Directional Lya equivalent boosting I. Spherically symmetric distributions of clumps
MMon. Not. R. Astron. Soc. , 1–10 (2014) Printed 15 October 2018 (MN L A TEX style file v2.2)
Directional Ly α equivalent boosting I. Sphericallysymmetric distributions of clumps M. Gronke (cid:63) and M. Dijkstra Institute of Theoretical Astrophysics, University of Oslo, Postboks 1029, 0315 Oslo, Norway
Accepted 2014 July 25. Received 2014 July 9; in original form 2014 June 12
ABSTRACT
We quantify the directional dependence of the escape fraction of Lyman- α (Ly α )and non-ionizing UV-continuum photons from a multiphase medium, and investi-gate whether there exist directional enhancements in the Ly α equivalent width (EW).Our multiphase medium consists of spherically symmetric distributions of cold, dustyclumps embedded within a hot dust-free medium. We focus on three models from theanalysis presented by Laursen et al. (2013). We find that for a Ly α and UV-continuumpoint source, it is possible to find an EW boost b ( θ, φ ) > b in a few per cent of sightlines, where ¯ b denotes the boost averaged over all photons. For spatially extendedsources this directional dependence vanishes quickly when the size of the UV emittingregion exceeds the mean distance between cold dusty clumps. Our analysis suggeststhat directional EW boosting can occur, and that this is mostly driven by reducedescape fractions of UV photons (which gives rise to UV-continuum ‘shadows’), andless due to an enhanced Ly α escape fraction (or beaming), in certain directions. Key words: radiative transfer – ISM: clouds – galaxies: ISM – line: formation –scattering – galaxies: high-redshift
Lyman- α emitting galaxies are an important probe of thehigh-redshift Universe. Their strong Ly α radiation makesthem easy to find in narrowband surveys, and allows foreasier spectroscopic follow-up in order to confirm their red-shift. The most distant Lyman- α emitting galaxies havebeen found out to z ∼ . α line is its equivalent width (EW), which is defined as theratio of the total line flux and the surrounding flux density.Ly α radiation from galaxies is predominantly emitted as hy-drogen recombination radiation inside HII regions surround-ing young, hot stars (Partridge & Peebles 1967). Syntheticstellar models can therefore be used to predict its strength(Schaerer 2003; Raiter et al. 2010). These models predictthat ‘conventional’ stellar populations can produce a max-imum EW max = 240 ˚A (Laursen et al. 2013, and referencestherein). Larger EW values have been observed (e.g. Daw-son et al. 2004; Nilsson et al. 2011; Kashikawa et al. 2012),which has led to speculation that these galaxies may harbour‘unusual’ (extremely metal-poor or even metal-free) stellarpopulations (Malhotra & Rhoads 2002; Dijkstra & Wyithe2007). (cid:63) E-mail: [email protected]
However, radiative transfer of Ly α photons through amultiphase medium is a complex process which can affect theEW in non-trivial ways. Neufeld (1991) argued that multi-phase gas, consisting of cold dusty clumps embedded with ahot tenuous medium, can facilitate the escape of Ly α pho-tons from dusty media, and ‘under special conditions’ boostthe EW. This boost requires an enhanced chance of escapefor Ly α photons compared to (non-ionizing) UV-continuumphotons. The so-called ‘Neufeld scenario/mechanism’ relieson the fact that hydrogen atoms can only (efficiently) scat-ter Ly α photons, which can prevent Ly α from penetrat-ing HI clouds. If dust resides in these clouds, Ly α photonscan effectively avoid being absorbed by dust grains. Conse-quently, Ly α photons propagate mainly through the inter-cloud medium (ICM) which contains little or no dust. TheUV-continuum radiation, on the other hand, is not affectedby the ‘shielding’ and instead penetrates deep into the cloudswhere the chance of destruction is much higher.Hansen & Oh (2006) explored the Neufeld-mechanismnumerically with a Ly α radiative transfer code, and showedthat quantitatively dusty clumps can lead to EW-boosting.Recent studies have explored under what physical con-ditions EW boosting occurs: Duval et al. (2014) foundseveral requirements had to be fulfilled for activating theNeufeld-mechanism: a practically empty ICM, slow & uni-form galactic outflows, high dust density within the clumpsand a large number/covering factor of the clumps, and c (cid:13) a r X i v : . [ a s t r o - ph . GA ] A ug M. Gronke and M. Dijkstra concluded that such conditions are “quite unlikely/difficultto find”. Laursen et al. (2013) (hereafter, L13) obtaineda similar conclusion after studying a wide range of morerealistic models. L13 conclude that they consider “theNeufeld model to be an extremely unlikely reason for theobserved high EWs”. Although L13 comment on directionalvariations in the EW-boost, b has so far always calculatedas an average value, denoted with ¯ b . This simplification isunderstandable, as resolving the directional dependence re-quires a much larger number of photons in the Monte Carlo(MC) simulation, which makes it practically impossible toexplore a large parameter space (which was the goal inLaursen et al. (2013) & Duval et al. (2014)). However, itis important to recall that observations do not measure anaveraged boost parameter per galaxy, but instead probe b ina specific direction. This means even when realistic modelspredict an average EW boost of ¯ b ∼
