Accretion onto Stars in the Disks of Active Galactic Nuclei
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Accretion onto Stars in the Disks of Active Galactic Nuclei
Alexander J. Dittmann,
1, 2
Matteo Cantiello,
1, 3 and Adam S. Jermyn Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA Department of Astronomy and Joint Space-Science Institute, University of Maryland, College Park, MD 20742-2421, USA Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA (Received ...; Revised ...; Accepted ...)
Submitted to ApJABSTRACTDisks of gas accreting onto supermassive black holes are thought to power active galactic nuclei(AGN). Stars may form in gravitationally unstable regions of these disks, or may be captured fromnuclear star clusters. Because of the dense gas environment, the evolution of such embedded starscan diverge dramatically from those in the interstellar medium. This work extends previous studiesof stellar evolution in AGN disks by exploring a variety of ways that accretion onto stars in AGNdisks may differ from Bondi accretion. We find that tidal effects from the supermassive black holesignificantly alter the evolution of stars in AGN disks, and that our results do not depend critically onassumptions about radiative feedback on the accretion stream. Thus, in addition to depending on ρ/c s ,the fate of stars in AGN disks depends sensitively on the distance to and mass of the supermassiveblack hole. This affects where in the disk stellar explosions occur, where compact remnants form andpotentially merge to produce gravitational waves, and where different types of chemical enrichmenttake place. Keywords:
Stellar physics (1621); Stellar evolutionary models (2046); Massive stars(732); Quasars(1319);Galactic center(565) INTRODUCTIONThe centers of most massive galaxies are thought toharbor supermassive black holes (SMBHs) (Kormendy& Ho 2013). These SMBHs can power quasars and ac-tive galactic nuclei (AGNs) through the release of grav-itational energy as matter spirals into their deep po-tential wells (Ho 2008; Lynden-Bell 1969). Analyses ofthe redshift-dependent AGN luminosity function haveinferred the efficiency, luminosity, and integrated dura-tion of SMBH accretion (So(cid:32)ltan 1982; Yu & Tremaine2002), suggesting that SMBHs typically accrete for a to-tal of tens to hundreds of millions of years in luminousquasars, although individual episodes may be as shortas ∼ years (King & Nixon 2015; Schawinski et al.2015). Corresponding author: Alexander J. [email protected]
AGN accretion has historically been modeled as occur-ring through geometrically thin, optically thick accre-tion disks (Novikov & Thorne 1973; Shakura & Sunyaev1973). Such disk models are thermally unstable at smallradii where radiation pressure dominates (Shakura &Sunyaev 1976; Shapiro et al. 1976), although radiation-magnetohydrodynamics simulations indicate that theymay be stabilized by magnetic pressure or convection-driven turbulence (Jiang & Blaes 2020; Jiang et al.2019a,b). Additionally, standard thin accretion diskmodels (Shakura & Sunyaev 1973) become gravitation-ally unstable at large radii, and multiple methods oftaking this into account have been suggested (Sirko &Goodman 2003; Thompson et al. 2005).When compared to microlensing observations, classi-cal thin disk models under-predict accretion disk sizesat optical wavelengths by factors of ∼ a r X i v : . [ a s t r o - ph . GA ] F e b Dittmann, Cantiello, and Jermyn hen 1983; Kelly et al. 2009; MacLeod et al. 2010), al-though phenomenological models have attempted to ac-count for such behavior (Dexter & Agol 2011; Lee &Gammie 2020).Other AGN observations are also incompatible withstandard models. Broad emission lines can suddenlyappear or disappear, accompanied by dramatic changesin luminosity (Frederick et al. 2019; LaMassa et al. 2015;MacLeod et al. 2019). Quasars exhibiting extreme vari-ability systematically accrete at lower rates than typicalquasars, although such quasars appear to belong to thetail of a continuous distribution, rather than to a dis-tinct population (Rumbaugh et al. 2018). Additionally,spectroscopic modeling of AGN broad line regions in-dicates that their metallicitiy is generally greater thansolar, roughly independent of redshift, and an increas-ing function of SMBH mass (Nagao et al. 2006; Xu et al.2018). The iron abundance in AGN disks may also besubstantial, evidenced by the observation of the iron Kα X-ray emission line (Nandra et al. 1997; Tanaka et al.1995), although abundance inferences are subject to sub-stantial uncertainties (Garc´ıa et al. 2018; Tomsick et al.2018).Furthermore quasars do not exist in isolation: al-though AGNs typically outshine their host galaxies, ob-servations suggest that nuclear star clusters are ubiq-uitous around SMBHs (Neumayer et al. 2020). Duringactive phases, these stars can be captured into the accre-tion disk even if initially on misaligned orbits (e.g. Arty-mowicz et al. 1993; Rauch 1995). Passages through thedisk act to circularize eccentric stellar orbits (MacLeod& Lin 2020), and torques can further align and circu-larize stellar orbits (Rauch 1995; Tanaka & Ward 2004).Stars may also form in the outer regions of AGN disks,where the disk is gravitationally unstable and the cool-ing timescale is short (Gammie 2001; Toomre 1964).Stars in AGN disks (AGN stars) and the compactobjects left behind at the end of their evolution canenhance AGN metallicities (Artymowicz et al. 1993),source LIGO/VIRGO events (e.g. Bartos et al. 2017;McKernan et al. 2018; Stone et al. 2017; Tagawa et al.2020), and contribute to the growth of high-redshiftSMBHs (Dittmann & Miller 2020). However, the evo-lution of AGN stars is much more exotic than the evo-lution of typical stars in the interstellar medium, dueto the high ambient density and temperature in AGNdisks. This alters the stellar boundary conditions andhence their structure and evolution.Previous works have studied stellar evolution subjectto AGN-like boundary conditions, ranging from starssubject to irradiation from an AGN (Tout et al. 1989)to those embedded within AGN disks and subject to Bondi-like spherically symmetric accretion and super-Eddington mass loss (Cantiello et al. 2020). In thiswork we consider alternative models of radiation feed-back on accretion, as well as deviations from sphericalsymmetry inherent to a disk environment, described inSection 2. We outline our numerical methods in Sec-tion 3, and present results in Section 4. We find thatthe assumptions of radiative feedback have minor effectson stellar evolution in AGN disks, and that tidal effectsare the most important of those considered in this work.We show how chemical enrichment varies as a functionof generic disk parameters, and give general formulasfor which accretion disk conditions lead AGN stars toend their lives as explosive transients, or live indefinitelywith mass loss and accretion balancing one another. ANALYTIC CONSIDERATIONSWhen the specific angular momentum of accreting ma-terial is low, and the radiation from the accreting staris sufficiently sub-Eddington, accretion may be mod-elled as a spherically symmetric process occurring at theBondi rate ˙ M B = ηπR B ρc s , (1)where η is an efficiency factor ( η (cid:46) ρ and c s are thedensity and sound speed of ambient gas, R B = 2 GM ∗ c s (2)is the Bondi radius, M ∗ is the mass of the star and G isthe gravitational constant.In addition to the Bondi radius, two other length-scales are important for stars in AGN disks. First, thescale height of the disk is given by H ≡ √ c s Ω , (3)where Ω = (cid:115) GM • r • (4)is the Keplerian angular velocity of the AGN disk, M • is the mass of the SMBH, and r • is the distance fromthe star to the SMBH. Secondly, the Hill radius is R H = r • (cid:18) M ∗ M • (cid:19) / = (cid:18) GM ∗ (cid:19) / . (5)This is the radius of a sphere within which the gravityof the star dominates that of the SMBH. Note that withthe above definitions R H /R B ∝ H /R H and H/R H ∝ hq − / , where q ≡ M ∗ /M • is the mass ratio and h ≡ H/r • . ccretion onto stars in AGN disks ρ = ρ ) or one thatvaries only as a function of height from the midplane( ρ = ρ f ( z )), where ρ is the midplane gas density. Al-though this approximation should break down in mostcases, especially as the length scale for accretion be-comes large, it enables us to keep our investigationlargely independent of the precise and highly uncertainstructure of AGN disks.2.1. Radiative feedback
As stars become more massive, their luminosity ( L ∗ )increases rapidly, L ∗ ∝ M ∼ ∗ (B¨ohm-Vitense 1992). Asthis radiation impinges on ambient gas, it causes a spe-cific force on gas a distance r away of f g = F κ/c , where κ is the opacity of the gas, c is the speed of light, and F is the radiative flux, given for a spherically symmetricradiation field by L ∗ / πr . The net acceleration of agas parcel due to radiation and gravity is thus f = 1 r (cid:18) L ∗ κ π − GM ∗ (cid:19) = − GM ∗ r (cid:18) − L ∗ κ πGM ∗ c (cid:19) . (6)The luminosity where the net acceleration becomes zerois the Eddington luminosity L Edd = 4 πGM ∗ c/κ. (7)Moreover, from Equation (6), it can be seen that the effective mass of the star as experienced by ambient gasis reduced by a factor of (1 − L ∗ /L Edd ). As R B ∝ M ∗ ,the effective Bondi radius is reduced by the same fac-tor. Similarly, the accretion rate onto a star is modifiedaccording to ˙ M = ˙ M B (cid:18) − L ∗ L Edd (cid:19) . (8)This picture is entirely one dimensional, which is adrastic simplification of reality. For example, rotatingstars are more luminous at their poles than their equa-tors (Lucy 1967; von Zeipel 1924), and density pertur-bations can lead to channels of accretion through theRayleigh-Taylor instability (Davis et al. 2014; Krumholz& Thompson 2013). Cantiello et al. (2020) attempted toaccount for deviations from spherical symmetry by us-ing, instead of Equation (8), the following phenomeno-logical prescription:˙ M = ˙ M B (1 − tanh | L ∗ /L Edd | ) . (9)This prescription is useful because it allows super-Eddington accretion, which can happen in geometries where radiation is able to escape in one direction whileaccretion primarily happens along another, among otherscenarios. We present results computed using each pre-scription (Equations (8) and (9)) and find that theylead to quantitative, but not qualitative, changes in theoverall picture. SMBH R B H R B R H Ω ( R + R B ) R B Ω ( R - R B ) A. Rarefication
C. TidesB. Shear
Figure 1.
