Stellar Evolution in the Disks of Active Galactic Nuclei Produces Rapidly Rotating Massive Stars
Adam S. Jermyn, Alexander J. Dittmann, Matteo Cantiello, Rosalba Perna
DDraft version March 1, 2021
Typeset using L A TEX twocolumn style in AASTeX63
Stellar Evolution in the Disks of Active Galactic Nuclei Produces Rapidly Rotating Massive Stars
Adam S. Jermyn , Alexander J. Dittmann ,
1, 2
Matteo Cantiello ,
1, 3 and Rosalba Perna
1, 4 Center for Computational Astrophysics, Flatiron Institute, 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 Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA
Submitted to ApJABSTRACTStars can either be formed in or captured by the accretion disks in Active Galactic Nuclei (AGN).These AGN stars are irradiated and subject to extreme levels of accretion, which can turn even low-mass stars into very massive ones (
M > (cid:12) ) whose evolution may result in the formation of massivecompact objects (
M > (cid:12) ). Here we explore the spins of these AGN stars and the remnants theyleave behind. We find that AGN stars rapidly spin up via accretion, eventually reaching near-criticalrotation rates. They further maintain near-critical rotation even as they shed their envelopes, becomecompact, and undergo late stages of burning. This makes them good candidates to produce high-spinmassive black holes, such as the ones seen by LIGO-Virgo in GW190521g, as well as long Gamma RayBursts (GRBs) and the associated chemical pollution of the AGN disk.
Keywords:
Stellar physics (1621); Stellar evolutionary models (2046); Massive stars(732);Quasars(1319) INTRODUCTIONActive Galactic Nuclei (AGN) are believed to be pow-ered by massive accretion disks draining into supermas-sive black holes (SMBHs) (Lynden-Bell 1969). Becauseof their crucial role in AGN, these accretion disks havebeen extensively studied since the pioneering work ontheir structure by Shakura & Sunyaev (1973).Recent years have seen a rekindled interest in AGNdisks, particularly in light of gravitational wave detec-tions by LIGO and Virgo. In particular, the observationof black holes (BHs) with masses above the maximummass allowed by pair instability in massive stars (Abbottet al. 2020), as well as in the lower mass gap (Abbottet al. 2020), finds a natural explanation in the environ-ments of AGN disks, where compact object mergers areenhanced, and neutron stars (NSs) and BHs grow byaccretion due to the very high disk gas densities (e.g.McKernan et al. 2012; Yang et al. 2019; Tagawa et al.2020).
Corresponding author: Adam S. [email protected]
The presence of stars and compact stellar remnants inAGN disks is not surprising. Stars can end up in thedisks of AGNs via at least two mechanisms: in-situ for-mation when disks become self-gravitating and unstableto fragmentation (e.g. Kolykhalov & Syunyaev 1980;Goodman 2003; Dittmann & Miller 2020), and capturefrom the nuclear star cluster surrounding the AGN as aresult of momentum and energy loss as the stars interactwith the disk (e.g. Artymowicz et al. 1993; MacLeod &Lin 2020; Fabj et al. 2020). Once in an AGN disk, starsmay evolve and thereby form compact objects such asblack holes and neutron stars.Stars in AGN disks (AGN stars) are believed to evolvequite differently from those in standard galactic envi-ronments. AGN disks are much hotter and denser thanthe interstellar medium, so AGN stars are subject tovery different boundary conditions than normal stars.The evolution of these stars has been recently studiedby Cantiello et al. (2020), who found that AGN starscan quickly become very massive (
M >
100 M (cid:12) ) dueto rapid accretion fueled by the large gas reservoir inthe AGN disk, and that chemical mixing plays a criticalrole. a r X i v : . [ a s t r o - ph . H E ] F e b Jermyn et al.
The study of Cantiello et al. (2020) did not, however,investigate the role of rotation in these stars and how itis affected by the special environment of an AGN disk.Rotation plays a very important role in stellar evolu-tion (e.g. Maeder 2009; Langer 2012) and in determiningwhether a massive star at the end if its life can produce along gamma-ray burst (long GRB) during core collapse(MacFadyen & Woosley 1999). Understanding the rota-tion of these stars is also crucial to further constrain theorigin of the LIGO/Virgo BHs via their inferred spins.Here we investigate the spin evolution of stars embed-ded in AGN disks (AGN stars). In Section 2 we brieflydescribe our model of stellar evolution in AGN disks, in-cluding two key improvements introduced by Dittmannet al. (2021) to the treatment of accretion and mass lossfrom the approach of Cantiello et al. (2020). In Section 3we present a model of the rotation rates of these stars,with a focus on the angular momentum changes associ-ated with mass gain/loss and on the stochastic natureof accretion in a turbulent medium. We analytically ex-plore the consequences of this model in Section 4, andfind that at least during the accretion stage many AGNstars ought to spin up to near-critical rotation rates.These rapid rotators represent most of the parameterspace we are able to explore in our models, and theseresults are confirmed by our numerical simulations ofthe coupled rotating stellar evolution of AGN stars inSection 5. We then discuss the implications of theserapid rotators for compact objects, gravitational waveobservations, and long GRBs in Section 6. We concludewith a brief summary of our results in Section 7. STELLAR MODELSWe model stellar evolution using revision 15140 of theModules for Experiments in Stellar Astrophysics (MESAPaxton et al. 2011, 2013, 2015, 2018, 2019) software in-strument. Details of the
