Lyman Α Radiation Hydrodynamics of Galactic Winds Before Cosmic Reionization
Lyman α Radiation Hydrodynamics of Galactic Winds Before Cosmic Reionization
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Citation Smith, Aaron, Volker Bromm, and Abraham Loeb. 2016. Lyman α Radiation Hydrodynamics of Galactic Winds Before Cosmic Reionization. Monthly Notices of the Royal Astronomical Society 464, no. 3 (October 10): 2963–2978.
Published Version doi:10.1093/mnras/stw2591
Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:42668793
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Lyman α radiation hydrodynamics of galactic winds before cosmic reionization
Aaron Smith,1? Volker Bromm1 and Abraham Loeb2 1Department of Astronomy, The University of Texas at Austin, Austin, TX 78712, USA 2Department of Astronomy, Harvard University, 60 Garden Street, Cambridge, MA 02138, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT The dynamical impact of Lyman α (Lyα) radiation pressure on galaxy formation depends on the rate and duration of momentum transfer between Lyα photons and neutral hydrogen gas. Although photon trapping has the potential to multiply the effective force, ionizing ra- diation from stellar sources may relieve the Lyα pressure before appreciably affecting the kinematics of the host galaxy or efficiently coupling Lyα photons to the outflow. We present self-consistent Lyα radiation-hydrodynamics simulations of high-z galaxy environments by coupling the Cosmic Lyα Transfer code (colt) with spherically symmetric Lagrangian frame hydrodynamics. The accurate but computationally expensive Monte-Carlo radiative transfer calculations are feasible under the one-dimensional approximation. The initial starburst drives an expanding shell of gas from the centre and in certain cases Lyα feedback significantly en- hances the shell velocity. Radiative feedback alone is capable of ejecting baryons into the 8 intergalactic medium (IGM) for protogalaxies with a virial mass of Mvir . 10 M . We com- pare the Lyα signatures of Population III stars with 105 K blackbody emission to that of direct collapse black holes with a nonthermal Compton-thick spectrum and find substantial differ- ences if the Lyα spectra are shaped by gas pushed by Lyα radiation-driven winds. For both sources, the flux emerging from the galaxy is reprocessed by the IGM such that the observed Lyα luminosity is reduced significantly and the time-averaged velocity offset of the Lyα peak is shifted redward. Key words: galaxies: formation – galaxies: high-redshift – cosmology: theory.
1 INTRODUCTION tral hydrogen is opaque to the Lyα line, photon trapping effec- tively acts as a force multiplier applied to gas surrounding H ii re- Radiation from the first stars and galaxies initiated a dramatic trans- gions. Indeed, the role of Lyα radiation pressure throughout the formation throughout the Universe, marking the end of the cos- galactic assembly process has been discussed extensively, particu- mic dark ages (Bromm & Yoshida 2011; Loeb & Furlanetto 2013). larly in the context of other feedback mechanisms (e.g. Cox 1985; The observational frontier for high-redshift galaxies has been ex- Haehnelt 1995; Oh & Haiman 2002; McKee & Tan 2008). Such tended into the epoch of reionization (Bouwens et al. 2011; Finkel- discussions tend to focus on order of magnitude estimates based stein et al. 2013; Oesch et al. 2015; Stark et al. 2015; Zitrin et al. on idealized radiative transfer calculations (Wise et al. 2012; Dijk- 2015; Oesch et al. 2016). Furthermore, next-generation observato- stra & Loeb 2008; Milosavljevic´ et al. 2009). Up to this point, ac- ries such as the James Webb Space Telescope (JWST; Gardner et al. arXiv:1607.07166v2 [astro-ph.GA] 25 Oct 2016 curate radiation-hydrodynamics (RHD) simulations incorporating 2006) will probe even deeper into the past and provide essential de- Lyα feedback have not been performed, either because the effects tails about cosmic history. The Lyα transition of neutral hydrogen are considered sub-dominant or the perceived computational costs (H i) plays a prominent role in spectral observations of high-z ob- prohibited such a treatment. However, the nature of this question jects. However, due to the high opacity of pre-reionized gas, direct requires full consideration of the dynamical coupling between mat- detection is challenging. Still, it may be possible to observe the in- ter and radiation. direct signatures of radiatively-driven outflows, including from Lyα radiation pressure which is more prominent in these conditions. The first galaxies were likely atomic cooling haloes whose 4 Within the first galaxies, up to two-thirds of the ionizing pho- virial temperatures activate Lyα line cooling, i.e. Tvir & 10 K tons from massive stars are reprocessed into Lyα radiation (Par- (Bromm & Yoshida 2011). In this framework, the first galaxies tridge & Peebles 1967; Dijkstra 2014). However, because neu- greatly impacted their surroundings as the initial drivers of reion- ization. Furthermore, radiative feedback from Population III or II stars dramatically altered the gas within these relatively low-mass ? E-mail: [email protected] systems. Feedback physics is crucial for understanding the multi-
c 2016 The Authors 2 A. Smith et al. scale connections of astrophysical phenomena and their observa- Much of the uncertainty regarding Lyα dynamics is related tional signatures. So far simulations have focused on thermally- to difficulties in numerical modeling. Monte-Carlo radiative trans- driven supernova (SN) feedback while the impact of Lyα radiation fer (MCRT) has emerged as the prevalent method for accurate Lyα pressure is still relatively unexplored. calculations (Ahn et al. 2002; Zheng & Miralda-Escudé 2002; Di- The physical processes that couple Lyα radiation to gas dy- jkstra et al. 2006). In many cases Lyα transfer codes are used to namics are continuum absorption by dust and momentum transfer post-process realistic hydrodynamical simulations (e.g. Tasitsiomi via multiple scattering with neutral hydrogen. These two mecha- 2006; Laursen et al. 2009; Verhamme et al. 2012; Smith et al. nisms are roughly independent of each other in the sense that high 2015). However, idealized models described by a few basic param- dust content reduces the Lyα escape fraction while an absence eters have also been widely used to study Lyα spectra from mod- of dust results in a pure scattering scenario. Wise et al. (2012) erate redshift galaxies (Ahn 2004; Verhamme et al. 2006; Gronke argue for the existence of a metallicity upper limit such that if et al. 2015; Gronke & Dijkstra 2016). Along these lines, Dijkstra Z & 0.05 Z then Lyα radiation pressure may be ignored because & Loeb (2008) performed the first, direct MCRT calculations of of the increasing impact of dust opacity (see also Henney & Arthur Lyα radiation pressure for various spherically symmetric models 1998). However, it is uncertain whether metallicity thresholds are representing different stages of galaxy formation. The authors de- universally applicable considering the nontrivial nature of Lyα ra- scribe scenarios for supersonic Lyα-driven outflows for ∼ 106 M diative transfer in inhomogeneous, dusty media. Throughout this minihalo environments. They argue that Lyα pressure is too weak paper we approximate first galaxies as metal-free environments so to affect larger (& 109 M ) high-redshift star-forming galaxies, ex- dust effects are not discussed in detail. Nonetheless, even without cept in the case of . 1 kpc galactic supershells in the interstellar absorption, Lyα photon trapping only affects the residual H i within medium as explored in greater detail by Dijkstra & Loeb (2009). ionized regions. Therefore, unless the gas remains neutral for a long We emphasize that each of the above estimates are based on non- enough duration even relatively strong sources are kinematically in- dynamical simulations. Still, we must consider the possibility that consequential. Furthermore, geometric effects such as gas clump- Lyα radiation can drive galactic winds and affect regions with neu- ing, rotation, and filamentary structure often lead to anisotropic es- tral gas throughout the galaxy formation process. In contrast to the cape, photon leakage, or otherwise altered dynamical impact. MCRT method, Latif et al. (2011) use the stiffened equation of state In regions dominated by Lyα radiation pressure we expect proposed by Spaans & Silk (2006) to mimic Lyα pressure effects expansion to set in and eventually limit the impact of subsequent in a cosmological simulation. However, it is unclear whether such feedback. Historically, this was recognized to be important in the an approach faithfully reproduces the complex physics involved context of planetary nebulae as far back as Ambarzumian (1932), with Lyα radiative transfer. The qualitative result is similar to other Zanstra (1934), Struve (1942), and Chandrasekhar (1945). The au- forms of radiation pressure, which collectively regulate star forma- thors found that expansion could lower the Lyα opacity to the ex- tion by driving turbulence and increase the efficiency of supernova- tent that the radiation pressures due to Lyα and Lyman continuum driven outflows (Wise et al. 2012). are the same order of magnitude. Later, Cox (1985) examined Lyα A substantial effort has recently been invested in RHD sim- pressure in the context of providing disc support during the forma- ulations using MCRT, which is often more accurate than other tion epoch of spiral galaxies. Bithell (1990) and Haehnelt (1995) methods but comes at a higher computational cost (e.g. Abdika- claimed further importance in galactic modeling by arguing that malov et al. 2012; Harries 2015; Roth & Kasen 2015; Tsang & fully ionized, self-gravitating