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T 1. 1 E / X 3 lc oepyis—hdoyais—sas rotation stars: — hydrodynamics — physics hole black 1 R INTRODUCTION twocolumn eateto hsc n srnm,Uiest fLeicest of University Astronomy, and Physics of Department 2 ∗ oubaAtohsc aoaoy oubaUiest,N University, Columbia Laboratory, Astrophysics Columbia hc stedsac twhich at distance the is which , es1988 Rees lnC .Golightly, A. C. Elen tl nAASTeX62 in style .Teapaac of appearance The ). ia irpineet:terl fselrspin stellar of role the events: disruption Tidal t − 5 3 ( es1988 Rees 1 rcR Coughlin R. Eric ABSTRACT ; ftebuddebris: bound the of hne 1989 Phinney ue ocm rmlrerdiadne orahperi- reach to as- need and are radii stars at large centre disrupted from come tidally to most sumed as orbit parabolic distribution energy specific the with and rameter, R ri,adtesa sol atal irpe if disrupted partially only is star the and orbit, hsipc aaee sflos h rtclpitof point critical on the depends at follows: event is as disruption disruption parameter the impact of this extent The tance. sdsrbtd50 material distributed bound/unbound is of amount the parabolic being unity: to close ob nfr.Ti hw h aeo alaki pro- is fallback of rate the shows This to portional uniform. be to n odsuto cusfor occurs disruption no and g Dsaeuulymdle ihtesa na on star the with modelled usually are TDEs h trrahstetdlrdu iha matpa- impact an with radius tidal the reaches star The M = ˙ fb GM , = 2, ∗ β ihrsett t ri h ia force tidal the orbit its to respect with ∼ dM BH r ecse,L17H UK 7RH, LE1 Leicester, er, dt wYr,N,107 USA 10027, NY, York, ew h ai ftdlrdu oprcnr dis- pericentre to radius tidal of ratio the , ot h canonical the hoots oee hntesa pn faster, spins star the when However . and tla pn eas n htin that find also We spin. stellar e 50 aeti ok ihtetdlforce tidal the with works this rade t /c ,drvdfo h eaieKpe energy Kepler negative the from derived ), − t asn eae n sometimes and delayed causing it, t E ∗ = − hc oesalmnu flare. luminous a powers which w R 2 3 5 bound e a asscoeeog nwhat in enough close passes hat .J Nixon J. C. 5 β iigteecnrct fteobtas orbit the of eccentricity the giving , . h ffc fselrrtto on rotation stellar of effect the : g a ean on oteblack the to bound remains tar − / dM = c where , dE mn h oe-a ne of index power-law the amine 3 re D lightcurves. TDE erved 0 ,weealarger a where 1, = 50. ae eteeoeconclude therefore We rate. . r r 01Ω a a = dE dt − + r r − p p breakup = 1 2 R ≈  g 1 dM dE .A euto h orbit the of result a As 1. 2 stegaiainlradius gravitational the is πGM β t rvd nya only provide ) − 1 3 5 ≪ t (2 / 3 BH πGM aebriefly rate .However, 1.  β 2 3 BH ie deeper a gives dM dE ) 3 2 t − assumed 5 3 < β β c (1) (2) 1, is 2 Golightly et al. dependent on stellar structure and can vary by a fac- and found a range of periods from 0.2 to 70 days. As- tor of 2 (Guillochon & Ramirez-Ruiz 2013). β would suming Sun-like mass and radius, this faster period cor- ∼ c also vary with the spin of the star depending on spin responds to a spin 0.057Ωbr. However, given the bias of magnitude and direction. TDE detections in post-starburst galaxies and, as young A star spins at a fraction of its break up velocity which stars rotate faster, we might expect many disruptions in addition to the black hole’s tidal force would quicken involving stars with higher spins. For example, a so- its disruption, provided they were in the same direction. lar mass Pre-Main Sequence star of age 1Myr would If the star is spinning against the direction of the tidal reach spins of 0.07 of break up (Bouvier 2013∼ ). A B-type force, it