A Relativistic Electron-Positron Outflow from a Tepid Fireball
aa r X i v : . [ a s t r o - ph ] D ec A Relativistic Electron-Positron Outflow from a Tepid Fireball
Katsuaki Asano and Fumio Takahara [email protected], [email protected] ABSTRACT
Through detailed numerical simulations, we demonstrate that relativistic out-flows (Lorentz factor Γ ∼
7) of electron-positron pairs can be produced by radia-tive acceleration even when the flow starts from a nearly pair equilibrium state atsubrelativistic temperatures. Contrary to the expectation that pairs annihilateduring an expansion stage for such low temperatures, we find that most pairscan survive for the situations obtained in our previous work. This is becausein the outflow-generating region the dynamical timescale is short enough eventhough the fireball is optically thick to scattering. Several problems that shouldbe solved to apply to actual active galactic nucleus jets are discussed.
Subject headings: plasmas — relativity — galaxies: jets
1. Introduction
One of the most challenging problems in astrophysics is the acceleration mechanism ofrelativistic jets from active galactic nuclei (AGNs) and Galactic black hole candidates. Thebulk Lorentz factor of these jets is above 10 and the kinetic power is almost comparableto the Eddington luminosity. Although various jet models have been proposed, we do notyet have a satisfactory solution. Recent remarkable progress in magnetohydrodynamical(MHD) simulations includes Poynting-dominated jet formation by a rapidly rotating blackhole with an accretion disk (e.g. Mckinney 2006; Hawley & Krolik 2006). Along the spinaxis a centrifugal funnel filled with magnetic field is formed. The magnetic field in the funnelis considered to be amplified by the differential rotation of the accretion disk and the frame-dragging effect of the black hole. However, in the simulation of Mckinney (2006), the total Interactive Research Center of Science, Graduate School of Science, Tokyo Institute of Technology, 2-12-1Ookayama, Meguro-ku, Tokyo 152-8550, Japan Department of Earth and Space Science, Graduate School of Science, Osaka University, Toyonaka 560-0043, Japan m e c . However, the char-acteristic size of AGNs is too large to achieve complete thermal equilibrium. The novelmodel proposed to overcome this problem is electron-positron pair outflow from a “Wienfireball” (Iwamoto & Takahara 2002, 2004), which is a photon-dominated, optically thickplasma, but the densities of photons and pairs are less than those in the complete thermalequilibrium fireballs. Pairs coupled with photons by scattering are thermally accelerated,expending the internal energy of the fireball. As is seen in the original model of the fire-ball, the Lorentz factor Γ = 1 / p − β increases with the radius R , and the temperaturedrops as T ∝ R − . Iwamoto & Takahara (2002, 2004) showed that a sufficient amount ofelectron-positron pairs survive as a relativistic outflow, if the temperature of the fireball atthe photosphere is relativistic ( & m e c ).When the temperature is relativistic, all the cross-sections of pair creation, annihilationand Compton scattering are of the same order. Therefore, a balance between pair creationand annihilation is realized, and the number densities of pairs and photons are of the sameorder as long as photons and pairs are coupled with each other. On the other hand, thecondition to make the pair-annihilation timescale shorter than the dynamical timescale, n + σ T R Γ β > (cid:20) θ ln (1 . θ + 1 . (cid:21) , (1)is almost the same as the condition of being optically thick to scattering, where n + , σ T , and θ = T /m e c are the positron density in the comoving frame, Thomson cross section, andthe normalized temperature, respectively. This implies that the number of pairs is almostconserved outside the photosphere, where photons and pairs are decoupled. Therefore, arelativistic temperature at the photosphere is required to obtain a sufficient amount of pairsavoiding annihilation.The temperature of fireballs is determined by microscopic processes in their formationsites, presumably accretion disks, such as radiative cooling, Compton scattering, and γ − γ pair production (see e.g. Svensson 1984). Assuming that pairs are confined with protons,several authors (see e.g. Kusunose 1987; Bj¨ornsson & Svensson 1992, and references there 3 –in) investigated pair equilibrium plasmas. However, a series of studies of pair equilibriumplasmas implies that the equilibrium temperature is too low for the Wien fireball model. Theplasma temperature typically becomes θ ∼ . ∼ cm and luminosity & erg s − , while the Wien fireball model requires a super-Eddington luminosity & erg s − (under the assumption of spherical symmetry) and a high temperature θ > . θ ∼ . R ∼ several times 10 cm for the luminosity L = 10 ergs − , though a considerable amount of pairs outflow with a mildly relativistic velocity. Inthe simulations of AT07, the acceleration of outflows is not finished within the simulationscale. We pessimistically supposed that a rapid extinction of pairs inside the photospheremay occur, since the obtained temperature is so low.However, the detailed situations in AT07 are different from the simplified inner boundaryconditions of the outflow in the simulations of Iwamoto & Takahara (2004). The pair-photon ratio at the outer boundary of the simulation region is smaller than that of theWien equilibrium plasma by a factor of ∼
