Tidal disruption and ignition of white dwarfs by moderately massive black holes
aa r X i v : . [ a s t r o - ph ] N ov Tidal disruption and ignition of white dwarfs by moderatelymassive black holes
S. Rosswog , E. Ramirez-Ruiz and W.R.Hix School of Engineering and Science, Jacobs University Bremen, Campus Ring 1, 28759Bremen, Germany Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA95064 Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN37831-6374
ABSTRACT
We present a numerical investigation of the tidal disruption of white dwarfs bymoderately massive black holes, with particular reference to the centers of dwarfgalaxies and globular clusters. Special attention is given to the fate of whitedwarfs of all masses that approach the black hole close enough to be disruptedand severely compressed to such extent that explosive nuclear burning can betriggered. Consistent modeling of the gas dynamics together with the nuclearreactions allows for a realistic determination of the explosive energy release. Inthe most favorable cases, the nuclear energy release may be comparable to thatof typical type Ia supernovae. Although the explosion will increase the massfraction escaping on hyperbolic orbits, a good fraction of the debris remains tobe swallowed by the hole, causing a bright soft X-ray flare lasting for about ayear. Such transient signatures, if detected, would be a compelling testimony forthe presence of a moderately mass black hole (below 10 M ⊙ ).
1. Introduction
White dwarfs, the end point of stellar evolution for stars with masses from about 0.07to 10 solar masses, are extremely common (Weidemann 1990). And we don’t have to lookfar away, either. There are several billions of them in the halo of our very own MilkyWay galaxy (Reid 2005). They are not only observed in isolation but in binary systems,with normal stellar companions, and, less frequently, with compact stellar companions. Ata distance of tens of kiloparsecs, a number of globular clusters (Brodie & Strader 2006;Moehler & Bono 2008) have a high enough central density to let white dwarfs interact andcollide with other stars, or, if developed, with a central massive black hole (Gerssen et al.2002, 2003; Gebhardt et al. 2002, 2005). And at a distance of less than ten kiloparsecs, our 2 –galactic nucleus, a central massive black hole of a few million solar masses (Ghez et al. 1998),is surrounded by swarms of all kind of stars, some of them white dwarfs and all of them proneto collisions. In such dense environments, stars of all varieties can exchange mass, disrupteach other or merge, and their merger products can get involved in similar interactions.Because of this reason, the innermost ∼ . R wd , the gravitational radius of the black hole, R g = 2 GM BH /c ≃ × M BH , cm, where M BH , denotes the hole’s mass in units of 10 M ⊙ , and the tidalradius, R τ . The tidal radius, defined as the distance within which a stars gets disrupted,obviously depends on the type of star being considered. For a white dwarf it is roughly givenby R τ ≃ . × M / , (cid:18) R wd cm (cid:19) (cid:18) M wd . M ⊙ (cid:19) − / cm . (1)Essentially, R τ is the distance from the hole at which M BH /R τ equals the mean density of thepassing star. R τ is, in order of magnitude, the same as the Roche radius, which is a preciselydefined quantity, but only strictly applicable to a star in a circular orbit with synchronizedspin. The strength of the tidal encounter is measured by the dimensionless parameter η = (cid:18) M wd M BH R p3 R wd3 (cid:19) / , (2)which is simply the square root of the ratio of the surface gravity of the star to the tidalgravity at the surface when the star is at pericenter distance R p . Most literature on tidaldisruption uses β = R τ /R p = η − / to measure the encounter strength. When η ≤
1, thestar is disrupted in a single flyby. The energy required to tear the star apart (that is, thestar’s self-binding energy) is supplied at the expense of the orbital energy, which at R τ islarger by ( M BH /M wd ) / .For white dwarfs, the ratio between the tidal disruption radius and the gravitational 3 –radius is β g = (cid:18) R τ R g (cid:19) ≃ . M − / , (cid:18) R wd cm (cid:19) (cid:18) M wd . M ⊙ (cid:19) − / . (3)Note that this is indeed inside the gravitational radius for black hole masses exceeding M BH , lim ≃ . × (cid:18) R wd cm (cid:19) / (cid:18) M wd . M ⊙ (cid:19) − / M ⊙ . (4)For this reason, white dwarfs only experience disruptive physical conditions when approach-ing a moderately massive black hole (Luminet & Pichon 1989). If M BH ≪ M BH , lim it ishowever, sufficiently far outside R g that the disruption can be approximated as a Newtonianprocess, and it makes little difference whether the hole is described by a Schwarzschild or aKerr metric. When the central hole has a mass M BH ≫ M wd , the size of the star remainssmaller than R τ . A white dwarf cannot thus be disrupted (Figure 1) without entering thestrong relativistic domain if β > min "(cid:18) M BH M wd (cid:19) / , (cid:18) M BH , lim M BH (cid:19) / . (5)The type of the black hole (Schwarzchild or Kerr?) then has an important quantitativeeffect, as does (for a Kerr hole) the orientation of the stellar orbit relative to the spin axis.Much of our effort in this paper is therefore dedicated to investigate the disruption ofwhite dwarfs by moderately massive black holes, with particular reference to the centers ofdwarf galaxies (e.g. Mathur et al. 2008), globular clusters (e.g. Portegies Zwart et al. 2004;Baumgardt et al. 2004) and the intermediate-mass black hole candidates in active galacticnuclei (Greene & Ho 2004). Special attention is given to the fate of white dwarfs of allmasses that approach a black hole close enough to be disrupted and severely compressedto such extent that explosive nuclear burning can be triggered. Tidal disruption of whitedwarfs has previously been difficult to model due to its three-dimensional nature and the del-icate interplay between gravity, gas dynamics and nuclear reactions. Crucial aspects of thephysics of white dwarf disruption and ignition were first understood by Luminet & Pichon(1989), mainly based on the affine model earlier introduced by Carter & Luminet (1983).Luminet & Pichon (1989) suggested that for very close encounters ( η ≤ §
2. De-tailed hydrodynamic simulations of the disruption of white dwarfs of various masses, initialcomposition and passage distances are presented in §§
3, 4 and 5, the role of the mass of theblack hole in shaping the evolution and ignition of the disrupted white dwarf is discussed in §
4. The resulting gravitational wave signals are shown in §
6. Discussion and conclusions arepresented in §
