text: 4 Dijkstra et al. 2. MODEL Rybicki & Lightman 1979, Eq 1.78) (cid:16) (cid:17) 2.1. Geometry hν B × g − κ = κ(ν)= ul ul u n n φ(ν), (3) Our analysis is limited to spherically symmetric gas ν 4π∆ν g l u ul l clouds with a uniform density. These simplifying as- hν sumptions oﬀer us a clear view on the relevant radiative jν = j(ν)= 4π∆uνl nuAulφ(ν), processes that determine whether the masing conditions ul exist. In particular, they allow us to treat the radiative where B = c2 A , A denotes the Einstein A- transfer of Lyα photons analytically, which represents a ul 2hν3 ul ul coeﬃcient for the transition from the higher energy state major computational advantage. Under these assump- ‘u’ to the lower energy state ‘l’. The energy diﬀerence tions the cloud is fully characterised by a single number between the two states is given by ∆E = hν . The fac- ul density, n. For a given gas mass M , this gives a cloud § gas tors gl and nl (gu and nu) denote the statistical weight radius R. We discuss in how our main results are ex- and number density of atoms in (cid:82)the lower (upper) en- pected to be aﬀected by these simplifying assumptions. ergy state. The function φ(ν) denotes the line proﬁle (cid:113) function (also known as the Voigt function), which is Hydrodynamical simulations indicate that that the gas density proﬁle is closer to isothermal (i.e ρ(r) ∝ r−2, e.g. norma≡lized to unity through ∆1νul dνφ(ν) = 1, in which Shang et al. 2010, Pacucci & Ferrara 2014). The gas ∆ν ν 2kT . density increases towards the centre of the collapsing gas ul ul mpc2 cloud, which may lead to the formation of a quasi-star, a → We consider the 2p 2s (for which g = 2, supermassive star, or a direct-collapse black hole in the × 1/2− − 1/2 →l g = 2, A = 1.60 10 9 s 1) and the 2s 2p centre of the cloud. We therefore also consider models in u ul 1/2 × 3−/2 which the gas cloud contains a central source of ionising tr−ansition (for which gl = 2, gu = 4, Aul = 8.78 10 7 radiation, which represents a scenario in which a central s 1). We assume that the atoms in the 2p state are black hole has already formed. Accretion rates onto the divided between 2p and 2p following their statisti- ∼ − 1/2 3/2 central black hole can be up to 0.1M(cid:12) yr 1 (e.g. Latif cal weight. That is, for the 9.9 GHz transition we have × & Volonteri 2015), which can power a central source with n = n and n = n 2/3, while for the 1.1 GHz we − l 2s ×u 2p a luminosity exceeding 1044 erg s 1. We do not consider have nl = n2p 1/3§ and nu = n2s. We justify adopting the hydrodynamic impact of the central source on the this assumption in . gas. Instead, we focus on the emission properties of gas We write the line center total optical depth τFS through with properties that were favourable for the DCBH sce- 0 the center of the collapsing gas cloud of radius R —in nario, i.e. the gas is pristine, no fragmentation occurred, cl both ﬁne structure transitions—as and no stars have formed. (cid:16) (cid:17) 2.2. The Fine Structure Signal ≡ λ2A × g − 1 τFS 2R κ = R ul u n n , (4) We study the impact of the collapsing halo on CMB 0 cl 0 cl π g l u A l α photons passing through them (see Fig 1). In general, a × − gas cloud with uniform density changes the CMB inten- where Aα →= 6.25 108 s 1 is the Einstein coeﬃcient sity, Iν, by an amount (cid:0) (cid:1) for the 2p 1s transition. This expression does not contain the temperature-dependent width ∆ν , because D the Voigt parameter5 that is present in the Voigt func- Iν(s) = Iν,0e−κνs + κjν 1 − e−κνs , (1) tion, φ(ν) is av = 4πA∆≈ανD ≈ 117(Tgas−/104)−1/2 for the ν 3cm-transition, and a 13(T /104) 1/2 for the 27cm- v gas at a distance s into the cloud. Here, I denotes the in- transition (Dijkstra et al. 2008b). The line proﬁle func- ν,0 tensity before entering the cloud, κ denotes the opacity tion evaluated at line center is (see e.g. Chluba & Sun- − ν of the cloud (in cm 1), and j denotes its volume emis- yaev 2009) − − − ν sivity (in erg s 1 cm 3 Hz 1). Eq (1) implicitly assumes that the cloud is static. To account for gas motion, we would need to Doppler boost into the frame of the gas √1 ≈ 1 4∆ν § φ(0) = exp(a2)erfc(a ) = ul , (5) at s. However, we will show in that the ﬁne structure π v v πa A v α cross-sections are extremely broad in frequency, and that this Doppler boost can be safely ignored. For the relevant where we have used that erfc(a ) → e−√a2v when a (cid:29) 1. results in this paper, the second term on the right-hand- v av π v The factor ∆ν that enters φ(0) cancels the one that is side of Eq (1) is much smaller than I (see Appendix ) ul ν present in the expression for j(ν). The opacity through in which case Eq 1 simpliﬁes to 5 Both ﬁne structure transitions involve transitions between 2s − and 2p states, bo(cid:113)th of which have ﬁnite lifetimes. Under these Iν(s) = Iν,0e κνs. (2) conditions, the relevant Voigt parameter equals av = A4απ+∆Aν2Ds1s , where ∆νD(cid:29)= νucl 2mkpT (see Eq 10.74, Rybicki & Lightman 1979). The opacity through any transition is given by (see e.g. Since Aα A2s1s, we have av = 4πA∆ανD to high accuracy. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Dijkstra

text: et

text: al.

text: MODEL

text: Rybicki

text: Lightman

text: Eq

label: equation

text: Geometry

text: hν

text: B

text: g

text: κ

text: κ(ν)=

text: ul

text: ul

text: u

text: n

text: n

text: φ(ν),

text: Our

text: analysis

text: is

text: limited

text: to

text: spherically

text: symmetric

text: gas

text: ν

text: 4π∆ν

text: g

text: l

text: u

text: ul

text: l

text: clouds

text: with

text: a

text: uniform

text: density.

text: These

text: simplifying

text: as-

text: hν

text: sumptions

text: oﬀer

text: us

text: a

text: clear

text: view

text: on

text: the

text: relevant

text: radiative

text: jν

text: j(ν)=

text: 4π∆uνl

text: nuAulφ(ν),

text: processes

text: that

text: determine

text: whether

text: the

text: masing

text: conditions

text: ul

text: exist.

text: In

text: particular,

text: they

text: allow

text: us

text: to

text: treat

text: the

text: radiative

text: where

text: B

text: A

text: A

text: denotes

text: the

text: Einstein

text: A-

text: transfer

text: of

text: Lyα

text: photons

text: analytically,

text: which

text: represents

text: a

text: ul

text: ul

text: ul

text: coeﬃcient

text: for

text: the

text: transition

text: from

text: the

text: higher

text: energy

text: state

text: major

text: computational

text: advantage.

