text: M31 escape velocity and dynamical mass modelling 9 R > 15 kpc, opposite is true. The observed H i rotation the main model parameters for these cases. Importantly, we curve also shows a similar behaviour with respect to our de- see that while the introduced systematic shifts in the dM31 rived parametric rotation curve (Fig. 6). This suggests that have as such negligible eﬀect on our ﬁnal results, the biases our parametric model is not ﬂexible enough to capture the introduced in the υM31 alters the |Mvir| estimates by ∼ 8%. full information available in the data. Since the systematic Furthermore, one of our assumption is that the Sun’s total deviations are small, we only consider the parametric form tangential velocity r−elative to the Galactic centre, V(cid:12)+VLSR so as to take advantage of the reduced variance and mod- sums to 251.5 km s 1 noting that it is approximately equal − elling complexity. to the literature averaged value of 248 km s 1 (for R(cid:12) = Unfortunately, we are unable to constrain some model 8.2 kpc) from Bland-Hawthorn & Gerhard (2016). Since it parameters that were kept free. Importantly, we ﬁnd that is apparent from equations 1 and 2 that υ(cid:107) ∝ (V(cid:12) + VLSR), although a large value is preferred for k, it remains uncon- we test our choice. For this we adopt two diﬀerent values − − strained. Unconstrained k parameters due to scant number 240 km s 1 and 260 km s 1 for the Sun’s total tangential of high velocity stars have remained the main limitation of velocity and re-run the analysis. The measurements are re- the undertaken approach, the fate that has been also shared ported in the bottom two rows of the Table 1, and it is evi- − by the MW high velocity works (Leonard & Tremaine 1990; dent that the systematic of ±10 km s 1 in V(cid:12)+VLSR does not Smith et al. 2007, refer to their ﬁgures 1 and 7 respec- alter our ﬁnal estimates, and likewise when we adopt slightly − − tively). Also, we ﬁnd that the logarithmic number density diﬀerent values of U(cid:12) = 10 km s 1 and W(cid:12) = 7 km s 1 slope favour shallow (α = 2) over steep model (α = 3) and from Bland-Hawthorn & Gerhard (2016). The null eﬀect is the outlier fraction η → 1 suggesting that eﬀectively only expected because the T term deﬁned in equation 2 that con- a tiny fraction of our tracers are actually drawn from the tains these constants, for a small angle Ω, largely cancels out background model. in equation 1. 4.2 Virial properties: inference Eﬀects of change in the threshold velocity (υmin) Here, we ﬁrst substitute the measured υesc(r) run into Equation 14 to derive the total potential of the galaxy. As already described, υmin is the lower bound on the Then we ﬁt the three component galaxy model to the de- magnitude of the υ(cid:107) that is used to discriminate the high ve- − rived potential and ultimately estimate the virial proper- locity stars. In our ﬁnal results, we adopt υmin = 300 km s 1, ties (Mvir and c) of the dark matter halo, keeping the which is consistent with the value adopted in the literature disk and bulge models ﬁxed with Equation 20. This exer- Leonard & Tremaine (1990); Smith et al. (2007); Piﬄ et al. cise is repeated separately for both the deﬁnitions of halo (2014). However, here we desire to investigate the eﬀect of virial over-density. In Fig. 4 inset we show the joint dis- change in υmin.