text: 2 where H(= a˙ ) is the Hubble parameter, ρ is the energy a density and the eﬀective pressure p can be expressed as Now(cid:18)using equations (7) and (8), we obtain t(cid:19)he solution follows 1 B √ C √ 1+α p = p + Π, (4) ρ = − + − , (9) d 1 + A 3 ξ0 a3(1+α)(1+A 3 ξ0) where C is a constant of integration. Now using the which is the sum of the equilibrium pressure p and the − − de redshift formula z = 1 1 (choosing a = 1), equation bulk pressure Π = ξuγ , where uγ is the four velocity of a 0 the ﬂuid and ξ is the co;γeﬃcient of bulk viscosity and is a (9) can be r(cid:104)e-written in the form: (cid:105) function of energy density. The ﬁrst attempts at creating √ a viscosity theory of relativistic ﬂuids were executed by − − 1 Eckart [26] and Landau and Lifshitz [27] who considered ρ(z) = ρ0 As + (1 As)(1 + z)3(1+α)(1+A 3ξ0) 1+α , only a ﬁrst-order deviation from equilibrium. The bulk (10) viscous pressure Π is represented by the Eckart’s expres- √ sion which is proportional to the Hubble parameter H where ρ is the present value of the density and A = 0 s with proportionality factor identiﬁed as the bulk viscos- −√B with 0 < As < 1 and 1 + A > 3ξ0. ity coeﬃcient ξ which is deﬁned by ξ = ξ ρν, ξ and ν (1+A 3ξ0)C+B 0 0 Hence the Hubble parameter is obtained as are constants. For simplicity, choosing ν = 1, Π can be 2 written as [15] { − × H(z) = H [Ω A + (1 A ) √ 0 0 s s − Π = 3ξ H ρ. (5) 0 (cid:111) (cid:21) The Chaplygin gas is generally characterized by the −√ 1 12 equation of state (pressure is inversely proportional to (1 + z)3(1+α)(1+A 3ξ0) 1+α (11) − the energy density): p = A/ρ. This equation of state has an interesting connection with the d-branes which where Ω = 8πGρ0 is the present value of dimension are expressed via Nambu-Goto action [20]. It also enjoys 0 3H02 ∼ − − connections with the Newtonian hydrodynamical equa- density parameter and H0 72km s 1Mpc 1. The tions. Further the Eddington-Born-Infeld model can be equation of state parameter is given by seen as an aﬃne connection version for the Chaplygin gas approach [28]. The Chaplygin gas has been extensively p − B − 3√ξ H studied within the uniﬁed dark energy-dark matter mod- w = = A 0 , (12) ρ ρ1+α ρ els as well [29]. Therefore we choose the equation of state for modiﬁed Chaplygin gas as [30] while the adiabatic sound speed reads − B pd = Aρ ρα , (6) c2 = ∂p = A + αB − ξ0√H . (13) s ∂ρ ρ1+α 2 ρ where A, B and α are constants which are constrained by the astrophysical data (for recent constraints, see [31]). Note that the Chaplygin gas model is a dynamical dark The modiﬁed Chaplygin gas satisfactorily accommodates energy model as well and dark energy perturbations play an accelerating phase followed by a matter dominated crucial role in dynamical dark energy models as well [34]. phase of the universe. It is also consistent with the ob- B. Spherical top-hat collapse model servational studies dealing with the large scale structure [31]. Hence combining the equations (4) to (6), we get The spherical collapse (SC) which provides a way to glimpse into the nonlinear regime of perturbation theory √ was introduced ﬁrstly by Gunn and Gott [33]. The SC − B − p = Aρ 3ξ H ρ, (7) describes the evolution of a spherically symmetric pertur- ρα 0 bation embedded in a homogeneous background, which can be static, expanding or collapsing. One assumes a which is called viscous modiﬁed Chaplygin gas spherical ‘top-hat’ proﬁle for the perturbed region, i.e., (VMCG) [32]. Hence by choosing the above equation a spherically symmetric perturbation in some region of of state, we extend previous works dealing with viscous space with constant density. Following the assumption linear equation of state (B = 0) and viscous generalized of a top-hat proﬁle, namely the density perturbation is Chaplygin gas (A = 0) [15, 16]. It may also be termed as uniform throughout the collapse, so the evolution of per- a ‘viscous uniﬁed dark ﬂuid’. The conservation equation turbation is only time-dependent. is given by The perturbed quantities ρc and pc are related to their background counterparts by ρ = ρ + δρ and p = p + c c δp. Now the perturbed equation of state w is given by c ρ˙ + 3H(ρ + p) = 0 (8) [15, 16, 23] ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: H(=

text: a˙

text: is

text: the

text: Hubble

text: parameter,

text: ρ

text: is

text: the

text: energy

text: a

text: density

text: and

text: the

text: eﬀective

text: pressure

text: p

text: can

text: be

text: expressed

text: as

text: equations

text: and

text: we

text: obtain

text: solution

text: follows

text: B

text: C

text: p

text: p

text: Π,

text: ρ

text: d

text: A

text: ξ0

text: where

text: C

text: is

text: a

text: constant

text: of

text: integration.

text: Now

text: using

text: the

text: which

text: is

text: the

text: sum

text: of

text: the

text: equilibrium

text: pressure

text: p

text: and

text: the

text: de

text: redshift

text: formula

text: z

text: (choosing

text: a

text: equation

text: bulk

text: pressure

text: Π

text: ξuγ

text: where

text: uγ

text: is

text: the

text: four

text: velocity

text: of

text: a

text: the

text: ﬂuid

text: and

text: ξ

text: is

text: the

text: co;γeﬃcient

text: of

text: bulk

text: viscosity

text: and

text: is

text: a

text: can

text: be

text: in

text: the

text: form:

text: function

text: of

text: energy

text: density.

text: The

text: ﬁrst

text: attempts

text: at

text: creating

text: a

text: viscosity

text: theory

text: of

text: relativistic

text: ﬂuids

text: were

text: executed

text: by

text: Eckart

text: and

text: Landau

text: and

text: Lifshitz

text: who

text: considered

text: ρ(z)

text: ρ0

text: As

text: only

text: a

text: ﬁrst-order

text: deviation

text: from

text: equilibrium.

text: The

text: bulk

text: viscous

text: pressure

text: Π

text: is

text: represented

text: by

text: the

text: Eckart’s

text: expres-

text: sion

text: which

text: is

text: proportional

text: to

text: the

text: Hubble

text: parameter

text: H

text: where

text: ρ

text: is

text: the

text: present

text: value

text: of

text: the

text: density

text: and

text: A

text: s

text: with

text: proportionality

text: factor

text: identiﬁed

text: as

text: the

text: bulk

text: viscos-

text: −√B

text: with

text: As

text: and

text: A

text: ity

text: coeﬃcient

text: ξ

text: which

text: is

text: deﬁned

text: by

text: ξ

text: ξ

text: ρν,

text: ξ

text: and

text: ν

text: Hence

text: the

text: Hubble

text: parameter

text: is

text: obtained

text: as

text: are

text: constants.

text: For

text: simplicity,

text: choosing

text: ν

text: Π

text: can

text: be

text: written

text: as

text: H(z)

text: H

text: [Ω

text: A

text: A

text: s

text: s

text: Π

text: 3ξ

text: H

text: ρ.

text: The

text: Chaplygin

text: gas

text: is

text: generally

text: characterized

text: by

text: the

text: equation

text: of

text: state

text: (pressure

text: is

text: inversely

text: proportional

text: to

text: the

text: energy

text: density):

text: p

text: A/ρ.

text: This

text: equation

text: of

text: state

text: has

text: an

text: interesting

text: connection

text: with

text: the

text: d-branes

text: which

text: where

text: Ω

text: is

text: the

text: present

text: value

text: of

text: dimension

text: are

text: expressed

text: via

text: Nambu-Goto

text: action

text: It

text: also

text: enjoys

text: connections

text: with

text: the

text: Newtonian

text: hydrodynamical

text: equa-

text: density

text: parameter

text: and

text: s

text: The

text: tions.

text: Further

text: the

text: Eddington-Born-Infeld

text: model

text: can

text: be

text: equation

text: of

text: state

text: parameter

text: is

text: given

text: by

text: seen

text: as

text: an

text: aﬃne

text: connection

text: version

text: for

text: the

text: Chaplygin

text: gas

text: approach

text: The

text: Chaplygin

text: gas

text: has

text: been

text: extensively

text: p

text: B

text: 3√ξ

text: H

text: studied

text: within

text: the

text: uniﬁed

text: dark

text: energy-dark

text: matter

text: mod-

text: w

text: A

text: ρ

text: ρ1+α

text: ρ

text: els

text: as

text: well

text: Therefore

text: we

text: choose

text: the

text: equation

text: of

text: state

text: for

text: modiﬁed

text: Chaplygin

text: gas

text: as

text: while

text: the

text: adiabatic

text: sound

text: speed

text: reads

text: B

text: pd

text: Aρ

text: ρα

text: ∂p

text: A

text: αB

text: ξ0√H

text: s

text: ∂ρ

text: ρ1+α

text: ρ

text: where

text: A,

text: B

text: and

text: α

text: are

text: constants

text: which

text: are

text: constrained

text: by

text: the

text: astrophysical

text: data

text: (for

text: recent

text: constraints,

text: see

text: Note

text: that

text: the

text: Chaplygin

text: gas

text: model

text: is

text: a

text: dynamical

text: dark

text: The

text: modiﬁed

text: Chaplygin

text: gas

text: satisfactorily

text: accommodates

text: energy

text: model

text: as

text: well

text: and

text: dark

text: energy

text: perturbations

text: play

text: an

text: accelerating

text: phase

text: followed

text: by

text: a

text: matter

text: dominated

text: crucial

text: role

text: in

text: dynamical

text: dark

text: energy

text: models

text: as

text: well

text: phase

text: of

text: the

text: universe.

text: It

text: is

text: also

text: consistent

text: with

text: the

text: ob-

text: B.

text: Spherical

text: top-hat

text: collapse

text: model

text: servational

text: studies

text: dealing

text: with

text: the

text: large

text: scale

text: structure

text: Hence

text: combining

text: the

text: equations

text: to

text: we

text: get

text: The

text: spherical

text: collapse

text: (SC)

text: which

text: provides

text: a

text: way

text: to

text: glimpse

text: into

text: the

text: nonlinear

text: regime

text: of

text: perturbation

text: theory

text: was

text: introduced

text: ﬁrstly

text: by

text: Gunn

text: and

text: Gott

text: The

text: SC

text: B

text: p

text: Aρ

text: 3ξ

text: H

text: ρ,

text: describes

text: the

text: evolution

text: of

text: a

text: spherically

text: symmetric

text: pertur-

text: ρα

text: bation

text: embedded

text: in

text: a

text: homogeneous

text: background,

text: which

text: can

text: be

text: static,

text: expanding

text: or

text: collapsing.