2. Numerical Methods and Initial Model
The observational consequences of stellar disruption depend on what happens to thedebris (Rees 1988; Evans & Kochanek 1989; Kobayashi et al. 2004). To quantify this, wehave performed detailed three-dimensional, hydrodynamical calculations. The gas dynamicsis coupled with a nuclear network to explore the effects of nuclear energy generation dur-ing the strong compression phase. The reader is referred to Rosswog et al. (2008a) for acomplementary description of the numerical methods employed in calculating the disruptiveevent. 5 –
The SPH-formulation used in this study follows closely the one described in Benz (1990),a brief derivation of the equations can be found in Rosswog & Price (2008). Forces fromself-gravity are calculated using a binary tree (Benz et al. 1990b) while the gravitationalforces from the central black hole are calculated using a Paczy´nski-Wiita pseudo potential(Paczy´nski & Wiita 1980) with a polynomial extension (Rosswog 2005) to avoid the singu-larity at the Schwarzschild radius. We have taken particular care to avoid artifacts from theuse of artificial viscosity (AV). The so-called Balsara-switch (Balsara 1995) is implemented toavoid spurious shear forces. More importantly, we use time dependent viscosity parameters(Morris & Monaghan 1997), so that far from shocks artificial viscosity is essentially absent,but near a shock front the associated viscous parameters rise to values that are able to avoidpost-shock oscillations (see Fig. 1 of (Rosswog et al. 2008a) for an illustration). Once theshocks has passed, the parameters decay again. Details of the AV implementation and testscan be found in (Rosswog et al. 2000).A MacCormack predictor-corrector method is used to evolve the fluid in time. Themethod is implemented with individual time steps, i.e. each particle i is advanced on its owntime step dt i = 2 n i × dt min , where dt min is the smallest step of all the particles. n i is chosento be the largest integer satisfying the condition dt i < dt i, des , where dt i, des is the desired timestep for particle i . With this time marching implementation the total energy is conservedto better than 4 × − and the total angular momentum to better than 2 × − . Notethat this could, in principle, be improved even further by taking into account the so-called“grad-h”-terms (Springel & Hernquist 2002; Monaghan 2002) and extra-terms arising fromadapting gravitational smoothing terms (Price & Monaghan 2007). A comparison betweenthe formulation used here and the one using “grad-h”-terms can be found in Rosswog & Price(2007). We use the HELMHOLTZ equation of state (EOS), developed by the Center for As-trophysical Thermonuclear Flashes at the University of Chicago. The EOS allows to freelyspecify the chemical composition of the gas and it can be coupled to a nuclear reactionnetwork. The electron-/positron equation of state is calculated without approximation.In other words, it makes no assumption about the degree of degeneracy or relativity andthe exact expressions are integrated numerically. The nuclei in the gas are treated as aMaxwell-Boltzmann gas, the photons as blackbody radiation. The EOS is used in tabularform with densities in the range 10 − ≤ ρY e ≤ g cm − and temperatures varying be- 6 –tween 10 and 10 K. A sophisticated, biquintic Hermite polynomial interpolation is used toenforce the thermodynamic consistency (i.e. the Maxwell-relations) at interpolated values(Timmes & Swesty 2000).
To account for the feedback onto the fluid from nuclear transmutations we use a minimalnuclear reaction network developed by Hix et al. (1998). It couples a conventional α -networkstretching from He to Si with a quasi-equilibrium-reduced α -network. The QSE-reducednetwork neglects reactions within small equilibrium groups that form at temperatures above3 . × K to reduce the number of abundance variables needed. Although a set of onlyseven nuclear species is used, this network reproduces all burning stages from He-burning toNSE accurately. For details and tests we refer to Hix et al. (1998).The network is coupled to the hydrodynamics in an operator splitting fashion, i.e.hydrodynamics and nuclear burning -which may in extreme cases require vastly different timesteps- are integrated separately. In a first step, the hydrodynamic equations are integratedwith the above described predictor-corrector scheme to obtain new quantities at time step t n +1 . In this step we ignore the nuclear source term in the energy equation, the new value forthe specific energy of particle a is denoted by ˜ u n +1 a . The specific energy has to be correctedfor the energy release that occurred between t n and t n +1 : ǫ a,n → n +1 = − N A X j m j c Z t n +1 t n dY j,a dt ( ρ a ( t ) , T a ( t ) , Y k,a ( t )) dt (6)= − N A X j =1 m j c ( Y n +1 j,a − Y nj,a ) , (7)where N A is Avogadro’s constant, m j c is the mass energy of nucleus j and Y j,a is theabundance of nucleus j in particle a . We use ρ a ( t ) ≈ ρ a ( t n ) + t − t n t n +1 − t n { ρ a ( t n +1 ) − ρ a ( t n ) } and T a ( t ) ≈ t − t n t n +1 − t n { ˜ T a ( t n +1 ) − T a ( t n ) } , where ˜ T a ( t n +1 ) is the temperature derived from˜ u n +1 a , to integrate the abundances Y j,a via the implicit backward Euler method (the networkintegration is described in detail in (Hix et al. 1998)) . The final value for the specific energyat time t n +1 is given by u n +1 a = ˜ u n +1 a + ǫ a,n → n +1 . (8) Note that the temperature evolution along a hydrodynamical time step is different from the descriptionin Rosswog et al. (2008a). In practice, we only see tiny differences between both approaches. u n +1 a . Once the derivatives have been updated, the procedure can be repeated for thenext time step. For details of the time step criteria we refer the reader to Rosswog et al.