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A New Model for Bulk and Thermal Comptonization in Accretion Powered X-Ray Pulsars

Explore a new model for Comptonization in X-ray pulsars, discussing observations, photon sources, mathematical formalism, and applications to known pulsars. Analyze extreme physics scenarios in high- and low-luminosity pulsars, gravitational energy balance, levitation conditions, and radiation pressure dynamics.

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A New Model for Bulk and Thermal Comptonization in Accretion Powered X-Ray Pulsars

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  1. A New Model for Bulk and Thermal Comptonization inAccretion Powered X-Ray Pulsars Peter A. Becker (GMU) Michael T. Wolff (NRL) Kenneth D. Wolfram (NRL) Brent West (GMU)

  2. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Talk Outline • Observations, standard model • Categories of sources, basic physical relations • Photon sources, accretion geometry • Bulk and thermal Comptonization • Low-luminosity pulsars, bulk Comptonization model • High-luminosity pulsars, static magnetic model • Photon polarization modes, approximate dynamics • Bulk and thermal Comptonization in high-luminosity pulsars • Mathematical formalism and analytical solution • Alfvén radius and constraints on accretion geometry • Applications to Her X-1, Cen X-3, LMC X-4, X Per • Phase-averaged spectra, parameter correlations • Preliminary results for millisecond pulsar SAX J1808.4-3658 • XSPEC analysis of 4U 0115+63 • Conclusions and plans for future work

  3. Observations: Pulsating X-Ray Sources

  4. 4U 1538-522 Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Observations: Basic Model

  5. standing shock fan beam Theory: Basic Model binary companion accretion flow neutron star accretion disk

  6. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Observations: Pulsar Categories Bright Pulsars Faint Pulsars • More than 100 sources now known: New discoveries by RXTE, INTEGRAL • Wide luminosity range: Lx ~ 1034-38 ergs sec-1 • Wide range of pulsation periods: P~ 10-3-103 sec • Phase-averaged spectra: power-laws with quasi-exponential cutoffs • Ironemission lines and cyclotron absorption features often observed • Spectra usually fitted using ad hoc functions (power laws, exponentials, lines) • Static magnetic models do not reproduce the spectra very well Her X-1 X Per

  7. . * * * * * * * . * Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Observations: Extreme Physics • High-speed inflow: kinetic-gravitational energy balance free-fall velocity • Gravitational potential energy powers X-ray pulsar emission:

  8. * Eddington luminosity . Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Observations: Extreme Physics • Eddington luminosity: “levitation” condition for spherical case * • Eddington limit prorated by column area: • Critical luminosity for deceleration at stellar surface:

  9. * Eddington luminosity . Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Observations: Extreme Physics • Eddington luminosity: “levitation” condition for spherical case • Critical luminosity for deceleration at stellar surface: • Radiation pressure is important in X-ray pulsars: Lx ~ 1034-38ergs sec-1

  10. magnetic-centripetal force balance particle gyration speed cyclotron gyration frequency cyclotron energy • Thermal energy in X-ray pulsars: • Kinetic energy in X-ray pulsars: Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Observations: Extreme Physics • Cyclotron energy: • Magnetic energy is quantized in X-ray pulsars

  11. 4U 1538-522 Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Observations: Extreme Physics accretion flow • Radiation pressure dominates flow dynamics in sources with luminosity • In this case the radiation flux is locally super-Eddington • Gas decelerates to rest through a radiative, radiation-dominated shock • Spectrum is Comptonized Blackbody, Cyclotron, and Bremsstrahlung

  12. White, Swank, & Holt (1983) Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Observations: Faint Pulsars • Relatively low luminosities: Lx ~ 1034-36 ergs sec-1 • Energy indices  1, without strong cutoffs • Radiative transfer dominated bybulk Comptonization GX 304-1 X Persei

  13. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Observations: Faint Pulsars • Relatively low luminosities: Lx ~ 1034-36 ergs sec-1 • Energy indices  1, without strong cutoffs • Radiative transfer dominated bybulk Comptonization 4U 0352+30 GX 304-1

  14. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars • Static magnetic models treat magnetic effects in detail • Angular and energy dependences of cross sections are fully implemented • These models neglect the dynamics in the rapid inflow • Static magnetic models and dynamic thermal models do not reproduce the observed spectra very well • The nonthermalpower-law continuum is not well fitted F()(sec-1 cm-2 keV-1) (Mészáros & Nagle 1985) Log()(keV)

  15. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars • Static magnetic models treat magnetic effects in detail • Angular and energy dependences of cross sections are fully implemented • These models neglect the dynamics in the rapid inflow • Static magnetic models and dynamic thermal models do not reproduce the observed spectra very well • The nonthermalpower-law continuum is not well fitted F()(sec-1 cm-2 keV-1) (Mészáros & Nagle 1985) Log()(keV)