would hinder the disruption and can leave be- MS star, HD43317, observed by CoRoT rotates at 50% hind an intact portion of the star. of its critical velocity which corresponds to 0.28Ωbr Lodato et al. (2009) showed that the lightcurve is only (Rieutord & Lara 2013). ∼ − 5 proportional to t 3 at late times and depends on stellar TDEs are produced and fill the loss cone via one of two structure during the early stage of the fallback. They regimes: the diffusive or pinhole. In the pinhole regime a show that more compressible stars (smaller values of γ, far away star experiences one large kick to send it on an the polytropic index) give a more gentle rise to the peak. orbital path passing within the tidal radius of a SMBH. − 5 They conclude that the t 3 decay only holds where the These stars pass within the loss cone multiple time in energy distribution dM/dE is constant which is approx- an orbit (Stone & Metzger 2016). For stars much closer, imately true only at late times. Given that most of the the loss cone is filled slowly by stars diffusing over many accreted material returns to the black hole before reach- stellar orbits. This diffusive regime can cause stars to be − 5 ing t 3 , it is not clear how many observed events would slowly pushed to smaller orbits and then the tidal field − 5 be characterised by a t 3 lightcurve. At late times the can spin up the star to larger prograde velocities before lightcurve is expected to change shape due to the vis- disruption occurs. Stars in the pinhole regime tend to cous timescale in the accretion disc (power law index have large impact parameters whereas diffusive regime stars have β 1 (Stone & Metzger 2016). 1.2; Cannizzo et al. 1990). ≈ ∼−Guillochon & Ramirez-Ruiz (2013) looked at the ef- With the first light of LSST due in 2021, which it fect of changing the impact parameter for cases span- will do all-sky surveys every 3 days (Collaboration et al. ning grazing encounters (low β) to deep plunges (high 2017), the catalogue of observed TDEs is expected to β) using γ = 4/3, 5/3 for high and low mass main se- rapidly increase. Therefore more accurate lightcurves quence stars. They showed that the most concentrated would be useful to classify new observations. In this stars drop in fallback rate quickly after the peak where work we look at the whether stellar spin has a signifi- only partial disruption of the star occurred. Their re- cant role in numerical TDE models. We look at both − 5 sults show much steeper decays than the expected t 3 prograde and retrograde spins and compare their fall- where some of the stellar core remains intact, which sug- back rate curves to the non-spinning case, as well as gests we can determine from observations if a flare has analysing how the decay of the curves compare with pre- produced a survived core. dictions. Previous models of tidal disruption events have A star will break up if it is spun up to the point neglected the use of spinning stars so it is important to test whether it is actually fair to exclude this. Fallback at which Fsg Frot where self-gravity of the star is 2 2 ∼ 2 GM /R and the centrifugal force is M∗Ω R∗, where curves are so far only constrained to a few parameters ∗ ∗ ± ∗ Ω∗ is the stellar rotation frequency. This gives a break that determine the stream evolution, such as impact pa- up angular velocity: rameter, mass ratio and polytropic index. With the in- clusion of other properties, such as stellar spin, we may GM∗ find that there is a larger error involved when constrain- Ωbr = 3 (3) ing these parameters. R∗ s Stone et al. (2013) show that stellar spin widens the

