text: 6 J. L. Ward & J. M. D. Kruijssen 400. A subset of OB stars is deﬁned within each model association ‘rotation’ ( 0). In all panels, positions and velocities are shown vt > by randomly drawing fractions of OB stars from a Gaussian distri- relative to the association centre deﬁned as the mean position and bution based on the observed median (0.11) and dispersion (0.10) velocity of all stars. The relative position and velocity vector of the of the OB star fractions in the observed sample. Each of the stars is centre deﬁned using only the OB stars is marked in grey in each assigned a random position and proper motion, drawn from Gaus- panel. sian distributions centred on zero using the randomly generated The general impression of these maps is one of largely stochas- dispersion values. tic motion, albeit with some substructure in many cases. In the re- Cases V and VI are representative of globally expanding con- mainder of this section, we employ a number of quantitative tests by ﬁgurations of the case IV models. In case V, one third of the stars considering the cumulative distributions of several key kinematic within each association are forced to exhibit outwards radial motion diagnostics across the sample of OB associations. This allows us to with respect to the association centre by taking to be the abso- assess the degree to which their kinematics are best described by a vr lute value of the originally generated . On average, this causes net expansion or by random motion. vr / 2 3 of the stars to have positive radial velocities. In case VI, one 3.1 Number ratios third of the stars within each association are assigned radial veloc- If there is a widespread, systematic expansion of present-day ities equal to the largest absolute value of the original and , vr vt (| | | |) OB associations, then it is likely to be evident in the ratio of stars i.e. vr = max vr , vt . The tangential velocity is set to the other, movingawayfromtheassociationcentretostarsmovingtowardsthe smaller component while keeping its original sign. This model en- association centre. Figure 4 shows the cumulative distribution of the sures that radial motion dominates over tangential motion. This ∼ number ratio of sources with positive (outwards) radial velocities reproduces the strong radial anisotropy ( 0 4) that is expected β . from a monolithic formation scenario. (Nvr>0) to the number of sources with negative (inward) radial The model cases can be summarised as follows: velocities (Nvr<0) for all OB associations. Also shown in this Figure are the normalised cumulative distributions for each of the six model (i) Observed stellar positions and absolute velocities with ran- cases described in the previous section. dom directions of motion (case I). / It is immediately clear from this ﬁgure that, based on num- (ii) Observed stellar positions and absolute velocities where 1 3 ber ratios alone, the observed distribution of OB associations are of the stars are guaranteed to have positive radial velocities with re- far better represented by purely random distributions of stellar ve- spect to the mean position and velocity of the 20 nearest neighbours locities rather than those distributions representing either locally (case II). / expanding or globally expanding associations. There is a departure (iii) Observed stellar positions and absolute velocities where 1 3 from the case IV curve tending slightly towards expansion; how- of the stars are guaranteed to have positive radial velocities and ever, this eﬀect is relatively small and the maximum observed ratio dominant radial motions with respect to the mean position and ∼ is 1.5, falling short of what may be expected from a widespread, velocity of the 20 nearest neighbours (case III). systematic expansion. Qualitatively, the case I distributions are the (iv) Random stellar positions and velocities drawn from a four- closest approximation to the observed distributions. When the cen- dimensional Gaussian distribution (case IV). tres of the OB star populations are used, the case I approach of (v) Random stellar positions and velocities drawn from a four- / randomising proper motion directions recreates the departure from dimensional Gaussian distribution where 1 3 of the stars are guar- the case IV distribution of random positions and velocities. The anteed to have positive radial velocities (case V). locally expanding case II and III model distributions fall between (vi) Random stellar positions and velocities drawn from a four- / the distributions based on random velocity ﬁelds and the two glob- dimensional Gaussian distribution where 1 3 of the stars are guar- ally expanding distributions. When the centres of the entire stellar anteed to have positive radial velocities and dominant radial motions populations are used (right panel), the observed systems show a (case VI). small expansion relative to the random motion models (case I and ∼ Together, these cases span a physically appropriate range of con- IV), but this small excess is reproduced by forcing just 5% of the ﬁgurations for comparison to the observed kinematic diagnostics stars to exhibit outward radial motion as in cases II, III, V, and VI discussed below. (rather than the 1/3 used there). This is in strong contrast with the expectation from monolithic models, in which most of the stellar population exhibits expansion (Baumgardt & Kroupa 2007). Based 3 QUANTIFYING THE KINEMATICS OF OB on these number ratios alone, the observed velocities are certainly ASSOCIATIONS consistent with a population of stars with randomly distributed ve- locities and inconsistent with both global and localised expansion Through analysis of the radial velocities of the member stars scenarios. of OB associations, it is possible to determine whether the associa- Whilst giving the qualitative impression of a random distri- tions are showing signs of strong radial expansion and, by extension, bution, the information gained from this simple analysis is limited, distinguish between the monolithic or hierarchical models of star providing no indication of the magnitudes of the radial velocities formation. The left panel of Figure 3 shows the positions and ve- and by itself it is not a deﬁnitive measure of the dynamical behaviour locities for the OB association Cas OB14. These velocities are then of the sample. separated into their radial and tangential components in the middle 3.2 Median velocity distributions and right panels of Figure 3. These diagrams are shown for all 18 OB associations in Figure A1. In the left panels of Figures 3 and A1, To quantify the typical expansion (or contraction) velocities the OB stars are marked in blue with all other stars marked in red. of the OB associations in our sample, we take the median value of In the centre panels, the points are colour-coded according to the the radial velocities of the stellar populations for each OB associ- radial velocity direction: blue for 0 and red for 0. In the ation. The cumulative distributions of the median radial velocities vr < vr > right hand panels the points are colour-coded by tangential motion are shown in Figure 5 relative to both the centre of the OB stars direction: blue for clockwise ( 0) and red for anti-clockwise and the centre of all stars, along with the six model cases. Uncer- vt < MNRAS 000, 1–18 (2017)

text: J.

text: L.

text: Ward

text: J.

text: M.

text: D.

text: Kruijssen

text: A

text: subset

text: of

text: OB

text: stars

text: is

text: deﬁned

text: within

text: each

text: model

text: association

text: ‘rotation’

text: In

text: all

text: panels,

text: positions

text: and

text: velocities

text: are

text: shown

text: vt

font: IYZZES+rntxmi

font: AIHHWM+rtxmi

text: by

text: randomly

text: drawing

text: fractions

text: of

text: OB

text: stars

text: from

text: a

text: Gaussian

text: distri-

text: relative

text: to

text: the

text: association

text: centre

text: deﬁned

text: as

text: the

text: mean

text: position

text: and

text: bution

text: based

text: on

text: the

text: observed

text: median

text: and

text: dispersion

text: velocity

text: of

text: all

text: stars.

text: The

text: relative

text: position

text: and

text: velocity

text: vector

text: of

text: the

text: of

text: the

text: OB

text: star

text: fractions

text: in

text: the

text: observed

text: sample.

text: Each

text: of

text: the

text: stars

text: is

text: centre

text: deﬁned

text: using

text: only

text: the

text: OB

text: stars

text: is

text: marked

text: in

text: grey

text: in

text: each

text: assigned

text: a

text: random

text: position

text: and

text: proper

text: motion,

text: drawn

text: from

text: Gaus-

text: panel.

text: sian

text: distributions

text: centred

text: on

text: zero

text: using

text: the

text: randomly

text: generated

text: The

text: general

text: impression

text: of

text: these

text: maps

text: is

text: one

text: of

text: largely

text: stochas-

text: dispersion

text: values.

text: tic

text: motion,

text: albeit

text: with

text: some

text: substructure

text: in

text: many

text: cases.

text: In

text: the

text: re-

text: Cases

text: V

text: and

text: VI

text: are

text: representative

text: of

text: globally

text: expanding

text: con-

text: mainder

text: of

text: this

text: section,

text: we

text: employ

text: a

text: number

text: of

text: quantitative

text: tests

text: by

text: ﬁgurations

text: of

text: the

text: case

text: IV

text: models.

text: In

text: case

text: V,

text: one

text: third

text: of

text: the

text: stars

text: considering

text: the

text: cumulative

text: distributions

text: of

text: several

text: key

text: kinematic

text: within

text: each

text: association

text: are

text: forced

text: to

text: exhibit

text: outwards

text: radial

text: motion

text: diagnostics

text: across

text: the

text: sample

text: of

text: OB

text: associations.

text: This

text: allows

text: us

text: to

text: with

text: respect

text: to

text: the

text: association

text: centre

text: by

text: taking

text: to

text: be

text: the

text: abso-

text: assess

text: the

text: degree

text: to

text: which

text: their

text: kinematics

text: are

text: best

text: described

text: by

text: a

text: vr

text: lute

text: value

text: of

text: the

text: originally

text: generated

text: On

text: average,

text: this

text: causes

text: net

text: expansion

text: or

text: by

text: random

text: motion.

text: vr

text: of

text: the

text: stars

text: to

text: have

text: positive

text: radial

text: velocities.

text: In

text: case

text: VI,

text: one

text: Number

text: ratios

text: third

text: of

text: the

text: stars

text: within

text: each

text: association

text: are

text: assigned

text: radial

text: veloc-

text: If

text: there

text: is

text: a

text: widespread,

text: systematic

text: expansion

text: of

text: present-day

text: ities

text: equal

text: to

text: the

text: largest

text: absolute

text: value

text: of

text: the

text: original

text: and

text: vr

text: vt

text: OB

text: associations,

text: then

text: it

text: is

text: likely

text: to

text: be

text: evident

text: in

text: the

text: ratio

text: of

text: stars

text: i.e.

