text: 26 MARIANNE AKIAN, RAVINDRA BAPAT, AND STE´PHANE GAUBERT Let (i, j) be an arc of Opt(MBN). Then, by deﬁnition, there exists an optimal (cid:48) − ◦ ◦ − permutation ν for MBN such that j = ν(i), hence ν = τ 1 ν σ 1 is an optimal − (cid:48) − permutation for B such that τ 1(j) = ν (σ(i)), which implies that (σ(i), τ 1(j)) is an arc of Opt(B), so (i, j) is an arc of Opt(B) . Since the reverse implication is σ,τ also true (replace M and N by their inverse matrices), this shows the ﬁrst assertion of the lemma. (cid:62) Now let (U, V ) be a Hungarian pair with respect to B. We have B U V for ∈ (cid:62) ij i j all i, j [n]. Hence, (MBN) = W B − X − W U V − X − = ij i σ(i),τ 1(j) τ 1(j) i σ(i) τ 1(j) τ 1(j) (MU) (NT V ) , and i j · · · · · · per(MBN) = W W X X per B 1 n 1 n · · · · · · · · · · · · = W W U U X X V V 1 n 1 n 1 n 1 n · · · · · · = (MU) (MU) (NT V ) (NT V ) , 1 n 1 n so that (MU, NT V ) is a Hungarian pair with respect to MBN. Finally, (i, j) is an arc of Sat(MBN, MU, NT V ) if and only if (MBN) = (MU) (NT V ) , which is ij i j − equivalent to B − = U V − , then to the property that (σ(i), τ 1(j)) is σ(i),τ 1(j) σ(i) τ 1(j) (cid:3) an arc of Sat(B, U, V ), hence to the one that (i, j) is an arc of Sat(B, U, V ) . σ,τ R We denote by 1 the vector of n with all its entries equal to 1 = 0. ∈ R × (cid:54) Corollary 9.2. Let B n n be such that per B = 0, and let (U, V ) be a min Hungarian pair with respect to B. Then, (1, 1) is a Hungarian pair with respect − − ∈ R × to the matrix C := d(U) 1B d(V ) 1 n n, Sat(B, U, V ) = Sat(C, 1, 1) and min (cid:62) ∈ Opt(B) = Opt(C). In particular, per C = 1, C 1 = 0 for all i, j [n] and (i, j) ij is an arc of Sat(B, U, V ) if and only if C = 1. ij Proof. Let B, U, V, C be as in the corollary. Then, by Lemma 9.1 applied to M = − − d(U) 1 and N = d(V ) 1, we obtain the ﬁrst assertion of the corollary, together with (cid:62) ∈ per C = 1. Then, C 1 = 0 for all i, j [n] and (i, j) is an arc of Sat(B, U, V ) if ij (cid:3) and only if C = 1. ij ∈ R × (cid:54) Corollary 9.3. Let B n n be such that per B = 0, and let (U, V ) be a Hungar- min ⊂ ian pair with respect to B. Then, Opt(B) Sat(B, U, V ). Moreover, the following ∈ are equivalent for σ S : (i) σ is a permutation of Sat(B, U, V ); (ii) σ is a n permutation of Opt(B); (iii) σ is an optimal permutation of B. Proof. Let C be as in Corollary 9.2. Then, if (i, j) is an arc of Opt(C), there ∈ | | exists σ S such that j = σ(i) and σ = per C = 1. Since all entries of n C C are greater or equal to 1, this implies that C = 1, and so (i, j) is an arc of ij Sat(C, 1, 1). Since Opt(C) = Opt(B) and Sat(C, 1, 1) = Sat(B, U, V ), this shows ⊂ that Opt(B) Sat(B, U, V ). ∈ ⇒ ⇒ ⇒ Let us show that for σ S , (i) (iii) (ii) (i). Let σ be a permutation of n ∈ Sat(B, U, V ), this means that (i, σ(i)) is an arc of Sat(B, U, V ), for all i [n]. ∈ | | Then C = 1, for all i [n], which implies that σ = 1 = per C, and so σ is iσ(i) C an optimal permutation of C, hence an optimal permutation of B, by Lemma 9.1,

text: MARIANNE

text: AKIAN,

text: RAVINDRA

text: BAPAT,

text: AND

text: STE´PHANE

text: GAUBERT

text: Let

text: (i,

text: j)

text: be

text: an

text: arc

text: of

text: Opt(MBN).

text: Then,

text: by

text: deﬁnition,

text: there

text: exists

text: an

text: optimal

text: permutation

text: ν

text: for

text: MBN

text: such

text: that

text: j

text: ν(i),

text: hence

text: ν

text: τ

text: ν

text: σ

text: is

text: an

text: optimal

text: permutation

text: for

text: B

text: such

text: that

text: τ

text: ν

text: (σ(i)),

text: which

text: implies

text: that

text: (σ(i),

text: τ

text: is

text: an

text: arc

text: of

text: Opt(B),

text: so

text: (i,

text: j)

text: is

text: an

text: arc

text: of

text: Opt(B)

text: Since

text: the

text: reverse

text: implication

text: is

text: σ,τ

text: also

text: true

text: (replace

text: M

text: and

text: N

text: by

text: their

text: inverse

text: matrices),

text: this

text: shows

text: the

text: ﬁrst

text: assertion

text: of

text: the

text: lemma.

text: Now

text: let

text: (U,

text: V

text: be

text: a

text: Hungarian

text: pair

text: with

text: respect

text: to

text: B.

text: We

text: have

text: B

text: U

text: V

text: for

text: ij

text: i

text: j

text: all

text: i,

text: j

text: [n].

text: Hence,

text: (MBN)

text: W

text: B

text: X

text: W

text: U

text: V

text: X

text: ij

text: i

text: σ(i),τ

text: τ

text: i

text: σ(i)

text: τ

text: τ

text: (MU)

text: (NT

text: V

text: and

text: i

text: j

text: per(MBN)

text: W

text: W

text: X

text: X

text: per

text: B

text: n

text: n

text: W

text: W

text: U

text: U

text: X

text: X

text: V

text: V

text: n

text: n

text: n

text: n

text: (MU)

text: (MU)

text: (NT

text: V

text: (NT

text: V

text: n

text: n

text: so

text: that

text: (MU,

text: NT

text: V

text: is

text: a

text: Hungarian

text: pair

text: with

text: respect

text: to

text: MBN.

text: Finally,

text: (i,

text: j)

text: is

text: an

text: arc

text: of

text: Sat(MBN,

text: MU,

text: NT

text: V

text: if

text: and

text: only

text: if

text: (MBN)

text: (MU)

text: (NT

text: V

text: which

text: is

text: ij

text: i

text: j

text: equivalent

text: to

text: B

text: U

text: V

text: then

text: to

text: the

text: property

text: that

text: (σ(i),

text: τ

text: is

text: σ(i),τ

text: σ(i)

text: τ

text: an

text: arc

text: of

text: Sat(B,

text: U,

text: V

text: hence

text: to

text: the

text: one

text: that

text: (i,

text: j)

text: is

text: an

text: arc

text: of

text: Sat(B,

text: U,

text: V

text: σ,τ

text: R

text: We

text: denote

text: by

text: the

text: vector

text: of

text: n

text: with

text: all

text: its

text: entries

text: equal

text: to

text: R

text: Corollary

text: Let

text: B

text: n

text: n

text: be

text: such

text: that

text: per

text: B

text: and

text: let

text: (U,

text: V

text: be

text: a

text: min

text: Hungarian

text: pair

text: with

text: respect

text: to

text: B.

text: Then,

text: is

text: a

text: Hungarian

text: pair

text: with

text: respect

text: R

text: to

text: the

text: matrix

text: C

text: d(U)

text: d(V

text: n

text: n,

text: Sat(B,

text: U,

text: V

text: Sat(C,

text: and

text: min

text: Opt(B)

text: Opt(C).

text: In

text: particular,

text: per

text: C

text: C

text: for

text: all

text: i,

text: j

text: [n]

text: and

text: (i,

text: j)

text: ij

text: is

text: an

text: arc

text: of

text: Sat(B,

text: U,

text: V

text: if

text: and

text: only

text: if

text: C

text: ij

text: Proof.

text: Let

text: B,

text: U,

text: V,

text: C

text: be

text: as

text: in

text: the

text: corollary.

text: Then,

text: by

text: Lemma

text: applied

text: to

text: M

text: d(U)

text: and

text: N

text: d(V

text: we

text: obtain

text: the

text: ﬁrst

text: assertion

text: of

text: the

text: corollary,

text: together

text: with

text: per

text: C

text: Then,

text: C

text: for

text: all

text: i,

text: j

text: [n]

text: and

text: (i,

text: j)

text: is

text: an

text: arc

text: of

text: Sat(B,

text: U,

text: V

text: if

text: ij

text: and

text: only

text: if

text: C

text: ij

text: R

text: Corollary

text: Let

text: B

text: n

text: n

text: be

text: such

text: that

text: per

text: B

text: and

text: let

text: (U,

text: V

text: be

text: a

text: Hungar-

text: min

text: ian

text: pair

text: with

text: respect

text: to

text: B.

text: Then,

text: Opt(B)

text: Sat(B,

text: U,

text: V

text: Moreover,

text: the

text: following

text: are

text: equivalent

text: for

text: σ

text: S

text: (i)

text: σ

text: is

text: a

text: permutation

text: of

text: Sat(B,

text: U,

text: V

text: (ii)

text: σ

text: is

text: a

text: n

text: permutation

text: of

text: Opt(B);

text: (iii)

text: σ

text: is

text: an

text: optimal

text: permutation

text: of

text: B.