MESA microphysics and of othersoftware tools used in our experiments are provided inAppendix A. Because the stars of interest are embed-ded in an AGN disk we implement modified surfaceboundary conditions, accretion, and mass loss follow-ing Cantiello et al. (2020).As argued by Dittmann et al. (2021), the most impor-tant correction to the Bondi-Hoyle accretion rate usedby Cantiello et al. (2020) comes from the gravitationalinfluence of the supermassive black hole (SMBH). Nearthe SMBH tidal forces truncate the radius at whichthe stellar gravity dominates the gas dynamics, limit-ing the accretion rate. A similar situation occurs whengiant planets accrete from a protoplanetary disk, and hy-drodynamic simulations have shown that prescriptionstruncating accretion to begin at the Hill Radius can ac- curately predict accretion rates in the viscosity regimerelevant to AGN disks (Rosenthal et al. 2020; Li et al.2021). Accordingly, we take into account tidal effectsfrom the SMBH when calculating accretion onto AGNstars by truncating accretion to the Hill Radius.Additionally, when computing mass loss rates westudy the effects of reducing the Eddington luminosity L Edd by a factor of 1 − Ω / Ω , whereΩ c ≡ (cid:114) GM (cid:63) R (cid:63) (1)is the critical angular velocity for a rigidly-rotatingsphere. This accounts for the fact that the centrifugalacceleration reduces the effective escape velocity of thestar and so reduces the radiative acceleration needed tounbind material from the surface (Maeder 2009; Sanyalet al. 2015). We refer to models computed using the un-modified Eddington luminosity as Γ models and thosecomputed using the rotationally-reduced Eddington lu-minosity as Γ − Ω models. We generally believe the Γ − Ωprescription is more physical, though we show modelscomputed with both prescriptions for comparison.We initialized our runs with a non-rotating zero-agemain sequence solar model. We then relaxed the bound-ary conditions and accretion rate over approximately10 yr from solar-like to those described by Cantielloet al. (2020). Some models fail during this relaxation.We have endeavoured to minimize the number of suchfailures, but given the radical adjustment in conditionsit is not surprising that some failed, especially at higherAGN densities.After relaxation we evolved our models until either anage of 10 yr, a core temperature of 3 × K, or failure toconverge. We imposed the age limit because AGN disklifetimes are believed to be of order 1 −
100 Myr (Khrykinet al. 2019; Martini & Weinberg 2001; Haiman & Hui2001), and because AGN stars which do not completetheir evolution in 10 yr generally follow standard stellarevolution. The core temperature limit allows us to haltmodels during Oxygen burning. Models reaching thesecore temperatures are expected to undergo core collapsewithin a timescale ∼ yr.AGN stars rapidly accrete and reach the Eddington lu-minosity. Because of this, their interiors have compara-ble radiation and gas pressures and hence follow γ = 4 / GN Stars Spin Fast ROTATION EVOLUTIONWe assume that AGN stars only change angularmomentum through mass loss or accretion. Follow-ing Cantiello et al. (2020) we allow AGN stars to bothaccrete ( ˙ M gain ) and lose mass ( ˙ M loss ) at the same time,imagining a multidimensional system with simultaneousinflows and radiation-driven outflows.Because AGN disks are turbulent environments theangular momentum they accrete is best treated as a ran-dom variable. In Appendix B we describe a method formodelling the total angular momentum of the AGN staras a normally-distributed random variable characterizedby a mean (cid:104) J (cid:105) and a variance σ J .In practice, however, the correlation timescale of ve-locities in the AGN disk is of order the dynamical timein the Bondi sphere, which is short compared with theevolutionary timescale of an embedded star. As a resultwe always find that σ J / (cid:104) J (cid:105) (cid:28)
1, such that the angu-lar momentum is well-characterized by its expectationvalue. While our calculations follow the full stochas-tic model described in Appendix B, in what follows weconsider only the mean angular momentum, which wedenote simply by J .We neglect torques due to magnetic coupling betweenthe star and its environment, such that the mean angu-lar momentum of the star evolves due to mass loss andaccretion as dJdt = ˙ M gain j gain , avg − ˙ M loss j loss , avg , (2)where ˙ M gain is the accretion rate, ˙ M loss is the rate ofmass loss, j gain , avg is the mean specific angular momen-tum of the accreting material, and j loss , avg is the meanspecific angular momentum of the lost material. Theremainder of this section computes the terms appearingin this equation. 3.1. Mass Loss
We assume that mass is lost preferentially at the equa-tor of the star and so carries the equatorial specific an-gular momentum j loss = Ω R . (3)This is a conservative choice in that it maximizes thelost angular momentum; any other choice would resultin a smaller j loss and hence faster-rotating stars. Forsimplicity we assume that R equator equals the mean ra-dius of the star, so that j loss = JM (cid:63) (cid:18) M (cid:63) R (cid:63) I (cid:19) = JkM (cid:63) , (4) where I ≡ (cid:90) M (cid:63) r dm (5)is the moment of inertia of the star and k ≡ IM (cid:63) R (cid:63) (6)is the gyration parameter (i.e. the non-dimensional gy-ration radius). 3.2. Mass Accretion
Next, we must calculate the specific angular momen-tum of the accreting mass. There are three relevantlength-scales for accretion onto an AGN star, namely theBondi radius, the Hill radius, and the disk scale height.These three are generally not independent, so we chooseto parameterize the accretion physics in terms of theBondi and Hill radii.In a stationary, infinite medium we expect accretiononto a lone star to begin at the Bondi radius R Bondi = 2 GM (cid:63) c , (7)because that is the radius at which the escape velocityof the gravitational potential of the star becomes com-parable to the sound speed of the medium (Bondi 1952).Here c s is the sound speed in the AGN disk and M (cid:63) isthe mass of the star.When the star orbits a SMBH, however, the star’sgravity competes not only with the disk pressure butalso with tidal forces from the SMBH. The radius atwhich the star’s gravity dominates over tides from theSMBH is the Hill radius R Hill = r orb (cid:18) M (cid:63) M BH (cid:19) / , (8)where r orb is the orbital radius of the star and M BH isthe mass of the central black hole. To account for thiseffect we model the accretion as beginning at the smallerof R Bondi and R Hill , which we identify as the accretionradius R acc ≡ min ( R Bondi , R
Hill ) . (9)Dittmann et al. (2021) studied a variety of other effects,including vertical density variations, and found them tobe comparatively unimportant.If there is no viscous or magnetic angular momentumtransport across the the accretion radius then the angu-lar momentum of accreting material is just that of thematerial at the accretion radius in the frame comoving Jermyn et al.
SMBH FrameCo-moving
Figure 1.