text: One

text: assumes

text: a

text: which

text: is

text: called

text: viscous

text: modiﬁed

text: Chaplygin

text: gas

text: spherical

text: ‘top-hat’

text: proﬁle

text: for

text: the

text: perturbed

text: region,

text: i.e.,

text: (VMCG)

text: Hence

text: by

text: choosing

text: the

text: above

text: equation

text: a

text: spherically

text: symmetric

text: perturbation

text: in

text: some

text: region

text: of

text: of

text: state,

text: we

text: extend

text: previous

text: works

text: dealing

text: with

text: viscous

text: space

text: with

text: constant

text: density.

text: Following

text: the

text: assumption

text: linear

text: equation

text: of

text: state

text: (B

text: and

text: viscous

text: generalized

text: of

text: a

text: top-hat

text: proﬁle,

text: namely

text: the

text: density

text: perturbation

text: is

text: Chaplygin

text: gas

text: (A

text: It

text: may

text: also

text: be

text: termed

text: as

text: uniform

text: throughout

text: the

text: collapse,

text: so

text: the

text: evolution

text: of

text: per-

text: a

text: ‘viscous

text: uniﬁed

text: dark

text: ﬂuid’.

text: The

text: conservation

text: equation

text: turbation

text: is

text: only

text: time-dependent.

text: is

text: given

text: by

text: The

text: perturbed

text: quantities

text: ρc

text: and

text: pc

text: are

text: related

text: to

text: their

text: background

text: counterparts

text: by

text: ρ

text: ρ

text: δρ

text: and

text: p

text: p

text: c

text: c

text: δp.

text: Now

text: the

text: perturbed

text: equation

text: of

text: state

text: w

text: is

text: given

text: by

text: c

text: ρ˙

text: p)

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text: (cid:48) solution, at the throat, b(r ) = r the condition b (r ) < 1 is imposed. In addition, to have a traversable 0 0 0 wormhole there must be no event horizon which implies Φ(r) must be ﬁnite everywhere. The Einstein’s ﬁeld equations for the perfect ﬂuid with the energy-momentum tensor given by Ti = − − j diag[ ρ(t, r), τ(t, r), p(t, r), p(t, r), p (t, r)], in the case of zero radial tidal force wormhole (Φ(r) = 0), y are given by (cid:18) (cid:19) (cid:48) − 3R˙ 2 3R˙ B˙ 1 b 8πρ(t, r) = Tt = + + 3k + , (3) t R2 RB R2 r2 (cid:18) (cid:19) 2R¨ R˙ 2 2R˙ B˙ B¨ 1 b 8πτ(t, r) = Tr = + + + + k + , (4) r R R2 RB B R2 r3 (cid:26) (cid:18) (cid:19)(cid:27) (cid:48) − 2R¨ R˙ 2 2R˙ B˙ B¨ 1 − b b 8πp(t, r) = Tθ = + + + + k + , (5) θ R R2 RB B R2 2r3 2r2 (cid:26) (cid:18) (cid:19)(cid:27) (cid:48) − 3R¨ 3R˙ 2 1 b 8πp (t, r) = Ty = + + 3k + . (6) y y R R2 R2 r2 Since, ρ, τ, p and p depend on both r and t, to solve the Einstein ﬁeld equations, we can choose y the following Ansatz R2(t)ρ(t, r) = R2(t)ρ (t) + ρ (r), c w R2(t)τ(t, r) = R2(t)τ (t) + τ (r), c w (7) R2(t)p(t, r) = R2(t)p (t) + p (r), c w R2(t)p (t, r) = R2(t)p (t) + p (r). y yc yw The subscript c refers to the cosmological part and w refers to the wormhole part. Indeed, this is due to the fact that we simply assume the wormhole is small compared to the universe. With the above Ansatz, the Einstein ﬁeld equ(cid:20)ations and the c(cid:21)onservation laws are separated as (cid:48) − 3 RB˙ R˙ − b R2ρ (t) R˙ 2 + + k = ρ (r) + = l, (8) c 8π B w 8πr2 (cid:20) (cid:21) − 1 2R˙ B˙ R B¨R2 − b R2τ (t) 2RR¨ + R˙ 2 + + + k = τ (r) + = m, (9) c 8π B B w 8πr3 (cid:20) (cid:21) (cid:20) (cid:21) − (cid:48) 1 2R˙ B˙ R B¨R2 − − 1 b b R2p (t) + 2RR¨ + R˙ 2 + + + k = p (r) + = α, (10) c 8π B B w 8π 2r3 2r2 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: solution,

text: at

text: the

text: throat,

text: b(r

text: r

text: the

text: condition

text: b

text: (r

text: is

text: imposed.

text: In

text: addition,

text: to

text: have

text: a

text: traversable

text: wormhole

text: there

text: must

text: be

text: no

text: event

text: horizon

text: which

text: implies

text: Φ(r)

text: must

text: be

text: ﬁnite

text: everywhere.

text: The

text: Einstein’s

text: ﬁeld

text: equations

text: for

text: the

text: perfect

text: ﬂuid

text: with

text: the

text: energy-momentum

text: tensor

text: given

text: by

text: Ti

text: j

text: diag[

text: ρ(t,

text: r),

text: τ(t,

text: r),

text: p(t,

text: r),

text: p(t,

text: r),

text: p

text: (t,

text: r)],

text: in

text: the

text: case

text: of

text: zero

text: radial

text: tidal

text: force

text: wormhole

text: (Φ(r)

text: y

text: are

text: given

text: by

label: equation

text: B˙

text: b

text: r)

text: Tt

text: t

text: RB

text: R˙

text: B˙

text: B¨

text: b

text: r)

text: Tr

text: k

text: r

text: R

text: RB

text: B

text: R˙

text: B˙

text: B¨

text: b

text: b

text: r)

text: Tθ

text: k

text: θ

text: R

text: RB

text: B

text: b

text: (t,

text: r)

text: Ty

text: y

text: y

text: R

text: Since,

text: ρ,

text: τ,

text: p

text: and

text: p

text: depend

text: on

text: both

text: r

text: and

text: t,

text: to

text: solve

text: the

text: Einstein

text: ﬁeld

text: equations,

text: we

text: can

text: choose

text: y

text: the

text: following

text: Ansatz

text: r)

text: (t)

text: ρ

text: (r),

text: c

text: w

text: r)

text: (t)

text: τ

text: (r),

text: c

text: w

text: r)

text: (t)

text: p

text: (r),

text: c

text: w

text: (t,

text: r)

text: (t)

text: p

text: (r).

text: y

text: yc

text: yw

text: The

text: subscript

text: c

text: refers

text: to

text: the

text: cosmological

text: part

text: and

text: w

text: refers

text: to

text: the

text: wormhole

text: part.

text: Indeed,

text: this

text: is

text: due

text: to

text: the

text: fact

text: that

text: we

text: simply

text: assume

text: the

text: wormhole

text: is

text: small

text: compared

text: to

text: the

text: universe.

text: With

text: the

text: above

text: Ansatz,

text: the

text: Einstein

text: ﬁeld

text: and

text: the

text: laws

text: are

text: separated

text: as

text: RB˙

text: R˙

text: b

text: (t)

text: R˙

text: k

text: ρ

text: (r)

text: l,

text: c

text: 8π

text: B

text: w

text: B˙

text: R

text: b

text: (t)

text: R˙

text: k

text: τ

text: (r)

text: m,

text: c

text: 8π

text: B

text: B

text: w

text: B˙

text: R

text: b

text: b

text: (t)

text: R˙

text: k

text: p

text: (r)

text: α,

text: c

text: 8π

text: B

text: B

text: w

text: 8π

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text: (cid:8) (cid:2) (cid:3) (cid:9) − − − ≡ κ2ρ (n 1)r P + r (Q + x)2 1 + x2 + Ω 1 m Λ Ω = , (4.12) m 3H2 Q2(n + r) ≡ κ2ρ −(cid:2) ∆ (cid:3) DE 2 Ω = , (4.13) DE 3H2 Q2(n + r) − ≡ − − H˙ (cid:2) r 1 (Q + x)2 (cid:0) (cid:1)(cid:3) q 1 = + 1, (4.14) H2 Q2 − − (n + r) Q2(2(cid:8)r 1) +(cid:2) 4Qrx + 2r x(cid:3)2 1 (cid:9) w = DE 3∆ − 2 − − w (n 1)r P + r (Q +(cid:2) x)2 1 + x(cid:3)2 + Ω 1 m Λ + , (4.15) ∆ − 2 − ≡ − − 2H˙ 2q 1 2r (Q + x)2 1 1 wtot 1 (cid:2) 3H2 = 3 = (cid:0) 3Q(cid:1)2 (cid:3) + 3, (4.16) − − − − where ∆ = (n 1)r P + 2Qrx + (r + 1) x2 1 + Ω + Q2(r + 1) [n(r 1) r]. 2 Λ The scenario of mimetic F (R) gravity with the power-law form (4.1), i.e. the system × (4.10), admits 14 2 + 1 = 29 isolated physical critical points and three curves of critical points (one of them, namely T , exist only for a speciﬁc value of the parameter w ), which 14 m are displayed in Table 3 along with their existence and stability conditions. The details of × the analysis and the calculation of the various eigenvalues of the 5 5 perturbation matrix are presented in Appendix . Furthermore, for each critical point we calculate the values of various observables, such as the density parameters, the deceleration parameter and the dark- energy and total equation-of-state parameters, given by (4.12)-(4.16), and we summarize the − results in Table 4. Observe that for some speciﬁc points having either Q = 0 or r = n the expressions (4.12)-(4.16) are not well deﬁned (NWD), since the involved limits depend on the limit order. 5 Dynamical analysis for general F (R) forms As we saw in the previous sections, in order to perform the stability analysis one needs to choose a speciﬁc F (R) ansatz. However, this is restricting since for diﬀerent F (R) forms one must repeat the whole analysis from the start. Hence, in the present section, for completeness, we extend the usual procedure in order to be able to perform the analysis for arbitrary F (R) forms. Following the generalized method of [100], the idea is to suitably parametrize an arbitrary F (R) function and perform the dynamical analysis in general. Therefore, after this general analysis one can just substitute the speciﬁc F (R) form in the obtained results, without the need to repeat the whole dynamical elaboration from the beginning. In order to parametrize the arbitrary F (R) functions, we introduce the auxiliary vari- ables [16, 100] ≡ −RF R r F ≡ RF RR m . (5.1) F R Furthermore, we introduce the normalization factor – 13 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: κ2ρ

text: (n

text: P

text: r

text: (Q

text: Ω

text: m

text: Λ

text: Ω

text: m

text: r)

text: κ2ρ

text: DE

text: Ω

text: DE

text: r)

text: H˙

text: r

text: (Q

text: q

text: (n

text: r)

text: w

text: DE

text: 3∆

text: w

text: (n

text: P

text: r

text: (Q

text: Ω

text: m

text: Λ

text: (Q

text: wtot

text: where

text: (n

text: P

text: (r

text: Ω

text: [n(r

text: r].

