaging. We use the normalized inverse Fourier transform of the extracted, sub-pixel aligned calibrator visibilities as our PSF (or ‘dirty beam’, B). This contrasts with the stan- dard radio strategy of using the inverse Fourier transform of the target visibilities, with (A, φ) = (1, 0) as the ‘dirty beam’ or PSF. Because we include u points away from splodge centers, our technique is equivalent to construct- ing a full Spatial Transfer Function. We use our PSF to clean the inverse Fourier transformed, extracted, sub- (cid:48) pixel aligned target visibilities (or ‘dirty map’, I ). fringe We choose the smallest useful clean box. For the cases we show this was a square region 32 pixels on a side, centered on the brightest pixel in the fringe image. If we wish to image larger regions, we can do so up to a region one-quarter the size of the PSF; alternatively, we can use multiple pointings to construct a mosaic of ex- Fig. 7.— As Fig. 6, except for phase splodges. Note that in tremely large ﬁelds. ground-based O/IR, fringe phases cannot be obtained, so closure phase and source priors must be used to obtain astrophysically We show the clean model and restored map in Fig. 11 interesting information. (see appendix for deﬁnitions). The clean model is the same as that shown in the bottom right panel of Fig. 5 and compares well with the input model shown in Fig. 4a. For NRM systems which capture fringes as gridded detec- tor images (like NIRISS), the most useful interpretation of the data is obtained by viewing and comparing both the clean model and the restored map. Other models: Fig. 12 shows image reconstructions for some other extended emission models, including short and asymmetric bars and a ring (using the same tech- niques and exposure times as above). The second row demonstrates the recovery of a 9 NIRISS pixel long bar, at an integrated ﬂux of 9.5 mag, or 2 mag integrated contrast, equivalent to 4.4 mag pixel-to-pixel contrast ∼ − ( 10 2), in 5.5 (3.5) min√at 3.80, 4.30, (4.80)µm. As- suming SNR improves as t (i.e. we are photon noise limited), we should reach 5 mag of contrast (pixel-to- pixel) imaging extended structures in less than 10 min- utes. All results presented thus far use clean, without the use of a model prior or selfcal (which would incorpo- rate closure phase and closure amplitude information–see appendix for deﬁnitions). Much of ground-based opti- cal interferometry is used to measure a small number of Fig. 8.— Greatest u plane coverage possible given 2 observations parameters e.g. separation, contrast and position angle separated by a roll angle of 90 degrees. Two sets of amplitude for two point sources. However, this approach will not splodges (as for ﬁgure 6) with the second rotated by 90 degrees to the ﬁrst. Extraction reduces the coverage further (see ﬁgure 9, last work for sources with complicated extended structure, panel, for comparison, and appendix ). such as AGNs, where prior models are extremely diverse and have large numbers of variables. We note however transform) calibrator and target data, as shown in Fig. 10 that more advanced image reconstruction techniques are (see appendix for details). We have recovered our origi- available (Baron et al. 2012, and references therein). We nal fringes, now properly aligned for deconvolution, and have not investigated their use on space-based data and weighted according to their u plane coverage (the weight- oﬀer our simulations as a conservative estimate of ex- ing is non-uniform due to the combination of multiple ori- tended source imaging capabilities. entations and the lower weighting of oﬀ-center extracted 3.5. Other strategies points). We have chosen an image plane pixel scale ap- We have investigated some extensions to the simplest proximately twice as ﬁne as our original detector images clean reductions, shown in Fig. 13. Selfcal enforces (35.6 mas). Pixel scale and signal-to-noise ratio (SNR) closure relations on visibility data when a model prior per pixel can be traded in any deconvolution scheme– allows calculation of ‘expected’ closure phases (CPs) this is one eﬀect of various weighting schemes (see Briggs ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine##

text: Interferometric

text: imaging

text: of

text: AGN

text: with

text: JWST

text: and

text: appendix

text: While

text: this

text: trade

text: also

text: exists

text: for

text: full-

text: aperture

text: imaging,

text: the

text: inherently

text: smaller

text: PSF

text: core

text: and

text: consequent

text: improvement

text: in

text: theoretical

text: angular

text: resolu-

text: tion

text: of

text: an

text: NRM

text: system

text: means

text: that

text: any

text: result

text: achievable

text: with

text: a

text: full-aperture

text: deconvolution

text: can

text: be

text: improved

text: upon

text: in

text: NRM

text: imaging.