text: Under

text: these

text: assump-

text: ‘u’

text: to

text: the

text: lower

text: energy

text: state

text: ‘l’.

text: The

text: energy

text: diﬀerence

text: tions

text: the

text: cloud

text: is

text: fully

text: characterised

text: by

text: a

text: single

text: number

text: between

text: the

text: two

text: states

text: is

text: given

text: by

text: ∆E

text: hν

text: The

text: fac-

text: ul

text: density,

text: n.

text: For

text: a

text: given

text: gas

text: mass

text: M

text: this

text: gives

text: a

text: cloud

text: gas

text: tors

text: gl

text: and

text: nl

text: (gu

text: and

text: nu)

text: denote

text: the

text: statistical

text: weight

text: radius

text: R.

text: We

text: discuss

text: in

text: how

text: our

text: main

text: results

text: are

text: ex-

text: and

text: number

text: density

text: of

text: atoms

text: in

text: lower

text: (upper)

text: en-

text: pected

text: to

text: be

text: aﬀected

text: by

text: these

text: simplifying

text: assumptions.

text: ergy

text: state.

text: The

text: function

text: φ(ν)

text: denotes

text: the

text: line

text: proﬁle

text: function

text: (also

text: known

text: as

text: the

text: Voigt

text: function),

text: which

text: is

text: Hydrodynamical

text: simulations

text: indicate

text: that

text: that

text: the

text: gas

text: density

text: proﬁle

text: is

text: closer

text: to

text: isothermal

text: (i.e

text: ρ(r)

text: e.g.

text: norma≡lized

text: to

text: unity

text: through

text: dνφ(ν)

text: in

text: which

text: Shang

text: et

text: al.

text: Pacucci

text: Ferrara

text: The

text: gas

text: ∆ν

text: ν

text: density

text: increases

text: towards

text: the

text: centre

text: of

text: the

text: collapsing

text: gas

text: ul

text: ul

text: cloud,

text: which

text: may

text: lead

text: to

text: the

text: formation

text: of

text: a

text: quasi-star,

text: a

text: We

text: consider

text: the

text: (for

text: which

text: g

text: supermassive

text: star,

text: or

text: a

text: direct-collapse

text: black

text: hole

text: in

text: the

text: →l

text: g

text: A

text: s

text: and

text: the

text: centre

text: of

text: the

text: cloud.

text: We

text: therefore

text: also

text: consider

text: models

text: in

text: u

text: ul

text: which

text: the

text: gas

text: cloud

text: contains

text: a

text: central

text: source

text: of

text: ionising

text: tr−ansition

text: (for

text: which

text: gl

text: gu

text: Aul

text: radiation,

text: which

text: represents

text: a

text: scenario

text: in

text: which

text: a

text: central

text: s

text: We

text: assume

text: that

text: the

text: atoms

text: in

text: the

text: state

text: are

text: black

text: hole

text: has

text: already

text: formed.

text: Accretion

text: rates

text: onto

text: the

text: divided

text: between

text: and

text: following

text: their

text: statisti-

text: central

text: black

text: hole

text: can

text: be

text: up

text: to

text: yr

text: (e.g.

text: Latif

text: cal

text: weight.

text: That

text: is,

text: for

text: the

text: GHz

text: transition

text: we

text: have

text: Volonteri

text: which

text: can

text: power

text: a

text: central

text: source

text: with

text: n

text: n

text: and

text: n

text: n

text: while

text: for

text: the

text: GHz

text: we

text: l

text: ×u

text: a

text: luminosity

text: exceeding

text: erg

text: s

text: We

text: do

text: not

text: consider

text: have

text: nl

text: and

text: nu

text: We

text: justify

text: adopting

text: the

text: hydrodynamic

text: impact

text: of

text: the

text: central

text: source

text: on

text: the

text: this

text: assumption

text: in

text: gas.

text: Instead,

text: we

text: focus

text: on

text: the

text: emission

text: properties

text: of

text: gas

text: We

text: write

text: the

text: line

text: center

text: total

text: optical

text: depth

text: τFS

text: through

text: with

text: properties

text: that

text: were

text: favourable

text: for

text: the

text: DCBH

text: sce-

text: the

text: center

text: of

text: the

text: collapsing

text: gas

text: cloud

text: of

text: radius

text: R

text: —in

text: nario,

text: i.e.

text: the

text: gas

text: is

text: pristine,

text: no

text: fragmentation

text: occurred,

text: cl

text: both

text: ﬁne

text: structure

text: transitions—as

text: and

text: no

text: stars

text: have

text: formed.

text: The

text: Fine

text: Structure

text: Signal

text: g

text: τFS

text: κ

text: R

text: ul

text: u

text: n

text: n

text: We

text: study

text: the

text: impact

text: of

text: the

text: collapsing

text: halo

text: on

text: CMB

text: cl

text: cl

text: π

text: g

text: l

text: u

text: A

text: l

text: α

text: photons

text: passing

text: through

text: them

text: (see

text: Fig

text: In

text: general,

text: a

text: gas

text: cloud

text: with

text: uniform

text: density

text: changes

text: the

text: CMB

text: inten-

text: where

text: Aα

text: s

text: is

text: the

text: Einstein

text: coeﬃcient

text: sity,

text: Iν,

text: by

text: an

text: amount

text: for

text: the

text: transition.

text: This

text: expression

text: does

text: not

text: contain

text: the

text: temperature-dependent

text: width

text: ∆ν

text: because

text: D

text: the

text: Voigt

text: that

text: is

text: present

text: in

text: the

text: Voigt

text: func-

text: Iν(s)

text: κjν

text: e−κνs

text: tion,

text: φ(ν)

text: is

text: av

text: 4πA∆≈ανD

text: for

text: the

text: ν

text: and

text: a

text: for

text: the

text: v

text: gas

text: at

text: a

text: distance

text: s

text: into

text: the

text: cloud.

text: Here,

text: I

text: denotes

text: the

text: in-

text: transition

text: (Dijkstra

text: et

text: al.

text: The

text: line

text: proﬁle

text: func-

text: tensity

text: before

text: entering

text: the

text: cloud,

text: κ

text: denotes

text: the

text: opacity

text: tion

text: evaluated

text: at

text: line

text: center

text: is

text: (see

text: e.g.

text: Chluba

text: Sun-

text: ν

text: of

text: the

text: cloud

text: (in

text: cm

text: and

text: j

text: denotes

text: its

text: volume

text: emis-

text: yaev

text: ν

text: sivity

text: (in

text: erg

text: s

text: cm

text: Hz

text: Eq

text: implicitly

text: assumes

text: that

text: the

text: cloud

text: is

text: static.

text: To

text: account

text: for

text: gas

text: motion,

text: we

text: would

text: need

text: to

text: Doppler

text: boost

text: into

text: the

text: frame

text: of

text: the

text: gas

text: √1

text: 4∆ν

text: ul

text: at

text: s.

