The onset of f-mode secular instabilities in such fizzlers is affected strongly by the h(mc), ... For a broad range of fizzler equations of state and the core h(mc),.
The Astrophysical Journal, 616:1095–1101, 2004 December 1 # 2004. The American Astronomical Society. All rights reserved. Printed in U.S.A.
f-MODE SECULAR INSTABILITIES IN DELEPTONIZING FIZZLERS James N. Imamura Institute of Theoretical Science and Department of Physics, University of Oregon, Eugene, OR 97403
and Richard H. Durisen Department of Astronomy, Indiana University, Bloomington, IN 47405 Receivved 2004 April 28; accepted 2004 Auggust 7
ABSTRACT Fizzlers are intermediate states that may form between white dwarf and neutron star densities during the collapse of massive rotating stars. This paper studies the gravitational radiation reaction (GRR) driven f-mode secular instabilities of fizzlers with angular momentum distributions h(mc ) appropriate to the core collapse of massive rotating stars, where h is the specific angular momentum and mc is the cylindrical mass fraction. For core collapses that maintain axial symmetry, the h(mc ) of the remnant reflects the conditions in the precollapse stellar core, and, thus, the h(mc ) will resemble that of a uniformly rotating star supported by the pressure of relativistically degenerate electrons. Such an h(mc ) concentrates most angular momentum toward the equatorial region of the object. The onset of f-mode secular instabilities in such fizzlers is affected strongly by the h(mc ), whereas instability depends only weakly on compressibility. For a broad range of fizzler equations of state and the core h(mc ), the f-mode secular instability thresholds drop to T =jW j 0:034 0:042, 0.019–0.021, and 0.012–0.0135, for m ¼ 2, 3, and 4, respectively. These same thresholds with the Maclaurin spheroid h(mc ) are T =jW j ¼ 0:13 0:15, 0.10–0.11, and 0.08–0.09, respectively. The growth times gw for GRR-driven m ¼ 2 modes are long. For fizzlers with specific angular momentum J =M 1:5 ; 1016 cm2 s1 and T =jW j P 0:24 (c P 1014 g cm3 ), gw > 400 s. For these fizzlers, gw 3 de , the deleptonization timescale, and GRR-driven secular instabilities will not grow along a deleptonizing fizzler sequence except, possibly, at T =jW j near the dynamic bar mode instability threshold, T =jW j 0:27. Subject headingg s: hydrodynamics — instabilities — stars: rotation — supernovae: general
1. INTRODUCTION
Hayashi et al. 1998, 1999; Lai 2001). In the former case, the deleptonizing fizzler loses pressure support, contracts, spins up, and, under certain conditions, becomes unstable to dynamic bar mode instabilities, producing a burst of GW radiation (Imamura & Durisen 2001; Imamura et al. 2003). In the latter case, secular instability leads quietly to the formation of a dynamically stable bar that may persist for thousands of dynamical times as a source of continuous GW radiation (Lai & Shapiro 1995; Lai 2001). Fizzler bars can, in principle, persist until the bar sheds enough angular momentum to reach a state stable to rotationdriven instabilities, easily long enough to make them attractive targets for LIGO (Lai & Shapiro 1995; Lai 2001; Imamura et al. 2003). In this paper, we investigate the f-mode secular instabilities of fizzlers further. The f-modes considered here are the analogs of the Kelvin modes of incompressible fluids (cf. Chandrasekhar 1969). The modes have azimuthal dependence exp (im), where m is a constant and is the azimuthal angle. An f-mode secular instability may set in whenever more than one equilibrium configuration exists for a given angular momentum J or a given circulation. In this case, one equilibrium state can evolve to another if the second state has lower total energy and an appropriate dissipation mechanism exists. Secular instabilities grow on the dissipation timescale that, generally, is much longer than the dynamic timescale for the configuration. The f-mode secular instabilities driven by GRR and viscous dissipation have been studied. For fizzlers, the relevant dissipation mechanism is GRR. The theory of f-mode secular instabilities in rotating, self-gravitating, incompressible fluids is well in hand (e.g., see Chandrasekhar 1969). For more realistic astrophysical fluid