text: vr

text: max

text: vr

text: vt

text: The

text: tangential

text: velocity

text: is

text: set

text: to

text: the

text: other,

text: smaller

text: component

text: while

text: keeping

text: its

text: original

text: sign.

text: This

text: model

text: en-

text: association

text: centre.

text: Figure

text: shows

text: the

text: cumulative

text: distribution

text: of

text: the

text: sures

text: that

text: radial

text: motion

text: dominates

text: over

text: tangential

text: motion.

text: This

text: number

text: ratio

text: of

text: sources

text: with

text: positive

text: (outwards)

text: radial

text: velocities

text: reproduces

text: the

text: strong

text: radial

text: anisotropy

text: that

text: is

text: expected

text: β

text: from

text: a

text: monolithic

text: formation

text: scenario.

text: to

text: the

text: number

text: of

text: sources

text: with

text: negative

text: (inward)

text: radial

text: The

text: model

text: cases

text: can

text: be

text: summarised

text: as

text: follows:

text: velocities

text: for

text: all

text: OB

text: associations.

text: Also

text: shown

text: in

text: this

text: Figure

text: are

text: the

text: normalised

text: cumulative

text: distributions

text: for

text: each

text: of

text: the

text: six

text: model

text: (i)

text: Observed

text: stellar

text: positions

text: and

text: absolute

text: velocities

text: with

text: ran-

text: cases

text: described

text: in

text: the

text: previous

text: section.

text: dom

text: directions

text: of

text: motion

text: (case

text: I).

text: It

text: is

text: immediately

text: clear

text: from

text: this

text: ﬁgure

text: that,

text: based

text: on

text: num-

text: (ii)

text: Observed

text: stellar

text: positions

text: and

text: absolute

text: velocities

text: where

text: ber

text: ratios

text: alone,

text: the

text: observed

text: distribution

text: of

text: OB

text: associations

text: are

text: of

text: the

text: stars

text: are

text: guaranteed

text: to

text: have

text: positive

text: radial

text: velocities

text: with

text: re-

text: far

text: better

text: represented

text: by

text: purely

text: random

text: distributions

text: of

text: stellar

text: ve-

text: spect

text: to

text: the

text: mean

text: position

text: and

text: velocity

text: of

text: the

text: nearest

text: neighbours

text: locities

text: rather

text: than

text: those

text: distributions

text: representing

text: either

text: locally

text: (case

text: II).

text: expanding

text: or

text: globally

text: expanding

text: associations.

text: There

text: is

text: a

text: departure

text: (iii)

text: Observed

text: stellar

text: positions

text: and

text: absolute

text: velocities

text: where

text: from

text: the

text: case

text: IV

text: curve

text: tending

text: slightly

text: towards

text: expansion;

text: how-

text: of

text: the

text: stars

text: are

text: guaranteed

text: to

text: have

text: positive

text: radial

text: velocities

text: and

text: ever,

text: this

text: eﬀect

text: is

text: relatively

text: small

text: and

text: the

text: maximum

text: observed

text: ratio

text: dominant

text: radial

text: motions

text: with

text: respect

text: to

text: the

text: mean

text: position

text: and

text: is

text: falling

text: short

text: of

text: what

text: may

text: be

text: expected

text: from

text: a

text: widespread,

text: velocity

text: of

text: the

text: nearest

text: neighbours

text: (case

text: III).

text: systematic

text: expansion.

text: Qualitatively,

text: the

text: case

text: I

text: distributions

text: are

text: the

text: (iv)

text: Random

text: stellar

text: positions

text: and

text: velocities

text: drawn

text: from

text: a

text: four-

text: closest

text: approximation

text: to

text: the

text: observed

text: distributions.

text: When

text: the

text: cen-

text: dimensional

text: Gaussian

text: distribution

text: (case

text: IV).

text: tres

text: of

text: the

text: OB

text: star

text: populations

text: are

text: used,

text: the

text: case

text: I

text: approach

text: of

text: (v)

text: Random

text: stellar

text: positions

text: and

text: velocities

text: drawn

text: from

text: a

text: four-

text: randomising

text: proper

text: motion

text: directions

text: recreates

text: the

text: departure

text: from

text: dimensional

text: Gaussian

text: distribution

text: where

text: of

text: the

text: stars

text: are

text: guar-

text: the

text: case

text: IV

text: distribution

text: of

text: random

text: positions

text: and

text: velocities.

text: The

text: anteed

text: to

text: have

text: positive

text: radial

text: velocities

text: (case

text: V).

text: locally

text: expanding

text: case

text: II

text: and

text: III

text: model

text: distributions

text: fall

text: between

text: (vi)

text: Random

text: stellar

text: positions

text: and

text: velocities

text: drawn

text: from

text: a

text: four-

text: the

text: distributions

text: based

text: on

text: random

text: velocity

text: ﬁelds

text: and

text: the

text: two

text: glob-

text: dimensional

text: Gaussian

text: distribution

text: where

text: of

text: the

text: stars

text: are

text: guar-

text: ally

text: expanding

text: distributions.

text: When

text: the

text: centres

text: of

text: the

text: entire

text: stellar

text: anteed

text: to

text: have

text: positive

text: radial

text: velocities

text: and

text: dominant

text: radial

text: motions

text: populations

text: are

text: used

text: (right

text: panel),

text: the

text: observed

text: systems

text: show

text: a

text: (case

text: VI).

text: small

text: expansion

text: relative

text: to

text: the

text: random

text: motion

text: models

text: (case

text: I

text: and

text: Together,

text: these

text: cases

text: span

text: a

text: physically

text: appropriate

text: range

text: of

text: con-

text: IV),

text: but

text: this

text: small

text: excess

text: is

text: reproduced

text: by

text: forcing

text: just

text: of

text: the

text: ﬁgurations

text: for

text: comparison

text: to

text: the

text: observed

text: kinematic

text: diagnostics

text: stars

text: to

text: exhibit

text: outward

text: radial

text: motion

text: as

text: in

text: cases

text: II,

text: III,

text: V,

text: and

text: VI

text: discussed

text: below.

text: (rather

text: than

text: the

text: used

text: there).

text: This

text: is

text: in

text: strong

text: contrast

text: with

text: the

text: expectation

text: from

text: monolithic

text: models,

text: in

text: which

text: most

text: of

text: the

text: stellar

text: population

text: exhibits

text: expansion

text: (Baumgardt

text: Kroupa

text: Based

text: QUANTIFYING

text: THE

text: KINEMATICS

text: OF

text: OB

text: on

text: these

text: number

text: ratios

text: alone,

text: the

text: observed

text: velocities

text: are

text: certainly

text: ASSOCIATIONS

text: consistent

text: with

text: a

text: population

text: of

text: stars

text: with

text: randomly

text: distributed

text: ve-

text: locities

text: and

text: inconsistent

text: with

text: both

text: global

text: and

text: localised

text: expansion

text: Through

text: analysis

text: of

text: the

text: radial

text: velocities

text: of

text: the

text: member

text: stars

text: scenarios.

text: of

text: OB

text: associations,

text: it

text: is

text: possible

text: to

text: determine

text: whether

text: the

text: associa-

text: Whilst

text: giving

text: the

text: qualitative

text: impression

text: of

text: a

text: random

text: distri-

text: tions

text: are

text: showing

text: signs

text: of

text: strong

text: radial

text: expansion

text: and,

text: by

text: extension,

text: bution,

text: the

text: information

text: gained

text: from

text: this

text: simple

text: analysis

text: is

text: limited,

text: distinguish

text: between

text: the

text: monolithic

text: or

text: hierarchical

text: models

text: of

text: star

text: providing

text: no

text: indication

text: of

text: the

text: magnitudes

text: of

text: the

text: radial

text: velocities

text: formation.

text: The

text: left

text: panel

text: of

text: Figure

text: shows

text: the

text: positions

text: and

text: ve-

text: and

text: by

text: itself

text: it

text: is

text: not

text: a

text: deﬁnitive

text: measure

text: of

text: the

text: dynamical

text: behaviour

text: locities

text: for

text: the

text: OB

text: association

text: Cas

text: These

text: velocities

text: are

text: then

text: of

text: the

text: sample.

text: separated

text: into

text: their

text: radial

text: and

text: tangential

text: components

text: in

text: the

text: middle

text: Median

text: velocity

text: distributions

text: and

text: right

text: panels

text: of

text: Figure

text: These

text: diagrams

text: are

text: shown

text: for

text: all

text: OB

text: associations

text: in

text: Figure

text: In

text: the

text: left

text: panels

text: of

text: Figures

text: and

text: To

text: quantify

text: the

text: typical

text: expansion

text: (or

text: contraction)

text: velocities

text: the

text: OB

text: stars

text: are

text: marked

text: in

text: blue

text: with

text: all

text: other

text: stars

text: marked

text: in

text: red.

text: of

text: the

text: OB

text: associations

text: in

text: our

text: sample,

text: we

text: take

text: the

text: median

text: value

text: of

text: In

text: the

text: centre

text: panels,

text: the

text: points

text: are

text: colour-coded

text: according

text: to

text: the

text: the

text: radial

text: velocities

text: of

text: the

text: stellar

text: populations

text: for

text: each

text: OB

text: associ-

text: radial

text: velocity

text: direction:

text: blue

text: for

text: and

text: red

text: for

text: In

text: the

text: ation.

