A new model of the Central Engine of GRB and the Cosmic Jets
AA new model of the Central Engine of GRB and the Cosmic Jets ∗ Plamen P. Fiziev † , Denitsa R. Staicova ‡ Department of Theoretical Physics, Sofia University “St. Kliment Ohridski”,5 James Bourchier Blvd., 1164 Sofia, Bulgaria
Abstract
Despite all the already existing observational data, current models still cannot explain completely the ex-cessive energy output and the time variability of GRB. One of the reasons for this is the lack of a good modelof the central engine of GRB. A major problem in the proposed models with a black hole (BH) in the center isthat they don’t explain the observed evidences of late time activity of the central engine.In this paper we are starting the search for a possible model of that central engine as a rotating compactbody of still unknown nature. The formation of jets in the new model lies entirely on the fundamental TeukolskyMaster Equation. We demonstrate that this general model can describe the formation of collimated GRB-jetsof various forms. Some preliminary results are presented.
Gamma-Ray bursts have been mystifying scientists since their discovery in 1969. Their spectral and temporalbehavior raised many questions offering the unique opportunity to confront our theories with reality. After allthose years of discoveries and innovations, one thing is clear – GRB cannot be properly understood until there is agood model of the central engine. In this paper, we are starting the study of a new model of the central engine ofGRB that can explain some of their most intriguing characteristics. We do not wont to presuppose a large numberof hypothetical and unconfirmed by observations specific properties of the central engine. Instead, we prefer tofocus our consideration on the minimal number of natural assumptions, which seem to be unavoidable in any modelof central engine, and to investigate their possible consequences.
Before we present our model, we offer a summary of the properties of GRB that will be important in our studies.Gamma-ray burst are explosions on cosmic distances (biggest measured redshift is z > .
3) that emit hugeamounts of energy ( ∼ erg) in very short periods of time ( T ∼ seconds) ([1]). They have two distinctphases – prompt emission and afterglow – produced by different mechanisms. An unexpected feature found by themission SWIFT is the existence of flares ([1], [2] and [3]) – yet not completely explained peaks in the light curvesuperimposed over the continuing afterglow emission. The mechanisms of radiation that are believed to cause theburst are synchrotron emission and inverse compton scattering.The widely accepted statement us that there exist 2 types of GRB – short GRBs ( T < T > ∗ Based on talk given on 15.06.2007 at The Advanced Workshop on Gravity, Astrophysics and Strings, GAS@BS07, 10-16.06.2007,Primorsko, Bulgaria. † 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 n the characteristics of the detectors, mostly on their energy range, a new space missions are being prepared –GLAST among them – that can observe GRB in higher energies which is important for the good resolution of theusually hard prompt emission. Fireball model is one of the most frequently used models in GRB physics [6]. In this model the central engineof GRB emits matter in series of shells with different Lorentz factors. When the faster shells catch up with theslower ones, the resulting collision called internal shock, produces the hard prompt emission. The deceleration ofthe shells due to the contact with the local medium is believed to produce the softer afterglow. Major set-backsof this model are that it cannot explain the late activity of the central engine of which the existence of flares is anevidence, nor the nature of those flares.
