Toward a New Model of the Central Engine of GRB
TToward a New Model of the Central Engine of GRB ∗ Plamen P. Fiziev † , Denitsa R. Staicova ‡ Department of Theoretical Physics, Sofia University “St. Kliment Ohridski”,5 James Bourchier Blvd., 1164 Sofia, Bulgaria
Abstract
We present new developments of the simple model of the central engine of GRB, proposed recently.The model is based on minimal assumptions: some rotating compact relativistic object at the centerand stable perturbations of its rotating gravitational field, described by Teukolsky Master Equation.We show that using nonstandard polynomial solutions to the angular Teukolsky equation we candescribe the formation of collimated jets of various forms. Appearance of imaginary part of thesuperradiance-like frequency is established for the first time for pure vacuum black hole jet solutionsof Teukolsky equation.
Gamma-ray bursts (GRB) are among the most powerful astrophysical objects in the Universe. They areconsidered as highly collimated ( θ jet ∼ ◦ − ◦ ) explosions on cosmic distances emitting huge amountsof energy in very short periods of time ( ∼ seconds) [1, 2, 3, 4]. Despite the increasing amount ofobservational data, the theory of GRB is still far from being clear. One of the major problems is thelack of understanding of the nature of the so called central engine – its physical nature and the processinvolved in the emission of huge amounts of energy ( ∼ − erg).Unexpected feature discovered by the mission SWIFT is the existence of flares – a surprising evidenceof late time activity of the central engine [3, 4, 5]. There are clear indications that in some cases the flaresare produced by energy injection by the central engine. Although number of theories and models havebeen suggested to explain the light curves of GRB, the process however remains a mystery that cannotbe solved without a good model of the central engine.The current focus of GRB physics is on the propagation of the emitted matter that can explain theobserved light curves. The most used is the fireball model (for details see [1]). That model, however,cannot explain the observed flares. Another basic model is the cannonball model of long GRBs [6].The central engine – the physical object producing the GRB – before SWIFT epoch usually wasconsidered to be a Kerr [7] black hole (BH) – a hypothetical result of the death of a massive star for thelong GRB ( T > s ) (see [3], [8]) or as part of a binary merger of compact objects (BH-BH, NS-BH,or NS-NS; where NS=neutron star) for short GRB ( T < s ). The first hypotheses seems to be hardlycompatible with the observed flares, produced via energy injection by the central engine. The secondhypotheses was recently refuted by the existing detectors of gravitational waves [9]. The latest analysisshows that the short and long GRB may have a similar central engine (except for its duration) [10].It was believed that one of the possible models for such engine could be a Kerr black hole (KBH) insuper-radiant mode [11, 12] – the wave analogue of the Penrose process. The late-time evolution of aperturbation of a Kerr metrics is governed by the quasi-normal modes (QNM) – oscillation with complexfrequencies that are determined only by the BH parameters (mass, charge and angular momentum) ∗ This article is based on the talk given on 18.09.2007 at the Fourth Aegean Summer School, 17-22 September 2007,Lesvos, Greece † E-mail: fi[email protected]fia.bg ‡ E-mail: [email protected] a r X i v : . [ a s t r o - ph . H E ] M a r
13, 11]. Those frequencies are obtained solving the Teukolsky Master Equations with certain boundaryconditions. According to the standard theory, supperradiance occurs in the process of scattering ofexterior waves on KBH for real frequencies ω < ω critical = am/ M r + ( m > r + > ∼ − erg) [15], as shown long time ago by Wald (see in [12]).In a series of talks [16], we have presented a mathematical description of relativistic jets that canhelp to penetrate into the mysterious physics behind the central engine. A simple model, based on novelsolutions of the angular Teukolsky equation, however, not regular, but singular ones [16, 17, 19] wasproposed. They seem to describe in natural way the most important feature of the relativistic jets – theircollimation. In this paper, we would like to announce the recent results of our numerical simulations.The main new result is the appearance of a complex critical frequency, which play the role of the wellknown supperradiance ones. Its imaginary part is of the same order of amplitude as the real part andyields an exponential abatement of the superradiance-like emission in jets, created by KBH. Our simplified model was already presented in [16, 17]. Thus, we will omit the details and give only asummary of the basic ideas behind it. In brief, the jets observed in GRBs require a rotating object – KBHor a compact massive matter object, to produce them. In the jet’s problem we can use the Kerr metric todescribe the gravitational field of that object at least in a very good approximation [17], since the visiblejets are formed at distances about 20-100 event-horizon-radii [15]. At such distances one practically isnot able to distinguish the exterior field of KBH from the exterior field of rotating objects of completedifferent kinds (see for example [17] and the references therein).As a result, the only working way to get information about the real nature of the central engine isto study jet’s spectra. The different objects yield different frequencies