We define the break-up fraction as λ = Ω∗/Ωbr where energy distribution of the star and note that the mis- Ω∗ > 0 is a prograde spin and Ω∗ < 0 is retrograde. alignment between large stellar spins and the orbital an- Current observations of spinning stars include the Sun gular momentum could have a larger effect on the energy which spins at 0.002 of its break up velocity (with an spread. We can estimate the tidal radius for a spinning equatorial period of 24 days) and Meibom et al. (2015) star (cf. Kesden 2012) by balancing the self-gravity with who found in a sample of 30 cool stars, from the NGC the tidal force and force due to spin: 6819 cluster observed by Kepler, they spun on aver- age with 0.035Ω (assuming a stellar radius, from a br F = F F (4) rotation period of 18.2 days) and Nascimento Jr et al. sg tidal ± rot (2014) found a mean rotation period of 19 days over 43 main-sequence stars but with some as slow as 27 days. McQuillan et al. (2014) determined the periods 2 GM∗ GMBHM∗R∗ 2 for 34, 030 Kepler MS stars of temperature < 6500K 2 = 3 M∗Ω∗R∗ (5) R∗ rt ± Disruption of Spinning Stars 3

The allows for prograde and retrograde spins where of the black hole, being the second term in parenthe- prograde± (+) helps the tidal force and retrograde ( ) ses. Equating these two yields the rotation rate of the hinders. − star that would match the spread in energy generated by the tidal force alone. Defining this rotation rate as 1 − 1 3 2 3 3 3 MBH Ω∗R∗ Ωeq = λeqΩbr, where Ωbr = GM∗/R∗ is the breakup rt(Ω) = R∗ 1 (6) M∗ ∓ GM∗ rotation rate of the star (M∗ and R∗ are the stellar mass     and radius, respectively), thenp performing some algebra We note that this estimate behaves appropriately in demonstrates that λeq 1. Therefore, if the star is ro- the correct limits. When Ω∗ = 0 it recovers the standard tating at only a small fraction≃ of its breakup speed, then tidal radius and when Ω∗ = +Ωbr the radius is infinite, the spread in energy imparted by rotation will always be and when Ω the radius is reduced appropriately. − br subdominant to that imparted by the tidal force. The In Section 2 we present analytical calculations. In Sec- nonlinear term in Equation (9) is, therefore, negligible tion 3 we present our numerical simulations, and present for all physical stars. our conclusions in Section 4. Second, if the angular velocity is exactly parallel to the 2. ANALYTICAL PREDICTIONS center of mass velocity, then there is no first order term in the correction to the energy spread and the effects of 2.1. Impulse approximation rotation on the evolution of the tidally-disrupted debris Assume that the tidal force acts impulsively (cf. Lodato et al.are much smaller. In this case, the centrifugal barrier 2009), meaning that the star is unperturbed and main- generated by the rotation would reduce the gravitational tains perfect hydrostatic balance prior to reaching the self-confinement of the tidally-disrupted debris stream. tidal radius, and is completely destroyed (self-gravity The ability of the stream to fragment under its own and pressure negligible) after the star passes through self-gravity, as seen in Coughlin & Nixon (2015) and the tidal radius. Under this “impulse approximation,” Coughlin et al. (2016), would then be inhibited, but the the energy of a given gas parcel within the star at the spread in the energy along the stream would be largely moment the stellar center of mass (COM) reaches the unaltered by the rotation. tidal radius is Third, if the star is spinning at an oblique angle rel- ative to the COM velocity, then Equation (9) becomes 1 GM ǫ = v2 , (7) (assuming that the rotation rate is sub-breakup and ig- 2 − r noring the nonlinear term): | | where v is the instantaneous velocity vector of the gas parcel and r is its vector displacement from the black GM hole. Since the star is assumed to be in hydrostatic ǫ = v∗Ω + sin θ cos φ v∗Ω cos θ R. (10) z r2 − x equilibrium, we can write  ∗   In this expression, we let the line connecting the black v = v∗ + Ω∗ R, (8) × hole and the stellar COM define the x-direction, the where v∗ is the stellar COM velocity, Ω∗ is the angu- stellar COM velocity is in the y-direction, and the z- lar velocity of the star, and R is the position of the gas direction is defined from these in a right-handed sense; parcel within the star measured from the stellar COM. the rotation rate of the star is then Ω∗ = Ω , Ω , Ω { x y z} Further writing r = r∗ + R, where r∗ is the position of in these coordinates. We also defined the position of the the COM, employing the tidal approximation (i.e., keep- gas parcel within the star in spherical coordinates, so ing only first-order terms in the small quantity R/r∗), R = R sin θ cos φ, R sin θ sin φ, R cos θ . and using the fact that the star is on a parabolic orbit, Interestingly,{ Equation (10) demonstrates} that rota- the energy becomes tion in the x-direction serves to tilt the location of the most-bound debris element out of the orbital plane of the star: differentiating Equation (10) with respect to GM 1 2 ǫ = v∗ Ω∗ + r∗ R + (Ω∗ R) (9) θ, where φ = π for the most bound material, the energy × r3 · 2 ×  ∗  is minimized at the angle This expression yields the (conserved) energy of a gas parcel as a function of its position within the star, R, v∗Ωx cot θm = GM . (11) v∗Ω + 2 following the disruption, and from it we can deduce a z r∗ number of important effects of rotation on the dynamics 2 Since v∗Ω GM/r when the star is rotating at sub- of the TDE. z ≪ ∗ For one, the lowest-order (in Ω∗) effect of rotation is breakup velocities, we can Taylor expand this equation given by the first term in parentheses in this equation, to give R which scales identically with as the standard (non- π spinning) energy spread induced from the tidal force θ √2λ , (12) m ≃ 2 − x 4 Golightly et al.