Material accretes from a differentially rotatingdisk onto a star, spinning the star up. This is shown inthe frame of the SMBH (upper) and the frame co-movingwith the star (lower). AGN stars obtain retrograde rotation(backwards with respect to their orbital motion). with the star (Figure 1). In that frame the average an-gular momentum within the Bondi radius is given bythe differential rotation of the AGN disk, such that j gain , avg ≈ R d ( a Ω AGN ) da ≈ Ω AGN R , (10)where a is the semi-major axis of the orbit of the AGNstar around the central SMBH. Note that this angularmomentum is oriented retrograde relative to the diskbecause material closer to the SMBH is moving fasterthan the star, and material further out is moving slower.So long as j gain , avg < j max = (cid:112) GM (cid:63) R (cid:63) (11)material can fall directly from the accretion radius ontothe star and it is likely a good approximation to ne-glect torques at R acc . However, when j gain , avg > j max material must shed angular momentum in order to ac-crete. The material likely forms an accretion disk whichthen transports angular momentum outwards towardsand beyond R acc via viscous and/or magnetic torques.We therefore truncate j gain , avg to be no greater than j max . Implicitly we therefore assume that excess angu-lar momentum beyond j max is lost via magnetic/viscoustorques across the accretion radius. Dittmann et al. (2021) investigated the effect of mod-ifications to the accretion rate when the accreting angu-lar momentum is large and found that these corrections(1) do not qualitatively change AGN star evolution be-yond the effects of tides and (2) are generally less im-portant than the tidal effects which we have included.We therefore do not reduce the accretion rate in thiscase, though it is likely that the formation of a disk isassociated with some reduction in ˙ M gain and that mayhave some quantitative effect on the resulting evolution.3.3. Truncation
The star cannot spin faster than its critical rotationrate and so its total angular momentum is limited to J (cid:46) J crit ≈ k (cid:112) GM (cid:63) R (cid:63) , (12)where we have ignored aspherical corrections in this ap-proximation. To impose the constraint in equation (12)we truncate J at the end of each time-step to lie between- J crit and + J crit . ANALYTIC PREDICTIONSBefore studying our numerical results it is worth de-riving some analytical predictions. For these purposeswe focus on AGN stars that undergo runaway accretion,and divide their evolution into three phases:1. Initial runaway accretion.2. Constant mass (the “Immortal Phase” of Cantielloet al. 2020).3. Late-stage super-Eddington mass loss.4.1.
Runaway Accretion
During runaway accretion the accretion rate is alwaysmuch greater than mass loss. Moreover, because thestar begins with J = 0 the mean angular momentumis increasing. Therefore, with j loss , avg ∝ Ω (cid:63) we see that j gain , avg is typically larger than j loss , avg . With these twoconsiderations we neglect angular momentum loss andassume that dJdt = ˙ M gain j gain , avg . (13)With some rearranging, we find d ln J/J crit d ln M (cid:63) = j gain , avg M (cid:63) J − d ln J crit d ln M (cid:63) . (14)Inserting equation (12) for J crit and treating k as a con-stant we find d ln J/J crit d ln M (cid:63) = j gain , avg M (cid:63) J − − d ln R (cid:63) d ln M (cid:63) , (15) GN Stars Spin Fast M (cid:63) / ˙ M ) d ln R (cid:63) /dt .We now estimate d ln R (cid:63) /d ln M (cid:63) . Cantiello et al.(2020) found that when AGN stars become massive theybecome radiation-dominated and thus approach γ = 4 / k .Because of their high accretion rates they continue toundergo core hydrogen burning throughout the accre-tion phase. The rate of nuclear burning is an extremelystrong function of core temperature, so this regulatestheir core temperatures to a narrow window aroundlog T / K ≈ .
5. Because AGN stars in this phase areradiation-dominated, we know that P c ≈ P rad = 13 aT , (16)where the subscript ‘c’ denotes a quantity evaluated inthe core and a is the radiation gas constant. For a poly-trope we also know from scaling considerations that P c ≈ GM (cid:63) R (cid:63) . (17)Combining equations (16) and (17) we find R (cid:63) ∝ M / (cid:63) T − , (18)so if T c is approximately constant then R (cid:63) ∝ M / (cid:63) . (19)Inserting equation (19) into equation (15) we find d ln J/J crit d ln M (cid:63) = j gain , avg M (cid:63) J − . (20)Expanding j gain , avg then yields d ln J/J crit d ln M (cid:63) = min (cid:18) M (cid:63) Ω AGN R J , J crit kJ (cid:19) − . (21)With increasing mass we see that this reaches a fixedpoint when JJ crit = 47 min (cid:18) M (cid:63) Ω AGN R J crit , k (cid:19) , (22)that is, the specific angular momentum approaches thesmaller of either 4 / k times critical or the specific an-gular momentum of the accreting material. k is only approximately fixed because the precise definition of R (cid:63) matters, and this is not necessarily a constant as our boundaryconditions do depend on M (cid:63) . Hence k does vary slightly as thestar accretes, though empirically (Section 5) we do find that k isapproximately constant during this phase of evolution. For many choices of AGN disk parameters, M (cid:63) Ω AGN R (cid:29) J crit /k , which means that d ln J/J crit d ln M (cid:63) = J crit kJ − . (23)Hence with increasing mass we see that J/J crit in-creases rapidly, asymptoting to the fixed point where J = (4 / k ) J crit . For a sphere of uniform density, k = 2 /
5. AGN stars have higher densities in their coresthan their envelopes and so have k < /
5. As a resultthe fixed point has J = (10 / J crit > J crit and so in ourmodels J grows to J crit and truncates there.4.2. Constant Mass
In the constant-mass regime, accretion is balanced bymass loss. The mean angular momentum evolves ac-cording to dJdt = ˙ M gain j gain , avg − ˙ M loss j loss (24)= ˙ M gain (cid:18) j gain , avg − JkM (cid:63) (cid:19) . (25)This has a fixed point when JkM (cid:63) = j gain , avg , (26)so stars in this evolutionary phase evolve to a specificangular momentum which is proportional to that whichthey accrete.When M (cid:63) Ω AGN R (cid:29) √ GM (cid:63) R (cid:63) = J crit /k we trun-cate j gain , avg to √ GM (cid:63) R (cid:63) = J crit /kM (cid:63) , in which casethe fixed point has J = J crit (27)and stars tend towards critical rotation. For M (cid:63) Ω AGN R (cid:28) J crit /k , AGN stars will evolve towardssub-critical rotation.4.3. Late-Stage Mass Loss
During late-stage mass loss the rate of mass loss isalways much greater than the accretion rate, so the an-gular momentum evolves according to dJdt = ˙ M loss j loss = J ˙ M loss kM (cid:63) . (28)Following the same reasoning as in Section 4.1 and againassuming constant k we obtain d ln J/J crit d ln M = 1 k −
32 + 12 d ln R (cid:63) d ln M (cid:63) ≈ k − . (29) Jermyn et al.