Accretion onto an AGN star (blue circle) orbit-ing a supermassive black hole can be limited by geometricand tidal effects. Geometric effects include rarefication (A),which is important when the Bondi radius R B becomes com-parable to or larger than the disk’s scale height H , and shear(B), which is due to the disk’s Keplerian rotation. Tidaleffects (C) are caused by the gravity of the SMBH, whichbecomes important when the Hill radius is smaller than theBondi radius. Vertical stratification
It is common to assume that accretion disks are ver-tically isothermal. In such disks the gas density variesvertically as ρ ( z ) = ρ exp (cid:2) − ( z/H ) (cid:3) . For a star inthe midplane of the disk this means that the averagedensity of gas at the Bondi radius decreases with in-creasing Bondi radius, and so generally decreases as thestar becomes more massive. Deviations from an isother-mal structure or a more realistic treatment of radiationtransport and opacities could reveal density inversionsor more shallow density gradients (e.g. Hubeny et al.2000; Jiang & Blaes 2020; Meyer & Meyer-Hofmeister1982; Milsom et al. 1994). Because these complicationsall reduce the effect of stratification, assuming a verti-cally isothermal disk allows us to gauge the maximum Dittmann, Cantiello, and Jermyn impact that vertical density variations could have on theevolution of AGN stars.To account for how changes in density vertically candecrease the accretion rate, we average the density inEquation (1) over a sphere with radius R B centered onthe star. We find that in this case, with z = R B cos θ ina polar coordinate system centered on the star, (cid:104) ρ (cid:105) ρ = 12 (cid:90) π sin ( θ ) exp (cid:20) − (cid:18) R B H (cid:19) cos ( θ ) (cid:21) dθ = √ π HR B erf (cid:18) R B H (cid:19) , (10)where (cid:104) ... (cid:105) denotes an average over the Bondi sphere, erfis the error function and ρ is the density of the midplaneof the disk. If we use (cid:104) ρ (cid:105) to compute the accretion ratewe then find ˙ M = ˙ M B √ π HR B erf (cid:18) R B H (cid:19) . (11)2.3. Shear
Shear in the accretion disk imbues accreting gas withnet angular momentum in the frame comoving with thestar, and can therefore limit accretion onto an embeddedstar. To leading order, at a distance ∆ s away from thestar the angular velocity of the disk is different from thatof the star by an amount∆Ω = Ω( r • ) − Ω( r • + ∆ s ) ≈
32 Ω( r • ) ∆ sr • , (12)where we have assumed a Keplerian rotation profile forsimplicity. The linear velocity of gas at the Bondi ra-dius relative to the star is ∆ v = R B ∆Ω. In a spher-ical coordinate system centered on the star, ∆ s = R B cos ( θ ) sin ( φ ) at a distance R B from the star. Av-eraging the specific angular momentum relative to thestar, l = ∆ s ∆ v , over the Bondi sphere, we find that theaverage specific angular momentum is (cid:104) l (cid:105) = R B r • )8 πr • (cid:90) π sin ( φ ) dφ (cid:90) π cos ( θ ) sin ( θ ) dθ = R B Ω( r • )4 r • . (13)Following Krumholz et al. (2005), we estimate the im-pact of angular momentum on the accretion. Let ω ∗ = (cid:104) l (cid:105) c s R b , (14)where ω ∗ is a measure of the vorticity of the flow. Thenthis angular momentum reduces the accretion rate by afactor of ˙ M ˙ M B = min { , f s ( ω ∗ ) } , (15) where f s ( ω ∗ ) = 2 πω ∗ sinh − (cid:104) (2 ω ∗ ) / (cid:105) , (16)Equation (A7) of Krumholz et al. (2005). Note that inthis case, ω ∗ = R B Ω / r • c s = R B / √ Hr • . Because R B /r • is usually very small, one can expect that othergeometric effects become important before shear unlessthe disk is very thick and H ∼ r • .2.4. Tidal effects
When the Bondi radius is much smaller than the diskscale height and Hill radius, disk geometry and the grav-ity of the SMBH have a negligible effect on accretiononto AGN stars. However, because R B /R H ∝ M / (cid:63) , R B naturally becomes larger than R H as stars growmore massive. The reason this alters the accretion rateonto stars can be understood by considering the casewhere R B (cid:38) R H . In this case, the motion of gas at R B > R > R H is controlled by the gravity of the SMBHwith only minor influences from the star. To incorpo-rate this effect we replace the Bondi radius with thesmaller out of the Hill and Bondi radii when calculatingaccretion rates˙ M = ˙ M B min (cid:40) , (cid:18) R H R B (cid:19) (cid:41) , (17)along the lines of Rosenthal et al. (2020).In disks with sufficiently low viscosity, accretion canbe further reduced by the so-called ‘tidal barrier’, andby the conservation of potential vorticity along flow lines(Dobbs-Dixon et al. 2007). Li et al. (2021) carried out asuite of 2D and 3D viscous hydrodynamics simulationsusing an α − viscosity prescription. For α (cid:38) − , theresults were better described by the scaling of Rosen-thal et al. (2020) (equation (17) above), while for lower α the scalings of Dobbs-Dixon et al. (2007) were a bet-ter fit. To the extent that an α -viscosity accurately de-scribes angular momentum transport in AGN disks, sim-ulations and comparisons with observations suggest ef-fective values of α in AGN disks larger than 10 − (Hogg& Reynolds 2016; King et al. 2007). Hence we expectthat (17) is appropriate for stars in AGN disks.If stars becomes sufficiently massive, they can opena gap in the disk due to strong nonlinear disk-star in-teractions (Duffell et al. 2014; Kanagawa et al. 2018;Ward 1997). Contrary to the classical picture of gapopening, where gas cannot flow into the gap and theobject is locked in step with the viscous evolution ofthe disk (Ward 1997), numerous numerical studies haveshown that gas can flow into the gap, albeit at a muchlower surface density, and that objects within the gapcan migrate faster or slower than the viscous speed of ccretion onto stars in AGN disks min Σ = 11 + 0 . K , (18)where K = q h − α − (Duffell 2015; Kanagawa et al.2015). Equation (18) follows from the assumptions thatthe gap depth is determined by the balance betweentorques from the disk and the viscous angular momen-tum flux, along with the assumption that the objectprimarily interacts with gas at the bottom of the gap,which was found to be consistent with the results ofvarious hydrodynamical simulations (Kanagawa et al.2017).For K (cid:38)
10, the surface density of the disk can be sig-nificantly depleted, further decreasing the accretion rateonto massive AGN stars. Because we expect modestvalues of α ( > − ) to be applicable to AGN disks, asopposed to α < − as may be applicable in planetarycontexts (Duffell & MacFadyen 2013), even for values of h ∼ − , q would need to be greater than 10 − for thedisk surface density to deplete significantly. Thus, evenfor low-mass SMBHs with M • ∼ M (cid:12) , stars wouldneed to reach M ∗ (cid:38) M (cid:12) before beginning to mean-ingfully influence the disk surface density. We find thatAGN stars typically reach masses at most ∼ M (cid:12) ,so we do not consider the effects of gap formation onaccretion. NUMERICAL METHODSWe model the evolution of AGN stars using revision15140 of the Modules for Experiments in Stellar Astro-physics (MESA Paxton et al. 2011, 2013, 2015, 2018,2019) software instrument. We implement mass loss andmodified boundary conditions following Cantiello et al.(2020). We consider accretion of a gas mixture withmass fractions X = 0 . , Y = 0 .