3 where λx is defined by Ωx = λx GM∗/R∗. It is also modify these results, as shown in the previous subsec- straightforward to show that the most-unbound debris tion. p element is located at an angle of π/2+√2λx. Therefore, From Equation (13), the energy of a given gas parcel rotation in the x-direction shifts the line defining the within the star at the time of disruption is purely a func- maximum energy spread from the x-axis to one that is tion of its linear position in the star, x. Furthermore, tilted from the x-axis by the small angle √2λx. the energy-period relation of a Keplerian orbit ensures Finally, if the angular velocity is purely in the z- that all gas parcels with the same energy return to the direction, then v∗ Ω r∗, and the expression for the black hole at the same time t. Therefore, to derive the × ∝ energy simplifies to total fallback mass Mfb that has yet to return to the black hole, one can parameterize the mass in the star at the time of disruption in terms of x and then use Equa- GM GMx ǫ = v∗Ω+ x = 1+ √2λ , (13) tion (13) to write x(ǫ)= x(ǫ(t)). It follows geometrically r2 r2  ∗  ∗ that this can be written (see also Lodato et al. 2009 and   Coughlin & Begelman 2014) where x = R∗ sin θ cos φ and the last line follows from 1/3 letting v∗ = 2GM/r∗, r∗ = R∗(M/M∗) , and R∗ Ω = λ GM∗/R3. This expression demonstrates that, dMfb(x)=2π ρ(R)RdRdx, (16) p∗ as for the non-rotating case, surfaces of constant energy Zx withinp the star coincide with surfaces of constant linear where ρ is the density of the star at the time of disrup- displacement in the direction connecting the black hole tion, assumed to be the original stellar density profile; and the stellar COM. We also see that, if Ω is positive, by writing ρ(R) we are ignoring any latitudinal varia- corresponding to alignment between the orbital angular tions induced by the stellar rotation. This is a good momentum vector of the stellar COM and the angu- approximation when the rotation rate is not too close to lar velocity of the star, the total spread in the energy the breakup velocity. Using the energy-period relation increases, while antialignment reduces the effective en- for a Keplerian orbit in Equation (16) then gives for the ergy spread. Thus, stars rotating in a prograde sense fallback rate – Ω aligned with the angular momentum of the star – are more easily disrupted than non-spinning stars, while −5/3 R∗ retrograde-rotating stars are less easily disrupted. dMfb 4π R∗ t M˙ fb = ρ(R)RdR, Equation (13) also shows that the most-bound gas dt ≡ 3 Tmb Tmb   Zx(t) parcel has an energy (17) −2/3 where x(t) = R∗ (t/Tmb) . Finally, if we define the GMR∗ ǫmb = 1+ √2λ . (14) dimensionless position within the star as η = R/R∗ and 2 3 − r∗ the average stellar density by ρ∗ = 3M∗/(4πR ), then   ∗ Using the energy-period relation for a Keplerian orbit this expression simplifies to then gives the return time of the most tightly bound −5/3 1 debris: M∗ t ρηdη M˙ fb = . (18) T T ρ∗ mb  mb  Zx(t)/R∗ 3/2 − R∗ 2πM 3/2 This expression clearly illustrates the rough magnitude Tmb = 1+ √2λ . (15) and asymptotic, t−5/3 scaling of the fallback rate. 2 M∗√GM     The integral in Equation (18) cannot, in general, be This shows that prograde-spinning stars (positive λ) re- done analytically so we numerically integrate it and plot sult in shorter return times of the most-bound debris, the solutions for several λ in Fig. 1. while retrograde spin yields a longer return time. 3. SIMULATIONS 2.2. Fallback model We present simulations using the 3D smoothed par- Here we construct a simple, analytic model for the ticle hydrodynamics (SPH) code phantom (Price et al. return of the debris to the black hole following the dis- 2017). We use a solar-type star modelled as a poly- 6 ruption of a spinning star analogous to that presented trope (γ = 5/3) and a 10 M⊙ black hole sink particle in Lodato et al. (2009). For the sake of simplicity and modelled with Newtionian gravity 1. The first step of because our numerical simulations are restricted to this case (see Section 3.1), we confine our calculations to the 1 scenario in which the stellar angular velocity is parallel Relativistic effects are small during the disruption phase for these parameters. It would be necessary to include relativistic to the orbital angular momentum of the star, so Equa- effects to model the circularisation process, and for very deep tion (13) describes the spread in the energy following plunges (Guillochon & Ramirez-Ruiz 2013; Gafton et al. 2015) ∼ R the disruption. Tilts to the rotation should only slightly with pericentre distances 1 g. Disruption of Spinning Stars 5

