Self-Similar Scaling of Solitons and Compactons in Relativistic Jets

1. HST Image of Quasar Jets Nonlinear Dispersion Relationship (3,4) Quasar 3C120 Keith Andrew, Michael Carini, Brett Bolen Keith.Andrew@wku.edu Mike.Carini@wku.edu Brett.Bolen@wku.edu Department of Physics and Astronomy Western Kentucky University Bowling Green KY Self-Similar Scaling of Solitons and Compactons in Relativistic Jets From Dr. Marsher BU websiteFrame from a conceptual animation of 3C 120 created by COSMOVISION                          Large Amplitude Nonlinear Fields-Solitons Nonlinear Schrödinger Equation ://rst.gsfc.nasa.gov/Sect20/h_accretion_disk_02.jpg&imgrefurl=http://rst.gsfc.nasa.gov/ Soliton- long lived nondissipative wave form where nonlinear amplitude growth is balanced by dissipative losses, need not be topological in origin Compacton-long lived wave form with well defined functional relationship between amplitude, width and speed of propagation, no exponential envelope (3,7) AbstractThe jet forming inner region of an object containing a massive Kerr black hole will contain a hot turbulent lepton plasma that can be modeled by a system of relativistic MHD-NPDE. The nonlinearities in these equations give rise to long lived localized soliton solutions and soliton like solutions known as compactons that exist at all length scales. These objects could give rise to structure formation at all locations along the jet that appear as shock bows, vortices or knots that would cause luminosity variations along the jet axis. Here we study the scaling behavior of these solutions in jet environments by using the dimensionless scaling rules from the Buckingham Pi Theorem with the self-similar scaling of nonlinear wavelets in a system of relativistic NPDE to estimate the resulting fractional change in jet luminosity. Buckingham’s Pi Theorem Only dimensionless quantities needed (1) The field components F are representative of vector potential components or electric field or magnetic field components. • Existence Requirements • Convection • Dispersion • Diffusion • Nonlinear G-Newton’s constant of gravitation B- magnetic field l- characteristic length of the jet -ρ-densityof surrounding medium -c-speed of light -v-jet’s ejection velocity M- core mass dM/dt- mass accretion rate L-jet luminosity L=L(l, v, c, G, M, dM/dt, B, ρ) Wavelet Scaling Rules Multiscale Wavelet Similarity Analysis For Nonlinear PDEs (1,5,6) Waves characterized by: 1. Amplitude A 2. Width w 3. Velocity v-limited by dispersion relationship Localized soliton and compacton solutions expanded with Gaussian Family Wavelets Odd power η=3 MHD Equations Even power η=4 Conclusions Fractional change in luminosity Relativistic Hydrodynamic Equations in terms of the potentials (3) Luminosity ~ Field Amplitude Squared Width constrained by jet diameter, velocity constrained by dispersion For a given scale, j, the similarity transformation Maps the NPDE->single scale algebraic constraint of the form F(A,w,v)=0 for localized soliton like solutions. 1. Espinosa, M. H., Mendoza, S. Hydrodynamical scaling laws for astrophysical Jets, arXiv.astro ph/0503336 v1, (Mar 2005) 2. Tevecchio, F., Jets at all scales, arXiv.astro-ph/0212254v1, (Dec. 2002) 3. Marklund, M., Tskhakaya, D.D, Shukla, P.K., Quantum Electrodynamical shocks and solitons in astrophysical plasmas, arXiv.astro-ph/0510485 v1 Jan. 2002) 4. Schwinger, J., On Gauge Invariance and Vacuum Polarization, Phys. Rev. 82, 664 (1951) 5. P G Kevrekidis, V V Konotop, A R Bishop and S Takeno 2002 J. Phys. A: Math. Gen.35 L641-L652 6. Ludu, A, O’Connell, R.F., Draayer, J.P., Nonlinear Equations and Wavelets, Mulit-Scale Analysis, arXiv.math-ph/0201043 v1(Oct 2005) 7. Tatsumo, T., Berezhiani, V. I., Mahajn, S. M., Vortex Solitons-Mass, Energy and Angular momentum bunching in relativistic electron-positron plasmas, arXiv.astro-ph/0008212 v1, (Aug 2000) From BU website: http://www.bu.edu/ blazars/research.html U-internal energy P-pressure -ρ-density Φ-external gravitational potential V-velocity vector field B-magneticfield