The gravitational wave (GW) signature of the core collapse of massive rotating stars has generated renewed interest recently with the emergence of several ground-based Michelson interferometer GW observatories (e.g., the Laser Interferometer Gravitational-Wave Observatory [LIGO], Abramovici et al. 1995; TAMA-300, Tsubono et al. 1997; GEO-600, Danzmann et al. 1995; and Virgo, Bradaschia et al. 1990). The suggestion that the core collapse of some massive rotating stars could produce significant GW emission is not new. It was first explored in the 1970s by Shapiro & Lightman (1976; see also Tohline 1980). The idea fell out of favor in the 1980s, when it was found that core collapse in typical massive rotating stars was only a weak source of GW emission (Finn & Evans 1990). The idea was revived in the 1990s with the suggestion that cores of massive stars may contain enough angular momentum to make them susceptible to f-mode instabilities (Lai & Shapiro 1995; Hayashi et al. 1998, 1999; Heger et al. 2000; Fryer & Heger 2000; Imamura & Durisen 2001; Fryer et al. 2002; Imamura et al. 2003). Core collapse in a rapidly rotating massive star hangs up before nuclear density is reached if the core contains sufficient angular momentum, where ‘‘sufficient’’ means that the centrifugal force overcomes the effects of gravity before nuclear density is reached. Such objects are referred to as ‘‘fizzlers.’’ The long-term evolution of a fizzler is driven by deleptonization and cooling (Imamura & Durisen 2001; Imamura et al. 2003) and/or through nonaxisymmetric gravitational radiation reaction (GRR)–driven secular instabilities (Lai & Shapiro 1995; 1095
1096
IMAMURA & DURISEN
configurations, the theory is not as well developed. Most work has centered on polytropes for which it has been shown that the properties of the f-mode instabilities depend on both the compressibility and angular momentum distribution of the object. The dependence on the angular momentum distribution was first pointed out by Imamura et al. (1985) and Managan (1985). Imamura et al. (1995) showed that, for angular momentum distributions that produced strong differential rotation, the secular stability limit for the m ¼ 2 bar mode could fall well below the Maclaurin spheroid limit T =jW j ¼ 0:14, where T is the rotational kinetic energy and W is the gravitational energy of the fizzler (see also Imamura & Durisen 2001; Yoshida et al. 2002; Karino & Eriguchi 2002). Further, analyses that take general relativistic effects into account show that the m ¼ 2 secular instability T =jW j threshold drops below 0.14 in rotating neutron stars (Stergioulas & Friedman 1998). These results are not recognized by many workers, who still often quote T =jW j 0:14 as the bar mode secular stability limit. In this paper, we use a Lagrangian variational principle (LVP; Friedman & Schutz 1978; Durisen & Imamura 1981; Clement 1979, 1989; Imamura et al. 1985, 1995; Imamura & Durisen 2001) to locate instability thresholds and the Lagrangian normal mode equation (Lynden-Bell & Ostriker 1967) to calculate approximate eigenfunctions and eigenvalues for fizzlers. We then determine growth rates for the GRR-driven f-mode secular instabilities using the approximate eigenfunctions and eigenvalues. Our techniques here are similar to those of a previous paper (Imamura & Durisen 2001), but here we provide a more thorough treatment of a realistic fizzler angular momentum distribution. The remainder of our paper is organized as follows. In x 2, the details of the physical models are discussed. In x 3, the results of our calculations are presented and discussed. In x 4, our principal conclusions are summarized. 2. PHYSICAL PICTURE 2.1. Fizzlers and f-Mode Instabilities Massive stars at the ends of their nuclear lifetimes have Fe-Ni cores of mass Mc 1:25 2:05 M (Timmes et al. 1996), central densities c 109 g cm3, and material properties Ye 0:4 0:49 and Sb =k 1 2 (see Hashimoto 1995). Here Ye is electron fraction per baryon and Sb is the entropy per baryon. The dynamic collapse of the core is triggered by electron captures and/or cooling of the hot lepton-rich material through the photodissociation of iron nuclei. The inner core collapses first while the outer core, to first approximation, remains stationary. The inner core collapses until either nuclear density is reached or it becomes centrifugally supported. After collapse, the core rebounds to an equilibrium configuration determined by its mass and angular momentum (Fryer & Heger 2000; Fryer & Warren 2004). During this collapse and settling, the core maintains a relatively high Ye (>0.3). After the inner core collapse, the outer core accretes onto the inner core on a timescale of 0.01–0.1 s. Imamura & Durisen (2001) argued that fizzlers are stable to secular and dynamic instabilities during the collapse and accretion phases. After inner core collapse and accretion, the fizzler evolves either through deleptonization and cooling and/or through the onset of GRR-driven secular instabilities. During this phase of evolution, Imamura & Durisen (2001) argued that, for any Ye , the instability growth time gw is greater than the deleptonization timescale de if c P nuc , i.e., for fizzlers. Only if c k nuc does gw < de occur during deleptonization. However,
Vol. 616