1

0.002 prograde 0.002 retrograde 0.01 prograde 0.01 retrograde 10-1 0.05 prograde 0.05 retrograde 0.2 prograde 0.2 retrograde no spin

/yr]

sun 10-2

dM/dt [M

10-3

10-4 10-1 1 10 t [yr]

Figure 1. The predicted fallback curves for different stellar spin fractions, λ. Compared to a non-spinning star, prograde spins (black) lead to earlier fallback with higher peaks and retrograde spins (red) fall back later with smaller peaks. The deviation from the non-spinning case (dash-dot) increases with spin magnitude. the simulation is to relax the polytrope to put the star outer parts of the star rotating at the same rate but in equilibrium. Once relaxed, the star is placed on a with different velocities. We look at break-up velocity parabolic orbit starting outside the tidal field of the hole fractions: λ = ( 0.002, 0.01, 0.05, 0.2), where posi- at 4.9r and with an impact parameter β = 1, where tive corresponds± to prograde± spin± and± negative is retro- ∼ t β = rt(Ω = 0)/rp. We do not use the tidal radius equa- grade. We chose these values to show that even a small tion affected by spin (Equation 6) as this allows us to spin, like the Sun (λ =0.002), can have an effect on the see the effect of the stellar spin in a controlled manner, fallback curve. Higher spins for solar type stars do not rather than changing the orbit and then trying to differ- fit with observations and some simulations of higher λ entiate between the two effects. The star is initially in do not get fully disrupted. Within this paper we refer hydrostatic equilibrium where pressure and self-gravity to each spin by its fraction λ and direction, and define are balanced. Disruption starts as the star passes within the spin of the star as: the tidal radius where the tidal force of the hole over- comes the self-gravity of the star and the pressure redis- GM∗ tributes the energy in the star and widens the specific Ω∗ = λΩbr = λ 3 (19) energy distribution (Lodato et al. 2009). The disrupted s R∗ star further stretches into a stream as it continues its orbit. We can follow the return of the bound mate- We relax the star before starting the simulations to rial to the hole to measure the fallback rate. The black remove initial perturbations (we relax the star for 10 hole is initially given a small accretion radius (20Rg) to rotation periods in the relevant corotating frame, with a prevent swallowing the star whole. Once the disrupted velocity damping. In practise the star relaxes after 1 ∼ star is sufficiently past the hole, the accretion radius rotation period). is increased to 3rt. We do not attempt to resolve the Spin effects the stellar structure by expanding the star circularisation and disc-forming process so any particle isotropically in the x and y directions but not in z as that crosses the accretion radius is removed from the there is more centrifugal support in the z = 0 plane. This causes the star to be non-spherical in the z x plane simulation and is considered accreted. We measure the − fallback rate from the rate of particles being removed. but for low spins there is little distortion so the star is We set the polytrope spinning with a corotation angu- approximately spherical with an aspect ratio of 1.03 for lar velocity, equation (19), at a fraction of its break-up a spin of λ =0.2. The distortion is more significant for velocity. The corotation corresponds to the inner and higher spins, e.g. for λ =0.4 (not simulated) the aspect ratio is 1.17. 6 Golightly et al.