−For this ﬁrst we lower the limit and set tributions of the Mvir(M200) and c resulted from this ex- υmin = 290 km s 1, this increases sample size to 111 as more ercise in blue(red) contours for a case with ‘default’ values stars are now classiﬁed as high velocity. With the new set of the model parameters. The marginalized distributions of of data we ﬁnd that the Mvir/1012 M(cid:12) reduces marginally the halo parameters are shown in the corresponding colours from 0.8 to 0.76. Second, when we increase the limit and set − at the top and the right sides of the inset-ﬁgure. Unfor- υmin = 310 km s 1 we ﬁnd that the sample size reduces to tunately, the marginalized distributions for the c hint that 65 and the Mvir/1012 M(cid:12) slightly increases from 0.8 to 0.9. within a large range of parameter space that we explore i.e. Moreover−, we ﬁnd that relaxing the assumption on υmin by [1,60], we are unable to constrain the c parameter, but we ±10 km s 1 leads to insigniﬁcant changes in the estimates are able to measure the virial masses regardless; our ﬁnal of other parameters, e.g., υesc,0, γ as well. Rows 6-7 in Ta- estimates for the Mvir (M200) (also shown in the ﬁrst row ble 1 provide the estimates of the model parameters for these in Table 1) are 0.8 ± 0.1 (0.7 ± 0.1) × 1012 M(cid:12). Substitut- cases. ing these values for the masses in equation 18, we determine rvir (r200) = 240 ± 10 (188+−711) kpc. As discussed earlier in Indiscernible eﬀects due to change in some parameters Section , we note that the Mvir or rvir values are always greater than for M200 or r200. There are also some model dependencies that we tested 4.3 Relaxing the assumptions but which led to negligible changes in our ﬁnal results, we brieﬂy discuss them here. Throughout the paper, to Finally, it is instructive to investigate repercussions of relaxing some of the key assumptions on our ﬁnal results, model the−outliers (Equation 7), we assume that the σg−= which we discuss below. 1000 km s 1 −and νg = 1. We test that adopting 800 km s 1 Eﬀects of change in M31 central properties or 1200 km s 1 for σg and/or varying νg ∈ [0.5, 2] eﬀec- tively do not result any changes in the ﬁnal Mvir measure- Arguably, the heliocentric distance and the line-of-sight ments. Similarly, given the large distance between M31-MW velocity of the M31 galaxy are known fairly accurately, with (dM31 (cid:29) R0) the assumed position of the Sun on the MW the scatter of roughly 1 − 2% amongst the literature (e.g., disk (R(cid:12) = 8.2 kpc versus 8.5 kpc) has negligible eﬀect in de Vaucouleurs et al. 1991; Stanek & Garnavich 1998; Mc- our analysis. Moreover, we check that our ﬁnal results are Connachie et al. 2005; van der Marel & Guhathakurta 2008; robust to the change of at-least 5% in the upper limits of Conn et al. 2012). Nevertheless, we systematically shift t−he the prior ranges assumed for υesc, γ, υ(cid:107)t and α. Similarly, re- adopted value of dM31 = 780 kpc an−d υM31 = −301 km s 1 ducing the lower limit of prior range for α parameter from (Section ) by ±10 kpc and ±5 km s 1 respectively and re- 2.0 to 1.0 do not modify our ﬁnal Mvir measurement. A run our analysis to understand their implications on our slight change of ±1 kpc in the adopted value of escape ve- ﬁnal results. Rows 2-5 in Table 1 provide the estimates of locity scale-length rs = 15 kpc has no discernible eﬀect in MNRAS 000, 1–13 (2018)

