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Transcript
The role of electromagnetism in tidal disruption events Andrej Čadež University in Ljubljana Why should global electomagnetic fields be important in large scale phenomena? • Freefall of pressureless fluid into a black hole. • Question: How can the tidal energy be “almost instantaneously” transformed into X rays? • An out of a sleeve proposal? Gravity directly couples to electromagnetism through dynamics. Pulsar slow down as an example of coupling dynamics to electromagnetism --- the case of Crab • It is often assumed that pulsars slow down as the result of electromagnetic radiation that they emit as rotating magnets and that their plerionic nebulae are illuminated by synchrotron radiation emitted in the hot corona of the neutron star. • However, detailed observations of the famous Crab nebula and its pulsar present some striking displays which, I believe, point to organized structures on a large scale and thus hint on the presence of the direct coupling of dynamics to electromagnetism. 3D Structure of Crab nebula (by spectroscopy) • Filaments, formed by line radiation from H, N and S (Hred, N-green, Sblue) form a regular cage spread over an almost spherical oval with a parsec across. The gravitational field alone can not form such a structure X-rays are emitted in a regular disk-jet pattern, completely contained within the line radiation cage Pulsar slow-down • If energy loss is magnetic dipole radiation then: • • Rotational phase residuals with respect to the braking law of the episode and braking index during the episode. The total number of turns during the ~30 years is 2 × 1010 . 𝑑𝐸𝑟𝑜𝑡 𝜔4 ∝− 𝑟𝑝 4 𝐵𝑠 2 so: 𝑑𝑡 4𝜋𝜇0 𝑑𝜔 = 𝜔 = −𝐾𝜔3 𝑑𝑡 𝜔𝜔 Braking index: 𝑛 = 2 = 3 𝜔 • But: Crab pulsar slow down closely follows a braking index law with the braking index staying constant for a few years and then abruptly changes to a different value between 2.2 and 2.6 • Braking index less than 3 can be understood, if the nebula has a resistive component Pulsar EM field interacting with nebula • Magnetic dipole radiation into dispersive medium described by conductivity and permittivity: 𝑗 = 𝜎𝐸 and 𝐷 = 𝜀𝐸 • The dispersion relation between 𝑘 and 𝜔 