0 0.2 retrograde merical simulations with our analytical predictions from Section 2. Both the analytical and numerical solutions show that the fallback occurs earlier for prograde spins and later for retrograde spins. They also both show -2×10-7 the same trend with peak fallback rate – the prograde case has a higher peak than the retrograde case. How- ever, the analytical and numerical curves for the same

[km/s]

phi spin value do not agree closely due to the simplifying as-

v -4×10-7 sumptions in the analytical model discussed at the start of Section 2. We show in Fig. 5 the velocity structure, without their orbital velocity i.e. only showing the effect of spin and -6×10-7 the tidal force, of the star at its initial position before orbiting and at pericentre. We see that prograde is spin- 0 0.5 1 ning in the direction of the orbit which causes it to r [R ] sun be disrupted quicker. For faster retrograde spins (e.g. Figure 2. A check that the spin of the polytrope (black par- 0.2) the stars remained partially intact, as a result of − ticles) maintains the correct velocity (red dashed) by taking the tidal forces having to spin down the star before dis- the polytrope out of the corotating frame and putting it in ruption can occur. We can see this highlighted in the isolation in its inertial frame. zoomed section of Fig. 6 and in the right panel of Fig. 7, where the spike in density is the bound core. This is We relax the spinning polytrope in the corotating expected as the retrograde case moves the tidal radius frame where it appears static. We can then check the inwards (Equation 6; recall we have used β(Ω=0)=1 to define the simulation orbits). While a retrograde spin polytrope is spinning at the correct Ω∗ by taking it out of the corotating frame after it has relaxed, and plac- hinders stellar disruption, a prograde spin will quicken ing it in isolation in its inertial frame. Fig. 2 shows the disruption and debris fallback as the stellar material is azimuthal velocities of the particles (black dots) as a slightly more bound so returns to the hole quicker. function of radius together with the exact solution (red We can see from Fig. 8 the stellar rotation does not dashed line), where it shows a good fit to the correct have a significant impact on the stream geometry, ex- spin. cept where a bound core survives for higher (retrograde) We can measure the fallback rates for each spin and spins as in Fig. 6. In Fig. 9 we show how spin affects compare these to the non-spinning case. We also look the energy distribution. As discussed in Section 2 and at the decay of the curves and whether they follow shown by Eq. 9, for small spins fractions the change in the predicted t−5/3 behaviour. We model the evolu- the energy distribution due to stellar rotation is small tion for 1.6yrs as usually most of the bound mate- compared to effect from the tidal force. rial will fall∼ back within this time ( 76% for the non- We measured the power law of the fallback decay to ∼ see whether the curves in the spinning cases follow the spinning case). If the power law index does not settle −5/3 to the expected 5/3 within early times, then it is un- expected t . We performed linear regression on sec- likely to ever be− observed. The luminosity, which fol- tions of the slope and found that the power law does vary lows the behaviour of the fallback curve if the viscous from the predicted value, see Fig. 10. The power law in- dex initially overshoots the 5/3 and more so in the timescale is much shorter than variations in the fallback − rate (Lodato 2012), will be too low to detect if t−5/3 is retrograde cases. This is due to the self-gravitating na- not reached until late times. ture of the stream (Coughlin & Nixon 2015). The power law may bounce back to a less steep gradient than 5/3 The results we present in this paper were done with − a million particles. We also ran the simulations with after the overshoot as material has been redistributed 104 and 105 particles and found no physical change in along the stream by self-gravity. At late times the pow- shape of the fallback curves, only a decrease in simula- erlaw index is roughly constant. We can also see that tion noise and error bar size with an increase in particle the power law indexes do not settle very well on the ex- pected n = 5/3, and generally stay above this until number which allows us to view any real features with − more clarity. late times. At late times the errors grow as the number of particles in the stream decreases. This is in agreement 3.1. Results with Lodato et al. (2009) who also found the power law − Fig. 3 shows the fallback rates, over a time of approxi- to be shallower and only reached t 5/3 at late times mately 1.6 years, for different stellar spin velocities. The for non-spinning polytropes. This has been found in curves for the retrograde spins show less and delayed fall- several other studies (Guillochon & Ramirez-Ruiz 2013; back of material to the hole compared to the prograde Coughlin & Nixon 2015; Wu et al. 2018). and non-spinning cases. In Fig. 4 we compare our nu- Disruption of Spinning Stars 7

Figure 3. Comparing fallback rates of bound stellar material for different stellar rotation velocities both prograde (black) and retrograde (red) with respect to the orbit (shown with error bars in the bottom panel). We see that stars with prograde spin disrupt sooner and with higher peaks compared to the non-spinning case (dash-dot), whereas a retrograde spin hinders disruption as the tidal forces need to spin down the star first. 8 Golightly et al.