objects can be supported by radia- Milosavljevic´ 2015). Indeed, a dynamical approach seems timely tion pressure for characteristic lengths of `α ∼ 100 pc − 3 kpc. for Lyα feedback because of increasingly powerful computational More recently, Dijkstra & Loeb (2008) found that multiple scatter- resources in conjunction with improved theoretical and numerical ing within high H i column density shells is capable of enhancing algorithms (e.g. Mihalas & Mihalas 1984; Mihalas & Auer 2001; the effective Lyα radiation pressure by one or two orders of mag- Castor 2004). Radiation hydrodynamics also provides a more gen- nitude. The static case sets the upper limit on force multiplication eral context for Lyα studies than allowed from post-processing sim- while subsequent acceleration renders an ever diminishing effective ulations. The dynamical context may also provide new information opacity (Dijkstra & Loeb 2009). The exact nature of the dynamics about the likelihood of observations at various stages of galaxy evo- of Lyα-driven winds and the extent of their impact is an intriguing lution, especially before and after reionization. Cosmological RHD yet challenging problem. simulations have undergone significant advances in recent years Still, there are other scenarios in which Lyα feedback may (Jeon et al. 2015; Norman et al. 2015; So et al. 2014). There are play a significant role. For example, Oh & Haiman (2002) argue hints that radiative feedback will boost the escape fraction of ion- that Lyα trapping can constrain the efficiency of star formation izing radiation in the shallow potential wells of the first galaxies, in high-z galaxies. This follows from a discussion by Rees & Os- which may significantly impact their visibility (Pawlik et al. 2013). triker (1977) in which cooling radiation attempts to unbind a sys- These lower mass systems may quickly respond to Lyα radiation, tem in free-fall collapse. In this picture, although radiation pres- ionizing radiation, supernova explosions, and other feedback mech- sure does not overcome collapse it increases the temperature and anisms which modify aspects of the standard picture of galaxy and the boosted Jeans mass is likely to inhibit fragmentation (see also star formation. Latif et al. 2011). Oh & Haiman (2002) conclude that Lyα photon The observed Lyα flux depends on properties of the host pressure is likely to be an important source of feedback until super- galaxy, the intervening IGM, and the external sight line to the detec- nova explosions become dominant. In fact, Lyα radiation pressure tor. Previous studies have shown that asymmetries in the gas distri- may contribute alongside other feedback mechanisms in the forma- bution introduce a directional dependence of the spectrum, which tion of intermediate-mass “seed” black holes (Dijkstra et al. 2008; can be strongly enhanced by the presence of significantly lower- Milosavljevic´ et al. 2009; Smith et al. 2016). Finally, Lyα radiation column density pathways as is the case for the models of Behrens pressure has also been studied in the context of the first stars, where et al. (2014) and Dijkstra et al. (2016). Still, the intrinsic escape of it may reverse outflow along the polar directions but is not likely to Lyα photons from individual galaxies likely deviates from isotropy be significant elsewhere (McKee & Tan 2008; Stacy et al. 2012). by at most a factor of a few, especially when considering Lyα ra-
MNRAS 000,1–16 (2016) Lyα radiation hydrodynamics of galactic winds before reionization 3
11 diative transfer effects due to the larger-scale environment (Smith 10− et al. 2015). Therefore, we focus our study of Lyα feedback on 12 10− spherically symmetric setups, which affects the robustness of obser- 13 ¢ 10 vational predictions but is also more computationally feasible be- 3 − − 10 14 cause the aggregate statistics converge with fewer photon packets. m − c 15 This paper is organized as follows. In Section2 we discuss the cos- g 10− r e 16 mological context and provide analytic estimates for Lyα feedback ¡ 10− 2 effects. In Section3 we present the radiative transfer methodology 5 , 17
α 10− to obtain Monte-Carlo estimates of local dynamical quantities, in- ˙ COLT N 18 cluding tests of our code against known solutions. In Section4 we / 10− U LR99 19 present the remaining hydrodynamics methodology along with ad- 10− L /4πr2c 20 α ditional physics required for our models. In Section5 we present 10−
Lyα RHD simulations of galaxies with different strengths for the 1 0 1 2 3 central starburst or black hole. This is intended to determine the 10− 10 10 10 10 dynamical impact of Lyα radiation pressure during the early stages r (kpc) of galaxy formation. Finally, in Section6 we conclude.
Figure 1. Radial profile of the energy density U(r) in the Lyα radiation field for an expanding neutral IGM. The central source is normalized to an 52 −1 2 THE COSMOLOGICAL CONTEXT emission rate of N˙α = 10 photons s . The blue dotted line represents the free-streaming limit while the red dashed line shows the analytic solution Lyα radiation pressure from the first stars likely had a substantial from Equation (2) which assumes the diffusion approximation at extremely impact on their host environments. Relatively low mass minihaloes low temperature. The increased energy density within r . 1 Mpc is due to 6 with Mvir . 10 M would have been especially susceptible to Lyα photon trapping. The black line is from a COLT simulation. Lyα scattering against the neutral gas surrounding the central H ii region. A similar scenario is also possible for H i “supershells” in interstellar environments within more massive galaxies later in cos- the characteristic scales discussed by Loeb & Rybicki (1999) the mic history. However, the overall impact of Lyα trapping dimin- comoving frequency shift from line centre at which the optical 13 3/2 ishes rapidly as ongoing star formation and AGN activity enlarge depth reaches unity is ν∗ ≈ 1.2 × 10 Hz [(1 + z)/11] so that the ionized bubbles and eventually reionize the Universe. In the re- τIGM(ν∗) ≡ 1. This corresponds to a proper radius r∗ ≈ 1 Mpc for mainder of this section we present analytical estimates of Lyα feed- which the Doppler shift due to Hubble expansion produces the crit- back effects to determine the relative importance in various con- ical frequency shift ν∗. Finally, the relative proper velocity at the −1 texts. In some cases these already serve as test cases of our code. critical radius is v∗ ≡ H(z)r∗|z=10 ≈ 1450 km s . Figure1 represents the energy density as a function of radius and demonstrates that the COLT simulation smoothly connects the 2.1 Sources in a neutral expanding IGM diffusion and free-streaming cases. The analytic expression for the We now consider the radiation pressure due to a Lyα point source angle-averaged intensity is at redshift z = 10 embedded in a neutral, homogeneous intergalac- !3/2 ( ) 1 9 9˜r2 tic medium undergoing Hubble expansion. As the photons scatter J˜ = exp − , (1) π πν3 ν3 and diffuse they eventually experience enough cosmological red- 4 4 ˜ 4˜ shifting that they move far into the red wing of the line profile and where the dimensionless radius, frequency, and intensity are given h 2 i free streaming becomes unavoidable. Loeb & Rybicki (1999) cal- byr ˜ ≡ r/r∗,ν ˜ ≡ ν/ν∗, and J˜ ≡ J/ Lα/(r∗ ν∗) , respectively. The culated an analytic solution for the angle-averaged intensity J(ν, r) energy density in the Lyα radiation field is as a function of radius valid in the diffusion limit. This is an ideal 4π Z ∞ test case because the initial setup is clear, analytic expressions for U(r) = J(ν, r)dν Lyα radiation quantities are readily available, and we may compare c 0 1/3 13 7/3 to previous results obtained by Dijkstra & Loeb (2008). 18 Γ( 6 ) Lα r∗ = 3/2 2 We briefly describe our setup parameters. The neutral hy- 7π c r∗ r −4 −3 drogen background number density is nH, IGM ≈ 2.5 × 10 cm −13 −3 −7/3 ≈ 4.3 × 10 erg cm N˙ α,52 r , (2) [(1 + z)/11]3 throughout the entire domain. The solution from Loeb kpc & Rybicki (1999) assumes a zero temperature limit despite the which is independent of redshift. We have introduced the dimen- 52 −1 higher cosmic background temperature of TCMB ≈ 30 K. Therefore, sionless quantity N˙ α,52 ≡ N˙ α/(10 photons s ) to normalize the in order to minimize thermal effects we use a uniform temperature emission rate of the central Lyα source and the quantity rkpc ≡ of T = 1 K. Our grid consists of concentric spherical shells spaced r/(1 kpc) to normalize the radius. We note that the r−7/3 scaling such that the thickness increases with radius according to ∆r ∝ r1/2. was also found by Chuzhoy & Zheng (2007), who provide an intu- The minimum radius rmin ≈ 50 pc is chosen to achieve a reasonable itive explanation for the exponent. This solution is plotted as the red resolution up to the maximum radius rmax ≈ 10 Mpc where the dashed line and is valid out to ∼ 1 Mpc where the redshifted pho- IGM is optically thin to redshifted Lyα photons. The velocity field tons transition to the optically thin limit. The free-streaming energy 2 follows from an isotropic expansion law of v(r) = H(z)r, where the density is given by Lα/(4πr c) and is plotted as the blue dotted line 1/2 3/2 redshift-dependent Hubble parameter is H(z) ≈ H0Ωm (z + 1) . in Fig.1. Scattering traps the photons and enhances their energy Here we assume a flat matter-dominated high-z Universe with a density at small radii compared to the optically thin case. −1 −1 present-day Hubble constant of H0 = 67.8 km s Mpc and Following Dijkstra & Loeb (2008) we also compare the ra- matter density parameter of Ωm ≈ 0.3. Specifically, in terms of diation force to the gravitational force as a function of radius in
MNRAS 000,1–16 (2016) 4 A. Smith et al.