text: Proof.

text: Let

text: C

text: be

text: as

text: in

text: Corollary

text: Then,

text: if

text: (i,

text: j)

text: is

text: an

text: arc

text: of

text: Opt(C),

text: there

text: exists

text: σ

text: S

text: such

text: that

text: j

text: σ(i)

text: and

text: σ

text: per

text: C

text: Since

text: all

text: entries

text: of

text: n

text: C

text: C

text: are

text: greater

text: or

text: equal

text: to

text: this

text: implies

text: that

text: C

text: and

text: so

text: (i,

text: j)

text: is

text: an

text: arc

text: of

text: ij

text: Sat(C,

text: Since

text: Opt(C)

text: Opt(B)

text: and

text: Sat(C,

text: Sat(B,

text: U,

text: V

text: this

text: shows

text: that

text: Opt(B)

text: Sat(B,

text: U,

text: V

text: Let

text: us

text: show

text: that

text: for

text: σ

text: S

text: (i)

text: (iii)

text: (ii)

text: (i).

text: Let

text: σ

text: be

text: a

text: permutation

text: of

text: n

text: Sat(B,

text: U,

text: V

text: this

text: means

text: that

text: (i,

text: σ(i))

text: is

text: an

text: arc

text: of

text: Sat(B,

text: U,

text: V

text: for

text: all

text: i

text: [n].

text: Then

text: C

text: for

text: all

text: i

text: [n],

text: which

text: implies

text: that

text: σ

text: per

text: C,

text: and

text: so

text: σ

text: is

text: iσ(i)

text: C

text: an

text: optimal

text: permutation

text: of

text: C,

text: hence

text: an

text: optimal

text: permutation

text: of

text: B,

text: by

text: Lemma


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text: NON-ARCHIMEDEAN

label: paragraph

text: VALUATIONS

text: OF

text: EIGENVALUES

text: OF

text: MATRIX

text: POLYNOMIALS

label: reference

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text: Bini,

text: V.

text: Noferini,

text: and

text: M.

text: Sharify.

text: Locating

text: the

text: Eigenvalues

text: of

text: Matrix

text: Polyno-

text: mials.

text: SIAM

text: J.

text: Matrix

text: Anal.

text: Appl.,

text: N.

text: Bourbaki.

text: Commutative

text: Algebra.

text: Springer,

text: R.

text: B.

text: Bapat

text: and

text: T.

text: E.

text: S.

text: Raghavan.

text: Nonnegative

text: Matrices

text: and

text: Application.

text: Cam-

text: bridge

text: University

text: Press,

text: R.

text: B.

text: Bapat,

text: D.

text: Stanford,

text: and

text: P.

text: van

text: den

text: Driessche.

text: Pattern

text: properties

text: and

text: spec-

text: tral

text: inequalities

text: in

text: max

text: algebra.

text: SIAM

text: Journal

text: of

text: Matrix

text: Analysis

text: and

text: Applications,

text: P.

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text: Max-linear

text: systems:

text: theory

text: and

text: algorithms.

text: Springer

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text: in

text: Mathematics.

text: Springer-Verlag,

text: London,

text: G.

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text: D.

text: Dubois,

text: J.P.

text: Quadrat,

text: and

text: M.

text: Viot.

text: Analyse

text: du

text: comportement

text: p´eriodique

text: des

text: syst`emes

text: de

text: production

text: par

text: la

text: th´eorie

text: des

text: dio¨ıdes.

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text: Le

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text: France,

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text: Cuninghame-Green.

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text: Algebra.

text: Number

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text: Lecture

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text: Springer,

text: R.A.

text: Cuninghame-Green

text: and

text: P.F.J

text: Meijer.

text: An

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text: RI,

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text: Einsiedler,

text: M.

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text: and

text: D.

text: Lind.

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text: amoebas

text: and

text: tropical

text: varieties.

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text: Angew.

text: Math.,

text: S.

text: Friedland.

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text: Linear

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text: Gassner

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text: A

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text: scaling

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text: polynomial

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text: In

text: Rafael

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text: Romero-Viv´o,

text: editors,

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text: third

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text: for

text: tropical

text: standard

text: and

text: polynomial

text: eigenvalue

text: problems.

text: In

text: preparation.

text: I.

text: Itenberg,

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text: and

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text: Berlin,

text: Reprint

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text: edition.

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text: Kolokoltsov

text: and

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text: Maslov.

text: Idempotent

text: analysis

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text: applications.

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text: Comput.

text: Math.

text: and

text: Math.

text: Phys.,,

text: (Z˘.

text: Vyˇcisl.

text: Mat.

text: i

text: Mat.

text: Fiz.

text: no.

text: G.

text: L.

text: Litvinov,

text: V.

text: P.

text: Maslov,

text: and

text: G.

text: B.

text: Shpiz.

text: Idempotent

text: functional

text: analysis:

text: An

text: algebraical

text: approach.

text: Mat.

text: Notes,

text: V.P.

text: Maslov.

text: M´ethodes

text: Operatorielles.

text: Mir,

text: Moscou,

text: French

text: Transl.

text: Julio

text: Moro,

text: James

text: V.

text: Burke,

text: and

text: Michael

text: L.

text: Overton.

text: On

text: the

text: Lidskii-Vishik-

text: Lyusternik

text: perturbation

text: theory

text: for

text: eigenvalues

text: of

text: matrices

text: with

text: arbitrary

text: Jordan

text: struc-

text: ture.

text: SIAM

text: J.

text: Matrix

text: Anal.

text: Appl.,

text: Yanyuan

text: Ma

text: and

text: Alan

text: Edelman.

text: Nongeneric

text: eigenvalue

text: perturbations

text: of

text: Jordan

text: blocks.

text: Linear

text: Algebra

text: Appl.,

text: A.W.

text: Marshall

text: and

text: Olkin.

text: Inequalities:

text: theory

text: of

text: majorization

text: and

text: its

text: applications.

text: Number

text: in

text: Serie

text: Math.

text: in

text: Science

text: and

text: Engineering.

text: Academic

text: Press,


text: where treatment assignment does not change. Finally, we need to argue that Property 4.12 holds with these data. In essence, Property 4.12 implies that the outcome variable value is independent of the threshold variable conditional on the assignment variable, treatment, regime indicator and confounders. In this example, the treatment rule (prescription of statins if the 10-year CVD risk score is 20% or greater) is determined by the UK government and is not patient-speciﬁc. Furthermore, since Z is determined solely by the 10-year CVD risk score, conditioning on this variable implies directly that further knowledge of Z is unnecessary to make inference regarding the LDL cholesterol level. Hence, we apply the result of Theorem 4.14 to estimate the causal eﬀect of statin therapy on LDL cholesterol level, using a fuzzy RDD and the THIN subset of data. 5.1 Models and Estimation ∈ A B For a chosen bandwidth ζ (ζ (0, 1)), we deﬁne and to be the sets of patients whose 10-year ζ ζ CVD risk scores lie above and below the threshold, respectively. Setting the treatment threshold to be − x = 0.2 and deﬁning XC = X 0.2 to be the centred 10-year CVD risk score for the ith patient, we 0 i i consider the following linear models: ∈ A Y = β + β XC + ω (for i ); i 0a 1a i i ζ ∈ B Y = β + β XC + ω (for i ). i 0b 1b i i ζ ∼ N ∈ A ∪ B with ω (0, σ2) for i . The function that we wish to evaluate is i ζ ζ E | − E | − (Y C, X = x + (cid:15), Z = 1) (Y C, X = x (cid:15), Z = 0) h(C) = P f | 0 − Pf | 0 − . (13) (T = 1 C, X = x + (cid:15), Z = 1) (T = 1 C, X = x (cid:15), Z = 0) f 0 f 0 − The numerator of (13) is estimated by βˆ βˆ . To estimate the denominator of (13) we consider the 0a 0b (cid:88) following estimates: P | ≈ 1 (T = 1 C, X = x + (cid:15), Z = 1) πˆ = |A | (cid:88) T f 0 a i ζ ∈A i ζ P | − ≈ 1 (T = 1 C, X = x (cid:15), Z = 0) πˆ = |B | T f 0 b i ζ ∈B i ζ Hence, the causal eﬀect estimate for the eﬀect of statins on LDL cholesterol level is given by − βˆ βˆ ηˆ = 0a − 0b πˆ πˆ a b where (βˆ , βˆ ) denote the maximum likelihood estimates of (β , β ). We consider the calculation of η 0a 0b 0a 0b using the THIN subset data and a choice of three RDD bandwidths (ζ = 0.05, 0.1 and 0.15). Estimated values, ηˆ, together with associated 95% conﬁdence intervals, are shown in Table 2. Examining Table 2, we see that for each chosen RDD bandwidth, the estimated causal eﬀect of statins on LDL cholesterol level appears to diﬀer from zero, suggesting that statins appear to reduce the LDL cholesterol level in general. This would be expected, especially in light of the many large-scale randomised trials that have provided substantial evidence in favour of the beneﬁcial eﬀect of statins with respect to LDL cholesterol level. We have used the arguments and results presented in earlier sections of the paper to justify and construct a regression discontinuity design on real observational data, through a decision theoretic approach. We note that the use of a restricted sample of patients for this demonstrative example implies that the results reported are not necessarily applicable to the general population, or of strict clinical/epidemiological signiﬁcance. 6 Discussion We have developed a formal, decision theoretic approach to the regression discontinuity design using the language of conditional independence. Conditions under which the causal eﬀect of a treatment on a 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: treatment

text: assignment

text: does

text: not

text: change.