2. The photon spectrum obtained numerically isalso different from the simple Wien spectrum. It may be valuable to simulate subsequentevolution of the outflows outside such a tepid fireball obtained in the simulations in AT07.In this Letter, we demonstrate that the tepid fireball obtained by the simulations ofAT07 can be accelerated to a relativistic velocity conserving the pair number, even thoughthe fireball temperature is not high enough. In §
2, we describe our simulation method, andshow our results in § §
2. Method2.1. Energy-Release Region
We numerically obtain spherically symmetric, steady solutions of radiation and pairoutflows. As the inner boundary condition, we employ the numerical values obtained at theouter boundary in the simulations of AT07. Namely, the simulations of AT07 provide uswith the formation stage of the fireball. First of all, let us review the physical situations in 4 –AT07. In AT07, the proton number density is assumed to be n p ( R ) = n exp (cid:2) − ( R/R ) (cid:3) , (2)where R and n are constant. In this region the plasma was assumed to be heated, whichmimics energy release via viscous heating. The heating rate of the plasma was proportionalto n p . The plasmas were divided into three fluids: proton ( p ), background electron ( e ),and pair ( e ± ) fluids. The background electrons and protons were assumed to be static; thegravitational force of the central black hole may arrest the background plasma. Althoughthe validity of this multifluid approximation is not always assured, we had adopted thisapproximation for simplicity. For the microphysics, Coulomb scattering, Compton scatter-ing, bremsstrahlung, and electron-positron pair annihilation and creation were taken intoaccount. The frictional forces and heating of the pair plasma due to Coulomb scatteringwith background fluids were calculated using numerical results from Asano et al. (2007).The effect of the frictional force was less important than the radiative force. The sphericalsymmetry in the geometry and the mildly relativistic outflow yield beamed photon distribu-tion. It is a well-known effect that too fast flows are decelerated even by a beamed radiationfield (Compton drag). The flow velocity was self-regulated by the beamed photon field.Radiative transfer was solved with the Monte Carlo method. Taking into account Comptonscattering and pair creation, the evolutions of energy and direction along their trajectoriesare fully solved for each photon. The energy range of photons x ≡ ε/m e c was from 10 − to10 , and the outer boundary was set at R = R out ≡ R . For the total heating rate L = 10 erg s − , AT07 obtained mildly relativistic ( U ≡ Γ β ∼
1) outflows. The ratios of the totalluminosity to the mass ejection rate, η ≡ L/ πR n ± ( R out ) U ( R out ) m e c , are ∼
30, but thetemperatures are relatively low ( θ ∼ . Using U , n ± , θ and beamed photon field at the outer boundary in AT07 as the innerboundary conditions, we simulate the behavior outside the energy-release region (the simu-lation region of AT07). The detailed method of the simulations is the same as that in AT07.However, we assume that protons do not exist anymore, so we switch off heating and theCoulomb friction due to background protons and electrons. Thus, we take into account onlyCompton scattering, bremsstrahlung, and electron-positron pair annihilation and creationfor the outer region. We solve hydrodynamics and radiative transfer only for pair fluids. 5 –
3. Results
In AT07 two sets of parameters were adopted: R = 10 cm with n = 10 cm − , and R = 3 × cm with n = × cm − . These values are characteristic of accretion plasmasin AGNs. In both cases, the Thomson scattering depth due to the e -fluid τ p ∼ n R σ T ∼ . n may not be a critical parameter. Although the results for the latter parameter set wereexplained in detail in AT07, we show the results employing the former compact case as anenergy-release region in Figs. 1 and 2. The outflow is accelerated mainly by a radiative force.While the photosphere is located at ∼ cm, the acceleration turns out to continue evenoutside the photosphere. Even though the fraction of photons interacting with the plasmais small, the influence of radiative interaction outside the photosphere cannot be neglectedbecause of the copious amount of photons. The final Lorentz factor becomes ∼ R = 2 R (the boundarybetween the energy-release region and the outer region), the heating from the backgroundprotons makes electron temperature increase with radial distance. The sudden shutdownof the heating at R = 2 R results in the discrete change of temperature gradient. Outsidethe energy-release region the temperature decreases monotonically, but the temperature dropbegins to slow down around the photosphere. Outside the photosphere, the simple analyticalformulae for fireballs ( U ∝ R and T ∝ R − ) are no longer applicable.If the plasma is in the Wien equilibrium, a drastic extinction of pairs should occur forsuch a low temperature. We also plot the positron number flux R n + U in Fig. 1. Thedashed line (result from AT07) shows that most of the outflowing material is provided inthe outer part of the energy-release region. Outside the energy-release region, even thoughthe temperature decreases from θ ∼ . .