Our toy model is extremely simple and based on the least possible assumptions. For a start, we have a Kerr blackhole or some other rotating compact massive object, both described by the Kerr metric ([7]): exactly – in the firstcase, and approximately – in the second case. In Boyer-Lindquist coordinates the metric is: ds = (1 − M r/ Σ) dt + 4 M ar sin ( θ ) / Σ dtdφ − (Σ / ∆) dr − Σ dθ − sin ( θ ) (cid:2) r + a + 2 M a r sin( θ ) / Σ (cid:3) dφ (1)with ∆ = r − M r + a , Σ = r + a cos θ .For this metric, Teukolsky studied the linearized perturbations of the Einstein equations and found that theequations describing different perturbations generalize to [8]: L = (cid:20) ( r + a ) ∆ − a sin θ (cid:21) d dt + 4 M ar ∆ d dtdφ + (cid:20) a ∆ − θ (cid:21) d dφ − ∆ − s ddr (cid:18) ∆ s +1 ddr (cid:19) − θ ) ddθ (cid:18) sin θ ddθ (cid:19) − s (cid:20) a ( r − M )∆ − i cos θ sin θ (cid:21) ddφ − s (cid:20) M ( r − a )∆ − r − ia cos θ (cid:21) ddt + (cid:0) s cot θ − s (cid:1) . (2)Following the procedure established by Teukolsky, we perform separation of the variables in the Teukolskyequations using the following substitution: Φ = e ( ωt + mφ ) i S ( θ ) R ( r ) , (3)where m=...-2, -1, 0, 1, 2... is an integer, ω is the frequency, and S ( θ ) and R ( r ) are the angular and the radial partof the equation. It is important to emphasize that the frequency ω is a complex number: ω = ω R + iω I . Noticethat we use different substitution for the time dependence: e iωt , not the original one: e − iωt , that Teukolsky used.One of the most important assumption we use is the stability condition ω I >
0, that ensures that the initialperturbation won’t become infinite with time and it will damp instead.Applying the operator (2) L on the function (3) Φ , Teukolsky found that the equations for θ and r separate[8] to an angular and a radial part. The angular Teukolsky Equation (TAE): (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)The Teukolsky radial equation TRE:∆ R lm,rr + 2( s + 1)( r − M ) R lm,r + V ( r ) R lm = 0 , (5)where u = cos( θ ), A is the constant of separation of the variables and the potential V(r) is: V ( r ) = − A − isω r + ω r (cid:0) r + a (cid:1) + 2 is (cid:0) ma ( r − M ) + ω M (cid:0) r − a (cid:1)(cid:1) + 2 ω M r (cid:0) ω a + 2 am (cid:1) + m a ∆ . the Gamma-ray Large Area Space Telescope(GLAST ) was renamed to Fermi Gamma-ray Space Telescope after the launch in 2008 | s | of the field. It isremarkable that all fields can be described with one set of equations. Also, it is important that s A lm and ω areindependent parameters.To solve that equation, we use the standard notations: r + = M + √ M − a is the event horizon and r − = M − √ M − a is the Cauchy horizon. It’s easy to see that there is a symmetry between r + and r − in the TREand that r + and r − are regular singular points of the TRE, while r = ∞ is an irregular singular point.Using the software package Maple to solve that linear differential equation we acquire two independent exactsolutions of the radial Teukolsky equation in the outer domain ( r > r + ) [9]: 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) (6)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) (7)where HeunC is the confluent Heun function (see [10] and [11]) and its parameters are : α = 2 i (cid:0) r + − r − (cid:1) ω, β = − i ( ω ( a + r + ) + am ) r + − r − − ,γ = − i ( ω ( a + r − ) + am ) r + − r − + 1 , δ = − (cid:0) r + − r − (cid:1) ω (cid:0) i + (cid:0) r − + r + (cid:1) ω (cid:1) .η = 12 1 (cid:0) r + − r − (cid:1) (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) . The angular equation has regular solutions studied by Teukolsky and Press([12]. Beside those solutions, onecan find polynomial solutions of TAE. As far as we know these polynomial solutions haven’t been studied, beingsingular. Explicitly, two solutions of TAE in terms of confluent Heun functions are: S ( − ± ,s,m ( θ ) = e ± Ω cos θ (cos ( θ/ | s − m | (sin ( θ/ −| s + m | × 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 | × HeunC (cid:18) ∓ , | s + m | , | s − m | , − s, m − s − s − Ω − A − s, sin θ (cid:19) , where Ω = aω . In general, these are two linearly independent solutions of the TAE. They arise from the propertiesof the Heun functions. A special attention is required in the case s ∈ Z – then s − m ∈ Z and the second linearlyindependent solution is not in the above form, but includes an integral of the confluent Heun function. Once equipped with the solutions, we proceed with numerical search for the explicit value of the frequency ω . Amajor technical problem in the use of the regular solutions of the angular equation is that we are trying to solveconnected spectral problem with two complex solutions. It has been solved for the first time