text: Λ

text: The

text: scenario

text: of

text: mimetic

text: F

text: (R)

text: gravity

text: with

text: the

text: power-law

text: form

text: i.e.

text: the

text: system

text: admits

text: isolated

text: physical

text: critical

text: points

text: and

text: three

text: curves

text: of

text: critical

text: points

text: (one

text: of

text: them,

text: namely

text: T

text: exist

text: only

text: for

text: a

text: speciﬁc

text: value

text: of

text: the

text: parameter

text: w

text: which

text: m

text: are

text: displayed

text: in

text: Table

text: along

text: with

text: their

text: existence

text: and

text: stability

text: conditions.

text: The

text: details

text: of

text: the

text: analysis

text: and

text: the

text: calculation

text: of

text: the

text: various

text: eigenvalues

text: of

text: the

text: perturbation

text: matrix

text: are

text: presented

text: in

text: Appendix

text: Furthermore,

text: for

text: each

text: critical

text: point

text: we

text: calculate

text: the

text: values

text: of

text: various

text: observables,

text: such

text: as

text: the

text: density

text: parameters,

text: the

text: deceleration

text: parameter

text: and

text: the

text: dark-

text: energy

text: and

text: total

text: equation-of-state

text: parameters,

text: given

text: by

text: and

text: we

text: summarize

text: the

text: results

text: in

text: Table

text: Observe

text: that

text: for

text: some

text: speciﬁc

text: points

text: having

text: either

text: Q

text: or

text: r

text: n

text: the

text: expressions

text: are

text: not

text: well

text: deﬁned

text: (NWD),

text: since

text: the

text: involved

text: limits

text: depend

text: on

text: the

text: limit

text: order.

text: Dynamical

text: analysis

text: for

text: general

text: F

text: (R)

text: forms

text: As

text: we

text: saw

text: in

text: the

text: previous

text: sections,

text: in

text: order

text: to

text: perform

text: the

text: stability

text: analysis

text: one

text: needs

text: to

text: choose

text: a

text: speciﬁc

text: F

text: (R)

text: ansatz.

text: However,

text: this

text: is

text: restricting

text: since

text: for

text: diﬀerent

text: F

text: (R)

text: forms

text: one

text: must

text: repeat

text: the

text: whole

text: analysis

text: from

text: the

text: start.

text: Hence,

text: in

text: the

text: present

text: section,

text: for

text: completeness,

text: we

text: extend

text: the

text: usual

text: procedure

text: in

text: order

text: to

text: be

text: able

text: to

text: perform

text: the

text: analysis

text: for

text: arbitrary

text: F

text: (R)

text: forms.

text: Following

text: the

text: generalized

text: method

text: of

text: the

text: idea

text: is

text: to

text: suitably

text: parametrize

text: an

text: arbitrary

text: F

text: (R)

text: function

text: and

text: perform

text: the

text: dynamical

text: analysis

text: in

text: general.

text: Therefore,

text: after

text: this

text: general

text: analysis

text: one

text: can

text: just

text: substitute

text: the

text: speciﬁc

text: F

text: (R)

text: form

text: in

text: the

text: obtained

text: results,

text: without

text: the

text: need

text: to

text: repeat

text: the

text: whole

text: dynamical

text: elaboration

text: from

text: the

text: beginning.

text: In

text: order

text: to

text: parametrize

text: the

text: arbitrary

text: F

text: (R)

text: functions,

text: we

text: introduce

text: the

text: auxiliary

text: vari-

text: ables

text: −RF

text: R

text: r

text: F

text: RF

text: RR

text: m

text: F

text: R

text: Furthermore,

text: we

text: introduce

text: the

text: normalization

text: factor

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: Name Ω Ω q w w m DE DE tot T (cid:15) NWD NWD NWD NWD NWD 1 T (cid:15) NWD NWD NWD w NWD 2 m T (cid:15) 0 1 1 1 1 3 3 3 T (cid:15) NWD NWD 1 NWD 1 4 3 T (cid:15) 0 1 1 1 1 5 3 3 T (cid:15) NWD NWD 1 NWD 1 6 3 − − T (cid:15) NWD NWD 3 1 NWD 1 + 1 7 2n n − − T (cid:15) NWD NWD −1 + −3 1 NWD (7 6−n)n+1 8 n 1 1 2n 6n2 9n+3 T (cid:15) 0 1 1 1 1 9 3 3 T (cid:15) NWD NWD 1 w 1 10 m 3 − − T (cid:15) NWD NWD 2n+3(wm+1) w 1 + wm+1 11 2n m n T (cid:15) NWD NWD NWD w NWD 12 m − − T (cid:15) NWD NWD 1 NWD 1 13 − T rc14(1 n) n(rc14+1) 1 1 1 14 n+rc14 n+rc14 3 3 T NWD NWD NWD NWD NWD 15 T 0 1 1 1 1 16 3 3 − − − T 0 1 1 1 1 17 T NWD NWD 1 NWD 1 18 3 Table 4. The real critical points and curves of critical points of the system (4.10) of mimetic F(R) gravity with the power-law form (4.1), and the corresponding values of the matter and dark energy density parameters, of the deceleration parameter, and of the dark-energy and total equation- ± of-state parameters, calculated through (4.12)-(4.16). We use the notation (cid:15) = 1, where (cid:15) = +1 − corresponds to expanding universe and (cid:15) = 1 to contracting one, with the stability conditions outside − parentheses corresponding to (cid:15) = +1 while those inside parentheses to (cid:15) = 1. NWD stands for “Not well-deﬁned”. while the deﬁnition of D gives rise to the additional constraint (Q + x)2 + y = 1. (5.5) Therefore, we can use the above two constraints in order to eliminate two variables, which for convenience are chosen to be y and z, through – 15 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Name

label: table

text: Ω

text: Ω

text: q

text: w

text: w

text: m

text: DE

text: DE

text: tot

text: T

text: NWD

text: NWD

text: NWD

text: NWD

text: NWD

text: T

text: NWD

text: NWD

text: NWD

text: w

text: NWD

text: m

text: T

text: T

text: NWD

text: NWD

text: NWD

text: T

text: T

text: NWD

text: NWD

text: NWD

text: T

text: NWD

text: NWD

text: NWD

text: n

text: T

text: NWD

text: NWD

text: −1

text: −3

text: NWD

text: n

text: T

text: T

text: NWD

text: NWD

text: w

text: m

text: T

text: NWD

text: NWD

text: w

text: m

text: n

text: T

text: NWD

text: NWD

text: NWD

text: w

text: NWD

text: m

text: T

text: NWD

text: NWD

text: NWD

text: T

text: n)

text: T

text: NWD

text: NWD

text: NWD

text: NWD

text: NWD

text: T

text: T

text: T

text: NWD

text: NWD

text: NWD

text: Table

text: The

text: real

text: critical

text: points

text: and

text: curves

text: of

text: critical

text: points

text: of

text: the

text: system

text: of

text: mimetic

text: F(R)

text: gravity

text: with

text: the

text: power-law

text: form

text: and

text: the

text: corresponding

text: values

text: of

text: the

text: matter

text: and

text: dark

text: energy

text: density

text: parameters,

text: of

text: the

text: deceleration

text: parameter,

text: and

text: of

text: the

text: dark-energy

text: and

text: total

text: equation-

text: of-state

text: parameters,

text: calculated

text: through

text: We

text: use

text: the

text: notation

text: where

text: corresponds

text: to

text: expanding

text: universe

text: and

text: to

text: contracting

text: one,

text: with

text: the

text: stability

text: conditions

text: outside

text: parentheses

text: corresponding

text: to

text: while

text: those

text: inside

text: parentheses

text: to

text: NWD

text: stands

text: for

text: “Not

text: well-deﬁned”.