03 at the photosphere, the number of pairs isalmost conserved. At R = 2 R , the pair-annihilation timescale is estimated as ∼ × s inthe comoving frame, while the timescale of density drop ( n + becomes half in this timescale)due to plasma expansion is ∼ s. Thus, the two timescales are already comparable eventhough 2 R is inside the photosphere. Since the pair-annihilation timescale ∝ n − , we canneglect the pair annihilation outside 2 R . Photons and pairs are not in the Wien equilibrium,though the plasma is optically thick to scattering. This is because the pair plasma keepson being heated during the initial acceleration, and is smoothly evacuated outward at amildly relativistic velocity. If the temperature drop begins further inside, lower electrontemperature may lead to rapid annihilation of pairs.A problem is that the energy outflow rate of the pair plasma is only ∼ /
60 of the 6 – D e n s it y Positron Density
R U [cm -3 ] UR n + U (Arbitrary Unit) × cm]10 Fig. 1.— Distributions of the positron density in the comoving frame n + (left axis), 4-velocity U (right axis), and R n + U (arbitrary units, see the left axis for reference). The dashed linesare the results from AT07.total luminosity L . The rest of the energy escapes as photons from the boundary. In Fig.3, we plot the energy spectrum of the photons escaping from the outer boundary of thissimulation L γ ( x ) dx ( x ≡ ε/m e c ). The spectrum has a broader feature than that in theWien spectrum. There are huge amounts of soft photons ( x < − ) in comparison withthe thermal spectrum. As shown in Fig. 4, the photon field is highly beamed. As for X-rayphotons ( x ∼ − ), the off-axis flux is diluted by a factor of ∼
100 compared with theon-axis flux.We also simulate an outflow for the broader profile of protons: R = 3 × cm with n = × cm − . The qualitative results are the same as the results described above,but the final Lorentz factor Γ ∼
5. Given the luminosity L , the radius of the photospheremay not change drastically. Therefore, considering the behavior Γ ∝ R , the larger initial 7 – θ R × cm]0.010.1 0.010.1 Fig. 2.— Temperature of the pair plasma. The dashed line is the result from AT07.radius (namely the radius at U ∼
1) results in a smaller Lorentz factor. On the other hand,the energy outflow rate of the pair plasma is 14 % of the total luminosity L , which is largerthan the former case. While a more compact energy release is preferable to increase the finalLorentz factor, conversely the efficiency of the energy conversion into pair outflow becomeslarger for a more capacious source.
4. Discussion
Our simulation shows that an energy release of 10 erg s − within a size of ∼ cm produces a pair outflow of Γ ∼
7. Even though the fireball temperature is tepid, rapidannihilation of pairs is avoided. If the entire system we have considered is moving with aLorentz factor Γ s ≥ .
06 ( β s ≥ . x L γ ( ) [s -1 ] = ε / m e c x -5 -4 -3 -2 -1 Fig. 3.— Energy spectrum of the photons escaping from the outer boundary.than 10. Such a nonrelativistic outflow of protons may be easily achieved, as various MHDsimulations show. Therefore, similar simulations to ours including the outflow of protons arevaluable, while we have assumed that the background protons are static.The reason why a rapid annihilation of pairs does not occur may be a slight differencein pair density from the prediction of the Wien equilibrium. This difference comes from thefinite size of the energy-release region. In the energy-release region the outflowing materialkeeps on being heated and is evacuated with a mildly relativistic velocity. The rapid ex-pansion makes the pair plasma escape smoothly from the optically thick region, avoidingpair annihilation. For such a marginal optical depth, the Wien equilibrium is no longer areasonable approximation. The pair densities obtained numerically are suitable for avoidingpair annihilation and keeping the plasma optically thick to scattering, which are opposing 9 –requirements in general.However, resultant densities are too low to convert the radiation energy into the kineticenergy of the pair plasma efficiently. The efficiency in our simulation is only 1/60 of the totalluminosity. The results may depend on the profiles of the plasma heating rate. Simulationsfor other types of profiles of the plasma heating rate are worth studying. We do not simulatefor
L > erg s − , because the higher pair density requires high computational cost, whilethe temperature tends to decrease with L . This is another challenge for us, though there isa possibility of rapid extinction of pairs due to the lower temperature.Of course, the most idealized method to efficiently produce relativistic outflows is tocreate fireballs with relativistic temperatures. Relativistic temperatures for optically thickplasmas assure a higher density of pairs, which is preferable for both the energy efficiency andthe final Lorentz factor. However, as AT07 showed for steady solutions, even if the effectsof mildly relativistic motion are taken into account, the temperature of fireballs is close tothat of static pair equilibrium plasmas obtained by past studies. It is valuable to seek amuch faster solution ( U >
1) than AT07. As is usually seen in solutions of the accretiondisk, by varying the boundary conditions we may obtain another type of solution with hightemperature and high outward advection. However, it is difficult to make
U >
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This preprint was prepared with the AAS L A TEX macros v5.2.
11 – µ x dn xxxx γ / d µ ( , µ x ) [cm -3 ]=10 -3 =10 -2 =10 -1 =10 Fig. 4.— Angular distribution of photons at the outer boundary, where µ is the cosinebetween the direction of the photon and the radial direction. The results are plotted frombinned data with ∆ µ = 0 . . ×9