of perturbations of space-timegeometry, because of the different boundary conditions on their surface (See the articles by Fiziev in [13]and [17]). Thus, measuring the real jet’s frequencies one can get indisputable evidences about the actualnature of the central engine.In the present article we are probing only the KBH model for generation of jets. The results forother possible models will be discussed elsewhere. Due to the well known ”no hair” theorems, the non-charged KBH solutions of Einstein equations, which potentially may be of astrophysical interest, dependon extremely small number of parameters – the mass and the angular momentum. The unique boundaryconditions on the event horizon yield unique robust spectra, defined only by the mass and the angularmomentum of the KBH. These spectra differ essentially from the spectra of relativistic jets, generatedby central engine of any other nature. Thus the KBH-based model of relativistic jets seems to be easilyrecognizable from observational point of view, at least up to the disguise effects by the KBH environment.Using the Teukolsky Master Equation (TME) we acquire the linearized perturbation of the Kerrmetric (see [11]). From there, we follow the procedure established by Teukolsky for separation of thethe variables in the TME using the substitution: Φ = e ( ωt + mφ ) i S ( θ ) R ( r ) where m = 0 , ± , . . . and ω = ω R + iω I is a complex frequency (with ω I > R ( r ):∆ d dr R ( r ) + 2 ( s + 1) ( r − M ) ddr R ( r ) + (cid:18) K − is ( r − M ) K ∆ − isω r − λ (cid:19) R ( r ) = 0 (1)can be expressed in terms of confluent Heun functions [16, 17, 18, 19]).2he most important for GRB physics seem to be electromagnetic waves with s = −
1. In this casetwo independent exact solutions in outer domain are: R ( r ) = C e − iω r (cid:0) r − r + (cid:1) i ω ( a r +2)+ am − r ++ r − (cid:0) r − r − (cid:1) i ω ( a r − am − r ++ r − +1 HeunC (cid:18) α, β, γ, δ, η, − r − r + r + − r − (cid:19) (2)and R ( r ) = C e − iω r (cid:0) r − r + (cid:1) − i ω ( a r +2)+ am − r ++ r − +1 (cid:0) r − r − (cid:1) i ω ( a r − am − r ++ r − +1 HeunC (cid:18) α, − β, γ, δ, η, − r − r + r + − r − (cid:19) , (3)where: α = 2 i (cid:0) r + − r − (cid:1) ω , β = − i ( ω ( a + r +2 )+ am ) r + − r − − γ = − i ( ω ( a + r − )+ am ) r + − r − + 1, δ = − i (cid:0) r + − r − (cid:1) ω (cid:0) − i (cid:0) r − + r + (cid:1) ω (cid:1) , η =
12 1 ( r + − r − ) (cid:104) ω r + + (cid:0) iω − ω r − (cid:1) r + + (cid:0) − aω m − ω a − A (cid:1) (cid:0) r + + r − (cid:1) + (cid:0) iω r − − iω r + + 4 A − ω a − (cid:1) r − r + − a ( m + ω a ) (cid:105) For the angular function S ( θ ), as an exact solution the angular equation: (cid:2)(cid:0) − u (cid:1) S lm,u (cid:3) ,u + (cid:20) ( aωu ) + 2 aωsu + s + s A lm − ( m + su ) − u (cid:21) S lm = 0 (4)we use polynomial solutions in terms of Heun polynomials ([16, 17, 18, 19]). In equation (4) we are usingthe variable u = cos θ . Putting Ω = aω we have explicit expressions: S ( − ± ,s,m ( θ ) = e ± Ω cos θ (cos ( θ/ | s − m | (sin ( θ/ −| s + m | × (5) HeunC (cid:18) ± , | s − m | , | s + m | , − s, m − s s − Ω − A − s, cos θ (cid:19) and S (1) ± ,s,m ( θ ) = e ± Ω cos( θ ) (cos ( θ/ | s − m | (sin ( θ/ −| s + m | × (6) HeunC (cid:18) ∓ , | s + m | , | s − m | , − s, m − s − s − Ω − A − s, sin θ (cid:19) . The polynomial solutions of TME describe one way waves and are most suitable for modeling of therelativistic jets [17]. Posing the polynomial condition on the angular solutions, we obtain for the one wayelectromagnetic waves on Kerr background the explicit formula A s = − ,m ( ω ) = − Ω − m ± (cid:112) Ω +Ω m. (7)This simple form of the separation constant A for polynomial solutions is the most significant mathemat-ical advantage of our model of jets, generated by electromagnetic perturbations of Kerr metric.The KBH boundary condition for the radial equation ca be obtained using the following assumptions:1. On the horizon we allow only incoming in the horizon waves. Then we obtain the different solutionworking in each interval of frequencies: a) for m = 0, only R ; b) for m > R , if ω R ∈ ( − ω cr ,
0) and R on the outside; for m < R when ω R ∈ (0 , ω cr ) and R outside.2. On infinity we allow only outgoing waves. In general, the function R is a linear combination of aningoing ( R ← ) and an outgoing ( R → ) wave: R = C ← R ← + C → R → with some constants C ← and C → . Inorder to have only outgoing waves, we need to have C ← = 0. This equation defines the spectral conditionfor the frequency ω . The main mathematical problem is that the explicit form of the constant C ← isnot known. Therefore to solve the spectral problem we use the around way, proposed by Fiziev in [13]:3he straightforward check shows that for solutions (2), (3) we have lim r →∞ R → R ← = 0 in the special direction arg ( r ) = 3 π/ − arg ( ω ) in complex plane C r . Taking lim r →∞ RR ← in this direction, we obviously obtain thespectral condition for ω in the form C ← = lim r →∞ RR ← = 0 . (8) The numerical evaluations with confluent Heun functions are generally too complicated for a numberof reasons. At present the only software package able to deal with them is Maple. Unfortunately, theexisting versions of Maple package require too much time for numerical evaluation, even on modern fastcomputers with large amount of memory. Additional problem is that the procedure calculating thosefunctions is not working well in the whole complex plane and special attention must be paid to theregions where the procedure becomes unstable.