If the star is spinning prograde with respect to its orbit 1 then it is spun up by the tidal forces and will fallback quicker and with more material for faster initial spins. Conversely, the black hole has to spin down a retrograde 10-1 spinning star leading to delayed, and sometimes only partial, disruptions.

/yr]

sun We also see some interesting features around the peak 10-2 of the fallback curve, in all but the faster prograde cases, where the fallback of bound material deviates from the

dM/dt [M −5/3 0.2 prograde numerical expected t decay by accreting extra mass in self- 10-3 0.2 prograde analytical gravitating clumps in the debris stream and then less 0.2 retrograde numerical mass afterwards. The total mass accreted still averages 0.2 retrograde analytical out to half of the original stellar mass (due to half of 10-4 the debris being bound, half unbound). We find that

-1 even after the overshoot occurs, the power law remains 10 1 −5/3 t [yr] shallower and does not settle to the expected t until late times, by which point the TDE is unlikely to be Figure 4. A comparison of the simulation fallback curves observable anyway. (solid lines) with analytically predicted curves (dashed) from We initially calculated predictions for the debris fall- Section 2. The analytical and numerical show the same back rate with the impulse approximation, which as- trends of a prograde spin giving a higher peak and earlier sumes the star is undisturbed until it reaches pericen- fallback compared to retrograde. However, the differences tre. Comparing to our numerical simulations, we see the show the need for numerical modelling of TDEs to correctly analytical solutions are not a perfect fit with smaller recover features in the curves from effects such ass changes peak fallback rates and longer material return times to the stellar structure due to spin. than the numerical. The impulse approximation also misses out features around the peak that should occur due to changes in the stellar structure before it reaches We ran simulations for misaligned spins for λ = 0.05 pericentre. However the analytical solutions recover the where we rotate the star to produce θ = 45◦ (where ◦ trends observed in the simulations, e.g. how the rise and the angular velocity vector of the star is rotated by 45 peak fallback rates change with stellar spin. around the x-axis for prograde; i.e. the angular velocity ◦ Our analytic arguments suggested, and our numerical vector has positive y and z components) and 90 (where simulations confirmed, that stellar spin is only impor- the angular velocity vector points in the direction of the tant for modifying the features of the fallback (e.g., the orbital velocity at pericentre). We see in Fig. 11 that return time of the most bound debris and the time to varying the spin angle in this way only changes the fall- peak fallback rate) when the star is spinning at a mod- back curve slightly compared to the aligned spins for the ◦ ◦ est fraction of its breakup velocity. One way of gener- 45 case. For the 90 case, we expect from Eq. 9 that ating such rapidly spinning stars prior to disruption is there is no first order effect from spin (and remembering if the disrupting SMBH is in a binary system: as shown that any second order effect is negligible) which is con- in Coughlin et al. (2017) a star can have a number of firmed by Fig. 11 where the fallback curves match the “close encounters” – where the star comes within at least non-spinning case. three tidal radii of either black hole – prior to being dis- rupted as it traces out a chaotic, three-body orbit in 4. DISCUSSION AND CONCLUSIONS the binary potential. By performing a statistical analy- We ran simulations of a solar mass star, modelled as sis of millions of three-body encounters, Coughlin et al. a polytrope of 1 million particles with index γ = 5/3, (2017) demonstrated that & 10% of all three-body or- 6 being tidally disrupted by a 10 M⊙ black hole with an bits resulting in disruptions had at least one close en- impact parameter of β = 1 for a just full disruption. We counter. In these close encounters, while the tidal field did this for a standard case of a non-spinning star and of the black hole may not be sufficient to completely black hole and then introduced stellar spins of values: unbind the star, the tidal torque will spin the star up 0.002, 0.01, 0.05 and 0.2 (fractions of the star’s to a significant fraction ( 10%) of its breakup velocity, ± ± ± ± 3 ∼ break up velocity, where Ω∗ = λ GM∗/R∗) where sign and repeated close encounters could push the fraction indicates direction of spin with respect to its orbit, tak- of breakup to near unity. One might therefore expect ing prograde as positive. p some stars disrupted by binaries to be rapidly rotating, When we include stellar rotation into the simulation, necessitating the inclusion of this effect on the predicted we get interesting changes in behaviour. The direction fallback curves. of the star’s spin will either help or hinder disruption The change in fallback rates, both shape and normal- as the tidal forces will also impart a spin onto the star. isation, suggest that stellar spin can play an important Disruption of Spinning Stars 9

Figure 5. Star at initial position on orbit (left) and then at pericentre (right) for prograde 0.2 (top), non-spinning (middle), retrograde 0.2 cases (bottom). Overlayed is the velocity structure of the star (not including orbital velocity) and the star spins prograde or retrograde with respect to its orbit. The length of the velocity arrows indicates the column integral of v dz 10 Golightly et al.