1, there may be distinctlines of sight with larger values of b . As a consequenceof this directional dependence, radiative-transfer inducedEW-boosting may still provide a relevant explanation forlarge EWs, even when the angle-averaged value suggeststhat no-boost occurs .Throughout this paper, we define the directional depen-dent boost factor b ( θ, φ ) as b ( θ, φ ) ≡ f esc ,Lyα ( θ, φ ) f esc ,UV ( θ, φ ) , (1)where f esc ,Lyα ( θ, φ ) denotes the escape fraction of Ly α pho-tons in direction ( θ, φ ), f esc ,UV ( θ, φ ) that of UV-continuumphotons . We refer to the “average” boost, ¯ b , as the boostcalculates using all photons. This ‘photon-weighted’ boosthas been the subject of previous studies. We stress that thisaverage quantity can be different from the ‘angle-averaged’boost – defined as (cid:104) b (cid:105) ≡ π (cid:82) d Ω b ( θ, φ ) – as ¯ b puts moreweight on sight lines along which more photons escape.In order to study the directional dependence of b ( θ, φ ),we bin the Ly α and UV-continuum escape fractions inequiareal bins using the HEALPix tessellation and obtaina measure for b ( θ, φ ). As mentioned previously, this task re-quires a large amount of photons per geometrical setup, andwe therefore focus on only three sets of parameters.In the current paper, we focus on spherical distribu-tions of clumps following Laursen et al. (2013, and alsoDuval et al. 2014). Clearly, anisotropic gas distributions(such as discs, biconical outflows, etc) will give rise tomore anisotropic escape of Ly α photons, and we will studythis in a forth-coming paper. This study contains the firstcalculations of Ly α and UV-continuum escape over thefull 2D plane of the sky. Previous studies either presentedazimuthally averaged quantities (Laursen & Sommer-Larsen2007; Verhamme et al. 2012; Zheng & Wallace 2013; Behrenset al. 2014) or focused solely on Ly α escape (Yajima et al.2012). Our study shows that generating 2D maps of Finkelstein et al. (2008, 2009) characterize the EW boost with a‘ q parameter’ which is defined as f esc ,Lyα ≡ exp( − qτ a , eff ). Here, τ a, eff is an ‘effective’ absorption optical depth given by f esc , UV ≡ exp( − τ a, eff ). The parameter q is therefore related to b via b =exp( − qτ a , eff ) / exp( − τ a , eff ), and b > b <
1] corresponds to q < q > http://healpix.sourceforge.net EW-boosting introduces new complications to the analysis.The benefit of using spherical clump distributions is that itallows us to ( i ) connect easily to previous studies (Laursenet al. 2013; Duval et al. 2014), and ( ii ) more easily identifykey quantities that are relevant for directional EW-boosting.The paper is organized as follows: in Sec. 2 we givean overview over the model parameters, present our mod-els, and, briefly outline our new radiative transfer code. Wepresent our results in Sec. 3 and discuss them in Sec. 4.Finally, we conclude in Sec. 5. To facilitate comparison with previous work, we closely fol-low the analysis of L13. In their analysis, a galaxy is rep-resented by a spherical distribution of clumps embeddedwithin a hot dust-free medium, both of which extend out toradius r gal . L13 provide an extensive analysis of what arereasonable values for the parameters describing the mul-tiphase interstellar medium of this galaxy. Specifically, weadopt the parameters shown in Table 1 and described insection 3 of their paper. These parameters fall in four cate-gories:(i) Inter-Cloud Medium (ICM) Parameters. Parametersdefining the hot ICM are the hydrogen number density n HI , ICM , and the temperature T ICM . The ICM extends outto r gal and is assumed to be dust-free.(ii) Emission Parameters. The initial distance from thecentre of the simulation box/galaxy is drawn from r ∼ exp( − r/H em ) distribution. Additionally, photons are emit-ted with a probability of P cl within a cloud. The emissionfrequency is drawn from a Gaussian with standard deviation σ em .(iii) Cloud geometry. L13 ran roughly half of the simu-lations with constant cloud sizes, while in the other half,cloud sizes were drawn from an exponential distribution. Inthe following, we focus solely on clouds wit a constraint size r cl . L13 found this choice had little impact on the radiativetransfer. A key parameter is the covering factor f c , definedas the average number of clouds from the centre to the edgeof the distribution (as in L13) . The volume filling factor F cl of clumps – which is the fraction of the galactic vol-ume covered by clouds – relates to the covering factor as F cl = 4 r cl f c / (3 r gal ).This category also contains the parameters for the HInumber density inside the cloud, n HI , cl , the gas tempera-ture in the clouds, T cl , and their dust content. We quantitythe dust content of a cloud via its ‘absorbing’ optical depthfrom the centre to the surface of a cloud, τ a, cl . This absorb-ing optical depth relates to the total optical depth in dust, τ d, cl , via τ a, cl = (1 − A ) τ d, cl , where A is the dust albedo.Throughout this work, we used A = 0 .