Because density increases towards the center of the starwe have k < /
5, which means that k < / J/J crit becomes smaller as the star loses mass. Note,however, the factor of J crit /J difference between equa-tion (21) and (29): during the accretion phase the starevolves towards critical faster than it loses angular mo-mentum in the mass loss phase. As a result we expectstars to net gain angular momentum during their evolu-tion.In particular, so long as M (cid:63) Ω AGN R (cid:29) J crit /k weexpect that most AGN stars eventually reach criticalrotation. When they subsequently lose mass they spindown. How far they fall below critical depends on howmuch mass they lose and on the gyration parameter k ,which we cannot estimate in an analytic fashion. RESULTSWe now turn to the results of our numerical simula-tions. 5.1.
Time Evolution
Figure 2 shows the evolution of mass ( M (cid:63) , upper), an-gular velocity relative to critical ( J/J c , middle), and gy-ration parameter ( k , lower) as functions of time for stel-lar models with both the Γ (left) and rotation-reduced(right, Γ − Ω) prescription for the Eddington lumi-nosity and associated mass loss. Evolutionary tracksare coloured by AGN density ρ AGN , which we sweptin the range 10 − − − g cm − . We prescribe theAGN temperature by specifying a disk sound speed c s = 10 km s − , which is the default used by Cantielloet al. (2020). The Keplerian angular velocity Ω AGN determines the distance from the SMBH and so thestrength of tidal effects, and was set to 10 − rad s − .For both the Γ and Γ − Ω L Edd prescriptions we seethat models above an AGN disk density of 10 − g cm − exhibit rapid runaway accretion, in agreement with thefindings of Cantiello et al. (2020) and Dittmann et al.(2021). In the case of the Γ prescription, models withdensity above 3 × − g cm − are then immortal, ex-hibiting a balance between accretion and mass losswhich replenishes fresh hydrogen in their cores. Mod-els below this critical density eventually deplete theircore hydrogen and evolve, rapidly losing mass to be-come high-metallicity compact objects. In the case ofthe Γ − Ω prescription, mass loss generally wins over ac-cretion in the end, and we see most models evolve intocompact objects.In each case the initial accretion causes the stars tospin up to critical rotation. Surprisingly, the stars whichlose mass remain critically rotating. This is not what wepredicted in Section 4.3, and the reason for the discrep- ancy is that there we assumed that the gyration param-eter is constant throughout the star’s evolution, whereaswe see from the lower row of Figure 2 that stars becomemuch more compact as they enter later stages of nuclearburning, resulting in a lower gyration parameter. Thus,even though their specific angular momentum falls, theybecome compact even faster and remain critically rotat-ing.The few models which do not end their lives as criti-cal rotators are those which remained at relatively lowmasses. These models are not chemically or quasi-chemically homogeneous, and proceed to evolve onto thered giant branch. The resulting large radial extensionimplies these objects rotate with very slow, sub-criticalrotation velocities. We expect such evolution to dom-inate at low densities of (cid:46) − g cm − for the Γ pre-scription and (cid:46) − g cm − for the Γ − Ω prescription.5.2.
Tidal Forces
We next study the role of tides, which we chooseto parameterize by Ω
AGN . Figure 3 shows the peakmass (upper) and final mass (lower) for models in agrid running over ρ AGN ∈ [10 − ... − ]g cm − andΩ AGN ∈ [10 − ... − ]rad s − . The range of angularvelocities corresponds to radial coordinates in the diskon the order of 10 ... gravitational radii, and the rangeof densities corresponds to of order 10 ... gravitationalradii. Here the gravitational radius is r g = 2 GM BH c , (30)where c is the speed of light.As before, we study models with both the Γ andΓ − Ω prescription for L Edd , which governs the rate ofmass loss. As a reminder the Γ − Ω prescription has arotationally-reduced L Edd and so undergoes rotationallyenhanced mass loss.With increasing Ω
AGN the Hill radius falls below theBondi radius and accretion onto the star becomes tidallylimited. Thus we see a trend towards decreasing peakand final mass with increasing Ω
AGN . For the same rea-son more models successfully ran at higher Ω
AGN be-cause they accreted more slowly during the initial modelrelaxation.Comparing the Γ and Γ − Ω prescriptions we see lowermasses in the models where rotation reduces L Edd . Thisis because as L Edd decreases the threshold for forming asuper-Eddington wind decreases, so significant mass lossbegins at lower masses than in the regular- L Edd models.Consistent with Dittmann et al. (2021) in the Γ gridwe see several different classes of evolution: • At low densities of 10 − g cm − or less stars do notaccrete beyond 10M (cid:12) and so follow ordinary mas- GN Stars Spin Fast M / M fl Γ Γ − Ω (reduced L Edd ) J / J c Age / yr k ≡ I / M R Age / yr log ρ AGN / g cm − -11.0-11.5-12.0 -12.5-13.0-13.5 -14.0-14.5-15.0 -15.5-16.0-16.5 -17.0-17.5-18.0 Figure 2.
The mass (upper),
J/J c (middle), and gyration parameter k (lower) are shown as functions of stellar age for Γ(left) and Γ − Ω (right) models as functions of the AGN density ρ AGN . The end of each evolutionary track is labelled by eithera star (likely core collapse), an upward arrow (ongoing accretion), or a right-going arrow (immortal phase). Recall that theΓ − Ω models have L Edd reduced to account for rotational effects, and so undergo more rapid mass loss. These models haveKeplerian angular velocity Ω
AGN = 10 − rad s − and AGN disk sound speed c s = 10 km s − . Age is referenced to the end ofthe atmospheric boundary condition blend period, near the start of accretion. Jermyn et al. -14 -12 -10 -8 Ω A G N / r a d s − Γ Γ − Ω (reduced L Edd ) M m a x / M fl -16 -13 ρ AGN / g cm − -14 -12 -10 -8 Ω A G N / r a d s − -16 -13 ρ AGN / g cm − M f i n a l / M fl Figure 3.