(2008a).As will be shown below, the peak compression occurs at a spatially fixed point (see densitypanels in Fig. 6). The white dwarf fuel is fed with free-fall velocity v ff = (2 GM BH /R p ) / =1 . · km s − ( M BH / ⊙ ) / ( R p / km) − / into this compression point. This ismany orders of magnitude larger than typical flame propagation speeds, therefore flamepropagation effects can be safely neglected for this investigation. It is a vital ingredient for any calculation to start out from initial conditions that areas accurate as possible. In order to build stars in hydrostatic equilibrium, it is thereforenecessary to find SPH particle distributions whose number density reflects the equilibriummass density distribution. To this end we solve the spherically symmetric Lane-Emdenequations for a star of given mass and composition to find a one dimensional density profile.For simplicity, and to start from a conservatively low value, we assume a uniform temperatureof T = 5 × K. This assumption will be relaxed in one run (run 10, see Table 1) to testfor the sensitivity to this initial condition. Once the solution to the Lane-Emden equationhas been obtained, we distribute the desired number of particles, N , inside a unit sphereaccording to a close-packed configuration. Subsequently, we map this distribution into thevolume of the star that is to be constructed. As an example, the particle distributions beforeand after mapping of a 0.2 M ⊙ white dwarf are shown in Figure 2.The mapping of the initial configuration is done so that the SPH-particle number densityin the star, n , yields ρ LE ( r ) = m · n ( r ) , (9)where m is the mass of each particle and ρ LE is the density profile obtained by solving theLane-Emden equation. We denote quantities referring to the star (unit sphere) with (un-)primed variables. If the unit sphere has a constant density ρ and contains N particles, eachwith mass m = M wd /N , the mass in the shell between r n − and r n is M n ≃ πr n − ∆ rρ . Asillustrated in Figure 3, the image of this shell (in the star) contains the same mass, but nowbetween r ′ n − and r ′ n , so that M ′ n ≃ πr ′ n − ∆ r ′ n ρ n − . The condition M ′ n = M n then yields∆ r ′ n = (cid:18) r n − r ′ n − (cid:19) (cid:18) ρ ρ ′ n − (cid:19) ∆ r. (10) 8 –The resultant configuration is very close to hydrostatic equilibrium. To find the “real”numerical equilibrium state we relax this configuration with the full hydrodynamics code byapplying an artificial, velocity dependent damping force as, for example, in Rosswog et al.(2004).We use this procedure for all but the heaviest white dwarfs (1.2 M ⊙ ). Since theiradiabatic exponent is already approaching the critical 4/3, these stars are very centrallycondensed and, as a result, the bulk of the SPH particles resides very close to each other inthe stellar center. This would, even for a moderate resolution of a few hundred thousandparticles, result in most of the particles running at time steps that are orders of magnitudesmaller than the dynamical time of the star. To alleviate this computational burden in theheaviest white dwarfs we allow for slightly different particle masses (about a factor of 10from the center to the stellar surface).Throughout this simulation set we assume a uniform nuclear composition accross the whitedwarfs. Stars with masses < ⊙ are instantiated as pure helium, more massive starsare modeled as 50% carbon and 50% oxygen. Note that helium core burning may producecomposition gradients throughout the stars with oxygen being more abundant in the stellarcore. Since it is mainly the core that becomes ignited during the compression, see Sect. 3,such a gradient may slightly reduce the nuclear energy that can be released in principle.Common to all calculations is the initiation of the calculations with the white dwarf beingplaced safely outside R τ . Initial separations are at least twice, in most cases three timesthe tidal radius to allow the initially spherical star to adjust properly to the changing tidalpotential as it approaches the black hole. All stars are set onto parabolic orbits so that R p is larger than R g .The performed runs are summarized in Table 1.
3. Events in the Life of a Tidally Disrupted White Dwarf
Since the focus of this study is closely related to the nuclear energy release during strongencounters, practically all following calculations refer to cases with penetration factors β clearly beyond unity, see Table 1. To illustrate the evolution of a white dwarf that is justmarginally disrupted, we show in Fig. 4 a 0.6 M ⊙ CO white dwarf that passes a 1000 M ⊙ black hole with a penetration factor of β = 0 . R τ . The lower panelshows late stages at t= 138.24 and 380.53 s. The circle indicates in both cases the location 9 –of the tidal radius. At about 1.5 R τ the star becomes noticeably deformed and substantiallyspun up by the time it reaches pericentre (snapshot 5, upper panel). While receeding fromthe black hole large, puffed-up lobes form at the extremes of the star. The inner lobe returnsto the black hole (see t=138.24 s, lower panel) while the outer is ejected to infinity, both areconnected by a well-defined and homogenous spaghetti-like tube of white dwarf debris.The opposite limit, an exremely strong encounter, is illustrated in Figure 5. The snapshotsare taken from our numerical simulations of a 0.2 M ⊙ white dwarf approaching a 10 M ⊙ blackhole on a parabolic orbit with pericenter distance well within the tidal radius ( R p = R τ / R ∼ R τ .When the star is still far from the black hole, it remains close to its initial stationaryequilibrium state as characterized by the usual virial relation. This is because the timescalecharacterizing the rate of change of the tidal force will initially be very long compared withthe intrinsic timescale characterizing the corresponding quadrupole oscillations of the star.The tidal bulge raised on the star by the black hole becomes an order unity distortionnear the tidal radius. The resultant gravitational torque spins it up to a good fraction ofits corotation angular velocity by the time it gets disrupted. The large surface velocitiesand the order unity tidal bulge combine to overcome the star’s self gravity and lead to thedisruption of the star.Our present investigation is concerned with cases of deeper penetration within thetidal radius so that the core as well as the envelope are affected. The behavior of a whitedwarf passing well within the tidal radius exhibits special features (Carter & Luminet 1983;Bicknell & Gingold 1983). As illustrated in Figure 6, the degenerate star is not only elon-gated along the orbital direction but also severely compressed perpendicular to the orbitalplane. This anisotropy can be understood as arising from the fact that the principal tidalaxes within the orbital plane will rotate prior to pericenter passage so that the correspondingeffects of elongation and compression will roughly cancel out, whereas the third principal axisretains a fixed direction so that compression orthogonal to the orbital plane is uncontested(see, e.g., the appendix of (Brassart & Luminet 2008)). Each section of the star is squeezedthrough a point of maximum compression at a fixed point on the star’s orbit. This takesplace on a timescale comparable to the crossing time of the star through periastron, δt ∼ R wd v p ≃ . (cid:18) M wd . M ⊙ (cid:19) − / (cid:18) R wd cm (cid:19) / (cid:18) M BH M ⊙ (cid:19) − / s , (11) 10 –where v p ∼ ( R g /R τ ) / c ≃ × (cid:18) M wd . M ⊙ (cid:19) / (cid:18) R wd cm (cid:19) − / (cid:18) M BH M ⊙ (cid:19) / cm s − , (12)is the orbital velocity at periastron.During this very short lived phase, the star attains its maximum degree of compression(here by a factor ∼ R wd /R wd ∼ R wd δt ∼ v p ≪ c s ∼ . × (cid:18) R wd cm (cid:19) − / (cid:18) M wd . M ⊙ (cid:19) / cm s − (13)during the drastic compression of the stellar material. This compression is thus halted bya shock (Kobayashi et al. 2004; Brassart & Luminet 2008), raising the matter, which thenrebounds perpendicular to the orbital plane, to a higher adiabat (Figure 7). As a result,the temperatures increase sharply and trigger explosive burning (of He for the case shownin Figure 6).The typical temperature of the shocked star and the thermal energy produced by shockheating can be roughly estimated from the virial theorem as T ∼ × (cid:18) R wd cm (cid:19) − (cid:18) M wd . M ⊙ (cid:19) K , (14)and E therm ∼ E G ∼ (cid:18) R wd cm (cid:19) − (cid:18) M wd . M ⊙ (cid:19) erg . (15)Adiabatic compression alone can only increase the stellar surface and interior layer temper-atures by a modest factor. If adiabatic compression was the only source of heating, theresponse to the flow to the varying potential would be ∼ R wd /c s ∼ × K, approach-ing but not quite reaching nuclear statistical equilibrium (NSE). During this brief periodof compression, nuclei up to and beyond Si are synthesized. The initial composition waspure He and the final mass fraction in iron-group nuclei is about 15%. This result should betaken as a modest underestimate, since the seven species nuclear network only provides an 11 –approximation to the detailed nuclear processes. Post-processing calculations, using a 300isotope nuclear network over thermodynamic particle histories resulting from these calcula-tions, show significant nuclear flow above silicon, for helium-rich portions of the gas withpeak temperatures above 2 × K. As a result, heavier elements (like calcium, titanium andchromium) would be made, accompanied by a modest increase in the energy generation. Itis thus safe to conclude that the white dwarf is tidally ignited and that a sizable mass ofiron-group nuclei is injected into the outflow.The variation of the specific energy in the released gas, in the absence of explosive energyinput, is determined mainly by the relative depth of a mass element across the disruptedstar in the potential well of the black hole (Rees 1988). This is much larger than the bindingenergy and the kinetic energy generated by spin-up near pericenter. Even though the meanspecific binding energy of the debris to the hole is comparable with the self-binding energyof the original star, the spread about this mean is larger by a factor ( M BH /M wd ) / . Nuclearenergy released during the drastic compression and distortion of the stellar material, furtherenhances the spread in specific energies at pericenter (Figure 9), and, as a result, the massfraction escaping on hyperbolic orbits is increased from ∼
50% to ∼
65% of the initial massof the white dwarf. The mass fraction that is ejected rather than swallowed, though lessspectacular than typical Type Ia supernovae (Hillebrandt & Niemeyer 2000), should havemany distinctive observational signatures (the reader is referred to Kasen et al. 2008 fora detailed description of the optical light curves and spectra resulting from the unbounddebris before it becomes translucent). First, the explosion itself should be different, sincethe disrupted, degenerate stars should be, on average, lighter than those exploding as typeIa supernovae. Second, the spectra should exhibit large Doppler shifts, as the ejected debriswould be expelled with speeds ≥ km/s. Finally , the optical light curve should be ratherunique as a result of the radiating material being highly squeezed into the orbital plane (onethus expects different timescales for conversion of nuclear energy to observable luminositywhen compared with normal type Ia events).Although the explosion will increase the fraction of ejected debris, a good fraction ( ∼ a ∼ (cid:18) M BH M ⊙ (cid:19) − / (cid:18) R wd cm (cid:19) (cid:18) M wd . M ⊙ (cid:19) − / R g , (16) 12 –and the period is only t a ∼ (cid:18) a R g (cid:19) / (cid:18) M BH M ⊙ (cid:19) − / s . (17)If the gaseous debris suffered no internal dissipation due to high viscosity or shocks, itwould, after one or two orbital periods, form a highly elliptical disc with a big spread inapocenter distances between the most and least bound orbits, but where at pericenter, R p ,the radial focusing of the orbits acts as an effective nozzle (Figure 4 and 10). After pericenterpassage, the outflowing gas is on orbits which collide with the infalling stream near theoriginal orbital plane at apocenter, giving rise to an angular momentum redistributing shock(Figure 10) much like those in cataclysmic variable systems. The debris raining down would,after little more than its free-fall time, settle into a disc. This orbiting debris starts to formwhen the most tightly bound debris falls back. The simulation shows that the first materialreturns at a time ≤ t a , with an infall rate of about 10 M ⊙ yr − (Figure 11). Such highinfall rates are expected to persist, relative steadily, for at least a few orbital periods, beforeall the highly bound material rains down. The vicinity of the hole would thereafter be fedsolely by injection of the infalling matter. The early mass infall rate is sensitive to the stellarstructure (Lodato et al. 2008; Ramirez-Ruiz & Rosswog 2008), at late times, t ≥ t fb ≈ t − / (Rees 1988; Phinney 1989). Once the torus is formed, it will evolveunder the influence of viscosity, radiatively cooling winds and time dependent mass inflow.A luminosity comparable to the Eddington value, ∼ L Edd = 10 ( M BH / M ⊙ ) erg s − ,can therefore only be maintained for at most a year; thereafter the flare would rapidly fade.It is clear that most of the debris would be fed to the hole far more rapidly than it couldbe accepted if the radiative efficiency were high; much of the bound debris must eitherescape in a radiatively-driven outflow or be swallowed inefficiently. The rise and the peakbolometric luminosity can be predicted with some confidence. However, the effective surfacetemperature (and thus the fraction of luminosity that emerges predominantly in the softX-ray band) is harder to predict, as it depends on the size of the effective photosphere thatshrouds the hole. Such transient signals, if detected, would be a compelling testimony forthe presence of moderately massive black holes in the centers of globular clusters and dwarfgalaxies.
