IonisingStellarFeedbackwithPhantomandCMacIonize

MayaPetkova

Supervisor:IanBonnellCollaborators:GuillaumeLaibe,BertVandenbroucke,

JimDale

SPHandMCRTHierarchical cluster formation 3

Figure 1. The stellar cluster forms through the hierarchical fragmentation of a turbulent molecular cloud. Each panel shows a region 1parsec on a side. The logarithm of the column density is plotted from a minimum of 0.025 (black) to a maximum of 250 (white) g cm−3.The stars are indicated by the white dots. The four panels capture the evolution of the 1000 M⊙ system at times of 1.0, 1.4, 1.8 and 2.4initial free-fall times, where the free-fall time for the cloud is tff = 1.9×105 years. The turbulence causes shocks to form in the molecularcloud, dissipating kinetic energy and producing filamentary structure which fragment to form dense cores and individual stars (panel A).The stars fall towards local potential minima and hence form subclusters (panel B). These subclusters evolve by accreting more starsand gas, ejecting stars, and by mergers with other subclusters (panel C). The final state of the simulation is a single, centrally condensedcluster with little substructure (panel D). The cluster contains more than 400 stars and has a gas fraction of approximately 16 per cent.

into the existing potential wells. This process repeats itselfuntil several hundred stars are formed and mostly containedin five subclusters. The further evolution is marked by a de-creasing star formation rate as the subclusters, aided by thedissipative effects of their embedded gas, sink towards eachother and finally merge to form one single cluster containingover 400 stars. The final cluster is approximately sphericalin shape with a centrally condensed core as is observed inyoung stellar clusters (Hillenbrand 1997; Lada 1999).

The hierarchical nature of the formation process is il-lustrated in figure 2, which shows the evolution of the localand global stellar number-densities for the cluster. The localstellar density is calculated for each star from the minimumvolume required to contain the star’s ten nearest neighbours.We use the median value of this distribution to quantify atypical local stellar density. In contrast, the global stellarnumber-density is calculated from the volume required tocontain half of the total number of stars. This typifies the

Bonnell,Bate&Vine(2003)

A&A 536, A79 (2011)

Herschel/PACS

0.2pc

Herschel/Spire

JHK Spitzer/IRAC

Fig. 9. Three-color synthetic images of the simulation, from the near-infrared (top left) and mid-infrared (top right) to the far-infrared (bottom leftand right). The images are binned to the pixel resolutions appropriate for ground based observations (JHK) and for Spitzer and Herschel obser-vations respectively. The images were convolved with the appropriate instrumental PSFs, and include Gaussian noise with realistic levels. Thewavelengths, stretches, and levels used for the images are identical to those used in Fig. 8.

A79, page 14 of 17

Robitaille(2011)

MonteCarloRadiaPveTransfer

SmoothedParPcleHydrodynamics

MCRTRecap

SPH MCRTMovesparPclestonewposiPonsbasedonforces.

Propagateslightthroughadensitygrid.

ParPcleposiPons,densitystructure

Thermalenergydepositedinthe

parPcles

SPHandMCRT

SPH MCRTMovesparPclestonewposiPonsbasedonforces.

Propagateslightthroughadensitygrid.

ParPcleposiPons,densitystructure

Thermalenergydepositedinthe

parPcles

SPHandMCRT

LagrangianvsEulerianCloud

SPH MCRT

VoronoitessellaPon

VoronoitessellaPonGas and dust with Voronoi MCRT 763

Figure 2. Number density slice (at z = 0.0) of atomic hydrogen (in cm−3) from the Dale et al. (2012) simulation described in Section 4 for the Cartesian(left-hand panel) and the Voronoi (right-hand panel) versions of MOCASSIN. We note the far higher resolution in the dense filamentary structures for the Voronoirendition for exactly the same number of cells. In contrast, the Cartesian version has unnecessary more resolution in the low-density expanses in between thedense structures. Positions are measured in pc.

from massive stars on turbulent molecular clouds of a range ofmasses and radii. The model clouds are initialized as smooth sphereswith a mild centrally condensed Gaussian density profile and animposed divergence-free supersonic turbulent velocity field with aBurgers power spectrum. The gas rapidly responds to the velocityfield by developing complex filamentary structure, with the densestregions eventually fragmenting to form stars. Clusters of stars areoften found at filament junctions accreting gas from the filaments,which serve as accretion flows.

Once a few O-type stars form, each calculation is forked into acontrol run which continues as before and a feedback run wherethe ionizing radiation from the massive stars is modelled using theStromgren volume algorithm described in Dale et al. (2007). Bothcalculations are then permitted to continue for as close to 3 Myr aspossible. The complex environment in which the ionizing sourcesare found is a challenging test for a radiative transport algorithm.