text: However,

text: we

text: will

text: show

text: in

text: that

text: the

text: ﬁne

text: structure

text: π

text: v

text: v

text: πa

text: A

text: v

text: α

text: cross-sections

text: are

text: extremely

text: broad

text: in

text: frequency,

text: and

text: that

text: this

text: Doppler

text: boost

text: can

text: be

text: safely

text: ignored.

text: For

text: the

text: relevant

text: where

text: we

text: have

text: used

text: that

text: erfc(a

text: when

text: a

text: results

text: in

text: this

text: paper,

text: the

text: second

text: term

text: on

text: the

text: right-hand-

text: v

text: av

text: π

text: v

text: The

text: factor

text: ∆ν

text: that

text: enters

text: cancels

text: the

text: one

text: that

text: is

text: side

text: of

text: Eq

text: is

text: much

text: smaller

text: than

text: I

text: (see

text: Appendix

text: ul

text: ν

text: present

text: in

text: the

text: expression

text: for

text: j(ν).

text: The

text: opacity

text: through

text: in

text: which

text: case

text: Eq

text: simpliﬁes

text: to

text: Both

text: ﬁne

text: structure

text: transitions

text: involve

text: transitions

text: between

text: and

text: states,

text: of

text: which

text: have

text: ﬁnite

text: lifetimes.

text: Under

text: these

text: Iν(s)

text: κνs.

text: conditions,

text: the

text: relevant

text: Voigt

text: parameter

text: equals

text: av

text: where

text: νucl

text: (see

text: Eq

text: Rybicki

text: Lightman

text: The

text: opacity

text: through

text: any

text: transition

text: is

text: given

text: by

text: (see

text: e.g.

text: Since

text: Aα

text: we

text: have

text: av

text: 4πA∆ανD

text: to

text: high

text: accuracy.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: ∼ Mapping the Hα emission of star forming galaxies at z 1 9 6 ACKNOWLEDGEMENTS 2015, MNRAS, 452, 486 D´ıaz-Santos, T., Charmandaris, V., Armus, L., et al. 2010, We would like to thank the referee for carefully reading ApJ, 723, 993 the manuscript and providing valuable comments and sug- D´ıaz-Santos, T., Armus, L., Charmandaris, V., et al. 2013, gestions. G.E.M. acknowledges support from STFC through ApJ, 774, 68 grant ST/K00106X/1, the John Fell Oxford University Elbaz, D., Daddi, E., Le Borgne, D., et al. 2007, A&A, 468, Press (OUP) Research Fund and the University of Ox- 33 ford. JPS, CMH and IRS acknowledge support from STFC Elbaz, D., Dickinson, M., Hwang, H.S., et al. 2011, A&A, (ST/I001573/1). JPS also acknowledges support from a 533, 119 Hintze Research Fellowship. IRS acknowledges support from Ellison, S. L., Patton, D. R., Simard, L., & McConnachie, an ERC Advanced Investigator programme DUSTYGAL A. W. 2008, ApJL, 672, L107 and a Royal Society/Wolfson Merit Award. AJB gratefully Finlator, K., & Dav´e, R. 2008, MNRAS, 385, 2181 acknowledges the hospitality of the Research School of As- Fo¨rster Schreiber, N. M., Genzel, R., Bouch´e, N., et al. tronomy & Astrophysics at the Australian National Univer- 2009, ApJ, 706, 1364 sity, Mount Stromlo, Canberra where some of this work was Garilli, B., Le F`evre, O., Guzzo, L., et al. 2008, A&A, 486, done under the Distinguished Visitor scheme. DS acknowl- 683 edges ﬁnancial support from the Netherlands Organisation Garilli, B., Guzzo, L., Scodeggio, M., et al. 2014, A&A, for Scientiﬁc research (NWO) through a Veni fellowship 562, A23 and from FCT through a FCT Investigator Starting Grant Genzel, R., Tacconi, L. J., Gracia-Carpio, J., et al. 2010, and Start-up Grant (IF/01154/2012/CP0189/CT0010). We MNRAS, 407, 2091 thank Ian Smail for co-ordinating the eﬀorts of the KROSS Genzel, R., Tacconi, L. J., Lutz, D., et al. 2015, ApJ, 800, survey and for stimulating discusssions. We thank Holly El- 20 bert and Timothy Green for their observations and Matthieu Gracia-Carpio, J., Sturm, E., Hailey-Dunsheath, S. et al. Schaller for providing the EAGLE simulation quantities. 2010, ApJ, 728, 7 GEM thanks Ryan Houghton and Borris Haussler for help- Gonza´lez, V., Labb´e, I., Bouwens, R. J., et al. 2010, ApJ, full discussions. Based on observations made with ESO Tele- 713, 115 scopes at the La Silla Paranal Observatory under the pro- Guzzo, L., Scodeggio, M., Garilli, B., et al. 2014, A&A, gramme IDs 60.A-9460, 092.B-0538, 093.B-0106 and 094.B- 566, A108 0061.This research uses data from the VIMOS VLT Deep H¨aussler, B., McIntosh, D. H., Barden, M., et al. 2007, Survey, obtained from the VVDS database operated by Ce- ApJS, 172, 615 sam, Laboratoire dstrophysique de Marseille, France.This Hung, C.-L., Sanders, D. B., Casey, C. M., et al. 2013, ApJ, paper uses data from the VIMOS Public Extragalactic Red- 778, 129 shift Survey (VIPERS). VIPERS has been performed us- Kennicutt, R. C., Jr. 1998, ARA&A, 36, 189 ing the ESO Very Large Telescope, under the “Large Pro- Kewley, L. J., Heisler, C. A., Dopita, M. A., & Lumsden, gramme” 182.A-0886. The participating institutions and S. 2001, ApJS, 132, 37 funding agencies are listed at http://vipers.inaf.it. This pa- Kewley, L. J., & Dopita, M. A. 2002, ApJS, 142, 35 per uses data from zCOSMOS which is based on observa- Kim J.-W., Edge A. C., Wake D. A., Stott J. P., 2011b, tions made with ESO Telescopes at the La Silla or Paranal MNRAS,410, 241 Observatories under programme ID 175.A-0839. We thank Larson, D., Dunkley, J.,Hinshaw, G., et al. 2011, ApJS, D. Sobral and M. Cirasuolo for helping with the survey. 192, 16 REFERENCES Lawrence, A., Warren, S. J., Almaini, O., et al. 2007, MN- Andrews, B. H., & Martini, P. 2013, ApJ, 765, 140 RAS, 379, 1599 Bothwell, M. 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label: paragraph

text: Mapping

text: the

text: Hα

text: emission

text: of

text: star

text: forming

text: galaxies

text: at

text: z

label: section

text: ACKNOWLEDGEMENTS

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text: T.,

text: Charmandaris,

text: V.,

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text: manuscript

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text: We

text: thank

text: Holly

text: El-

text: bert

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text: Timothy

text: Green

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text: observations

text: and

text: Matthieu

text: Gracia-Carpio,

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text: ApJ,

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text: V.,

text: Labb´e,

text: I.,

text: Bouwens,

text: R.