  16. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars • I had experience dealing with radiation-dominated shocks around black holes • Classical Rankine-Hugoniot shocks cannot describe stagnation on the neutron star • Following Basko & Sunyaev, I studied the effect of a radiative radiation dominated shock in the accretion column • The kinetic energy of the bulk inflow is efficiently transferred to the radiation field via electron scattering Velocity Strong, conservative (RH), radiation-dominated shock Strong, radiative, radiation-dominated shock • The kinetic energy is radiated away so that the gas can settle onto the stellar surface

  17. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Photon Sources Accretion Geometry: • Seed photons are produced via bremsstrahlung, cyclotron, and blackbody emission cyclotron • Photons are redistributedin energy via collisions with electrons bremsstrahlung blackbody • Electrons have bulk and stochastic (thermal) motion

  18. B e e p Inflow Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Photon Sources • Seed photons are produced via bremsstrahlung, cyclotron, and blackbody emission Bremsstrahlung Thermal Mound Polar Cap Neutron Star

  19. B e e p Inflow Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Photon Sources • Seed photons are produced via bremsstrahlung, cyclotron, and blackbody emission n=0 Cyclotron n=1 Thermal Mound Polar Cap Neutron Star

  20. B e Inflow Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Photon Sources • Seed photons are produced via bremsstrahlung, cyclotron, and blackbody emission Blackbody Thermal Mound Polar Cap Neutron Star

  21. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Comptonization • What is Comptonization? The equilibration of a photon distribution via electron scattering • Bulk Comptonization: The up-scattering of photons in a converging electron flow • Thermal Comptonization: The up-or down-scattering of photons in a stochastic (thermal) electron distribution • Bulk vs. Thermal Comptonization: Both processes can contribute (bulk gives power-law, thermal gives Wien bump and cutoff)

  22. . N0= monochromatic photon injection rate Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars This transport equation governs the distribution function, not the specific intensity Theory: Faint Pulsars Transport Equation Formalism (Becker & Wolff 2005): • The pure-bulk Comptonization transport equation is: time derivative escape through walls photon source bulk Compton diffusion along axis where fG= Green’sfunction v= plasma accretion velocity (v < 0) tesc= local timescale for escape through column walls ||= electron scattering cross section parallel to B field energy averaged z= height above stellar surface  = photon energy

  23. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars Gravity is considered in later models “Exact” Velocity Profile (Becker, ApJ, January 1, 1998): • Following Basko & Sunyaev (1975) and Davidson (1973), we can model the variation of the inflow velocity in a cylindrical) geometry. • Gravity is neglected, so that the momentum flux is conserved. • The kinetic energy is radiated away through the column walls. • The photon diffusion (escape) timescale is comparable to the dynamical (accretion) timescale (this is an eigenvalue problem). • The resulting “exact” velocity profile is given by: “exact”

  24. . M0= mass accretion rate Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Photon Escape Transport Equation Formalism (Becker & Wolff 2005): • The escape of photons through the column walls is modeled using an escape-probability formalism: or where r0= accretion column radius ┴= electron scattering cross section perpendicular to B field ne = electron number density

  25. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Faint Pulsars • Becker & Wolff (2005) obtained an exact solution for the Green’s function describingbulk Comptonizationin“cold” X-ray pulsars • Blackbody seed photons are produced in the dense thermal mound • Photons are upscattered due to collisions with electrons in shock • Spectra haveenergy indices1, because no cutoff is included • Bulk Comptonization model agrees with observations of faint, steep-spectrum pulsars X Persei GX 304-1

  26. White, Swank, & Holt (1983) Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Observations: Bright Pulsars • Relatively high luminosities: Lx ~ 1036-38 ergs sec-1 • Flatter spectra: energy indices <1 • Spectra show high-energy quasi-exponential cutoffs GX 304-1 X Persei

  27. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars • The bulk Comptonizationmodel cannot produce flat spectra or high-energy quasi-exponentialturnovers • This suggests thatthermal Comptonizationneeds to be included • Compton scatteringtransfers energyfrom high to low frequencies, producing the cutoffs and flattening the spectra • We have incorporatedthermalandbulk Comptonizationinto our current radiative transfer model

  28. B E e p Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Photon Polarization • Photons have either ordinary or extraordinary polarization Ordinary Mode • Ordinary photon E-field is in the plane of photon propagation and the magnetic field • No electron excitation is possible • Cross section isnon-resonant n=0 n=1 Thermal Mound Polar Cap Neutron Star

  29. B E e p Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Photon Polarization • Photons have either ordinary or extraordinary polarization Extraordinary Mode • Extraordinary photon E-field is perpendicular to plane of photon propagation and the magnetic field • Electron excitation is possible • Cross section is resonant n=0 n=1 Thermal Mound Polar Cap Neutron Star