text: while

text: the

text: deﬁnition

text: of

text: D

text: gives

text: rise

text: to

text: the

text: additional

text: constraint

text: (Q

text: y

text: Therefore,

text: we

text: can

text: use

text: the

text: above

text: two

text: constraints

text: in

text: order

text: to

text: eliminate

text: two

text: variables,

text: which

text: for

text: convenience

text: are

text: chosen

text: to

text: be

text: y

text: and

text: z,

text: through

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text: 6 BODAGHEE ET AL. Figure 5. Infrared K -band spectra of two stars (labeled “2” and “3” s in Fig.4) in the YMC VVV CL077 from observations gathered with NOAO/OSIRIS. ∼ × − − − ual ﬂux of 7 10 14 erg cm 2 s 1 can be attributed to the diffuse cluster emission. For a distance of 11 kpc (assumed), − this translates to an X-ray luminosity of 1034 erg s 1. Flux − ± densities reported for IRAS 16311 4726 are 20.1 0.6 Jy at ± 12 µm and 153 5 Jy at 25 µm (Helou & Walker 1988). Ex±- trapolating a power law with a photon in×dex−of Γ = 2.8 0.1 l×eads−to an X-r−ay −ﬂux density of 1.3 10 10 Jy at 1 keV (3.0 10 15 erg cm 2 s 1 over 0.5–10 keV). This is an order of magnitude smaller than the lower 90%-conﬁdence boundary × − of the intrinsic ﬂux density (3 10 9 Jy at 1 keV). In order to match the X-ray ﬂux, the photon index should be no steeper than 2.5, which is consistent with what we found in Section . Figure 4. Image of VVV CL077 in the infrared Ks-band from A∼ simp×le estimatio−n of the cluster mass density yields NOAO/NEWFIRM. Circles denote the positional uncertainty of the X-ray ρ (1–4) 103 M(cid:12) pc 3. This assumes that all 21 visually- point sources described in this work (Table2), and the visual extent of the identiﬁed members of the cluster are similar to the sources YMC according to Borissova et al. (2011). Infrared objects targeted for we identiﬁed as O type stars with masses in the range of follow-up spectroscopy are numbered 1–5. 10–50 M(cid:12). While the mass density is consistent with those Carina Nebula (Muno et al. 2006a; Townsley et al. 2011a,c). of other YMCs,∼the cluster mass implied by this ﬁrst-order In addition to the combined emission from unresolved stars, approximation ( 103 M(cid:12)) is an order of magnitude below the the diffuse component could originate from putative collisions inferred masses of other YMCs (Figer 2008). This indicates between the winds of massive stars, although the temperature a large population of unresolved stars hiding in and around and absorption in VVV CL077 are higher than expected for the cluster which is not surprising given the extinction in this such processes (e.g., Stevens & Hartwell 2003). direction. − ± Two of the stars VVV CL077 display infrared spectral sig- From the observed infrared excess E(H Ks) = 1.5 0±.2, we natures typical of late-type O stars (Hanson et al. 1996). estimate the extinction in the optical band as AV−= 15 2 as- There is good overall agreement between the outlines of suming a near-infrared extinction law with α = 2.1 (Stead & the emission as seen in the X-rays and infrared, and this Hoare 2009). Applying the relation of Gu¨ver & O¨ zel (2009) × − mwhueltrie-wtahveelXen-grathy caonndsiisntefrnacryedcoenmtiinsuseiosninctoointhceidreadsipoatbiaalnldy ythieeldcsonatcriobluutmionndfernosmitymooflNecHul=ar(3h.y3d+−11r..o10g)en1i0s22inccmlud2.edW, htehne with a local maximum in the 843 MHz continuum map from total column density of the interstellar medium towards the × − × − MGPS-2 (Murphy et al. 2007). The coincidence of the multi- Norma region ranges from 4.9 1022 cm 2 to 9.4 1022 cm 2 wavelength emission from VVV CL077 could indicate a com- (Rahoui et al. 2014). This is consistent with our X-ray-derived × − mon physical origin, however, the observed X-ray ﬂux of the value of (6.4+−3.7) 1022 cm 2. diffuse component is larger than expected from an extrapola- Plasma tem2p.9eratures of around 1–2 keV are found for the tion of the infrared ﬂux density (see below). X-ray emitting point sources in the cluster which is higher − The nearest catalogued ultra-compact H II region is IRAS 16311 t4h7a2n6average for massive stars (e.g., Rauw et al. 2014). The (cid:48)(cid:48) (Helou & Walker 1988; Bronfman et al. 1996) which is 6 X-ray source located in the outskirts of the cluster (CXOU − from the infrared position of VVV CL077. It has a veloc- J163448.5 473229) has a temperature that is statistically − ity of 121 km s 1 with respect to the local standard of rest, compatible (at 90% conﬁdence) with 0.7 keV which is a more which translates to a source distance of approximately 11 kpc typical value. As we move towards the center of the cluster, (e.g., Brand & Blitz 1993). There is also a catalogued H II we ﬁnd that the X-ray derived temperatures of these stars (cid:48) region 7 away from VVV CL077 with a listed distance of are higher, possibly due to increasing column density (which ± 10.9 0.2 kpc (Russeil 2003). If these regions are related, then preferentially absorbs low-temperature photons). With its − we can assign a distance of around 11 kpc to VVV CL077. proximity to IRAS 16311 4726, this could indicate that the This would make it one of the most distant Galactic YMCs stars in the core of VVV CL077 are still embedded within known, comparable to Mercer 81 (Davies et al. 2012), at a their dusty natal cocoons; similar to the hot, young stars of location consistent with the far Norma Arm. W49A (Conti & Blum 2002), a cluster that resides within Taking the visual extent of the bulk of the stellar population an ultra-compact H II region. However, we can not exclude (cid:48)(cid:48) in the infrared as the angular scale (15 : Borissova et al. 2011) that the unusual spectral values are the result of poor photon gives a cluster diameter of 0.8 pc for an assumed distance of statistics. 11 kpc. This value is consistent with the intrinsic sizes of Based on the IR spectra, we narrowed the range of spectral other Galactic YMCs (Figer 2008, and references therein). types to O5–O9 giants for two of the stars in the cluster. We − − Corrected for absorption, the diff±use e×miss−ion from −VV−V derived the distance modulus m M = 5 log(d)+A −5 using the CL077 has an intrinsic ﬂux of (9 3) 10 14 erg cm 2 s 1 observed J-band magnitude of CXOU J163448.0 473246 in − (0.5–10-keV). This is more than an order of magnitude above Table 4, the absolute J-band magnitude of 4.6 for an O giant the ﬂux expected from the Galactic Ridge diffuse emission (Martins & Plez 2006), a±nd AJ = 0.28AV. This gives a dis- (Ebisawa et al. 2001, 2005). If we assume, nevertheless, that tance to the object of 10 3 kpc, consistent with the distance − the Ridge emission accounts for 20% of the ﬂux from VVV assumed based on its possible link with IRAS 16311 4726. CL077, as it does for Wd 1 (Muno et al. 2006a), then the resid- In massive stars, the ratio between the X-ray and bolo- ##LTFigure## ##LTFigure##

text: BODAGHEE

text: ET

text: AL.

text: Figure

text: Infrared

label: caption

text: K

text: -band

text: spectra

text: of

text: two

text: stars

text: (labeled

text: “2”

text: and

text: “3”

text: s

text: in

text: in

text: the

text: YMC

text: VVV

text: from

text: observations

text: gathered

text: with

text: NOAO/OSIRIS.

font: WYQTVI+txsy

text: ual

text: ﬂux

text: of

text: erg

text: cm

text: s

text: can

text: be

text: attributed

text: to

text: the

text: diffuse

text: cluster

text: emission.

text: For

text: a

text: distance

text: of

text: kpc

text: (assumed),

text: this

text: translates

text: to

text: an

text: X-ray

text: luminosity

text: of

text: erg

text: s

text: Flux

text: densities

text: reported

text: for

text: IRAS

text: are

text: Jy

text: at

text: µm

text: and

text: Jy

text: at

text: µm

text: (Helou

text: Walker

text: Ex±-

text: trapolating

text: a

text: power

text: law

text: with

text: a

text: photon

text: in×dex−of

text: Γ

text: l×eads−to

text: an

text: X-r−ay

text: −ﬂux

text: density

text: of

text: Jy

text: at

text: keV

text: erg

text: cm

text: s

text: over

text: keV).

text: This

text: is

text: an

text: order

text: of

text: magnitude

text: smaller

text: than

text: the

text: lower

text: boundary

text: of

text: the

text: intrinsic

text: ﬂux

text: density

text: Jy

text: at

text: keV).

text: In

text: order

text: to

text: match

text: the

text: X-ray

text: ﬂux,

text: the

text: photon

text: index

text: should

text: be

text: no

text: steeper

text: than

text: which

text: is

text: consistent

text: with

text: what

text: we

text: found

text: in

text: Section

text: Figure

text: Image

text: of

text: VVV

text: in

text: the

text: infrared

text: Ks-band

text: from

text: A∼

text: simp×le

text: estimatio−n

text: of

text: the

text: cluster

text: mass

text: density

text: yields

text: NOAO/NEWFIRM.

text: Circles

text: denote

text: the

text: positional

text: uncertainty

text: of

text: the

text: X-ray

text: ρ

text: pc

text: This

text: assumes

text: that

text: all

text: visually-

text: point

text: sources

text: described

text: in

text: this

text: work

text: and

text: the

text: visual

text: extent

text: of

text: the

text: identiﬁed

text: members

text: of

text: the

text: cluster

text: are

text: similar

text: to

text: the

text: sources

text: YMC

text: according

text: to

text: Borissova

text: et

text: al.

text: Infrared

text: objects

text: targeted

text: for

text: we

text: identiﬁed

text: as

text: O

text: type

text: stars

text: with

text: masses

text: in

text: the

text: range

text: of

text: follow-up

text: spectroscopy

text: are

text: numbered

text: While

text: the

text: mass

text: density

text: is

text: consistent

text: with

text: those

text: Carina

text: Nebula

text: (Muno

text: et

text: al.

text: Townsley

text: et

text: al.

text: of

text: other

text: YMCs,∼the

text: cluster

text: mass

text: implied

text: by

text: this

text: ﬁrst-order

text: In

text: addition

text: to

text: the

text: combined

text: emission

text: from

text: unresolved

text: stars,

text: approximation

text: is

text: an

text: order

text: of

text: magnitude

text: below

text: the

text: the

text: diffuse

text: component

text: could

text: originate

text: from

text: putative

text: collisions

text: inferred

text: masses

text: of

text: other

text: YMCs

text: (Figer

text: This

text: indicates

text: between

text: the

text: winds

text: of

text: massive

text: stars,

text: although

text: the

text: temperature

text: a

text: large

text: population

text: of

text: unresolved

text: stars

text: hiding

text: in

text: and

text: around

text: and

text: absorption

text: in

text: VVV

text: are

text: higher

text: than

text: expected

text: for

text: the

text: cluster

text: which

text: is

text: not

text: surprising

text: given

text: the

text: extinction

text: in

text: this

text: such

text: processes

text: (e.g.,

text: Stevens

text: Hartwell

text: direction.

text: Two

text: of

text: the

text: stars

text: VVV

text: display

text: infrared

text: spectral

text: sig-

text: From

text: the

text: observed

text: infrared

text: excess

text: E(H

text: Ks)

text: we

text: natures

text: typical

text: of

text: late-type

text: O

text: stars

text: (Hanson

text: et

text: al.

text: estimate

text: the

text: extinction

text: in

text: the

text: optical

text: band

text: as

text: AV−=

text: as-

text: There

text: is

text: good

text: overall

text: agreement

text: between

text: the

text: outlines

text: of

text: suming

text: a

text: near-infrared

text: extinction

text: law

text: with

text: α

text: (Stead

text: the

text: emission

text: as

text: seen

text: in

text: the

text: X-rays

text: and

text: infrared,

text: and

text: this

text: Hoare

text: Applying

text: the

text: relation

text: of

text: Gu¨ver

text: O¨

text: zel

text: mwhueltrie-wtahveelXen-grathy

text: htehne

text: with

text: a

text: local

text: maximum

text: in

text: the

text: MHz

text: continuum

text: map

text: from

text: total

text: column

text: density

text: of

text: the

text: interstellar

text: medium

text: towards

text: the

text: (Murphy

text: et

text: al.

text: The

text: coincidence

text: of

text: the

text: multi-

text: Norma

text: region

text: ranges

text: from

text: cm

text: to

text: cm

text: wavelength

text: emission

text: from

text: VVV

text: could

text: indicate

text: a

text: com-

text: (Rahoui

text: et

text: al.

text: This

text: is

text: consistent

text: with

text: our

text: X-ray-derived

text: mon

text: physical

text: origin,

text: however,

text: the

text: observed

text: X-ray

text: ﬂux

text: of

text: the

text: value

text: of

text: cm

text: diffuse

text: component

text: is

text: larger

text: than

text: expected

text: from

text: an

text: extrapola-

text: Plasma

text: of

text: around

text: keV

text: are

text: found

text: for

text: the

text: tion

text: of

text: the

text: infrared

text: ﬂux

text: density

text: (see

text: below).