32 (Li & Draine 2001).The total dust optical depth relates to gas metallicity Z as This differs from the definition for f DK12c ≡ n cl πr adoptedin Dijkstra & Kramer (2012), in which n cl denotes the numberdensity of clouds. The two covering factors are related via f c = (cid:82) r gal d s f DK12c ( s ). c (cid:13)000
32 (Li & Draine 2001).The total dust optical depth relates to gas metallicity Z as This differs from the definition for f DK12c ≡ n cl πr adoptedin Dijkstra & Kramer (2012), in which n cl denotes the numberdensity of clouds. The two covering factors are related via f c = (cid:82) r gal d s f DK12c ( s ). c (cid:13)000 , 1–10 irectional Ly α Equivalent Boosting τ d,cl = n HI ,cl r cl σ d Z/Z (cid:12) . Here, Z (cid:12) and σ d denote the Sun’smetallicity and the dust cross-section, respectively.(iv) Cloud motion. Cloud motion plays a crucial role inhow deep the Ly α photons travel into the clouds and, hence,their chance of absorption. As L13, we allow two types ofcloud motion: ( a ) a symmetric outflow with the radial de-pendent velocity. The magnitude of the outflow is v ( r ) = v ∞ (cid:40) − (cid:18) rr min (cid:19) − α (cid:41) / (2)for r > r min and otherwise zero. This three parameter out-flow model follows from the assumption that the acceler-ation of the cloud decreases proportional to r − α (as intro-duced by Steidel et al. 2010). Instead of using V out = v ( r gal ),we stick to the notation by Dijkstra & Kramer (2012) with v ∞ = v ( r → ∞ ); ( b ) a superimposed velocity dispersionwith standard deviation σ cl .The twelve variables described above span a vast parameterspace which has been explored intensively by L13. Instead oftrying to cover as much of the parameter space as possible,here we focus on three “cornerstone” parameter sets. • First, we study the fiducial model from L13. The fidu-cial model contains 6500 clouds with r cl = 100 pc out to r gal = 5 kpc. This results in a volume filling factor of F cl = 0 .
052 and a covering factor f c = 2. The clouds con-tain a hydrogen number density n HI , cl = 1 cm − , and dustresulting in τ a, cl = 0 .
48. This model was designed to be assimple as possible, and we assume that the ICM is empty,that the clouds are static, and that all photons are emittedfrom a central source with an initial frequency of x i = 0.Furthermore, we set the temperatures to T cl = 10 K and T ICM = 10 K. The photon-averaged EW boost for the fidu-cial model is ¯ b ≈ • We chose the second set of parameters to lie withina range which L13 describes as ‘realistic’. This rangeis selected to represent a model that is closer to real-ity. This model has ( n HI , cl , T cl , τ a, cl , F cl , n HI , ICM , T
ICM ) =(0 .
35 cm − , . × K , . × − , . , − cm − , K),which lies in the centre of the parameter range quoted tobe reasonable by L13. The parameters r cl and r gal are un-changed compared to the fiducial model. The main differ-ences are that the clouds have a velocity dispersion of σ cl =40 km s − , a “momentum based outflow” with parameters( v ∞ , r min , α ) = (40 km s − , , . H em = 1 kpc,and roughly a third of the photons were forced to start theirpropagation within a cloud, i.e., P cl = 0 .
35. Moreover, thephoton is not emitted with x i = 0 but instead the emissionline width is σ em = 40 km s − . These values lead to a valueof ¯ b (cid:46) • The third parameter set lies in the range which L13label as “extreme, but possibly conceivable” and is, there-fore, labelled as ‘extreme model’. The cloud motion modeas well as the emission site distribution is the same as in the‘realistic model’ but the parameters are altered to ( n HI , cl ,T cl , τ a, cl , F cl , n HI , ICM , T
ICM , σ cl , v ∞ , σ em , P cl ) = (1 . − , K , . , . , − cm − , K ,
14 km s − ,
23 km s − ,
15 km s − , . b > b ( θ, φ ) for individ-ual sight lines, and the directional binning is of impor-tance. Since the solid angle of a telescope as seen from theLy α emitting galaxy is basically zero, the number of pixelsshould be as high as possible. On the other hand, to preventPoisson-noise from dominating our statistics, we would likethe number of photons per directional bin to be (cid:38) ∼ Ly α and ∼ UV photons and increase the number of binsuntil the b distribution reaches convergence (when this ispossible: we will see below that this is not always the case). The computations were performed with a newly createdMonte Carlo (MC) radiative transfer code tlac . The codewas developed after the ingredients described, e.g., inLaursen (2010) and Dijkstra (2014). We adopt the stan-dard convention of expressing frequencies in terms of x ≡ ( ν − ν ) / ∆ ν D = c ( ν/ν − /v th – where v th is the thermalvelocity of the hydrogen atoms and ν the Ly α line centre.We emit a photon in a random direction (cid:126)k . We thengenerate τ from the distribution P ( τ ) = exp( − τ ). Weconvert τ into the distance d the photon propagates be-fore interacting with a hydrogen or dust particle via τ = (cid:82) d d s [ σ HI n HI ( s ) + σ d n d ( s )]. We decide whether a photoninteracts with HI or dust by comparing a uniform ran-dom variable R to the probability for interaction with dust P dust = σ d n d / ( σ HI n HI + σ d n d ). In case of dust-interaction,we compare a second uniform random variable to the dustalbedo A to determinate if a scattering or absorption occurs.We generate a new random direction of the photon fromthe proper ‘phase function’ P ( µ )d µ , which gives the proba-bility that the cosine between the old and the new direction µ ≡ k in · k out lies in the range µ ± d µ/
2. Here, k in ( k out )denotes the propagation direction before (after) scattering.For resonant scattering via the 2 P / state (probability of1 /
3) gives a rise to uniform scattering, whereas scatteringvia the 2 P / state (probability of 2 /
3) results in a scat-tering with P core , / ( µ ) = 7 / µ / P wing ( µ ) = 3 / µ ). Dust scattering isimplemented using the Henyey & Greenstein (1941) phasefunction with an asymmetry parameter of g = 0 . x → x (cid:48) accordingto x (cid:48) = x − u (cid:107) + u (cid:107) µ + u ⊥ (cid:112) − µ . (3)We choose a reference frame so that the velocity of the atomin units of v th is ( u (cid:107) , u ⊥ ,
0) and u (cid:107) is aligned to the incomingphoton’s direction. Furthermore, u (cid:107) is generated from u (cid:107) ∼ e − u (cid:107) / (cid:0) ( x − u (cid:107) ) + a (cid:1) (4)and u ⊥ is drawn from a centreed Gaussian with standarddeviation of σ = 1 / √
2. Using a truncated Gaussianinstead speeds up the code tremendously (Ahn et al. 2002).However, care should be exercised when applying this “core c (cid:13) , 1–10 M. Gronke and M. Dijkstra
Table 1.