is 𝑘2 𝜔 2 𝑐 =𝜀 + 𝑖𝜇0 𝜎𝜔 • Poynting flux from pulsar: magnetic moment 𝑝𝑚 , angular velocity 𝜔 if dispersive shell is at radius 𝑟𝑠 is: • 𝑖𝑓 𝑘𝑟𝑝 ≪ 1 3 2 𝜔2 𝑐3 • 𝜀 > 0: 𝐿1 = 𝜇0 2 𝜔 6𝜋 • 𝜀 < 0: 𝐿1 = 𝜇0 2 𝜇0 𝜎 𝜔 𝑝𝑚 2 6𝜋 𝑟𝑠 𝜀 + 𝜇0 𝜎 𝑟𝑠 𝑝𝑚 2 𝜔𝜔 𝜔2 ⟹ 𝑛= ⟹ 𝑛=1 =3− 2 𝜔 2 𝑟𝑠 3/2 1+𝜀 𝑐 𝜇0 𝜎𝑐 (can not explain Crab braking) What is needed to explain average Crab braking index 𝑛 ≈ 2.5? 𝑛 =3− 1. 𝜎 = 𝑛𝑒 𝑒 2 ; 𝜈𝑐 𝑚𝑒 𝜈𝑐 2 2 3 𝜔 𝑟𝑠 1+𝜀 2 𝑐 𝜇0 𝜎𝑐 = 2 .5 requires 𝜎 = 1 3/2 𝜔 2 𝑟𝑠 𝜀 3 𝑐 𝜇0 𝑐 … electron collision frequency ⟹ 𝜈𝑐 = 2. 𝜈𝑐 = 𝜈𝑒𝑒 + 𝜈𝑒𝑖 + 𝜈𝑒𝛾 = 𝑒2 𝑛𝑒 3 3 4𝜋𝜀0 3 𝑚𝑒 𝑘 𝑇 4 𝜋 2 𝑛𝑒 3 𝜀2 = 3.5 × 10−12 𝐴 𝑉𝑚 𝜀 3/2 𝑟𝑠 𝑟𝑝 × 8.1 × 109 𝑐𝑚3 𝑠 −1 very high! 2 𝑛𝑒 + 2𝑍 𝑛𝑖 𝐿𝑜𝑔 Λ + 𝜎𝑇 𝐵2 𝛾 𝑚𝑒 𝑐 𝜇0 𝑒 Assume 𝑟𝑠 = 𝑟𝑝 , 𝜀~1, magnetic field on pulsar surface 𝐵0 = 108 T. 1. If 𝜈𝑐 = 𝑊𝐵 = 𝐵2 2 2 𝜈𝑒𝛾 then 𝐿𝑛𝑒𝑏 = 𝑚𝑒 𝑐 𝛾𝑒 𝜈𝑐 𝑛𝑒 𝑑𝑉 ≈ 2 𝛾𝑒 𝑛𝑒 𝜎𝑇 𝑐 𝑑𝑉 ; magnetic energy 2𝜇0 𝐵2 𝐵0 𝑟 8𝜋 3 𝑩0 2 3 𝑑𝑉; if 𝐵 is dipole 𝐵 = 𝑟𝑝 − 3𝑟 ∙ 𝐵0 5 , then 𝑊𝐵 = 𝑟𝑝 ~7 × 1034 𝐽 3 2𝜇0 𝑟 𝑟 3 𝜇0 𝑛𝑒 𝛾𝑒 2 𝑒𝑟𝑔 37 38 [𝑒𝑟𝑔 ] 𝐿𝑛𝑒𝑏 ~3 × 10 [ , 𝐿 ~2 × 10 𝑠] 𝑠 𝑠𝑙𝑜𝑤𝑑𝑜𝑤𝑛 1000𝑐𝑚−3 1000 and Can the nebular plasma satisfy the calculated conditions? • i.e. 1. 2. 3. 4. 5. 𝛾𝑒 2 ~3, 𝜀 > 0 1000 𝑛𝑒 ~0.3𝑐𝑚−3 , 𝛾𝑒 ~1 × 105 , 𝜀~1 𝑛𝑒 1000𝑐𝑚−3 𝑛𝑒 is consistently lower than electron density in filaments 𝛾𝑒 ~1 × 105 is not inconsistent with energy spectrum 𝜀~1 ??? Neutral plasma 𝑘 2 𝑐 2 = 𝜔2 − 𝜔𝑝 2 , so for 𝜔 < 𝜔𝑝 : 𝜀 = 0. 𝑛𝑒 𝑒 2 Plasma frequency 𝜔𝑝 = 𝜀 𝑚 ∗ is higher than the frequency of pulsar 0 𝑒 EM wave, so wave can not propagate and heat plasma, only locally non-neutral plasma allows propagation. To keep the population of high energy electrons tied to the nebula, long range electric and magnetic fields are required. Plasma can not be locally neutral! 1 𝜀 Size of heating region 𝑙ℎ ~ ℐ𝑚[𝑘] = 𝜇 𝜎𝑐 ~7.6 × 108 𝑚: about solar 0 size!!! Kohri, Ohira, Ioka: MNRAS2012 The role of the nebula and assumed structure • Reprocessing the power emitted by the central source • Providing electromagnetic field to prevent electron depletion in the central source Structure: • no line radiation detected inside the “cage” of filaments, but gamma emission present • high energy electrons must be present at very low density • Collision frequencies : • • 1 𝑛𝑖 𝑐 3 2 6 = 75 × 10 𝑍𝑖 𝐿𝑜𝑔 𝜈𝑒𝑖 13 𝑐𝑚−3 𝑣𝑒 1 𝑛𝑖 𝑐 