at high c (under high-kT conditions), dissipation due to bulk viscosity is efficient, and the GRR-driven secular instability may be prevented from growing (Lindblom & Detweiler 1977; Yoshida & Eriguchi 1995; however, see Lai & Shapiro 1995; Lai 2001). Imamura & Durisen (2001) concluded that fizzlers were neither secularly nor dynamically unstable to nonaxisymmetric modes during and immediately following the core collapse phase. An important point, however, is that Imamura & Durisen considered precollapse cores with the angular momentum distribution of a uniform density, uniformly rotating spheroid, i.e., a Maclaurin spheroid (see also Lai & Shapiro 1995; Lai 2001). A more appropriate angular momentum distribution for the collapsing core is that of a uniformly rotating spheroid supported by degenerate electrons (Liu & Lindblom 2001; New & Shapiro 2001). Here we investigate fizzlers with this angular momentum distribution. 2.2. Stability Analysis The GRR-driven f-mode secular instability is excited when the retrograde m-mode (in the fluid frame) is dragged forward by the rotation of the body into prograde motion with respect to an observer at infinity. This occurs at the neutral points along sequences of rotating equilibrium models, i.e., at the points along the equilibrium sequences where the retrograde m-mode’s eigenfrequency passes through zero as measured in the stationary reference frame. In this paper, equilibrium model sequences are produced using a modified version of the Hachisu (1986) selfconsistent-field method (Imamura & Durisen 2001). As shown by Friedman & Schutz (1978), the neutral points are located where the canonical energy of the perturbation Z Z Ec ¼ 0:5! 2 x = AðxÞ d 3 x þ 0:5 x = CðxÞ d 3 x ð1Þ passes through 0. Here, x is the Lagrangian displacement vector, ! is the eigenfrequency, and AðxÞ and CðxÞ are linear operators (Schutz 1979). The Ec forms the basis for an LVP (see Friedman & Schutz 1978). We locate neutral points using the LVP. See Imamura et al. (1995) for a detailed description of our numerical method. Here we give a brief discussion of our implementation of the LVP. We evaluate the canonical energy Ec using x of the form m1 exp (im) ð2Þ x / f A($); i½A($) þ B($); 0g $=Req (Bardeen et al. 1977, hereafter BFSS77), where A($) and B($) are arbitrary functions, $ is the radial coordinate, and Req is the equatorial radius of the fizzler. For theRpurpose of the neutral point analysis, we normalize x so that x x d 3 x ¼ 1, when integrated over the fizzler. The x may contain a component that fails to conserve circulation, the so-called ‘‘trivials’’ (e.g., see Friedman & Schutz 1978). Trivial perturbations simply relabel fluid elements and so do not correspond to physical perturbations. They do, however, affect Ec . We eliminate trivials by explicitly forcing the trial x to conserve circulation. Generalizing the expression for the circulation constraint in BFSS77 leads to ! ½$ðA 0 þ B 0 Þ þ mB $A 0 0:5 m þ
$(ln ) 0 ½(1 m)A mB þ mB ¼ 0: ð3Þ We find a best A($) and B($) in the neighborhood of neutral points in the following manner.
No. 2, 2004
f -MODE INSTABILITIES IN DELEPTONIZING FIZZLERS 1097 j We represent A($) as a power law in $, Aj ($) ¼ ($=Req ) j,Aj ($) ¼ To$=R evaluate gw , eigenfunctions and eigenvalues are needed eq ) where j is an arbitrary integer, and calculate the function Bj ($) away from the neutral points. We find approximate eigenvalues using the circulation constraint with ! ¼ 0. The Aj and Bj are and eigenfunctions using the Lagrangian normal mode equathen used to construct a Lagrangian displacement vector, j . tion (Lynden-Bell & Ostriker 1967), ; General x are then formed as linear combinations of the j Z Z Z P x ¼ Nj¼1 aj j and Ec takes the form 2 3 3 x = AðxÞ d x i! x = BðxÞ d x x = CðxÞ d 3 x ¼ 0: ! Ec ¼ 0:5
N X
aj ak Cjk
j¼1; k¼1
N . X
aj ak Fjk ;
j¼1; k¼1
where Z Cjk ¼
j C ðk Þ d 3 x
ð5Þ
j k d 3 x:
ð6Þ
and Z Fjk ¼
We minimize Ec with respect to the ai . This yields the N constraints N X aj Ci j þ Cji Fi j þ Fji Ec ¼ 0;
i ¼ 1 N:
ð7Þ
j¼1
The above eigenvalue problem is solved using standard techniques for N ¼ 7. The resultant eigenvectors, when inserted into Ec , place upper limits on the neutral point locations in terms of T =jW j. Despite the rather restrictive form assumed for A($), i.e., a power-law dependence on $ and independence of z, the chosen form for A($) likely allows us to locate neutral point locations to within two digits in T =jW j based on our experience with earlier polytrope calculations (Imamura et al. 1985, 1995). The growth time for a GRR-driven mode is given by E c ð8Þ gw ¼ 2 ; E˙ gw where E˙ gw , the gravitational radiation loss rate, is found from the quadrupole gravitational radiation torque in the slow motion approximation (e.g., Finn & Evans 1990). For rotation about the z-axis and pure m eigenfunctions, 1G E˙ gw ¼ 5 c5
d 3 Ii j d 3 Ii j ; d 3t d 3t
ð9Þ
where Ii j is the trace reduced quadrupole moment tensor with components Ix x ¼ I 0 ¼ Iy y ; Ix y ¼ iI 0 ¼ Iy x ; and all other Ii j ¼ 0. Here
Z 1 $ 3 d$ dz exp (i!t); I0 ¼ 2 where 1 is the magnitude of the density eigenfunction.