Overview of the results from the three models. N p,Lyα N p,UV ¯ f esc ,Lyα ¯ f esc ,UV ¯ b fiducial 1 . × . × .
93 0 .
38 2 . . × . × .
74 0 .
89 0 . . × . × .
33 0 .
21 1 . skipping technique”, as it may force photons unnaturallydeep into the clouds which leads to lower values of b (see L13 for details). To be conservative we completelyrelinquish the acceleration scheme.In order to calculate radiative transfer within a clumpymedium, several algorithms have been developed and em-ployed previously. One approach is to treat the propagationthrough the ICM exactly and whenever a cloud is encoun-tered, the cloud is treated as a virtual particle with its ownphase function and redistribution function. This methodwas used successfully by Hansen & Oh (2006). In this casephoton trajectories are followed only in the hot interclumpmedium which keeps the computational cost low. The down-side of this approach is that the phase-and-redistributionfunctions depend on the clump properties (e.g. whetherthere are density and/or velocity gradients), and would needto be evaluated first. L13 took a different approach and mod-eled the path of the photons within the clouds too, as thecode utilized by L13 supports a refined mesh structure. Thismade it possible to model spherical clouds with a high refine-ment level at the their surfaces. A third possibility for cloudtreatment is to handle clouds as virtual particles, similarto the first approach, but to explicitly track photons insideclumps. Thus, whenever a photon encounters a cloud, thecloud is represented by concentric spherical shells throughwhich we propagate the photon . This gives the full solutionwithout the memory requirement needed in order to resolveeach cloud separately. This approach was used by Dijkstra& Kramer (2012) and will be also employed in this work.Our code was tested intensively, and was successfullycompared against analytic solutions for the Ly α spectraemerging from extremely optically thick slabs (as in Neufeld1990; Harrington 1973) and spheres (as in Dijkstra et al.2006). It passed the ICM propagation tests described in Di-jkstra & Kramer (2012). We also computed N – the averagenumber of spherical clumps encountered in the absence ofdust – as a function of f c and found that it agreed perfectlywith the calculations by Hansen & Oh (2006), who foundthat N = f + f c (see Eq. 62 in Hansen & Oh 2006).Finally, we compared the results obtained for the fiducialmodel to the code developed for the analysis in (Dijkstra &Kramer 2012), and found good agreement. Table 1 shows the total number of photons used and theresulting escape fractions. In the following, we will presentand discuss the results from each model individually. This allows us to include velocity and/or density gradients in-side clouds, but these were absent in our current analysis.
Our fiducial model yields escape fractions of 0 .
925 and 0 . α and UV-photons, respectively. This results in aphoton averaged boost factor of ¯ b = ¯ f esc ,Lyα / ¯ f esc ,UV ≈ . ∼ . Also, the application of anacceleration scheme by L13 could lead to slight variationsas discussed above. However, because we study fluctuationsaround ¯ b its numerical value is not very important.We show our main results in Fig. 1. The upper pan-els show the number of Ly α ( left panel ) and UV-photons( right panel ) escaping in a given direction. More specifically,we show the deviation from uniform escape, ∆ f , which isdefined as ∆ f esc ( θ, φ ) ≡ n p ( θ, φ )¯ n p − , (5)where n p ( θ, φ ) is the number of photons received in a givendirectional bin, and ¯ n p ≡ ¯ f esc N p /N bins in which N p the totalnumber of photons emitted, and ¯ f esc the photon averagedescape fraction. This definition allows the use of the samecolour coding while keeping the proportionality to n p ( θ, φ )and N p .The Mollweide projections in Fig. 1 are obtained usingthe HEALPix parameter n sides = 128. This means that wehave N bins = 12 n = 196608 equiareal directional bins.This choice leaves the number of Ly α -photons per bin inthe range [301 , , , α photons and (372 , , α escapes more homogeneously than the UV-continuum. Interms of ∆ f esc , the quantiles are ( − . , − . , .
96) forLy α and ( − . , − . , .
7) for the UV-continuum.The lower left panel of Fig. 1 shows the spher-ical distribution of the EW boost factor b ( θ, φ ) = n p,Lyα ( θ, φ ) N p,UV / ( n p,UV ( θ, φ ) N p,Lyα ). Clearly, areas witha relatively high escape fraction for Ly α photons(∆ f esc ,Lyα ∼
0) and a low one for their UV counterparts(∆ f esc ,UV (cid:46) − .
5) results in a high boost factor of b (cid:38) b = 2 . f esc ,Lyα yield mostly a moderate boost of b ∼ f esc ,UV is large.The most prominent area with a large boost is clearlydue to the cloud located closest to the source. In the pro-jection maps of Fig. 1, its centre and contour are markedwith a black cross and black dashed line, respectively. Thiscloud is located in the direction ( θ, φ ) cloud ≈ (2 . , .