text: Finally,

text: we

text: need

text: to

text: argue

text: that

text: Property

text: holds

text: with

text: these

text: data.

text: In

text: essence,

text: Property

text: implies

text: that

text: the

text: outcome

text: variable

text: value

text: is

text: independent

text: of

text: the

text: threshold

text: variable

text: conditional

text: on

text: the

text: assignment

text: variable,

text: treatment,

text: regime

text: indicator

text: and

text: confounders.

text: In

text: this

text: example,

text: the

text: treatment

text: rule

text: (prescription

text: of

text: statins

text: if

text: the

text: CVD

text: risk

text: score

text: is

text: or

text: greater)

text: is

text: determined

text: by

text: the

text: UK

text: government

text: and

text: is

text: not

text: patient-speciﬁc.

text: Furthermore,

text: since

text: Z

text: is

text: determined

text: solely

text: by

text: the

text: CVD

text: risk

text: score,

text: conditioning

text: on

text: this

text: variable

text: implies

text: directly

text: that

text: further

text: knowledge

text: of

text: Z

text: is

text: unnecessary

text: to

text: make

text: inference

text: regarding

text: the

text: LDL

text: cholesterol

text: level.

text: Hence,

text: we

text: apply

text: the

text: result

text: of

text: Theorem

text: to

text: estimate

text: the

text: causal

text: eﬀect

text: of

text: statin

text: therapy

text: on

text: LDL

text: cholesterol

text: level,

text: using

text: a

text: fuzzy

text: RDD

text: and

text: the

text: THIN

text: subset

text: of

text: data.

text: Models

text: and

text: Estimation

text: A

text: B

text: For

text: a

text: chosen

text: bandwidth

text: ζ

text: (ζ

text: we

text: deﬁne

text: and

text: to

text: be

text: the

text: sets

text: of

text: patients

text: whose

text: ζ

text: ζ

text: CVD

text: risk

text: scores

text: lie

text: above

text: and

text: below

text: the

text: threshold,

text: respectively.

text: Setting

text: the

text: treatment

text: threshold

text: to

text: be

text: x

text: and

text: deﬁning

text: XC

text: X

text: to

text: be

text: the

text: centred

text: CVD

text: risk

text: score

text: for

text: the

text: ith

text: patient,

text: we

text: i

text: i

text: consider

text: the

text: following

text: linear

text: models:

text: A

text: Y

text: β

text: β

text: XC

text: ω

text: (for

text: i

text: i

text: i

text: i

text: ζ

text: B

text: Y

text: β

text: β

text: XC

text: ω

text: (for

text: i

text: i

text: i

text: i

text: ζ

text: N

text: A

text: B

text: with

text: ω

text: for

text: i

text: The

text: function

text: that

text: we

text: wish

text: to

text: evaluate

text: is

text: i

text: ζ

text: ζ

text: E

text: E

text: (Y

text: C,

text: X

text: x

text: Z

text: (Y

text: C,

text: X

text: x

text: Z

text: h(C)

text: P

text: f

text: Pf

text: (T

text: C,

text: X

text: x

text: Z

text: (T

text: C,

text: X

text: x

text: Z

text: f

text: f

text: The

text: numerator

text: of

text: is

text: estimated

text: by

text: βˆ

text: βˆ

text: To

text: estimate

text: the

text: denominator

text: of

text: we

text: consider

text: the

text: following

text: estimates:

text: P

text: (T

text: C,

text: X

text: x

text: Z

text: πˆ

text: |A

text: T

text: f

text: a

text: i

text: ζ

text: ∈A

text: i

text: ζ

text: P

text: (T

text: C,

text: X

text: x

text: Z

text: πˆ

text: |B

text: T

text: f

text: b

text: i

text: ζ

text: ∈B

text: i

text: ζ

text: Hence,

text: the

text: causal

text: eﬀect

text: estimate

text: for

text: the

text: eﬀect

text: of

text: statins

text: on

text: LDL

text: cholesterol

text: level

text: is

text: given

text: by

text: βˆ

text: βˆ

text: ηˆ

text: πˆ

text: πˆ

text: a

text: b

text: where

text: (βˆ

text: βˆ

text: denote

text: the

text: maximum

text: likelihood

text: estimates

text: of

text: (β

text: β

text: We

text: consider

text: the

text: calculation

text: of

text: η

text: using

text: the

text: THIN

text: subset

text: data

text: and

text: a

text: choice

text: of

text: three

text: RDD

text: bandwidths

text: (ζ

text: and

text: Estimated

text: values,

text: ηˆ,

text: together

text: with

text: associated

text: conﬁdence

text: intervals,

text: are

text: shown

text: in

text: Table

text: Examining

text: Table

text: we

text: see

text: that

text: for

text: each

text: chosen

text: RDD

text: bandwidth,

text: the

text: estimated

text: causal

text: eﬀect

text: of

text: statins

text: on

text: LDL

text: cholesterol

text: level

text: appears

text: to

text: diﬀer

text: from

text: zero,

text: suggesting

text: that

text: statins

text: appear

text: to

text: reduce

text: the

text: LDL

text: cholesterol

text: level

text: in

text: general.

text: This

text: would

text: be

text: expected,

text: especially

text: in

text: light

text: of

text: the

text: many

text: large-scale

text: randomised

text: trials

text: that

text: have

text: provided

text: substantial

text: evidence

text: in

text: favour

text: of

text: the

text: beneﬁcial

text: eﬀect

text: of

text: statins

text: with

text: respect

text: to

text: LDL

text: cholesterol

text: level.

text: We

text: have

text: used

text: the

text: arguments

text: and

text: results

text: presented

text: in

text: earlier

text: sections

text: of

text: the

text: paper

text: to

text: justify

text: and

text: construct

text: a

text: regression

text: discontinuity

text: design

text: on

text: real

text: observational

text: data,

text: through

text: a

text: decision

text: theoretic

text: approach.

text: We

text: note

text: that

text: the

text: use

text: of

text: a

text: restricted

text: sample

text: of

text: patients

text: for

text: this

text: demonstrative

text: example

text: implies

text: that

text: the

text: results

text: reported

text: are

text: not

text: necessarily

text: applicable

text: to

text: the

text: general

text: population,

text: or

text: of

text: strict

text: clinical/epidemiological

text: signiﬁcance.

text: Discussion

text: We

text: have

text: developed

text: a

text: formal,

text: decision

text: theoretic

text: approach

text: to

text: the

text: regression

text: discontinuity

text: design

text: using

text: the

text: language

text: of

text: conditional

text: independence.

text: Conditions

text: under

text: which

text: the

text: causal

text: eﬀect

text: of

text: a

text: treatment

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text: a

text: ##LTLine##

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text: Improper Gaussian Signaling in Full-Duplex Relay Channels with Residual Self-Interference Mohamed Gaafar, Mohammad Galal Khafagy, Osama Amin, and Mohamed-Slim Alouini Computer, Electrical, and Mathematical Sciences and Engineering (CEMSE) Division King Abdullah University of Science and Technology (KAUST), Thuwal, Makkah Province, Saudi Arabia. { } } Email: mohamed.gaafar, mohammad.khafagy, osama.amin, slim.alouini @kaust.edu.sa @kaust.edu.sa Abstract— We study the potential employment of improper means to mitigate it represent the main challenge in full-duplex Gaussian signaling (IGS) in full-duplex cooperative settings communications, especially with the fact that its adverse effect with residual self-interference (RSI). IGS is recently shown can typically be an increasing function of the relay power. to outperform traditional proper Gaussian signaling (PGS) in Therefore, increasing the relay power no longer guarantees an several interference-limited channel settings. In this work, IGS is employed in an attempt to alleviate the RSI adverse effect enhanced end-to-end performance. For instance, by increasing in full-duplex relaying (FDR). To this end, we derive a tight the relay power in a ﬁxed-rate transmission scheme, the relay upper bound expression for the end-to-end outage probability in may forward more reliably to the destination in the second terms of the relay signal parameters represented in its power and hop. However, it also increases the RSI level which negatively circularity coefﬁcient. We further show that the derived upper affects the reliability in the ﬁrst hop. Hence, higher relay power bound is either monotonic or unimodal in the relay’s circularity coefﬁcient. This result allows for easily locating the global optimal budgets cannot be always utilized beyond a certain threshold. point using known numerical methods. Based on the analysis, Consequently, employing interference mitigation strategies in IGS allows FDR systems to operate even with high RSI. It is FDR is crucial to accomplish a satisfactory performance of shown that, while the communication totally fails with PGS as full-duplex transmissions. the RSI increases, the IGS outage probability approaches a ﬁxed Improper Gaussian signaling (IGS) has been recently stud- value that depends on the channel statistics and target rate. The obtained results show that IGS can leverage higher relay power ied in [5], where higher degrees of freedom for the 3-user budgets than PGS to improve the performance, meanwhile it single-input single-output interference channel were shown to relieves its RSI impact via tuning the signal impropriety. be achievable. This comes in contrary to other communication settings with interference-free channels where proper Gaussian I. INTRODUCTION signaling (PGS) is the optimal choice. PGS assumes the zero- Contrary to a long-held acceptance that radio front-ends mean complex Gaussian transmit signal to be statistically cannot simultaneously transmit and receive, a truly promising circularly symmetric with uncorrelated real and imaginary potential for full-duplex communications has been shown by components. On the other hand, IGS is a general class of recent hardware developments [1], [2]. Indeed, by multiplexing signals where circularity and uncorrelatedness conditions can inbound and outbound trafﬁc over the same channel resource, a be relaxed [6]. The results in [5] motivate the need to further full-duplex radio can recover the spectral efﬁciency loss known study the potential gains of IGS in communication scenarios to be encountered by its half-duplex counterpart. Performance where interference imposes a noticeable limitation. merits of full-duplex radio have been recently investigated in For Gaussian channels, and within the class of Gaussian different communication settings, including full-duplex bidi- signals, IGS has been recently adopted to improve the per- rectional communication, full-duplex base stations, and full- formance of different interference channel communication duplex relaying (FDR) [3], with the latter being the focus of systems [7]–[10]. In general, recent work on the interference this work. These merits have qualiﬁed full-duplex communi- channel showed that IGS can actually support higher rates cation to be considered as a candidate technology for future in certain interference-limited scenarios [7]. In an underlay ﬁfth generation (5G) wireless networks [4]. cognitive radio setting, IGS increases the spectrum sharing FDR allows a relay node to listen to an information source opportunity for secondary users by relieving the interference and simultaneously forward to its intended destination. Theo- impact on the authorized users [8]–[10]. retically, this simultaneous transmission/reception doubles the The potential gains of IGS have been also recently stud- spectral efﬁciency in the relay channel. However, in practice, ied in [11] for the multiple-input multiple-output (MIMO) this comes at the expense of a self-interference level intro- relay channel when a partial decode-and-forward strategy is duced at the receiver of the relay node from its own transmit- adopted. In such a relaying strategy, the relay only decodes a ter. Even with the application of advanced self-interference part of the message, while the rest of the message is treated isolation and cancellation techniques, a level of residual self- as an additional interference term. It was shown in [11] that interference (RSI) persists. Such a persistent RSI link and the PGS is optimal within the class of Gaussian signals. However, the work in [11] assumed an ideal full-duplex relay channel, The research reported in this publication was supported by funding from King Abdullah University of Science and Technology (KAUST). where the self-interference imposed by the relay’s transmitter