4. Influence of Black Hole’s Mass on Disruption
The characteristic tidal radius, R τ , for a given white dwarf star is solely determinedby the black hole’s mass, while the strength of the tidal encounter is traditionally measured 13 –by the dimensionless parameter β = R τ /R p (provided β ≤ β g ). The aim of this section isto illustrate, with the help of a few specific calculations, the role of the black hole’s massin shaping the evolution and ignition of the disrupted white dwarf. To quantify this, wehave performed detailed three-dimensional, hydrodynamical calculations of the dynamics ofa 0 . M ⊙ white dwarf approaching black holes of various masses on parabolic orbits with β = 5. For black hole masses 10 M ⊙ ≤ M BH ≤ M ⊙ and β = 5, a 0 . M ⊙ white dwarfcan be disrupted without entering the strong relativistic domain (Figure 1). However, com-plete disruption occurs sufficiently close to the hole for a Newtonian approximation to beinadequate (Figure 12).The behavior of a 0 . M ⊙ white dwarf approaching black holes of various masses isillustrated in Figure 13. As discussed previously, the effects of the black hole’s mass (fora fixed β and M wd ) will be to move the critical pericentric distance by M / and changethe star’s crossing time through the maximum compression point by M − / . As a result,stars approaching black holes of increasing mass will appear more elongated along the orbitalplane, as each section of the star is squeezed faster through a point of maximum compressionat a fixed point on the star’s increasingly extended orbit. Such stars are not only moreelongated but are even more severely compressed into a prolate shape. If the initial whitedwarf is made of pure He (as in the 0.2 M ⊙ case), the combustion rate will be determinedby the 3 α reaction, on time scale approximately given by (Khokhlov & Ergma 1986) t b ≈ . × − T exp(4 . /T ) ρ − , (18)where T is the temperature in units 10 K and ρ the density in 10 gcm − .If the time scale on which the white dwarf can react, its dynamical time scale t G =( G ¯ ρ wd ) − / , is much shorter than the burning time scale t b , the star can expand rapidlyenough to quench burning by a reduction of temperature and density. Appreciable burningwill therefore only take place if t b ≪ t G . As illustrated in Figure 14, this comparison of timescales can be used as a simple estimate for whether or not substantial He-burning will occur.Promising models for thermonuclear explosions are those that reach high temperatures athigh densities (upper right corner). For a fixed β , increasing M BH raises the nuclear energyrelease. For black hole masses M BH > M ⊙ , nuclear energy released during the drasticcompression and distortion of a 0.2 M ⊙ white dwarf approaching moderately massive blackholes with β = 5 is fast enough to release energy in excess of that required to tear the starapart: E G (Figure 15). When the black hole mass is ≫ M ⊙ most white dwarfs would be,however, swallowed whole ( R τ < R g ). 14 –
5. Tidal Disruption for a Variety of White Dwarfs
White dwarfs are thought to be the final evolutionary state of all stars whose mass isbelow ∼
10 M ⊙ (Weidemann 1990). Stars with masses less than about half a solar massbecome degenerate before helium ignition, and therefore will end their lifes as helium whitedwarfs. In isolation, such low mass stars have life times much longer than the present age ofthe Universe. Still, such He white dwarfs are observed, they are thought to be the result of theevolution of a close binary system (e.g. Heber 2002). Stars of low or medium mass, achievehelium burning and become carbon-oxygen white dwarfs (perhaps the commonest sort).The observed white dwarf mass distribution is strongly peaked around 0.6 M ⊙ (Kepler et al.2007).When a white dwarf is subject to strong tidal compression, triggering of nuclear processesin the stellar cores depends on initial composition. To investigate this dependence, we haveperformed calculations with various white dwarf masses/compositions. As outlined above,our white dwarf models are initialized with pure He-composition for M wd < . ⊙ and 50%carbon and 50% oxygen otherwise.Although the thermodynamical evolution and the nuclear energy release are very sensi-tive to the initial stellar composition (in this text, we have so far considered only a pure He0.2 M ⊙ white dwarf), we found that ignition is in fact a natural outcome for white dwarfs ofall masses passing well with in the tidal radius. For example, a C/O 1 . M ⊙ white dwarf (hereassumed to be composed of 50% carbon and 50% oxygen throughout the star) approaching a500 M ⊙ black hole on a parabolic orbit with pericenter distance β = 3 . M ⊙ of Fe are synthesised in the flow (Figure 17). Similarly, atleast 0.66 M ⊙ of Fe are synthesised when a C/O 1 . M ⊙ white dwarf approaches a 10 M ⊙ black hole with β = 3. By contrast with the previous two cases, explosive energy release istoo slow to release much energy on a dynamical timescale when a C/O 1 . M ⊙ white dwarfapproaches a 10 M ⊙ black hole with β = 1 . . × − M ⊙ of Fe aresynthesised.0.18 M ⊙ of Fe are synthesised when a typical 0.6 M ⊙ C/O white dwarf approachesa 500 M ⊙ black hole with β = 5. To explore how sensitive our models are to the initialtemperature, we have re-run this simulation, but now with a hot white dwarf. The initialspecific energy of a cold ( T = 5 × K) equilibrium model was adjusted in a way thatthe temperature varied linearly between 5 × K in the centre to 5 × K at the stellarsurface. In the subsequent relaxation the star expanded slightly, since the degeneracy waslifted to a small extent. As a result of the reduced degeneracy, lower peak temperaturesand densities are reached (see Figure 18), reducing slightly the amount of nucleosynthesiswhich takes place. We consider the shown results as representative, but the actual range of 15 –behaviour for given sets of masses and orbits may be wider due to gradients in temperatureand nuclear composition which have not been explored systematically here. To summarize,in the most favorable cases, the nuclear energy release, is comparable to that of typical TypeIa supernovae.In the tidal pinching process, explosive nucleosynthesis is likely to proceed only forwhite dwarfs passing well within the tidal radius (Figure 19). Explosive energy release,as calculated here, appears to be a natural outcome for 0.6 M ⊙ (1.2 M ⊙ ) white dwarfsapproaching moderately massive black holes with β ≥ β ≥
6. Gravitational Waves
A white dwarf approaching the tidal radius will be disrupted in a single flyby. Theresulting stellar debris trail is not compact enough to emit strong gravitational waves afterleaving R p . The detectable gravitational wave signal will therefore have a burst-like behavior,roughly characterized by an amplitude h and a duration t ∼ /f , where h ≈ Gc ¨ QD ∼ GM wd R g c R p D ∼ × − β (cid:18) D (cid:19) − (cid:18) M wd . M ⊙ (cid:19) / (cid:18) R wd cm (cid:19) − M / , , (19)and f ∼ (cid:18) GM BH R (cid:19) / ∼ . β / (cid:18) M wd . M ⊙ (cid:19) / (cid:18) R wd cm (cid:19) − / Hz . (20)LISA will be able to detect gravitational waves of amplitude h ∼ − for burst sources inthe frequency range f ∼ − − − Hz (Danzmann 2003). Gravitational waves from whitedwarf stellar disruption could thus be detectable if M wd ≥ . M ⊙ and the source distance D ≤
10 kpc (Figure 20).The gravitational wave amplitudes h + and h × shown in Figure 21 are calculated in thequadrupole approximation. The reduced quadrupole moments can be written in terms ofthe SPH-particle properties (Centrella & McMillan 1993) I jk = X i m i ( x ji x ki − δ jk r i ) . (21)The second time derivatives, ¨ I jk , can then be expressed in terms of the particle propertiesby simple, direct differentiation. The retarded gravitational wave amplitudes for a distantobserver along the z -axis at distance D are given by D h + = Gc ( ¨ I xx − ¨ I yy ) (22) 16 –and D h × = 2 Gc ¨ I xy . (23)Both duration and amplitudes of the gravitational wave bursts that are produced by thedisruption calculations (Figure 21) are in agreement with the simple estimates given above.