102 104 Υ ) α = 10 2 Fα > Fgrav if r < req
2 −
5 1 3 , 10 10 α z = 10 ) ˙ N /
grav CR7 9 0 2 , 10 10 Υα F = 1 p L α M
= = 10
( 11 1 α 1 10 L L
10 F − α × ( = 10
v Υ α 9 a = 10 2
r COLT
vir L L g 2 0 10 α F 10− 1 dU = 10 /r
/ Fα = 3nH dr 7 eq α L L r F LR99 1 α = 10 3 10− 10− 5 LR99 L 1 0 1 2 3 10− 10 10 10 10 10 2 r (kpc) −106 107 108 109 1010 1011 1012 Mvir (M )
Figure 2. Ratio of the Lyα radiation force to the gravitational force on a hydrogen atom. The vertical axis is normalized for a point mass of Figure 3. Ratio of the equilibrium radius to the virial radius as given by 9 ˙ 52 −1 Mp = 10 M and an emission rate of Nα = 10 photons s . The red Equation (5). The Lyα radiation force overwhelms gravity for neutral gas dashed line represents the analytic expression from Equation (4) (see also within req. The solid lines have a constant mass to (Lyα) light ratio in so- Fig.1). The black curve is calculated from a COLT simulation via the mo- lar units, i.e. Υα ≡ (Mvir/M )/(Lα/L ), while the dashed lines represent mentum transfer methodology. The grey curve assumes the Eddington ap- fixed luminosity. The contours are meant to guide the eye and represent the proximation to obtain the force from the energy density estimator (see Sec- product of the RMS amplitude of linearly extrapolated density fluctuations tion3). σ and a normal distribution of N(log f?, µlog f? = −3, σlog f? = 0.75).
Fig.2. For simplicity and consistency with the simulation setup we of the equilibrium radius to the virial radius is assume a point mass Mp for the central source, which yields an in- 2 !−5/4 verse square law for the gravitational acceleration, agrav = GMp/r . r 1 + z eq ≈ N˙ 3/4 M−13/12 . The radiation force may be calculated directly by Monte-Carlo es- 14 α,52 p,9 (5) rvir 11 timators as shown by the solid black curve. Additionally, in regions where the Eddington approximation holds we may calculate the ac- Thus, assuming a constant mass to light ratio the potential for Lyα- celeration due to Lyα feedback on the gas as driven outflows is more substantial in less massive haloes. This is illustrated in Fig.3 for haloes at a redshift of z = 10. To bet- !−3 U z 1 d −1 −1 ˙ −10/3 1 + ter interpret lines of constant Υα we include contours of proba- aα ≈ ≈ 80 km s Myr Nα,52rkpc . (3) 3ρH dr 11 ble values based on a universal star formation efficiency f?. If we assume the mass in baryons is M ≈ M Ω /Ω and the mass Thus, the ratio of the radiation force to the gravitational force is b vir b m in Pop III stars is M? ≈ f? Mb then Equ. (5) may be written as !−3 −3 3/4 −1/3 −5/4 F 1 + z req/rvir ≈ 82 [ f?/(10 )] M [(1 + z)/11] . The contours rep- α ≈ ˙ −1 −4/3 p,9 18 Nα,52 Mp,9 rkpc resent the product of the root-mean-square amplitude of linearly Fgrav 11 !−3 extrapolated density fluctuations σ (see Loeb & Furlanetto 2013) −1 −4/3 1 + z − ≈ 4.3 Υ r , (4) and a normal distribution on log f? with mean µlog f? = 3 and α,2 kpc 11 standard deviation σlog f? = 0.75 in the vertical direction. 9 Although the Lyα force can have a greater relative impact than where we have introduced Mp,9 ≡ Mp/(10 M ) as a normaliza- gravity it is important to realize the effect is predicated on the pres- tion for the central point mass and Υα ≡ (Mp/M )/(Lα/L ) as the mass to (Lyα) light ratio in solar units; furthermore, we de- ence of neutral hydrogen gas. Thus, kinematic changes to galactic assembly may be mitigated if the ionization timescale tion is shorter fine Υα,2 ≡ Υα/100. Equation (4) is plotted as the red dashed line in Fig.2. The analytic and simulation curves in figure 2 of Di- than the minimum feedback timescale tα. A simple order of magni- jkstra & Loeb (2008) bend downward at small radii compared to tude estimate for tα is the interval required to accelerate the gas to our case because the authors considered a modified NFW density a velocity of vα, which according to Equation (3) is profile for the dark matter halo. At large radii the Eddington ap- 10/9 !10/3 !−1/3 vα M9 vα,10 r 1 + z proximation used to obtain the grey curve becomes increasingly tα ∼ ≈ 27 kyr , (6) aα N˙ r 11 unreliable. However, the different methods of estimating the Lyα α,54 vir −1 radiation force are consistent with both the analytic expression and where the velocity normalization is vα,10 ≡ vα/(10 km s ), which each other, in their region of respective validity. is comparable to the thermal velocity of the gas. In comparison Finally, we investigate this model in the context of differ- the competing ionization timescale from the source is roughly ent galaxy scenarios. The Lyα radiation force overwhelms grav- tion ∼ Ne/N˙ ion. Here we approximate the number of available elec- ity within a characteristic radius corresponding to the equilibrium trons in primordial gas by Ne ≈ (X + Y/2)Mb/mH, where X ≈ 0.75 point where Frad/Fgrav = 1. In the case of Fig.2 the equilib- and Y ≈ 0.25 represent the mass fraction of hydrogen and helium, ≈ rium radius is req 9 kpc as calculated by the general expression respectively, and Mb is the total mass of baryons being ionized, i.e. ≈ −3/4 −9/4 req 3 kpc Υα,2 [(1 + z)/11] . The virial radius for a halo of Mb ≈ MpΩb/Ωm. The rate of ionizing photons is related to the rate 1/3 −1 ˙ ˙ mass Mvir scales as rvir ∝ Mvir (1 + z) so it follows that the ratio of Lyα photons by Nα ≈ 0.68(1 − fesc)Nion, where fesc is the escape
MNRAS 000,1–16 (2016) Lyα radiation hydrodynamics of galactic winds before reionization 5 fraction of ionizing photons (Dijkstra 2014). The above arguments which is imparted by the source under single scattering. Therefore combine to give the change in velocity is approximately ! N f MFLα∆t e ˙ −1 esc v ≈ tion ∼ ≈ 327 kyr Nα,52 Mp,9 1 − , (7) ∆ 2 N˙ ion 0.08 4πr mHNH ic ≈ −1 −1 −2 which demonstrates that it is possible for Lyα feedback to have 10 km s MF,50 Lα,8 ∆tkyr NH i,19 r100pc , (11) a dynamical impact before the gas becomes ionized by the same where we have used the following for notational convenience: central source. The Lyα timescale is about an order of magnitude 8 MF,50 ≡ MF/50, Lα,8 ≡ Lα/(10 L ), ∆tkyr ≡ ∆t/(1 kyr), NH i,19 ≡ shorter with strong radial dependence but exhibits only weak scal- 19 −2 NH i/(10 cm ), and r100pc ≡ r/(100 pc). Even in the relatively ing with mass and redshift: short time of ∆t ≈ 1 kyr the shell can accelerate to speeds compa- 1/2 −1 10/3 −1/3 ! ! rable to the thermal velocity vth = 12.85 T4 km s and affect the tα 1/9 r 1 + z ≈ 0.084 Mp,9 vα,10 . (8) local hydrodynamics. tion rvir 11 The above calculation neglects recombinations which would slow the propagation of the ionization front. Also, the mass contained 2.4 Impact of ionizing radiation pressure within a given radius could be significantly smaller than the halo We now compare the impact of Lyα radiation pressure to that of mass. Still, as the estimates are based on a post-processing model, ionizing radiation. To simplify the calculation we assume black- we defer further discussion until self-consistent dynamics are con- body emission from a point source in the shell model considered sidered in Section5. in Section 2.3. This spectrum may be parametrized by the bolo- metric luminosity L? and effective temperature Teff. Thus, the flux 2.2 Overcoming the gravitational binding energy through a shell of radius r is given by (Krumholz et al. 2007; Greif et al. 2009)
While the previous discussion applies to neutral gas in the inter- −τν L?e galactic medium we now consider a similar argument that con- Fν = Bν (12) 4σ T 4 r2 nects to the virial halo itself. Rees & Ostriker (1977) considered SB eff a scenario in which Lyα cooling radiation overcomes the grav- where σSB is the Stephan-Boltzmann constant, Bν is the Planck R r itational binding energy for a virialized system. The condition function, and τ ≈ σ n d` = σ N (r) is the optical depth to ν 0 ν H i ν H i 2 Lαttrap & GM /R is roughly equivalent to the requirement that the ionizing photons. If we also employ the nebular approximation then 2 trapping time be longer than the dynamic time, i.e. ttrap > tdyn, each photoionization occurs from the 1 S ground state of H i and 1/3 1/6 6 where ttrap ≈ 15 tlightτ6 T4 for optical depths of τ6 ≡ τ/10 & 1, the frequency-dependent cross section becomes (see equation 2.4 4 and where tlight is the light crossing time with T4 ≡ T/(10 K) of Osterbrock & Ferland 2006) (Adams 1975; Neufeld 1991). This leads to a critical Lyα lumi- !4 13.6 eV exp[4 − 4 ε−1 tan−1 ε] nosity of (Loeb & Furlanetto 2013) σ = σ , (13) ν 0 hν 1 − exp(−2π/ε) !4/3 !2 ! Mvir 1 + z 15 tlight √ L ∼ 39 −1 , −18 2 α,crit 10 erg s 6 (9) where ε ≡ hν/13.6 eV − 1 and σ0 ≈ 6.3 × 10 cm . For the 10 M 11 ttrap gas coupling the absorption coefficient for ionizing radiation is which is roughly equivalent to the luminosity generated by a kν = ρσν/µH, where the mean molecular weight is roughly the 3 10 M Population III star cluster with an escape fraction for ion- mass of hydrogen, i.e. µH ≈ mH. Putting this together we calcu- izing photons of fesc = 0.1. A lower escape fraction implies that late the acceleration due to ionizing radiation Lyα photons are more efficiently produced while a higher escape 1 Z fraction reduces the Lyα luminosity for the same star formation aγ ≡ kνFνdν efficiency f . The value of f = 0.1 is a typical time-averaged es- cρ ? esc Z ∞ 9 L cape fraction in 10 M haloes, although fesc can be much higher ≈ ? σ B e−τν dν 4 2 ν ν for minihaloes (∼ 0.5) or much smaller for more mature haloes (. 