text: We

text: use

text: the

text: normalized

text: inverse

text: Fourier

text: transform

text: of

text: the

text: extracted,

text: sub-pixel

text: aligned

text: calibrator

text: visibilities

text: as

text: our

text: PSF

text: (or

text: ‘dirty

text: beam’,

text: B).

text: This

text: contrasts

text: with

text: the

text: stan-

text: dard

text: radio

text: strategy

text: of

text: using

text: the

text: inverse

text: Fourier

text: transform

text: of

text: the

text: target

text: visibilities,

text: with

text: (A,

text: φ)

text: as

text: the

text: ‘dirty

text: beam’

text: or

text: PSF.

text: Because

text: we

text: include

text: u

text: points

text: away

text: from

text: splodge

text: centers,

text: our

text: technique

text: is

text: equivalent

text: to

text: construct-

text: ing

text: a

text: full

text: Spatial

text: Transfer

text: Function.

text: We

text: use

text: our

text: PSF

text: to

text: clean

text: the

text: inverse

text: Fourier

text: transformed,

text: extracted,

text: sub-

text: pixel

text: aligned

text: target

text: visibilities

text: (or

text: ‘dirty

text: map’,

text: I

text: fringe

text: We

text: choose

text: the

text: smallest

text: useful

text: clean

text: box.

text: For

text: the

text: cases

text: we

text: show

text: this

text: was

text: a

text: square

text: region

text: pixels

text: on

text: a

text: side,

text: centered

text: on

text: the

text: brightest

text: pixel

text: in

text: the

text: fringe

text: image.

text: If

text: we

text: wish

text: to

text: image

text: larger

text: regions,

text: we

text: can

text: do

text: so

text: up

text: to

text: a

text: region

text: one-quarter

text: the

text: size

text: of

text: the

text: PSF;

text: alternatively,

text: we

text: can

text: use

text: multiple

text: pointings

text: to

text: construct

text: a

text: mosaic

text: of

text: ex-

text: Fig.

text: As

text: Fig.

text: except

text: for

text: phase

text: splodges.

text: Note

text: that

text: in

text: tremely

text: large

text: ﬁelds.

text: ground-based

text: O/IR,

text: fringe

text: phases

text: cannot

text: be

text: obtained,

text: so

text: closure

text: phase

text: and

text: source

text: priors

text: must

text: be

text: used

text: to

text: obtain

text: astrophysically

text: We

text: show

text: the

text: clean

text: model

text: and

text: restored

text: map

text: in

text: Fig.

text: interesting

text: information.

text: (see

text: appendix

text: for

text: deﬁnitions).

text: The

text: clean

text: model

text: is

text: the

text: same

text: as

text: that

text: shown

text: in

text: the

text: bottom

text: right

text: panel

text: of

text: Fig.

text: and

text: compares

text: well

text: with

text: the

text: input

text: model

text: shown

text: in

text: Fig.

text: For

text: NRM

text: systems

text: which

text: capture

text: fringes

text: as

text: gridded

text: detec-

text: tor

text: images

text: (like

text: NIRISS),

text: the

text: most

text: useful

text: interpretation

text: of

text: the

text: data

text: is

text: obtained

text: by

text: viewing

text: and

text: comparing

text: both

text: the

text: clean

text: model

text: and

text: the

text: restored

text: map.

text: Other

text: models:

text: Fig.

text: shows

text: image

text: reconstructions

text: for

text: some

text: other

text: extended

text: emission

text: models,

text: including

text: short

text: and

text: asymmetric

text: bars

text: and

text: a

text: ring

text: (using

text: the

text: same

text: tech-

text: niques

text: and

text: exposure

text: times

text: as

text: above).

text: The

text: second

text: row

text: demonstrates

text: the

text: recovery

text: of

text: a

text: NIRISS

text: pixel

text: long

text: bar,

text: at

text: an

text: integrated

text: ﬂux

text: of

text: mag,

text: or

text: mag

text: integrated

text: contrast,

text: equivalent

text: to

text: mag

text: pixel-to-pixel

text: contrast

text: in

text: min√at

text: As-

text: suming

text: SNR

text: improves

text: as

text: t

text: (i.e.