text: The

text: cumulative

text: distributions

text: of

text: the

text: median

text: radial

text: velocities

text: vr

text: vr

text: right

text: hand

text: panels

text: the

text: points

text: are

text: colour-coded

text: by

text: tangential

text: motion

text: are

text: shown

text: in

text: Figure

text: relative

text: to

text: both

text: the

text: centre

text: of

text: the

text: OB

text: stars

text: direction:

text: blue

text: for

text: clockwise

text: and

text: red

text: for

text: anti-clockwise

text: and

text: the

text: centre

text: of

text: all

text: stars,

text: along

text: with

text: the

text: six

text: model

text: cases.

text: Uncer-

text: vt

text: MNRAS


text: 4 C.Wei et al. (cid:28) κ, γ 1) and to the linear order, the components of the from the non-linear P (k). Therefore, the weak lensing δ matrix are related to the second derivatives of the grav- survey can be used to probe the gravitational growth of itational potential as (Bartelmann & Schneider 2001; the density structure. Hilbert et al. 2009; (cid:90)Kilbinger 2015) While dealing with full-sky measurements, it is use- ful to note that the angular power spectrum of weak − (cid:48) (cid:48) lensing convergence κ and complex shear γ can be de- A − 2 χ (cid:48) r(χ χ )r(χ ) (cid:48) (cid:48) (θ, χ) = δ dχ Φ (r(χ )θ, χ ), rived from the spin-s spherical harmonics Y m (Stebbins ij ij c2 r(χ) ,ij s (cid:96) 0 1996). The relations of power spectra between the con- (2) vergence, shear E- and B-mode (Schneider et al. 2002; Bunn 2003; Bunn et al. 2003; Zhao & Baskaran 2010) where δ is the Kronecker delta, c is the speed of ij have been derived by Hu (2000) for an all-sky lensing light, χ is the comoving distance and r(χ) the comov- deformation tensor ﬁeld. Here we brieﬂy summarize the ing angular diameter distance. According to the Pois- spin-s spherical harmonic decomposition of the full-sky son equation, gravitational potential Φ can be related lensing, referring the reader to Hu (2000) for detailed to the density contrast δ. Hence, convergence κ can be discussions of the power spectrum in weak lensing. expressed as a weighted integral of the over-density δ As reviewed in the appendix of Becker (2013), the along the line of sig(cid:90)ht, convergence, lensing shear and rotation in the distor- tion matrix (Eq. 1) can be decomposed by the spherical 3H2Ω χ (cid:48) r(χ − χ(cid:48))r(χ(cid:48)) δ(r(χ(cid:48))θ, χ(cid:48)) harmonics (Hu 2000; B(cid:88)ecker 2013), κ(θ, χ) = 0 m dχ (cid:48) , 2c2 r(χ) a(χ ) 0 −1 (3) κ(nˆ) = (cid:96)((cid:96) + 1)φ Y m(nˆ) (7) 2 (cid:96)m (cid:96) (cid:96)m where H0 is the Hubble constant, Ωm is the(cid:48) matter (cid:115) density in units of the critical density, and a(χ ) is the (cid:88) (cid:48) scale factor at χ . 2.2. Power Spectrum of Weak Lensing Field ± 1 ((cid:96) + 2)! ± γ (nˆ) iγ (nˆ) = − (φ iΩ )± Y m(nˆ) 1 2 2 ((cid:96) 2)! (cid:96)m (cid:96)m 2 (cid:96) In the ﬂat-sky limit, the power spectrum of the conver- (cid:96)m gence Cκκ((cid:96)) on the modulus (cid:96) is known as the two-point (8) (cid:88) correlation in Fourier space, (cid:104) ∗ (cid:48) (cid:105) − (cid:48) −1 κ˜((cid:96))κ˜ ((cid:96) ) = (2π)2δ ((cid:96) (cid:96) )Cκκ((cid:96)), (4) ω(nˆ) = (cid:96)((cid:96) + 1)Ω Y m(nˆ), (9) D 2 (cid:96)m (cid:96) (cid:96)m where δD((cid:96)) is the Dirac delta function. Using equa- where φ is the lensing deﬂection potential, Ω is the tion (3), one can derive the angular power spectrum of pseudo-scalar potential (as described by Stebbins 1996), the convergenc(cid:90)e ﬁeld in the Limb(cid:18)er approxima(cid:19)tion, and nˆ denotes a given position on the sky. Consequently, the diﬀerent power spectra can be related as, χH W(χ)2 (cid:96) Cκκ((cid:96)) = dχ P k = , χ , (5) 1 r(χ)2 δ r(χ) Cκκ((cid:96)) = (cid:96)2((cid:96) + 1)2Cφφ((cid:96)) (10) 0 4 where P (k, χ) is the 3-D power spectrum of the mat- δ ter distribution at the given comoving distance χ, and 1 Cωω((cid:96)) = (cid:96)2((cid:96) + 1)2CΩΩ((cid:96)) (11) the integral is calculated along the line of sight to the 4 comoving horizon distance χ . Here the weight function H W(χ) is deﬁned as, 1 ((cid:96) + 2)! − CEE((cid:96)) = − Cκκ((cid:96)) (12) W(χ) = 3H02Ωm r(χH χ)r(χ) 1 . (6) (cid:96)2((cid:96) + 1)2 ((cid:96) 2)! 2c2 r(χ ) a(χ) H From the non-linear theoretical models, such as the 1 ((cid:96) + 2)! CBB((cid:96)) = − Cωω((cid:96)). (13) Haloﬁt model (Smith et al. 2003; Takahashi et al. 2012), (cid:96)2((cid:96) + 1)2 ((cid:96) 2)! one can predict the convergence power spectrum Cκκ((cid:96)) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: C.Wei

text: et

text: al.

text: κ,

text: γ

text: and

text: to

text: the

text: linear

text: order,

text: the

text: components

text: of

text: the

text: from

text: the

text: non-linear

text: P

text: (k).

text: Therefore,

text: the

text: weak

text: lensing

text: δ

text: matrix

text: are

text: related

text: to

text: the

text: second

text: derivatives

text: of

text: the

text: grav-

text: survey

text: can

text: be

text: used

text: to

text: probe

text: the

text: gravitational

text: growth

text: of

text: itational

text: potential

text: as

text: (Bartelmann

text: Schneider

text: the

text: density

text: structure.

text: Hilbert

text: et

text: al.

text: While

text: dealing

text: with

text: full-sky

text: measurements,

text: it

text: is

text: use-

text: ful

text: to

text: note

text: that

text: the

text: angular

text: power

text: spectrum

text: of

text: weak

label: equation

text: lensing

text: convergence

text: κ

text: and

text: complex

text: shear

text: γ

text: can

text: be

text: de-

text: A

text: χ

text: r(χ

text: χ

text: )r(χ

text: (θ,

text: χ)

text: δ

text: dχ

text: Φ

text: (r(χ

text: )θ,

text: χ

text: rived

text: from

text: the

text: spin-s

text: spherical

text: harmonics

text: Y

text: m

text: (Stebbins

text: ij

text: ij

text: r(χ)

text: ,ij

text: s

text: The

text: relations

text: of

text: power

text: spectra

text: between

text: the

text: con-

text: vergence,

text: shear

text: E-

text: and

text: B-mode

text: (Schneider

text: et

text: al.

text: Bunn

text: Bunn

text: et

text: al.

text: Zhao

text: Baskaran

text: where

text: δ

text: is

text: the

text: Kronecker

text: delta,

text: c

text: is

text: the

text: speed

text: of

text: ij

text: have

text: been

text: derived

text: by

text: Hu

text: for

text: an

text: all-sky

text: lensing

text: light,

text: χ

text: is

text: the

text: comoving

text: distance

text: and

text: r(χ)

text: the

text: comov-

text: deformation

text: tensor

text: ﬁeld.

text: Here

text: we

text: brieﬂy

text: summarize

text: the

text: ing

text: angular

text: diameter

text: distance.

text: According

text: to

text: the

text: Pois-

text: spin-s

text: spherical

text: harmonic

text: decomposition

text: of

text: the

text: full-sky

text: son

text: equation,

text: gravitational

text: potential

text: Φ

text: can

text: be

text: related

text: lensing,

text: referring

text: the

text: reader

text: to

text: Hu

text: for

text: detailed

text: to

text: the

text: density

text: contrast

text: δ.

text: Hence,

text: convergence

text: κ

text: can

text: be

text: discussions

text: of

text: the

text: power

text: spectrum

text: in

text: weak

text: lensing.

text: expressed

text: as

text: a

text: weighted

text: integral

text: of

text: the

text: over-density

text: δ

text: As

text: reviewed

text: in

text: the

text: appendix

text: of

text: Becker

text: the

text: along

text: the

text: line

text: of

text: convergence,

text: lensing

text: shear

text: and

text: rotation

text: in

text: the

text: distor-

text: tion

text: matrix

text: (Eq.