7. Discussion
This paper presents a computational investigation of the mechanical and nuclear evo-lution of white dwarfs passing well within the tidal radius of a moderately massive blackhole. A comprehensive top to bottom approach is adopted: we follow the tidal disruptionand compression leading up to the ignition of the white dwarf; the complex propagation ofthe nuclear energy release through the star; the resultant gravitational wave signal and thesubsequent accretion of the bound debris.This paper has outlined several potentially observable effects. The detection of a pecu-liar, underluminous thermonuclear explosion (Rosswog et al. 2008b; Kasen et al. 2008) ac-companied by a thermal transient signal of predominantly soft X-rays with a peak luminosity L ∼ L Edd = 10 M BH, erg/s, fading within a year would, if detected, be a compelling testi-mony for the existence of a new mechanism by which white dwarf ignition can be achieved.Although the thermodynamical and nuclear energy of the star is sensitive to the initial com-position, we found that thermonuclear ignition is a natural outcome for white dwarfs of allmasses passing well within the tidal radius, with lighter stars requiring deeper penetrationinto the tidal radius due to their lower densities. For simplicity, we have instantiated ourinitial carbon oxygen white dwarf models (M WD ≥ . ⊙ ) as homogeneously mixed starswith a 50% mass fraction of each nucleus. While such internal chemical profiles are likelyaccurately realized in nature in very massive white dwarfs ( ∼ ⊙ ) (Mazzitelli & Dantona1986), for lower masses the gravothermal adjustment of the interior during the cooling phaseproduces oxygen-enhanced stellar cores surrounded by very carbon-rich mantles ( X C ∼ . C ( α, γ ) O rate and the detailsof how convection proceeds (Mazzitelli & Dantona 1986; Salaris et al. 1997; Straniero et al.2003), but this general stratification tendency is well-established. Thus, the disruption of astandard 0.6 M ⊙ white dwarf should produce a highly carbon-enriched remnant atmosphere.Maybe, the recently detected carbon-rich transient SCP 06F6 accompanied by an X-raysignal (Gaensicke et al. 2008) is already the first example for this class of object.The gravitational forces from the central black hole are currently calculated using aPaczy´nski-Wiita pseudo potential but our goal is to incorporate the effects of general rel-ativity, as white dwarfs passing well within the tidal radius cannot be disrupted without 17 –entering the strong relativistic regime. The form of the black hole (Schwarzschild or Kerr?)then has an important quantitative effect, as does (for a rotating Kerr hole) the orientationof the stellar orbit relative to the black hole spin axis. The orbits are then not ellipses, butmay turn through 2 π or even more (Luminet & Pichon 1989). A fluid element, which in thecase of an elliptical orbit would cross the orbital plane just once, may then have two or moretraversals. This opens up the possibility of multiple shocks.For a white dwarf which does not pass close enough to the hole to release much energy ona dynamical timescale, adiabatic cooling would severely reduce the internal radiative contentbefore the debris became translucent (just as a supernova would be optically inconspicuousin the absence of continuing energy injection in the months after the explosion). There wouldbe no transient until, as discussed above, the bound debris fell back onto the hole after t fb .The integrated output from this flare could, in principle, amount to a few per cent of thewhite dwarf’s rest mass, but would probably be significantly less, because most of the debriswould be fed to the black hole far more rapidly than it could be accepted if the radiativeefficiency were high; much would then be swallowed inefficiently or most likely escape inradiatively-driven directed outflow: its ram pressure and subsequent heating could inhibitthe steady accretion that would otherwise be inevitable in any galaxy or globular clusterharboring a moderately massive black hole.We thank Holger Baumgardt, Peter Goldreich, Jim Gunn, Piet Hut, Dan Kasen, Bron-son Messer and Martin Rees for very useful discussions. E. R. acknowledges support fromthe DOE Program for Scientific Discovery through Advanced Computing (SciDAC; DE-FC02-01ER41176). The simulations presented in this paper were performed on the JUMPcomputer of the H¨ochstleistungsrechenzentrum J¨ulich. Oak Ridge National Laboratory ismanaged by UT-Battelle, LLC, for the U.S. Department of Energy under contract DE-AC05-00OR22725. REFERENCES
Alexander, T. 2005, Phys. Rep., 419, 65Balsara, D. S. 1995, J. Comp. Phys., 121, 357Baumgardt, H., Makino, J., & Ebisuzaki, T. 2004, ApJ, 613, 1143Brassart, M., & Luminet, J.-P. 2008, A&A, 481, 259 18 –Benz, W. 1990, Numerical Modelling of Nonlinear Stellar Pulsations Problems and Prospects,269Benz, W., et al. 1990, ApJ, 348, 647Bicknell, G. V., & Gingold, R. A. 1983, ApJ, 273, 749Brodie, J. P., & Strader, J. 2006, ARA&A, 44, 193Carter, B., & Luminet, J.