text: X-ray

text: emitting

text: point

text: sources

text: in

text: the

text: cluster

text: which

text: is

text: higher

text: The

text: nearest

text: catalogued

text: ultra-compact

text: H

text: II

text: region

text: is

text: IRAS

text: for

text: massive

text: stars

text: (e.g.,

text: Rauw

text: et

text: al.

text: The

text: (Helou

text: Walker

text: Bronfman

text: et

text: al.

text: which

text: is

text: X-ray

text: source

text: located

text: in

text: the

text: outskirts

text: of

text: the

text: cluster

text: (CXOU

text: from

text: the

text: infrared

text: position

text: of

text: VVV

text: It

text: has

text: a

text: veloc-

text: has

text: a

text: temperature

text: that

text: is

text: statistically

text: ity

text: of

text: km

text: s

text: with

text: respect

text: to

text: the

text: local

text: standard

text: of

text: rest,

text: compatible

text: (at

text: conﬁdence)

text: with

text: keV

text: which

text: is

text: a

text: more

text: which

text: translates

text: to

text: a

text: source

text: distance

text: of

text: approximately

text: kpc

text: typical

text: value.

text: As

text: we

text: move

text: towards

text: the

text: center

text: of

text: the

text: cluster,

text: (e.g.,

text: Brand

text: Blitz

text: There

text: is

text: also

text: a

text: catalogued

text: H

text: II

text: we

text: ﬁnd

text: that

text: the

text: X-ray

text: derived

text: temperatures

text: of

text: these

text: stars

text: region

text: away

text: from

text: VVV

text: with

text: a

text: listed

text: distance

text: of

text: are

text: higher,

text: possibly

text: due

text: to

text: increasing

text: column

text: density

text: (which

text: kpc

text: (Russeil

text: If

text: these

text: regions

text: are

text: related,

text: then

text: preferentially

text: absorbs

text: low-temperature

text: photons).

text: With

text: its

text: we

text: can

text: assign

text: a

text: distance

text: of

text: around

text: kpc

text: to

text: VVV

text: proximity

text: to

text: IRAS

text: this

text: could

text: indicate

text: that

text: the

text: This

text: would

text: make

text: it

text: one

text: of

text: the

text: most

text: distant

text: Galactic

text: YMCs

text: stars

text: in

text: the

text: core

text: of

text: VVV

text: are

text: still

text: embedded

text: within

text: known,

text: comparable

text: to

text: Mercer

text: (Davies

text: et

text: al.

text: at

text: a

text: their

text: dusty

text: natal

text: cocoons;

text: similar

text: to

text: the

text: hot,

text: young

text: stars

text: of

text: location

text: consistent

text: with

text: the

text: far

text: Norma

text: Arm.

text: (Conti

text: Blum

text: a

text: cluster

text: that

text: resides

text: within

text: Taking

text: the

text: visual

text: extent

text: of

text: the

text: bulk

text: of

text: the

text: stellar

text: population

text: an

text: ultra-compact

text: H

text: II

text: region.

text: However,

text: we

text: can

text: not

text: exclude

text: in

text: the

text: infrared

text: as

text: the

text: angular

text: scale

text: Borissova

text: et

text: al.

text: that

text: the

text: unusual

text: spectral

text: values

text: are

text: the

text: result

text: of

text: poor

text: photon

text: gives

text: a

text: cluster

text: diameter

text: of

text: pc

text: for

text: an

text: assumed

text: distance

text: of

text: statistics.

text: kpc.

text: This

text: value

text: is

text: consistent

text: with

text: the

text: intrinsic

text: sizes

text: of

text: Based

text: on

text: the

text: IR

text: spectra,

text: we

text: narrowed

text: the

text: range

text: of

text: spectral

text: other

text: Galactic

text: YMCs

text: (Figer

text: and

text: references

text: therein).

text: types

text: to

text: giants

text: for

text: two

text: of

text: the

text: stars

text: in

text: the

text: cluster.

text: We

text: Corrected

text: for

text: absorption,

text: the

text: diff±use

text: e×miss−ion

text: from

text: −VV−V

text: derived

text: the

text: distance

text: modulus

text: m

text: M

text: log(d)+A

text: −5

text: using

text: the

text: has

text: an

text: intrinsic

text: ﬂux

text: of

text: erg

text: cm

text: s

text: observed

text: J-band

text: magnitude

text: of

text: CXOU

text: in

text: This

text: is

text: more

text: than

text: an

text: order

text: of

text: magnitude

text: above

text: Table

text: the

text: absolute

text: J-band

text: magnitude

text: of

text: for

text: an

text: O

text: giant

text: the

text: ﬂux

text: expected

text: from

text: the

text: Galactic

text: Ridge

text: diffuse

text: emission

text: (Martins

text: Plez

text: a±nd

text: AJ

text: This

text: gives

text: a

text: dis-

text: (Ebisawa

text: et

text: al.

text: If

text: we

text: assume,

text: nevertheless,

text: that

text: tance

text: to

text: the

text: object

text: of

text: kpc,

text: consistent

text: with

text: the

text: distance

text: the

text: Ridge

text: emission

text: accounts

text: for

text: of

text: the

text: ﬂux

text: from

text: VVV

text: assumed

text: based

text: on

text: its

text: possible

text: link

text: with

text: IRAS

text: as

text: it

text: does

text: for

text: Wd

text: (Muno

text: et

text: al.

text: then

text: the

text: resid-

text: In

text: massive

text: stars,

text: the

text: ratio

text: between

text: the

text: X-ray

text: and

text: bolo-

text: ##LTFigure##

text: ##LTFigure##


text: Fig. 3.— Predicted radio surface brightness contours for the SC disk+halo+source model, showing the eﬀects of beam smearing. Contours are shown at 0.03, 0.1, 0.3, 1, and 3 K as seen by an external observer at 1500 MHz for diﬀerent beam widths. The beam full width at half maximum (FWHM) is shown at the bottom left in each panel. (Left) Model smoothed with projected beam width of 1.5 kpc. The plane and halo components are readily distinguished. The thick contour at 0.1 K highlights the halo component. (Middle) Same as left panel, but with the thick contour at 1 K highlighting that at this level the contour is dominated by plane component. Contours near peak brightness are insensitive to the halo component regardless of beam smearing. (Right) Model smoothed with projected beam width of 5 kpc with thick contour highlighted at 0.1 K. Beam smearing in this case prevents identiﬁcation of the halo component even at faint contours. frequency5. radio map, and compares the observed axial ratio of that We then create a contour map of the resulting model contour to the same absolute contour in the model. Since (including the eﬀects of galaxy inclination and observing the halo emission extends far from the plane, this method beam width) and measure the axial ratio of the model is insensitive to details of emission near the plane, but at the same contour as the faintest contour in the radio assumes consistent halo brightness for all galaxies. The data, thereby providing a direct comparison of the model second method scales the contours by the peak surface to observations. brightness, comparing the faintest contour in each ra- Not all of the radio maps are sensitive to extended dio map to the same fractional contour in the model. diﬀuse emission, although in all maps the faintest contour This method allows the halo brightness to vary from one is the contour most likely to be sensitive. As seen in galaxy to another, but introduces some potential depen- Figure 3, when the faintest contour is not at a suﬃciently dence on the details of peak emission in the plane. low surface brightness it is dominated by emission in the Both methods yield similar results (Table 1). Ten of plane regardless of the presence or absence of a radio the radio maps have absolute contours comparable to the ∼ halo. In maps with signiﬁcant beam smearing, even the predicted halo surface brightness ( 0.1 K at 1500 MHz) faintest contour cannot distinguish between (convolved) and have beam width small compared to the galaxy ma- planar emission and diﬀuse extra-planar emission. For jor axis. Allowing the halo temperature to scale with the each radio map, we generate the corresponding model observed peak surface brightness yields 20 maps with a map with and without the halo component, and compare contour capable of detecting a scaled radio halo. Figure the axial ratios of the resulting models evaluated at the 4 shows the distribution of observed and model axial ra- faintest contour level of the radio map. Radio maps for tios. After accounting for the spread in inclination angle, which the faintest contour fails to distinguish between beam smearing, and observing frequency, the SC model ± the presence or absence of the SC model halo component predicts axial ratios 0.23 0.02 for the faintest absolute ± are not used in this analysis. contours and 0.24 0.09 for the faintest fractional con- We would like to search for extended halos of the type tours, consistent with the expected spherical halo compo- predicted by the SC model in a way that is not biased nent. The radio data, in contrast, show strongly elliptical ± ± by galaxies’ diﬀering absolute luminosities or the details contours, with mean axial ratios 1.4 0.7 and 1.5 1.1 for of planar emission. We test for these biases based on the absolute and fractional contours, respectively. All of gaTlaoxyAplpuemarinIonsiAtypJbLy repeating the axial ratio analysis the observed galaxies had an axial ratio larger than the usPinregprtiwntotydpieﬀseetreunsitngmLAeTtEhXosdtsy.le Temhuelaﬁterasptj vm. e1t2h/1o6d/1u1ses the corresponding model prediction. absolute surface brightness of the faintest contour in each AXIAL RATIO OF EDGE-ON SPIRAL GALAXIES AS A TEST FOR EXTENDED BRIGHT RADIO HALOS J. Singal1, A. Kogut2, E. Jones1, H. Dunlap1 Subject headings: Galaxy:halo - radio continuum:galaxies 5 Since all but one of the maps are observed at frequencies very ##LTFigure## ##LTFigure##

text: Fig.

text: Predicted

text: radio

text: surface

text: brightness

text: contours

text: for

text: the

text: SC

text: disk+halo+source

text: model,

text: showing

text: the

text: eﬀects

text: of

text: beam

text: smearing.

text: Contours

text: are

text: shown

text: at

text: and

text: K

text: as

text: seen

text: by

text: an

text: external

text: observer

text: at

text: MHz

text: for

text: diﬀerent

text: beam

text: widths.

text: The

text: beam

text: full

text: width

text: at

text: half

text: maximum

text: (FWHM)

text: is

text: shown

text: at

text: the

text: bottom

text: left

text: in

text: each

text: panel.

text: (Left)

text: Model

text: smoothed

text: with

text: projected

text: beam

text: width

text: of

text: kpc.

text: The

text: plane

text: and

text: halo

text: components

text: are

text: readily

text: distinguished.

text: The

text: thick

text: contour

text: at

text: K

text: highlights

text: the

text: halo

text: component.

text: (Middle)

text: Same

text: as

text: left

text: panel,

text: but

text: with

text: the

text: thick

text: contour

text: at

text: K

text: highlighting

text: that

text: at

text: this

text: level

text: the

text: contour

text: is

text: dominated

text: by

text: plane

text: component.