28, and Z = 0. Al-though this is not realistic for gas in most AGN disks,it highlights metal production by AGN stars and allowsus to successfully evolve stars without incurring numer-ical issues. The dependence of AGN star evolution onaccreted composition is left for the future.In addition to the accretion prescription used inCantiello et al. (2020), we implement in MESA thevarious processes described in Section 2. One of them,Equation (11), requires special treatment to ensure re-producibility and make use of the auto-differentiationmodule within MESA. We approximate the error func-tion by a B¨urmann series (Sch¨opf & Supancic 2014; Whittaker & Watson 1963)erf( x ) ≈ √ π x | x | (cid:112) − e − x (cid:18) √ π e − x − e − x (cid:19) , (19)which has a maximum relative error (cid:46) .
37% andproduces, bit-for-bit, the same result on all computa-tional platforms owing to the use of the CR-LIBM li-brary Daramy-Loirat et al. (2006).As shown by Cantiello et al. (2020), many AGN starsevolve with L ∗ ∼ L Edd , such that super-Eddingtoncontinuum-driven winds are likely to dominate theirmass loss. Therefore, we do not include line-drivenmass loss (but see Smith 2014, for a review of the dif-ferent ways massive stars can lose their mass). Sim-ilar to other works (e.g. Paxton et al. 2011), we as-sume a super-Eddington outflow at the escape veloc-ity v esc = (2 GM ∗ /R ∗ ) / , and following Cantiello et al.(2020) we compute an associated mass-loss rate˙ M loss = − L ∗ v (cid:20) (cid:18) L ∗ − L Edd . Edd (cid:19)(cid:21) , (20)where the hyperbolic tangent acts to smooth the onsetof mass loss and help our calculations converge.For models approaching the Eddington luminosity wealso assume an enhancement of compositional mixingin radiative regions. This is justified by the fact thatthe threshold for vertical instability decreases as starsbecome radiation dominated and the adiabatic indexΓ approaches 4 /
3. A number of processes can trig-ger instabilities in the radiative envelope, including rota-tion (Eddington 1929) and locally super-Eddington lu-minosities (Jiang et al. 2018). Details of the mixingimplementation are discussed in Cantiello et al. (2020).We initialized our runs with a zero-age main sequencesolar model. We then relaxed the boundary conditionsand accretion rate over approximately 10 yr from solar-like to those described by Cantiello et al. (2020). Thisrelaxation happens sequentially, with the boundary con-ditions fully relaxed before the accretion rate relaxationbegins. The total relaxation time is kept short so thatit does not amount to a significant fraction of the totalevolutionary time of our initial 1 M (cid:12) models. RESULTS4.1.
Archetypal Models
It is useful to examine a few archetypal evolutionarytracks before studying whole populations.
Dittmann, Cantiello, and Jermyn
In cases where the accretion rate is very small , stellarevolution is largely unaffected and stars reach the endof their lives with a similar mass to that at which theybegan. However, stars subject to higher accretion ratescan have significantly different evolutions. This differ-ence can be seen in Figures 2 and 3, which show theevolution of a 1 M (cid:12) AGN star embedded in gas with asound speed of 10 km s − and ambient gas densities of2 × − g cm − and 8 × − g cm − respectively. m a ss ( M ) Y m a ss ( M ) Mass accretedMass lost
120 125 130 135 140time (Myr)01 X c Y c Figure 2.
The evolution of a typical model in the interme-diate accretion regime is shown. The ambient gas density is2 × − g cm − and the sound speed is 10 km s − . The toppanel shows the helium mass fraction in the star as a func-tion of time and stellar mass coordinate. Convective regionsare marked by light blue hatching. The middle panel dis-plays the mass accreted and mass lost over time. The lowerpanel shows the core hydrogen and helium mass fractionsthroughout the star’s life. The stars begin with a steep composition gradient,but become chemically homogeneous as they increase inmass, accreting fresh hydrogen and helium. Through-out most of its life, the ρ = 2 × − g cm − model hasa convective core and radiative envelope. From about ∼ −
137 Myr, the hydrogen mass fraction in thecore increases due to mixing of accreted material. Afterthis, accretion is effectively halted by radiative feedbackand the star loses its source of fresh hydrogen. The he-lium mass fraction then increases throughout the star Accretion timescale much longer than the stellar nuclear burn-ing timescale. m a ss ( M ) Y m a ss ( M ) Mass accretedMass lost X c Y c Figure 3.
The evolution of a typical model in the runawayaccretion regime, leading to an “immortal” AGN star. Theambient gas density is 8 × − g cm − and the sound speedis 10 km s − . The top panel shows the helium mass fractionin the star as a function of time and stellar mass coordinate.Convective regions are marked by light blue dashed lines.The middle panel displays the mass accreted and mass lostover time. The lower panel shows the core hydrogen andhelium mass fractions throughout the star’s life. until it is eventually depleted through fusion into heav-ier elements. This general trend is followed by stars inthe ‘intermediate’ accretion regime, where the nuclearburning timescale is shorter than, but comparable to,the accretion timescale (Cantiello et al. 2020).Unlike the ρ = 2 × − g cm − case, in the ρ =8 × − g cm − model mass loss and accretion comeinto an approximate balance once the star nears the Ed-dington limit. As a consequence the mass of the modelreaches an approximate steady state. The star is almostfully convective, so accreted material is rapidly mixedthroughout. For the same reason, mass lost from thestar is chemically enriched by the ashes of nuclear fu-sion occurring in the stellar core. Examining the massloss and accretion budget, it is evident that this singlestar that began at 1 M (cid:12) is able to process well above10 M (cid:12) worth of gas in ∼
10s of Myr (Figure 3, middlepanel). Stars with higher initial masses or in higher-density (or lower-sound speed) environments are able toreach this evolutionary stage much more quickly, since M/ ˙ M B ∝ M − ρ − . Because these models can persist athigh masses indefinitely in the appropriate conditions,we refer to these models as ‘immortal’ for simplicity. ccretion onto stars in AGN disks H He C N O10 M / M / M y r YIELDS
AGN = 2 × 10 g cm C s , AGN = 10 km s H He C N O10 M / M / M y r YIELDS
AGN = 8 × 10 g cm C s , AGN = 10 km s Figure 4.