text: J.,

text: et

text: al.

text: ApJ,

text: full

text: discussions.

text: Based

text: on

text: observations

text: made

text: with

text: ESO

text: Tele-

text: scopes

text: at

text: the

text: La

text: Silla

text: Paranal

text: Observatory

text: under

text: the

text: pro-

text: Guzzo,

text: L.,

text: Scodeggio,

text: M.,

text: Garilli,

text: B.,

text: et

text: al.

text: A&A,

text: gramme

text: IDs

text: and

text: research

text: uses

text: data

text: from

text: the

text: VIMOS

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text: et

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text: Survey,

text: obtained

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text: the

text: VVDS

text: database

text: operated

text: by

text: Ce-

text: ApJS,

text: sam,

text: Laboratoire

text: dstrophysique

text: de

text: Marseille,

text: France.This

text: Hung,

text: C.-L.,

text: Sanders,

text: D.

text: B.,

text: Casey,

text: C.

text: M.,

text: et

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text: paper

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text: data

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text: the

text: VIMOS

text: Public

text: Extragalactic

text: Red-

text: shift

text: Survey

text: (VIPERS).

text: VIPERS

text: has

text: been

text: performed

text: us-

text: Kennicutt,

text: R.

text: C.,

text: Jr.

text: ARA&A,

text: ing

text: the

text: ESO

text: Very

text: Large

text: Telescope,

text: under

text: the

text: “Large

text: Pro-

text: Kewley,

text: L.

text: J.,

text: Heisler,

text: C.

text: A.,

text: Dopita,

text: M.

text: A.,

text: Lumsden,

text: gramme”

text: The

text: participating

text: institutions

text: and

text: S.

text: ApJS,

text: funding

text: agencies

text: are

text: listed

text: at

text: This

text: pa-

text: Kewley,

text: L.

text: J.,

text: Dopita,

text: M.

text: A.

text: ApJS,

text: per

text: uses

text: data

text: from

text: zCOSMOS

text: which

text: is

text: based

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text: observa-

text: Kim

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text: Edge

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text: C.,

text: Wake

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text: Perturbed distribution functions 5 − We can now rewrite f (Eq. 17) as R = R (1 e cos θ ) , (22) 1 g R f = f + f , (28) where (cid:113) 1 1,R 1,z (cid:40) (cid:41) ≡ where (cid:88) e(J , J ) 2J /(κR2) (23) R φ R g is the eccentricity of the orbit. We consider orbits with f1,R ≡ Re 1 cRjm0Fjm0ei[jθR+m(θφ−Ωpt)] , (29) − low e, for which the epicyclic approximation holds. Using (cid:40) j= 1 (cid:41) (cid:88) the deﬁnition of e and the mapping of Eq. (4) and Eq. (5), we can rewrite Φ and expand it in powers of e, dropping all the terms that(cid:110)1,aRre O(e2), to obtain (e.g(cid:111)., Weinberg 1994) f ≡ Re 1 cz F ei[jθR+m(θφ−Ωpt)+2lθz] , 1,z jm2l jm2l − (cid:40)(cid:34) j,l= 1 − − Φ = Re Φ (R, 0)eim(θφ ∆φ Ωpt) (30) 1,R a ≈ (cid:35) (cid:41) where the Fourier coeﬃcients cR and cz are given by Re Φ (R , 0)(1 + imeγ sin θ )+ jkl jkl a g R Eq. (25), Eq. (27), and − ∂∂ΦRa (Rg, 0)e cos θR eim(θφ−Ωpt) , (24) ≡ j ∂∂Jf0 + k∂∂Jf0 + l∂∂Jf0 F (J , J , J ) R −φ z . (31) jkl R φ z jκ + k (ω Ω ) + lν φ p where γ is deﬁned as in Eq. (8). Note that the function − h(t) in Eq. (14) is just h(t) = exp( imΩpt) in thi−s case, 3.3 Moments of the distribution function and the frequency ω in Eqs. (16)-(17) is thus ω = mΩ . p p p One of the main motivation of the present work is to We can now evaluate the Fourier coeﬃcients for Φ in the 1,R traditional way (cid:90) (cid:90) understand the present response of a disc stellar population, represented by a DF f in an axisymmetric potential, to a 0 quasi-static perturbing non-axisymmetric potential in terms (cid:90) 1 2π 2π cR (J , J , J ) = dθ dθ of radial and vertical mean motions (e.g., Faure et al. 2014). jkl R φ z (2π)3 0 R 0 φ Such mean motions can be computed through the zeroth and × 2π (cid:40)(cid:34) − −(cid:35) ﬁrst order moments of the perturbed DF f = f + (cid:15)f . Here dθ Φ (R, 0)eim(θφ ∆φ)e i(jθR+kθφ+lθz) 0 1 z a we will assume a given form for f , namely the Schwarzschild 0 0 DF of Eq. (10). ≈ k (cid:41) δ δ δ + δ| | sgn(j)γe Φ (R , 0) We will focus on the mean motions projected on the km l0 j0 j 1 2 a g plane (for the radial motion) and on both sides of the plane (for the vertical motion). Indeed, it was already shown nu- − R ∂Φ δ|j|1 2g e ∂Ra (Rg, 0) , (25) merically (Faure et al. 2014; Monari et al. 2015) that spi- ral or bar perturbations typically lead to a breathing mode response of the disc, i.e. a density response that has even where δ is the Kronecker delta. We can now also parity with respect to the Galactic plane (i.e., is plane- treat Φ in the same way, replacing Φ (R, 0) herabove by 1,z a symmetric), and a mean vertical velocity ﬁeld that has odd ∂2Φa(R,0) z2 . From Eq. (5), we note that z2 can be expressed ∂z2 2 parity. Hence we will concentrate hereafter on the projected as (cid:88) surface density Σ(R, φ), the projected mean radial velocity (cid:104) (cid:105) ﬁeld v (R, φ), and the diﬀerence between the mean verti- R 2J J 1 ei2lθz cal velocity ﬁeld above and below the plane z2 = νz cos2 θz = νz − 2|l| . (26) l= 1 (cid:104) (cid:105) ≡ (cid:104) (cid:105) − (cid:104) (cid:105) ∆ v (R, φ) v (z > 0) v (z < 0). (32) z z z The Fourier coeﬃcien(cid:90)ts for Φ (cid:90) are then 1,z These can be computed by integrating the perturbed (cid:90) 1 2π 2π cz (J , J , J ) = dθ dθ DF over all z (or half of them in the case of the vertical jkl R φ z (2π)3 0 R 0 φ motion) and all velocities(cid:90), i.e., (cid:90) × 2π (cid:18) ∂2Φ (R, 0(cid:19)) z2 (cid:40)(cid:34) − − (cid:35) dθ a eim(θφ ∆φ)e i(jθR+kθφ+lθz) ∞ 0 z ∂z2 2 Σ(R, φ) ≡ (cid:90) dz (cid:90) d3v(f + (cid:15)f ), (33a) −∞ 0 1 ≈ 12δkm δl0 + δ|2l|2 Jνz δj0 + δ|j|1 k2sgn(cid:41)(j)γe Σ(R, φ)(cid:104)v (cid:105)(R, φ) ≡ (cid:90)∞ dz (cid:90)d3vv (f + (cid:15)f ), (33b) R −∞ R 0 1 × ∂∂2zΦ2a (Rg, 0) − δ|j|1 R2g e∂∂R3∂Φza2 (Rg, 0) . (27) Σ(R, φ)∆(cid:104)vz(cid:105)(R, φ) ≡ 4 ∞ dz d3vvz(f0 + (cid:15)f1), (33c) 0 (cid:13) c 2015 RAS, MNRAS 000, 1–15 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Perturbed

text: distribution

text: functions

text: We

text: can

text: now

text: rewrite

text: f

text: (Eq.