45) ata distance of ∼
152 pc.The histogram in the lower right panel of Fig. 1 showsthe exact b occurrences. We plot the cumulative fraction ofsight lines that have a boost greater than b . Different linesrepresent different angular resolution in the sky. The curves L13 noticed significant variations between different realizations(Laursen, private communication)c (cid:13)000
152 pc.The histogram in the lower right panel of Fig. 1 showsthe exact b occurrences. We plot the cumulative fraction ofsight lines that have a boost greater than b . Different linesrepresent different angular resolution in the sky. The curves L13 noticed significant variations between different realizations(Laursen, private communication)c (cid:13)000 , 1–10 irectional Ly α Equivalent Boosting -1.0 -0.5 ∆ f esc Ly α -1.0 -0.5 ∆ f esc UV b EW boost b − − − Ω ( > b ) / π N p,Lyα = 1 . × N p,UV = 1 . × Figure 1.
Results from the fiducial model. We show the deviation from uniform escape, ∆ f esc (defined in Eq. 5), for Ly α ( top left panel )and for the (non-ionizing) UV-continuum photons ( top right panel ). The black cross and the black dashed line denote the centre andcontour of the closest cloud, respectively. Lower left panel:
Map of the boost factor b . The nearest cloud is marked as described above. Lower right panel:
Cumulative distribution for b , i.e. the fraction of the sky that contains a EW-boost > b . From light to dark blue the HEALPix parameter n sides is varied from 4 to 512 which corresponds to ∼ (14 , , , , , , , ) directional bins.The red lines mark the binning used for the spherical maps, n sides = 128 (solid) and its average standard deviation for a given [ b, b + d b ]interval (dashed). Additionally, the vertical, black line denotes the photon-weighted ¯ b . evolve as we increase resolution: this quantifies that increas-ing angular resolution gives rise to larger fluctuations in in-dividual pixels. This is partly because of the reduced numberof photons in each pixel, which increases Poisson noise, andpartly physical. In the Appendix we show a Figure whichshows the same curves, in case where they are determinedcompletely by Poisson fluctuations (Fig. A1). These plotsprovide a quick check that our results are statistically ro-bust.Specifically, from light blue to dark blue, the n sides pa-rameter is increased in powers of two from 4 to 512. Thiscorresponds to N bins ≈ (192 , , × , . × , . × , . × , . × , . × ). The red curve high-lights the n sides = 128 or N bins = 1 . × case which isused in the projection maps. For this curve we show also theaverage Poisson uncertainty of all directional bins within agiven [ b, b + d b ]. We give a more detailed explanation of theerror estimation in the Appendix.In addition, a black vertical line indicates the photon-weighted ¯ b = ¯ f esc ,Lyα / ¯ f esc ,UV which is equivalent to N bins =1. Note, that the n sides = 512 and n sides = 256 are basicallyidentical from the red curve which we interpret as signal of convergence. We tested this by using only a tenth of theavailable data which resulted in indistinguishably curves for n sides = 4 to n sides = 256. The highest binned set ( n sides =512) possessed more fluctuations due to a lower signal-to-noise ratio.Focusing on the red curve in the histogram of Fig. 1,it is clear that ∼
10% of the sight lines have a boost factorof b (cid:38)
8. A small percentage of them even reaches b (cid:38) b ≈ .
4, these largevalues of b are surprising given the spherically symmetricsetup of our problem. For this model we used ∼ . × Ly α photons and ∼ . × UV photons. The photon-weighted escape fractions forthe two species are ¯ f esc ,Lyα ≈ .
74 and ¯ f esc ,UV ≈ .
89 whichyields a boost factor of ¯ b ≈ .
84. This is in agreement withL13 where most of the realistic runs are in 0 . (cid:46) ¯ b (cid:46) . c (cid:13) , 1–10 M. Gronke and M. Dijkstra b EW boost . . . . . . b − − − Ω ( > b ) / π N p,Lyα = 2 . × N p,UV = 6 . × Figure 2.
Results from the realistic model.
Left:
Directional dependence of the boost factor b . Right:
Cumulative distribution functionof b . From light to dark blue the HEALPix parameter n sides is varied from 32 to 512 ( ∼ (111 , , , , ) directional bins).The red solid line marks the binning used for the spherical maps, n sides = 128. Again, the red dashed lines mark the b ± (cid:104) σ b (cid:105) for a given[ b, b + d b ] range. Additionally, the vertical, black line denotes the photon-weighted ¯ b . lines greater than b . For this plot we varied the HEALPix parameter n sides from 32 to 512 resulting in approximately(1 . × , . × , . × , . × , . × ) di-rectional bins. Again, the n sides = 128 curve is highlightedwith red. Also as before, the red dashed lines mark the meanstandard deviation per b -bin, σ b .From these plots, it is clear that the EW boost is muchmore isotropic in the realistic model than in the fiducialmodel. In fact, the b -distribution is indistinguishable frompure Poisson noise around ¯ b . For example, the probability P ( b > ¯ b + σ b ) = P ( b − σ b > ¯ b ) ∼ .
16 which is what wouldbe expected for pure Gaussian noise . Moreover, we foundthat the Ω( > b ) distribution emerging from an empty sim-ulation box - from which directional dependent fluctuationsarise entirely because of Poisson noise- is close to identicalin shape (this distribution is shown in Appendix A).Another striking feature is the lack of convergence ofthe curves displayed in the right panel of Fig. 2. This re-flects the lack of real anisotropic features, and that thesecurves are Poisson-noise-dominated: increasing the numberof directional bins leads to few photons per bin, and hence tolarger uncertainties on the value of b . We find the same lackof convergence in the empty test case shown in AppendixA). For the extreme model we use ∼
30 million Ly α and ∼ ∼
21% UV and ∼
33% Ly α photons escaped the simulation box. This yields Another way to see this is by looking at the (0 . , . , . , , α and (2860 , , , , α and (2870 , , a average boost factor of ¯ b ≈ .