Chemical yields for archetypal ‘intermediate’ (fig-ure 2, upper) and ‘immortal’ (figure 3, lower) models. Forboth cases, we have excluded the first 4 × years of thestar’s lives. Because the immortal model could fuse indefi-nitely, we have presented yields in terms of the mass lost permillion years. Note that the yields from immortal stars con-tain very little metal content because these stars primarilyfuse hydrogen, although their metal content is significantlyenriched in nitrogen. As mortal stars lose mass and reachlater stages of evolution they eventually produce significantamounts of carbon and oxygen as well. The chemical yields of mass lost from these models areshown in Figure 4. Although the immortal model losesmass at a rate ∼ times larger than the intermediatemodel, this material has a much lower metal content and so the overall yield of metals is lower. Thus, mass lossfrom immortal stars matters less for enriching the diskwith metals than that lost from mere mortals, althoughthe large mass loss rates could play an important rolein determining the overall structure of the AGN disk.Additionally, stars may migrate through the disk, andthe disk itself will dissipate over time. Thus, eventually‘immortal’ stars will find themselves in lower-density en-vironments and begin to evolve similarly to those in theintermediate accretion regime, reaching later stages ofburning and losing most of their nuclear-processed ma-terial (e.g. Figure 10 and 11 in Cantiello et al. 2020).4.2. Model Grids
With these general types of evolution in mind, we aimto understand how each modification to accretion dis-cussed in Section 2 affects the evolution of AGN stars.We present in Figure 5 a series of M ∗ ( t ) diagrams foreach of the accretion prescriptions discussed in Section2. The prescriptions used by Cantiello et al. (2020) aretaken as a baseline and shown in the first row. Each sub-sequent row demonstrates exactly one modification frombaseline, with the exception of the fourth row which in-cludes two modifications. In each panel we show a vari-ety of Ω, ρ (the angular velocity and density of the gas atthe stellar location in the AGN disk). Colors show den-sity, with lighter and yellower colors indicating stars inlower-density environments and darker and bluer colorsindicating those in higher-density environments. Themodels in the left column are embedded in gas witha sound speed of 3 km s − , while the models in theright column are embedded in gas with a sound speedof 10 km s − . Line styles indicate the angular velocityΩ about the SMBH, and where necessary M • = 10 M (cid:12) is used to calculate r • . Note that some models do notdepend on Ω. This sparse grid of models elucidates theimpact of the different modifications to accretion ontoAGN stars.We begin by considering the first and second rows,which show models computed using our baseline pre-scription and those computed using stronger radiativefeedback (Equation (8)). The more severe reduction inaccretion rate imposed by Equation 8 extends the rangeof densities leading to intermediate stellar evolution, asopposed to runaway, by about an order of magnitude.Of the effects we consider, tidal forces from the SMBHare generally the most significant. Specifically, tidaleffects slow accretion onto AGN stars enough to shiftmany from the runaway regime to the intermediateregime, and to shift many from the intermediate regimeinto the regime of fairly standard stellar evolution. Be-cause of its independence from disk structure, support Dittmann, Cantiello, and Jermyn m a ss ( M ) c s = 3 × 10 cm s base c s = 1 × 10 cm s = 1e-19= 2e-19= 5e-19= 1e-18= 2e-18= 5e-18= 1e-17= 2e-17= 5e-17= 1e-1610 m a ss ( M ) radiative feedback m a ss ( M ) tides = 1e-19= 2e-19= 5e-19= 1e-18= 2e-18= 5e-18= 1e-17= 2e-17= 5e-17= 1e-16= 1e-12= 1e-10= 1e-0810 m a ss ( M ) tides+radiation m a ss ( M ) rarefication = 1e-19= 2e-19= 5e-19= 1e-18= 2e-18= 5e-18= 1e-17= 2e-17= 5e-17= 1e-16= 1e-12= 1e-10= 1e-0810 age (yrs) m a ss ( M ) age (yrs)shear Figure 5.
Evolutionary tracks of stellar mass over time for various accretion prescriptions. On the x-axes, the reported ageincludes the ∼ years over which the boundary conditions for each model were relaxed. We begin plotting results at theend of the relaxation process for each run. Each column corresponds to a different ambient sound speed, as indicated by thecolumn headers. Each row presents results for a different accretion prescription, as indicated by the text boxes in each row.Line colors correspond to different ambient gas densities (given in g cm − ), and line styles correspond to different AGN diskangular frequencies (given in s − ) at the star’s location. Stars indicate the final masses of stars at the end of their lives. Recallthat for M • = 10 M (cid:12) , Ω = { − , − , − } s − corresponds to distances from the SMBH r • ≈ { . , . , . } pc. ccretion onto stars in AGN disks R H /R B ,their impact may become less significant if R B is re-duced by radiation from the star. To test the extentto which this matters, we carried out another suite ofsimulations using both modifications. For small Ω, tidaleffects are minimal and stars evolve essentially identi-cally to the models without tidal effects. However, athigher Ω, tidal effects dominate and evolution is largelythe same as when only considering tidal effects. Thus wefind that the evolution of AGN stars only depends signif-icantly on raditative feedback assumptions for stars thatare far from the SMBH, e.g. ∼ .
36 pc for a 10 M (cid:12) black hole.Vertical stratification in the disk can also decrease theaccretion rate onto AGN stars enough to cause qual-itative deviations in their evolution. However, its ef-fects are less significant than tidal effects because rar-efication only operates in one dimension in our models.This can be understood by considering the growth of theBondi radius as the star accretes: although gas directlyabove and below the star decreases in density signifi-cantly, gas near the midplane is relatively unchanged.Still, smaller disk scale heights can lead to greatly re-duced accretion. We find that rarefication is also moresignificant for models accreting cooler gas. This canbe understood by recalling that H ∝ c s Ω − , so whileΩ = 10 − s − and Ω = 10 − s − models evolve verysimilarly at c s = 10 cm s − , model tracks diverge sig-nificantly at c s = 3 × cm s − .We expect, however, that this investigation likely over-estimates the effects of disk rarefication on the evolu-tion of AGN stars, since AGN disks are not perfectlyvertically isothermal, which would lead to a less steepdecline in density vertically. Similarly, radiation-MHDsimulations of AGN accretion disks tend to develop ver-tical profiles that decline less rapidly from the midplanethan would a vertically isothermal disk (e.g. Jiang et al.2019a). Additionally, when taking into account realis-tic opacities, some disk models (Hubeny et al. 2000) andsimulations (Jiang & Blaes 2020) have shown that in thepresence of opacity bumps, density inversions can occurin the disk, leading to an increase in density away fromthe midplane. Thus, the effects of rarefication here arelikely to be unrealistically strong.We find shear to have a fairly negligible effect on theevolution of AGN stars. As the effects of shear dependon the disk scale height, it leads to minor changes in theevolution of some stars at c s = 10 cm s − only fairly b a s e r a d i a t i v e f e e d b a c k t i d e s t i d e s + r a d i a t i o n r a r e fi c a t i o n s h e a r − − − − ρ ( g c m − ) M m a x ( M (cid:12) ) Figure 6.
The maximum mass reached for a subset of themodels in Figure 5 with c s = 10 cm s − and, where appli-cable, Ω = 10 − s − . The y-axis indicates the disk densityused for each model. Vertical black lines separate the resultsbased on the accretion prescription used, indicated on thex-axis. close to the SMBH, Ω (cid:38) − s − . In cooler disks, withcommensurately smaller scale heights, shear can be sig-nificant over a somewhat larger range in Ω. However,shear is naturally less important for determining the ac-cretion rate onto AGN stars, since its importance scalesas R B /r • , whereas rarefication and tidal effects scale as R B /H and R B /R H , and r • > H, R H .A summary of the maximum masses reached in the c s = 10 cm s − , Ω = 10 − s − runs is resented inFigure 6. At this Ω, models accounting for shear havealmost no deviation from baseline calculations. Simi-larly, runs accounting for the rarefication of the diskreach maximum masses in between the masses reachedby models using the base assumptions and those ac-counting for tidal effects. At this Ω, the models account-ing for reductions in the R B and R H due to radiation(as outlined in section 2.1) as well as tidal effects reachmaximum masses that are lower than those of modelsthat reduced R B but did not include tidal effects, andmaximum masses that are higher than when accountingfor tidal effects without reducing R H and R B .4.3. Tide-mediated stellar evolution
Having identified tides as the most significant mod-ification to Bondi accretion, of those considered here,we now turn to study their effects on the evolution ofAGN stars. Recall that the Hill radius of a star can bewritten in terms of its mass and the angular velocity ofits Keplerian orbit. Similarly, the Bondi accretion ratedepends only on ρ/c s and the mass of the star. Thus,0 Dittmann, Cantiello, and Jermyn − − − − − Ω ( s − ) − − − − − − ρ ( g c m − ) M (cid:12) fit 10 M m a x ( M (cid:12) ) Figure 7.