3 3 = 41 × 10 𝐿𝑜𝑔 Λ 𝜈𝑒𝑒 13 𝑐𝑚−3 𝑣𝑒 Λ year year • Conclude: mean free path of both ions and electrons is longer than the size of the “cage”, electrons and ions effectively do not exchange energy. • Assume that the “cage” represents a cold (~10.000K) neutral plasma confinement Boltzmann and generation of electric field • Consider the ensemble of electrons and ions inside the “cage” as a two component ideal gas of particles acted on only by the electric field that they mutually create and is constrained to move inside the “cage”. The central engine is very small and heats electrons to very high temperature, while ions remain relatively cold because of their small 𝑒 𝑚 ratio. The distributions of electrons and ions in the phase space are described by their respective distribution functions 𝑓𝑒 𝑟, 𝑝 and 𝑓𝑖 𝑟, 𝑝 . In stationary state the distribution functions are constants of motion, so their Poisson brackets with the system Hamiltonian vanishes: 𝑓𝑒 , ℋ = 0, 𝑓𝑖 , ℋ = 0. 𝑁 𝑁 𝑒 𝑖 • System Hamiltonian: ℋ = 𝑖=1 𝐻𝑒 𝑟𝑖 , 𝑝𝑖 + 𝑖=1 𝐻𝑖 𝑟𝑖 , 𝑝𝑖 , here 𝐻𝑒 and 𝐻𝑖 are single particle Hamiltonians for electrons and ions (protons) respectively: • 𝐻𝑒(𝑖) = ±𝑒 𝑐𝐴0 − 𝑐𝛽 𝑖 𝑝𝑖 ± 𝑒𝐴𝑖 + 𝛼𝑔 𝑐 𝛾 𝑖𝑗 𝑝𝑖 ± 𝑒𝐴𝑖 𝑝𝑗 ± 𝑒𝐴𝑗 + 𝑚𝑒(𝑖) 2 𝑐 2 Electromagnetic 4-potential: 𝐴 = 𝐴0 , 𝐴1 , 𝐴2 , 𝐴3 𝑔00 𝑔𝑜𝑗 −𝛼𝑔 2 + 𝛽 𝑘 𝛽𝑘 𝛽𝑗 Metric tensor: 𝑔 𝑔𝑖𝑗 = 𝑖0 𝛽𝑖 𝛾𝑖𝑗 Neglecting gravity (𝛼𝑔 =1, 𝛽 = 0,0,0 , 𝛾𝑖𝑗 = 𝛿𝑖𝑗 ) and magnetic field (𝐴 = 0,0,0 ) as well as taking the nonrelativistic limit (even if not justified it does not but essentially change the result in this case) one can write the distribution functions as: 𝑝2 2𝑚𝑒 𝑖 𝑓𝑒(𝑖) = 𝐸𝑥𝑝 −𝛼𝑒 𝑖 − 𝛽𝑒 𝑖 ∓ 𝑒Φ , where α and β are the Lagrange multipliers eventually determining the numbers of electrons and ions and their energy. The charge density is: 𝜚𝑒 = 𝑒 𝑓𝑖 − 𝑑3 𝑝 𝑓𝑒 ℎ3 ; Coulomb equation ΔΦ = 𝜌 𝜀0 . A similar development as in Debye-Hückel theory leads to: Electric field in rarefied plasma of electrons and ions of different temperature • Field equation: Δ𝛿 = 𝜅 𝑒 −𝛿 − 𝑒 𝜏+ 𝜏− 𝛿 , 𝛼𝑓 𝜆𝑐 ℕ ), 2𝜋𝜏+ 𝑅 0 𝑚𝑒 𝑐 2 , 𝛽± 𝑒(Φ−Φ ) 0 • where: 𝜅 … (electrostatic energy/thermal energy) (κ = 𝜏± = 𝛿= , 𝑅 … radius of 𝑚𝑒 𝑐 2 𝜏+ the “cage”, ℕ0 … total number of particles to within a numerical factor, 𝛼𝑓 … fine structure constant, 𝜆𝑐 … 𝑟 Compton wavelength, Δ dimensionless Laplacian with respect to coordinates 𝑟, 𝜗, 𝜑 ⟶ 𝜉 = , 𝜗, 𝜑 . 