ð13Þ
ð4Þ
ð10Þ ð11Þ
ð12Þ
See Schutz (1979) for the definition of the linear operator B(x). The normal mode equation is solved for x and ! as described briefly below; for a more detailed discussion of our numerical technique, see Imamura & Durisen (2001) and references therein. In the absence of dissipation, fizzlers are not dynamically unstable for T =jW j P 0:27 and ! is purely real. With dissipation, ! becomes complex. However, the imaginary component is small compared to the real frequency for secular instability. Purely real ! is then an appropriate approximation to take. A complication arises because x and ! must simultaneously satisfy the circulation constraint and the normal mode equation. Consequently, an iterative approach must be used to determine the eigenfunctions and eigenvalues. We use the fol- j $=R an inlowing scheme: (1) We set Aj ($) / ($=Req ) j,Awhere j ð$Þ /j is eq teger, and guess a trial !. We then calculate Bj ($) from the circulation constraint and form j . (2) We use the j to evaluate the normal mode equation and find !. (3) We compare the guessed and calculated values for !. If they agree to within one part in 104, then the j is deemed to be self-consistent. If not, the trial ! is adjusted using the Newton-Raphson method. Steps 1 through 3 are repeated until self-consistency is attained. (4) Steps 1 to 3 are repeated for j ¼ 0 to 10. We adopt the powerlaw j that minimizes the eigenfrequency. 3. RESULTS 3.1. Fizzler Formation Let h be the specific angular momentum and mc the cylindrical mass fraction about the rotation axis. Fizzler formation depends on the form of the angular momentum distribution h(mc ), as well as on the fizzler total angular momentum (Fryer & Heger 2000; Imamura & Durisen 2001). For consistency with earlier works, we parameterize the angular momentum distributions as follows. A spherical, nonrotating polytrope with index n 0 is calculated. Then, assuming that the polytrope rotates uniformly and does not deform, the specific angular momentum as a function of the cylindrical mass fraction mc is calculated. This defines the specific angular momentum distribution h(mc ) as a function of the polytropic index n 0 . For a uniformly rotating sphere supported by the pressure of relativistically degenerate electrons (the initial precollapse core), h(mc ) is defined by n 0 ¼ 3, and for uniform density spheroids, the Maclaurin spheroids, h(mc ) is defined by n 0 ¼ 0 (e.g., Bodenheimer & Ostriker 1973; Pickett et al. 1996). The critical angular momentum Jf for fizzler formation is defined such that when J > Jf , Ye ¼ 0:4 objects have central densities c P 1:1 ; 1014 g cm3. This density limit is chosen because it is the at which the Lattimer & Swesty (1991) equation of state (EOS) stiffens. For smaller Ye , the Jf drop to smaller angular momentum. For M ¼ 1:2, 1.4, and 1.6 M, Jf 1:3 ; 1049 , 2:4 ; 1049 , and 4 ; 1049 g cm2 s1, respectively, for n 0 ¼ 3 fizzlers. The models are rapidly rotating with T =jW j ¼ 0:077, 0.11, and 0.15, respectively. The limiting Jf for n 0 ¼ 3 are higher than for n 0 ¼ 0 fizzlers, where the Jf are only 1049 and 2 ; 1049 g cm2 s1 for 1.2 and 1.6 M fizzlers,
1098
IMAMURA & DURISEN
Fig. 1.—A 1.38 M n 0 ¼ 0 fizzler with J ¼ 1:74 ; 1049 g cm2 s1. The fizzler has c ¼ 1014 g cm3, Req ¼ 93:1 km , T =jW j ¼ 0:111, and central angular frequency 0 ¼ 2419 rad s1. The contour levels are =c ¼ 0:9, 0.75, 0.5, 0.25, 0.1, 0.01, 0.001, 0.0001, 0.00001, 0.