text: can

text: be

text: decomposed

text: by

text: the

text: spherical

text: χ

text: r(χ

text: harmonics

text: (Hu

text: κ(θ,

text: χ)

text: m

text: dχ

text: r(χ)

text: a(χ

text: −1

text: κ(nˆ)

text: Y

text: m(nˆ)

text: where

text: is

text: the

text: Hubble

text: constant,

text: Ωm

text: is

text: matter

text: density

text: in

text: units

text: of

text: the

text: critical

text: density,

text: and

text: a(χ

text: is

text: the

text: scale

text: factor

text: at

text: χ

text: Power

text: Spectrum

text: of

text: Weak

text: Lensing

text: Field

text: γ

text: (nˆ)

text: iγ

text: (nˆ)

text: (φ

text: iΩ

text: Y

text: m(nˆ)

text: In

text: the

text: ﬂat-sky

text: limit,

text: the

text: power

text: spectrum

text: of

text: the

text: conver-

text: gence

text: on

text: the

text: modulus

text: is

text: known

text: as

text: the

text: two-point

text: correlation

text: in

text: Fourier

text: space,

text: −1

text: ω(nˆ)

text: Y

text: m(nˆ),

text: D

text: where

text: is

text: the

text: Dirac

text: delta

text: function.

text: Using

text: equa-

text: where

text: φ

text: is

text: the

text: lensing

text: deﬂection

text: potential,

text: Ω

text: is

text: the

text: tion

text: one

text: can

text: derive

text: the

text: angular

text: power

text: spectrum

text: of

text: pseudo-scalar

text: potential

text: (as

text: described

text: by

text: Stebbins

text: the

text: ﬁeld

text: in

text: the

text: and

text: nˆ

text: denotes

text: a

text: given

text: position

text: on

text: the

text: sky.

text: Consequently,

text: the

text: diﬀerent

text: power

text: spectra

text: can

text: be

text: related

text: as,

text: χH

text: dχ

text: P

text: k

text: χ

text: δ

text: r(χ)

text: where

text: P

text: (k,

text: χ)

text: is

text: the

text: power

text: spectrum

text: of

text: the

text: mat-

text: δ

text: ter

text: distribution

text: at

text: the

text: given

text: comoving

text: distance

text: χ,

text: and

text: the

text: integral

text: is

text: calculated

text: along

text: the

text: line

text: of

text: sight

text: to

text: the

text: comoving

text: horizon

text: distance

text: χ

text: Here

text: the

text: weight

text: function

text: H

text: W(χ)

text: is

text: deﬁned

text: as,

text: W(χ)

text: r(χH

text: χ)r(χ)

text: r(χ

text: a(χ)

text: H

text: From

text: the

text: non-linear

text: theoretical

text: models,

text: such

text: as

text: the

text: Haloﬁt

text: model

text: (Smith

text: et

text: al.

text: Takahashi

text: et

text: al.

text: one

text: can

text: predict

text: the

text: convergence

text: power

text: spectrum

text: ##LTLine##

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text: 8 C.Wei et al. 1.0 by including additional physics about cold gas stripping z = 0.0 and an analytical modeling of orphan galaxies. In this z = 0.5 0.9 z = 1.0 model, the stellar mass function, HI and H mass func- 2 tions are tuned to match the observational data (Keres 0.8 et al. 2003; Zwaan et al. 2005; Baldry et al. 2008; Li & 0.7 White 2009). The fraction and spatial distributions of central versus satellite galaxies are reproduced roughly 0.6 correctly by the model, as shown in Luo et al. (2016) and Guo et al. (2013). 0.5 To describe the morphology of a galaxy, we use the ra- tio between the bulge and the total mass (B/T), which 0.4 Late-type Early-type can be predicted from the semi-analytical model, as the classiﬁcation of ‘early-type’ and ‘late-type’ galaxies 0.0 0.2 0.4 0.6 0.8 1.0 (Parry et al. 2009). Following Joachimi et al. (2013a), B/T we adopt B/T = 0.6 to classify the model galaxies into early or late types. Fig. 3 shows the cumulative prob- Figure 3. Cumulative probability distributions of the bulge- to-total mass ratio of model galaxies at three diﬀerent red- ability distribution of B/T of our simulated galaxies at shifts. three diﬀerent redshifts. More than 80% of all galax- ies are late-types, and the fraction is slightly higher at and shear power spectra are more accurate extending to high redshifts. This fraction of early/late-types is con- higher (cid:96). sistent with that found in other studies (e.g., Guo et al. 4. GALAXY MODELING 2013). However, compared to hydro-dynamical simula- tions, semi-analytical models are less powerful in pre- One important merit of our work is to include model dicting the shapes of galaxies. To proceed, we have to galaxies in the N-body simulations and to predict shear assign shapes to galaxies and their images with some correlation functions that can be compared directly to simpliﬁed prescriptions, as described below. observations. In this section, we describe how we model 4.2. Galaxy Shape Measurement the physical properties of galaxies and show the intrinsic alignment of the model galaxies. A common assumption is that the shape of an elliptical 4.1. Semi-analytic Models galaxy roughly follows that of its host dark matter halo, while the rotation axis of a spiral galaxy is determined The model galaxies are produced using the semi- by the spin of its halo (e.g., Kang et al. 2007; Okumura analytical model of Luo et al. (2016) which is based on et al. 2009; Agustsson & Brainerd 2010). Joachimi et the Guo et al. (2013) model, one version of the Munich al. (2013b, hereafter ’J13’) used this assumption and Semi-analytical model which is called L-Galaxies. In studied the alignment of galaxies from the Millennium a semi-analytical model, galaxy population is assigned Simulation (Springel et al. 2005). Here we follow J13 to to dark matter haloes on the basis of simple assump- assign shapes to model galaxies. tions of many physical processes. As a ﬁrst step in our To assign a shape to a model galaxy using the mass implementation of L-Galaxies, dark matter haloes are distribution of its dark matter halo, we need to distin- identiﬁed in the N-body simulation using the standard guish between central and satellite galaxies. A central Friends-of-Friends (FOF) algorithm. Only haloes that galaxy is assumed to be located at the center of a dark contain at least 20 particles are used. The subhaloes matter halo and its shape may be related to that of the within each FOF halo are identiﬁed with the SUBFIND host halo. The shape of a dark matter halo is usually algorithm (Springel et al. 2001, 2005). Merger trees deﬁned using the inertia tensor I (Bailin & Steinmetz of these dark matter (sub)haloes can be constructed ij 2005), (cid:88) by linking progenitors of a halo in diﬀerent snapshots. Galaxies are assumed to form at the centers of the dark Np matter haloes according to analytical prescriptions of I = m x , x , (19) ij p i,n j,n the relevant physical processes, such as gas cooling, star n=1 formation, supernova and black hole feedback. For the details of L-Galaxies, we refer the reader to Guo et where Np denotes the particle number of the FOF al. (2013). Luo et al. (2016) improved the prescrip- halo, and xn is the position of the n-th particle with re- tion for low-mass galaxies, especially satellite galaxies, spect to the center of the halo. By diagonalizing the in- (cid:54) (cid:54) ertia tensor I, one can get the eigenvalues λ λ λ 1 2 3 FDC ##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##

text: C.Wei

text: et

text: al.

font: default

text: by

text: including

text: additional

text: physics

text: about

text: cold

text: gas

text: stripping

text: z

text: and

text: an

text: analytical

text: modeling

text: of

text: orphan

text: galaxies.

text: In

text: this

text: z

text: z

text: model,

text: the

text: stellar

text: mass

text: function,

text: HI

text: and

text: H

text: mass

text: func-

text: tions

text: are

text: tuned

text: to

text: match

text: the

text: observational

text: data

text: (Keres

text: et

text: al.

text: Zwaan

text: et

text: al.

text: Baldry

text: et

text: al.

text: Li

text: White

text: The

text: fraction

text: and

text: spatial

text: distributions

text: of

text: central

text: versus

text: satellite

text: galaxies

text: are

text: reproduced

text: roughly

text: correctly

text: by

text: the

text: model,

text: as

text: shown

text: in

text: Luo

text: et

text: al.

text: and

text: Guo

text: et

text: al.

text: To

text: describe

text: the

text: morphology

text: of

text: a

text: galaxy,

text: we

text: use

text: the

text: ra-

text: tio

text: between

text: the

text: bulge

text: and

text: the

text: total

text: mass

text: (B/T),

text: which

text: Late-type

text: Early-type

text: can

text: be

text: predicted

text: from

text: the

text: semi-analytical

text: model,

text: as

text: the

text: classiﬁcation

text: of

text: ‘early-type’

text: and

text: ‘late-type’

text: galaxies

text: (Parry

text: et

text: al.

text: Following

text: Joachimi

text: et

text: al.

text: B/T

text: we

text: adopt

text: B/T

text: to

text: classify

text: the

text: model

text: galaxies

text: into

text: early

text: or

text: late

text: types.

text: Fig.

text: shows

text: the

text: cumulative

text: prob-

text: Figure

text: Cumulative

text: probability

text: distributions

text: of

text: the

text: bulge-

text: to-total

text: mass

text: ratio

text: of

text: model

text: galaxies

text: at

text: three

text: diﬀerent

text: red-

text: ability

text: distribution

text: of

text: B/T

text: of

text: our

text: simulated

text: galaxies

text: at

text: shifts.

text: three

text: diﬀerent

text: redshifts.

text: More

text: than

text: of

text: all

text: galax-

text: ies

text: are

text: late-types,

text: and

text: the

text: fraction

text: is

text: slightly

text: higher

text: at

text: and

text: shear

text: power

text: spectra

text: are

text: more

text: accurate

text: extending

text: to

text: high

text: redshifts.