text: Improper

label: title

text: Gaussian

text: Signaling

text: in

text: Full-Duplex

text: Relay

text: Channels

text: with

text: Residual

text: Self-Interference

text: Mohamed

label: author

text: Gaafar,

text: Mohammad

text: Galal

text: Khafagy,

text: Osama

text: Amin,

text: and

text: Mohamed-Slim

text: Alouini

text: Computer,

text: Electrical,

text: and

text: Mathematical

text: Sciences

text: and

text: Engineering

text: (CEMSE)

text: Division

text: King

text: Abdullah

text: University

text: of

text: Science

text: and

text: Technology

text: (KAUST),

text: Thuwal,

text: Makkah

text: Province,

text: Saudi

text: Arabia.

text: Email:

text: mohamed.gaafar,

text: mohammad.khafagy,

text: osama.amin,

text: slim.alouini

text: @kaust.edu.sa

text: @kaust.edu.sa

text: Abstract—

text: We

label: abstract

text: study

text: the

text: potential

text: employment

text: of

text: improper

text: means

text: to

text: mitigate

text: it

text: represent

text: the

text: main

text: challenge

text: in

text: full-duplex

text: Gaussian

text: signaling

text: (IGS)

text: in

text: full-duplex

text: cooperative

text: settings

text: communications,

text: especially

text: with

text: the

text: fact

text: that

text: its

text: adverse

text: effect

text: with

text: residual

text: self-interference

text: (RSI).

text: IGS

text: is

text: recently

text: shown

text: can

text: typically

text: be

text: an

text: increasing

text: function

text: of

text: the

text: relay

text: power.

text: to

text: outperform

text: traditional

text: proper

text: Gaussian

text: signaling

text: (PGS)

text: in

text: Therefore,

text: increasing

text: the

text: relay

text: power

text: no

text: longer

text: guarantees

text: an

text: several

text: interference-limited

text: channel

text: settings.

text: In

text: this

text: work,

text: IGS

text: is

text: employed

text: in

text: an

text: attempt

text: to

text: alleviate

text: the

text: RSI

text: adverse

text: effect

text: enhanced

text: end-to-end

text: performance.

text: For

text: instance,

text: by

text: increasing

text: in

text: full-duplex

text: relaying

text: (FDR).

text: To

text: this

text: end,

text: we

text: derive

text: a

text: tight

text: the

text: relay

text: power

text: in

text: a

text: ﬁxed-rate

text: transmission

text: scheme,

text: the

text: relay

text: upper

text: bound

text: expression

text: for

text: the

text: end-to-end

text: outage

text: probability

text: in

text: may

text: forward

text: more

text: reliably

text: to

text: the

text: destination

text: in

text: the

text: second

text: terms

text: of

text: the

text: relay

text: signal

text: parameters

text: represented

text: in

text: its

text: power

text: and

text: hop.

text: However,

text: it

text: also

text: increases

text: the

text: RSI

text: level

text: which

text: negatively

text: circularity

text: coefﬁcient.

text: We

text: further

text: show

text: that

text: the

text: derived

text: upper

text: affects

text: the

text: reliability

text: in

text: the

text: ﬁrst

text: hop.

text: Hence,

text: higher

text: relay

text: power

text: bound

text: is

text: either

text: monotonic

text: or

text: unimodal

text: in

text: the

text: relay’s

text: circularity

text: coefﬁcient.

text: This

text: result

text: allows

text: for

text: easily

text: locating

text: the

text: global

text: optimal

text: budgets

text: cannot

text: be

text: always

text: utilized

text: beyond

text: a

text: certain

text: threshold.

text: point

text: using

text: known

text: numerical

text: methods.

text: Based

text: on

text: the

text: analysis,

text: Consequently,

text: employing

text: interference

text: mitigation

text: strategies

text: in

text: IGS

text: allows

text: FDR

text: systems

text: to

text: operate

text: even

text: with

text: high

text: RSI.

text: It

text: is

text: FDR

text: is

text: crucial

text: to

text: accomplish

text: a

text: satisfactory

text: performance

text: of

text: shown

text: that,

text: while

text: the

text: communication

text: totally

text: fails

text: with

text: PGS

text: as

text: full-duplex

text: transmissions.

text: the

text: RSI

text: increases,

text: the

text: IGS

text: outage

text: probability

text: approaches

text: a

text: ﬁxed

text: Improper

text: Gaussian

text: signaling

text: (IGS)

text: has

text: been

text: recently

text: stud-

text: value

text: that

text: depends

text: on

text: the

text: channel

text: statistics

text: and

text: target

text: rate.

text: The

text: obtained

text: results

text: show

text: that

text: IGS

text: can

text: leverage

text: higher

text: relay

text: power

text: ied

text: in

text: where

text: higher

text: degrees

text: of

text: freedom

text: for

text: the

text: budgets

text: than

text: PGS

text: to

text: improve

text: the

text: performance,

text: meanwhile

text: it

text: single-input

text: single-output

text: interference

text: channel

text: were

text: shown

text: to

text: relieves

text: its

text: RSI

text: impact

text: via

text: tuning

text: the

text: signal

text: impropriety.

text: be

text: achievable.

text: This

text: comes

text: in

text: contrary

text: to

text: other

text: communication

text: settings

text: with

text: interference-free

text: channels

text: where

text: proper

text: Gaussian

text: I.

text: INTRODUCTION

text: signaling

text: (PGS)

text: is

text: the

text: optimal

text: choice.

text: PGS

text: assumes

text: the

text: zero-

text: Contrary

text: to

text: a

text: long-held

text: acceptance

text: that

text: radio

text: front-ends

text: mean

text: complex

text: Gaussian

text: transmit

text: signal

text: to

text: be

text: statistically

text: cannot

text: simultaneously

text: transmit

text: and

text: receive,

text: a

text: truly

text: promising

text: circularly

text: symmetric

text: with

text: uncorrelated

text: real

text: and

text: imaginary

text: potential

text: for

text: full-duplex

text: communications

text: has

text: been

text: shown

text: by

text: components.

text: On

text: the

text: other

text: hand,

text: IGS

text: is

text: a

text: general

text: class

text: of

text: recent

text: hardware

text: developments

text: Indeed,

text: by

text: multiplexing

text: signals

text: where

text: circularity

text: and

text: uncorrelatedness

text: conditions

text: can

text: inbound

text: and

text: outbound

text: trafﬁc

text: over

text: the

text: same

text: channel

text: resource,

text: a

text: be

text: relaxed

text: The

text: results

text: in

text: motivate

text: the

text: need

text: to

text: further

text: full-duplex

text: radio

text: can

text: recover

text: the

text: spectral

text: efﬁciency

text: loss

text: known

text: study

text: the

text: potential

text: gains

text: of

text: IGS

text: in

text: communication

text: scenarios

text: to

text: be

text: encountered

text: by

text: its

text: half-duplex

text: counterpart.

text: Performance

text: where

text: interference

text: imposes

text: a

text: noticeable

text: limitation.

text: merits

text: of

text: full-duplex

text: radio

text: have

text: been

text: recently

text: investigated

text: in

text: For

text: Gaussian

text: channels,

text: and

text: within

text: the

text: class

text: of

text: Gaussian

text: different

text: communication

text: settings,

text: including

text: full-duplex

text: bidi-

text: signals,

text: IGS

text: has

text: been

text: recently

text: adopted

text: to

text: improve

text: the

text: per-

text: rectional

text: communication,

text: full-duplex

text: base

text: stations,

text: and

text: full-

text: formance

text: of

text: different

text: interference

text: channel

text: communication

text: duplex

text: relaying

text: (FDR)

text: with

text: the

text: latter

text: being

text: the

text: focus

text: of

text: systems

text: In

text: general,

text: recent

text: work

text: on

text: the

text: interference

text: this

text: work.