𝑅 • Domain: 𝜉𝑚𝑖𝑛 < 𝜉 < 1 … 𝜉𝑚𝑖𝑛 𝑅 is the radius at which the central heating source takes over and 𝛿 reaches a constant value 𝛿2 . 𝑑𝛿 • Boundary conditions: total charge inside “cage”=0 (lim = 0), electric potential of “cage” with respect to ∞ 𝜉→1 𝑑𝜉 𝑒Φ0 is 0 (lim 𝛿 = − = 𝛿1 ). 2 𝜉→1 𝑚𝑒 𝑐 𝜏+ • Parameters determining solution: number of particles, total energy, ratio 𝜏+ 𝜏− , value 𝛿1 (related to potential difference between “cage” and central engine 𝛿2 ) must be determined a posteriori as the value at which the system reaches highest entropy for given volume energy and number of particles 15 Equation of state for 10 particles in R=1m sphere: 𝜏− = 𝜏+ a) Entropy (𝑆 𝑘𝑁𝑎𝑙𝑙) as function of energy (ℰ 𝑚𝑒 𝑐 2𝑁𝑎𝑙𝑙) and the electric parameter 𝛿0 . Entropy has local maxima for a fixed energy with respect to variation of 𝛿0 (marked by black), but, the maximum at 𝛿0 = 0 is higher than any other local maximum. The preferred thermodynamic state of equal electron-ion temperature plasma is locally neutral plasma with equation of state described by the Sackur-Tetrode equation 𝑉 4𝜋𝑚𝑒 ℇ𝑒 𝑆 = 𝑘𝑁𝑒 𝐿𝑜𝑔 𝑁𝑒 3ℎ2 𝑁𝑒 3/2 5 𝑉 4𝜋𝑚𝑝 ℇ𝑝 + + 𝑘𝑁𝑝 𝐿𝑜𝑔 2 𝑁𝑝 3ℎ2 𝑁𝑝 𝑉 4𝜋 𝑚𝑝 𝑚𝑒 ℇ 𝑆 = 𝑘𝑁 𝐿𝑜𝑔 𝑁 3ℎ2 𝑁 3 2 + 5 2 3/2 + 5 2 15 Equation of state for 10 particles in R=1m sphere: 𝜏− = 2𝜏+ States colored with colors of the bottom code all have higher entropy than that of the electrically neutral state at the same energy , many local entropy maxima generate states that are more probable than the state of local charge neutrality. Properties of non-neutral states • Charge density as a function of radius (for spherically symmetric boundary conditions!) The higher temperature component yields some entropy to allow the low temperature component to occupy more phase space, which makes the entropy of the complete system to increase (A process similar to crystallization?). The deep electrostatic well allows electrons in the central source to have very high kinetic energy, even if the pressure at 𝜉 = 1 is very low. Boltzmann and generation of magnetic field • The angular momentum lost by rotating pulsar is first absorbed by the