4cµHσSBTeffr νmin a few per cent) (see e.g. Gnedin et al. 2008; Wise & Cen 2009; Ya- 15σ L Z ∞ x4 exp[4 − 4 χ−1 tan−1 χ] ≈ 0 ? min e−τx dx jima et al. 2011; Ferrara & Loeb 2013; Paardekooper et al. 2015; 4π5µ r2c x (ex − 1) 1 − exp(−2π/χ) Faisst 2016; Xu et al. 2016). Thus, the maximum star formation H xmin 3 efficiency to avoid unbinding via Lyα feedback is 15σ0L? xmin ≈ Tγ 5 2 xmin − !1/3 !2 ! 4π µHr c e 1 M M 1 + z 15 t f ≡ ? −3 vir light . ≈ 260 km s−1 kyr−1 r−2T , (14) ? . 10 6 (10) pc γ Mgas 10 M 11 ttrap √ where χ ≡ x/xmin − 1 and xmin ≡ 13.6 eV/(kBTeff). The first line is the general definition while the second line incorporates 2.3 Lyα-driven galactic supershells the blackbody approximation from Equation 12. The third line em- As previously mentioned, Lyα radiation pressure does not appre- ploys a change of variables into nondimensional frequency and the ciably affect gas within ionized regions. We may estimate the kine- cross section from Equation 13 for direct integration. The fourth matic effects of photon trapping in the context of the shell model line takes advantage of the fact that at low NH i(r) the integrand is by considering the following parameters: (i) the Lyα luminosity sharply peaked at xmin so we apply a delta function approximation Lα, (ii) the local H i column density NH i, (iii) the shell radius r, and to obtain the coefficient and introduce an ionizing radiation force (iv) the duration that the gas remains neutral ∆t. For simplicity we transmission function Tγ ∈ [0, 1] which we discuss later in this consider a geometry dependent force multiplier, MF, as a way to section. The last line has been evaluated for a 100 M Pop III star 6.095 4.975 absorb any uncertainty in the enhancement to the total force, Lα/c, with L? = 10 L and Teff = 10 K (Schaerer 2002). Finally,
MNRAS 000,1–16 (2016) 6 A. Smith et al. we may relate this to the Lyα radiation pressure by considering the this paper. In Sections 3.1, 3.2, and 3.3 we describe methods for ionizing photon rate (Greif et al. 2009) calculating the Lyα energy density, force, and radiation pressure, respectively. πL Z ∞ B 15L Z ∞ x2dx N˙ = ? ν dν = ? . (15) ion 4 hν π4k T ex − σSBTeff νmin B eff xmin 1
In connection to Equation (11), the acceleration aα = ∆v/∆t and 3.1 Energy density due to photon trapping Lyα photon rate N˙ α ≈ 0.68(1 − fesc)N˙ ion provide the relative impact for a massive Pop III star: The Monte-Carlo method naturally tracks the time photons spend within each cell, which may be used as a measure of the energy a α −1 −1 density in the Lyα radiation field. This is because the total num- ≈ 0.94 MF,50NH i,19Tγ . (16) aγ ber of photons in a cell at a given time is reasonably approximated The Lyα force multiplier is certain to depend on the column density by the Lyα emission rate multiplied by the average time spent in ˙ of the shell. However, the exact relationship may be complicated by the cell, i.e. Ncell = Nαhticell, which is related to the Lyα source lu- ˙ several other factors such as geometry, bulk velocity, dust content, minosity via Lα = hνNα. For convenience we assume all photons 1 temperature, emission spectrum, and various three-dimensional ef- are relatively close in energy, i.e. ∆ν να, which is straightfor- fects such as turbulence and low-opacity holes. Analytic estimates ward to relax if desired. In this framework we keep track of the total suggest that M ∼ t /t ≈ 15 (τ /105.5)1/3 for a uniform slab length traversed by all paths of all photons within the particular cell F trap light 0 P (Adams 1975). In the context of Equations 11 and 16 we use the and define `cell ≡ cNphhticell where Nph is the number of photon computed results from Dijkstra & Loeb (2008) for static shells (see packets in the simulation. The energy density is obtained after inte- 0.44 grating over all independent photon paths up to the point of escape: their figure 6). The data follow MF ≈ 10 NH i,19 over the range 19 21 −2 NH i ∈ (10 , 10 ) cm , indicating that the effective force ratio is −0.56 bounded by aα/aγ & 0.19 NH i,19 . Finally, we emphasize that this hνNcell Lα X Ucell = = `cell . (17) is the total column density of the shell while the final transmission Vcell cNphVcell term Tγ uses the cumulative radial column density (discussed be- low), therefore the ratio may change across the shell itself. The summation is over all paths of all photons within the particular In the idealized case of a thin, dense shell with no interior H i cell and the volume depends on the geometry under consideration, 4π 3 − 3 it is possible that Lyα photons and ionizing radiation provide equal e.g. Vcell = ∆x∆y∆z for Cartesian cells and Vcell = 3 (router rinner) contributions to the acceleration. However, recombinations within for spherical shells. Finally, the sum over the average residence time is equivalent to the total trapping time over the cells in a do- the H ii region will absorb ionizing photons throughout the volume, P yielding an effective ionizing radiation force transmission function main D, i.e. ttrap = Nph htiD. that depends on the radial column density NH i(r). The exact form of Tγ may be calculated according to Equation 14, however for intu- ition it may be thought of as a step function that rapidly decreases 3.2 Radiation force due to scattering events beyond a critical column density of N ≈ 3 × 10−17 cm−2. If H i,crit Monte-Carlo radiative transfer also lends itself to tracking momen- the vanishing transmission is modeled as an effective optical depth tum transfer from Lyα photon packets to hydrogen atoms during we find T ≈ exp[−2.3 × 10−18 cm2 N ] but this falls off much γ H i each scattering event. The exchange of momentum is given by faster than the exact solution, so another reasonable approxima- ∆p = hν (n − n ), where the unit vectors n and n represent the tion for N ∈ (1017, 1021) cm−2 is the complementary error func- c i f i f H i initial and final directions of the scattered photon. For convenience tion T ≈ erfc[1.15 log(N /N )]/2. As the ionization front ad- γ H i H i,crit we track the overall momentum exchange which is related to the vances this recombination opacity increases until it balances the average by P ∆p = N h∆pi. To good approximation the momen- ionization rate. Thus, as long as the shell remains neutral we ex- ph tum rate p˙ ≡ ∆p/∆t may be written in terms of the Lyα emission pect the acceleration to be dominated by Lyα radiation pressure. rate N˙ . However, the effect of each scattering event must be ap- For concreteness, under Case B recombination we may estimate the α plied to the entire cell as a single fluid element. Therefore, the force optical depth as τ ∼ (α n2dt)σ d` ∼ 10 d`2(N˙ n )2/3. Here, R ν B 0 S ion,50 10 is diluted by the inertial mass of the cell, i.e. m = ρdV. The ac- N˙ ≡ N˙ /(1050 s−1) and n ≡ n/(10 cm−3), where we have cell ion,50 ion 10 celeration due to Lyα scattering is used the light crossing time dt ∼ d`/c. Furthermore, the distance is ` ≡ `/R scaled to the Strömgren radius, i.e. d S d S. With these values N˙ αh∆pi Lα X Lyα momentum transfer exceeds that of ionizing radiation at the acell = = ni − nf , (18) ρVcell cNphρVcell ionization front by several orders of magnitude. where the summation is over all scatterings of all photons within the particular cell. The geometry determines the direction of inter- est for the contribution from each scattering event. For example, 3 METHODOLOGY: RADIATIVE TRANSFER the radial component of momentum is ∆pr ∝ (ni − nf ) · rˆ. In prac- In this section we discuss the post-processing methodology that tice we employ a Monte-Carlo estimator to reduce noise in regions enables us to accurately calculate the Lyα radiation pressure. For with fewer scatterings. The idea is to use continuous momentum details regarding the MCRT method employed in the Cosmic Lyα deposition to determine the gas coupling. The photon contribution Transfer code (COLT) the reader is referred to Smith et al. (2015). The numerical methods outlined below introduce additional func- tionality for COLT that is then tested in preparation for fully cou- 1 For line transfer this is justified because even photons in the extreme wing pled radiation hydrodynamics. Although COLT is capable of calcu- of the profile differ from the median energy by a small fraction, e.g. a ve- lating radiation pressure for general three-dimensional grids with locity offset of ∆v ≈ 3000 km s−1 corresponds to a one per cent deviation mesh refinement we focus on spherically symmetric profiles for from the assumed energy hνα.