text: This

text: fraction

text: of

text: early/late-types

text: is

text: con-

text: higher

text: sistent

text: with

text: that

text: found

text: in

text: other

text: studies

text: (e.g.,

text: Guo

text: et

text: al.

text: GALAXY

text: MODELING

text: However,

text: compared

text: to

text: hydro-dynamical

text: simula-

text: tions,

text: semi-analytical

text: models

text: are

text: less

text: powerful

text: in

text: pre-

text: One

text: important

text: merit

text: of

text: our

text: work

text: is

text: to

text: include

text: model

text: dicting

text: the

text: shapes

text: of

text: galaxies.

text: To

text: proceed,

text: we

text: have

text: to

text: galaxies

text: in

text: the

text: N-body

text: simulations

text: and

text: to

text: predict

text: shear

text: assign

text: shapes

text: to

text: galaxies

text: and

text: their

text: images

text: with

text: some

text: correlation

text: functions

text: that

text: can

text: be

text: compared

text: directly

text: to

text: simpliﬁed

text: prescriptions,

text: as

text: described

text: below.

text: observations.

text: In

text: this

text: section,

text: we

text: describe

text: how

text: we

text: model

text: Galaxy

text: Shape

text: Measurement

text: the

text: physical

text: properties

text: of

text: galaxies

text: and

text: show

text: the

text: intrinsic

text: alignment

text: of

text: the

text: model

text: galaxies.

text: A

text: common

text: assumption

text: is

text: that

text: the

text: shape

text: of

text: an

text: elliptical

text: Semi-analytic

text: Models

text: galaxy

text: roughly

text: follows

text: that

text: of

text: its

text: host

text: dark

text: matter

text: halo,

text: while

text: the

text: rotation

text: axis

text: of

text: a

text: spiral

text: galaxy

text: is

text: determined

text: The

text: model

text: galaxies

text: are

text: produced

text: using

text: the

text: semi-

text: by

text: the

text: spin

text: of

text: its

text: halo

text: (e.g.,

text: Kang

text: et

text: al.

text: Okumura

text: analytical

text: model

text: of

text: Luo

text: et

text: al.

text: which

text: is

text: based

text: on

text: et

text: al.

text: Agustsson

text: Brainerd

text: Joachimi

text: et

text: the

text: Guo

text: et

text: al.

text: model,

text: one

text: version

text: of

text: the

text: Munich

text: al.

text: hereafter

text: used

text: this

text: assumption

text: and

text: Semi-analytical

text: model

text: which

text: is

text: called

text: L-Galaxies.

text: In

text: studied

text: the

text: alignment

text: of

text: galaxies

text: from

text: the

text: Millennium

text: a

text: semi-analytical

text: model,

text: galaxy

text: population

text: is

text: assigned

text: Simulation

text: (Springel

text: et

text: al.

text: Here

text: we

text: follow

text: to

text: to

text: dark

text: matter

text: haloes

text: on

text: the

text: basis

text: of

text: simple

text: assump-

text: assign

text: shapes

text: to

text: model

text: galaxies.

text: tions

text: of

text: many

text: physical

text: processes.

text: As

text: a

text: ﬁrst

text: step

text: in

text: our

text: To

text: assign

text: a

text: shape

text: to

text: a

text: model

text: galaxy

text: using

text: the

text: mass

text: implementation

text: of

text: L-Galaxies,

text: dark

text: matter

text: haloes

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text: 4 Joshi, R. et al. Figure 3. The [O ii] emission line proﬁle seen in SDSS-DR7 (left) and DR12 (right) stacked spectra for various W2796 bins. The solid line shows the best ﬁt double Gaussian to the data. The continuum-subtracted spectra are shifted by a constant oﬀset in luminosity for display purpose. Figure 4. Left panel: The [O ii] luminosity of Mg ii absorbers for various W2796 bins for SDSS-DR7 (red circles) and DR12 (blue triangles). Right panel: The [O ii] luminosity surface density (Σ[O II]) as a function of W2796 for SDSS-DR7 (red circles) and DR12 (blue triangles). The dotted line shows the best ﬁt for Σ[O II] versus W2796 from M´enard et al. (2011) while shaded region show the 1σ uncertainty (see Appendix for more discussion on this diﬀerence). The dashed and dot-dashed line are the best-ﬁtting power law for our measurements based on SDSS-DR7 and DR12 data set. 20bp145bp589bp712bp datasets for each W bin are listed in column 4 and 7 of 2796 Figure 5. Lower panel: The distribution of σ as a function of Table 1, respectively. We ﬁrst compare our results based [OII] W2796 for SDSS-DR7 (diamonds) and SDSS-DR12 (stars). Upper on SDSS-DR7 stacked spectra with that of M´enard et al. panel: shows the [O ii] line proﬁle for various W2796 bins for SDSS- (2011), based on SDSS-DR4, as spectra were obtained using DR7 and DR12. 3 arcsec ﬁbre in both cases. Note that our redshift range (cid:54) (cid:54) of 0.55 z 1.3 is slightly diﬀerent from that of M´enard (cid:54) (cid:54) et al. (2011) of 0.36 z 1.3. As suggested by M´enard et al. R tra to study the eﬀect of parameter on the average red- (2011) we model this relationship using a powerlaw of the (cid:104) (cid:105) dening induced by Mg ii absorbers (York et al. 2006; Khare form Σ = AWα and compute the best ﬁt parameters [O I±I] 0 ± × − − et al. 2012). For this, we have shifted each spectrum to the of α = 1.69 0.11 and A = (2.55 0.27) 1037erg s 1 kpc 2 absorber rest frame and computed their geometric mean. for SDSS-DR7. While comparing our best ﬁt parameters ± Here, we do not use the normalized or continuum subtracted with M´enard et al. (2011), i.e., α = 1.75 0.11 and A = ± × − − spectrum to preserve the average continuum shape which is (1.48 0.18) 1037erg s 1 kpc 2, the α is found to be same ∼ important for determining the characteristic extinction law whereas normalization factor is found to be diﬀerent (i.e. ∼ (see also, York et al. 2006). To compute the relative extinc- 1.7 times higher) at a signiﬁcance level of 3.3σ. We note tion we have generated the stacked spectra for a control sam- that this diﬀerence in the normalization factor is mainly ± ± ple of quasars, within ∆z = 0.05 of zem and ∆rmag = 0.5 due to diﬀerence in the way quasar continuum is modelled. of rmag, without absorbers in their spectra. We will discuss M´enard et al. (2011) have modelled the continuum with an the results of this analysis in Section . iterative running median of sizes ranging from 500 to 15 4 RESULTS pixels. This basically smooths all small scale ﬂuctuations. In fact, we get the similar values of Σ if we use the con- We generate several composite spectra in various W [O II] 2796 tinuum ﬁtting procedure adopted by M´enard et al. (2011) and redshift bins. Since, we are interested in measuring the [see Fig. A2 and related discussions in the Appendix]. How- average [O ii] nebular emission line luminosity (or surface ever, for rest of the paper we will present result from data brightness) associated with the Mg ii absorbers, the direct using our continuum ﬁtting procedure. detection of [O ii] nebular emission reported in Paper 1 are It is clear from the ﬁgure that Σ measured from also included in most of our analysis. We also present the [O II] the SDSS-DR7 data are higher than those from SDSS-DR12 results when these systems are excluded from the analysis. data (apart from the ﬁrst equivalent width bin). The diﬀer- 4.1 Average [O ii] emission and ﬁbre eﬀects ence in 3 out of the 5 equivalent width bin considered here is To quantify the ﬁbre loss eﬀect we have constructed more than 2σ (see column 4 and 7 of Table 1). The best ﬁtted composite−spectra b−y dividing−our samp−le into ﬁve(cid:62)W2796 relationship between Σ[O II] and W2796(cid:104) for th(cid:105)e SDSS-DR12 bins of 1 1.5, 1.5 2.0, 2.0 2.5, 2.5 3.0 and 3 ˚A data are shown in dot-dashed line (i.e., Σ = AWα with ± ± × [O II] − 0− for SDSS-DR7 and DR12. The stacked proﬁles for various α = 1.28 0.11 and A = (2.95 0.29) 1037erg s 1 kpc 2) in W bins and the number of systems used to construct the ﬁgure. While this ﬁt is not as good as that for DR7 data, 2796 the stacked spectra are shown in Fig. 3. It is apparent that what is evident is that the ﬁt to DR12 data is consistently for a given equivalent width bin the [O ii] line luminosity lower than that for the DR7 data in all but one equivalent found for SDSS-DR7 is higher than that of SDSS-DR12. In width bin. This clearly conﬁrms that the observed [O ii] the left panel of Fig. 4, we show the median [O ii] line lu- surface brightness is aﬀected by ﬁbre size eﬀects even after minosity as a function of W for the Mg ii systems in normalizing the ﬂux with the projected ﬁbre size. In particu- 2796 SDSS-DR7 (circles) and DR12 (triangles). We ﬁnd a clear lar our results are consistent with the fact that (1) there is a increasing trend of L[O II] with increasing W2796. This is con- large scatter in the W2796 vs impact parameter relationship sistent with the trend found by Noterdaeme et al. (2010) in even at high equivalent widths (see ﬁgure 1 of Nielsen et al. a stacking analysis of Mg ii systems they have found from 2013; Lan et al. 2014; Huang et al. 2016) and (2) gradients SDSS-DR7 quasars. We note that median L we detect are known to be present in the surface brightness proﬁles of [O II] in SDSS-DR7 sample is similar to that obtained by Noter- galaxies which results in lower measured [O ii] luminosities daeme et al. (2010, see also their table 5). when smaller ﬁbres probe predominately the outer parts of In right panel of Fig. 4, we show Σ as a function of galaxies. [O II] W . We conﬁrm the strong correlation between W and In the lower panel of Fig. 5 we also show the depen- 2796 2796 Σ in both the SDSS-DR7 (circles) and DR12 (triangles) dence of velocity width (deconvolved for the instrumental [O II] datasets. The Σ measurement for SDSS-DR7 and DR12 broadening) of [O ii] line (σ ) on the W of Mg ii sys- [O II] [OII] 2796 MNRAS 000, 1–9 (2018)

text: Joshi,

text: R.