60 which is consistent withthe prediction of L13.We present the results in Fig. 3 as before. At firstglance, the existence of patches in the projection map sug-gests that the EW boost in this case is neither as anisotropicas in the fiducial model, nor as isotropic as in the realis-tic model. More quantitatively, the (0 . , . , .
95) quan-tiles are (30 , ,
63) for Ly α and (3588 , , , , α and (4041 , , > b compared to 10% of thesight lines have (cid:38) b in the fiducial case). Also, we also donot find convergence when we increase the number of bins,in contrast to the fiducial model. We investigated the directional dependence of the EWboost factor b in three models labelled as fiducial, realisticand extreme. The fiducial contains strong anisotropiesin the emerging EW boost factor. The other two yielda more homogeneous b -distribution. Moreover, in thefiducial case the b -distribution converges as we increasethe angular resolution (see Fig. 1). We found over 10% ofsight lines with a boost factor which is three times largerthan the averaged value of ¯ b ≈ .
4. In the realistic modelthis convergence was absent, and consistent with Poissonfluctuations on a uniform distribution. The extreme modelyields an almost-isotropic b -distribution (although here theaveraged EW boost ¯ b > α radiative transfer problems is the hydrogen column c (cid:13)000
4. In the realistic modelthis convergence was absent, and consistent with Poissonfluctuations on a uniform distribution. The extreme modelyields an almost-isotropic b -distribution (although here theaveraged EW boost ¯ b > α radiative transfer problems is the hydrogen column c (cid:13)000 , 1–10 irectional Ly α Equivalent Boosting b EW boost b − − − Ω ( > b ) / π N p,Lyα = 2 . × N p,UV = 4 . × Figure 3.
Same as Figure 3.2 but for the ‘extreme model’ (see text). density, N HI , along a certain line of sight. We therefore com-pute this quantity for each directional bin for n sides = 128( N bins = 12 n ≈ × ) and 10 million randomlydrawn sight lines. These sight lines start at the photons’emission sites, i.e., in the fiducial case from the centreof the simulation box and in the other two models at aposition drawn the spatial distribution (see Sec. 2). For thelatter two models a single directional bin would on averagesample ∼ / (2 × ) = 50 emission locations within thesimulation box in the absence of scattering. Throughout ourdiscussion, we convert the hydrogen column density N HI tothe dust column density or its optical depth. We chooseto use the optical depth of the absorbing dust τ a becausethis parameter controls directly the absorption probabilityand has, hence, a strong impact on the boost value. Thereexists a one-to-one relation between τ a and N HI .Each directional bin has therefore one associated τ a (forthe fiducial model) or (cid:104) τ a (cid:105) (for the realistic and extrememodels). We compare this quantity to the directionalescape fraction f esc ( θ, φ ) = n p / ( N p /N bins ) for each bin(see § θ, φ ). In the fiducial model, (cid:104) τ a (cid:105) = τ a , because allphotons are emitted in the same location. However, in therealistic and extreme models we have (cid:104) τ a (cid:105) is the mean of(one average) 51 sight lines.The fiducial model data (red in Fig. 4) presents theNeufeld mechanism in an exemplary way: while the escapefraction of the UV photons follows closely the theoreticallyexpected value of exp( − τ a ) (solid black line), the Ly α escape rate is unaffected by the directional optical depthand instead constant at f esc ,Lyα ≈ ¯ f esc ,Lyα . In other words,the Ly α photons do not penetrate as deeply into thehydrogen clouds lying on the direct sight line. Instead, theirpath mainly traces the low-density ICM, which results ina higher escape fraction independent of the direction inwhich these photons were first emitted . Consequently, sightlines with high τ a values possess a large EW boost as the difference between f esc ,Lyα and f esc ,UV in Fig. 4 shows .The sight lines of the realistic and extreme model covera much smaller range of (cid:104) τ a (cid:105) in Fig. 4 (shown in blue andgreen, respectively). This is because each (cid:104) τ a (cid:105) represents themean of on average N sight lines /N bins ≈
51 sight lines. Whilein the fiducial case each sight line starts at the identical posi-tion, for a spatially extended source one particular directionsamples ∼
51 randomly selected emission sites throughoutthe cloud, each of which contain their own τ a .This difference introduces two additional distinctions: (i) The scatter of (cid:104) τ a (cid:105) for a given f esc is much larger withan extended source: the 1 − σ dispersion on (cid:104) τ a (cid:105) is only0 .
09 in the fiducial data compared to 0 .
48 and 0 .
57 in therealistic and extreme data, respectively. (ii)
The resulting f esc ,UV depends very weakly on (cid:104) τ a (cid:105) . This is most evidentfor the extreme model (green in Fig. 4), as the (cid:104) τ a (cid:105) -rangeis too small for the realistic model (shown in blue). The ex-treme model typically fulfills f esc ,UV (cid:38) exp( − τ a ), as f esc , UV in a certain direction (and hence a given (cid:104) τ a (cid:105) ) is weightedtowards sight lines that pass though low N HI (we refer tothis as ‘statistical favouring’). That is, UV photons emerg-ing the simulation box in a certain direction are more likelyto originate from a location with τ a (cid:46) (cid:104) τ a (cid:105) . This character-istic favouring of lower density trajectories plays a key rolein the disappearance of the b anisotropy when introducingan extended source. Two departures from this simple correlation are for ( i ) largecolumn densities ( τ a (cid:38) .