text: as

text: R

text: R

text: e

text: cos

text: θ

text: g

text: R

text: f

text: f

text: f

text: where

text: where

text: e(J

text: J

text: R

text: φ

text: R

text: g

text: is

text: the

text: eccentricity

text: of

text: the

text: orbit.

text: We

text: consider

text: orbits

text: with

text: Re

text: low

text: e,

text: for

text: which

text: the

text: epicyclic

text: approximation

text: holds.

text: Using

text: j=

text: the

text: deﬁnition

text: of

text: e

text: and

text: the

text: mapping

text: of

text: Eq.

text: and

text: Eq.

text: we

text: can

text: rewrite

text: Φ

text: and

text: expand

text: it

text: in

text: powers

text: of

text: e,

text: dropping

text: all

text: the

text: terms

text: to

text: obtain

text: Weinberg

text: f

text: Re

text: cz

text: F

text: j,l=

text: Φ

text: Re

text: Φ

text: (R,

text: ∆φ

text: Ωpt)

text: a

text: where

text: the

text: Fourier

text: coeﬃcients

text: cR

text: and

text: cz

text: are

text: given

text: by

text: Re

text: Φ

text: (R

text: imeγ

text: sin

text: θ

text: jkl

text: jkl

text: a

text: g

text: R

text: Eq.

text: Eq.

text: and

text: ∂∂ΦRa

text: (Rg,

text: cos

text: θR

text: eim(θφ−Ωpt)

text: j

text: F

text: (J

text: J

text: J

text: R

text: −φ

text: z

text: jkl

text: R

text: φ

text: z

text: jκ

text: k

text: (ω

text: Ω

text: lν

text: φ

text: p

text: where

text: γ

text: is

text: deﬁned

text: as

text: in

text: Eq.

text: Note

text: that

text: the

text: function

text: h(t)

text: in

text: Eq.

text: is

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text: 10 G. Monari et al. (cid:0) (cid:1) 10 We compare Eq. (50) with the predictions of Eq. (41) in Fig. 6. This comparison is made by averaging u (R, φ, z) − z 5 over z with weight exp ν2z2/2/σ˜2 (i.e., in the case z )1 where the vertical density is isothermal like in the case of - ms 0 Schwarzschild’s DF). The predictions of Eq. (50) are an or- 〉(kR - der of magnitude larger than the predictions of Eq. (41) and 〈v 5 the simulation (so much that we do not show, for readabil- - ity, the complete range of Fig. 6). The phases are instead in prefect agreement. A more sophisticated (albeit not fully dy- 10 namical) approach was taken by Monari et al. (2015), relat- ∅( ) ing the radial and tangential motions for a very cold stellar 0 1 2 3 4 5 6 disc or ﬂuid to the vertical motions via the continuity equa- rad tion. The predictions of Monari et al. (2015) (again, averaged along z with weight the isothermal density distribution) are 10 also shown on Fig. 6, allowing to show the typical reduction factor (as well as some phase-shift related to missing terms 5 in the cold ﬂuid approximation). We note that the breathing )1 modes are qualitatively similar to those observed in the ex- - ms 0 tended solar neighbourhood (Williams et al. 2013), but that 〉(kR - the amplitude of these motions is much lower than observed. 〈v 5 It nevertheless remains to be seen how the coupling of mul- - tiple perturbers will aﬀect these vertical motions (Monari et al. in prep.). 10 ∅( ) 4.3 Distribution function at a point in 0 1 2 3 4 5 6 conﬁguration space rad Our computation of the exact form of the perturbed DF 10 away from the main resonances also allows us to study the detailed behaviour of f = f +(cid:15)f at a given point in conﬁg- 0 1 5 uration space (R, φ, z), in terms of the actions and angles, and compare it with the unperturbed version of the DF, )1 -ms 0 f0. First, let us note that the dimensions of phase-space, 〉(kvR - g(Riv,eφn,tzh)e=cocnosntrsatainntt,ofdeacrﬁexaesde pfrooimnt 6intcoon3ﬁ, geuvreantiwonhesnpatchee, 〈 5 DF depends both on actions and angles. We focus on the - case (R, φ, z) = (8 kpc, 0, 0) (i.e, the typical position of the 10 Sun in our model) and we add the constraint J = 0, ad- z ditionally decreasing the dimensionality of phase-space to 2 ∅( ) 0 1 2 3 4 5 6 dimensions. (cid:112) rad The two variables that we choose to display are (θ , J ). The other angles and actions are constrained by R R − Figure 3. Several predictions for the response of mean vR of a R = Rg(Jφ) 2JR/κ(Jφ) cos θR, Jz = 0, θφ = φ + stellar system or a cold ﬂuid to the potential model used in this ∆φ(J , J , θ ), and θ = π/2 (because z = 0). In practice φ R R z work. Predictions by (predictions by Lin & Shu 1964, 1966; Lin we solve numerically the ﬁrst of this constraints for each et al. 1969): red dashed line Eq. (47), red solid line Eq. (48), green F pair (θ , J ) to get J , and it is then trivial to get θ . In line Eq. (48) multiplied by the reduction factor by Binney & R R φ φ the case of the unperturbed DF, f , the true dependence is Tremaine 2008. Blue dashed line Eq. (40), blue solid line simula- 0 tion. Top panel: R = 7 kpc. Central panel: R = 8 kpc. Bottom obviously on JR and Jφ, but we can translate it in terms of panel: R = 9 kpc. (θR, JR) in terms of the above constraints at a ﬁxed point in conﬁguration space. (cid:110) (cid:111) The comparison between f (θ , J ) and f(θ , J ) is 0 R R R R shown in Fig. 7. As is apparent from this ﬁgure, both f and 0 − (cid:15)uz,1(R, φ, z) = Re uaz(R, z)eim(φ Ωpt) , (49) f decrease with JR, but f0 is symmetric wi±th respect to θR = π while f not, which is due to the exp( iθ ) terms. The R asymmetries in Fig. 7 can be directly translated to features the third of Euler’s equation (without any reduction in the (v , v ) velocity space. To visualize this transforma- R φ factor) to the ﬁrst order in (cid:15) leads to (as ﬁrst shown in tion, we are helped by the map in Fig. 8, which shows how to Faure et al. 2014) associate (θ , J ) with (v , v ) at (R, φ, z) = (8 kpc, 0, 0). R R R φ This ﬁgure displays curves of constant J and θ in the R R ua(R, z) = i ∂Φa(R, z−)/∂z . (50) (vR, vφ) space (see also McMillan 2011). Each of the central z m [Ω(R) Ωp] circular curves represents a value of JR, while the lines that (cid:13) c 2015 RAS, MNRAS 000, 1–15 ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: G.