We chose the end state of the Run I calculation, a 104 M⊙ cloudevolved under the influence of ionizing radiation for ≈2.2 Myr andhosting, by the end of the simulation, six ionizing stars. Four ofthe massive stars are located in a dense cluster. They have largelydestroyed the accretion flows feeding the cluster, eroding them intoconical inward-pointing pillar-like formations. The expanding H II

regions have also excavated an irregularly shaped bubble occupyinga large fraction of the cloud volume.

The original SPH calculation initially used 106 particles. Wechoose a simulation snapshot 2.2 Myr after the initiation of ioniza-tion and discard the low-density gas at the edges of the simulatedcloud, focusing on a cube centred on the origin and side length 30 pcwhich contains all the massive ionizing sources and the dense coldgas swept up by the expanding H II region. This region contains≈6.59 × 105 SPH particles (some of the original 106 have beeninvolved in star formation and some have been expelled from thecubic volume by expanding ionized bubbles), which were then usedto define the centres of the Voronoi grid. The density inside eachVoronoi element was calculated by simply dividing the mass of theparticle by the volume of the Voronoi element.

The Cartesian density grid was constructed by imposing a uni-form 873 grid inside the 30 pc box, resulting in ≈6.59 × 105 cells,so that the number of resolution elements in the two representa-tions of the density field was the very close. For each SPH particle,

a list of all the grid cell centres overlapped by the particle wasgenerated and a standard SPH density sum using the cubic splinekernel (Monaghan & Lattanzio 1985) was performed to computethe contribution of the particle to the density inside each cell in theCartesian grid. The contributions were normalized to ensure thatthe total mass given to all cells by each particle was equal to theparticle mass. In the cases where particles were smaller than thegrid size and did not overlap any cell centres, the particle’s entiremass was smeared out over the volume of the cell containing theparticle’s centre.

In the left-hand panel of Fig. 2, we show a surface density plot ofall gas in the 3D Cartesian grid, showing that the density structureis complex, but that the Cartesian representation results in clearpixelization. The right-hand panel of Fig. 2 shows the same for theVoronoi density field derived from constructing a Voronoi tessella-tion directly on the SPH particle locations. It is very clear that thelatter is able to resolve much finer detail, despite using exactly thesame number of resolution elements, owing to the adaptive spatialresolution offered by the Voronoi technique.

The loss of resolution in the Cartesian case has significant conse-quences for the temperature structure derived by the photoionizationmodelling. This is demonstrated in Fig. 3, showing a slice in elec-tron temperature through the xy plane (z = 0) of the 3D Cartesian(left-hand panel) and a slice through the equivalent location in theVoronoi mesh (right-hand panel). In the next section, we will dis-cuss the implications on synthetic spectra, particularly with regardto commonly used emission line diagnostics.

4.1 Emission lines

The results for a number of selected emission lines together with thevolume-averaged mean electron temperature weighted by the protonand electron densities and the mean fractional He+ to fractionalH+, ⟨He+⟩/⟨H+⟩, are compared for the classic and Voronoi runsin Table 3. As expected, the total Hβ line luminosity in the twocalculations, which have the same input ionizing luminosity andthe same total mass, is very similar. However, the temperature inthe ionizing region and the ionization level differ somewhat. TheVoronoi run is hotter on average and shows a higher ionization level.This is shown by the (T[Np Ne]) value quoted in Table 3, which

MNRAS 456, 756–766 (2016)

at University of St A

ndrews on A

pril 15, 2016http://m

nras.oxfordjournals.org/D

ownloaded from

Hubber,Ercolano&Dale(2016)

AnSPHparPcleanditskernel

W (r,h) = 1h3π

1−1.5 rh⎛

⎝⎜⎞

⎠⎟2

+ 0.75 rh⎛

⎝⎜⎞

⎠⎟3

, r ≤ h

0.25 2− rh

⎛

⎝⎜

⎞

⎠⎟3

,h ≤ r ≤ 2h

0, r ≥ 2h

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

0.5 1.0 1.5 2.0 2.5 3.0

r

h

0.05

0.10

0.15

0.20

0.25

0.30

W

SPHdensitysum

ρ(!r ) = miW (!r − !ri,h)

i=1

N

∑

Voronoicelldensity

HowdoweintegrateasphericallysymmetricfuncPonoverthe

volumeofanyrandompolyhedron?

HowdoweintegrateasphericallysymmetricfuncPonoverthe

volumeofanyrandompolyhedron?

Can

(analy&cally)

Yes.Andthisishow.