The peak mass (upper) and final mass (lower) in M (cid:12) is shown for each model with the Γ (left) and Γ − Ω (right)prescriptions as functions of the AGN density ρ AGN and Keplerian angular velocity Ω
AGN . GN Stars Spin Fast
R > (cid:12) (blue region in Figure 4). • At higher densities and Ω / /ρ AGN (cid:46) cm g − s − / tidal effects are unimportant.Stars rapidly gain mass and reach 10 − M (cid:12) .At that point either the models remain at thathigh mass in the immortal state or the MESA mod-els fail to converge. In the latter case we believethat if they converged they would enter the im-mortal state at an even higher mass (purple regionin left panel). • At higher Ω / /ρ AGN (cid:38) cm g − s − / the Hillradius falls below the Bondi radius for a wide rangeof stellar masses. This limits the accretion rateand thereby limits the peak mass. However, theseobjects do not undergo ordinary stellar evolution,because they are near the Eddington limit andrapidly exchange material with the AGN disk viaaccretion and mass loss (green region). • Models at very high ρ AGN and Ω
AGN accreterapidly but
MESA fails to converge too early intotheir evolution for us to even guess as to what willhappen at late times (red region).In addition to the above, a small fraction of models onthe boundary between regions (2) and (3) peak in massaround 100M (cid:12) , burn through all of their Hydrogen, andundergo rapid mass loss down to 10M (cid:12) , eventually be-coming compact helium/carbon/oxygen stars. A moredetailed characterization of this behaviour is providedby Dittmann et al. (2021).Because many stars become rapid rotators, when wereduce L Edd to account for rotational effects (Γ − Ω) wesee the boundaries between these classes shift consider-ably: • Rather than an unusual edge-case, a majority ofΓ − Ω models with Ω / /ρ AGN (cid:46) cm g − s − / peaks around 30 − , (cid:12) . This accretionspins them to near-critical rotation and enhancesmass loss. They then deplete in hydrogen andrapidly shrink to become compact 10M (cid:12) he-lium/carbon/oxygen stars (purple region on rightin Figure 4). • At ρ (cid:38) − g cm − and Ω AGN (cid:46) − rad s − the accreting angular velocity is not enough tospin stars up to critical. The resulting reductionin L Edd is small, so these objects accrete with-out bound until
MESA fails to converge. We expectthese would become immortal if
MESA were able to follow their evolution further region (dark blueregion on right in Figure 4).A schematic summarizing the mass evolution we seefor both the Γ and Γ − Ω grids is shown in Figure 4.Figure 5 shows the corresponding angular momentumevolution. The upper row shows the critical ratio
J/J c at the time of peak mass and the lower row shows thesame at the end of the run. In nearly all cases the mod-els reach critical rotation by the time their masses peakand remain critical through the end of the run. Themodels which do not reach critical rotation at any pointare almost all at very low Ω AGN and high ρ AGN . Theseaccrete quickly, but the infalling material has low spe-cific angular momentum and so does not spin them upto critical.
MESA then fails to converge when the runsreach very large masses, so we never see if these modelsare immortal or become compact objects. Interestinglymodels at the same Ω
AGN but lower density do becomecritical, because while they accrete to a sub-critical rota-tion rate they subsequently become compact stars whichultimately makes them critical.A number of models reach critical rotation at somepoint but end their evolution sub-critical. In the Γ gridthese follow the line Ω / /ρ AGN ≈ cm g − s − / .Models in this population accrete slowly because tidaleffects truncate R acc to be less than R Bondi . This slowaccretion means that models only make it up to a massof roughly 10M (cid:12) , at which point they undergo rela-tively normal stellar evolution and run up the Red GiantBranch (RGB). This causes them to inflate substantially,raising J c and thereby lowering J/J c .In the Γ − Ω grid the same thing happens but overa wider range of parameter space, extending down toΩ
AGN ≈ − rad s − at ρ AGN (cid:46) − g cm − . Moremodels end up on the RGB in this grid because theyexperienced rotationally-enhanced mass loss.In both grids, models at higher Ω AGN than this pop-ulation accrete slower and so end up with M (cid:63) ≈ (cid:12) after 10 yr. Because the specific angular momentum ofthe infalling material is high and k (cid:46) / k at peak mass (upper) and the end of the run(lower). Comparing with Figure 3 we see that models0 Jermyn et al. Q u a s i n o r m a l M a ss i v e S t a r E v o l u t i o n High Mass,
Immortal StateTidally Limited,
Low-Mass
Evolution Models Fail
Early Q u a s i n o r m a l M a ss i v e S t a r E v o l u t i o n High Mass, Immortal State,
Subcritical Rotation B o r d e r : C o m p ac t O b j ec t s Models Fail
EarlyTidally Limited,
Low-Mass
Evolution Peak at high mass,
Evolve to Compact Object
Γ Γ − Ω (Reduced L
Edd ) Figure 4.
The boundaries between different kinds of evolution discussed in the text are shown for the grids with the Γ (left)and Γ − Ω (right) prescriptions as functions of the AGN density ρ AGN and Keplerian angular velocity Ω
AGN . which lose substantial amounts of mass are much morecompact at the end of their mass loss than at their peakmass (i.e. k falls with time). This is particularly evidentin the runs with Γ − Ω prescription, most of which shed90% of their mass from peak to end.In both grids the models which do not end in a criticalstate are split into two populations. There is a cluster athigh densities and low Ω
AGN for which
MESA fails to con-verge before they reach their maximum masses. How-ever, we believe by analogy with models at slightly lowerdensities which do converge that these models wouldbecome critical if we were able to follow their evolutionthrough into the compact object state. The second pop-ulation are at low densities and high Ω AGN . These starsaccrete slowly enough that they become giants ratherthan reaching extreme masses and then forming com-pact objects. These giant models have enormous mo-ments of inertia and so even though they spend mostof their lives with near-critical rotation they die as slowrotators. ASTROPHYSICAL IMPLICATIONSTo summarize our findings, in regions of the AGNdisk where ρ AGN > ρ crit ≈ − g cm − we expect cap-tured stars to undergo rapid accretion. This densityscale increases towards the SMBH as tidal effects be- come more important, and decreases with sound speedas ρ crit ∝ c − s (Dittmann et al. 2021).Because there is an angular velocity gradient in theAGN disk, the disk has a net vorticity in the frame ofa co-orbiting star. On scales of the Bondi radius thisvorticity means that gas in the disk has a large, typ-ically super-Keplerian, specific angular momentum inthe frame of the star. When that gas accretes, even iflimited to the Keplerian angular velocity, it serves torapidly spin the star up to critical even very far out inthe disk where Ω AGN is as low as 3 × − rad s − .Depending on the exact physical prescriptions used,many of these stars then evolve through later stages ofnuclear burning, undergo rapid mass loss, and becomecompact 10M (cid:12) high-metallicity objects. We expect corecollapse to occur soon after, though we have not triedto follow the collapse process in MESA (Cantiello et al.2020). Because these stars become more compact asthey lose mass they remain critical rotators through theend despite shedding a large fraction of their peak an-gular momentum. They are therefore good candidatesfor producing long GRBs and fast-spinning black holes.The stars which do not lose mass enter an immortalphase (Cantiello et al. 2020) where fresh hydrogen-richmaterial is accreted fast enough to continuously replen-ish the core. This phase ends when these stars either
GN Stars Spin Fast -14 -12 -10 -8 Ω A G N / r a d s − Γ Γ − Ω (reduced L Edd ) J m a x / J c -16 -13 ρ AGN / g cm − -14 -12 -10 -8 Ω A G N / r a d s − -16 -13 ρ AGN / g cm − J f i n a l / J c Figure 5.