respectively (Imamura et al. 2003). This is easily understood. The n 0 ¼ 3 h(mc ) concentrates more J toward the equatorial radius of the core than does the n 0 ¼ 0 h(mc ), so the central region of the core collapses more freely to higher density. This increases the J required to halt the collapse before nuclear density is reached. The density structure, mass distribution mc ($), and rotational angular frequency ($) as a function of cylindrical radius $ for 1.4 M n 0 ¼ 0 and 3 fizzlers for J near Jf are shown in Figures 1 and 2, respectively. The effect of h(mc ) is seen clearly in the structures of the fizzler envelopes. The inner 50% of the mass is distributed roughly spherically for the fizzlers. Outside the inner region, an extensive low- Keplerian envelope forms in the n 0 ¼ 3 fizzler. The structure of the Keplerian envelope of the n 0 ¼ 3 fizzler is uncertain, however. At low , the fizzler EOS becomes poorly known. For 107 g cm3, the EOS is taken to be an n ¼ 1:5 polytrope with K set so that the pressure is the same as for the fizzler EOS at 107 g cm3. The second-to-last contour is =c ¼ 105 and the outer contour is ¼ 0, the surface.
Fig. 2.—A 1.42 M n 0 ¼ 3 fizzler with J ¼ 2:51 ; 1049 g cm2 s1. The fizzler has c ¼ 1014 g cm3, Req ¼ 770 km , T =jW j ¼ 0:106, and central angular frequency 0 ¼ 1320 rad s1. The contour levels are =c ¼ 0:9, 0.75, 0.5, 0.25, 0.1, 0.01, 0.001, 0.0001, 0.00001, 0.
The effect of compressibility is demonstrated using polytropic models. For polytropic indices n ¼ 3=2 and 5=2, the neutral points for m ¼ 2, 3, and 4 fall at T =jW j ¼ 0:029, 0.025, and 0.020 and T =jW j ¼ 0:033, 0.023, and 0.017, respectively. The thresholds fall with n, but not greatly. Implicit in this result is that the form of the angular velocity distribution ($) for given h(mc) does not strongly affect the instability thresholds. This is interesting because the equilibrium structures for polytropes depend strongly on n and n 0 . The structures of polytropes with n much less than n 0 are quite different from those with n roughly n 0 or greater than n 0 . For an n = 3=2 polytrope with n 0 = 3, the ($) distribution increases over the central region of the polytrope, reaches a peak, and then falls off, forming a roughly Keplerian envelope in the outer region (e.g., Liu & Lindblom 2001; New & Shapiro 2001). Objects with typical fizzler properties will thus have T =jW j well beyond the neutral points. Furthermore, the neutral points in Table 1 are at lower T =jW j than those of many of the core collapse remnants found by Fryer & Warren (2004) in TABLE 1 Neutral Points for n 0 ¼ 3 Fizzlers
3.2. Secular Instability The neutral points for n 0 ¼ 3 fizzlers fall at much lower T =jW j than for n 0 ¼ 0 fizzlers, as first reported by Imamura & Durisen (2001), who analyzed low-mass, Mc < 1 M, n 0 ¼ 3 fizzlers with Ye ¼ 0:2 and 0.3. We extend Imamura & Durisen’s results by determining the stability properties of n 0 ¼ 3 fizzlers for parameters that include the expected fizzler mass range, Mc ¼ 1 2 M (see Table 1). The quantity 12 in Table 1 is defined as 12 ¼ c =ð1012 g cm3 Þ. The m ¼ 2 instability threshold falls at T =jW j 0:034 0:042 for the Ye and c considered, far below the typical threshold for n 0 ¼ 0 fizzlers of 0.14 (see Imamura & Durisen 2001).
Vol. 616
Ye
12
m=2
m=3
m=4
m=5
0.4....................
100 10 1 100 10 1 100 10 1
0.0375 0.0389 0.0423 0.0363 0.0358 0.0377 0.0351 0.0339 0.0338
0.0197 0.0198 0.0214 0.0196 0.0193 0.0198 0.0199 0.0194 0.0195
0.0119 0.0126 0.0135 0.0126 0.0125 0.0125 0.0130 0.0127 0.0129
0.0088 0.0090 0.0093 0.0089 0.0089 0.0089 0.0092 0.0087 0.0087
0.3....................
0.2....................