text: Monari

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text: 〉(kR

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text: red

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text: R

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text: φ

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text: the

text: case

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text: the

text: unperturbed

text: DF,

text: f

text: the

text: true

text: dependence

text: is

text: Tremaine

text: Blue

text: dashed

text: line

text: Eq.

text: blue

text: solid

text: line

text: simula-

text: tion.

text: Top

text: panel:

text: R

text: kpc.

text: Central

text: panel:

text: R

text: kpc.

text: Bottom

text: obviously

text: on

text: JR

text: and

text: Jφ,

text: but

text: we

text: can

text: translate

text: it

text: in

text: terms

text: of

text: panel:

text: R

text: kpc.

text: (θR,

text: JR)

text: in

text: terms

text: of

text: the

text: above

text: constraints

text: at

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text: A&A proofs: manuscript no. sdss_sf_v8_anda ≥ condition on the luminosity of the starburst L L , where observed on kpc scales (Wills et al. 1999; Lehnert & Heckman M ≥ L is an “Eddington-like” luminosity threshold. If L L , 1996). M M a momentum-driven wind develops, with velocity vout (cid:38) 2σ∗ FoLllowing fromLEq. 8, by (simplistically) approximating (where σ∗ is the stellar velocity dispersion; Murray et al. both virial(v) and outﬂow(v) with Gaussian distributions, the 2005). line-of-sight velocity dispersion of the gas corresponds to: § In we have identiﬁed the galaxy stacks displaying the sig- nature of galactic outﬂows as those for which the excess of gas σ2 = σ2 + σ2 , (9) gas virial outﬂow velocity with respect to the stellar velocity, as estimated from the high-velocity tail of the LoSVD, is signiﬁcant. More specif- | | − | | ically, we have considered values of v0.1 gas v99.9 stars signif- where σvirial can be approx≈imated by the stellar line-of-sight icantly greater than zero as a clear signature of outﬂows. The velocity dispersion, i.e. σvirial σstars. As a result, the “outﬂow” next step would be comparing the scaling relationships obtained line-of-sight velocity dispersion would be: for the observed outﬂows with the expectations of theoretical | | − | | msuoitdaebllse. fHoorwideevnetri,fytihnegqouuatnﬂtoitwys,v0is.1ngoast appvr9o9.p9rsitaartse,fworhtihleebceoimng- σoutﬂow = (σ2gas − σ2stars)1/2. (10) parison with models, because it has no clear interpretation in terms of the physical quantities usually employed by theorists. We note that Eq. 10 can be extended also to the percentile ve- Unfortunately, because of the information lost with the stacking, locities as, in the approximation of Gaussian distributions, they it is not possible to estimate the mass entrained in the observed correspond to multiples of the velocity dispersion. Therefore, ac- outﬂows, and therefore to calculate some important parameters cording to this line of reasoning, the outﬂow velocity can be ap- such as mass-loss rate, kinetic power and momentum rate of the − outﬂows, which would help to discriminate between diﬀerent proximated by the quantity (v2N,gas v2N,stars)1/2, where vN indicates the N-th percentile velocity. feedback models. Most importantly, spatially resolved spectro- As we commented earlier, because of the observed blue scopic information are needed to infer these quantities (Rupke asymmetries observed in the stellar LoSVDs of some galaxy & Veilleux 2013; Cazzoli et al. 2014; Arribas et al. 2014). How- § stacks ( ), the percentile velocities on the blue side of the stellar ever, we can obtain a proxy of the outﬂow velocity, and this can LoSVD, i.e. the 15.9th, 2.3th and 0.1th, may be not appropriate be exploited to shed light on the physical mechanism powering to trace virial motions in the galaxy. Therefore, in the follow- the ionised outﬂows. § ing, we adopt the same strategy as in and deﬁne the outﬂow The LoSVD of ionised gas traces a combination of dy- velocity as: namical motions of gas clouds within the gravitational poten- tial of the galaxy and of perturbed motions due to star forma- tion feeedback-related processes such as outﬂows and turbulence − in the disk. In the following analysis, since we consider only vout = (v2N,gas v2100−N,stars)1/2, (11) the high-velocity tail of the LoSVD, we assume that outﬂows dominate over turbulence, and so, simplistically, that any non- where the percentile velocity of the gas is chosen on the blue gravitational motions of the gas at such high velocities are due side of the LoSVD (because it is less aﬀected by dust), i.e. N= − to outﬂows and not to turbulence. Therefore, the LoSVD of the 0.1, 2.3 or 15.9, and, for the stars, we use 100 N, which gives the gas may be regarded as the convolution of two line-of-sight ve- corresponding percentile velocity on the red side of the LoSVD. locity distributions, one tracing the virial motions, and the other We note that vout can be inferred from Eq. 11 only in those cases one tracing the outﬂow: for which vN,gas > v100−N,stars: this means that vout cannot be calcu- lated for quite a few bins below the MS, as illustrated by Fig. 18. Lgas(v) = Lvirial(v) ∗ Loutﬂow(v). (8) tionFoigfugraelsax1y9 panrodp2e0rtisehso(we.tgh.eSoFuRtﬂ, oσw∗ vaenldocsiStyFR(v)o,uft)orastwaofudnicf-- ferent estimates of the outﬂow velocity, indicated as vout,low and We note that such approximation reﬂects the fact that the typ- vout,high, probing respectively the low and the high≡velocity ga−s iaacnnadaltimonugotsﬂftroolwimkselsdyteatcreocfnotesridmstiinnogfolruoercgaigolainslasexdsypbrsueabambdlpealsceraoansrdse tcrhahetihmsetnreelmlyasor,ddeeimsstk-, wv28i4t.1hisntartsh)1e/2ioannidsevdouot,uhtigﬂhow≡s,(va20n.1dgdaseﬁ−nve29d9.9asstavros)u1t,/lo2w. We(vn21o5.t9egaths at in the middle panels of Fig. 19 and 20 the outﬂow velocity is (see also the case of NGC 1569 discussed by Martin et al. 2002). shown as a function of the stellar velocity dispersion instead of In this scenario, the outﬂowing material participates, at least to the stellar mass, so that these trends can be readily compared to some extent, also to the virial (rotational) motions in the galaxy. the predictions of models of star formation-driven feedback. Fig- However, we stress that Eq. 8 is not universally applicable. In ures 19 and 20 indicate that the outﬂow velocity scales with SFR particular, Eq. 8 does not hold if the kinematics of the high ve- locity gas is largely dominated by a large-scale “bulk outﬂow” and sSFR. In particular, vout,high, shows a very tight (ρ > 0.8) cor- relation with the SFR. Previous studies found a similar increase expanding perpendicular to the disk. This is the case of pow- of outﬂow speed with SFR (e.g. Martin 2005 using a sample of erful starburst-driven superwinds (e.g. M 82, NGC 253) if they ∼ IR luminous galaxies at z 010; Arribas et al. 2014 for outﬂows are probed on scales of at least a few kpc, i.e. suﬃciently larger ∼ detected in z 0 (U)LIRGs; Chisholm et al. 2014 in local star than the typical vertical scale height of the stellar disk, where forming and starburst galaxies; Heckman et al. 2011 and Heck- the motions of the gas in outﬂow is clearly distinct from the man et al. 2015 using a a sample of UV-luminous starbursts and disk motions. Indeed, even the strong “bulk outﬂows” that de- velop in powerful starbursts such as M 82 appear to originate as 10 Although the correlation in their Fig. 6 is mainly driven by the three a collection of individual chimneys expanding out from the disk dwarf starbursts with detected NaID outﬂows from Schwartz & Martin that subsequently merge to form the classical biconical outﬂows (2004). Article number, page 14 of 20 ##LTLine##