The peak ratio
J/J c (upper) and at the end of the run (lower) is shown for each model with the Γ (left) and Γ − Ω(right) prescriptions as functions of the AGN density ρ AGN and Keplerian angular velocity Ω
AGN . enter a low-density pocket of the AGN disk or when thedisk itself dissipates. In either case mass loss then comesto overwhelm accretion and the immortal phase ends,with stars rapidly evolving towards the same compact10M (cid:12) state as before (Cantiello et al. 2020). We expectthis to again end in core collapse of a critically-rotatingstar.We now examine the prospects for producing longGRBs and rapidly rotating black holes. 6.1. Production of GRBs, and their observability inAGN disks
Our results, and in particular the fact that massivestars in AGNs are found to be fast rotators, bear im-portant implications for long GRBs. These are foundto be a fraction of ∼ . −
4% of SNe-Ibc in the localUniverse (e.g. Della Valle 2006), and are known to beassociated with very energetic supernovae from the col-lapse of massive stars (Hjorth et al. 2003; Stanek et al.2003), as suggested by theoretical models (MacFadyen& Woosley 1999; Woosley & Heger 2006; Yoon et al.2
Jermyn et al. -14 -12 -10 -8 Ω A G N / r a d s − Γ Γ − Ω (reduced L Edd ) -2 -1 k p e a k m a ss -16 -13 ρ AGN / g cm − -14 -12 -10 -8 Ω A G N / r a d s − -16 -13 ρ AGN / g cm − -2 -1 k f i n a l Figure 6.
The gyration parameter k = I/M (cid:63) R (cid:63) at the time of peak mass (upper) and the end of the run (lower) is shownfor each model with the Γ (left) and Γ − Ω (right) prescriptions as functions of the AGN density ρ AGN and Keplerian angularvelocity Ω
AGN . γ -ray emission is believed to be producedwithin a relativistic jet (e.g. Piran 1999), and a key ele-ment to launch a jet is believed to be an hyperaccretingdisk around a BH (MacFadyen et al. 2001).For an accretion disk to be formed, a fraction of gasmust remain bound after the supernova explosion, andit must posses a specific angular momentum j m at least Note however some models assume a magnetar engine (Thomp-son et al. 2004; Metzger et al. 2011). as large as the specific angular momentum of the laststable orbit, j lso . Our results (cfr. upper panel ofFig. 7) show that close to core collapse the envelopesof AGN stars are endowed with enough angular mo-mentum to produce an accretion disk around a newlyformed BH ( j m > j lso virtually for any BH with non-zero spin). Hence we draw the important conclusionthat, upon their death, massive stars in AGN disks aretypically expected to produce long GRBs. It is interest-ing to compare these stars to the Wolf-Rayet progenitorsof standard long GRBs from field stars (MacFadyen &
GN Stars Spin Fast O B u r n C B u r n j L S O ( a = ) j L S O ( a = ) H e B u r n j / c m s O B u r n C B u r n H e B u r n m / M t ff / s Figure 7. (Upper) The specific angular momentum profilein an AGN star model is shown as a function of enclosedmass coordinate m . Also shown are the specific angular mo-menta of least-stable orbits of a black hole of mass m andspins a = 0 and a = 1. (Lower) The free-fall time is shown asa function of enclosed mass. Three models are shown from asingle evolutionary track computed with the Γ − Ω prescrip-tion, ρ AGN = 10 − g cm − , and Ω AGN = 10 − rad s − . Themodels were chosen at the onset of helium burning, the onsetof carbon burning, and late stages of oxygen burning. Woosley 1999; Yoon et al. 2006; Woosley & Heger 2006;Cantiello et al. 2007). The similarity is not surprising,since here we assume that AGN stars accreting largeamounts of mass are well mixed (Cantiello et al. 2020).Hence they evolve quasi-chemically homogeneously, sim-ilarly to rapidly rotating long GRB progenitors.The amount of mass which is available for accretion isgiven by the fraction which remains bound after the su-pernova explosion (examples are given in Fig.6 of Pernaet al. 2014). For weak explosions, most of the materialfalls back.The material that remains bound falls back ona timescale which is on the order of the free-falltime (Woosley & Heger 2012) t ff ( r ) = 1 √ G ¯ ρ , (31) where ¯ ρ is the mean density of the star. The precisedistribution of the initial fallback radii for all the boundparticles will clearly depend on the details of the su-pernova explosion, but a minimum value is given by thelocation of the particles prior to the explosion. The free-fall time of the envelope (bottom panel of Fig. 7) is onthe order of a few tens of seconds, as in the bulk of thetypical range of durations of long GRBs.After the bound material falls back, it circularizes at aradius R circ determined by the condition j ( R circ ) = j m .Subsequently, the evolution of the disk is determined bythe viscous timescale t ( R circ ) = R H α Ω K ∼ . α − − m − / R / (cid:18) RH (cid:19) s , (32)where m = M/ (3 M (cid:12) ), R = R/ (10 cm), Ω K isthe Keplerian velocity of the gas in the disk, H thedisk scale-height, and α the viscosity parameter, writ-ten in units of α − ≡ α/ . t ff and t (R circ ). In the inner parts of the disk, up tohundreds of gravitational radii, the scale is set by thefree-fall one, yielding accretion rates on the order of˙ m acc ∼ . − . (cid:12) s − over several tens of seconds,as typical of long GRBs.With massive stars in AGN disks possessing the keyelements to power a long GRB upon their death, thenext question to address is the likelihood to observe suchsources as they emerge from the dense environments ofAGN disks. The question of the observability of rel-ativistic, electromagnetic transients produced in AGNdisks was recently addressed by Perna et al. (2021), con-sidering two specific models for the disk structure, theone by Sirko & Goodman (2003) (SG in the following)and the one by Thompson et al. (2005) (TQM in thefollowing). The location of the sources was assumed tobe in the disk’s mid-plane, which is the most pessimisticcase in terms of observability. This turns out to be alsothe most likely occurrence, since it is expected that mostof the stars interacting with the disk should end up inits mid-plane (e.g. Tanaka & Ward 2004).The analysis by Perna et al. (2021) showed that theoutcome is quite dependent on the disk model and on theSMBH mass, which relates to disk properties, such asthe density and the radial extent. Long GRBs occurringin disks around SMBHs of mass ∼ M (cid:12) , are expectedto appear as typical transients (that is similar to the onesoccurring in standard galactic environments) for mostlocations of the disk, except for some regions, ∼ -a few × R g ( R g being the gravitational radius) inthe SG disk, and between a few × − R g in the4 Jermyn et al.