text: These

text: merits

text: have

text: qualiﬁed

text: full-duplex

text: communi-

text: channel

text: showed

text: that

text: IGS

text: can

text: actually

text: support

text: higher

text: rates

text: cation

text: to

text: be

text: considered

text: as

text: a

text: candidate

text: technology

text: for

text: future

text: in

text: certain

text: interference-limited

text: scenarios

text: In

text: an

text: underlay

text: ﬁfth

text: generation

text: wireless

text: networks

text: cognitive

text: radio

text: setting,

text: IGS

text: increases

text: the

text: spectrum

text: sharing

text: FDR

text: allows

text: a

text: relay

text: node

text: to

text: listen

text: to

text: an

text: information

text: source

text: opportunity

text: for

text: secondary

text: users

text: by

text: relieving

text: the

text: interference

text: and

text: simultaneously

text: forward

text: to

text: its

text: intended

text: destination.

text: Theo-

text: impact

text: on

text: the

text: authorized

text: users

text: retically,

text: this

text: simultaneous

text: transmission/reception

text: doubles

text: the

text: The

text: potential

text: gains

text: of

text: IGS

text: have

text: been

text: also

text: recently

text: stud-

text: spectral

text: efﬁciency

text: in

text: the

text: relay

text: channel.

text: However,

text: in

text: practice,

text: ied

text: in

text: for

text: the

text: multiple-input

text: multiple-output

text: (MIMO)

text: this

text: comes

text: at

text: the

text: expense

text: of

text: a

text: self-interference

text: level

text: intro-

text: relay

text: channel

text: when

text: a

text: partial

text: decode-and-forward

text: strategy

text: is

text: duced

text: at

text: the

text: receiver

text: of

text: the

text: relay

text: node

text: from

text: its

text: own

text: transmit-

text: adopted.

text: In

text: such

text: a

text: relaying

text: strategy,

text: the

text: relay

text: only

text: decodes

text: a

text: ter.

text: Even

text: with

text: the

text: application

text: of

text: advanced

text: self-interference

text: part

text: of

text: the

text: message,

text: while

text: the

text: rest

text: of

text: the

text: message

text: is

text: treated

text: isolation

text: and

text: cancellation

text: techniques,

text: a

text: level

text: of

text: residual

text: self-

text: as

text: an

text: additional

text: interference

text: term.

text: It

text: was

text: shown

text: in

text: that

text: interference

text: (RSI)

text: persists.

text: Such

text: a

text: persistent

text: RSI

text: link

text: and

text: the

text: PGS

text: is

text: optimal

text: within

text: the

text: class

text: of

text: Gaussian

text: signals.

text: However,

text: the

text: work

text: in

text: assumed

text: an

text: ideal

text: full-duplex

text: relay

text: channel,

text: The

text: research

text: reported

text: in

text: this

text: publication

text: was

text: supported

text: by

text: funding

text: from

text: King

text: Abdullah

text: University

text: of

text: Science

text: and

text: Technology

text: (KAUST).

text: where

text: the

text: self-interference

text: imposed

text: by

text: the

text: relay’s

text: transmitter


text: Mon. Not. R. Astron. Soc. 000, 1–13 (2002) Printed 28 May 2020 (MN LATEX style ﬁle v2.2) The evolution of the Sun’s birth cluster and the search for the solar siblings with Gaia C.A. Mart´ınez-Barbosa,1(cid:63) A.G.A Brown,1† T. Boekholt,1 S. Portegies Zwart,1 E. Antiche2 and T. Antoja3 1 Leiden Observatory, Leiden University, P.B. 9513, Leiden 2300 RA, the Netherlands 2 Departament d’ Astronomia i Meteorologia, Institut de Cie`ncies del Cosmos, Universitat de Barcelona, IEEC, Mart´ı Franque`s 1, E-08028 Barcelona, Spain 3 Scientiﬁc Support Ofﬁce, Directorate of Science and Robotic Exploration, European Space Research and Technology Centre (ESA/ESTEC), Keplerlaan 1, NL-2201 AZ Noordwijk, the Netherlands Accepted XXXXXXXXXXX. Received XXXXXXXXXXXX; in original form XXXXXXXXXXXXX ABSTRACT We use self-consistent numerical simulations of the evolution and disruption of the Sun’s birth cluster in the Milky Way potential to investigate the present-day phase space distribution of the sun’s siblings. The simulations include the gravitational N-body forces within the clus- ter and the effects of stellar evolution on the cluster population. In addition the gravitational forces due to the Milky Way potential are accounted for in a self-consistent manner. Our aim is to understand how the astrometric and radial velocity data from the Gaia mission can be used to pre-select solar sibling candidates. We vary the initial conditions of the Sun’s birth cluster, as well as the parameters of the Galactic potential. In particular, we use different conﬁgura- tions and strengths of the bar and spiral arms. We show that the disruption time-scales of the cluster are insensitive to the details of the non-axisymmetric components of the Milky Way model and we make predictions, averaged over the different simulated possibilities, about the number of solar siblings that should appear in surveys such as Gaia or GALAH. We ﬁnd a large variety of present-day phase space distributions of solar siblings, which depend on the cluster initial conditions and the Milky Way model parameters. We show that nevertheless ro- bust predictions can be made about the location of the solar siblings in the space of parallaxes ((cid:36)), proper motions (µ) and radial velocities (V ). By calculating the ratio of the number r of simulated solar siblings to that of the number of stars in a model Galactic disk, we ﬁnd (cid:62) (cid:54) (cid:54) − that this ratio is above 0.5 in the region given by: (cid:36) 5 mas, 4 µ 6 mas yr 1, and (cid:54) (cid:54) − −2 V 0 km s 1. Selecting stars from this region should increase the probability of r success in identifying solar siblings through follow up observations. However the proposed pre-selection criterion is sensitive to our assumptions, in particular about the Galactic poten- tial. Using a more realistic potential (e.g., including transient spiral structure and molecular clouds) would make the pre-selection of solar sibling candidates based on astrometric and ra- dial velocity data very inefﬁcient. This reinforces the need for large scale surveys to determine precise astrophysical properties of stars, in particular their ages and chemical abundances, if we want to identify the solar family. Key words: Galaxy: kinematics and dynamics — open clusters and associations: general — solar neighbourhood — Sun: general 1 INTRODUCTION (open clusters) supply new stars to the disk of the Galaxy through several processes, such as shocks from encounters with spiral arms Since most of the stars are born in star clusters (Lada & Lada and Giant Molecular Clouds (Gieles et al. 2006, 2007). 2003), these systems are considered the building blocks of galaxies. In the Milky Way star clusters located in the Galactic halo (Globu- The dynamical evolution of star clusters involves several phys- lar clusters) populate the Galactic disk through mergers (Lee et al. ical mechanisms. At earlier stages of their evolution, star clusters 2013). On the other hand star clusters formed in the Galactic disk lose mass mainly due to stellar evolution and two-body relaxation processes, which in turn, enlarge the size of star clusters (Taka- (cid:63) E-mail: cmartinez@strw.leidenuniv.nl hashi & Portegies Zwart 2000; Baumgardt & Makino 2003; Madrid † E-mail: brown@strw.leidenuniv.nl et al. 2012). This evolutionary stage is called the expansion phase (cid:13)c 2002 RAS

text: Mon.

text: Not.

text: R.

text: Astron.

text: Soc.

text: Printed

text: May

text: (MN

text: LATEX

text: style

text: ﬁle

text: The

text: evolution

text: of

text: the

text: Sun’s

text: birth

text: cluster

text: and

text: the

text: search

text: for

text: the

text: solar

text: siblings

text: with

text: Gaia

text: C.A.

text: A.G.A

text: T.

text: S.

text: Portegies

text: E.

text: and

text: T.

text: Leiden

text: Observatory,

text: Leiden

text: University,

text: P.B.

text: Leiden

text: RA,

text: the

text: Netherlands

text: Departament

text: d’

text: Astronomia

text: i

text: Meteorologia,

text: Institut

text: de

text: Cie`ncies

text: del

text: Cosmos,

text: Universitat

text: de

text: Barcelona,

text: IEEC,

text: Mart´ı

text: Franque`s

text: Barcelona,

text: Spain

text: Scientiﬁc

text: Support

text: Ofﬁce,

text: Directorate

text: of

text: Science

text: and

text: Robotic

text: Exploration,

text: European

text: Space

text: Research

text: and

text: Technology

text: Centre

text: (ESA/ESTEC),

text: Keplerlaan

text: AZ

text: Noordwijk,

text: the

text: Netherlands

text: Accepted

text: XXXXXXXXXXX.

text: Received

text: XXXXXXXXXXXX;

text: in

text: original

text: form

text: XXXXXXXXXXXXX

text: ABSTRACT

text: We

text: use

text: self-consistent

text: numerical

text: simulations

text: of

text: the

text: evolution

text: and

text: disruption

text: of

text: the

text: Sun’s

text: birth

text: cluster

text: in

text: the

text: Milky

text: Way

text: potential

text: to

text: investigate

text: the

text: present-day

text: phase

text: space

text: distribution

text: of

text: the

text: sun’s

text: siblings.

text: The

text: simulations

text: include

text: the

text: gravitational

text: N-body

text: forces

text: within

text: the

text: clus-

text: ter

text: and

text: the

text: effects

text: of

text: stellar

text: evolution

text: on

text: the

text: cluster

text: population.