nebula, so on must expect that in stationary state the nebula must have some constant angular momentum, which may generate magnetic field. The Boltzmann distribution functions must contain an additional vector Lagrange multiplier 𝛾 related to the total angular momentum. To make further calculations manageable we again take the nonrelativistic limit and write: • 𝑓𝑒(𝑖) = 𝐸𝑥𝑝 −𝛼𝑒 𝑖 − 𝛽𝑒 𝑝−𝑒 𝐴 𝑝2 𝑖 2𝑚𝑒 𝑖 ∓ 𝑒Φ − 𝛾 ∙ 𝑟 × 𝑝 Field equations • 𝜌𝑒(𝑖) = 𝒆 • 𝑗𝑥 𝒆(𝒊) = 𝒆 • 𝑗𝑦 𝒆(𝒊) =𝒆 𝑑3 𝑝 𝑓𝑒(𝑖) 3 ℎ = ∓𝒆 𝑝𝑥 −𝒆𝐴𝑥 𝑑 3 𝑝 𝑓 3 𝑚 ℎ 𝑝𝑦 −𝒆𝐴𝑦 𝑚 𝑑3 𝑝 𝑓 3 ℎ 𝟑 2𝜋𝑚𝑒(𝑖) 𝟐 𝛽𝑒(𝑖) ℎ2 = ∓𝒆 = ±𝒆 𝑚𝑒(𝑖) 𝛾2 ∓𝛾 𝑦𝐴𝑥 −𝑥𝐴𝑦 + 2𝛽 𝑥 2 +𝑦 2 𝑒(𝑖) 𝑒 −𝛼𝑒(𝑖) ±𝛽𝑒(𝑖) 𝒆Φ𝑒 𝑒 𝟑 2𝜋𝑚𝑒(𝑖) 𝟐 𝛽𝑒(𝑖) ℎ2 𝟑 2𝜋𝑚𝑒(𝑖) 𝟐 𝛽𝑒(𝑖) ℎ2 𝑚𝑒(𝑖) 𝛾2 𝒆𝛾 𝑦𝐴𝑥 −𝑥𝐴𝑦 + 2𝛽 𝑥 2 +𝑦 2 𝑒(𝑖) 𝑒 −𝛼𝑒(𝑖) ±𝛽𝒆Φ𝑒 𝑒 • 𝐴=𝑊 𝑥 2 + 𝑦 2 , 𝑧 {𝑦, −𝑥, 0} (since j is toroidal) • Maxwell: • ∆Φ𝑒 = 𝜌𝑒 +𝜚𝑖 𝜀0 • 𝛻 × 𝛻 × 𝐴 = 𝜇𝑜 𝑗𝑒 + 𝑗𝑖 𝑚𝑒(𝑖) 𝛾2 𝒆𝛾 𝑦𝐴𝑥 −𝑥𝐴𝑦 + 2𝛽 𝑥 2 +𝑦 2 𝑒(𝑖) 𝑒 −𝛼𝑒(𝑖) ±𝛽𝒆Φ𝑒 𝑒 • 𝑗𝑧 𝒆(𝒊) =0 𝛾 𝛾 𝑦 𝛽𝑒(𝑖) 𝑥 𝛽𝑒(𝑖) Simplified field equations and solutions • Boundary conditions all positive charge at center, total charge in spherical enclosure zero, enclosure “superconductive” • ∆Φ𝑒 = • ∆𝑊 𝑊 = − 𝒆 2𝜋𝑚 𝜀0 𝛽ℎ2 𝜇0 𝒆 2𝜋𝑚 𝛽 𝛽ℎ2 𝟑 𝟐 𝟑 𝟐 𝑒 −𝛼+𝛽𝒆Φ𝑒 𝑒 𝑒 −𝛼+𝛽𝒆Φ𝑒 𝑒 𝑚𝛾 𝛾𝒆 𝑊+2𝒆𝛽 𝑟 2 𝑆𝑖𝑛2 𝜃 + 𝑞 3 𝛿 𝜀0 𝑟 Δ = 𝜕2 𝜕𝑟 2 + 2 𝜕 𝑟 𝜕𝑟 + 1 𝜕2 𝑟2 𝜕𝜃2 + 𝐶𝑜𝑡 𝜃 𝜕 𝜕𝜃 𝑚𝛾 𝛾𝒆 𝑊+2𝒆𝛽 𝑟 2 𝑆𝑖𝑛2 𝜃 𝑞 𝛾 𝜏 60 0.133 0.05 0.1950.25 0.4 20.25 0.25 60 0.1 𝛾 Δ𝑊 = 𝜕2 𝜕𝑟 2 + 4 𝜕 𝑟 𝜕𝑟 + 1 𝜕2 𝑟2 𝜕𝜃2 + 3 𝐶𝑜𝑡 𝜃 𝜕 𝜕𝜃 Magnetic field behavior Increasing angular momentum makes magnetic stress stronger at equator 𝑞 𝛾 𝜏 60 0.05 0.25 𝑞 𝛾 𝜏 60 0.1 0.25 Increasing temperature with high angular momentum increases the pressure of the jet 𝑞 𝛾 𝜏 60 0.4 0.25 𝑞 𝛾 𝜏 60 0.4 0.5 Conclusions • The preliminary results suggest that very rarified collision-less plasma segregates into regions with alternating density of electrons and ions • Such plasma can exist in cavities behind shock fronts in interstellar space. • The potential difference between the center of the cavity and its surface grows with disparity between ion and electron temperature and allows electrons at the center to have large kinetic energy. • If angular momentum of the heating source is fed to the plasma, it naturally induces magnetic field and forms jets (and probably synchrotron disk as well)