TQM disk model, when the prompt emission and earlyafterglow emerge on a timescale set by the diffusion time t diff ≈ H ρ σ T m p c , (33)where H is the scale height of the disk and ρ the densityin the disk mid-plane, σ T the Thomson cross section,and m p the proton mass.For AGNs with SMBHs of larger masses, the regionsin which both the prompt GRB emission and the af-terglow appear normal are gradually reduced: the in-creasing opacity of the disk causes the transients to bediluted on the diffusion timescale. The magnitude ofthis timescale varies from minutes to several years, be-ing generally smaller in the inner disk regions and forless massive AGN disks (see Fig.5 in Perna et al. 2021,for quantitative details).6.2. Black Hole Spins
Our models predict that stars embedded in AGNdisks of densities ρ AGN > − g cm − typically growto large masses ( M max > (cid:12) ). The exceptions arethose very near the SMBH with Ω AGN > − rad s − ,where tidal effects slow accretion. Even these lessmassive stars, however, typically evolve into mas-sive helium/carbon/oxygen stars of roughly 10M (cid:12) (seelower row of Figure 3) which, owing to their com-pactness, likely undergo core collapse and form blackholes (Cantiello et al. 2020).Our models also suggest that AGN stars end theirlives with near-critical rotation. This then seems to bethe typical fate of AGN stars, to form black holes withmass M BH ≈ (cid:12) and spin a ≈ M (cid:38) (cid:12) and a ≈
1. This channel ofblack hole formation is of particular interest given therecent observation of GW190521, a merger of two massgap black holes ( M ≈ (cid:12) , M ≈ (cid:12) ) consistentwith large spins ( a ≈ . +0 . − . , a ≈ . +0 . − . ) (Ab-bott et al. 2020), though see also Nitz & Capano (2021).While we do not have estimates of the formation ratesof such objects, these black holes could be the descen-dants of low-mass stars embedded in an AGN disk. Af-ter accreting and becoming massive rapid-rotators, theylost part of their mass and formed ≈ (cid:12) black holes. These black holes further accreted up from the disk ,or they migrated and merged with other compact rem-nants, reaching their pre-GW190521 masses (e.g. McK-ernan et al. 2012; Secunda et al. 2019; Yang et al. 2019;Tagawa et al. 2020).A challenge to this picture is that the observed mergerwas most consistent with spins misaligned with the or-bital plane (Abbott et al. 2020), while a strong predic-tion of our models is that the black hole spins shouldbe aligned with the rotation of the AGN disk. While itis certainly possible for the orbital plane of the merg-ing binary to be misaligned with the disk, this does notseem to be the most likely scenario, which suggests thatother physics we have not considered could be at workin GW190521. SUMMARYStars in AGN disks are thought to evolve in a widevariety of unusual ways. Depending on the AGN diskdensity and sound speed and the strength of tidal forceswe expect AGN stars to • Cease to age, • Accrete up to masses over 10 M (cid:12) , • Undergo quasi-chemically homogeneous evolution, • Shed the vast majority of their mass, • Spin up to critical rotation, and/or • Form compact helium/carbon/oxygen stars.All of these behaviours are exhibited by stellar modelsthat began with the same zero-age main-sequence 1M (cid:12) initial condition, and the variation we see is entirely afunction of the conditions in the AGN disk.Here we studied the spin evolution of AGN stars, aswell as its impact on the accretion and evolution of theseobjects. At densities ρ AGN (cid:38) − g cm − AGN starsaccrete rapidly. When that gas accretes it serves torapidly spin AGN stars up to critical, even very far outin the disk where Ω
AGN is as low as 3 × − s − . Wefind that rotational enhancement of mass loss is then im-portant, causing most of these stars to then rapidly losemass and evolve into compact, critically-rotating 10M (cid:12) objects made of helium and heavier elements. At Eddington-limited accretion rates this process would take oforder 100 − GN Stars Spin Fast
Software:
MESA (Paxton et al. 2011, 2013,2015, 2018, 2019, http://mesa.sourceforge.net),
MESASDK matplotlib (Hunter 2007),
NumPy (van der Walt et al. 2011),
RemoteExperiments https://github.com/adamjermyn/remote experiments6
Jermyn et al.