text: we

text: are

text: photon

text: noise

text: limited),

text: we

text: should

text: reach

text: mag

text: of

text: contrast

text: (pixel-to-

text: pixel)

text: imaging

text: extended

text: structures

text: in

text: less

text: than

text: min-

text: utes.

text: All

text: results

text: presented

text: thus

text: far

text: use

text: clean,

text: without

text: the

text: use

text: of

text: a

text: model

text: prior

text: or

text: selfcal

text: (which

text: would

text: incorpo-

text: rate

text: closure

text: phase

text: and

text: closure

text: amplitude

text: information–see

text: appendix

text: for

text: deﬁnitions).

text: Much

text: of

text: ground-based

text: opti-

text: cal

text: interferometry

text: is

text: used

text: to

text: measure

text: a

text: small

text: number

text: of

text: Fig.

text: Greatest

text: u

text: plane

text: coverage

text: possible

text: given

text: observations

text: parameters

text: e.g.

text: separation,

text: contrast

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text: angle

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text: a

text: roll

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text: degrees.

text: Two

text: sets

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text: amplitude

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text: sources.

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text: We

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text: We

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text: ##LTLine##


text: 2 ≈ 2. SFR - DENSITY RELATION ON VARIOUS SCALES 2013) and high redshift (τSF 0.7 Gyr Genzel et al. 2010; Daddi et al. 2010; Tacconi et al. 2013). In this paper we are It is useful to think about relationship between star forma- going to overlook these important details, since we are only tion rate and gas densities (on any scale) in terms of the time- interested in the broad-brush eﬀect of varying spatial scale on scale for gas consumption, the so-called “depletion time” τSF, the SFR - density relation. Hence, for our purposes it is suﬃ- ρ˙∗ = ρ . cient to consider τSF as constant on large scales. τSF preLsasdead aest al. (2010) observations can now be simply ex- The above equation is, however, an incorrect way of consid-  (cid:104) (cid:105) ering star formation, because the density is only deﬁned on a (cid:104) (cid:105) ρ 1pc particular scale - we need to explicitly consider the range of ρ˙∗ = 20 Myr, ρ > ρmin, 1pc spatial scales that is relevant for our problem. 0, ρ < ρmin, Let us take some spatial scale L. One can imagine a region of interest (be it the whole universe or a single galaxy) divided with ρmin = 700M(cid:12)/ pc3. For this equation to be consistent into boxes of size L. If we average gas densities on scale L, with the Kennicutt-Schmidt relation on large (say, 500 pc) they become meaningfully deﬁned. Thus, the above equation scale, (cid:104) (cid:105) could be replaced with (cid:104) (cid:105) (cid:104)ρ˙∗(cid:105)500pc = ρ2 G50y0rpc , (cid:104)ρ˙∗(cid:105)L = ρ L , exactly 1% of the molecular g(cid:68)as must(cid:69)sit above the small-scale τSF density threshold ρmin - after all, but even that is not general enough. Star formation rate is (cid:104) (cid:105) (cid:104) (cid:105) not a deterministic function of gas density, but dependent on ρ˙∗ = ρ˙∗ . 