by Press & Teukolsky([12]) and after that developed trough the mechanism of continued fractions by Leaver [13].3igure 1: Few examples of jets that we obtained from the solutions of the angular equation.In our work, the case is being simplified drastically by the consideration of polynomial solutions, i.e., by theassumption that the confluent Heun functions in the angular equation are polynomials. From here, we manage toobtain a simple analytic relation A ( ω ). Using the explicit form of A ( ω ) we can plot the angular part of the solution(3) for certain values of ω .An important feature of the singular polynomial solutions is that they provide a natural explanation for theexistence of jets. Examples of such jets in the case s=-1 can be seen on figure 1. In that figure, one can see bothlimited and unlimited solutions, although the second one can be constrained adding a form factor with knownvalue. Also, we see that different form of the jets arise for different m. Note that the case | s | = 1, presented inthis talk, corresponds to perturbations, which describe electromagnetic waves. An analogous treatment of the case | s | = 2, which is more complicated and corresponds to gravitational waves, is presented in [14] and [15].The general solution we use to obtain those plots for | s | = 1 is: S ( θ ) = (cos θ +1) m (1 − cos θ ) m − e Ω cos θ HeunC (cid:18) , m +1 , m − , , m − A − − Ω + 12 , cos θ (cid:19) . Using the properties of the confluent Heun function and imposing the polynomial conditions on the solutionsof the angular equation, we obtain the following polynomials, summarized in Table 1.The radial equation is much harder to deal with. Using A ( ω ) and the right boundary conditions for a blackhole, we want to look for zeroes of the resulting functions. This task is hard considering the profile of the confluentHeun functions.We use the distributions of the eigenvalues in the complex plane for the singular case: s = − , m = 1 for certain A ( ω ) (cid:0) explicitly A s = − ,m ( ω ) = − Ω − m ± √ Ω +Ω m (cid:1) to find points resembling to zeroes of the function. Afterthat we need an algorithm that can prove they are exactly a zeros of the function and not just a minimum.Such new algorithms for searching for the complex roots of a complex function are being developed by the team.The results will be reported in the following paper.Another problem is that the numerical calculations are additionally burdened by the CPU time required formaking a complex plot of the solutions by the package Maple. Although in very preliminary stage, our model of central engine seems to be able to produce qualitatively someof the basic features observed in GRB. The formation of relativistic jets is supposed to be caused by the rotationof a compact central body of any nature. At least in a good approximation its exterior gravitational field can bedescribed by Kerr solution. The Teukolsky Master Equation (with appropriate boundary conditions) is fundamental4igure Formula Parametersa), b) e − Ω cos θ (cid:0) (cid:0) − ∓ √ Ω − Ω (cid:1) cos θ (cid:1) sin θ L = − m = 1c) e − Ω cos θ cot θ m = L = 0d) e − Ω cos θ (cid:18) cos θ (cid:19) − ( m +1) (cid:18) sin θ (cid:19) m − (cid:16) (cid:112) Ω (Ω + m ) (cid:17) sin θ m L = 1, m = 6e), f) e − Ω cos θ (cid:18) θ (cid:16) ∓ (cid:112) Ω + Ω (cid:17)(cid:19) L = m = 1g) e − Ω cos θ tan θ m = L = 0h) e − Ω cos θ (cid:18) cos θ (cid:19) − m − (cid:18) sin θ (cid:19) m − (cid:16) (cid:112) Ω (Ω + m ) (cid:17) (cid:0) cos θ (cid:1) m L = − m = 1Table 1: A table of the functions used to plot Figure 3enough to account for both types of GRB with their maybe different origin. The essential assumption we usedis that the imaginary part of the frequency should be positive. This provides stability of the solutions. Differentobjects can be described by different boundary conditions, though in our case, we used the standard black holeboundary conditions for the radial Teukolsky equation: on the horizon we have only entering waves and on infinitywe have only outgoing waves. In contrast, we used a novel boundary conditions for the angular Teukolsky equationand this enables us to describe mathematically the collimated jets of different forms. Our preliminary results showthe potential of this mathematical model for description of the central engine as a rotating relativistic compactobject of any nature. The development of our study will be published in the papers to follow. Acknowledgements
This paper made use of data supplied by the UK Swift Science Data Centre at the University of Leicester.Our work is supported by the Foundation ”Theoretical and Computational Physics and Astrophysics” and byBulgarian National Scientific Fund under contracts DO-1-872, DO-1-895 and DO-02-136.