text: et

text: al.

text: Figure

text: The

label: caption

text: [O

text: ii]

text: emission

text: line

text: proﬁle

text: seen

text: in

text: (left)

text: and

text: (right)

text: stacked

text: spectra

text: for

text: various

text: bins.

text: The

text: solid

text: line

text: shows

text: the

text: best

text: ﬁt

text: double

text: Gaussian

text: to

text: the

text: data.

text: The

text: continuum-subtracted

text: spectra

text: are

text: shifted

text: by

text: a

text: constant

text: oﬀset

text: in

text: luminosity

text: for

text: display

text: purpose.

text: Figure

text: Left

text: panel:

text: The

text: [O

text: ii]

text: luminosity

text: of

text: Mg

text: ii

text: absorbers

text: for

text: various

text: bins

text: for

text: (red

text: circles)

text: and

text: (blue

text: triangles).

text: Right

text: panel:

text: The

text: [O

text: ii]

text: luminosity

text: surface

text: density

text: (Σ[O

text: II])

text: as

text: a

text: function

text: of

text: for

text: (red

text: circles)

text: and

text: (blue

text: triangles).

text: The

text: dotted

text: line

text: shows

text: the

text: best

text: ﬁt

text: for

text: Σ[O

text: II]

text: versus

text: from

text: M´enard

text: et

text: al.

text: while

text: shaded

text: region

text: show

text: the

text: 1σ

text: uncertainty

text: (see

text: Appendix

text: for

text: more

text: discussion

text: on

text: this

text: diﬀerence).

text: The

text: dashed

text: and

text: dot-dashed

text: line

text: are

text: the

text: best-ﬁtting

text: power

text: law

text: for

text: our

text: measurements

text: based

text: on

text: and

text: data

text: set.

text: datasets

text: for

text: each

text: W

text: bin

text: are

text: listed

text: in

text: column

text: and

text: of

text: Figure

text: Lower

text: panel:

text: The

text: distribution

text: of

text: σ

text: as

text: a

text: function

text: of

text: Table

text: respectively.

text: We

text: ﬁrst

text: compare

text: our

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text: [OII]

text: for

text: (diamonds)

text: and

text: (stars).

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text: on

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text: spectra

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text: that

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text: M´enard

text: et

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text: panel:

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text: various

text: bins

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text: SDSS-

text: based

text: on

text: as

text: spectra

text: were

text: obtained

text: using

text: and

text: arcsec

text: ﬁbre

text: in

text: both

text: cases.

text: Note

text: that

text: our

text: redshift

text: range

text: of

text: z

text: is

text: slightly

text: diﬀerent

text: from

text: that

text: of

text: M´enard

text: et

text: al.

text: of

text: z

text: As

text: suggested

text: by

text: M´enard

text: et

text: al.

text: R

text: tra

text: to

text: study

text: the

text: eﬀect

text: of

text: parameter

text: on

text: the

text: average

text: red-

text: we

text: model

text: this

text: relationship

text: using

text: a

text: powerlaw

text: of

text: the

text: dening

text: induced

text: by

text: Mg

text: ii

text: absorbers

text: (York

text: et

text: al.

text: Khare

text: form

text: Σ

text: AWα

text: and

text: compute

text: the

text: best

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text: parameters

text: [O

text: I±I]

text: et

text: al.

text: For

text: this,

text: we

text: have

text: shifted

text: each

text: spectrum

text: to

text: the

text: of

text: α

text: and

text: A

text: s

text: kpc

text: absorber

text: rest

text: frame

text: and

text: computed

text: their

text: geometric

text: mean.

text: for

text: While

text: comparing

text: our

text: best

text: ﬁt

text: parameters

text: Here,

text: we

text: do

text: not

text: use

text: the

text: normalized

text: or

text: continuum

text: subtracted

text: with

text: M´enard

text: et

text: al.

text: i.e.,

text: α

text: and

text: A

text: spectrum

text: to

text: preserve

text: the

text: average

text: continuum

text: shape

text: which

text: is

text: s

text: kpc

text: the

text: α

text: is

text: found

text: to

text: be

text: same

text: important

text: for

text: determining

text: the

text: characteristic

text: extinction

text: law

text: whereas

text: normalization

text: factor

text: is

text: found

text: to

text: be

text: diﬀerent

text: (i.e.

text: (see

text: also,

text: York

text: et

text: al.

text: To

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text: the

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text: times

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text: note

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text: have

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text: the

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text: spectra

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text: that

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text: the

text: normalization

text: factor

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text: mainly

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text: ∆z

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text: zem

text: and

text: ∆rmag

text: due

text: to

text: diﬀerence

text: in

text: the

text: way

text: quasar

text: continuum

text: is

text: modelled.

text: of

text: rmag,

text: without

text: absorbers

text: in

text: their

text: spectra.

text: We

text: will

text: discuss

text: M´enard

text: et

text: al.

text: have

text: modelled

text: the

text: continuum

text: with

text: an

text: the

text: results

text: of

text: this

text: analysis

text: in

text: Section

text: iterative

text: running

text: median

text: of

text: sizes

text: ranging

text: from

text: to

text: RESULTS

text: pixels.

text: This

text: basically

text: smooths

text: all

text: small

text: scale

text: ﬂuctuations.

text: In

text: fact,

text: we

text: get

text: the

text: similar

text: values

text: of

text: Σ

text: if

text: we

text: use

text: the

text: con-

text: We

text: generate

text: several

text: composite

text: spectra

text: in

text: various

text: W

text: [O

text: II]

text: tinuum

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text: procedure

text: adopted

text: by

text: M´enard

text: et

text: al.

text: and

text: redshift

text: bins.

text: Since,

text: we

text: are

text: interested

text: in

text: measuring

text: the

text: [see

text: Fig.

text: and

text: related

text: discussions

text: in

text: the

text: Appendix].

text: How-

text: average

text: [O

text: ii]

text: nebular

text: emission

text: line

text: luminosity

text: (or

text: surface

text: ever,

text: for

text: rest

text: of

text: the

text: paper

text: we

text: will

text: present

text: result

text: from

text: data

text: brightness)

text: associated

text: with

text: the

text: Mg

text: ii

text: absorbers,

text: the

text: direct

text: using

text: our

text: continuum

text: ﬁtting

text: procedure.

text: detection

text: of

text: [O

text: ii]

text: nebular

text: emission

text: reported

text: in

text: Paper

text: are

text: It

text: is

text: clear

text: from

text: the

text: ﬁgure

text: that

text: Σ

text: measured

text: from

text: also

text: included

text: in

text: most

text: of

text: our

text: analysis.

text: We

text: also

text: present

text: the

text: [O

text: II]

text: the

text: data

text: are

text: higher

text: than

text: those

text: from

text: results

text: when

text: these

text: systems

text: are

text: excluded

text: from

text: the

text: analysis.

text: data

text: (apart

text: from

text: the

text: ﬁrst

text: equivalent

text: width

text: bin).

text: The

text: diﬀer-

text: Average

text: [O

text: ii]

text: emission

text: and

text: ﬁbre

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text: in

text: out

text: of

text: the

text: equivalent

text: width

text: bin

text: considered

text: here

text: is

text: To

text: quantify

text: the

text: ﬁbre

text: loss

text: eﬀect

text: we

text: have

text: constructed

text: more

text: than

text: 2σ

text: (see

text: column

text: and

text: of

text: Table

text: The

text: best

text: ﬁtted

text: composite−spectra

text: b−y

text: dividing−our

text: samp−le

text: into

text: relationship

text: between

text: Σ[O

text: II]

text: and

text: for

text: bins

text: of

text: and

text: ˚A

text: data

text: are

text: shown

text: in

text: dot-dashed

text: line

text: (i.e.,

text: Σ

text: AWα

text: with

text: [O

text: II]

text: 0−

text: for

text: and

text: The

text: stacked

text: proﬁles

text: for

text: various

text: α

text: and

text: A

text: s

text: kpc

text: in

text: W

text: bins

text: and

text: the

text: number

text: of

text: systems

text: used

text: to

text: construct

text: the

text: ﬁgure.

text: While

text: this

text: ﬁt

text: is

text: not

text: as

text: good

text: as

text: that

text: for

text: data,

text: the

text: stacked

text: spectra

text: are

text: shown

text: in

text: Fig.

text: It

text: is

text: apparent

text: that

text: what

text: is

text: evident

text: is

text: that

text: the

text: ﬁt

text: to

text: data

text: is

text: consistently

text: for

text: a

text: given

text: equivalent

text: width

text: bin

text: the

text: [O

text: ii]

text: line

text: luminosity

text: lower

text: than

text: that

text: for

text: the

text: data

text: in

text: all

text: but

text: one

text: equivalent

text: found

text: for

text: is

text: higher

text: than

text: that

text: of

text: In

text: width

text: bin.