MNRAS 000,1–16 (2016) Lyα radiation hydrodynamics of galactic winds before reionization 7 is weighted by the traversed optical depth so the acceleration be- 2 3 comes T = 1 K X Lα U 1 acell = dτν n, (19) / 3 cNphρVcell P 2 4 6 8 τ0, edge = 10 = 10 = 10 = 10 where the sum is over all photon paths within the cell. 0 1/4
t τ0 40 τ6 h No core skipping g
i ≈ l t 10 3.3 Radiation pressure due to the rate of momentum flux / p a r t The final quantity of interest is Lyα radiation pressure, which dif- t T = 1 K fers from the energy density and radiation force as it applies to cell 1 0. 3 t τ0 τ 6 boundaries rather than interiors. Therefore, we consider the rate of h 13 g i l ≈ momentum flux as measured by the stream of photons propagat- t 10 / ing through a given cell surface. Similar to the force calculation the p a r
hν t 4
t T = 10 K momentum of the photon packet is p = c n, which we can imag- ine is transferred to a theoretical barrier representing the surface. 1 1 101 102 103 104 105 106 107 108 Therefore, the Lyα radiation pressure tensor is τ0 N˙ αhp ⊗ ni Lα X Pcell = = n ⊗ n, (20) Acell cNphAcell Figure 4. Ratio of pressure to energy density at a given radial line centre op- where the tensor product of the direction vector with itself is a sym- tical depth τ0 in uniform spheres at low and high temperatures, i.e. T = 1 K n ⊗ n nnT A 4 1 metric matrix, i.e. = . Here cell is the area of the cell and T = 10 K. In the diffusion limit the ratio approaches P/U = 3 . boundary, e.g. a Cartesian cell has Ax = ∆y∆z and a spherical shell The colours distinguish simulations with centre-to-edge optical depths of 2 2 4 6 8 has Ar = 4πr . The summation is over all crossings of all pho- τ0,edge = {10 , 10 , 10 , 10 }. The solid (dotted) lines are with (without) the tons through the particular interface. In practice we again employ a core skipping acceleration scheme. Core skipping results in smaller trap- Monte-Carlo estimator to calculate the continuous momentum flux ping times for the innermost shells. However, this represents a small frac- tion of the overall volume. The grey curves are approximate fits throughout in a particular direction. The pressure tensor becomes 6 1/4 the optically thick regime. At T = 1 K we find ttrap/tlight ≈ 40(τ0/10 ) L X for 103 < τ < 106 although the slope is shallower (≈ 1/8) for τ > 106. At α ⊗ 0 0 Pcell = `celldτν n n, (21) T = 104 K we find t /t ≈ 13(τ /106)0.3 for τ 106. cNphVcell trap light 0 0 & where the sum is over all photon paths within the cell and `cell is the 2 4 6 8 distance traversed. For concreteness, to determine the radial pres- optical depths τ0,edge = {10 , 10 , 10 , 10 } at low and high tem- sure in spherical shells the projected outer product terms reduce to peratures, i.e. T = 1 K and T = 104 K. The pressure to en- 2 2 ∆Prr ∝ (n ⊗ n)rr = nr = (n · rˆ) . ergy density ratio and trapping time normalized to the light cross- ing time are shown in Fig.4. The solid (dotted) lines are sim- ulations with (without) the core skipping scheme, which signif- 3.4 Test Cases icantly improves the computational efficiency but underestimates the trapping time for the innermost radial shells. The trapping time Before proceeding further we verify COLT against known solu- is approximately t ≈ {1.5, 3.9, 6.3, 7.5, 13, 23, 35, 52, 70} and tions. This was done in Section 2.1 for the case of sources in an ex- trap,1 K t 4 ≈ {1.6, 4.2, 8.2, 12, 12, 9.1, 12, 25, 50} light crossings at the panding IGM. For completeness we also compare the Monte-Carlo trap,10 K radial optical depths log(τ0) = {0, 1, 2, 3, 4, 5, 6, 7, 8} for T = 1 K estimators for U and P in the diffusion and free streaming limits. and T = 104 K, respectively. At T = 1 K this is approximately 6 1/4 3 6 fit by ttrap/tlight ≈ 40(τ0/10 ) over the interval 10 < τ0 < 10 6 although the slope becomes more shallow (≈ 1/8) for τ0 > 10 . At 3.4.1 A point source in vacuum 4 6 0.3 T = 10 K the fit is roughly given by ttrap/tlight ≈ 13(τ0/10 ) for 6 A point source in vacuum is not expected to undergo any scattering τ0 & 10 , although we note it is robustly & 10 for τ0 & 100. events. Thus, a free streaming situation arises where the energy in a It is particularly interesting that the ratio of ttrap/tlight scales shell at radius r = ct remains constant in time but is spread through more slowly than predicted by analytic arguments (Adams 1975). 1/3 an ever increasing volume of ∆V = 4πr2∆r, where ∆r is the shell The standard τ0 scaling assumes that Lyα escapes in a ‘single excursion’ during which there is a ‘restoring force’ that pushes thickness. The shell flash has an energy of ∆E = Lα∆t = Lα∆r/c 2 Lyα wing photons back to the core by an amount hdx|xi ≈ −1/|x| and thus an energy density of U(r) = Lα/(4πr c), as verified by COLT. Also, with no scattering events the force is zero and the for each scattering event, where x ≡ (ν − να)/∆νD is the rela- relation to pressure is P = U as expected in this special case. tive frequency from line centre in units of Doppler widths ∆νD ≡ 11 1/2 (vth/c)να ≈ 10 Hz T4 (see Adams 1972; Harrington 1973). However, in cold media and far in the wing of the line, this restor- ing force becomes smaller than the ‘force’ exerted by atomic re- 3.4.2 Energy density in the diffusion limit coil. Specifically, recoil pushes Lya photons consistently√ to the red Photons in optically thick environments are expected to undergo by an amount dx ≈ g ≡ h∆νD/2kBT ≈ 0.02536/ T/K, where numerous scattering events. In these conditions the isotropic pres- g is known as the ‘recoil parameter’√ (see e.g. Field 1959; Adams sure and energy density are related as P = U/3. We perform 1971). For x & g−1 ≈ 39.44 T/K we expect recoil to overwhelm this test on uniform density spheres with various centre-to-edge the standard ‘restoring force’. In comparison, the typical frequency
MNRAS 000,1–16 (2016) 8 A. Smith et al. with which photons escape from a medium with a given optical We note that the non-equilibrium chemistry and cooling discussed 6 1/3 −1/6 depth and temperature is xpeak ≈ 33.6 (τ0/10 ) (T/K) , see the in Section 4.4 does include radiative processes involving the gas at discussion following equation 34 in Smith et al. (2015) based on grid scales, such as collisional (de)excitation. the analytic solution for a static sphere derived by Dijkstra et al. 6 2 (2006). In short, for τ0 & 1.62 × 10 (T/K) photons need to diffuse so far into the wing of the line in order to escape that recoil prevents 4.2 Lab frame radiation hydrodynamics them from returning to the core, which further facilitates their es- The equations governing non-relativistic hydrodynamics are often cape and likely explains the flattening of the trapping time at higher written in an Eularian reference frame as a set of conservation laws. 4 τ0. Interestingly, at T = 10 K we find the trapping time is closer We quote the conservation of mass, momentum, and total energy to the analytic estimate, although the scaling exponent is still only with source terms related to gravity and radiation as follows: ∼ 0.3. We also note that this result depends on other environmental ∂ρ properties such as geometry. + ∇ · (ρv) = 0 , (25) ∂t ∂ρv v + ∇ · (ρv ⊗ v) + ∇P = −ρ∇Φ + G − G0 , (26) ∂t 0 c 4 METHODOLOGY: RADIATION HYDRODYNAMICS ∂ρe 0 + ∇ · (ρe + P0) v = −ρv · ∇Φ + cG . (27) Although there are several descriptions detailing the theory of radi- ∂t ation hydrodynamics, our treatment closely follows the conventions Here and in the following, all quantities with the subscript 0 are found in Mihalas & Mihalas (1984), Mihalas & Auer (2001), and evaluated in the comoving fluid reference frame while all other Castor (2004). We also benefited from the work of Abdikamalov quantities are in the lab frame, or the Eularian frame of the fixed et al. (2012), Harries (2015), Roth & Kasen (2015), and Tsang & coordinate system. Therefore, ρ is the lab frame density, v is the Milosavljevic´ (2015). lab frame velocity, P0 is the