500pc 1pc 500pc other physical parameters (previous history, dynamic state of the gas, etc); it needs to be treated as a stochastic process, In the language of depletion time the (cid:115)“constant eﬃciency described by a probability distribution of having a particular per free-fall time” ansatz becomes value of the instantaneous star(cid:104)fo(cid:105)rm|(cid:104)at(cid:105)ion rate density for a (cid:104) (cid:105) givFeonr gaansydfeonrmsityofatps,chaolewLev,epr,(iρt˙∗isLpoρssLib).le to deﬁne the av- τSF(L, (cid:104)ρ(cid:105)L, ...) = τﬀ((cid:15)SρF L) = (cid:15)S−F1 32G3π(cid:104)ρ(cid:105)L , erage instantaneous star form(cid:90)ation rate density at a given gas density, or, in a more familiar form, (cid:104) (cid:105) (cid:104) (cid:105) (cid:104)ρ˙∗(cid:105)L ((cid:104)ρ(cid:105)L) ≡ xp (x|(cid:104)ρ(cid:105)L) dx. (1) (cid:104)ρ˙∗(cid:105)L = (cid:15)SF τρﬀL = (cid:15)SF √3πρ/(3L3/22G). (3) Notation here becomes complex, so we use angular brack- This ansatz is consistent with the observational constraints (cid:104)(cid:105) ets to indicate averaging over a given spatial scale (a single presented by Krumholz & Tan (2007). However, those con- L box of size L), and the over-bar to mark the average over all straints sample not only various densities, but also various (cid:104) (cid:105) − boxes with the same ρ L; to sharpen the terminology, we also spatial scales. Indeed, the lowest densities nH (cid:38) 10 cm 3 in call the latter average ”the expectation value of the instan- Krumholz & Tan (2007) are sampled by CO measurements∼, taneous star formation rate surface density on scale L” (EV- that−typically include whole mole∼cular clo−uds on scales L ISFRD). With this notation and terminology, we can ﬁnally 10 30 pc; densities around nH 103 cm 3 are ob−served in relate the EVISFRD on spatial scale L to the gas density on Infrared Dark Clouds (IRDC), which have sizes 1 3 pc; ﬁ- ∼ − the same scale, nally the highest densities of n 105 cm 3 are only present H − (cid:104) (cid:105) in HCN-emitting star-forming cores with sizes around 0.1 (cid:104)ρ˙∗(cid:105) = ρ L , (2) 0.3 pc or below. Hence, the observational constraints used L by Krumholz & Tan (2007) all fall along a particular track τSF × (cid:104) (cid:105) ∼ − × − (cid:104) (cid:105) (cid:104) (cid:105) L2 ρ L(cid:104) (cid:105) (1 10) 103 cm 3 pc2 in the two-dimensional (or, equivalently, Σ∗ L = Σ L/τSF when considering sur- plane (L, ρ L). In other words, observational constraints tha∝t face densities). In that case depletion time becomes the func- s(cid:104)u(cid:105)p−port the “constant eﬃciency per free-fall time” (τSF tion of other gas proτpSeFrt=iesτSoFn(Lsc, a(cid:104)ρle(cid:105)LL,,...). sρcaInLle1f”/a2()cτte,SqFthu∝ealflLyo)l.wloewllinsgupt(cid:40)wpoortmthoede“lcsofnosrt(cid:16)athnet edﬃ(cid:17)epclieetniocnytpimereu:nit In other words, we need to explicitly consider the star for- : “linear” mplaaItnnieotnh(Lirs,e(cid:104)lfaρot(cid:105)riLom)n.aalissm(attlheeaslta)rgaet-wscoa-dle(cid:12)(cid:12)imKeennsnioicnuatlt-rSelcahtmioindtonretlhae- τSF(L, (cid:104)ρ(cid:105)L, ...) ∼ 2∞G, y(cid:26)r × min 1, LL0 , (cid:104)(cid:104)ρρ(cid:105)(cid:105)LL >< ρρmmiinn,, (cid:12) tion can be expressed as : “3/2” (cid:104) (cid:105) τSF = (2 ± 1) Gyr L(cid:29)100pc , τSF(L, (cid:104)ρ(cid:105)L(cid:16), ...) ∼ 12(cid:17)0G0yτrﬀ,( ρ L), LL <> LL00,, with substantial intrinsic (i.e. exceeding the formal observa- ∼ × tional error) scatter an≈d the actual value for τSF noticeably with ρmin ρ0 min 1, L02/L2 , cannot be distinguished at diﬀerent at z = 0 (τSF 2 Gyr Leroy et al. 2008; Bigiel et al. present without additional observational constraints. The pa- 2008; Bolatto et al. 2011; Bigiel et al. 2011; Leroy et al. 2012, rameter L in both cases must be in the range of a few hundred 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: MIUBVL+txsy

text: SFR

text: DENSITY

text: RELATION

text: ON

text: VARIOUS

text: SCALES

text: and

text: high

text: redshift

text: (τSF

text: Gyr

text: Genzel

text: et

text: al.

text: Daddi

text: et

text: al.

text: Tacconi

text: et

text: al.