text: In

text: addition

text: the

text: gravitational

text: forces

text: due

text: to

text: the

text: Milky

text: Way

text: potential

text: are

text: accounted

text: for

text: in

text: a

text: self-consistent

text: manner.

text: Our

text: aim

text: is

text: to

text: understand

text: how

text: the

text: astrometric

text: and

text: radial

text: velocity

text: data

text: from

text: the

text: Gaia

text: mission

text: can

text: be

text: used

text: to

text: pre-select

text: solar

text: sibling

text: candidates.

text: We

text: vary

text: the

text: initial

text: conditions

text: of

text: the

text: Sun’s

text: birth

text: cluster,

text: as

text: well

text: as

text: the

text: parameters

text: of

text: the

text: Galactic

text: potential.

text: In

text: particular,

text: we

text: use

text: different

text: conﬁgura-

text: tions

text: and

text: strengths

text: of

text: the

text: bar

text: and

text: spiral

text: arms.

text: We

text: show

text: that

text: the

text: disruption

text: time-scales

text: of

text: the

text: cluster

text: are

text: insensitive

text: to

text: the

text: details

text: of

text: the

text: non-axisymmetric

text: components

text: of

text: the

text: Milky

text: Way

text: model

text: and

text: we

text: make

text: predictions,

text: averaged

text: over

text: the

text: different

text: simulated

text: possibilities,

text: about

text: the

text: number

text: of

text: solar

text: siblings

text: that

text: should

text: appear

text: in

text: surveys

text: such

text: as

text: Gaia

text: or

text: GALAH.

text: We

text: ﬁnd

text: a

text: large

text: variety

text: of

text: present-day

text: phase

text: space

text: distributions

text: of

text: solar

text: siblings,

text: which

text: depend

text: on

text: the

text: cluster

text: initial

text: conditions

text: and

text: the

text: Milky

text: Way

text: model

text: parameters.

text: We

text: show

text: that

text: nevertheless

text: ro-

text: bust

text: predictions

text: can

text: be

text: made

text: about

text: the

text: location

text: of

text: the

text: solar

text: siblings

text: in

text: the

text: space

text: of

text: parallaxes

text: proper

text: motions

text: (µ)

text: and

text: radial

text: velocities

text: (V

text: By

text: calculating

text: the

text: ratio

text: of

text: the

text: number

text: r

text: of

text: simulated

text: solar

text: siblings

text: to

text: that

text: of

text: the

text: number

text: of

text: stars

text: in

text: a

text: model

text: Galactic

text: disk,

text: we

text: ﬁnd

text: that

text: this

text: ratio

text: is

text: above

text: in

text: the

text: region

text: given

text: by:

text: mas,

text: µ

text: mas

text: yr

text: and

text: −2

text: V

text: km

text: s

text: Selecting

text: stars

text: from

text: this

text: region

text: should

text: increase

text: the

text: probability

text: of

text: r

text: success

text: in

text: identifying

text: solar

text: siblings

text: through

text: follow

text: up

text: observations.

text: However

text: the

text: proposed

text: pre-selection

text: criterion

text: is

text: sensitive

text: to

text: our

text: assumptions,

text: in

text: particular

text: about

text: the

text: Galactic

text: poten-

text: tial.

text: Using

text: a

text: more

text: realistic

text: potential

text: (e.g.,

text: including

text: transient

text: spiral

text: structure

text: and

text: molecular

text: clouds)

text: would

text: make

text: the

text: pre-selection

text: of

text: solar

text: sibling

text: candidates

text: based

text: on

text: astrometric

text: and

text: ra-

text: dial

text: velocity

text: data

text: very

text: inefﬁcient.

text: This

text: reinforces

text: the

text: need

text: for

text: large

text: scale

text: surveys

text: to

text: determine

text: precise

text: astrophysical

text: properties

text: of

text: stars,

text: in

text: particular

text: their

text: ages

text: and

text: chemical

text: abundances,

text: if

text: we

text: want

text: to

text: identify

text: the

text: solar

text: family.

text: Key

text: words:

text: Galaxy:

text: kinematics

text: and

text: dynamics

text: open

text: clusters

text: and

text: associations:

text: general

text: solar

text: neighbourhood

text: Sun:

text: general

text: INTRODUCTION

text: (open

text: clusters)

text: supply

text: new

text: stars

text: to

text: the

text: disk

text: of

text: the

text: Galaxy

text: through

text: several

text: processes,

text: such

text: as

text: shocks

text: from

text: encounters

text: with

text: spiral

text: arms

text: Since

text: most

text: of

text: the

text: stars

text: are

text: born

text: in

text: star

text: clusters

text: (Lada

text: Lada

text: and

text: Giant

text: Molecular

text: Clouds

text: (Gieles

text: et

text: al.

text: these

text: systems

text: are

text: considered

text: the

text: building

text: blocks

text: of

text: galaxies.

text: In

text: the

text: Milky

text: Way

text: star

text: clusters

text: located

text: in

text: the

text: Galactic

text: halo

text: (Globu-

text: The

text: dynamical

text: evolution

text: of

text: star

text: clusters

text: involves

text: several

text: phys-

text: lar

text: clusters)

text: populate

text: the

text: Galactic

text: disk

text: through

text: mergers

text: (Lee

text: et

text: al.

text: ical

text: mechanisms.

text: At

text: earlier

text: stages

text: of

text: their

text: evolution,

text: star

text: clusters

text: On

text: the

text: other

text: hand

text: star

text: clusters

text: formed

text: in

text: the

text: Galactic

text: disk

text: lose

text: mass

text: mainly

text: due

text: to

text: stellar

text: evolution

text: and

text: two-body

text: relaxation

text: processes,

text: which

text: in

text: turn,

text: enlarge

text: the

text: size

text: of

text: star

text: clusters

text: (Taka-

text: E-mail:

text: cmartinez@strw.leidenuniv.nl

text: hashi

text: Portegies

text: Zwart

text: Baumgardt

text: Makino

text: Madrid

text: E-mail:

text: brown@strw.leidenuniv.nl

text: et

text: al.