We already reported that plotting the solutions of the angular equations with relation (7), we obtainfigures that resemble jets (see [16, 18]). Illustrations of the jet features of the model can be seen fromthe animations of such oscillating solutions on our site [20].An interesting comparison with Nature can be obtained looking at the picture of the discovered byNASA’s Spitzer Space Telescope “tornado-like“ object Herbig-Haro 49/50, created from the shock wavesof powerful protostellar jet hitting the circum-stellar medium. ”More observations should help us tounravel its mysterious nature” [21]. Without any doubts in this case the jet is not related with BH. Thisone, as well as many other real observations prove the presence of BH to be not necessary for generationof jets. The collimated jets of GRB are another example of a possible application of our simple model.There the main problem still remains the physical nature of the central engine, too. Clearly, if possible,a common model of jets of different scales is most desirable.Figure 1: Some of the shapes of jets that our model can represent. Compare it with the picture ofHerbig-Haro 49/50 (see [21])
The study of the solutions of radial Teukolsky equation (1) is much more complicated from computationalpoint of view. Even after fixing the relation (7) between A and ω , solving the KBH boundary condition (8)for radial equation is not a straight-forward process, because of the analytical and numerical problemsconnected with the evaluation of the confluent Heun function. Our method consisted in plotting andexamining the spectral condition in the complex plane C ω to find points that resemble roots of thetranscendental equation (8). Then we tested those points with more precise root-finding algorithms.4hus we found for the case s = −
1, 2 M = 1, a/M = 0 . | m | = 1 two roots: ω , = ± . i . . For the case m = 0 we obtained another pair of roots ω , = ± . i . . To confirm those roots, we plotted the studied function in small regions around the zeroes and obtainedthe expected conus-like form of the modulus of the function in the complex domain and the completerotation of it’s phases in angle 2 π around it’s simple zeroes.Interesting in these roots is that the absolute value of their real part is precisely the known criticalfrequency ω cr = am/ M r + , m > | ω R | crosses the critical value am/ M r + . The complexity of the critical frequencyshows an essential difference between our jets-from-KBH solutions and standard QNM, obtained usingregular solutions of the angular equation (4) [11]. It is also important that the real and the imaginaryparts of the critical frequency are with the same magnitude.The lack of zeros with ω I < Our simple toy model of central engine seems to be able to produce some of the basic features observedin GRB. The Teukolsky Master Equation with the correct boundary conditions is fundamental enoughto account for all types of GRB. The polynomial angular solutions of TME show a jet-like structure. Inthe radial equation, the essential assumptions we used is that the imaginary part of the frequency withBH boundary conditions (only entering waves on the horizon and only going to infinity waves) shouldbe positive. This provides stability of the solutions in direction of time-future infinity and indicates anexplosion in direction of time-past infinity. The roots we found numerically agree with our assumption.Interesting new result is the complex critical frequency ω jetc = ω R + iω I ( ω I ∼ ω R ) of ”superradiance” thatappeared, showing that jet’s to-be-superradiance modes decrease exponentially in time in the directionof the future and blow up in direction of the past. This article was supported by the Foundation ”Theoretical and Computational Physics and Astrophysics”and by the Bulgarian National Scientific Found under contracts DO-1-872, DO-1-895 and DO-02-136.One of us (DS) is grateful to the organizers of the Fourth Aegean Summer School, 17-22 September2007, Lesvos, Greece for financial support of the participation in this School.
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