References [1] Zhang B., Meszaros P.,
Gamma-Ray Bursts with Continuous Energy Injection and Their Afterglow Signature ,ApJ, , 712-722, 2002[2] Burrows D. N. et al.,
Bright X-ray Flares in Gamma-Ray Burst Afterglows , Science, , 1833-1835, 2005[3] Falcone A. D. et. al.,
The Giant X-ray Flare of GRB 050502B: Evidence for Late-Time Internal Engine Activity ,ApJ, , 1010-1017, 2005 54] N. Mirabal et al.,
GRB 060218/SN 2006AJ: a gamma-ray burst and prompt supernova at z = 0.0335 , ApJ ,L99-L102, 2006[5] Campana S. et al.,
The shock break-out of GRB 060218/SN 2006aj , Nature , 1008-1010, 2006[6] Piran T.,
Gamma-ray bursts and the fireball model , Physics Reports, , p. 575-667, 1999[7] Kerr R. P. , PRL, , 237, 1963[8] Teukolsky S. A., Perturbations of a rotating black hole I Fundamental Equations for Gravitational, Electromag-netic and Neutrino-field Perturbations , ApJ, , 635-473, 1973[9] Fiziev P. P., Class. Quant. Grav. , 2447-2468, 2006[10] Decarreau A., Dumont-Lepage M.-Cl., Maroni P., Robert A. et Ronveaux A., Formes canoniques des equationsconfluentes de l’equation de Heun , Anales de la Societe Scientifique de Bruxelles, , 53-78, 1978[11] Decarreau A., Maroni P. et Robert A., Sur les equations confluentes de l’equation de Heun , Anales de laSociete Scientifique de Bruxelles, , 151-189, 1978[12] Press W. H. & Teukolsky S. A., Perturbations of a rotating black hole II Dynamical Stability of the Kerrmetric , ApJ, , 649-673, 1973[13] Leaver E W 1985 Proc. Roy. Soc. London A , 012016, 2007[15] Fiziev P P, 2007 Exact Solutions of Regge-Wheeler and Teukolsky Equations , talk given on 23 May 2007at the seminar of the Astrophysical Group of the Uniwersytet Jagiellonski, Institut, Fizyki, Cracow, Poland,http://tcpa.uni-sofia.bg/research/Fiziev P P, 2007
Exact Solutions of Teukolsky Equations , talk given at the Conference Grav-ity, Astrophysics and Strings at Black Sea, 10-16 June 2007, Primorsko, Bulgaria, http://tcpa.uni-sofia.bg/conf/GAS/files/Plamen Fiziev.pdfFiziev P P, Staicova D R, 2007 A new model of the Central Engine of GRB and the Cosmic Jets, talkgiven at the Conference Gravity, Astrophysics and Strings at Black Sea 10-16 June 2007, Primorsko, Bulgaria,http://tcpa.uni-sofia.bg/conf/GAS/files/GRB Central Engine.pdfFiziev P P, Staicova D R, 2007 A new model of the Central Engine of GRB, talk given at the Fourth AegeanSummer School, 17-22 September 2007, Lesvos, GreeceFiziev P P, 2007
Exact Solutions of Regge-Wheeler and Teukolsky Equations , talk given on 28 December 2007 atthe seminar of the Department of Physics, University of in Nis, Serbia, http://tcpa.uni-sofia.bg/research/[16] Evans et al., A&A,469