text: escape

text: velocity

text: and

text: dynamical

text: mass

text: modelling

text: R

text: kpc,

text: opposite

text: is

text: true.

text: The

text: observed

text: H

text: i

text: rotation

text: the

text: main

text: model

text: parameters

text: for

text: these

text: cases.

text: Importantly,

text: we

text: curve

text: also

text: shows

text: a

text: similar

text: behaviour

text: with

text: respect

text: to

text: our

text: de-

text: see

text: that

text: while

text: the

text: introduced

text: systematic

text: shifts

text: in

text: the

text: rived

text: parametric

text: rotation

text: curve

text: (Fig.

text: This

text: suggests

text: that

text: have

text: as

text: such

text: negligible

text: eﬀect

text: on

text: our

text: ﬁnal

text: results,

text: the

text: biases

text: our

text: parametric

text: model

text: is

text: not

text: ﬂexible

text: enough

text: to

text: capture

text: the

text: introduced

text: in

text: the

text: alters

text: the

text: |Mvir|

text: estimates

text: by

text: full

text: information

text: available

text: in

text: the

text: data.

text: Since

text: the

text: systematic

text: Furthermore,

text: one

text: of

text: our

text: assumption

text: is

text: that

text: the

text: Sun’s

text: total

text: deviations

text: are

text: small,

text: we

text: only

text: consider

text: the

text: parametric

text: form

text: tangential

text: velocity

text: r−elative

text: to

text: the

text: Galactic

text: centre,

text: so

text: as

text: to

text: take

text: advantage

text: of

text: the

text: reduced

text: variance

text: and

text: mod-

text: sums

text: to

text: km

text: s

text: noting

text: that

text: it

text: is

text: approximately

text: equal

text: elling

text: complexity.

text: to

text: the

text: literature

text: averaged

text: value

text: of

text: km

text: s

text: (for

text: Unfortunately,

text: we

text: are

text: unable

text: to

text: constrain

text: some

text: model

text: kpc)

text: from

text: Bland-Hawthorn

text: Gerhard

text: Since

text: it

text: parameters

text: that

text: were

text: kept

text: free.

text: Importantly,

text: we

text: ﬁnd

text: that

text: is

text: apparent

text: from

text: equations

text: and

text: that

text: VLSR),

text: although

text: a

text: large

text: value

text: is

text: preferred

text: for

text: k,

text: it

text: remains

text: uncon-

text: we

text: test

text: our

text: choice.

text: For

text: this

text: we

text: adopt

text: two

text: diﬀerent

text: values

text: strained.

text: Unconstrained

text: k

text: parameters

text: due

text: to

text: scant

text: number

text: km

text: s

text: and

text: km

text: s

text: for

text: the

text: Sun’s

text: total

text: tangential

text: of

text: high

text: velocity

text: stars

text: have

text: remained

text: the

text: main

text: limitation

text: of

text: velocity

text: and

text: re-run

text: the

text: analysis.

text: The

text: measurements

text: are

text: re-

text: the

text: undertaken

text: approach,

text: the

text: fate

text: that

text: has

text: been

text: also

text: shared

text: ported

text: in

text: the

text: bottom

text: two

text: rows

text: of

text: the

text: Table

text: and

text: it

text: is

text: evi-

text: by

text: the

text: MW

text: high

text: velocity

text: works

text: (Leonard

text: Tremaine

text: dent

text: that

text: the

text: systematic

text: of

text: km

text: s

text: in

text: does

text: not

text: Smith

text: et

text: al.

text: refer

text: to

text: their

text: ﬁgures

text: and

text: respec-

text: alter

text: our

text: ﬁnal

text: estimates,

text: and

text: likewise

text: when

text: we

text: adopt

text: slightly

text: tively).

text: Also,

text: we

text: ﬁnd

text: that

text: the

text: logarithmic

text: number

text: density

text: diﬀerent

text: values

text: of

text: km

text: s

text: and

text: km

text: s

text: slope

text: favour

text: shallow

text: (α

text: over

text: steep

text: model

text: (α

text: and

text: from

text: Bland-Hawthorn

text: Gerhard

text: The

text: eﬀect

text: is

text: the

text: outlier

text: fraction

text: η

text: suggesting

text: that

text: eﬀectively

text: only

text: expected

text: because

text: the

text: T

text: term

text: deﬁned

text: in

text: equation

text: that

text: con-

text: a

text: tiny

text: fraction

text: of

text: our

text: tracers

text: are

text: actually

text: drawn

text: from

text: the

text: tains

text: these

text: constants,

text: for

text: a

text: small

text: angle

text: Ω,

text: largely

text: cancels

text: out

text: background

text: model.

text: in

text: equation

text: Virial

text: properties:

text: inference

text: Eﬀects

text: of

text: change

text: in

text: the

text: threshold

text: velocity

text: (υmin)

text: Here,

text: we

text: ﬁrst

text: substitute

text: the

text: measured

text: υesc(r)

text: run

text: into

text: Equation

text: to

text: derive

text: the

text: total

text: potential

text: of

text: the

text: galaxy.