The maximum mass achieved by each model isshown as a function of ρ and Ω. Gray squares indicate mod-els that failed during boundary condition relaxation. Thedashed black line is the power-law fit to M max = 8 M (cid:12) overthis range, given by Equation 21. Note that all runs arecalculated with c s = 10 km s − . to the extent that tides and accretion govern the evolu-tion of AGN stars, we can study their evolution across avariety of disk characteristics and SMBH masses usingonly two parameters. This is not completely right, sincechanges in c s do affect the stellar atmosphere even atfixed ρ/c s , but the dependence on c s alone is weak com-pared with that on ρ/c s , which governs the accretionrate onto AGN stars. For this reason we perform thesimulations in this section at a single AGN disk soundspeed, c s = 10 km s − and assume that outcomes ofstellar evolution such as final masses and rate of massloss scale with ρ/c s .One of the most important features of the evolutionof AGN stars is that initially low-mass stars can accreteenough gas from the disk and become massive stars, end-ing their lives as compact objects. Additionally, sincethese stars reach the Eddington luminosity, they alsolose large amounts of processed material before reach-ing the end of their evolution. Thus, in Figure 7 wepresent the maximum masses achieved by AGN stars inour models.To a good degree of accuracy, quantities such as max-imum and final masses can be determined based on themass-independent factors that control the accretion rateonto AGN stars. For sufficiently large Ω, such that tidaleffects are significant, ˙ M ∝ ρ c − s Ω − / . Thus, a corre-sponding power-law contour in the ρ − Ω plane is givenby ρ = A c s Ω / for some constant A . One such curvefor, M max = 8 M (cid:12) is marked by a black dashed line in − − − − − Ω ( s − ) − − − − − − ρ ( g c m − ) yr10 yrIntermediate / immortalboundary fit 10 M fi n a l ( M (cid:12) ) Figure 8.
The final mass achieved by each model is shownas a function of ρ and Ω. Gray squares indicate models thatfailed during boundary condition relaxation. The dashedblack line is the power-law fit given by Equation 22 for theintermediate-immortal transition in ρ and Ω. The dashedthin blue line indicates models with final ages of 10 years,and the dot-dashed thin blue line indicates models with finalages of 10 years. Here, the reported age does not includetime over which the boundary conditions were relaxed foreach model. Note that all runs are calculated with c s =10 km s − . Figure 7 and is given by (cid:18) ρ g cm − (cid:19) (cid:38) . × − (cid:16) c s cm s − (cid:17) (cid:18) Ωs − (cid:19) / . (21)In the low-Ω limit, as seen in Figure 5, quantitiessuch as the maximum and final mass over the courseof an AGN star’s life become independent of Ω, and ρ (cid:38) × − g cm − at c s = 10 cm s − is sufficientfor stars to reach greater than 8 M (cid:12) . Thus, Equation(21) can be used to predict in which regions of a diskstars will become massive before the end of their lives, atwhich point they will form compact objects. This resultmay be useful for studies of gravitational waves involv-ing AGN disks, as well as interpretations of anomalousAGN flares.The disk conditions where stars become ‘immortal’can be seen in the high- ρ low-Ω area of Figure 7 wheremaximum masses begin to plateau. However, this dis-tinction is easier to see in terms of the final mass ofa star, or in the case of stars in the immortal regime,their asymptotic stellar mass. This quantity is plot-ted in Figure 8, which includes a power-law fit to theintermediate-immortal boundary, given by (cid:18) ρ g cm − (cid:19) (cid:38) . × − (cid:16) c s cm s − (cid:17) (cid:18) Ωs − (cid:19) / . (22) ccretion onto stars in AGN disks − − − − − Ω ( s − ) − − − − − − ρ ( g c m − ) − M loss (M (cid:12) Myr − ) − − − − − Ω ( s − ) − − M loss , CNO (M (cid:12) Myr − ) Figure 9.
Average rates of mass (left: total, right: carbon, nitrogen, oxygen) loss from each model as a function of ρ andΩ. Gray squares indicate models that failed during boundary condition relaxation. The thee models at high ρ and low Ωwith unusually large CNO mass loss failed before becoming chemically homogeneous. Note that all runs are calculated with c s = 10 km s − In the low-Ω limit, one can estimate based on Figure 5that ρ (cid:38) × − − × − g cm − at c s = 10 cm s − gives the location of the boundary, depending on onesassumption about radiative feedback. This expressionmay be useful for predicting the upper(lower) limits on ρ (Ω) where AGN stars may undergo supernova or othertransient events at the end of their lives. We also includecontours for stellar ages in Figure 8, which can be usedto estimate whether or not a given AGN star would beable to reach its final mass within a disk lifetime. Fordisk lifetimes of ∼ −
100 Myr, most intermediatestars will have time to reach the final stages of theirevolution, even if initailly only 1 M (cid:12) . Additionally, asshown in Figure 4, both the rate of stellar mass loss andits composition can vary significantly for immortal stars.We further investigate the mass loss rates of AGNstars and the metal content of the lost mass. From theleft panel of Figure 9, we see that at low ρ/c s or high Ωthe total mass lost from a model is insignificant, whilethe opposite is true for stars that accrete more rapidly.This may be useful for gauging the kinematic impact ofwinds from AGN stars on accretion disks. For exam-ple, in the disk models of Thompson et al. (2005), thereis a contribution to the pressure support from stellarfeedback that is independent of disk opacity, there at-tributed to supernovae. However, the extreme mass loss rates from immortal stars could also provide significantpressure support and alter the disk commensurately. Forexample, consider an immortal star with M ∗ ∼
400 M (cid:12) , losing mass at a rate of ∼ M (cid:12) / Myr. We find thattypical immortal stars have escape velocities in excessof 10 cm s − , not unlike massive OB stars (Lamers &Cassinelli 1999; Smith 2014). Using the escape veloc-ity as a rough estimate of the outflow speed, this givesa ram pressure at the Hill radius ( R H ∼ cm for M ∗ ∼
400 M (cid:12) ) of ∼ − erg cm − . Depending on theaccretion disk conditions, this can be well in excess ofthe ambient pressure of the AGN disk, ρc s , which rangesfrom 10 − to 10 − erg cm − for the disk conditions ex-plored in this work. Depending on the number of AGNstars within a given accretion disk, they may provide asignificant fraction of the total pressure support.Only the ejecta from intermediate stars is particularlymetal rich, since those stars are able to progress throughlater stages of evolution, as demonstrated in the rightpanel of Figure 9. On the other hand, immortal starshave ejecta that is overall lower in metals, but is rel-atively more nitrogen rich (Figure 4), as their energygeneration is dominated by the CNO cycle. Stars in theintermediate regime may also lose a significant amountof metal-enriched material during a supernova, but we2 Dittmann, Cantiello, and Jermyn have not included this contribution in our current anal-ysis. ASTROPHYSICAL IMPLICATIONSThe results of stellar evolution in AGN disks dependstrongly not only on the ambient gas density and tem-perature, but also the strength of the gravity of theSMBH on the surrounding gas. This can lead to a com-plex radial dependence on the effects of stars on a givenaccretion disk. One such example is sketched in Figure10, which illustrates how at large distances accretionmay be slow and stellar evolution normal due to thelow gas density, but closer to the SMBH accretion maybe staved off by the gravity of the black hole. Betweenthese extremes, AGN stars may result in a number ofastrophysical phenomena, from explosive transients toenriching the metallicity of the AGN disk.Massive stars in AGN disks, and the near-Eddingtonmass loss they experience, can alter the accretion diskcomposition and structure significantly. As shown inFigure 9, AGN stars in the intermediate accretionregime can supply the disk with metals at rate of ∼ . − M (cid:12) per Myr, even when accreting a mixturecontaining only hydrogen and helium. Additionally, asshown in Figure 8, many of these stars end their liveswith masses (cid:38)