No. 2, 2004
f -MODE INSTABILITIES IN DELEPTONIZING FIZZLERS
Fig. 3.—GRR timescales gw for Ye ¼ 0:4, n 0 ¼ 3 fizzlers and fixed c . The curves for c ¼ 1012 g cm3 are marked with bird’s feet, for c ¼ 1013 g cm3 with open triangles, and for c ¼ 1014 g cm3 with filled triangles. Top: The gw in units of s. The solid line segments are for gw > 0 (growth) and the dashed line segments are for gw < 0 (damping). Middle: The masses Mc in M for the fizzlers shown in the top panel. Bottom: The specific angular momenta J =Mc for the fizzlers shown in the top panel.
their detailed numerical study of the collapse of massive rotating stars. Fizzlers may thus be susceptible to GRR-driven secular instabilities. A key issue, then, is how fast GRR-driven instabilities grow. To find the growth rate, we determine the eigenfunctions and eigenvalues of m ¼ 2 barlike modes for n 0 ¼ 3 fizzlers that are then used to find gw (see x 2.2). To test the accuracy of the method, we consider an uniformly rotating M ¼ 1:39 M polytropic star with EOS defined by K ¼ G ; 1012 and n ¼ 1, parameters chosen so that the star is the same as the neutron star studied by Yoshida & Eriguchi (1995). For T =jW j ¼ 0:095, the m ¼ 3 mode is unstable. Our growth rate is 1 ¼ 6:6. Yoshida & Eriguchi’s Figure 1 gives a growth log gw 1 ¼ 6:2, a factor of 2.5 smaller than our result. rate of log gw The gw values we compute are within a factor of a few of the precise values. This is sufficient for our subsequent arguments. The difference in the gw values is probably due to inaccuracies in our eigenfunctions. In Figures 3 and 4, the m ¼ 2 gw , Mc , and J are shown for some of the fizzler sequences used to determine the neutral point locations given in Table 1. The gw are long, increase with decreasing c , and are fairly insensitive to Ye for given c . Overall, the gw > 106 s for T =jW j < 0:12 even when c ¼ 1014 g cm3. Further, we note that from our comparison with Yoshida & Eriguchi (1995) that the calculated gw are probably lower limits to the instability growth time. Viscous dissipation increases the growth times and may damp the growth of the GRR-driven secular instability if it is efficient enough (Lindblom & Detweiler 1977; Detweiler & Lindblom 1977). We consider these issues further in the next
1099
Fig. 4.—GRR timescales gw for Ye ¼ 0:2, 0.3, and 0.4, n 0 ¼ 3 fizzlers and c ¼ 1014 g cm3. The curves for Ye ¼ 0:4 are marked with bird’s feet, for Ye ¼ 0:3 with open triangles, and for Ye ¼ 0:2 with filled triangles. Top: The gw in units of s. The solid line segments are for gw > 0 (growth) and the dashed line segments are for gw < 0 (damping). Middle: The masses Mc in M for the fizzlers shown in the top panel. Bottom: The specific angular momenta J =Mc for the fizzlers shown in the top panel.
section, when we investigate a deleptonizing fizzler as it evolves to high T =jW j. 3.3. DeleptonizinggFizzlers Using an approach similar to that in Imamura & Durisen (2001) for n 0 ¼ 0 fizzlers, we model the evolution to high T =jW j of a deleptonizing 1.4 M , n 0 ¼ 3 fizzler with J 4 ; 1049 g cm2 s1. We approximate the deleptonization evolution using a set of fizzlers with Ye ¼ 0:4, 0.35, 0.3, and 0.25. The properties of the equilibrium models are given in Table 2. The fizzlers have c < nuc except for Ye ¼ 0:25. The Mc and J for the fizzlers in the sequence are only roughly the same because the numerical equilibrium code holds both the average density in the central cell and the ratio of the polar-toequatorial radii Rp =Req fixed as a model is converged on a finite grid. Because the average density of the central cell depends on the numerical resolution and Rp =Req is based on the ratio of integers, it is not possible, in general, to produce a sequence of models with precisely the same Mc and J. The number of cells used for each computation is given in Table 2 and ranges from Jreq ¼ 128 to 966. The Ye ¼ 0:30 fizzler is calculated for Jreq ¼ 512 and 966. Although the two models are not based on precisely the same parameters, their global properties are the same to within 5%. The GRR-driving timescales gw and oscillation frequencies for the four models are given in Table 3. The gw are much longer than de , which is expected to be on the order of seconds (Imamura & Durisen 2001). If we extrapolate gw to
1100
IMAMURA & DURISEN
Vol. 616
TABLE 2 Deleptonizing Fizzler Equilibrium Models
Ye
Jreq
(1012 g cm3)
T/|W |
W (1052 ergs)
M (M)
J (1049 g cm2 s1)
0
R eq ( km)
Rp /R eq
0.4................... 0.35................. 0.3................... 0.3................... 0.25.................