comoving pressure, and the last equa- tion is written in terms of the total specific energy, or the sum of comoving specific internal energy and lab frame specific kinetic 4.1 Lyα radiative transport ≡ 1 | |2 0 energy: e 0 + 2 v . The radiation source terms G and cG are re- The lab frame Lyα radiative transfer equation is spectively the momentum and energy components of the lab frame force four-vector, which directly correspond to the source terms of 1 ∂Iν + n · ∇Iν = kν (Jν − Iν) + S ν(r) , (22) Equations (23) and (24). c ∂t In this paper we assume an ideal gas equation of state so that where I is the specific intensity of the radiation, ν is the frequency, ν R the comoving pressure is isotropic and completely specified by n is a unit vector in an arbitrary direction, and J ≡ 1 I dΩ is ν 4π ν − the intensity averaged over the solid angle dΩ. The source terms on P0 = (γad 1)ρ0 , (28) the right are represented by S ν(r), the emission function for newly where γad ≡ CP/CV is the adiabatic index, or ratio of specific heat created photons at the position r, and kν ≡ nH iσν, the absorption co- at constant pressure to that at constant volume. efficient which is equivalent to the neutral hydrogen number den- The final source term we have included is gravity, which is sity multiplied by the frequency-dependent Lyα scattering cross- represented by the scalar potential Φ and may be a function of space section. Additional terms may be included to describe absorption and time. However, in spherical symmetry we may utilize Newton’s and scattering of Lyα photons by dust as well as other thermal pro- shell theorem. Thus, the acceleration depends on the mass enclosed 2 cesses. However, in this paper we consider only ‘dust-free’ envi- within the radius of the shell, i.e. agrav = −∇Φ = −GM MNRAS 000,1–16 (2016) Lyα radiation hydrodynamics of galactic winds before reionization 9 approximation using the source intensity Jν and photoionization 4.5 Timestep criteria cross-sections σ , σ , and σ . The rate of ionizing photons H i He i He ii Because photoionization can rapidly affect the chemical and ther- in each frequency range is (Jeon et al. 2014) mal state of the gas we employ timestep sub-cycling ∆tsub to accu- Z νmax,i 4πJν rately follow the evolution. We employ an explicit algorithm with N˙ ion,i ≡ dν , (29) νmin,i hν timesteps limited to a small fraction of the cooling timescale and depletion timescales for individual abundances: where the subscript i ∈ {1, 2, 3} denotes the particular band and ˙ P ˙ ( ) Nion = Nion,i. The photoionization rate is defined by 0ρ nx ∆tsub = sub min , , (37) Z νmax,i x ˙ ˙ n˙ 4πJν Γ − Λ x Γx,i ≡ dν σx , (30) νmin,i hν where sub . 0.1 is the sub-cycling factor. It is also important to en- t while the photoheating rate is given by sure the sub-cycling stops after the hydrodynamical time ∆ . Also the number of absorptions cannot exceed the number of available Z ν max,i 4πJ ˙ E ≡ dν ν σ (hν − hν ) , (31) ionizing photons, i.e. ∆tsub ≤ Nion,eff,i/Nion,abs,i. Therefore, the (vol- x,i x x ˙ νmin,i hν ume) specific heating and cooling in each sub-cycle are Γ = Γ∆tsub ˙ t where the subscript x ∈ {H i, He i, He ii} denotes the species. There- and Λ = Λ∆ sub, respectively. To ensure the accuracy of the chemi- fore, the average photoionization cross-section and average energy cal and thermodynamic state of the gas we update the species abun- imparted to the gas per ionizing photon is dances and internal energy throughout each sub-cycle. However, we apply the cumulative force from the absorption of ionizing pho- Γx,i Ex,i tons at the end of the hydrodynamical timestep. Therefore, the ac- hσxii ≡ and hεxii ≡ . (32) N˙ ion,i Γx,i celeration due to ionizing radiation is the weighted average from P To guarantee conservation of photons over a hydrodynami- each sub-timestep, i.e. aγ = aγ,sub∆tsub/∆t. cal timestep ∆t we calculate the total number of ionizing photons emitted by the source as N = N˙ ∆t. However, as photons ion,i ion,i 4.6 Monte-Carlo estimators are absorbed near the source the effective number of ionizing pho- 0 tons Nion,eff,i is reduced to represent the remaining portion. There- The final ingredient is to evaluate G and G in terms of the Monte- fore, due to absorption in interior cells the effective photon rate is Carlo estimators discussed in Section3. The MCRT calculations N˙ ion,eff,i = Nion,eff,i/∆t. For a shell of thickness ∆r the optical depth take place in the comoving frame and are related to quantities in the for a given species and frequency band is τx,i = nxhσxii∆r while lab frame by the Lorentz transformation (Mihalas & Auer 2001): P the total optical depth from all species is τi = τx,i. Therefore, the 0 0 v 0 v absorption rate in the cell is G = γ G + · G0 ≈ G + · G0 , (38) 0 c 0 c ! N˙ = N˙ 1 − e−τi . (33) v γ v v ion,abs,i ion,eff,i G = G + γ G0 + · G ≈ G + · G , (39) 0 c 0 γ + 1 c 0 0 c 0 The rate of change in number density is where γ ≡ (1 − |v|2/c2)−1/2. Although Lyα scattering is only ap- X X τx,i N˙ ion,abs,i n˙ion,x = n˙ion,x,i = , (34) proximately isotropic the anisotropic corrections are small com- τ V i i i cell pared to the Lyα radiation pressure. Therefore, because kν is only weakly dependent on the scattering angle we may evaluate the the rate of change in specific internal energy by volume is R R X energy component as G0 ≈ c−1 dνk dΩ(I − J ) = 0. The ˙ 0 ν ν ν Γ = hεxiin˙ion,x,i , (35) isotropic approximation also simplifies the momentum calculation: x,i −1 R R −1 R G0 ≈ c dνkν dΩ(Iν − Jν)n = c dνkν Fν. Putting this to- and the acceleration along the ray due to ionizing momentum trans- gether with the ionizing radiation gives the following: fer is 0 Γ − Λ X hhν i n˙ G0 = ρ(aα + aγ) and G0 = , (40) a = x i ion,x,i . (36) c γ c ρ x,i where the accelerations are based on Equations (19) and (36). where hhνxii ≡ hνx + hεxii is the average photon energy. 4.7 Lagrangian radiation hydrodynamics 4.4 Non-equilibrium chemistry and cooling We may now convert Equations (25)-(27) into the Lagrangian RHD We self-consistently solve the rate equations for a primordial chem- framework. Derivatives in this frame are denoted by the uppercase istry network consisting of H, H+, He, He+, He++, and e− (Bromm differential operator D(•)/Dt ≡ ∂(•)/∂t + v · ∇(•). The RHD equa- et al. 2002). Reactions affecting these abundances include H and tions are as follows: He collisional ionization and recombination from Cen (1992). Af- Dρ + ρ∇ · v = 0 , (41) ter incorporating the photoionization rates we obtain overall num- Dt ber density rates, i.e.n ˙ x = n˙ion,x + n˙chem,x. Meanwhile the cooling Dv ∇P + 0 = −∇Φ + a + a , (42) mechanisms include collisional ionization, collisional excitation, Dt ρ α γ recombination cooling, bremsstrahlung, and inverse Compton cool- D0 D(1/ρ) Γ − Λ ing. Collectively, the cooling provides the rate of change in specific + P0 = , (43) internal energy by volume Λ˙ . For computational efficiency these Dt Dt ρ rates are tabulated as a function of the logarithmic temperature and accurate to second order in (v/c). The Lagrangian formulation has linear interpolation is employed throughout the simulation. the advantage of substantially simplifying the numerical setup for MNRAS 000,1–16 (2016) 10 A. Smith et al. one-dimensional problems. Our spherically symmetric hydrody- 10 namics solver is based on the von Neumann-Richtmyer staggered 6 8 Mvir = 10 M Mvir = 10 M mesh scheme described by von Neumann & Richtmyer (1950), ¯ ¯ 7 9 Mvir = 10 M Mvir = 10 M Mezzacappa & Bruenn (1993), and Castor (2004). This algorithm 8 ¯ ¯ 10 Mvir = 10 M has the advantage of adaptive resolution which is useful for tracking ¯ shock waves by allowing the grid to follow the motion of the gas. Escape : rsh > rvir c 6 s With this approach we must include artificial viscosity to damp nu- e v merical oscillations near shocks. We use both linear and quadratic / h s f = 10 2 f = 10 3 viscosity in the references above. v 4 − − 2 