-P. 1983, A&A, 121, 97Centrella, J. M., & McMillan, S. L. W. 1993, ApJ, 416, 719Danzmann, K. 2003, Advances in Space Research, 32, 7, 1233Dearborn, D.S.P., Wilson, J.R. and Mathews, G.J. 2005, ApJ, 630, 309Evans, C. R., & Kochanek, C. S. 1989, ApJ, 346, L13Gaensicke, B.T., Levan, A.J., Marsh, T.R. & Wheatley, P.J. 2008, 2008arXiv0809.2562Gebhardt, K., Rich, R. M., & Ho, L. C. 2002, ApJ, 578 L41Gebhardt, K., Rich, R. M., & Ho, L. C. 2005, ApJ, 634, 1093Gerssen, J., et al. 2002, AJ, 124, 124, 3270Gerssen, J., et al. 2003, AJ, 125, 376Ghez, A. M., Klein, B. L., Morris, M., & Becklin, E. E. 1998, ApJ, 509, 678Greene, J.E & Ho, L.C 2004, ApJ, 610, 722Heber, U., Science 2002, 296, 2344Hillebrandt, W., & Niemeyer, J. C. 2000, ARA&A, 38, 191Hix, W. R., et al. 1998, ApJ, 503, 332Kasen , D., Ramirez-Ruiz, E. & Rosswog, S. 2008, submitted to ApJKepler, S.O. et al., MNRAS 2006, 375, 1315Khokhlov, A. M. & Ergma, E. V. 1986, Soviet Astronomy Letters 12, 152Kobayashi, S., Laguna, P., Phinney, E. S., & M´esz´aros, P. 2004, ApJ, 615, 855 19 –Lodato, G., King, A.R. & Pringle, J.E. 2008, arXiv:0810.1288Luminet, J.-P., & Pichon, B. 1989, A&A, 209, 103Mathur, S., Ghosh, H., Ferrarese, L., & Fiore, F. 2008, arXiv:0807.0422Mazzitelli, I. & Dantona, F., ApJ 1986, 311 762Moehler, S., & Bono, G. 2008, arXiv:0806.4456Morris, J., & Monaghan, J. 1997, J. Comp. Phys., 136, 41Monaghan, J.J. 2002, MNRAS, 335, 843Paczy´nski, B., & Wiita, P. 1980, A&A, 88, 23Phinney, E.S. 1989, IAU Symp. 136: The Center of the Galaxy, 543Portegies Zwart, S. F., et al. 2004, Nature, 428, 724Price, D.J. & Monaghan, J.J., MNRAS 2007, 374, 1347Ramirez-Ruiz, E. & Rosswog, S. 2008, arXiv:0808.3847Reid, I. N. 2005, ARA&A, 43, 247Rees, M. J. 1988, Nature, 333, 523Rosswog, S., et al. 2000, A&A, 360, 171Rosswog, S., Speith, R., & Wynn, G. A. 2004, MNRAS, 351, 1121Rosswog, S., ApJ 2005, 634, 1202Rosswog, S. & Price, D.J., MNRAS 2007, 379, 915Rosswog, S., Ramirez-Ruiz, E., Hix, W. R., & Dan, M. 2008, Computer Physics Communi-cations, 179, 184Rosswog, S., Ramirez-Ruiz, E., & Hix, W. R. 2008, ApJ, 679, 1385Rosswog, S. & Price, D., Springer Lecture Notes in Computational Science and Engineering,in press (2008), arXiv0802.0418RSalaris, M. et al. 1997, ApJ, 486, 413Springel, V. & Hernquist, L. MNRAS 2002, 333, 649 20 –Straniero, O. and Dom´ınguez, I. and Imbriani, G. & Piersanti, L. 2003, ApJ, 583, 878Timmes, F. X. & Swesty, F. D. 2000, ApJS, 126, 501Weidemann, V. 1990, ARA&A, 28, 103Wilson, J. R., & Mathews, G. J. 2004, ApJ, 610, 368
This preprint was prepared with the AAS L A TEX macros v5.2.
21 –Table 1: Summary of the performed runs. M wd and M BH are the masses of the white dwarfand the black hole, respectively, β is the ratio of tidal radius and pericentre distance. Thetype of gravity is indicated by N (Newtonian) or PW (Paczy´nski-Wiita). Column six statesthe number of SPH particles used in the simulation, E burn is the energy generated in burningprocesses. To put this number into context we briefly state the gravitational binding energies(in ergs) of the different white dwarfs: log( E bin , . M ⊙ ) = 49 .
13, log( E bin , . M ⊙ ) = 50 . E bin , . M ⊙ ) = 50 .
98. “Fe” labels the mass in iron-group elements, “expl.” in the commentcolumn indicates that the produced nuclear energy exceeds the WD gravitational bindingenergy.run M wd M BH β grav SPH part. log( E burn ) “Fe” [M ⊙ ] comments1 0.2 1000 12 N 4034050 50.46 0.025 expl.2 0.2 1000 12 PW 4034050 50.44 0.034 expl.3 0.2 1000 12 PW 200452 50.44 Γ = 5 / < − explore BH influence, expl.5 0.2 500 5 PW 100027 49.64 < − explore BH influence, expl.6 0.2 1000 5 PW 100027 49.76 < − explore BH influence, expl.7 0.2 5000 5 PW 100027 49.93 < − explore BH influence, expl.8 0.6 500 5 N 502479 50.68 0.18 expl.9 0.6 500 5 N 502479 50.62 0.13 hot, initial WD10 0.6 1000 0.9 N 1006446 0.00 0. no nuclear burning11 0.6 1000 5 PW 502479 50.43 3 × −
12 0.6 10000 1.5 PW 502479 45.07 < −
13 1.2 100 3.5 N 100027 51.01 0.58 expl.14 1.2 500 2.6 PW 502479 51.16 0.66 expl.15 1.2 1000 1.5 PW 502479 49.63 0 . β is plotted as a function ofthe black hole mass M BH . Note that a black hole of ≫ M ⊙ can swallow all white dwarfswithout first disrupting them (adapted from Luminet & Pichon 1989). 23 –Fig. 2.— Building initial conditions by solving the spherically symmetric Lane-Emden equa-tions for a star of given mass and composition to find a one dimensional density profile. Theparticles are then distributed inside a unit sphere according to a close-packed prescription. Left Panel:
Close-packed particle distributions in a unit sphere. This distribution is thenmapped into the volume of the star that is to be constructed.
Right Panel:
Particle dis-tribution after mapping onto the density profile of a 0.2 M ⊙ white dwarf. Shown are theprojections of the particle positions onto the xy-plane. This particle distribution is thenfinlly ’relaxed’ into its true numerical equilibrium. 24 –Fig. 3.— Mapping between unit sphere and star. We denote quantities referring to the star(unit sphere) with (un-)primed variables. 25 –Fig. 4.— Evolution of a 0.6 M ⊙ white dwarf passing a 1000 M ⊙ black hole with a penetrationfactor β of only 0.9. The snapshots show t= 0.34, 3.43, 6.86, 10.29, 13.72, 17.15, 20.58 and24.01 s after the simulation start in the upper, and 138.24 and 380.53 s in the lower panel. 26 –Fig. 5.— A 0.2 M ⊙ white dwarf (modeled with more than 4 × SPH particles) approachinga 10 M ⊙ black hole on a parabolic orbit with pericenter distance R p = R τ /
12 is distorted,spun up during infall and then tidally disrupted. Shown are density cuts at various instantsalong the orbital (xy-) plane. Color bar gives the amplitude of log ρ = [log ρ max , log ρ min ] incgs units: [4 . , .
3] for t =6.2 s, [3 . , .
4] for t =6.8 s, and [0 , .