text: This

text: evolutionary

text: stage

text: is

text: called

text: the

text: expansion

text: phase

text: RAS


text: 2 C.A. Mart´ınez-Barbosa et al. (Gieles et al. 2011), which takes about 40% of the star cluster’s life- disk. For small birth clusters (few thousand stars with a total mass time. Once star clusters overcome the expansion phase, the effects of the order of 1000 M(cid:12)) such as employed by Brown et al. (2010) of the external tidal ﬁeld of the Galaxy become important, depend- and Portegies Zwart (2009), Mishurov & Acharova (2011) predict ing on their location with respect to the Galactic centre. This stage that practically no solar siblings will currently be located within is called the evaporation phase (Gieles et al. 2011) and it is charac- 100 pc from the sun. However, for larger birth clusters (of order 104 terized by the gradual dissolution of star clusters in the Galaxy. stars, in line with predictions from e.g. Dukes & Krumholz 2012) The dissolution rate of star clusters depends on their Galacto- one can still expect to ﬁnd a good number of siblings presently or- centric distance (Madrid et al. 2012), orbit (Baumgardt & Makino biting the Galaxy within 100 pc from the Sun. 2003), orbital inclination (Webb et al. 2014) and on Galaxy prop- Ongoing surveys of our galaxy, in particular the Gaia mis- erties such as the mass and size of the Galactic disk (Madrid sion (Lindegren et al. 2008) and the GALAH survey (De Silva et al. et al. 2014). Additionally, open clusters in the Milky Way are 2015), will provide large samples of stars with accurately deter- also dissolved due to non axisymmetric perturbations such as bars mined distances, space motions, and chemical abundance patterns, (Berentzen & Athanassoula 2012), spiral arms (Gieles et al. 2007) thus enabling a much improved search for the sun’s siblings. In this and giant molecular clouds (Gieles et al. 2006; Lamers & Gieles paper we investigate the potential of the Gaia astrometric and ra- 2006). The strongest tidal stripping occurs at times when open clus- dial velocity data to narrow down the selection of candidate solar ters cross regions of high density gas, for instance, during spiral siblings for which detailed chemical abundance studies should be arms passages (Gieles et al. 2007; Kruijssen et al. 2011) or during undertaken in order to identify the true siblings. Our investigation collisions with giant molecular clouds (Gieles et al. 2006). Open is done by performing simulations of the evolution and disruption clusters can also radially migrate over distances of up to 1 kpc in of the Sun’s birth cluster in a realistic (although static) Galactic po- a short time scale (∼ 100 Myr) when the Galactic spiral struc- tential, including the bar and spiral arms. The aim is to predict the ture is transient (Fujii & Baba 2012). This radial migration pro- present-day phase space distribution of the siblings and simulate cess can also be efﬁcient in the absence of transient structure if the astrometric and radial velocity data collected by Gaia. We in- the resonances due the bar and spiral structure overlap (Minchev clude the internal N−body processes in the cluster to account for & Famaey 2010). Radial migration affects the orbits of open clus- the disruption time scale. We use a full stellar mass spectrum and ters in the Galaxy, increasing or decreasing their perigalacticon a parametrized stellar evolution code to make accurate predictions distance, which in turn inﬂuences their dissolution times (see e.g. of how the solar siblings are observed by Gaia. To this end we also J´ılkova´ et al. 2012). account for the effects of extinction and reddening. The high eccentricities and inclinations observed in the The rest of this paper is organized as follows. In Sect. we Edgeworth-Kuiper belt objects together with the discovery of de- describe the simulations. In Sect. we explore the evolution and cay products of 60Fe and other radioactive elements in the mete- disruption of the Sun’s birth cluster due to the bar and spiral arms orite fossil record, suggest that the Sun was born in an open cluster of the Galaxy. In Sect. we present the current phase-space dis- 4.6 Gyr ago (Portegies Zwart 2009, and references therein). Iden- tribution of solar siblings obtained from the simulations. In Sect. tifying the stars that were formed together with the Sun (the solar we make use of the simulated positions and motions of the solar siblings) would enable the determination of the Galactic birth ra- siblings to investigate the robustness of the selection criterion pro- dius of the Sun as well as further constrain the properties of its posed by Brown et al. (2010) to the uncertainties in the present-day birth cluster (Bland-Hawthorn et al. 2010; Adams 2010). The birth phase space distribution of the solar siblings. An updated set of se- radius affects the evolution of the solar system, and in particular lection criteria based on parallax, proper motion and radial velocity the Oort cloud, which is sensitive to the Galactic environment the information is presented. In Sect. we use these criteria to examine Sun passes through along its orbit (e.g. Portegies Zwart & J´ılkova´ stars that were previously suggested as solar siblings candidates 2015). and further discuss our results. In Sect. we summarize. The Sun’s birth cluster will undergo all the disruptive pro- 2 SIMULATION SET-UP cesses described above and thus dissolve, leading to the spreading out of its stars over the Galactic disk. The subsequent distribution The goals of the simulations of the Sun’s birth cluster are to of the solar siblings was studied by Portegies Zwart (2009), who predict the present-day phase space distribution of the solar sib- evolved the Sun’s birth cluster in an axisymmetric model for the lings and how these are expected to appear in the Gaia catalogue. Galactic potential and concluded that tens of solar siblings might In particular we wish to account for the uncertainties in the initial still be present within a distance of 100 pc from the Sun. Several conditions of the birth cluster and the parameters of the Milky Way attempts have since been made to ﬁnd solar siblings (e.g. Brown potential. The predictions of the Gaia observations require the use et al. 2010; Bobylev et al. 2011; Liu et al. 2015); however, only of a realistic mass spectrum for the siblings, and accounting for four plausible candidates have been identiﬁed so far (Batista & Fer- stellar evolution and extinction and interstellar reddening effects. nandes 2012; Batista et al. 2014; Ram´ırez et al. 2014). This small We thus employ the following elements in the simulations: number of observed solar siblings might be a consequence of the lack of accurate predictions of the present-day phase space distri- Galactic model The Milky Way potential is described by an ana- bution of solar siblings together with insufﬁciently accurate stellar lytic model containing a disk, bulge and halo, as well as a bar and kinematic data. spiral arms. The parameters of the bar and spiral arms are varied in Brown et al. (2010) used test particle simulations to predict the the simulations to account for uncertainties in their strengths and current distribution of solar siblings in the Milky Way. They con- pattern speeds (Sect. ). (cid:62) cluded that stars with parallaxes ((cid:36)) 10 mas and proper motions Cluster model The Sun’s birth cluster is modelled with a mass (cid:54) − (µ) 6.5 mas yr 1, should be considered solar sibling candidates. spectrum for the stars and we account for the gravitational N-body Their conclusions were criticised by Mishurov & Acharova (2011) effects within the cluster as well as the effect of the Galaxy’s grav- who pointed out that in more realistic Galactic potentials the solar itational ﬁeld on the cluster stars. The use of N-body models for siblings are expected to be much more spread out over the Galactic the birth cluster is motivated by the desire to account for the dis- (cid:13)c 2002 RAS, MNRAS 000, 1–13

text: C.A.

text: Mart´ınez-Barbosa

text: et

text: al.

text: (Gieles

text: et

text: al.

text: which

text: takes

text: about

text: of

text: the

text: star

text: cluster’s

text: life-

text: disk.

text: For

text: small

text: birth

text: clusters

text: (few

text: thousand

text: stars

text: with

text: a

text: total

text: mass

text: time.

text: Once

text: star

text: clusters

text: overcome

text: the

text: expansion

text: phase,

text: the

text: effects

text: of

text: the

text: order

text: of

text: such

text: as

text: employed

text: by

text: Brown

text: et

text: al.

text: of

text: the

text: external

text: tidal

text: ﬁeld

text: of

text: the

text: Galaxy

text: become

text: important,

text: depend-

text: and

text: Portegies

text: Zwart

text: Mishurov

text: Acharova

text: predict

text: ing

text: on

text: their

text: location

text: with

text: respect

text: to

text: the

text: Galactic

text: centre.

text: This

text: stage

text: that

text: practically

text: no

text: solar

text: siblings

text: will

text: currently

text: be

text: located

text: within

text: is

text: called

text: the

text: evaporation

text: phase

text: (Gieles

text: et

text: al.

text: and

text: it

text: is

text: charac-

text: pc

text: from

text: the

text: sun.

text: However,

text: for

text: larger

text: birth

text: clusters

text: (of

text: order

text: terized

text: by

text: the

text: gradual

text: dissolution

text: of

text: star

text: clusters

text: in

text: the

text: Galaxy.

text: stars,

text: in

text: line

text: with

text: predictions

text: from

text: e.g.

text: Dukes

text: Krumholz

text: The

text: dissolution

text: rate

text: of

text: star

text: clusters

text: depends

text: on

text: their

text: Galacto-

text: one

text: can

text: still

text: expect

text: to

text: ﬁnd

text: a

text: good

text: number

text: of

text: siblings

text: presently

text: or-

text: centric

text: distance

text: (Madrid

text: et

text: al.

text: orbit

text: (Baumgardt

text: Makino

text: biting

text: the

text: Galaxy

text: within

text: pc

text: from

text: the

text: Sun.

text: orbital

text: inclination

text: (Webb

text: et

text: al.

text: and

text: on

text: Galaxy

text: prop-

text: Ongoing

text: surveys

text: of

text: our

text: galaxy,

text: in

text: particular

text: the

text: Gaia

text: mis-

text: erties

text: such

text: as

text: the

text: mass

text: and

text: size

text: of

text: the

text: Galactic

text: disk

text: (Madrid

text: sion

text: (Lindegren

text: et

text: al.

text: and

text: the

text: GALAH

text: survey

text: (De

text: Silva

text: et

text: al.

text: et

text: al.

text: Additionally,

text: open

text: clusters

text: in

text: the

text: Milky

text: Way

text: are

text: will

text: provide

text: large

text: samples

text: of

text: stars

text: with

text: accurately

text: deter-

text: also

text: dissolved

text: due

text: to

text: non

text: axisymmetric

text: perturbations

text: such

text: as

text: bars

text: mined

text: distances,

text: space

text: motions,

text: and

text: chemical

text: abundance

text: patterns,

text: (Berentzen

text: Athanassoula

text: spiral

text: arms

text: (Gieles

text: et

text: al.

text: thus

text: enabling

text: a

text: much

text: improved

text: search

text: for

text: the

text: sun’s

text: siblings.

text: In

text: this

text: and

text: giant

text: molecular

text: clouds

text: (Gieles

text: et

text: al.

text: Lamers

text: Gieles

text: paper

text: we

text: investigate

text: the

text: potential

text: of

text: the

text: Gaia

text: astrometric

text: and

text: ra-

text: The

text: strongest

text: tidal

text: stripping

text: occurs

text: at

text: times

text: when

text: open

text: clus-

text: dial

text: velocity

text: data

text: to

text: narrow

text: down

text: the

text: selection

text: of

text: candidate

text: solar

text: ters

text: cross

text: regions

text: of

text: high

text: density

text: gas,

text: for

text: instance,

text: during

text: spiral

text: siblings

text: for

text: which

text: detailed

text: chemical

text: abundance

text: studies

text: should

text: be

text: arms

text: passages

text: (Gieles

text: et

text: al.

text: Kruijssen

text: et

text: al.

text: or

text: during

text: undertaken

text: in

text: order

text: to

text: identify

text: the

text: true

text: siblings.

text: Our

text: investigation

text: collisions

text: with

text: giant

text: molecular

text: clouds

text: (Gieles

text: et

text: al.

text: Open

text: is

text: done

text: by

text: performing

text: simulations

text: of

text: the

text: evolution

text: and

text: disruption

text: clusters

text: can

text: also

text: radially

text: migrate

text: over

text: distances

text: of

text: up

text: to

text: kpc

text: in

text: of

text: the

text: Sun’s

text: birth

text: cluster

text: in

text: a

text: realistic

text: (although

text: static)

text: Galactic

text: po-

text: a

text: short

text: time

text: scale

text: Myr)

text: when

text: the

text: Galactic

text: spiral

text: struc-

text: tential,

text: including

text: the

text: bar

text: and

text: spiral

text: arms.

text: The

text: aim

text: is

text: to

text: predict

text: the

text: ture

text: is

text: transient

text: (Fujii

text: Baba

text: This

text: radial

text: migration

text: pro-

text: present-day

text: phase

text: space

text: distribution

text: of

text: the

text: siblings

text: and

text: simulate

text: cess

text: can

text: also

text: be

text: efﬁcient

text: in

text: the

text: absence

text: of

text: transient

text: structure

text: if

text: the

text: astrometric

text: and

text: radial

text: velocity

text: data

text: collected

text: by

text: Gaia.