128 512 512 966 900
1.05 32.6 110 103 262
0.0814 0.150 0.194 0.193 0.242
2.04 5.70 6.67 6.69 12.16
1.398 1.439 1.380 1.383 1.437
3.94 3.98 4.16 4.10 3.95
144.7 1137 2432 2317 4011
1944 1814 2263 2174 2100
0.127 0.031 0.014 0.015 0.007
T =jW j 0:27, we find that gw 100 s and gw may be comparable to de , given the uncertainty in our calculated gw . However, because viscous effects can suppress the growth of the GRR-driven modes (Lindblom & Detweiler 1977; Detweiler & Lindblom 1977), there is even a question of whether the GRRdriven f-modes are unstable. GRR-driven modes can grow only if gw T , the viscous damping timescale (Lindblom & Detweiler 1977; Detweiler & Lindblom 1977). In the vicinity of T =jW j ¼ 0:27 for c 1014 g cm3, the material temperature is high, kT > 5 MeV, and bulk viscosity dominates shear viscosity. The energy loss rate due to bulk viscosity is given by E˙ ¼
Z
d 3 x
ð14Þ
(Ipser & Lindblom 1991), where is the kinematic bulk viscosity coefficient and ¼ : = v:
ð15Þ
The for fizzler material is not known. For dense matter ( > nuc ), bulk viscosity coefficients have been calculated by Sawyer (1989) and Haensel & Shaeffer (1992) for the modified Urca process and the direct Urca process, respectively, for material transparent to neutrinos. At the high temperatures associated with fizzlers, dense material is not transparent to neutrinos. For this case, Lai & Shapiro (1995; see also Lai 2001) estimated bulk viscosity coefficients that were many orders of magnitude smaller for both the direct Urca and modified Urca processes. Evaluating E˙ using our approximate eigenfunctions and eigenvalues and the published kinematic bulk viscosity coefficients, we find that E˙ 3 E˙ gw for the case E˙ T in which the material is transparent to neutrinos and E˙ gw for the case in which the material is opaque to neutrinos. GRR-driven f-modes may thus grow near T =jW j 0:27, where gw may be comparable to de . However, this conclusion depends on the estimates for the bulk viscosity of fizzler material. The n 0 ¼ 3 deleptonizing fizzler may be more stable to the dynamic barlike instability than n 0 ¼ 0 fizzlers. The Ye ¼
0:25 n 0 ¼ 3 fizzler in Table 2 has T =jW j 0:24. Imamura & Durisen (2001) showed that Ye ¼ 0:25 n 0 ¼ 0 fizzlers had T =jW j 0:27. We do not test the Ye ¼ 0:25 fizzler for dynamic barlike instabilities because of the poor numerical resolution of the model. For larger Ye (lower T =jW j), however, the n 0 ¼ 3 fizzlers are stable to dynamic barlike instabilities. 4. CONCLUSIONS We studied the f-mode secular stabilities in fizzlers for an angular momentum distribution (n 0 ¼ 3) appropriate to stellar core collapse. The n 0 ¼ 3 angular momentum distribution is that of a uniformly rotating sphere supported by the pressure of relativistic degenerate electrons. We located the neutral points for a range of EOSs and calculated the growth rates for f-mode secular instabilities excited by the coupling of the star and the gravitational radiation wavefield. The secular instability thresholds drop to very low rotation rates compared to fizzler models when we use the angular momentum distribution (n 0 ¼ 0) of the Maclaurin spheroids. For the n 0 ¼ 3 fizzlers, the secular instability thresholds ranged over T =jW j ¼ 0:034 0:042, 0.019–0.021, 0.012–0.0135, and 0.0087–0.0093 for m ¼ 2, 3, 4, and 5, respectively. The onset of secular instability depends only weakly on compressibility, so it is not affected by uncertainties in the fizzler EOS. For n 0 ¼ 3 fizzlers, GRR-driven secular instabilities grow only very slowly. For the models tested, the growth time gw > 400 s for T =jW j < 0:24. For these fizzlers, the deleptonization timescales are generally much shorter than gw . Consequently, even though the f-mode secular instability may set in at very low rotation so that all fizzlers are potentially unstable, GRRdriven f-mode secular instabilities are not likely to play a role during the fizzler secular evolution (deleptonization) phase, because the instability growth time is long. The situation is exacerbated by the high bulk viscosity of hot fizzler matter. Altogether, this reinforces the conclusion of Imamura & Durisen (2001) that f-mode secular instabilities do not play an important role in the evolution of fizzlers.
Imamura thanks the National Aeronautics and Space Administration for support.
TABLE 3 Bar Mode Frequencies and Growth Times for the Deleptonizing Fizzlers
Ye
(1012 g cm3)
T/|W |
( Hz)
0.4........................................... 0.35......................................... 0.3........................................... 0.25.........................................