5 SIMULATION RESULTS 5.1 Models 0 0 2 4 6 8 10 5.1.1 Galaxy models t (Myr) In order to determine the dynamical impact of Lyα feedback on galaxy formation we explore a series of idealized model galaxies parametrized by the total mass and source luminosity. For simplic- Figure 5. The ratio of the shell velocity vsh to the local escape velocity 1/2 ity we assume an NFW dark matter halo profile and thereby calcu- vesc ≡ (2GM MNRAS 000,1–16 (2016) Lyα radiation hydrodynamics of galactic winds before reionization 11 5 140 10 8 10 Mvir = 10 M Mvir = 10 M Mvir = 10 M ¯ ¯ 3 ¯ 120 M = 109 M M = 107 M 4 f = 10− 9 vir vir Mvir = 10 M ) ¯ ¯ ¯ s 6 / 100 Mvir = 10 M o 8 ¯ n M = 10 M vir m , ¯ 3 k h ( s 7 80 Mvir = 10 M v 3 ¯ k / f = 10− 6 a α Mvir = 10 M e 60 , p h 2 ¯ v s v ∆ 40 1 20 0 0 120 4 4 2 1 8 f = 10− f = 10− f = 10− ) 100 Mvir = 10 M s 3 1 2 ¯ / f = 10− f = 10− f = 10− o m 80 n 3 k , 3 8 ( h f = 10− Mvir = 10 M s v ¯ 4 k 60 a / f = 10− e α 2 p , v 40 h s ∆ v 1 20 0 0 2 4 6 8 10 0 0 2 4 6 8 10 t (Myr) t (Myr) Figure 7. Top panel: The time evolution of the velocity offset of the red peak 6−10 Figure 6. Top panel: The time evolution of the ratio of the shell velocity of the Lyα spectrum for different values of halo mass Mvir = 10 M at −3 for simulations with and without Lyα radiation pressure for different halo a fixed star formation efficiency of f? = 10 . Bottom panel: The velocity 6−10 −3 −1−(−4) masses Mvir = 10 M at a fixed star formation efficiency of f? = 10 . offset for different values of f? = 10 at a fixed halo mass of Mvir = −1−(−4) 8 Bottom panel: The shell velocity ratio for different values of f? = 10 10 M . The location of the emerging red peak is fairly constant throughout 8 at a fixed halo mass of Mvir = 10 M . the simulations with a slight decrease in time. low gravitational potential wells of minihalos allow the shell ve- susceptible to shell ejection than the more massive galaxies. We 2 1/2 locity to exceed the escape velocity vesc ≡ (2GM MNRAS 000,1–16 (2016) 12 A. Smith et al. 140 4 4 200 No IGM f = 10− f = 10− 120 red 3 Pop III 3 exp( τ ) f = 10− f = 10− − GP 2 ) CR7 ) 100 f = 10− s 150 Laursen et al. (2011) s / 2 / 1 f = 10− f = 10− m m 80 k k 1 ( f = 10 ( − k a k 100 e a 60 e p p v v Pop III ∆ ∆ 40 50 20 0 0 106 107 108 109 1010 37 38 39 40 41 42 43 44 45 log Lα (erg/s) Mvir (M ) ¯ Figure 8. Time-averaged velocity offset for different values of halo mass Figure 9. Time-averaged velocity offset for stellar sources as a function M = 106−10 M and star formation efficiency f = 10−1−(−4). The veloc- of intrinsic and observed Lyα luminosity for different values of halo mass vir ? 6−10 −1−(−4) ity offset has strong dependence on the efficiency of forming stars while the Mvir = 10 M and star formation efficiency f? = 10 . The shaded mass mostly affects the luminosity. regions have been corrected for transmission through the IGM based on two different models represented by diamonds and stars (see text). The dark grey arrow represents the overall shift from intrinsic to observed Lyα properties, i.e. the IGM produces a redder and fainter Lyα signature. For reference, we halo mass in Fig.8. Although there is some degeneracy at di fferent include the bright Lyα source CR7 at z = 6.6 with a measured offset of −1 times between these parameters the qualitative behavior is appar- +160 km s , discussed in Section6 (large grey star symbol). ent. The efficiency of star formation has a strong impact on the velocity offset while the mass affects the luminosity. However, as dius of r ≈ 500 kpc and that reionization is complete by redshift the mass of the galaxy is not a direct observable, we also provide H ii z ≈ 6. Any IGM model should be considered in the context of the velocity offsets as a function of the observed Lyα luminosity re statistical variations depending on the particular galaxy and sight- in Fig.9. The intrinsic (circles) and observed (diamonds and stars) line. Therefore, the observables presented in this paper represent time-averaged velocity offsets follow similar trends as those dis- optimistic yet plausible expectations of averaged IGM effects. See cussed above. However, the Lyα luminosity that survives transmis- Dijkstra (2014) for a review containing additional discussion and sion through the neutral IGM at z = 10 may be significantly re- related references. duced. Furthermore, the observed velocity offset may also appear much redder than the spectra emerging from the galaxy. This over- all shift from intrinsic to observed Lyα characterization is essen- 5.4 Stellar vs. black hole source spectra tial to interpreting theoretical predictions and observational bias of high-z Lyα (non)detections. We now explore black hole sources as an alternative production It may be difficult to disentangle the effects of intrinsic galaxy mechanism for Lyα radiation pressure. We discuss this scenario parameters and IGM models, but larger samples of high-z galax- in the context of so-called “direct collapse” black holes (DCBHs) ies with measurements for Lα,obs and ∆vpeak may provide insights (Bromm & Loeb 2003; Johnson & Haardt 2016), invoked as seeds regarding the natures of Lyα emitting galaxies. For example, we for the growth of supermassive black holes (SMBHs) of a few expect Lyα detections to be biased towards bright galaxies within billion solar masses observed in z > 6 quasars (Fan et al. 2006; & 1 Mpc ionized patches of the Universe or galaxies with a signif- Mortlock et al. 2011; Wu et al. 2015). The DCBH model in this icant flux emerging & 100 km s−1 redward of the Lyα line centre. study employs the nonthermal Compton-thick spectrum presented We include two treatments of frequency dependent transmission by Pacucci et al. (2015). The stellar and DCBH SEDs differ sig- through the IGM in Fig.9 to demonstrate a range of Ly α repro- nificantly in their effective photoionization cross-sections and en- cessing scenarios. The model labeled as Laursen et al. (2011) is ergy transfer per ionizing photon as calculated according to Equa- represented by stars and employs their “benchmark” Model 1 curve tions (29)–(32). The Compton-thick environment reprocesses the at z ≈ 6.5 as described in relation to their figures 2 and 3. The curve broadband spectrum such that only X-ray and non-ionizing pho- is a statistical average of a large number of sightlines (& 103) cast tons remain. This DCBH model is qualitatively different than a typ- from several hundreds of galaxies through a simulated cosmologi- ical quasar spectrum, which retains the UV ionizing photons and is cal volume. A region of the Universe undergoing early reionization therefore more similar to the stellar (blackbody) source except the at higher redshifts may have a qualitatively similar transmission Lyα line profile is significantly broader. For example, Vanden Berk red curve. The model labeled exp −τGP is based on the analytic pre- et al. (2001) find that the composite quasar spectrum from the Sloan scription of Madau & Rees (2000), which accounts for the optical Digital Sky Survey (SDSS) has an average Lyα emission-line pro- −1 depth of the red damping wing of the Gunn-Peterson (GP) trough. file width of σλ,α ≈ 19.46 Å ≈ 4800 km s . After removing all Lyα photons blueward of the circular velocity We present a select cases to match the most common phys- 8−9 to account for cosmological infall (see Dijkstra et al. 2007), we ical conditions for DCBH formation, i.e. Mvir ≈ 10 M and 5−6 attenuate the remaining Lyα profile based on the likelihood of un- M• ≈ 10 M . Specifically, we use the supplementary data pro- dergoing a scattering event in a uniform, neutral IGM. We assume vided by Pacucci et al. (2015) to calculate typical ionization rates the galaxies reside within a local ionized bubble with a physical ra- for their low-density profile, standard accretion scenario at three MNRAS 000,1–16 (2016) Lyα radiation hydrodynamics of galactic winds before reionization 13 Table 1. Summary of the source parameters for the DCBH models. The 8 stages represent early (5 Myr), intermediate (75 Myr), and late (115 Myr) 143 km/s hydrodynamical response during the growth of the black hole. The black • 6 hole escape fraction fesc ≈ 0.01 accounts for three-dimensional effects al- c s lowing some of the ionizing photons to escape without significant repro- e 91 km/s v cessing. / 4 h s DCBH (Stage II) v 108 km/s Model Stage I Stage II Stage III 2 99 km/s t• [Myr] 5 75 115 5 5 6 M• [M ] 1.33 × 10 6.52 × 10 1.48 × 10 0 −1 × 42 × 43 × 43 Lα [erg s ] 8.72 10 3.63 10 6.43 10 8 M = 109 M −1 42 42 42 vir fe•sc = 0. 