text: Contours

text: near

text: peak

text: brightness

text: are

text: insensitive

text: to

text: the

text: halo

text: component

text: regardless

text: of

text: beam

text: smearing.

text: (Right)

text: Model

text: smoothed

text: with

text: projected

text: beam

text: width

text: of

text: kpc

text: with

text: thick

text: contour

text: highlighted

text: at

text: K.

text: Beam

text: smearing

text: in

text: this

text: case

text: prevents

text: identiﬁcation

text: of

text: the

text: halo

text: component

text: even

text: at

text: faint

text: contours.

text: radio

text: map,

text: and

text: compares

text: the

text: observed

text: axial

text: ratio

text: of

text: that

text: We

text: then

text: create

text: a

text: contour

text: map

text: of

text: the

text: resulting

text: model

text: contour

text: to

text: the

text: same

text: absolute

text: contour

text: in

text: the

text: model.

text: Since

text: (including

text: the

text: eﬀects

text: of

text: galaxy

text: inclination

text: and

text: observing

text: the

text: halo

text: emission

text: extends

text: far

text: from

text: the

text: plane,

text: this

text: method

text: beam

text: width)

text: and

text: measure

text: the

text: axial

text: ratio

text: of

text: the

text: model

text: is

text: insensitive

text: to

text: details

text: of

text: emission

text: near

text: the

text: plane,

text: but

text: at

text: the

text: same

text: contour

text: as

text: the

text: faintest

text: contour

text: in

text: the

text: radio

text: assumes

text: consistent

text: halo

text: brightness

text: for

text: all

text: galaxies.

text: The

text: data,

text: thereby

text: providing

text: a

text: direct

text: comparison

text: of

text: the

text: model

text: second

text: method

text: scales

text: the

text: contours

text: by

text: the

text: peak

text: surface

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text: X - 12 DORELLI ET AL.: HALL MHD GANYMEDE SIMULATIONS y J V in B + - Vout JxB force out δi" δe" of the plane - + x Δ " e B Δ " i J out-of-plane magnetic field, B z Figure 11. The acceleration mechanism responsible for the current sheet ion jets is a simple con- sequence of the Hall ﬁelds within the ion diﬀusion regions. In the left panel, the cyan box is the ion diﬀusion region, and the yellow box is the electron diﬀusion region. The blue and red ovals show the quadrupolar out-of-plane magnetic ﬁeld pattern, supported by the in-plane Hall currents. These in-plane × currents produce an out-of-plane J B acceleration, shown schematically in the right panel. plasma with the upstream magnetopause ion jet, e) an comparisons between simulations of other ﬂybys and Galileo Alfv´enic ion jet consisting of Jovian magnetospheric plasma data will be pursued in future work. extending across the central wake region, f) Harang-like dis- We note that multi-ﬂuid simulations by Benna et al. continuities in the ionospheric convection pattern, and g) the [2010] have produced ion drift belts very similar to those presence of Jovian magnetospheric plasma in a thin d scale found in our Ganymede simulations. However, while Benna i et al. [2010] invoke electron pressure gradient drifts to ex- layer (centered on Z = 0 close to the moon in the wake plain the ion belts, our simulations did not include the elec- region. We have argued in this work that some of these consequences (the large out-of-plane magnetic ﬁeld pertur- tron pressure gradient in Ohm’s law. The similarity of ×our results to those of Benna et al. [2010] suggest that Hall J B bations and the asymmetric pattern of Kelvin-Helmholtz forces can explain the ion drift belts. We further specu- waves) are already apparent in the G8 magnetometer obser- late that the “double magnetopause” structure observed in vations. Some of the other consequences should be testable our Ganymede simulations is the same structure observed at using the other Galileo ﬂybys and including other Galileo Mercury by Slavin et al. [2008] in the MESSENGER data: instruments (plasma and waves), but such an investigation a thickened dusk-side magnetopause boundary layer driven is beyond the scope of the present paper. More extensive by Hall-mediated reconnection (rather than pressure gradi- ent drifts, as argued by Mu¨ller et al×. [2012]). Further simu- lation work that isolates the Hall J B force from pressure gradient drifts is required to deﬁnitively answer this ques- tion. Since the eﬀects observed in our simulations are directly driven by magnetic reconnection, they should be present in all magnetospheres. The large scale consequences and ob- servable signatures, however, will depend on a number of factors: a) the size of the intrinsic magnetic ﬁeld (which in- ﬂuences the size of the magnetosphere), b) the role of mag- netic reconnection in driving global magnetospheric convec- tion, c) the nature of the magnetized body’s atmosphere and surface (which inﬂuences how ﬁeld-aligned currents close). We expect the Hall eﬀect to have global inﬂuence in small magnetospheres for which convection is driven primarily by magnetic reconnection. For example, the Hall eﬀect likely plays a negligible role in the large scale structure of Jupiter’s co-rotating magnetosphere. Mercury, on the other hand, possesses a very small magnetosphere with convection driven by magnetic reconnection, so that Hall-mediated re- connection should produce global ﬁeld-aligned current and ion drift belt structures. Mercury’s surface conductivity is very small, however, and it is not clear how the Hall-induced ﬁeld-aligned currents will close. Earth lies somewhere in be- tween: its magnetosphere is relatively large (several orders Figure 12. A cut along the X axis crossing the upstream of magnitude relative to the ion inertial length), and its magnetopause conﬁrms that the ions are accelerated to convection pattern transitions from an outer reconnection ≈ dominated region to an inner co-rotation dominated region. the local Alfv´en speed (in this case, 140 km/sec) in Understanding the role of ion-scale reconnection in global the out-of-plane direction. The upper panel shows the magnetospheric structure is thus a compelling problem that three components of the magnetic ﬁeld across the up- will beneﬁt greatly from future comparative simulation stud- stream magnetopause; the bottom panel shows the three ies. components of the bulk velocity. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: X

text: DORELLI

text: ET

text: AL.:

text: HALL

text: MHD

text: GANYMEDE

text: SIMULATIONS

text: y

font: default

text: J

text: V

text: in

text: B

text: Vout

text: JxB

text: force

text: out

text: δi"

text: δe"

text: of

text: the

text: plane

text: x

text: Δ

text: e

text: B

text: Δ

text: i

text: J

text: out-of-plane

text: magnetic

text: field,

text: B

text: z

text: Figure

text: The

text: acceleration

text: mechanism

text: responsible

text: for

text: the

text: current

text: sheet

text: ion

text: jets

text: is

text: a

text: simple

text: con-

text: sequence

text: of

text: the

text: Hall

text: ﬁelds

text: within

text: the

text: ion

text: diﬀusion

text: regions.

text: In

text: the

text: left

text: panel,

text: the

text: cyan

text: box

text: is

text: the

text: ion

text: diﬀusion

text: region,

text: and

text: the

text: yellow

text: box

text: is

text: the

text: electron

text: diﬀusion

text: region.

text: The

text: blue

text: and

text: red

text: ovals

text: show

text: the

text: quadrupolar

text: out-of-plane

text: magnetic

text: ﬁeld

text: pattern,

text: supported

text: by

text: the

text: in-plane

text: Hall

text: currents.

text: These

text: in-plane

text: currents

text: produce

text: an

text: out-of-plane

text: J

text: B

text: acceleration,

text: shown

text: schematically

text: in

text: the

text: right

text: panel.

text: plasma

text: with

text: the

text: upstream

text: magnetopause

text: ion

text: jet,

text: e)

text: an

text: comparisons

text: between

text: simulations

text: of

text: other

text: ﬂybys

text: and

text: Galileo

text: Alfv´enic

text: ion

text: jet

text: consisting

text: of

text: Jovian

text: magnetospheric

text: plasma

text: data

text: will

text: be

text: pursued

text: in

text: future

text: work.

text: extending

text: across

text: the

text: central

text: wake

text: region,

text: f)

text: Harang-like

text: dis-

text: We

text: note

text: that

text: multi-ﬂuid

text: simulations

text: by

text: Benna

text: et

text: al.

text: continuities

text: in

text: the

text: ionospheric

text: convection

text: pattern,

text: and

text: g)

text: the

text: have

text: produced

text: ion

text: drift

text: belts

text: very

text: similar

text: to

text: those

text: presence

text: of

text: Jovian

text: magnetospheric

text: plasma

text: in

text: a

text: thin

text: d

text: scale

text: found

text: in

text: our

text: Ganymede

text: simulations.

text: However,

text: while

text: Benna

text: i

text: et

text: al.

text: invoke

text: electron

text: pressure

text: gradient

text: drifts

text: to

text: ex-

text: layer

text: (centered

text: on

text: Z

text: close

text: to

text: the

text: moon

text: in

text: the

text: wake

text: plain

text: the

text: ion

text: belts,

text: our

text: simulations

text: did

text: not

text: include

text: the

text: elec-

text: region.

text: We

text: have

text: argued

text: in

text: this

text: work

text: that

text: some

text: of

text: these

text: consequences

text: (the

text: large

text: out-of-plane

text: magnetic

text: ﬁeld

text: pertur-

text: tron

text: pressure

text: gradient

text: in

text: Ohm’s

text: law.

text: The

text: similarity

text: of

text: ×our

text: results

text: to

text: those

text: of

text: Benna

text: et

text: al.

text: suggest

text: that

text: Hall

text: J

text: B

text: bations

text: and

text: the

text: asymmetric

text: pattern

text: of

text: Kelvin-Helmholtz

text: forces

text: can

text: explain

text: the

text: ion

text: drift

text: belts.

text: We

text: further

text: specu-

text: waves)

text: are

text: already

text: apparent

text: in

text: the

text: magnetometer

text: obser-

text: late

text: that

text: the

text: “double

text: magnetopause”

text: structure

text: observed

text: in

text: vations.

text: Some

text: of

text: the

text: other

text: consequences

text: should

text: be

text: testable

text: our

text: Ganymede

text: simulations

text: is

text: the

text: same

text: structure

text: observed

text: at

text: using

text: the

text: other

text: Galileo

text: ﬂybys

text: and

text: including

text: other

text: Galileo

text: Mercury

text: by

text: Slavin

text: et

text: al.

text: in

text: the

text: MESSENGER

text: data:

text: instruments

text: (plasma

text: and

text: waves),

text: but

text: such

text: an

text: investigation

text: a

text: thickened

text: dusk-side

text: magnetopause

text: boundary

text: layer

text: driven

text: is

text: beyond

text: the

text: scope

text: of

text: the

text: present

text: paper.

text: More

text: extensive

text: by

text: Hall-mediated

text: reconnection

text: (rather

text: than

text: pressure

text: gradi-

text: ent

text: drifts,

text: as

text: argued

text: by

text: Mu¨ller

text: et

text: al×.