DivergenceTheorem

Green’sTheorem

f (x)dx = F(b)−F(a)a

b

∫

DivergenceTheorem

Fr =1r2

r2W (r)dr∫ =1r2

1h3π

13r3 − 3

10h2r5 + 1

8h3r6, r ≤ h

1483r3 − 3

hr4 + 6

5h2r5 − 1

6h3r6 − h

3

15⎛

⎝⎜

⎞

⎠⎟,h ≤ r ≤ 2h

h3

4, r ≥ 2h

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

W (r) = 1h3π

1−1.5 rh⎛

⎝⎜⎞

⎠⎟2

+ 0.75 rh⎛

⎝⎜⎞

⎠⎟3

, r ≤ h

0.25 2− rh

⎛

⎝⎜

⎞

⎠⎟3

,h ≤ r ≤ 2h

0, r ≥ 2h

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

∇⋅!F dV

V∫ =!F ⋅ n dA

∂V∫

W dVV∫ = ∇⋅

!F dV

V∫

!F = Frr

Green’sTheorem

HR =1R

Fr sinθ dR∫ =1R

r03

h3π

16µ−2 −

340

r0h⎛

⎝⎜

⎞

⎠⎟2

µ−4 −140

r0h⎛

⎝⎜

⎞

⎠⎟3

µ−5 +B1r03 ,µ ≥

r0h

1443µ−2 −

r0h⎛

⎝⎜

⎞

⎠⎟µ−3 +

310

r0h⎛

⎝⎜

⎞

⎠⎟2

µ−4 −130

r0h⎛

⎝⎜

⎞

⎠⎟3

µ−5 +115

r0h⎛

⎝⎜

⎞

⎠⎟−3

µ⎛

⎝⎜⎜

⎞

⎠⎟⎟+

B2r03 ,r02h

≤ µ ≤r0h

−14r0h⎛

⎝⎜

⎞

⎠⎟−3

µ +B3r03 ,µ ≤

r02h

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

∇⋅!H dA

A∫ =!H

∂A"∫ ⋅ mdl

!F ⋅ n dA

∂V∫ = ∇⋅!H dA

A∫!H = HR

!R

µ = cosθ = r0r

FinalsoluPonI0 =ϕ +C

I1 = −sin−1

1+ r02

R02 cos

2ϕ

1+ r02

R02

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

+C

I−2 =ϕ +r02

R02 tanϕ +C

I−4 =ϕ + 2r02

R02 tanϕ +

13r04

R04 tanϕ sec2ϕ + 2( )+C

α =R0r0

µ =

r0R0cosϕ

1+ r02

R02 cos

2ϕ

u = 1− (1+α 2 )µ 2

I−3 =α(1+α 2 )

42u1−u2

+ log(1+u)− log(1−u)⎛

⎝⎜

⎞

⎠⎟+

α2log(1+u)− log(1−u)( )+ tan−1 u

α

⎛

⎝⎜

⎞

⎠⎟+C

I−5 =α(1+α 2 )2

1610u− 6u3

(1−u2 )2+3 log(1+u)− log(1−u)( )