text: This

text: clearly

text: conﬁrms

text: that

text: the

text: observed

text: [O

text: ii]

text: the

text: left

text: panel

text: of

text: Fig.

text: we

text: show

text: the

text: median

text: [O

text: ii]

text: line

text: lu-

text: surface

text: brightness

text: is

text: aﬀected

text: by

text: ﬁbre

text: size

text: eﬀects

text: even

text: after

text: minosity

text: as

text: a

text: function

text: of

text: W

text: for

text: the

text: Mg

text: ii

text: systems

text: in

text: normalizing

text: the

text: ﬂux

text: with

text: the

text: projected

text: ﬁbre

text: size.

text: In

text: particu-

text: (circles)

text: and

text: (triangles).

text: We

text: ﬁnd

text: a

text: clear

text: lar

text: our

text: results

text: are

text: consistent

text: with

text: the

text: fact

text: that

text: there

text: is

text: a

text: increasing

text: trend

text: of

text: L[O

text: II]

text: with

text: increasing

text: This

text: is

text: con-

text: large

text: scatter

text: in

text: the

text: vs

text: impact

text: parameter

text: relationship

text: sistent

text: with

text: the

text: trend

text: found

text: by

text: Noterdaeme

text: et

text: al.

text: in

text: even

text: at

text: high

text: equivalent

text: widths

text: (see

text: ﬁgure

text: of

text: Nielsen

text: et

text: al.

text: a

text: stacking

text: analysis

text: of

text: Mg

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text: In

text: right

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text: we

text: show

text: Σ

text: as

text: a

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text: MNRAS


text: Optimizing deep-space optical communication under power constraints Marcin Jarzynaa, Wojciech Zwolin´skib, Micha(cid:32)l Jachurab, and Konrad Banaszeka,b aCentre of New Technologies, University of Warsaw, Banacha 2c, 02-097 Warszawa, Poland bFaculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland ABSTRACT We investigate theoretically the eﬃciency of deep-space optical communication in the presence of background noise. With decreasing average signal power spectral density, a scaling gap opens up between optimized simple- decoded pulse position modulation and generalized on-oﬀ keying with direct detection. The scaling of the latter follows the quantum mechanical capacity of an optical channel with additive Gaussian noise. Eﬃcient communication is found to require a highly imbalanced distribution of instantaneous signal power. This condition can be alleviated through the use of structured receivers which exploit optical interference over multiple time bins to concentrate the signal power before the detection stage. Keywords: Optical communication, photon-starved channels, on-oﬀ keying, pulse position modulation 1. INTRODUCTION Using optical- rather than radio-frequency range for satellite communication prospectively oﬀers higher band- widths for data transfer, reduces diﬀractive losses in signal transmission, and additionally alleviates certain reg- ulatory issues. However, in the deep-space regime the diminishing power of the received signal requires diﬀerent signal modulation methods compared e.g. to conventional ﬁber-optic communication. A widely used format in deep-space communication is pulse position modulation (PPM), which encodes information in the temporal position of a light pulse within a frame of otherwise empty time bins.1,2 The standard technique to detect the PPM signal relies on time-resolved photon counting. Registering a photocount identiﬁes the position of a pulse, while no photocount event within the frame implies erasure of the prepared PPM symbol. Such erasures can be dealt with eﬃciently using suitable forward error correcting codes. The purpose of this paper is to identify the attainable eﬃciency of the PPM format under the constraint of a speciﬁed average signal power. The analysis includes counts generated by background noise assuming an arbitrary relation between the signal and the noise power, which goes beyond previously obtained results for noiseless or moderate-noise scenarios.3–6 Optimization is carried out with respect to the length of the PPM frame. Two statistical models for background counts motivated by diﬀerent physical photodetection setups are considered. The Poissonian model is applicable to standard photodetection, while the Gaussian model would be relevant to more advanced mode-selective detection. In the latter scheme the temporal mode carrying the modulated signal is separated prior to direct detection using e.g. nonlinear optical frequency conversion.7–11 The eﬃciency limits are quantiﬁed with the help of Shannon mutual information which sets the theoretical upper bound on the attainable transmission rate for a given modulation/detection technique.12 Shannon mutual information can be in principle attained with the help of forward error correction, although the eﬃciency of practical codes usually remains below that value. In the presence of background noise the timing of a photocount does not necessarily match the position of a pulse. Furthermore, photocounts may occur in multiple time bins within a single PPM frame. The simple decoding strategy extracts information only from events when one photocount has been registered over the PPM frame, treating multiple photocounts as erasures. In this scenario mutual information exhibits unfavorable, Further author information: (Send correspondence to K.B.) M.J.: E-mail: m.jarzyna@cent.uw.edu.pl K.B.: E-mail: k.banaszek@cent.uw.edu.pl ##LTLine##

text: Optimizing

label: title

text: deep-space

text: optical

text: communication

text: under

text: power

text: constraints

text: Marcin

label: author

text: Jarzynaa,

text: Wojciech

text: Zwolin´skib,

text: Jachurab,

text: and

text: Konrad

text: Banaszeka,b

text: aCentre

text: of

text: New

text: Technologies,

text: University

text: of

text: Warsaw,

text: Banacha

text: Warszawa,

text: Poland

text: bFaculty

text: of

text: Physics,

text: University

text: of

text: Warsaw,

text: Pasteura

text: Warszawa,

text: Poland

text: ABSTRACT

text: We

label: abstract

text: investigate

text: theoretically

text: the

text: eﬃciency

text: of

text: deep-space

text: optical

text: communication

text: in

text: the

text: presence

text: of

text: background

text: noise.

text: With

text: decreasing

text: average

text: signal

text: power

text: spectral

text: density,

text: a

text: scaling

text: gap

text: opens

text: up

text: between

text: optimized

text: simple-

text: decoded

text: pulse

text: position

text: modulation

text: and

text: generalized

text: on-oﬀ

text: keying

text: with

text: direct

text: detection.

text: The

text: scaling

text: of

text: the

text: latter

text: follows

text: the

text: quantum

text: mechanical

text: capacity

text: of

text: an

text: optical

text: channel

text: with

text: additive

text: Gaussian

text: noise.

text: Eﬃcient

text: communication

text: is

text: found

text: to

text: require

text: a

text: highly

text: imbalanced

text: distribution

text: of

text: instantaneous

text: signal

text: power.

text: This

text: condition

text: can

text: be

text: alleviated

text: through

text: the

text: use

text: of

text: structured

text: receivers

text: which

text: exploit

text: optical

text: interference

text: over

text: multiple

text: time

text: bins

text: to

text: concentrate

text: the

text: signal

text: power

text: before

text: the

text: detection

text: stage.

text: Keywords:

text: Optical

text: communication,

text: photon-starved

text: channels,

text: on-oﬀ

text: keying,

text: pulse

text: position

text: modulation

text: INTRODUCTION

text: Using

text: optical-

text: rather

text: than

text: radio-frequency

text: range

text: for

text: satellite

text: communication

text: prospectively

text: oﬀers

text: higher

text: band-

text: widths

text: for

text: data

text: transfer,

text: reduces

text: diﬀractive

text: losses

text: in

text: signal

text: transmission,

text: and

text: additionally

text: alleviates

text: certain

text: reg-

text: ulatory

text: issues.

text: However,

text: in

text: the

text: deep-space

text: regime

text: the

text: diminishing

text: power

text: of

text: the

text: received

text: signal

text: requires

text: diﬀerent

text: signal

text: modulation

text: methods

text: compared

text: e.g.

text: to

text: conventional

text: ﬁber-optic

text: communication.

text: A

text: widely

text: used

text: format

text: in

text: deep-space

text: communication

text: is

text: pulse

text: position

text: modulation

text: (PPM),

text: which

text: encodes

text: information

text: in

text: the

text: temporal

text: position

text: of

text: a

text: light

text: pulse

text: within

text: a

text: frame

text: of

text: otherwise

text: empty

text: time

text: The

text: standard

text: technique

text: to

text: detect

text: the

text: PPM

text: signal

text: relies

text: on

text: time-resolved

text: photon

text: counting.

text: Registering

text: a

text: photocount

text: identiﬁes

text: the

text: position

text: of

text: a

text: pulse,

text: while

text: no

text: photocount

text: event

text: within

text: the

text: frame

text: implies

text: erasure

text: of

text: the

text: prepared

text: PPM

text: symbol.

text: Such

text: erasures

text: can

text: be

text: dealt

text: with

text: eﬃciently

text: using

text: suitable

text: forward

text: error

text: correcting

text: codes.

text: The

text: purpose

text: of

text: this

text: paper

text: is

text: to

text: identify

text: the

text: attainable

text: eﬃciency

text: of

text: the

text: PPM

text: format

text: under

text: the

text: constraint

text: of

text: a

text: speciﬁed

text: average

text: signal

text: power.

text: The

text: analysis

text: includes

text: counts

text: generated

text: by

text: background

text: noise

text: assuming

text: an

text: arbitrary

text: relation

text: between

text: the

text: signal

text: and

text: the

text: noise

text: power,

text: which

text: goes

text: beyond

text: previously

text: obtained

text: results

text: for

text: noiseless

text: or

text: moderate-noise

text: Optimization

text: is

text: carried

text: out

text: with

text: respect

text: to

text: the

text: length

text: of

text: the

text: PPM

text: frame.