APPENDIX A. SOFTWARE DETAILSWe performed calculations using revision 15140 of the Modules for Experiments in Stellar Astrophysics (MESAPaxton et al. 2011, 2013, 2015, 2018, 2019) software instrument.The MESA EOS is a blend of the OPAL (Rogers & Nayfonov 2002), SCVH (Saumon et al. 1995), FreeEOS (Irwin2004), 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); Oda et al. (1994); Langanke & Mart´ınez-Pinedo (2000). (For MESA versions before 11701):Screening is included via the prescriptions of Salpeter (1954); Dewitt et al. (1973); Alastuey & Jancovici (1978); Itohet al. (1979). (For MESA versions 11701 or later): Screening is included via the prescription of Chugunov et al. (2007).Thermal neutrino loss rates are from Itoh et al. (1996).We performed extensive convergence testing on both the Γ and Γ − Ω model grids using more than 1,600 modelsto study the dependence of our results on time-step and mesh resolution. We determined that our results are in-dependent of resolution for spatial resolution parameter mesh delta coeff up to 1 and time resolution parameter time delta coeff up to 0.2. Note that this time resolution requires the use of our custom time-step controls.The final configuration files and code used in our model grids are available in Jermyn et al. (2021). These are givenin
Python git commits which can be found in thegit repository stored in Jermyn et al. (2021). Each such commit corresponds to a single
MESA run directory used toperform one of our runs, including the full configuration files and ‘run star extras’ code used. We further provide thePickle files specifying the configurations and short-sha’s of commits we used in the final set of convergence tests whichdemonstrate that our results are converged.The same git repository contains a history of nearly all
MESA runs used to develop this work contributed to thiswork. These are commits whose messages contain the word ‘patch’ and which do not lie on any branch. These git experiments were performed using the
RemoteExperiments software package, details of which may be found athttps://github.com/adamjermyn/remote experiments. B. STOCHASTIC ANGULAR MOMENTUM EVOLUTIONThe gas in AGN disks is believed to be turbulent, with a characteristic length-scale of the disk scale height H andcharacteristic velocity-scale of the sound speed c s . This turbulence imparts a random additional component to theangular momentum which accretes onto a star embedded in the disk.To model this system we treat the total angular momentum of the AGN star as a normally-distrubuted randomvariable with mean (cid:104) J (cid:105) and variance σ J . The mean evolves according to the differential equation d (cid:104) J (cid:105) dt = ˙ M gain j gain , avg − ˙ M loss j loss , avg , (B1)which just says that the mean angular momentum increases according to the mean accreted angular momentum j gain , avg and decreases according to the mean lost angular momentum j loss , avg . The variance evolves according to asimilar equation: dσ J dt = ˙ M j , std τ turb − k ˙ M loss M (cid:63) σ J . (B2)We obtain the first term by assuming that the accreted material follows a random walk in j with characteristic time-scale τ turb and step size j gain , std set by the structure of turbulence in the disk. We obtain the second term by assumingthat the lost material has the specific angular momentum of the surface j loss = J/kM (cid:63) , where k ≡ IM (cid:63) R (cid:63) (B3) GN Stars Spin Fast I is the moment of inertia of the star.Equation (B1) is the same as equation (2), just with a different notation to emphasize the fact that J is a randomvariable. We evaluated all terms in this equation in Section 3, so all that remains is to evaluate the terms τ turb and j gain , std which appear in equation (B2).The angular momentum within the accretion radius varies stochastically due to turbulence in the disk, giving rise tothe term j gain , std . This is just the standard deviation of the specific angular momentum within the accretion stream,which we approximate by j gain , std ≈ v τ turb , (B4)where v turb is the turbulent velocity at the scale of the accretion radius and τ turb is the characteristic time-scale of theturbulence at that length-scale. We can estimate τ turb ≈ R acc /v turb and v turb ≈ c s ( R acc /H ) n for R acc < H , where H ≈ c s Ω AGN , (B5) H is the scale height of the disk, n is an index which depends on the nature of the turbulence, and Ω AGN ≡ (cid:112) GM BH /a is the angular velocity of the orbit of the star. In the inertial range n = 1 / n = 2 / v turb only equals c s on the outermost scale of H , we use the Kolmogorov scaling and write v turb ≈ c s min (cid:34) , (cid:18) R acc H (cid:19) / (cid:35) (B6)and τ turb ≈ min( H, R acc ) v turb . (B7)We can then evaluate equation j gain , std using equation (B4).Finally recall that we cannot allow the star to rotate super-critically. To incorporate this constraint, after eachtime-step we check if (cid:104) J (cid:105) is super-critical. If it is, we truncate it to the nearer of ± J crit and set σ J = 0. Moresophisticated mappings are possible (e.g. Trenkler 1996), but in testing we found that when our models attain criticalrotation the variance σ J rapidly diminishes, making this approach a good approximation.REFERENCES Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2020,ApJL, 892, L3, doi: 10.3847/2041-8213/ab75f5Abbott, R., Abbott, T. D., Abraham, S., et al. 2020, Phys.Rev. Lett., 125, 101102,doi: 10.1103/PhysRevLett.125.101102Alastuey, A., & Jancovici, B. 1978, ApJ, 226, 1034,doi: 10.1086/156681Artymowicz, P., Lin, D. N. C., & Wampler, E. J. 1993,ApJ, 409, 592, doi: 10.1086/172690Bondi, H. 1952, MNRAS, 112, 195,doi: 10.1093/mnras/112.2.195Buchler, J. R., & Yueh, W. R. 1976, ApJ, 210, 440,doi: 10.1086/154847 Burgers, J. 1948, in Advances in Applied Mechanics, Vol. 1,A Mathematical Model Illustrating the Theory ofTurbulence, ed. R. Von Mises & T. Von K´arm´an(Elsevier), 171 – 199,doi: https://doi.org/10.1016/S0065-2156(08)70100-5Cantiello, M., Jermyn, A. S., & Lin, D. N. C. 2020, arXive-prints, arXiv:2009.03936.https://arxiv.org/abs/2009.03936Cantiello, M., Yoon, S. C., Langer, N., & Livio, M. 2007,A&A, 465, L29, doi: 10.1051/0004-6361:20077115Cassisi, S., Potekhin, A. Y., Pietrinferni, A., Catelan, M., &Salaris, M. 2007, ApJ, 661, 1094, doi: 10.1086/516819Chugunov, A. I., Dewitt, H. E., & Yakovlev, D. G. 2007,PhRvD, 76, 025028, doi: 10.1103/PhysRevD.76.025028Cyburt, R. H., Amthor, A. M., Ferguson, R., et al. 2010,ApJS, 189, 240, doi: 10.1088/0067-0049/189/1/240 Jermyn et al.
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GN Stars Spin Fast19