4] for t =10.3 s. 27 –Fig. 6.— Tidal deformation of the white dwarf before and after passage through pericenter,as the star attains its maximum degree of compression (same simulation as referred to inFigure 5). The panels in the upper rows show cuts of temperature (in units of 10 K) anddensity (in cgs units) through the orbital (xy-) plane. Color bar gives the values of T =[ T min , T max ] (log ρ = [log ρ min , log ρ max ]): [0 , . , . t =6.8 s, [0 , . , . t =7.0 s, and [0 , . , . t =7.2 s. The panels in the lower two rows showthe temperature (in units of 10 K) and column density (in cgs units) distributions in thexz-pane (averaged along the y-direction). Color bar gives the values of T = [ T min , T max ](log Σ = [log Σ min , log Σ max ]): [0 , . , . t =6.8 s, [0 , . , . t =7.0s, and [0 , . , . t =7.2 s. The dimension of the bar scale is 10 cm. 28 –Fig. 7.— Evolution of the entropy ε of the central portion of the disrupted white dwarfjust after it attains its maximum degree of compression (same simulation as referred to inFigures 5 and 6). This compression is halted by a shock, raising the matter to a higheradiabat. The panels show entropy (in cgs units) cuts through the orbital (xy-) plane. 29 –Fig. 8.— Evolution of the abundances during the disruption and ignition of a 0.2 M ⊙ whitedwarf passing a 10 M ⊙ black hole (same simulation as referred to in Figures 5–7). 30 –Fig. 9.— Differential mass distributions in specific energy for the 0.2 M ⊙ white dwarf debris(same simulation as referred to in Figures 5–8). 31 –Fig. 10.— Column density (in cgs units) and temperature (in units of 10 K) distributionsin the orbital plane of the bound debris minutes after disruption. The most tightly bounddebris would transverse an elliptical orbit with major axis ∼ R g before returning to R ≈ R τ (same simulation as referred to in Figures 5–9). 32 –Fig. 11.— The rate at which the 0.2 M ⊙ white dwarf debris returns to the vicinity of theblack hole (same simulation as referred to in Figures 5–10). 33 –Fig. 12.— The importance of relativistic effects from the central black hole in determiningthe behavior of white dwarfs passing within the tidal radius. The trajectory of the center ofmass of a 0.2 M ⊙ white dwarf approaching a 10 M ⊙ black hole on a parabolic orbit withimpact parameter β are shown using Newtonian gravity (N; black) and the pseudo-Newtonianrelativistic potential of Paczy´nski-Wiita (PW, blue). Each time, the star is modeled withmore than 10 partices. 34 –Fig. 13.— A 0.2 M ⊙ white dwarf approaching black holes of various masses on a parabolicorbit with β = 5. The panels in the show density (in cgs units) cuts through the orbital(xy-) plane. Color bar gives the values of log ρ = [log ρ min , log ρ max ]. For M BH = 10 M ⊙ ,log ρ = [4 . , .
3] for t= 14 . . , .
4] for t= 15 .
4, and [3 . , .
8] for t= 16 .
1. For M BH =5 × M ⊙ , log ρ = [4 . , .
3] for t= 15 . . , .
65] for t= 16 .
4, and [3 . , .
8] for t= 17 . M BH = 10 M ⊙ , log ρ = [4 . , .
4] for t= 15 . . , .
65] for t= 16 .
3, and [3 . , .
6] fort= 17 .
2. For M BH = 5 × M ⊙ , log ρ = [4 . , .
4] for t= 15 . . , .
75] for t= 16 .
0, and[3 . , .
8] for t= 16 .
3. 35 –Fig. 14.— The importance of the central black hole’s mass in determining the behavior ofa 0.2 M ⊙ white dwarf passing within β = 5 (same simulations as referred to in Figure 13).The evolution of the compressed, and tidally disrupted white dwarf in the ρ − T plane. Thehottest 10% of the particles are identified and their average temperature (in units of 10 K)is plotted as a function of their average density (in cgs units). These trajectories alwaysstart cold and dense (right lower corner) and become hot and during the black hole flyby. Ifthe time scale on which the white dwarf can react (dynamical time scale t G ) is longer thanthe burning time scale ( t b , see Eq. (18)), the star cannot expand rapidly enough to quenchburning. 36 –Fig. 15.— Energy generated in nuclear burning (in units of the star’s binding energy) asa function of time (same simulations as referred to in Figures 13 and 14). Here time ismeasured in units of the dynamical timescale ( t G ) of the initial 0.2 M ⊙ white dwarf. Inaddition, the time axis has been shifted so that the maximum E b occurs at t = 0. 37 –Fig. 16.— A 1.2 M ⊙ white dwarf (modeled with more than 5 × SPH particles) approachinga 500 M ⊙ black hole on a parabolic orbit with pericenter distance r min = r T / .
2. Shown aretemperature (in units of 10 K) and surface density (in cgs units) cuts at various instantsalong the orbital (xy-) plane. 38 –Fig. 17.— Evolution of the abundances during the disruption and ignition of a 1.2 M ⊙ whitedwarf approaching a 500 M ⊙ black hole (same simulation as referred to in Figure 16). 39 –Fig. 18.— Comparison of the average properties of the hottest 10% of the SPH particles foran initially cold and warm star (runs 8 and 9 in Table 1). 40 –Fig. 19.— The evolution of the compressed, and tidally disrupted white dwarfs of variousmasses in the ρ − T plane: A 0.2 He M ⊙ white dwarf approaching a 10 M ⊙ black hole with β = 12; a 0.6 C/0 M ⊙ white dwarf approaching a 10 M ⊙ black hole with β = 1 . M ⊙ white dwarf approaching a 10 M ⊙ black hole with β = 1 .
5. The hottest 10%of the particles are identified and their average temperature (in units of 10 K) is plottedas a function of their average density (in cgs units). These trajectories always start coldand dense (right lower corner) and become hot and during the black hole flyby. For the 1.2 M ⊙ and 0.6 M ⊙ cases, the time scale on which the white dwarf can react is similar to theburning time scale and the star expands rapidly enough to quench explosive energy release. 41 –Fig. 20.— Frequency and amplitude of gravitational radiation produced by the disruptionof white dwarfs approaching an intermediate mass black hole, assuming β = 1 and that thecluster hosting the black hole is at D = 8kpc. 42 –Fig. 21.— Gravitational wave amplitudes. Top panel: gravitational radiation producedby the disruption of a 0.2 M ⊙ white dwarf approaching a 10 M ⊙ black hole with β = 12. Bottom panel: gravitational radiation produced by the disruption of a 0.6 M ⊙ white dwarfapproaching a 10 M ⊙ black hole with β = 1 ..