10 M (cid:12) , and may further enrich the diskand surrounding regions, as suggested by Artymowiczet al. (1993). Metallicity enrichment has a sensitivedependence on both ρ/c s and Ω. This could be furthercomplicated by migration through the disk, (e.g. Lyraet al. 2010; Paardekooper et al. 2010; Tanaka et al.2002), which can also be altered by the near-Eddingtonluminosities of AGN stars (e.g. Hankla et al. 2020). De-tailed modeling along those lines is beyond the scopeof the present work, but Equations (21) and (22) mayprove useful to such efforts.Despite these difficulties, AGN stars are a promisingchannel for producing the supersolar metallicities com-monly observed in both high- and low-redshift AGN (e.g.Hamann et al. 2002; Maiolino & Mannucci 2019; StorchiBergmann & Pastoriza 1989). Notably, AGN metallici-ties have been measured to be larger than that of theirhost galaxies, roughly independent of redshift, but cor-related with SMBH mass (Xu et al. 2018). Additionally,if these stars undergo supernova explosions, they couldsignificantly enhance the iron abundance in the disk, andcould be partially responsible for the large iron abun-dances inferred from X-ray emission line analyses (e.g.Nandra et al. 1997; Tanaka et al. 1995).Although the mass lost from immortal AGN stars isnot as metal-rich as that from their intermediate coun-terparts, nitrogen makes up a much larger fraction of the metal content of their winds, as shown in Figure 4.A large population of such stars may lead to an over-abundance of nitrogen with comparatively little metal-licity enhancement. This scenario bears some resem-blance to nitrogen-rich quasars, a subset ( ∼ ∼ years, sometimes attributed to tidal disruptions ofevolved stars (Kochanek 2016; Liu et al. 2018). The por-tion of the population not displaying variability may belinked to immortal stars within the accretion disk.As shown in Figure 8, there is a window in ρ/c s − Ωwhere AGN stars are both in the intermediate accre-tion regime, and can potentially reach the end of theirlives within an AGN disk lifetime. To a good approx-imation, since ˙ M B ∝ M ∗ , the lifetimes of AGN starsare inversely proportional to their initial masses, so thecontours shown in Figure 8 can be extrapolated to otherinitial stellar masses. We did not evolve our modelsfar enough to accurately predict compactnesses, andthus stellar fates (O’Connor & Ott 2013; Sukhbold &Woosley 2014), although the preliminary core compact-ness calculations of Cantiello et al. (2020) suggest thatAGN stars might preferentially form black holes. Re-gardless, these massive stars are expected to leave be-hind neutron stars and stellar-mass black holes withinthe disk, and can potentially produce electromagneticsignatures. Supernovae may transport significant an-gular momentum through the disk (Moranchel-Basurtoet al. 2021), and are a possible source of transients ob-served in optical time-domain surveys (Frederick et al.2020; Graham et al. 2017). Many AGN stars are ex-pected to reach the end of their lives rapidly rotatingdue to the angular momentum gained as they accrete,so black holes formed in the collapse of AGN stars couldgenerate gamma-ray bursts, which could in turn take ona number of appearances depending on their location inthe accretion disk (Jermyn et al. in prep.; Perna et al.2021).Compact objects remaining in the disk migrate dueto various torques (Hankla et al. 2020; Lyra et al. 2010;Paardekooper et al. 2010; Tanaka et al. 2002), a numberof which depend quadratically on the mass of the objectmass. Thus, heavier objects can migrate faster, leadingto mergers within the disk, which could make up a largefraction of LIGO sources (e.g. Bartos et al. 2017; McK-ernan et al. 2020; Stone et al. 2017). These objects mayalso migrate inward through the disk, eventually merg- ccretion onto stars in AGN disks Runaway Accretion Region Intermediate Accretion Region Slow Accretion RegionSMBH
AGN Disk
Supermassive Star (M ≫
100 M ⊙ )Massive Star (M ≈ ⊙ )Low-Mass Star (M < 8 M ⊙ )Compact Remnants Figure 10.
Illustration of different regimes of accretion in an AGN disk. Stellar seeds are expected to become massive andsupermassive in the intermediate and runaway accretion region, respectively. These stars pollute the disk with their super-Eddington winds and stellar explosions (SNe and GRBs). They also leave behind compact remnants that can dynamicallyinteract and merge. In the slow accretion region, stars only gain modest amounts of material. These stars are expected tooutlive the AGN phase. Note that the location of these different regions depends on the specific disk model. The accretionrate decreases towards the SMBH because of the increasing importance of tidal effects and, to a smaller extent, shear. Thesuccession shown in this illustration assumes a constant sound speed through the disk and a monotonic density increase towardsthe super massive black hole (SMBH). Realistic disk models might lead to different ordering. ing with the SMBH, leading to extreme- or intermediate-mass inspirals detectable by LISA (Derdzinski et al.2021, 2019; Yunes et al. 2011), and facilitating SMBHgrowth without being subject to the Eddington limit(Dittmann & Miller 2020).It is believed that the Milky Way experienced an AGNphase about 2-8 Myr ago (Bland-Hawthorn et al. 2019;Su et al. 2010). With its directly available observationsof stellar populations and stellar remnants, the GalacticCenter (GC) is a prime target for testing the possibleimpact of AGN stars evolution (Cantiello et al. 2020).The central parsec contains an unexpected large num-ber of young massive stars (Alexander 2005; Ghez et al.2003), with O/WR stars confined to the inner 0.5 pcregion (Bartko et al. 2010; Paumard et al. 2006). Thepresent day mass function of these stars is top-heavy(Genzel et al. 2010). Contrary to theoretical expecta-tions (Bahcall & Wolf 1976, 1977), the relative fractionof low-mass stars decreases moving towards the GC (Doet al. 2017; Genzel et al. 2010). Spectroscopy revealsthat some of the stars in the GC may be He-rich (Doet al. 2018; Habibi et al. 2017; Martins et al. 2008). Fi-nally, Hailey et al. (2018) found that low-mass X-raybinary candidates appear to be segregated to the inner1 pc. In the context of AGN stars evolution, low-mass starsare expected to become massive via accretion in the in-ner, denser regions of an AGN disk. Stars survivingthe AGN phase might still carry the signature of a diskchemically enriched via AGN star evolution. Compactremnants are also expected to be radially segregated, al-though migration effects might play an important roleas well (Tagawa et al. 2020). While there are a numberof possible competing explanations, it is intriguing thatAGN stars evolution could simultaneously account forthese puzzling observations of the GC. CONCLUSIONSStellar evolution in AGN disks primarily depends onthe properties of the disk around the star and the bal-ance between the gravity of the star and that of theSMBH. Uncertainty in the effects of radiative feedbackon the accretion stream only significantly affects ourmodels far from the SMBH (Ω (cid:46) − s − , r • (cid:38) .