5) UV photons escape more efficiently,which arise because UV-continuum photons actually do scatterand can consequently escape in these high- τ a directions; and ( ii )for very small optical depths ( τ a (cid:46) .
3) the Ly α photons areaffected by a similar enhanced f esc . The reason for the formercomes from the characteristic ‘weighting’ of lower-density direc-tions after a scattering event. This ‘weighting’ of trajectories willbe explained in detail below. The rise of f esc ,Lyα is also due tophotons that are scattered into the virtually empty direction –again with the statistically favouring of these sight lines. A mathematical way of phrasing this is via the ‘triangle in-equality’ which states that (cid:104) exp( − τ a ) (cid:105) > exp( −(cid:104) τ a (cid:105) ) (also seeHansen & Oh 2006).c (cid:13) , 1–10 M. Gronke and M. Dijkstra . . . . . . . . . h τ a i f e s c fiducial, Ly α fiducial, UVrealistic, Ly α realistic, UVextreme, Ly α extreme, UV Figure 4.
Directional escape fraction f esc = n p / ( N p /N bins ) ver-sus the average optical depth in this direction (cid:104) τ a (cid:105) . The red scat-ter points show the fiducial, the blue mark the realistic and greendenote the extreme model. Also, the “plus” (in darker colours)and “x” (lighter colours) symbols display the escape fraction ofthe Ly α and UV radiation in this direction, respectively. Theblack line highlights the exp( −(cid:104) τ a (cid:105) ) line. The Ly α data of the realistic and extreme are orderedin a fundamentally different way. In the extreme model,both the Neufeld effect and statistical favouring cause thedata points (shown in dark green in Fig. 4) to lie mainlyabove the exp( −(cid:104) τ a (cid:105) ) line. In the realistic model (shownin dark blue), f esc ,Lyα stretches far below this line. Thisfurther shows that the Neufeld mechanism is not active inthis case. That is, the Ly α photons’ trajectories reach farinto the clouds and a long path is required for eventualescape.In summary, for spherical distributions of dusty clumpstwo requirements have to be fulfilled in order to give rise toan anisotropic boost factor:(i) The photon emitting source may not be too extended.Otherwise, anisotropies that may exist in the UV luminosity– and as a consequence the EW boost – are washed out.(ii) The so called “Neufeld mechanism” has to be active.Without it, the directional distribution of the Ly α photonsfollows that of UV-continuum too closely, and no sight linepossesses an exceptional boost factor.In order to verify these two requirements, we first ran thefiducial model with an extended source and several valuesfor H em . Doing this, we find that already 100 pc < H em <
500 pc is sufficient to wipe out all anisotropies existent forsmaller scale lengths. We expect the limiting value to beroughly the inter-cloud spacing which is ∼
330 pc in thiscase.Secondly, we reversed the setup and simulated a pointsource in the realistic and extreme models. As a result, theaveraged boosts ¯ b increase to ¯ b ∼ . b ∼ .
4, respec-tively. Also, as expected, the b distribution of the modifiedextreme model yields big anisotropies with Ω( > b ) / π ≈ . We have explored the directional dependence of the escape ofLy α and (non-ionizing) UV-continuum from a ‘multi-phase’medium, and have quantified what fluctuations in the Ly α equivalent width (EW) this may introduce. The goal of thisanalysis was to address whether directional fluctuations inLy α EW can lead to substantial departures from the photon-averaged value, and whether this may help to explain ‘un-usually’ large observed values of the EW.Our multiphase medium consisted of spherical distribu-tions of dusty clumps within a hot dust-free medium. Ourmodels are taken from a recent analysis by Laursen et al.(2013), who provide a detailed analysis of what are phys-ically reasonable ranges of the parameters describing sucha clumpy medium. We focus on three models: ( i ) the ‘fidu-cial’ model corresponds to the fiducial model in Laursenet al. (2013), which facilitates comparison with their previ-ous work; ( ii ) the ‘realistic’ model corresponds to a modelwhich contains physically reasonable parameters for themultiphase gas, and which have an angle-averaged EW boost (cid:104) b (cid:105) <
1; ( iii ) the ‘extreme’ model corresponds to one of thefew models in Laursen et al. (2013) in which the Neufeld-mechanism is active, i.e. ¯ b > α Monte-Carlo radiative transfer cal-culations with ∼ − Ly α photons – with the stan-dard acceleration schemes turned off – and ∼ − UV-continuum photons for these three models . We stored thepropagation directions of each Ly α and UV-photon as theyescape from the multiphase gas, and use this information tomake 2D EW-boost maps (shown in Fig. 1– 3).We found that directional variations were very large inthe fiducial model, with ∼
10% [ ∼ b > b ∼ b > b ∼ b ∼ .