text: We

text: in-

text: the

text: resonances

text: due

text: the

text: bar

text: and

text: spiral

text: structure

text: overlap

text: (Minchev

text: clude

text: the

text: internal

text: N−body

text: processes

text: in

text: the

text: cluster

text: to

text: account

text: for

text: Famaey

text: Radial

text: migration

text: affects

text: the

text: orbits

text: of

text: open

text: clus-

text: the

text: disruption

text: time

text: scale.

text: We

text: use

text: a

text: full

text: stellar

text: mass

text: spectrum

text: and

text: ters

text: in

text: the

text: Galaxy,

text: increasing

text: or

text: decreasing

text: their

text: perigalacticon

text: a

text: parametrized

text: stellar

text: evolution

text: code

text: to

text: make

text: accurate

text: predictions

text: distance,

text: which

text: in

text: turn

text: inﬂuences

text: their

text: dissolution

text: times

text: (see

text: e.g.

text: of

text: how

text: the

text: solar

text: siblings

text: are

text: observed

text: by

text: Gaia.

text: To

text: this

text: end

text: we

text: also

text: J´ılkova´

text: et

text: al.

text: account

text: for

text: the

text: effects

text: of

text: extinction

text: and

text: reddening.

text: The

text: high

text: eccentricities

text: and

text: inclinations

text: observed

text: in

text: the

text: The

text: rest

text: of

text: this

text: paper

text: is

text: organized

text: as

text: follows.

text: In

text: Sect.

text: we

text: Edgeworth-Kuiper

text: belt

text: objects

text: together

text: with

text: the

text: discovery

text: of

text: de-

text: describe

text: the

text: simulations.

text: In

text: Sect.

text: we

text: explore

text: the

text: evolution

text: and

text: cay

text: products

text: of

text: and

text: other

text: radioactive

text: elements

text: in

text: the

text: mete-

text: disruption

text: of

text: the

text: Sun’s

text: birth

text: cluster

text: due

text: to

text: the

text: bar

text: and

text: spiral

text: arms

text: orite

text: fossil

text: record,

text: suggest

text: that

text: the

text: Sun

text: was

text: born

text: in

text: an

text: open

text: cluster

text: of

text: the

text: Galaxy.

text: In

text: Sect.

text: we

text: present

text: the

text: current

text: phase-space

text: dis-

text: Gyr

text: ago

text: (Portegies

text: Zwart

text: and

text: references

text: therein).

text: Iden-

text: tribution

text: of

text: solar

text: siblings

text: obtained

text: from

text: the

text: simulations.

text: In

text: Sect.

text: tifying

text: the

text: stars

text: that

text: were

text: formed

text: together

text: with

text: the

text: Sun

text: (the

text: solar

text: we

text: make

text: use

text: of

text: the

text: simulated

text: positions

text: and

text: motions

text: of

text: the

text: solar

text: siblings)

text: would

text: enable

text: the

text: determination

text: of

text: the

text: Galactic

text: birth

text: ra-

text: siblings

text: to

text: investigate

text: the

text: robustness

text: of

text: the

text: selection

text: criterion

text: pro-

text: dius

text: of

text: the

text: Sun

text: as

text: well

text: as

text: further

text: constrain

text: the

text: properties

text: of

text: its

text: posed

text: by

text: Brown

text: et

text: al.

text: to

text: the

text: uncertainties

text: in

text: the

text: present-day

text: birth

text: cluster

text: (Bland-Hawthorn

text: et

text: al.

text: Adams

text: The

text: birth

text: phase

text: space

text: distribution

text: of

text: the

text: solar

text: siblings.

text: An

text: updated

text: set

text: of

text: se-

text: radius

text: affects

text: the

text: evolution

text: of

text: the

text: solar

text: system,

text: and

text: in

text: particular

text: lection

text: criteria

text: based

text: on

text: parallax,

text: proper

text: motion

text: and

text: radial

text: velocity

text: the

text: Oort

text: cloud,

text: which

text: is

text: sensitive

text: to

text: the

text: Galactic

text: environment

text: the

text: information

text: is

text: presented.

text: In

text: Sect.

text: we

text: use

text: these

text: criteria

text: to

text: examine

text: Sun

text: passes

text: through

text: along

text: its

text: orbit

text: (e.g.

text: Portegies

text: Zwart

text: J´ılkova´

text: stars

text: that

text: were

text: previously

text: suggested

text: as

text: solar

text: siblings

text: candidates

text: and

text: further

text: discuss

text: our

text: results.

text: In

text: Sect.

text: we

text: summarize.

text: The

text: Sun’s

text: birth

text: cluster

text: will

text: undergo

text: all

text: the

text: disruptive

text: pro-

text: SIMULATION

text: SET-UP

text: cesses

text: described

text: above

text: and

text: thus

text: dissolve,

text: leading

text: to

text: the

text: spreading

text: out

text: of

text: its

text: stars

text: over

text: the

text: Galactic

text: disk.

text: The

text: subsequent

text: distribution

text: The

text: goals

text: of

text: the

text: simulations

text: of

text: the

text: Sun’s

text: birth

text: cluster

text: are

text: to

text: of

text: the

text: solar

text: siblings

text: was

text: studied

text: by

text: Portegies

text: Zwart

text: who

text: predict

text: the

text: present-day

text: phase

text: space

text: distribution

text: of

text: the

text: solar

text: sib-

text: evolved

text: the

text: Sun’s

text: birth

text: cluster

text: in

text: an

text: axisymmetric

text: model

text: for

text: the

text: lings

text: and

text: how

text: these

text: are

text: expected

text: to

text: appear

text: in

text: the

text: Gaia

text: catalogue.

text: Galactic

text: potential

text: and

text: concluded

text: that

text: tens

text: of

text: solar

text: siblings

text: might

text: In

text: particular

text: we

text: wish

text: to

text: account

text: for

text: the

text: uncertainties

text: in

text: the

text: initial

text: still

text: be

text: present

text: within

text: a

text: distance

text: of

text: pc

text: from

text: the

text: Sun.

text: Several

text: conditions

text: of

text: the

text: birth

text: cluster

text: and

text: the

text: parameters

text: of

text: the

text: Milky

text: Way

text: attempts

text: have

text: since

text: been

text: made

text: to

text: ﬁnd

text: solar

text: siblings

text: (e.g.

text: Brown

text: potential.

text: The

text: predictions

text: of

text: the

text: Gaia

text: observations

text: require

text: the

text: use

text: et

text: al.

text: Bobylev

text: et

text: al.

text: Liu

text: et

text: al.

text: however,

text: only

text: of

text: a

text: realistic

text: mass

text: spectrum

text: for

text: the

text: siblings,

text: and

text: accounting

text: for

text: four

text: plausible

text: candidates

text: have

text: been

text: identiﬁed

text: so

text: far

text: (Batista

text: Fer-

text: stellar

text: evolution

text: and

text: extinction

text: and

text: interstellar

text: reddening

text: effects.

text: nandes

text: Batista

text: et

text: al.

text: Ram´ırez

text: et

text: al.

text: This

text: small

text: We

text: thus

text: employ

text: the

text: following

text: elements

text: in

text: the

text: simulations:

text: number

text: of

text: observed

text: solar

text: siblings

text: might

text: be

text: a

text: consequence

text: of

text: the

text: lack

text: of

text: accurate

text: predictions

text: of

text: the

text: present-day

text: phase

text: space

text: distri-

text: Galactic

text: model

text: The

text: Milky

text: Way

text: potential

text: is

text: described

text: by

text: an

text: ana-

text: bution

text: of

text: solar

text: siblings

text: together

text: with

text: insufﬁciently

text: accurate

text: stellar

text: lytic

text: model

text: containing

text: a

text: disk,

text: bulge

text: and

text: halo,

text: as

text: well

text: as

text: a

text: bar

text: and

text: kinematic

text: data.

text: spiral

text: arms.

text: The

text: parameters

text: of

text: the

text: bar

text: and

text: spiral

text: arms

text: are

text: varied

text: in

text: Brown

text: et

text: al.

text: used

text: test

text: particle

text: simulations

text: to

text: predict

text: the

text: the

text: simulations

text: to

text: account

text: for

text: uncertainties

text: in

text: their

text: strengths

text: and

text: current

text: distribution

text: of

text: solar

text: siblings

text: in

text: the

text: Milky

text: Way.

text: They

text: con-

text: pattern

text: speeds

text: (Sect.

text: cluded

text: that

text: stars

text: with

text: parallaxes

text: mas

text: and

text: proper

text: motions

text: Cluster

text: model

text: The

text: Sun’s

text: birth

text: cluster

text: is

text: modelled

text: with

text: a

text: mass

text: (µ)

text: mas

text: yr

text: should

text: be

text: considered

text: solar

text: sibling

text: candidates.

text: spectrum

text: for

text: the

text: stars

text: and

text: we

text: account

text: for

text: the

text: gravitational

text: N-body

text: Their

text: conclusions

text: were

text: criticised

text: by

text: Mishurov

text: Acharova

text: effects

text: within

text: the

text: cluster

text: as

text: well

text: as

text: the

text: effect

text: of

text: the

text: Galaxy’s

text: grav-

text: who

text: pointed

text: out

text: that

text: in

text: more

text: realistic

text: Galactic

text: potentials

text: the

text: solar

text: itational

text: ﬁeld

text: on

text: the

text: cluster

text: stars.

text: The

text: use

text: of

text: N-body

text: models

text: for

text: siblings

text: are

text: expected

text: to

text: be

text: much

text: more

text: spread

text: out

text: over

text: the

text: Galactic

text: the

text: birth

text: cluster

text: is

text: motivated

text: by

text: the

text: desire

text: to

text: account

text: for

text: the

text: dis-

text: RAS,

text: MNRAS