36 pcfor a 10 M (cid:12) black hole). Additionally, other effects thatcan reduce accretion onto AGN stars are less significantthan tidal forces from the SMBH. Therefore, account-ing for tidal forces alone is likely sufficient to study theevolution of stars throughout AGN disks.Accordingly, we performed a survey in ρ/c s , Ω to mapthese key parameters to the outcomes of stellar evolution4
Dittmann, Cantiello, and Jermyn in AGN disks. All of our models were initially 1 M (cid:12) ,although a different initial mass would simply changethe time required for each star to accrete significantly( M ∗ / ˙ M ∝ /M ∗ ). Using this survey, we have deter-mined mappings between ρ/c s and Ω that can be usedto predict, given an accretion disk model, whether ex-plosive transients and compact remnants are expected(Figure 7, Equation(21)) or whether stars are expectedto becomes super-massive ( M ∗ > M (cid:12) ) and live in-definitely (Figure 8, Equation (22)). Depending on thestructure of a given accretion disk and the surroundingstellar population, a variety of outcomes are possible,such as that illustrated by Figure 10. AGN stars can, forexample, enhance the metallicity of the accretion disk,lead to mergers of objects within the disk and accom-panying gravitational waves, cause luminous transients,lend significant pressure support to the accretion disk,and leave behind stellar population with a top-heavymass function after the dispersal of the disk. SOFTWARE MESA (Paxton et al. 2011, 2013, 2015, 2018, 2019, http://mesa.sourceforge.net),
MESASDK (Townsend 2019), matplotlib (Hunter 2007), numpy (van der Walt et al.2011) ACKNOWLEDGMENTSWe thank Cole Miller for useful discussions. The Cen-ter for Computational Astrophysics at the Flatiron In-stitute is supported by the Simons Foundation. Compu-tations were performed using the Rusty cluster, of theFlatiron Institute; and the YORP cluster, administeredby the Center for Theory and Computation within theDepartment of Astronomy at the University of Mary-land.APPENDIX A. SOFTWARE DETAILSCalculations were carried out using MESA version 15140. The MESA EOS is a blend of the OPAL (Rogers &Nayfonov 2002), SCVH (Saumon et al. 1995), FreeEOS (Irwin 2004), HELM (Timmes & Swesty 2000), and PC(Potekhin & Chabrier 2010) EOSes.Radiative opacities are primarily from OPAL (Iglesias & Rogers 1993, 1996), with low-temperature data fromFerguson et al. (2005) and the high-temperature, Compton-scattering dominated regime by Buchler & Yueh (1976).Electron conduction opacities are from Cassisi et al. (2007).Nuclear reaction rates are from JINA REACLIB (Cyburt et al. 2010) plus additional tabulated weak reaction ratesFuller et al. (1985); Langanke & Mart´ınez-Pinedo (2000); Oda et al. (1994). Screening is included via the prescriptionof Chugunov et al. (2007). Thermal neutrino loss rates are from Itoh et al. (1996).We adopted a 21-isotope nuclear network (approx21.net). We used the Schwarzschild criterion to determine convec-tive boundaries and did not include convective overshooting.REFERENCES
Alexander, T. 2005, PhR, 419, 65Antonucci, R. R. J., & Cohen, R. D. 1983, ApJ, 271, 564Artymowicz, P., Lin, D. N. C., & Wampler, E. J. 1993,ApJ, 409, 592Bahcall, J. N., & Wolf, R. A. 1976, ApJ, 209, 214—. 1977, ApJ, 216, 883Bartko, H., Martins, F., Trippe, S., et al. 2010, ApJ, 708,834Bartos, I., Kocsis, B., Haiman, Z., & M´arka, S. 2017, ApJ,835, 165Batra, N. D., & Baldwin, J. A. 2014, MNRAS, 439, 771Bentz, M. C., Hall, P. B., & Osmer, P. S. 2004, AJ, 128, 561Bentz, M. C., & Osmer, P. S. 2004, AJ, 127, 576Blackburne, J. A., & Kochanek, C. S. 2010, ApJ, 718, 1079 Bland-Hawthorn, J., Maloney, P. R., Sutherland, R., et al.2019, ApJ, 886, 45B¨ohm-Vitense, E. 1992, Introduction to StellarAstrophysicsBuchler, J. R., & Yueh, W. R. 1976, ApJ, 210, 440Cantiello, M., Jermyn, A. S., & Lin, D. N. C. 2020, arXive-prints, arXiv:2009.03936Cassisi, S., Potekhin, A. Y., Pietrinferni, A., Catelan, M., &Salaris, M. 2007, ApJ, 661, 1094Chugunov, A. I., Dewitt, H. E., & Yakovlev, D. G. 2007,PhRvD, 76, 025028Cyburt, R. H., Amthor, A. M., Ferguson, R., et al. 2010,ApJS, 189, 240Dai, X., Kochanek, C. S., Chartas, G., et al. 2010, ApJ,709, 278 ccretion onto stars in AGN disks Daramy-Loirat, C., Defour, D., de Dinechin, F., et al. 2006,CR-LIBM A library of correctly rounded elementaryfunctions in double-precision, Research report, LIP,.https://hal-ens-lyon.archives-ouvertes.fr/ensl-01529804Davis, S. W., Jiang, Y.-F., Stone, J. M., & Murray, N.2014, ApJ, 796, 107Derdzinski, A., D’Orazio, D., Duffell, P., Haiman, Z., &MacFadyen, A. 2021, MNRAS, 501, 3540Derdzinski, A. M., D’Orazio, D., Duffell, P., Haiman, Z., &MacFadyen, A. 2019, MNRAS, 486, 2754Dexter, J., & Agol, E. 2011, ApJL, 727, L24Dittmann, A. J., & Miller, M. C. 2020, MNRAS, 493, 3732Do, T., Kerzendorf, W., Konopacky, Q., et al. 2018, ApJL,855, L5Do, T., Ghez, A., Morris, M., et al. 2017, in IAUSymposium, Vol. 322, The Multi-Messenger Astrophysicsof the Galactic Centre, ed. R. M. Crocker, S. N.Longmore, & G. V. Bicknell, 222–230Dobbs-Dixon, I., Li, S. L., & Lin, D. N. C. 2007, ApJ, 660,791Duffell, P. C. 2015, ApJL, 807, L11Duffell, P. C., Haiman, Z., MacFadyen, A. I., D’Orazio,D. J., & Farris, B. D. 2014, ApJL, 792, L10Duffell, P. C., & MacFadyen, A. I. 2013, ApJ, 769, 41D¨urmann, C., & Kley, W. 2015, A&A, 574, A52Eddington, A. S. 1929, MNRAS, 90, 54Ferguson, J. W., Alexander, D. R., Allard, F., et al. 2005,ApJ, 623, 585Frederick, S., Gezari, S., Graham, M. J., et al. 2019, ApJ,883, 31—. 2020, arXiv e-prints, arXiv:2010.08554Fuller, G. M., Fowler, W. A., & Newman, M. J. 1985, ApJ,293, 1Gammie, C. F. 2001, ApJ, 553, 174Garc´ıa, J. A., Kallman, T. R., Bautista, M., et al. 2018, inAstronomical Society of the Pacific Conference Series,Vol. 515, Workshop on Astrophysical Opacities, 282Genzel, R., Eisenhauer, F., & Gillessen, S. 2010, Reviews ofModern Physics, 82, 3121Ghez, A. M., Duchˆene, G., Matthews, K., et al. 2003,ApJL, 586, L127Graham, M. J., Djorgovski, S. G., Drake, A. J., et al. 2017,MNRAS, 470, 4112Habibi, M., Gillessen, S., Martins, F., et al. 2017, ApJ, 847,120Hailey, C. J., Mori, K., Bauer, F. E., et al. 2018, Nature,556, 70Hamann, F., Korista, K. T., Ferland, G. J., Warner, C., &Baldwin, J. 2002, ApJ, 564, 592 Hankla, A. M., Jiang, Y.