4. However, in both extreme andrealistic models the fluctuations were significantly reduced,and especially in the realistic model the fluctuations wereconsistent with Poisson noise. The main reason for this dif-ference is the spatial extend of the Ly α and UV-source inthese models: the fiducial model contained a central pointsource, while the other two models contained sources thatwere spatially extended. This has major implications: weshowed that in the fiducial model the nearest dusty clumpeffectively blocked UV-continuum photons from escaping.This clump thus casts a UV-continuum ‘shadow’ on the sky.In contrast, we found that Ly α photons escaped (much)more isotropically, which therefore enhanced the Ly α EWin directions corresponding to the UV-continuum shadow.Each point source within a simulation induces a similar pat-tern of UV-continuum shadows on the sky. When averagedover a large number of sources, the directionally dependenceof the EW is reduced. We found that the EW-anisotropiesvanish rapidly when the spatial extend of the Ly α and UV-continuum source starts to exceed the mean clump sep- We performed these calculations on ∼
200 cores. Since elevated EW regions correspond to regions of suppressedUV-continuum escape, it is really the spatial extend of the UV-continuum source that is relevant: to boost the EW in regions ofsuppressed UV-continuum escape we need isotropic escape of Ly α photons, which is more easily achieved for a spatially extendedLy α source. In this work however, we have assumed for simplicityc (cid:13)000
200 cores. Since elevated EW regions correspond to regions of suppressedUV-continuum escape, it is really the spatial extend of the UV-continuum source that is relevant: to boost the EW in regions ofsuppressed UV-continuum escape we need isotropic escape of Ly α photons, which is more easily achieved for a spatially extendedLy α source. In this work however, we have assumed for simplicityc (cid:13)000 , 1–10 irectional Ly α Equivalent Boosting aration. This presents a key quantity determining in thedirectional-dependence of the EW.Our results can be translated into a physical picturethat directionally-dependent EW boosting can be importantin cases where the source that dominates the UV-continuumluminosity of a galaxy is obscured from us by an HI cloudthat contains dust. In such a scenario, the dusty cloud sup-presses the observed UV-continuum flux. Scattering of Ly α photons off HI atoms in the ISM can cause Ly α photons toescape from the galaxy more isotropically. Large directionalEW boosts would likely still require an efficient (isotropic)escape of Ly α photons, and would thus benefit from havinga multiphase ISM. However, this multiphase ISM would nothave to produce ¯ b > – to give rise to elevated EWvalues in certain directions. In future work, we will explorethis directional EW-boosting in more realistic models of themultiphase ISM, starting with non-spherical clump distri-butions and clump distributions that have a covering fac-tor f c <
1, which appears favoured by observations of low-ionisation UV metal absorption lines (e.g. Heckman et al.2011; Jones et al. 2013). Anisotropic EW boosting should ob-servationally be very similar to isotropic EW boosting in thesense that large EWs correspond to low values for f esc , UV ,and therefore the amount of dust extinction/reddening. Wedefer a detailed study of observational signatures of direc-tional dependent EW-boosting to this future work. ACKNOWLEDGMENTS
We thank Peter Laursen for helpful correspondence, and forproviding details about the work in L13. We also thank thereviewer for the prompt and constructive feedback. Some ofthe results in this paper have been derived using the
HEALPix (Gorski et al. 2005) package.
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APPENDIX A: EMPTY SIMULATION BOX
In order to test the code and the statistical significance ofour findings, we repeated our analysis on an empty sim-ulation box. In this model, Ly α and UV-continuum pho-tons escape isotropically. However, the number of Ly α pho-tons and UV-continuum photons that strike a certain binon the sky fluctuates due to the finite number of photonsutilised in the simulation. For example, for N p,Lyα = 10 Ly α photons and n sides = 128 we expect ab average of¯ n Ly α = 10 / (12 × ) ≈
509 photons per bin and a boostaverage ¯ b = ¯ f esc ,Lyα / ¯ f esc ,UV = 1. The variance on the boost c (cid:13) , 1–10 M. Gronke and M. Dijkstra . . . . . . b − − − Ω ( > b ) / π N p,Lyα = 1 . × N p,UV = 1 . × Figure A1.
The cumulative histogram displayed also in Fig. 1–3,however, for an empty simulation box. b can be estimated from σ = (cid:16) ∂b∂f esc , UV (cid:17) σ f esc , UV + (cid:16) ∂b∂f esc ,Lyα (cid:17) σ f esc ,Lyα (A1)= b (cid:18) n Lyα + 1 n UV (cid:19) , (A2)where f esc ,UV and n UV denote the escape fraction andnumber of photons of UV-continuum photons in one direc-tion, respectively. The escape fraction is given by f esc ,UV = n UV / ( N p,UV /N bins ) with N p,UV being the total number ofUV photons emitted. Naturally, f esc ,Lyα and n Lyα have thesame meaning for Ly α -photons.Because n Lyα and n UV are random Poisson variablesand b is a ratio of both, the probability density function of b is more complicated. However, when assuming a large num-ber of photons per bin ( (cid:38) b -distribution can be found. The resulting dis-tribution is – to leading order – again a Gaussian centreedat ¯ b = 1 with variance σ b = ¯ b (¯ n − Lyα + ¯ n − UV ). For the abovementioned ¯ n Lyα ≈
509 this yields σ ¯ b ≈ .
057 which agreesvery well with our numerical results for the empty box.These considerations allow us to quantitatively under-stand the plot in Fig. A1 in the same fashion as used inFig. 1–3. For instance, we expect 16% [2 . σ ¯b [2 σ ¯b ] above the average. As we increase n sides , weincrease σ ¯b and our plots ‘widen’. Here, the rise in Ω( > b )with increasing number of bins is merely due to the Poissonnoise. Figure A1 nicely illustrates that the curves do notconverge as we increase the angular resolution. More quantitatively, the distribution of b in an empty boxcan be written as b ≈ ¯ b X Lyα / ¯ n Lyα X UV / ¯ n UV ≈ (1 + X Lyα / ¯ n Lyα − X UV / ¯ n UV ) where X ≡ n − ¯ n denote normal distributed variableswith zero mean, and a variance of ¯ n . We further used that X (cid:28) ¯ n .The parameter b can thus be expressed as the sum of the twoGaussian variables, and is thus Gaussian itself with its variancebeing the sum of the two variances (i.e. ¯ n − UV + ¯ n − α ). c (cid:13)000