1.05 32.6 110 262
0.0814 0.150 0.194 0.242
0.777 15.9 41.8 60.0
gw (s) 7.68 1.08 1.10 4.38
; ; ; ;
1010 106 10 3 10 2
No. 2, 2004
f -MODE INSTABILITIES IN DELEPTONIZING FIZZLERS
1101
REFERENCES Abramovici, A., et al. 1995, in Particle and Nuclear Astrophysics and CosImamura, J. M., & Durisen, R. H. 2001, ApJ, 549, 1062 mology in the Next Millennium, ed. E. W. Kolb & R. Peccei (Singapore: Imamura, J. N., Friedman, J. L., & Durisen, R. H. 1985, ApJ, 294, 474 World Scientific), 398 Imamura, J. N., Pickett, B. K., & Durisen, R. H. 2003, ApJ, 587, 341 Bardeen, J. M., Friedman, J. L., Schutz, B. F., & Sorkin, R. 1977, ApJ, 217, Imamura, J. N., Toman, J., Durisen, R. H., Pickett, B., K., & Yang, S. 1995, L49 ApJ, 444, 363 Bodenheimer, P., & Ostriker, J. P. 1973, ApJ, 180, 159 Ipser, J. R., & Lindblom, L. 1991, ApJ, 373, 213 Bradaschia, C., et al. 1990, Nucl. Instrum. Methods Phys. Res. A, 289, 518 Karino, S., & Eriguchi, Y. 2002, ApJ, 578, 413 Chandrasekhar, S. 1969, Ellipsoidal Figures of Equilibrium (New Haven: Yale Lai, D. 2001, in AIP Conf. Proc. 575, Astrophysical Sources for Ground-based Univ. Press) Gravitational Wave Detectors (Melville: AIP), 246 Clement, M. J. 1979, ApJ, 230, 230 Lai, D., & Shapiro, S. L. 1995, ApJ, 442, 259 ———. 1989, ApJ, 339, 1022 Lattimer, J., & Douglas Swesty, F. 1991, Nucl. Phys. A., 535, 331 Danzmann, K., et al. 1995, in Proc. First Eduardo Amaldi Conf. on Grav. Wave Lindblom, L., & Detweiler, S. 1977, ApJ, 211, 565 Experiments, ed. E. Coccia, G. Pizella, & F. Ronga (Singapore: World SciLiu, Y. T., & Lindblom, L. L. 2001, MNRAS, 324, 1063 entific), 100 Lynden-Bell, D., & Ostriker, J. P. 1967, MNRAS, 136, 293 Detweiler, S., & Lindblom, L. 1977, ApJ, 213, 193 Managan, R. 1985, ApJ, 294, 463 Durisen, R. H., & Imamura, J. N. 1981, ApJ, 243, 612 New, K. C. B., & Shapiro, S. L. 2001, ApJ, 548, 439 Friedman, J. L., & Schutz, B. F. 1978, ApJ, 221, 937 Pickett, B. K., Durisen, R. H., & Davis, G. A. 1996, ApJ, 458, 714 Finn, L. S., & Evans, C. R. 1990, ApJ, 351, 588 Sawyer, R. F. 1989, Phys. Rev. D, 39, 3804 Fryer, C. L., & Heger, A. 2000, ApJ, 541, 1033 Schutz, B. F. 1979, ApJ, 232, 874 Fryer, C. L., Holz, D. E., & Hughes, S. A. 2002, ApJ, 565, 430 Shapiro, S. L., & Lightman, A. P. 1976, ApJ, 207, 263 Fryer, C. L., & Warren, M. S. 2004, ApJ, 601, 391 Stergioulas, N., & Friedman, J. L. 1998, ApJ, 492, 301 Hachisu, I. 1986, ApJS, 61, 479 Timmes, F. X., Woosley, S. E., & Weaver, T. A. 1996, ApJ, 457, 834 Haensel, P., & Shaeffer, R. 1992, Phys. Rev. D, 45, 4708 Tohline, J. E. 1980, ApJ, 235, 866 Hashimoto, M. 1995, Prog. Theor. Phys., 94, 663 Tsubono, K., et al. 1997, in Gravitational Wave Detection, ed. K. Tsubono, Hayashi, A., Eriguchi, Y., & Hashimoto, M. 1998, ApJ, 492, 286 M. K. Fujimoto, & K. Kuroda (Tokyo: Universal Academy Press, Inc.), 183 ———. 1999, ApJ, 521, 376 Yoshida, S., & Eriguchi, Y. 1995, ApJ, 438, 830 Heger, A., Langer, N., & Woosley, S. E. 2000, ApJ, 528, 368 Yoshida, S., Rezzolla, L., Karino, S., & Eriguchi, Y. 2002, ApJ, 568, L41