⎛

⎝⎜

⎞

⎠⎟+

α(1+α 2 )4

2u1−u2

+ log(1+u)− log(1−u)⎛

⎝⎜

⎞

⎠⎟+

α2log(1+u)− log(1−u)( )+ tan−1 u

α

⎛

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⎞

⎠⎟+C

B1 =r03

4−23+310

r0h⎛

⎝⎜

⎞

⎠⎟2

−110

r0h⎛

⎝⎜

⎞

⎠⎟3⎛

⎝⎜⎜

⎞

⎠⎟⎟

B2 =r03

4

−23+310

r0h⎛

⎝⎜

⎞

⎠⎟2

−110

r0h⎛

⎝⎜

⎞

⎠⎟3

−15r0h⎛

⎝⎜

⎞

⎠⎟−2

, r0 ≤ h

−43+r0h⎛

⎝⎜

⎞

⎠⎟−

310

r0h⎛

⎝⎜

⎞

⎠⎟2

+130

r0h⎛

⎝⎜

⎞

⎠⎟3

−115

r0h⎛

⎝⎜

⎞

⎠⎟−3

,h ≤ r0 ≤ 2h

⎧

⎨

⎪⎪

⎩

⎪⎪

B3 =r03

4

−23+310

r0h⎛

⎝⎜

⎞

⎠⎟2

−110

r0h⎛

⎝⎜

⎞

⎠⎟3

+75r0h⎛

⎝⎜

⎞

⎠⎟−2

, r0 ≤ h

−43+r0h⎛

⎝⎜

⎞

⎠⎟−

310

r0h⎛

⎝⎜

⎞

⎠⎟2

+130

r0h⎛

⎝⎜

⎞

⎠⎟3

−115

r0h⎛

⎝⎜

⎞

⎠⎟−3

+85r0h⎛

⎝⎜

⎞

⎠⎟−2

,h ≤ r0 ≤ 2h

r0h⎛

⎝⎜

⎞

⎠⎟−3

, r0 ≥ 2h

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

HRRdϕ∫ =r03

h3π

16I−2 −

340

r0h⎛

⎝⎜

⎞

⎠⎟2

I−4 −140

r0h⎛

⎝⎜

⎞

⎠⎟3

I−5 +B1r03 I0,µ ≥

r0h

1443I−2 −

r0h⎛

⎝⎜

⎞

⎠⎟I−3 +

310

r0h⎛

⎝⎜

⎞

⎠⎟2

I−4 −130

r0h⎛

⎝⎜

⎞

⎠⎟3

I−5 +115

r0h⎛

⎝⎜

⎞

⎠⎟−3

I1⎛

⎝⎜⎜

⎞

⎠⎟⎟+

B2r03 I0,

r02h

≤ µ ≤r0h

−14r0h⎛

⎝⎜

⎞

⎠⎟−3

I1 +B3r03 I0,µ ≤

r02h

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

GraphicRepresentaPonoftheSoluPon Petkovaetal.2018

Voronoicellvertex

Voronoicellwall

ParPcleposiPon

r0

R0φ

GraphicRepresentaPonoftheSoluPon Petkovaetal.2018

ParPcleposiPon

Voronoicellwall

GraphicRepresentaPonoftheSoluPon Petkovaetal.2018

ParPcleposiPon

Voronoicellwall

KernelIntegraPonin2DPetkovaetal.2018

KernelIntegraPonin3DPetkovaetal.2018

h\ps://github.com/mapetkova/kernel-integraPon

NumericalTestsPetkovaetal.2018

ComparisonwiththeCommonDensityCalculaPonMethods

SNshock

Uniformcube

Clumpycloud

Diskgalaxy

Petkovaetal.2018

DensityCalculaPonTimingTestsPetkovaetal.2018

SPH MCRTMovesparPclestonewposiPonsbasedonforces.

Propagateslightthroughadensitygrid.

ParPcleposiPons,densitystructure

Thermalenergydepositedinthe

parPcles

LiveradiaPonhydrodynamics

SPH:Phantom(Priceetal.2017)+

MCRT:CMacIonize(Vandenbroucke&Wood,inpress)

+

Densitymapping:Petkovaetal.2018

LiveradiaPonhydrodynamics(test):D-typeexpansionofanHIIregion

Bisbasetal.2015

LiveradiaPonhydrodynamics(test):D-typeexpansionofanHIIregion

Bisbasetal.2015

LiveradiaPonhydrodynamics(test):D-typeexpansionofanHIIregion

Bisbasetal.2015

LiveradiaPonhydrodynamics(test):D-typeexpansionofanHIIregion

Bisbasetal.2015

LiveradiaPonhydrodynamics(test)

LiveradiaPonhydrodynamics(test)

Soontocome…

Daleetal.2012

MulPplesources

OpenQuesPon:ResoluPon8 Koepferl, Robitaille, Dale, Biscani

Fig. 4.— 2-d density slices of the Voronoi tessellation with di↵erent site methods (s1: top, s2: middle, s3: bottom); (white stars) sinkparticles with ID; (red stars) accreting sink particles and (blue stars) sink particles in the ionizing bubble within a box of 1 pc length.

3.3. Density and Temperature Mapping onto new Sites

In Figure 4, we showed slices of the density for dif-ferent site distribution methods. To make the plot weused for the sites coinciding with SPH particle positionsthe peak density ⇢

i,SPH at that position of the SPH par-ticle. When adding sites to a Voronoi mesh, where noSPH particles were located before, we need to evaluatethe properties such as density or temperature from theSPH distribution. We need to preserve the density andtemperature structure as precisely as possible becauselater those property meshes will be passed to the radia-tive transfer set-up. In this section, we will investigatethree techniques to map the density distribution onto aVoronoi mesh:

3.3.1. Method p1 — Evaluation of SPH Kernel Function

To evaluate the density at a new site j at the positionof the sink particle or at a circumstellar site around asink particle, we compute the SPH density function (fora detailed review, see Springel 2010b) for the N closestSPH particles:

⇢

j,p1=NX

i=1

m

i,SPHWij

(rij

, h

i

). (13)

Wheremi,SPH is the constant SPH particle mass with the

value m

i,SPH = 10�2M� in run I of the D14 SPH sim-

ulations and the kernel function W

ij

(rij

, h

i

) from Eq. 1.We estimate the temperature at a new site j by weight-

ing (Press et al. 1992) the SPH particle temperature

Koepferlet.al(2016)

SPH MCRTMovesparPclestonewposiPonsbasedonforces.

Propagateslightthroughadensitygrid.

ParPcleposiPons,densitystructure

Thermalenergydepositedinthe

parPcles

Summary