text: As

text: already

text: described,

text: υmin

text: is

text: the

text: lower

text: bound

text: on

text: the

text: Then

text: we

text: ﬁt

text: the

text: three

text: component

text: galaxy

text: model

text: to

text: the

text: de-

text: magnitude

text: of

text: the

text: that

text: is

text: used

text: to

text: discriminate

text: the

text: high

text: ve-

text: rived

text: potential

text: and

text: ultimately

text: estimate

text: the

text: virial

text: proper-

text: locity

text: stars.

text: In

text: our

text: ﬁnal

text: results,

text: we

text: adopt

text: υmin

text: km

text: s

text: ties

text: (Mvir

text: and

text: c)

text: of

text: the

text: dark

text: matter

text: halo,

text: keeping

text: the

text: which

text: is

text: consistent

text: with

text: the

text: value

text: adopted

text: in

text: the

text: literature

text: disk

text: and

text: bulge

text: models

text: ﬁxed

text: with

text: Equation

text: This

text: exer-

text: Leonard

text: Tremaine

text: Smith

text: et

text: al.

text: Piﬄ

text: et

text: al.

text: cise

text: is

text: repeated

text: separately

text: for

text: both

text: the

text: deﬁnitions

text: of

text: halo

text: However,

text: here

text: we

text: desire

text: to

text: investigate

text: the

text: eﬀect

text: of

text: virial

text: over-density.

text: In

text: Fig.

text: inset

text: we

text: show

text: the

text: joint

text: dis-

text: change

text: in

text: υmin.−For

text: this

text: ﬁrst

text: we

text: lower

text: the

text: limit

text: and

text: set

text: tributions

text: of

text: the

text: and

text: c

text: resulted

text: from

text: this

text: ex-

text: υmin

text: km

text: s

text: this

text: increases

text: sample

text: size

text: to

text: as

text: more

text: ercise

text: in

text: blue(red)

text: contours

text: for

text: a

text: case

text: with

text: ‘default’

text: values

text: stars

text: are

text: now

text: classiﬁed

text: as

text: high

text: velocity.

text: With

text: the

text: new

text: set

text: of

text: the

text: model

text: parameters.

text: The

text: marginalized

text: distributions

text: of

text: of

text: data

text: we

text: ﬁnd

text: that

text: the

text: reduces

text: marginally

text: the

text: halo

text: parameters

text: are

text: shown

text: in

text: the

text: corresponding

text: colours

text: from

text: to

text: Second,

text: when

text: we

text: increase

text: the

text: limit

text: and

text: set

text: at

text: the

text: top

text: and

text: the

text: right

text: sides

text: of

text: the

text: inset-ﬁgure.

text: Unfor-

text: υmin

text: km

text: s

text: we

text: ﬁnd

text: that

text: the

text: sample

text: size

text: reduces

text: to

text: tunately,

text: the

text: marginalized

text: distributions

text: for

text: the

text: c

text: hint

text: that

text: and

text: the

text: slightly

text: increases

text: from

text: to

text: within

text: a

text: large

text: range

text: of

text: parameter

text: space

text: that

text: we

text: explore

text: i.e.

text: Moreover−,

text: we

text: ﬁnd

text: that

text: relaxing

text: the

text: assumption

text: on

text: υmin

text: by

text: we

text: are

text: unable

text: to

text: constrain

text: the

text: c

text: parameter,

text: but

text: we

text: km

text: s

text: leads

text: to

text: insigniﬁcant

text: changes

text: in

text: the

text: estimates

text: are

text: able

text: to

text: measure

text: the

text: virial

text: masses

text: regardless;

text: our

text: ﬁnal

text: of

text: other

text: parameters,

text: e.g.,

text: γ

text: as

text: well.

text: Rows

text: in

text: Ta-

text: estimates

text: for

text: the

text: Mvir

text: (also

text: shown

text: in

text: the

text: ﬁrst

text: row

text: ble

text: provide

text: the

text: estimates

text: of

text: the

text: model

text: parameters

text: for

text: these

text: in

text: Table

text: are

text: Substitut-

text: cases.