text: A&A

text: proofs:

text: manuscript

text: no.

label: list

font: UWIUWT+txsy

text: condition

text: on

text: the

text: luminosity

text: of

text: the

text: starburst

text: L

text: L

text: where

text: observed

text: on

text: kpc

text: scales

text: (Wills

text: et

text: al.

text: Lehnert

text: Heckman

text: M

text: L

text: is

text: an

text: “Eddington-like”

text: luminosity

text: threshold.

text: If

text: L

text: L

text: M

text: M

text: a

text: momentum-driven

text: wind

text: develops,

text: with

text: velocity

text: vout

font: MBVNDY+txsya

text: 2σ∗

text: FoLllowing

text: fromLEq.

text: by

text: (simplistically)

text: approximating

text: (where

text: σ∗

font: LBJEPR+rtxmi

text: is

text: the

text: stellar

text: velocity

text: dispersion;

text: Murray

text: et

text: al.

text: both

text: virial(v)

text: and

text: outﬂow(v)

text: with

text: Gaussian

text: distributions,

text: the

text: line-of-sight

text: velocity

text: dispersion

text: of

text: the

text: gas

text: corresponds

text: to:

text: In

text: we

text: have

text: identiﬁed

text: the

text: galaxy

text: stacks

text: displaying

text: the

text: sig-

text: nature

text: of

text: galactic

text: outﬂows

text: as

text: those

text: for

text: which

text: the

text: excess

text: of

text: gas

text: σ2

text: σ2

text: σ2

text: gas

text: virial

text: outﬂow

text: velocity

text: with

text: respect

text: to

text: the

text: stellar

text: velocity,

text: as

text: estimated

text: from

text: the

text: high-velocity

text: tail

text: of

text: the

text: LoSVD,

text: is

text: signiﬁcant.

text: More

text: specif-

text: ically,

text: we

text: have

text: considered

text: values

text: of

text: gas

text: stars

text: signif-

text: where

text: σvirial

text: can

text: be

text: approx≈imated

text: by

text: the

text: stellar

text: line-of-sight

text: icantly

text: greater

text: than

text: zero

text: as

text: a

text: clear

text: signature

text: of

text: outﬂows.

text: The

text: velocity

text: dispersion,

text: i.e.

text: σvirial

text: σstars.

text: As

text: a

text: result,

text: the

text: “outﬂow”

text: next

text: step

text: would

text: be

text: comparing

text: the

text: scaling

text: relationships

text: obtained

text: line-of-sight

text: velocity

text: dispersion

text: would

text: be:

text: for

text: the

text: observed

text: outﬂows

text: with

text: the

text: expectations

text: of

text: theoretical

text: msuoitdaebllse.

text: σoutﬂow

text: parison

text: with

text: models,

text: because

text: it

text: has

text: no

text: clear

text: interpretation

text: in

text: terms

text: of

text: the

text: physical

text: quantities

text: usually

text: employed

text: by

text: theorists.

text: We

text: note

text: that

text: Eq.

text: can

text: be

text: extended

text: also

text: to

text: the

text: percentile

text: ve-

text: Unfortunately,

text: because

text: of

text: the

text: information

text: lost

text: with

text: the

text: stacking,

text: locities

text: as,

text: in

text: the

text: approximation

text: of

text: Gaussian

text: distributions,

text: they

text: it

text: is

text: not

text: possible

text: to

text: estimate

text: the

text: mass

text: entrained

text: in

text: the

text: observed

text: correspond

text: to

text: multiples

text: of

text: the

text: velocity

text: dispersion.

text: Therefore,

text: ac-

text: outﬂows,

text: and

text: therefore

text: to

text: calculate

text: some

text: important

text: parameters

text: cording

text: to

text: this

text: line

text: of

text: reasoning,

text: the

text: outﬂow

text: velocity

text: can

text: be

text: ap-

text: such

text: as

text: mass-loss

text: rate,

text: kinetic

text: power

text: and

text: momentum

text: rate

text: of

text: the

text: outﬂows,

text: which

text: would

text: help

text: to

text: discriminate

text: between

text: diﬀerent

text: proximated

text: by

text: the

text: quantity

text: where

text: vN

text: indicates

text: the

text: N-th

text: percentile

text: velocity.

text: feedback

text: models.

text: Most

text: importantly,

text: spatially

text: resolved

text: spectro-

text: As

text: we

text: commented

text: earlier,

text: because

text: of

text: the

text: observed

text: blue

text: scopic

text: information

text: are

text: needed

text: to

text: infer

text: these

text: quantities

text: (Rupke

text: asymmetries

text: observed

text: in

text: the

text: stellar

text: LoSVDs

text: of

text: some

text: galaxy

text: Veilleux

text: Cazzoli

text: et

text: al.

text: Arribas

text: et

text: al.

text: How-

text: stacks

text: the

text: percentile

text: velocities

text: on

text: the

text: blue

text: side

text: of

text: the

text: stellar

text: ever,

text: we

text: can

text: obtain

text: a

text: proxy

text: of

text: the

text: outﬂow

text: velocity,

text: and

text: this

text: can

text: LoSVD,

text: i.e.

text: the

text: and

text: may

text: be

text: not

text: appropriate

text: be

text: exploited

text: to

text: shed

text: light

text: on

text: the

text: physical

text: mechanism

text: powering

text: to

text: trace

text: virial

text: motions

text: in

text: the

text: galaxy.

text: Therefore,

text: in

text: the

text: follow-

text: the

text: ionised

text: outﬂows.