-F., & Armitage, P. J. 2020, ApJ,902, 50Ho, L. C. 2008, ARA&A, 46, 475Hogg, J. D., & Reynolds, C. S. 2016, ApJ, 826, 40Hubeny, I., Agol, E., Blaes, O., & Krolik, J. H. 2000, ApJ,533, 710Hunter, J. D. 2007, Computing in Science Engineering, 9,90Iglesias, C. A., & Rogers, F. J. 1993, ApJ, 412, 752—. 1996, ApJ, 464, 943Irwin, A. W. 2004, The FreeEOS Code for Calculating theEquation of State for Stellar Interiors, , .http://freeeos.sourceforge.net/Itoh, N., Hayashi, H., Nishikawa, A., & Kohyama, Y. 1996,ApJS, 102, 411Jermyn, A. S., Dittmann, A. J., Cantiello, M., & Perna, R.in prep.Jiang, L., Fan, X., & Vestergaard, M. 2008, ApJ, 679, 962Jiang, Y.-F., & Blaes, O. 2020, ApJ, 900, 25Jiang, Y.-F., Blaes, O., Stone, J. M., & Davis, S. W. 2019a,ApJ, 885, 144Jiang, Y.-F., Cantiello, M., Bildsten, L., et al. 2018, arXive-prints, arXiv:1809.10187Jiang, Y.-F., Stone, J. M., & Davis, S. W. 2019b, ApJ, 880,67Kanagawa, K. D., Tanaka, H., Muto, T., & Tanigawa, T.2017, PASJ, 69, 97Kanagawa, K. D., Tanaka, H., Muto, T., Tanigawa, T., &Takeuchi, T. 2015, MNRAS, 448, 994Kanagawa, K. D., Tanaka, H., & Szuszkiewicz, E. 2018,ApJ, 861, 140Kelly, B. C., Bechtold, J., & Siemiginowska, A. 2009, ApJ,698, 895King, A., & Nixon, C. 2015, MNRAS, 453, L46King, A. R., Pringle, J. E., & Livio, M. 2007, MNRAS, 376,1740Kochanek, C. S. 2016, MNRAS, 458, 127Kormendy, J., & Ho, L. C. 2013, ARA&A, 51, 511Krumholz, M. R., McKee, C. F., & Klein, R. I. 2005, ApJ,618, 757Krumholz, M. R., & Thompson, T. A. 2013, MNRAS, 434,2329LaMassa, S. M., Cales, S., Moran, E. C., et al. 2015, ApJ,800, 144Lamers, H. J. G. L. M., & Cassinelli, J. P. 1999,Introduction to Stellar WindsLanganke, K., & Mart´ınez-Pinedo, G. 2000, NuclearPhysics A, 673, 481Lee, D., & Gammie, C. F. 2020, arXiv e-prints,arXiv:2011.07151 Dittmann, Cantiello, and Jermyn
Li, Y.-P., Chen, Y.-X., Lin, D. N. C., & Zhang, X. 2021,ApJ, 906, 52Liu, X., Dittmann, A., Shen, Y., & Jiang, L. 2018, ApJ,859, 8Lucy, L. B. 1967, ZA, 65, 89Lynden-Bell, D. 1969, Nature, 223, 690Lyra, W., Paardekooper, S.-J., & Mac Low, M.-M. 2010,ApJL, 715, L68MacLeod, C. L., Ivezi´c, ˇZ., Kochanek, C. S., et al. 2010,ApJ, 721, 1014MacLeod, C. L., Green, P. J., Anderson, S. F., et al. 2019,ApJ, 874, 8MacLeod, M., & Lin, D. N. C. 2020, ApJ, 889, 94Maiolino, R., & Mannucci, F. 2019, A&A Rv, 27, 3Martins, F., Gillessen, S., Eisenhauer, F., et al. 2008, ApJL,672, L119Matsuoka, K., Nagao, T., Maiolino, R., Marconi, A., &Taniguchi, Y. 2009, A&A, 503, 721McKernan, B., Ford, K. E. S., & O’Shaughnessy, R. 2020,MNRAS, 498, 4088McKernan, B., Ford, K. E. S., Bellovary, J., et al. 2018,ApJ, 866, 66Meyer, F., & Meyer-Hofmeister, E. 1982, A&A, 106, 34Milsom, J. A., Chen, X., & Taam, R. E. 1994, ApJ, 421, 668Moranchel-Basurto, A., S´anchez-Salcedo, F. J., Chametla,R. O., & Vel´azquez, P. F. 2021, ApJ, 906, 15Morgan, C. W., Kochanek, C. S., Morgan, N. D., & Falco,E. E. 2010, ApJ, 712, 1129Nagao, T., Maiolino, R., & Marconi, A. 2006, A&A, 459, 85Nandra, K., George, I. M., Mushotzky, R. F., Turner, T. J.,& Yaqoob, T. 1997, ApJL, 488, L91Neumayer, N., Seth, A., & B¨oker, T. 2020, A&A Rv, 28, 4Novikov, I. D., & Thorne, K. S. 1973, in Black Holes (LesAstres Occlus), 343–450O’Connor, E., & Ott, C. D. 2013, ApJ, 762, 126Oda, T., Hino, M., Muto, K., Takahara, M., & Sato, K.1994, Atomic Data and Nuclear Data Tables, 56, 231Paardekooper, S. J., Baruteau, C., Crida, A., & Kley, W.2010, MNRAS, 401, 1950Paumard, T., Genzel, R., Martins, F., et al. 2006, ApJ, 643,1011Paxton, B., Bildsten, L., Dotter, A., et al. 2011, ApJS, 192,3Paxton, B., Cantiello, M., Arras, P., et al. 2013, ApJS, 208,4Paxton, B., Marchant, P., Schwab, J., et al. 2015, ApJS,220, 15Paxton, B., Schwab, J., Bauer, E. B., et al. 2018, ApJS,234, 34 Paxton, B., Smolec, R., Schwab, J., et al. 2019, ApJS, 243,10Perna, R., Lazzati, D., & Cantiello, M. 2021, ApJL, 906, L7Pooley, D., Blackburne, J. A., Rappaport, S., & Schechter,P. L. 2007, ApJ, 661, 19Potekhin, A. Y., & Chabrier, G. 2010, Contributions toPlasma Physics, 50, 82Rauch, K. P. 1995, MNRAS, 275, 628Rogers, F. J., & Nayfonov, A. 2002, ApJ, 576, 1064Rosenthal, M. M., Chiang, E. I., Ginzburg, S., &Murray-Clay, R. A. 2020, MNRAS, 498, 2054Rumbaugh, N., Shen, Y., Morganson, E., et al. 2018, ApJ,854, 160Saumon, D., Chabrier, G., & van Horn, H. M. 1995, ApJS,99, 713Schawinski, K., Koss, M., Berney, S., & Sartori, L. F. 2015,MNRAS, 451, 2517Sch¨opf, H. M., & Supancic, P. H. 2014, The MathematicaJournal, 16, doi:10.3888/tmj.16–11Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 500, 33—. 1976, MNRAS, 175, 613Shapiro, S. L., Lightman, A. P., & Eardley, D. M. 1976,ApJ, 204, 187Sirko, E., & Goodman, J. 2003, MNRAS, 341, 501Smith, N. 2014, ARA&A, 52, 487So(cid:32)ltan, A. 1982, MNRAS, 200, 115Stone, N. C., Metzger, B. D., & Haiman, Z. 2017, MNRAS,464, 946Storchi Bergmann, T., & Pastoriza, M. G. 1989, ApJ, 347,195Su, M., Slatyer, T. R., & Finkbeiner, D. P. 2010, ApJ, 724,1044Sukhbold, T., & Woosley, S. E. 2014, ApJ, 783, 10Tagawa, H., Haiman, Z., & Kocsis, B. 2020, ApJ, 898, 25Tanaka, H., Takeuchi, T., & Ward, W. R. 2002, ApJ, 565,1257Tanaka, H., & Ward, W. R. 2004, ApJ, 602, 388Tanaka, Y., Nandra, K., Fabian, A. C., et al. 1995, Nature,375, 659Thompson, T. A., Quataert, E., & Murray, N. 2005, ApJ,630, 167Timmes, F. X., & Swesty, F. D. 2000, ApJS, 126, 501Tomsick, J. A., Parker, M. L., Garc´ıa, J. A., et al. 2018,ApJ, 855, 3Toomre, A. 1964, ApJ, 139, 1217Tout, C. A., Eggleton, P. P., Fabian, A. C., & Pringle, J. E.1989, MNRAS, 238, 427Townsend, R. H. D. 2019, MESA SDK for Linux,v20190503, Zenodo, doi:10.5281/zenodo.2669541.https://doi.org/10.5281/zenodo.2669541 ccretion onto stars in AGN disks17