text: ing

text: these

text: values

text: for

text: the

text: masses

text: in

text: equation

text: we

text: determine

text: rvir

text: kpc.

text: As

text: discussed

text: earlier

text: in

text: Indiscernible

text: eﬀects

text: due

text: to

text: change

text: in

text: some

text: parameters

text: Section

text: we

text: note

text: that

text: the

text: Mvir

text: or

text: rvir

text: values

text: are

text: always

text: greater

text: than

text: for

text: or

text: There

text: are

text: also

text: some

text: model

text: dependencies

text: that

text: we

text: tested

text: Relaxing

text: the

text: assumptions

text: but

text: which

text: led

text: to

text: negligible

text: changes

text: in

text: our

text: ﬁnal

text: results,

text: we

text: brieﬂy

text: discuss

text: them

text: here.

text: Throughout

text: the

text: paper,

text: to

text: Finally,

text: it

text: is

text: instructive

text: to

text: investigate

text: repercussions

text: of

text: relaxing

text: some

text: of

text: the

text: key

text: assumptions

text: on

text: our

text: ﬁnal

text: results,

text: model

text: the−outliers

text: (Equation

text: we

text: assume

text: that

text: the

text: σg−=

text: which

text: we

text: discuss

text: below.

text: km

text: s

text: −and

text: νg

text: We

text: test

text: that

text: adopting

text: km

text: s

text: Eﬀects

text: of

text: change

text: in

text: central

text: properties

text: or

text: km

text: s

text: for

text: σg

text: and/or

text: varying

text: νg

text: eﬀec-

text: tively

text: do

text: not

text: result

text: any

text: changes

text: in

text: the

text: ﬁnal

text: Mvir

text: measure-

text: Arguably,

text: the

text: heliocentric

text: distance

text: and

text: the

text: line-of-sight

text: ments.

text: Similarly,

text: given

text: the

text: large

text: distance

text: between

text: velocity

text: of

text: the

text: galaxy

text: are

text: known

text: fairly

text: accurately,

text: with

text: the

text: assumed

text: position

text: of

text: the

text: Sun

text: on

text: the

text: MW

text: the

text: scatter

text: of

text: roughly

text: amongst

text: the

text: literature

text: (e.g.,

text: disk

text: kpc

text: versus

text: kpc)

text: has

text: negligible

text: eﬀect

text: in

text: de

text: Vaucouleurs

text: et

text: al.

text: Stanek

text: Garnavich

text: Mc-

text: our

text: analysis.

text: Moreover,

text: we

text: check

text: that

text: our

text: ﬁnal

text: results

text: are

text: Connachie

text: et

text: al.

text: van

text: der

text: Marel

text: Guhathakurta

text: robust

text: to

text: the

text: change

text: of

text: at-least

text: in

text: the

text: upper

text: limits

text: of

text: Conn

text: et

text: al.

text: Nevertheless,

text: we

text: systematically

text: shift

text: t−he

text: the

text: prior

text: ranges

text: assumed

text: for

text: υesc,

text: γ,

text: and

text: α.

text: Similarly,

text: re-

text: adopted

text: value

text: of

text: kpc

text: an−d

text: km

text: s

text: ducing

text: the

text: lower

text: limit

text: of

text: prior

text: range

text: for

text: α

text: parameter

text: from

text: (Section

text: by

text: kpc

text: and

text: ±5

text: km

text: s

text: respectively

text: and

text: re-

text: to

text: do

text: not

text: modify

text: our

text: ﬁnal

text: Mvir

text: measurement.

text: A

text: run

text: our

text: analysis

text: to

text: understand

text: their

text: implications

text: on

text: our

text: slight

text: change

text: of

text: ±1

text: kpc

text: in

text: the

text: adopted

text: value

text: of

text: escape

text: ve-

text: ﬁnal

text: results.

text: Rows

text: in

text: Table

text: provide

text: the

text: estimates

text: of

text: locity

text: scale-length

text: rs

text: kpc

text: has

text: no

text: discernible

text: eﬀect

text: in

text: MNRAS