text: In

text: this

text: paper

text: we

text: are

text: It

text: is

text: useful

text: to

text: think

text: about

text: relationship

text: between

text: star

text: forma-

text: going

text: to

text: overlook

text: these

text: important

text: details,

text: since

text: we

text: are

text: only

text: tion

text: rate

text: and

text: gas

text: densities

text: (on

text: any

text: scale)

text: in

text: terms

text: of

text: the

text: time-

text: interested

text: in

text: the

text: broad-brush

text: eﬀect

text: of

text: varying

text: spatial

text: scale

text: on

text: scale

text: for

text: gas

text: consumption,

text: the

text: so-called

text: “depletion

text: time”

text: τSF,

text: the

text: SFR

text: density

text: relation.

text: Hence,

text: for

text: our

text: purposes

text: it

text: is

text: suﬃ-

text: ρ˙∗

text: ρ

text: cient

text: to

text: consider

text: τSF

text: as

text: constant

text: on

text: large

text: scales.

text: τSF

text: preLsasdead

text: aest

text: al.

text: observations

text: can

text: now

text: be

text: simply

text: ex-

text: The

text: above

text: equation

text: is,

text: however,

text: an

text: incorrect

text: way

text: of

text: consid-

text: ering

text: star

text: formation,

text: because

text: the

text: density

text: is

text: only

text: deﬁned

text: on

text: a

text: ρ

text: particular

text: scale

text: we

text: need

text: to

text: explicitly

text: consider

text: the

text: range

text: of

text: ρ˙∗

text: Myr,

text: ρ

text: ρmin,

text: spatial

text: scales

text: that

text: is

text: relevant

text: for

text: our

text: problem.

text: ρ

text: ρmin,

text: Let

text: us

text: take

text: some

text: spatial

text: scale

text: L.

text: One

text: can

text: imagine

text: a

text: region

text: of

text: interest

text: (be

text: it

text: the

text: whole

text: universe

text: or

text: a

text: single

text: galaxy)

text: divided

text: with

text: ρmin

text: For

text: this

text: equation

text: to

text: be

text: consistent

text: into

text: boxes

text: of

text: size

text: L.

text: If

text: we

text: average

text: gas

text: densities

text: on

text: scale

text: L,

text: with

text: the

text: Kennicutt-Schmidt

text: relation

text: on

text: large

text: (say,

text: pc)

text: they

text: become

text: meaningfully

text: deﬁned.

text: Thus,

text: the

text: above

text: equation

text: scale,

text: could

text: be

text: replaced

text: with

text: ρ2

text: ρ

text: L

text: exactly

text: of

text: the

text: molecular

text: above

text: the

text: small-scale

text: τSF

text: density

text: threshold

text: ρmin

text: after

text: all,

text: but

text: even

text: that

text: is

text: not

text: general

text: enough.

text: Star

text: formation

text: rate

text: is

text: not

text: a

text: deterministic

text: function

text: of

text: gas

text: density,

text: but

text: dependent

text: on

text: ρ˙∗

text: ρ˙∗

text: other

text: physical

text: parameters

text: (previous

text: history,

text: dynamic

text: state

text: of

text: the

text: gas,

text: etc);

text: it

text: needs

text: to

text: be

text: treated

text: as

text: a

text: stochastic

text: process,

text: In

text: the

text: language

text: of

text: depletion

text: time

text: the

text: eﬃciency

text: described

text: by

text: a

text: probability

text: distribution

text: of

text: having

text: a

text: particular

text: per

text: free-fall

text: time”

text: ansatz

text: becomes

text: value

text: of

text: the

text: instantaneous

text: rate

text: density

text: for

text: a

text: givFeonr

text: to

text: deﬁne

text: the

text: av-

text: τSF(L,

text: L)

text: erage

text: instantaneous

text: star

text: rate

text: density

text: at

text: a

text: given

text: gas

text: density,

text: or,

text: in

text: a

text: more

text: familiar

text: form,

text: xp

text: dx.