text: Further

text: simu-

text: lation

text: work

text: that

text: isolates

text: the

text: Hall

text: J

text: B

text: force

text: from

text: pressure

text: gradient

text: drifts

text: is

text: required

text: to

text: deﬁnitively

text: answer

text: this

text: ques-

text: tion.

text: Since

text: the

text: eﬀects

text: observed

text: in

text: our

text: simulations

text: are

text: directly

text: driven

text: by

text: magnetic

text: reconnection,

text: they

text: should

text: be

text: present

text: in

text: all

text: magnetospheres.

text: The

text: large

text: scale

text: consequences

text: and

text: ob-

text: servable

text: signatures,

text: however,

text: will

text: depend

text: on

text: a

text: number

text: of

text: factors:

text: a)

text: the

text: size

text: of

text: the

text: intrinsic

text: magnetic

text: ﬁeld

text: (which

text: in-

text: ﬂuences

text: the

text: size

text: of

text: the

text: magnetosphere),

text: b)

text: the

text: role

text: of

text: mag-

text: netic

text: reconnection

text: in

text: driving

text: global

text: magnetospheric

text: convec-

text: tion,

text: c)

text: the

text: nature

text: of

text: the

text: magnetized

text: body’s

text: atmosphere

text: and

text: surface

text: (which

text: inﬂuences

text: how

text: ﬁeld-aligned

text: currents

text: close).

text: We

text: expect

text: the

text: Hall

text: eﬀect

text: to

text: have

text: global

text: inﬂuence

text: in

text: small

text: magnetospheres

text: for

text: which

text: convection

text: is

text: driven

text: primarily

text: by

text: magnetic

text: reconnection.

text: For

text: example,

text: the

text: Hall

text: eﬀect

text: likely

text: plays

text: a

text: negligible

text: role

text: in

text: the

text: large

text: scale

text: structure

text: of

text: Jupiter’s

text: co-rotating

text: magnetosphere.

text: Mercury,

text: on

text: the

text: other

text: hand,

text: possesses

text: a

text: very

text: small

text: magnetosphere

text: with

text: convection

text: driven

text: by

text: magnetic

text: reconnection,

text: so

text: that

text: Hall-mediated

text: re-

text: connection

text: should

text: produce

text: global

text: ﬁeld-aligned

text: current

text: and

text: ion

text: drift

text: belt

text: structures.

text: Mercury’s

text: surface

text: conductivity

text: is

text: very

text: small,

text: however,

text: and

text: it

text: is

text: not

text: clear

text: how

text: the

text: Hall-induced

text: ﬁeld-aligned

text: currents

text: will

text: close.

text: Earth

text: lies

text: somewhere

text: in

text: be-

text: tween:

text: its

text: magnetosphere

text: is

text: relatively

text: large

text: (several

text: orders

text: Figure

text: A

text: cut

text: along

text: the

text: X

text: axis

text: crossing

text: the

text: upstream

text: of

text: magnitude

text: relative

text: to

text: the

text: ion

text: inertial

text: length),

text: and

text: its

text: magnetopause

text: conﬁrms

text: that

text: the

text: ions

text: are

text: accelerated

text: to

text: convection

text: pattern

text: transitions

text: from

text: an

text: outer

text: reconnection

text: dominated

text: region

text: to

text: an

text: inner

text: co-rotation

text: dominated

text: region.

text: the

text: local

text: Alfv´en

text: speed

text: (in

text: this

text: case,

text: km/sec)

text: in

text: Understanding

text: the

text: role

text: of

text: ion-scale

text: reconnection

text: in

text: global

text: the

text: out-of-plane

text: direction.

text: The

text: upper

text: panel

text: shows

text: the

text: magnetospheric

text: structure

text: is

text: thus

text: a

text: compelling

text: problem

text: that

text: three

text: components

text: of

text: the

text: magnetic

text: ﬁeld

text: across

text: the

text: up-

text: will

text: beneﬁt

text: greatly

text: from

text: future

text: comparative

text: simulation

text: stud-

text: stream

text: magnetopause;

text: the

text: bottom

text: panel

text: shows

text: the

text: three

text: ies.

text: components

text: of

text: the

text: bulk

text: velocity.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: DORELLI ET AL.: HALL MHD GANYMEDE SIMULATIONS X - 5 Figure 3. In resistive MHD (left panel), the magnetospheric convection and ﬁeld-aligned currents (FAC) are symmetric about the Jovian magnetospheric inﬂow direction, exhibiting the classic Dungey convection (green arrows on the density plane and purple arrows on the inner sphere) and region-1 FAC (blue-white colorbar on the inner sphere) patterns. In constract, the Hall simulation (right panel) shows strong asymmetries in both the convection and FAC patterns. In the Hall simulation, Jovian magneto- spheric plasma appears to enter Ganymede’s magnetosphere along the sub-Jovian ﬂank, populating the plasma sheet with Jovian magnetospheric plasma (in contrast to the resistive MHD case, in which the wake region is more or less devoid of Jovian magnetospheric plasma). Note the pronounced bending of reconnected ﬁeld lines (green tubes) out of the reconnection plane in the Hall case. cell is advanced at the largest time step possible for numer- which produces the classic quadrupolar out-of-plane mag- ical stability) until a quasi-steady state was reached; then, netic ﬁeld pattern shown in the lower right panel of Figure “time-accurate” time stepping was used. For the Hall MHD 2 – generates a new system of ﬁeld-aligned currents that runs, we used two diﬀerent initial states: 1) local time step- are most intense near the upstream and downstream edges ping as for the MHD simulations, 2) the ﬁnal time-accurate of the polar cap. The global convection pattern also shows MHD state. We ran the simulations in time∼-accurate mode large asymmetries, with Jovian magnetospheric plasma ﬂow- long enough to reach a quasi-steady state ( 10 minutes of ing into Ganymede’s wake region through the sub-Jovian simulated time was found to be suﬃcient for Ganymede’s ﬂank. We demonstrate below that all of these eﬀects are small magnetosphere). We obtained similar results in both directly driven by Hall electric ﬁelds within the upstream cases, giving us conﬁdence that the ﬁnal Hall MHD state magnetopause and downstream tail current sheets and, thus, presented below was not inﬂuenced by the initial condition. should be present to some degree in all planetary magneto- spheres (e.g., Mercury and Earth) for which convection is 3. Results driven by collisionless magnetic reconnection. Figure 4 shows a closer view of the FAC and convection Figure 2 shows an overview of the Hall and resistive patterns in the polar cap. The resistive MHD case shows an MHD simulation results. The two upper ﬁgures show the anti-symmetric pattern of FAC into the moon’s atmosphere, − X (GphiO) component of the bulk velocity for the resis- with parallel current densities of about 0.05µ A/m2 (con- tive (left) and Hall (right) runs, showing that the large scale sistent with those reported by Jia et al. [2008, 2009]). In Alfv´en wing structure is not signiﬁcantly modiﬁed by the the Hall case, the FAC pattern is distorted by the reconnec- Hall eﬀect. Reconnection at the upstream magnetopause tion FAC system, and now the most intense current densities and downstream tail current sheets, however, produces the appear in bands centered at the upstream and downstream characteristic quadrupolar out-of-plane magnetic ﬁeld pat- edges of the polar cap. The polar cap convection pattern tern (lower right panel). The out-of-plane ﬁeld is supported in the Hall case shows localized acceleration channels near by a system of intense ﬁeld-aligned currents (FAC) that the upstream and downstream polar cap boundaries, both of merge into the large scale Alfv´en wings far from the moon, which are associated with the bending of the magnetic ﬁeld but these currents also extend all the way down to the lines out of the reconnection planes in the magnetopause and moon’s surface. − magnetotail current sheets (labelled “JxB” in the ﬁgure). Looking at the X Y plane (Figure 3), however, reveals We will refer to these acceleration channels as “Harang-like that the Hall eﬀect introduces signiﬁcant asymmetries into discontinuities,” since they are similar in appearance to the the global convection and FAC patterns. In resistive MHD, Harang Discontinuity observed in the evening sector convec- Jovian magnetospheric plasma ﬂows around the upstream tion electric ﬁeld of Earth’s ionosphere. magnetopause symmetrically, producing a density cavity de- The Hall reconnection physics is conﬁned to d scale lay- void of Jovian magnetospheric plasma in Ganymede’s wake. i ers and, as shown in Figure 2, has little impact on the large This is the classic Dungey [Dungey, 1961, 1963] convec- scale Alfv´en wing structure. However, Hall electric ﬁelds tion cycle, driven by magnetic reconnection at the upstream magnetopause and downstream tail current sheets. Asso- appear to have a d−ramatic impact on the global convection ciated with this symmetric convection pattern is an anti- pattern in the X Y plane. Figure 5 shows cuts of cur- symmetric pattern of FAC, most pronounced on the ﬂanks rent density magnitude and−bulk velocity streamlines (green and vanishing at the upstream and downstream edges of the lines and arrows) in the X Y plane for the resistive MHD polar cap. run (left panel) and the Hall run (right panel). The MHD In contrast, in the Hall MHD simulation Jovian magne- case shows the classic Dungey convection pattern, with tail tospheric plasma enters Ganymede’s magnetosphere along reconnection driving ﬂow jets moonward and tailward from the sub-Jovian ﬂank, ﬁlling Ganymede’s wake with Jovian an X line oriented more or less along the Y axis about 2RG magnetospheric plasma. The bending of the reconnected downstream of the moon. Note the symmetric pattern of ﬁeld lines (green tubes) out of the reconnection plane – Kelvin-Helmholtz (KH) waves on the ﬂanks, driven by the ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: DORELLI

text: ET

text: AL.:

text: HALL

text: MHD

text: GANYMEDE

text: SIMULATIONS

text: X

text: Figure

text: In

text: resistive

text: MHD

text: (left

text: panel),

text: the

text: magnetospheric

text: convection

text: and

text: ﬁeld-aligned

text: currents

text: (FAC)

text: are

text: symmetric

text: about

text: the

text: Jovian

text: magnetospheric

text: inﬂow

text: direction,

text: exhibiting

text: the

text: classic

text: Dungey

text: convection

text: (green

text: arrows

text: on

text: the

text: density

text: plane

text: and

text: purple

text: arrows

text: on

text: the

text: inner

text: sphere)

text: and

text: FAC

text: (blue-white

text: colorbar

text: on

text: the

text: inner

text: sphere)

text: patterns.

text: In

text: constract,

text: the

text: Hall

text: simulation

text: (right

text: panel)

text: shows

text: strong

text: asymmetries

text: in

text: both

text: the

text: convection

text: and

text: FAC

text: patterns.