Figure 6. For the 0.2 retrograde disrupted star after passing pericentre we can see the clear effect that a retrograde spin has on the extent of disruption from pericentre and onwards, particularly where we see a small dense core still intact.

Figure 7. The density structures of the disrupted stars as a function of distance from the black hole. The left and right tails of each plot show the bound and unbound parts of the debris stream. In the retrograde plot the vertical peak in the centre, highlighted within the box, shows the intact stellar core from it being harder to disrupt. Disruption of Spinning Stars 11

-8 2000

]

g

y [R 1000

log column density 0.05 prograde 0

-10 2000

]

g

y [R 1000

no spin 0 -12

2000

]

g

y [R 1000

0.05 retrograde 0 -14 -8000 -6000 -4000

x [Rg]

Figure 8. The disrupted debris streams for zero spin (middle panel) and λ = 0.05 prograde (top) and retrograde (bottom). The stream is plotted at a time of t = 17 days post pericentre. Stellar spin does not significantly affect the geometry of the stream. There is a detectable difference in the streams lengths, which is due to the quickened/delayed disruption from prograde/retrograde spins.

1 0.05 prograde no spin 0.05 retrograde

10-1

dm/de

10-2

10-3

-2 -1 0 1 2 e

Figure 9. The energy distribution for the zero spin (short dashed) and 0.05 spin (prograde: solid and retrograde: long dashed) 2 simulations shown in Fig 8, where e = E/∆E, ∆E = GMBHR∗/Rperi and dm = dM/M∗. These differences in the energy distribution result in differences in the fallback rates. 12 Golightly et al.

Figure 10. Power law evolution for different spins plotted with the expected −5/3 (dashed), where n = d log M/d˙ log t is the − power law index. The initial overshoots of the expected t 5/3 are due to accreting clumps which are larger for retrograde spins. The larger error bars towards the end are due to increased noise in the later parts of the fallback curves. We see that the power − law does not converge on −5/3 for most of the TDE evolution. It is unlikely that a t 5/3 lightcurve would be observed at late times. Disruption of Spinning Stars 13

Figure 11. Fallback curves for aligned (solid) and misaligned (dashed), prograde (black) and retrograde (red) spins and for ◦ the non-spinning case (blue solid). For the simulated 45 angles we find only a small variation in the fallback curves compared to an aligned spin with the same magnitude. However, for both the 90◦ cases, where the angular velocity vector is parallel to the orbital velocity vector at pericentre, the stellar spin has no effect on the stream and these fallback curves match the non-spinning case. role in defining the energy distribution of the stream and ity at the University of Leicester. The figures were thus the observable properties of the event. It may be made using splash (Price 2007), a visualisation tool for necessary to include such details to accurately determine SPH data. This work was performed using the DiRAC system parameters from observed data. Data Intensive service at Leicester, operated by the University of Leicester IT Services, which forms part of the STFC DiRAC HPC Facility (www.dirac.ac.uk). We thank the referee for useful and constructive com- The equipment was funded by BEIS capital funding via ments. We thank Giuseppe Lodato and Andrea Sacchi STFC capital grants ST/K000373/1 and ST/R002363/1 for interesting discussions on this topic. CJN is sup- and STFC DiRAC Operations grant ST/R001014/1. ported by the Science and Technology Facilities Coun- DiRAC is part of the National e-Infrastructure. cil (STFC) (grant number ST/M005917/1). ERC ac- knowledges support from NASA through the Einstein Software: Fellowship Program, grant PF6-170150. This research phantom(Price et al.2017,http://arxiv.org/abs/1702.03930) used the ALICE High Performance Computing Facil- Splash (Price 2007, http://arxiv.org/abs/0709.0832)

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