text: τρﬀL

text: Notation

text: here

text: becomes

text: complex,

text: so

text: we

text: use

text: angular

text: brack-

text: This

text: ansatz

text: is

text: consistent

text: with

text: the

text: observational

text: constraints

text: ets

text: to

text: indicate

text: averaging

text: over

text: a

text: given

text: spatial

text: scale

text: (a

text: single

text: presented

text: by

text: Krumholz

text: Tan

text: However,

text: those

text: con-

text: L

text: box

text: of

text: size

text: L),

text: and

text: the

text: over-bar

text: to

text: mark

text: the

text: average

text: over

text: all

text: straints

text: sample

text: not

text: only

text: various

text: densities,

text: but

text: also

text: various

text: boxes

text: with

text: the

text: same

text: ρ

text: L;

text: to

text: sharpen

text: the

text: terminology,

text: we

text: also

text: spatial

text: scales.

text: Indeed,

text: the

text: lowest

text: densities

text: nH

text: cm

text: in

text: call

text: the

text: latter

text: average

text: ”the

text: expectation

text: value

text: of

text: the

text: instan-

text: Krumholz

text: Tan

text: are

text: sampled

text: by

text: CO

text: measurements∼,

text: taneous

text: star

text: formation

text: rate

text: surface

text: density

text: on

text: scale

text: L”

text: (EV-

text: that−typically

text: include

text: whole

text: mole∼cular

text: clo−uds

text: on

text: scales

text: L

text: ISFRD).

text: With

text: this

text: notation

text: and

text: terminology,

text: we

text: can

text: ﬁnally

text: pc;

text: densities

text: around

text: nH

text: cm

text: are

text: ob−served

text: in

text: relate

text: the

text: EVISFRD

text: on

text: spatial

text: scale

text: L

text: to

text: the

text: gas

text: density

text: on

text: Infrared

text: Dark

text: Clouds

text: (IRDC),

text: which

text: have

text: sizes

text: pc;

text: ﬁ-

text: the

text: same

text: scale,

text: nally

text: the

text: highest

text: densities

text: of

text: n

text: cm

text: are

text: only

text: present

text: H

text: in

text: HCN-emitting

text: star-forming

text: cores

text: with

text: sizes

text: around

text: ρ

text: L

text: pc

text: or

text: below.

text: Hence,

text: the

text: observational

text: constraints

text: used

text: L

text: by

text: Krumholz

text: Tan

text: all

text: fall

text: along

text: a

text: particular

text: track

text: τSF

text: ρ

text: cm

text: in

text: the

text: two-dimensional

text: (or,

text: equivalently,

text: Σ∗

text: L

text: Σ

text: L/τSF

text: when

text: considering

text: sur-

text: plane

text: (L,

text: ρ

text: L).

text: In

text: other

text: words,

text: observational

text: constraints

text: tha∝t

text: face

text: densities).

text: In

text: that

text: case

text: depletion

text: time

text: becomes

text: the

text: func-

text: the

text: “constant

text: eﬃciency

text: per

text: free-fall

text: time”

text: (τSF

text: tion

text: of

text: other

text: gas

text: proτpSeFrt=iesτSoFn(Lsc,

text: In

text: other

text: words,

text: we

text: need

text: to

text: explicitly

text: consider

text: the

text: star

text: for-

text: “linear”

text: τSF(L,

text: min

text: ρρmmiinn,,

text: tion

text: can

text: be

text: expressed

text: as

text: τSF

text: Gyr

text: τSF(L,

text: ρ

text: L),

text: LL

text: with

text: substantial

text: intrinsic

text: (i.e.

text: exceeding

text: the

text: formal

text: observa-

text: tional

text: error)

text: scatter

text: an≈d

text: the

text: actual

text: value

text: for

text: τSF

text: noticeably

text: with

text: ρmin

text: ρ0

text: min

text: cannot

text: be

text: distinguished

text: at

text: diﬀerent

text: at

text: z

text: (τSF

text: Gyr

text: Leroy

text: et

text: al.

text: Bigiel

text: et

text: al.

text: present

text: without

text: additional

text: observational

text: constraints.

text: The

text: pa-

text: Bolatto

text: et

text: al.

text: Bigiel

text: et

text: al.

text: Leroy

text: et

text: al.

text: rameter

text: L

text: in

text: both

text: cases

text: must

text: be

text: in

text: the

text: range

text: of

text: a

text: few

text: hundred

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