text: Two

text: statistical

text: models

text: for

text: background

text: counts

text: motivated

text: by

text: diﬀerent

text: physical

text: photodetection

text: setups

text: are

text: considered.

text: The

text: Poissonian

text: model

text: is

text: applicable

text: to

text: standard

text: photodetection,

text: while

text: the

text: Gaussian

text: model

text: would

text: be

text: relevant

text: to

text: more

text: advanced

text: mode-selective

text: detection.

text: In

text: the

text: latter

text: scheme

text: the

text: temporal

text: mode

text: carrying

text: the

text: modulated

text: signal

text: is

text: separated

text: prior

text: to

text: direct

text: detection

text: using

text: e.g.

text: nonlinear

text: optical

text: frequency

text: The

text: eﬃciency

text: limits

text: are

text: quantiﬁed

text: with

text: the

text: help

text: of

text: Shannon

text: mutual

text: information

text: which

text: sets

text: the

text: theoretical

text: upper

text: bound

text: on

text: the

text: attainable

text: transmission

text: rate

text: for

text: a

text: given

text: modulation/detection

text: Shannon

text: mutual

text: information

text: can

text: be

text: in

text: principle

text: attained

text: with

text: the

text: help

text: of

text: forward

text: error

text: correction,

text: although

text: the

text: eﬃciency

text: of

text: practical

text: codes

text: usually

text: remains

text: below

text: that

text: value.

text: In

text: the

text: presence

text: of

text: background

text: noise

text: the

text: timing

text: of

text: a

text: photocount

text: does

text: not

text: necessarily

text: match

text: the

text: position

text: of

text: a

text: pulse.

text: Furthermore,

text: photocounts

text: may

text: occur

text: in

text: multiple

text: time

text: bins

text: within

text: a

text: single

text: PPM

text: frame.

text: The

text: simple

text: decoding

text: strategy

text: extracts

text: information

text: only

text: from

text: events

text: when

text: one

text: photocount

text: has

text: been

text: registered

text: over

text: the

text: PPM

text: frame,

text: treating

text: multiple

text: photocounts

text: as

text: erasures.

text: In

text: this

text: scenario

text: mutual

text: information

text: exhibits

text: unfavorable,

text: Further

text: author

text: information:

text: (Send

text: correspondence

text: to

text: K.B.)

text: M.J.:

text: E-mail:

text: m.jarzyna@cent.uw.edu.pl

text: K.B.:

text: E-mail:

text: k.banaszek@cent.uw.edu.pl

text: ##LTLine##


text: M31 escape velocity and dynamical mass modelling 7 Figure 4. Joint posterior distributions of the model parameters obtained from the escape velocity and mass modellings, showing the − − − − default case, where υmin = 300 kms 1, rs = 15 kpc, dM31 = 780kpc, vM31,h = 301kms 1, σg = 1000 kms 1 and νg = 1 are assumed. Panel in the inset shows the joint distributions of concentration and virial mass obtained from the mass modelling for the corresponding case, where blue and red distributions show results for two diﬀerent deﬁnitions of the halo masses as labelled. the heliocentric distance and true peculiar velocity, υ(cid:107)t, of the ﬁnal inference, and also where needed we propagate them to tracers. We must marginalize the distribution function over derive physical quantities of the interest. these latent variables before we can infer anything about the Similarly, in the separate case of ﬁtting the mass model model parameters. Unfortunately, the expression for the like- via equation 21, we again make use of the bmcmc. In this lihood function given in Equation 9 is non-analytical mean- case there is no latent variable and hence, bmcmc only uses ing marginalization must be be done numerically. There- the general Metropolis Hastings algorithm to sample the dis- fore, we set-up a multi-level (hierarchical) inference where tribution for the model parameters {Mvir, c}. For all the un- we could directly consider the latent variables as hyper- dertaken experiments, we run MWG for suﬃcient autocor- parameters and sample them for each proposal of the model relation time to ensure that the distributions of parameters parameters. For this we employ the bmcmc that uses Gibbs are stabilized around certain values. We report the median of sampling method for the marginal inference. In short, given the probability distribution of a model parameter as its best a set of observables, (α, δ, R, υ(cid:107), συ), for each tracer along estimate and 16th and 84th percentiles of the distribution as with the proposed values for (υ(cid:107)t, s) and assuming a galacto- measure of its uncertainty. centric number density distribution (equation 10), bmcmc 4 RESULTS yields the posterior distributions in the form of walks/chains of the model parameters Θ and ancillary data such as the In this work, we present new measurements of the es- log-likelihood value at each walk. We use these chains for the cape velocity curve υesc(r) and the virial properties of the MNRAS 000, 1–13 (2018) ##LTFigure## ##LTFigure## ##LTFigure##

text: escape

text: velocity

text: and

text: dynamical

text: mass

text: modelling

text: Figure

text: Joint

text: posterior

text: distributions

text: of

text: the

text: model

text: parameters

text: obtained

text: from

text: the

text: escape

text: velocity

text: and

text: mass

text: modellings,

text: showing

text: the

text: default

text: case,

text: where

text: υmin

text: kms

text: rs

text: kpc,

text: σg

text: kms

text: and

text: νg

text: are

text: assumed.

text: Panel

text: in

text: the

text: inset

text: shows

text: the

text: joint

text: distributions

text: of

text: concentration

text: and

text: virial

text: mass

text: obtained

text: from

text: the

text: mass

text: modelling

text: for

text: the

text: corresponding

text: case,

text: where

text: blue

text: and

text: red

text: distributions

text: show

text: results

text: for

text: two

text: diﬀerent

text: deﬁnitions

text: of

text: the

text: halo

text: masses

text: as

text: labelled.

text: the

text: heliocentric

text: distance

text: and

text: true

text: peculiar

text: velocity,

text: of

text: the

text: ﬁnal

text: inference,

text: and

text: also

text: where

text: needed

text: we

text: propagate

text: them

text: to

text: tracers.

text: We

text: must

text: marginalize

text: the

text: distribution

text: function

text: over

text: derive

text: physical

text: quantities

text: of

text: the

text: interest.

text: these

text: latent

text: variables

text: before

text: we

text: can

text: infer

text: anything

text: about

text: the

text: Similarly,

text: in

text: the

text: separate

text: case

text: of

text: ﬁtting

text: the

text: mass

text: model

text: model

text: parameters.

text: Unfortunately,

text: the

text: expression

text: for

text: the

text: like-

text: via

text: equation

text: we

text: again

text: make

text: use

text: of

text: the

text: bmcmc.

text: In

text: this

text: lihood

text: function

text: given

text: in

text: Equation

text: is

text: non-analytical

text: mean-

text: case

text: there

text: is

text: no

text: latent

text: variable

text: and

text: hence,

text: bmcmc

text: only

text: uses

text: ing

text: marginalization

text: must

text: be

text: be

text: done

text: numerically.

text: There-

text: the

text: general

text: Metropolis

text: Hastings

text: algorithm

text: to

text: sample

text: the

text: dis-

text: fore,

text: we

text: set-up

text: a

text: multi-level

text: (hierarchical)

text: inference

text: where

text: tribution

text: for

text: the

text: model

text: parameters

text: {Mvir,

text: c}.

text: For

text: all

text: the

text: un-

text: we

text: could

text: directly

text: consider

text: the

text: latent

text: variables

text: as

text: hyper-

text: dertaken

text: experiments,

text: we

text: run

text: MWG

text: for

text: suﬃcient

text: autocor-

text: parameters

text: and

text: sample

text: them

text: for

text: each

text: proposal

text: of

text: the

text: model

text: relation

text: time

text: to

text: ensure

text: that

text: the

text: distributions

text: of

text: parameters

text: parameters.

text: For

text: this

text: we

text: employ

text: the

text: bmcmc

text: that

text: uses

text: Gibbs

text: are

text: stabilized

text: around

text: certain

text: values.

text: We

text: report

text: the

text: median

text: of

text: sampling

text: method

text: for

text: the

text: marginal

text: inference.

text: In

text: short,

text: given

text: the

text: probability

text: distribution

text: of

text: a

text: model

text: parameter

text: as

text: its

text: best

text: a

text: set

text: of

text: observables,

text: (α,

text: δ,

text: R,

text: συ),

text: for

text: each

text: tracer

text: along

text: estimate

text: and

text: and

text: percentiles

text: of

text: the

text: distribution

text: as

text: with

text: the

text: proposed

text: values

text: for

text: s)

text: and

text: assuming

text: a

text: galacto-

text: measure

text: of

text: its

text: uncertainty.

text: centric

text: number

text: density

text: distribution

text: (equation

text: bmcmc

text: RESULTS

text: yields

text: the

text: posterior

text: distributions

text: in

text: the

text: form

text: of

text: walks/chains

text: of

text: the

text: model

text: parameters

text: Θ

text: and

text: ancillary

text: data

text: such

text: as

text: the

text: In

text: this

text: work,

text: we

text: present

text: new

text: measurements

text: of

text: the

text: es-

text: log-likelihood

text: value

text: at

text: each

text: walk.

text: We

text: use

text: these

text: chains

text: for

text: the

text: cape

text: velocity

text: curve

text: υesc(r)

text: and

text: the

text: virial

text: properties

text: of

text: the

text: MNRAS

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##