01 LHe ii [erg s ] 1.54 × 10 6.76 × 10 5.75 × 10 ¯ −1 51 51 52 8 N˙ [s ] 1.13 × 10 5.52 × 10 1.26 × 10 Mvir = 10 M fe•sc = 0. 1 ion,1 o ¯ −1 51 51 52 n 6 N˙ion,2 [s ] 1.53 × 10 7.52 × 10 1.71 × 10 , h −1 51 51 52 s N˙ion,3 [s ] 1.53 × 10 7.95 × 10 2.60 × 10 v / α 4 , h s v stages representing early (5 Myr), intermediate (75 Myr), and late 2 (115 Myr) hydrodynamical response during the growth of the black hole. This is done to cover a broader range of DCBH properties, 0 0 2 4 6 8 10 analogous to the star formation efficiency parameter. We also add an additional parameter which we call the black hole escape frac- t (Myr) • tion fesc ≈ 0.01 in order to account for three-dimensional effects. In principle this allows partial leakage of ionizing photons from Figure 10. Top panel: The ratio of the shell velocity vsh to the local es- the Compton-thick reprocessing region at . 1 pc. For our purposes 1/2 cape velocity vesc ≡ (2GM MNRAS 000,1–16 (2016) 14 A. Smith et al. 350 ies of Lyα radiative transfer in similar contexts inform us about I I 9 No IGM Mvir = 10 M the impact of three-dimensional effects (Dijkstra & Kramer 2012; I ¯ 300 red M = 108 M Behrens et al. 2014; Duval et al. 2014; Zheng & Wallace 2014; exp( τ GP ) vir − ¯ Smith et al. 2015). For example, geometry, gas clumping, rotation, Laursen fe•sc = 0. 01 ) 250 s filamentary structure, and anisotropic emission from the source of- et al. (2011) II f = 0. 1 / e•sc II ten lead to anisotropic escape, photon leakage, or otherwise altered m 200 I k ( II dynamical impact. Lyα observables such as the equivalent width III I III Stages I III k III IIIII − a 150 and escape fraction may also be affected. For additional discussion e III p I v II regarding the caveats of these models and methods see Smith et al. I DCBH ∆ 100 (2016), which is also useful when applying these results to obser- II I I II vations of high-z Lyα emitting galaxies. III IIIII III 50 III I II II III I I II III We also note that there are other ways to trigger outflows I II III II III which may lead to different results for the Lyα signatures. In star- 0 41 42 43 44 forming galaxies supernova explosions could lead to even faster log Lα (erg/s) winds leading to larger Lyα velocity offsets until the velocity gra- dient facilitates escape from Doppler shifting. Additionally, these environments are likely to already be metal enriched such that Figure 11. Time-averaged velocity offset for direct-collapse black holes as the Lyα luminosity could be several times fainter due to dust ab- a function of intrinsic and observed Lyα luminosity for different values of sorption. However, some of these uncertainties could be disentan- 8−9 • halo mass Mvir = 10 M and black hole escape fraction fesc = 0.01−0.1. gled by comparing the Lyα properties with complementary multi- 5 The black hole mass is M• = {1.33, 6.52, 14.8} × 10 M for Stage I, frequency observations. Additionally, our theoretical framework re- II, and III, respectively (see Table1). The shaded regions have been cor- lies on the ability to measure the velocity offset of high-z galaxies, rected for transmission through the IGM based on two different models (see which is only possible if there is a nonshifted line to compare with. Section 5.3). The resulting observations are characterized by a redder and If the He ii 1640 Å line is not observed then it may still be pos- fainter Lyα signature. The velocity offset is generally larger for the DCBH spectrum than equivalent stellar models. sible to observe Balmer lines. The JWST is capable of detecting the Hα,Hβ,Hγ, and Hδ lines with the Near-Infrared Spectrograph (NIRSpec) at a medium spectral resolution of R = 2700 out to red- these calculations indicate that multiple scattering within high H i shifts of z = 6.6, 9.3, 10.5, and 11.2 respectively. Beyond that the column density shells is capable of significantly enhancing the ef- lines fall into the wavelength coverage of the Mid-Infrared Instru- fective Lyα force (Dijkstra & Loeb 2008, 2009). Furthermore, Lyα ment (MIRI), which is an order of magnitude less sensitive than radiation pressure may contribute alongside other feedback mecha- the NIRSpec for line flux detection at comparable spectral resolu- nisms such as supernova explosions. Understanding the role of Lyα tion. Even though the ratio of Balmer-line intensities implies dimin- trapping in galaxies is a challenging but important problem. ishing returns, i.e. the Balmer decrement is typically Hα:Hβ:Hγ = We have studied Lyα radiation pressure in the context of 2.86:1:0.47 (Osterbrock & Ferland 2006), depending on the source galaxies at z ≈ 10 with different halo mass Mvir and star forma- redshift it may still be advantageous to look for a higher-order line, tion efficiency f? defined as the fraction of baryonic mass that is considering the relative spectroscopic performance gain of NIR- in stars. We have also considered Lyα feedback within the context Spec over MIRI. Therefore, we expect selected samples of high-z of DCBH formation. The central starburst or black hole emission Lyα emitters will also include the measured velocity offset similar drives an expanding shell of gas from the centre. The results of our to current lower-redshift surveys. one-dimensional models may be summarized as follows: Recently, the luminous COSMOS redshift 7 (CR7) Lyα emit- ter at z = 6.6 was confirmed to have strong He ii emission with no (1) Strong radiative feedback in minihaloes can launch supersonic detection of metal lines from the UV to the near infrared (Matthee outflows into the IGM and Lyα radiation pressure may enhance this et al. 2015; Sobral et al. 2015). As a result several groups have effect, considered the CR7 source in the context of a young primordial (2) Including Lyα feedback has a greater relative impact on the starburst or direct collapse black hole (Pallottini et al. 2015; Agar- shell velocity with higher f or larger M , ? vir wal et al. 2016; Hartwig et al. 2015; Visbal et al. 2016; Dijkstra (3) The velocity offset ∆v of the Lyα flux depends on the H i peak et al. 2016; Smidt et al. 2016). CR7 and similar galaxies therefore optical depth, therefore a higher neutral fraction due to a lower f ? represent an ideal application of the methodology presented herein or larger M translates to a greater velocity offset, vir as they allow for direct comparison with current and upcoming ob- (4) We provide quantitative estimates for the extent to which the servations. In Smith et al. (2016), we closely examined and repro- scattering of Lyα photons in the IGM leads to a reprocessing of the duced several Lyα signatures of the CR7 source under a DCBH observed flux, i.e. the observed Lyα luminosity is reduced and the model, including the velocity offset between the Lyα and He ii line velocity offset undergoes a redward shift, peaks. We also found that Lyα radiation pressure turns out to be (5) Lyα radiation pressure may have a significant dynamical im- dynamically important in the case of CR7. In the near future, other pact on gas surrounding DCBHs, with Lyα signatures typically sources similar to CR7 may provide additional constraints on early characterized by larger velocity offsets than stellar counterparts if galaxy and quasar formation. Indeed, Pacucci et al. 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