text: ing,

text: we

text: adopt

text: the

text: same

text: strategy

text: as

text: in

text: and

text: deﬁne

text: the

text: outﬂow

text: The

text: LoSVD

text: of

text: ionised

text: gas

text: traces

text: a

text: combination

text: of

text: dy-

text: velocity

text: as:

text: namical

text: motions

text: of

text: gas

text: clouds

text: within

text: the

text: gravitational

text: poten-

text: tial

text: of

text: the

text: galaxy

text: and

text: of

text: perturbed

text: motions

text: due

text: to

text: star

text: forma-

text: tion

text: feeedback-related

text: processes

text: such

text: as

text: outﬂows

text: and

text: turbulence

text: in

text: the

text: disk.

text: In

text: the

text: following

text: analysis,

text: since

text: we

text: consider

text: only

text: vout

text: the

text: high-velocity

text: tail

text: of

text: the

text: LoSVD,

text: we

text: assume

text: that

text: outﬂows

text: dominate

text: over

text: turbulence,

text: and

text: so,

text: simplistically,

text: that

text: any

text: non-

text: where

text: the

text: percentile

text: velocity

text: of

text: the

text: gas

text: is

text: chosen

text: on

text: the

text: blue

text: gravitational

text: motions

text: of

text: the

text: gas

text: at

text: such

text: high

text: velocities

text: are

text: due

text: side

text: of

text: the

text: LoSVD

text: (because

text: it

text: is

text: less

text: aﬀected

text: by

text: dust),

text: i.e.

text: N=

text: to

text: outﬂows

text: and

text: not

text: to

text: turbulence.

text: Therefore,

text: the

text: LoSVD

text: of

text: the

text: or

text: and,

text: for

text: the

text: stars,

text: we

text: use

text: N,

text: which

text: gives

text: the

text: gas

text: may

text: be

text: regarded

text: as

text: the

text: convolution

text: of

text: two

text: line-of-sight

text: ve-

text: corresponding

text: percentile

text: velocity

text: on

text: the

text: red

text: side

text: of

text: the

text: LoSVD.

text: locity

text: distributions,

text: one

text: tracing

text: the

text: virial

text: motions,

text: and

text: the

text: other

text: We

text: note

text: that

text: vout

text: can

text: be

text: inferred

text: from

text: Eq.

text: only

text: in

text: those

text: cases

text: one

text: tracing

text: the

text: outﬂow:

text: for

text: which

text: vN,gas

text: this

text: means

text: that

text: vout

text: cannot

text: be

text: calcu-

text: lated

text: for

text: quite

text: a

text: few

text: bins

text: below

text: the

text: MS,

text: as

text: illustrated

text: by

text: Fig.

text: Lgas(v)

text: Lvirial(v)

text: Loutﬂow(v).

text: oσw∗

text: ferent

text: estimates

text: of

text: the

text: outﬂow

text: velocity,

text: indicated

text: as

text: vout,low

text: and

text: We

text: note

text: that

text: such

text: approximation

text: reﬂects

text: the

text: fact

text: that

text: the

text: typ-

text: vout,high,

text: probing

text: respectively

text: the

text: low

text: and

text: the

text: high≡velocity

text: ga−s

text: tcrhahetihmsetnreelmlyasor,ddeeimsstk-,

text: at

text: in

text: the

text: middle

text: panels

text: of

text: Fig.

text: and

text: the

text: outﬂow

text: velocity

text: is

text: (see

text: also

text: the

text: case

text: of

text: NGC

text: discussed

text: by

text: Martin

text: et

text: al.

text: shown

text: as

text: a

text: function

text: of

text: the

text: stellar

text: velocity

text: dispersion

text: instead

text: of

text: In

text: this

text: scenario,

text: the

text: outﬂowing

text: material

text: participates,

text: at

text: least

text: to

text: the

text: stellar

text: mass,

text: so

text: that

text: these

text: trends

text: can

text: be

text: readily

text: compared

text: to

text: some

text: extent,

text: also

text: to

text: the

text: virial

text: (rotational)

text: motions

text: in

text: the

text: galaxy.

text: the

text: predictions

text: of

text: models

text: of

text: star

text: formation-driven

text: feedback.

text: Fig-

text: However,

text: we

text: stress

text: that

text: Eq.

text: is

text: not

text: universally

text: applicable.

text: In

text: ures

text: and

text: indicate

text: that

text: the

text: outﬂow

text: velocity

text: scales

text: with

text: SFR

text: particular,

text: Eq.

text: does

text: not

text: hold

text: if

text: the

text: kinematics

text: of

text: the

text: high

text: ve-

text: locity

text: gas

text: is

text: largely

text: dominated

text: by

text: a

text: large-scale

text: “bulk

text: outﬂow”

text: and

text: sSFR.

text: In

text: particular,

text: vout,high,

text: shows

text: a

text: very

text: tight

text: (ρ

text: cor-

text: relation

text: with

text: the

text: SFR.

text: Previous

text: studies

text: found

text: a

text: similar

text: increase

text: expanding

text: perpendicular

text: to

text: the

text: disk.

text: This

text: is

text: the

text: case

text: of

text: pow-

text: of

text: outﬂow

text: speed

text: with

text: SFR

text: (e.g.

text: Martin

text: using

text: a

text: sample

text: of

text: erful

text: starburst-driven

text: superwinds

text: (e.g.

text: M

text: NGC

text: if

text: they

text: IR

text: luminous

text: galaxies

text: at

text: z

text: Arribas

text: et

text: al.

text: for

text: outﬂows

text: are

text: probed

text: on

text: scales

text: of

text: at

text: least

text: a

text: few

text: kpc,

text: i.e.

text: suﬃciently

text: larger

text: detected

text: in

text: z

text: (U)LIRGs;

text: Chisholm

text: et

text: al.

text: in

text: local

text: star

text: than

text: the

text: typical

text: vertical

text: scale

text: height

text: of

text: the

text: stellar

text: disk,

text: where

text: forming

text: and

text: starburst

text: galaxies;

text: Heckman

text: et

text: al.

text: and

text: Heck-

text: the

text: motions

text: of

text: the

text: gas

text: in

text: outﬂow

text: is

text: clearly

text: distinct

text: from

text: the

text: man

text: et

text: al.

text: using

text: a

text: a

text: sample

text: of

text: UV-luminous

text: starbursts

text: and

text: disk

text: motions.

text: Indeed,

text: even

text: the

text: strong

text: “bulk

text: outﬂows”

text: that

text: de-

text: velop

text: in

text: powerful

text: starbursts

text: such

text: as

text: M

text: appear

text: to

text: originate

text: as

text: Although

text: the

text: correlation

text: in

text: their

text: Fig.

text: is

text: mainly

text: driven

text: by

text: the

text: three

text: a

text: collection

text: of

text: individual

text: chimneys

text: expanding

text: out

text: from

text: the

text: disk

text: dwarf

text: starbursts

text: with

text: detected

text: NaID

text: outﬂows

text: from

text: Schwartz

text: Martin

text: that

text: subsequently

text: merge

text: to

text: form

text: the

text: classical

text: biconical

text: outﬂows

text: Article

text: number,

text: page

text: of

text: ##LTLine##