text: In

text: the

text: Hall

text: simulation,

text: Jovian

text: magneto-

text: spheric

text: plasma

text: appears

text: to

text: enter

text: Ganymede’s

text: magnetosphere

text: along

text: the

text: sub-Jovian

text: ﬂank,

text: populating

text: the

text: plasma

text: sheet

text: with

text: Jovian

text: magnetospheric

text: plasma

text: (in

text: contrast

text: to

text: the

text: resistive

text: MHD

text: case,

text: in

text: which

text: the

text: wake

text: region

text: is

text: more

text: or

text: less

text: devoid

text: of

text: Jovian

text: magnetospheric

text: plasma).

text: Note

text: the

text: pronounced

text: bending

text: of

text: reconnected

text: ﬁeld

text: lines

text: (green

text: tubes)

text: out

text: of

text: the

text: reconnection

text: plane

text: in

text: the

text: Hall

text: case.

text: cell

text: is

text: advanced

text: at

text: the

text: largest

text: time

text: step

text: possible

text: for

text: numer-

text: which

text: produces

text: the

text: classic

text: quadrupolar

text: out-of-plane

text: mag-

text: ical

text: stability)

text: until

text: a

text: quasi-steady

text: state

text: was

text: reached;

text: then,

text: netic

text: ﬁeld

text: pattern

text: shown

text: in

text: the

text: lower

text: right

text: panel

text: of

text: Figure

text: “time-accurate”

text: time

text: stepping

text: was

text: used.

text: For

text: the

text: Hall

text: MHD

text: generates

text: a

text: new

text: system

text: of

text: ﬁeld-aligned

text: currents

text: that

text: runs,

text: we

text: used

text: two

text: diﬀerent

text: initial

text: states:

text: local

text: time

text: step-

text: are

text: most

text: intense

text: near

text: the

text: upstream

text: and

text: downstream

text: edges

text: ping

text: as

text: for

text: the

text: MHD

text: simulations,

text: the

text: ﬁnal

text: time-accurate

text: of

text: the

text: polar

text: cap.

text: The

text: global

text: convection

text: pattern

text: also

text: shows

text: MHD

text: state.

text: We

text: ran

text: the

text: simulations

text: in

text: time∼-accurate

text: mode

text: large

text: asymmetries,

text: with

text: Jovian

text: magnetospheric

text: plasma

text: ﬂow-

text: long

text: enough

text: to

text: reach

text: a

text: quasi-steady

text: state

text: minutes

text: of

text: ing

text: into

text: Ganymede’s

text: wake

text: region

text: through

text: the

text: sub-Jovian

text: simulated

text: time

text: was

text: found

text: to

text: be

text: suﬃcient

text: for

text: Ganymede’s

text: ﬂank.

text: We

text: demonstrate

text: below

text: that

text: all

text: of

text: these

text: eﬀects

text: are

text: small

text: magnetosphere).

text: We

text: obtained

text: similar

text: results

text: in

text: both

text: directly

text: driven

text: by

text: Hall

text: electric

text: ﬁelds

text: within

text: the

text: upstream

text: cases,

text: giving

text: us

text: conﬁdence

text: that

text: the

text: ﬁnal

text: Hall

text: MHD

text: state

text: magnetopause

text: and

text: downstream

text: tail

text: current

text: sheets

text: and,

text: thus,

text: presented

text: below

text: was

text: not

text: inﬂuenced

text: by

text: the

text: initial

text: condition.

text: should

text: be

text: present

text: to

text: some

text: degree

text: in

text: all

text: planetary

text: magneto-

text: spheres

text: (e.g.,

text: Mercury

text: and

text: Earth)

text: for

text: which

text: convection

text: is

text: Results

text: driven

text: by

text: collisionless

text: magnetic

text: reconnection.

text: Figure

text: shows

text: a

text: closer

text: view

text: of

text: the

text: FAC

text: and

text: convection

text: Figure

text: shows

text: an

text: overview

text: of

text: the

text: Hall

text: and

text: resistive

text: patterns

text: in

text: the

text: polar

text: cap.

text: The

text: resistive

text: MHD

text: case

text: shows

text: an

text: MHD

text: simulation

text: results.

text: The

text: two

text: upper

text: ﬁgures

text: show

text: the

text: anti-symmetric

text: pattern

text: of

text: FAC

text: into

text: the

text: moon’s

text: atmosphere,

text: X

text: (GphiO)

text: component

text: of

text: the

text: bulk

text: velocity

text: for

text: the

text: resis-

text: with

text: parallel

text: current

text: densities

text: of

text: about

text: (con-

text: tive

text: (left)

text: and

text: Hall

text: (right)

text: runs,

text: showing

text: that

text: the

text: large

text: scale

text: sistent

text: with

text: those

text: reported

text: by

text: Jia

text: et

text: al.

text: In

text: Alfv´en

text: wing

text: structure

text: is

text: not

text: signiﬁcantly

text: modiﬁed

text: by

text: the

text: the

text: Hall

text: case,

text: the

text: FAC

text: pattern

text: is

text: distorted

text: by

text: the

text: reconnec-

text: Hall

text: eﬀect.

text: Reconnection

text: at

text: the

text: upstream

text: magnetopause

text: tion

text: FAC

text: system,

text: and

text: now

text: the

text: most

text: intense

text: current

text: densities

text: and

text: downstream

text: tail

text: current

text: sheets,

text: however,

text: produces

text: the

text: appear

text: in

text: bands

text: centered

text: at

text: the

text: upstream

text: and

text: downstream

text: characteristic

text: quadrupolar

text: out-of-plane

text: magnetic

text: ﬁeld

text: pat-

text: edges

text: of

text: the

text: polar

text: cap.

text: The

text: polar

text: cap

text: convection

text: pattern

text: tern

text: (lower

text: right

text: panel).

text: The

text: out-of-plane

text: ﬁeld

text: is

text: supported

text: in

text: the

text: Hall

text: case

text: shows

text: localized

text: acceleration

text: channels

text: near

text: by

text: a

text: system

text: of

text: intense

text: ﬁeld-aligned

text: currents

text: (FAC)

text: that

text: the

text: upstream

text: and

text: downstream

text: polar

text: cap

text: boundaries,

text: both

text: of

text: merge

text: into

text: the

text: large

text: scale

text: Alfv´en

text: wings

text: far

text: from

text: the

text: moon,

text: which

text: are

text: associated

text: with

text: the

text: bending

text: of

text: the

text: magnetic

text: ﬁeld

text: but

text: these

text: currents

text: also

text: extend

text: all

text: the

text: way

text: down

text: to

text: the

text: lines

text: out

text: of

text: the

text: reconnection

text: planes

text: in

text: the

text: magnetopause

text: and

text: moon’s

text: surface.

text: magnetotail

text: current

text: sheets

text: (labelled

text: “JxB”

text: in

text: the

text: ﬁgure).

text: Looking

text: at

text: the

text: X

text: Y

text: plane

text: (Figure

text: however,

text: reveals

text: We

text: will

text: refer

text: to

text: these

text: acceleration

text: channels

text: as

text: “Harang-like

text: that

text: the

text: Hall

text: eﬀect

text: introduces

text: signiﬁcant

text: asymmetries

text: into

text: discontinuities,”

text: since

text: they

text: are

text: similar

text: in

text: appearance

text: to

text: the

text: the

text: global

text: convection

text: and

text: FAC

text: patterns.

text: In

text: resistive

text: MHD,

text: Harang

text: Discontinuity

text: observed

text: in

text: the

text: evening

text: sector

text: convec-

text: Jovian

text: magnetospheric

text: plasma

text: ﬂows

text: around

text: the

text: upstream

text: tion

text: electric

text: ﬁeld

text: of

text: Earth’s

text: ionosphere.

text: magnetopause

text: symmetrically,

text: producing

text: a

text: density

text: cavity

text: de-

text: The

text: Hall

text: reconnection

text: physics

text: is

text: conﬁned

text: to

text: d

text: scale

text: lay-

text: void

text: of

text: Jovian

text: magnetospheric

text: plasma

text: in

text: Ganymede’s

text: wake.

text: i

text: ers

text: and,

text: as

text: shown

text: in

text: Figure

text: has

text: little

text: impact

text: on

text: the

text: large

text: This

text: is

text: the

text: classic

text: Dungey

text: [Dungey,

text: convec-

text: scale

text: Alfv´en

text: wing

text: structure.

text: However,

text: Hall

text: electric

text: ﬁelds

text: tion

text: cycle,

text: driven

text: by

text: magnetic

text: reconnection

text: at

text: the

text: upstream

text: magnetopause

text: and

text: downstream

text: tail

text: current

text: sheets.

text: Asso-

text: appear

text: to

text: have

text: a

text: d−ramatic

text: impact

text: on

text: the

text: global

text: convection

text: ciated

text: with

text: this

text: symmetric

text: convection

text: pattern

text: is

text: an

text: anti-

text: pattern

text: in

text: the

text: X

text: Y

text: plane.

text: Figure

text: shows

text: cuts

text: of

text: cur-

text: symmetric

text: pattern

text: of

text: FAC,

text: most

text: pronounced

text: on

text: the

text: ﬂanks

text: rent

text: density

text: magnitude

text: and−bulk

text: velocity

text: streamlines

text: (green

text: and

text: vanishing

text: at

text: the

text: upstream

text: and

text: downstream

text: edges

text: of

text: the

text: lines

text: and

text: arrows)

text: in

text: the

text: X

text: Y

text: plane

text: for

text: the

text: resistive

text: MHD

text: polar

text: cap.

text: run

text: (left

text: panel)

text: and

text: the

text: Hall

text: run

text: (right

text: panel).

text: The

text: MHD

text: In

text: contrast,

text: in

text: the

text: Hall

text: MHD

text: simulation

text: Jovian

text: magne-

text: case

text: shows

text: the

text: classic

text: Dungey

text: convection

text: pattern,

text: with

text: tail

text: tospheric

text: plasma

text: enters

text: Ganymede’s

text: magnetosphere

text: along

text: reconnection

text: driving

text: ﬂow

text: jets

text: moonward

text: and

text: tailward

text: from

text: the

text: sub-Jovian

text: ﬂank,

text: ﬁlling

text: Ganymede’s

text: wake

text: with

text: Jovian

text: an

text: X

text: line

text: oriented

text: more

text: or

text: less

text: along

text: the

text: Y

text: axis

text: about

text: magnetospheric

text: plasma.

text: The

text: bending

text: of

text: the

text: reconnected

text: downstream

text: of

text: the

text: moon.

text: Note

text: the

text: symmetric

text: pattern

text: of

text: ﬁeld

text: lines

text: (green

text: tubes)

text: out

text: of

text: the

text: reconnection

text: plane

text: Kelvin-Helmholtz

text: (KH)

text: waves

text: on

text: the

text: ﬂanks,

text: driven

text: by

text: the

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##