  30. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Photon Polarization Ordinary Mode • Ordinary mode photons have a continuous (non-resonant) scattering cross section: • Above the cyclotron energy, the scattering cross section is Thomson • Parallel to the field direction (φ=0), the cross section is reduced by (ε/ εc)2for photons with energy ε < εc • Perpendicular to the field (φ=90), the cross section is Thomson for all photon energies

  31. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Photon Polarization Extraordinary Mode • Extraordinary mode photons have a resonant scattering cross section: • Above the cyclotron energy, the scattering cross section is Thomson except at the resonant energy ε < εc • Resonance is due to the change in the Landau level of the electron • For non-resonant photons, the cross section is reduced by (ε/ εc)2for energy ε < εc • The two modes can exchange photons via mode conversion • Outside the resonance, the cross section has no directional dependence

  32. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Photon Polarization Scattering Cross Sections • Most of the photons emitted by bright X-ray pulsars have energy ε < εc • Extraordinary mode photons with ε < εc see cross section T (ε/ εc)2 • Ordinary mode photons with ε < εc andφ=0 see cross section T (ε/ εc)2 • Ordinary mode photons with φ=90 see the Thomson cross section • The two modes can exchange photons via mode conversion • Hence the cross section parallel to the field is T (ε/ εc)2and therefore <<T • The dominant cross section perpendicular to the fieldis the ordinary one • We use a simplified two-dimensional model for the scattering cross section

  33. = angle-averaged electron scattering cross section _  . N0= monochromatic photon injection rate The cylindrical geometry could be replaced with a hollow or filled cone Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars Transport Equation Formalism (Becker & Wolff, ApJ, January 1, 2007): • The thermal+bulk Comptonization transport equation is: Bulk Kompaneets operator Thermal where fG= Green’sfunction v= plasma accretion velocity (v < 0) tesc= local timescale for escape through column walls ||= electron scattering cross section parallel to B field energy averaged energy averaged Te= electron temperature

  34. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Photon Sources Bremsstrahlung and Cyclotron: • In strong magnetic field, bremsstrahlung and cyclotron can be treated as a single process called magnetobremsstrahlung • Magnetobremsstrahlung photons are created in a dipole pattern, mainly as ordinaryphotons emitted perpendicular to the magnetic field (Lauer et al. 1983)

  35. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Photon Sources Bremsstrahlung and Cyclotron: • The magnetobremsstrahlung source can be approximated as the sum of bremsstrahlung and cyclotron sources • Bremsstrahlung emission is described by the spatially distributed, continuum, ordinary-mode source term includes low-energy cutoff • The low-energy self-absorption cutoff εabsis set using electron scattering energizes photons

  36. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Photon Sources Bremsstrahlung and Cyclotron: • The magnetobremsstrahlung source can be approximated as the sum of bremsstrahlung and cyclotron sources • Bremsstrahlung emission is described by the spatially distributed, continuum, ordinary-mode source term includes low-energy cutoff • The low-energy self-absorption cutoff εabsis set using • Cyclotron emission is described by the spatially distributed, monochromatic (resonant), extraordinary-mode source term

  37. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars Transport Equation Formalism (Becker & Wolff, ApJ, January 1, 2007): • The transport equation can be solved analytically using the approximatevelocity profile (Lyubarski & Sunyaev 1982) where v = 0 at surface • The approximate velocity profile can be stated in terms of z using: approximate • The exact velocity profile is given by: exact • The constant α is determined by setting the two profiles equal at the sonic point :

  38. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars Transport Equation Formalism (Becker & Wolff, ApJ, January 1, 2007): • The exact and approximate velocity profiles agree fairly well: approximate exact stagnation at stellar surface sonic point

  39. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars Transport Equation Formalism (Becker & Wolff, ApJ, January 1, 2007): • Using the approximate velocity profile, the transport equation can be rewritten as: where Measures escape time relative to accretion time • This equation is separable using the functions:

  40. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars Transport Equation Formalism (Becker & Wolff, ApJ, January 1, 2007): • The separation functionsg and h satisfy the differential equations: where Determines relative importance of bulk and thermal Comptonization • This eigenvalues for λ are determined by imposing boundary conditions on the spatial function g

  41. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars Transport Equation Formalism (Becker & Wolff, ApJ, January 1, 2007): • The spatial functiong satisfies the differential equation: This BC could be replaced with a free-surface condition that would allow a pencil beam with the general solution confluent hypergeometric functions • We require that g → 0 in the limit →  because of the strong advection by the fast inflow • We also require that dg/d→ 0 in the limit → 0 because the photon flux must vanish at the stellar surface

  42. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars Transport Equation Formalism (Becker & Wolff, ApJ, January 1, 2007): • The eigenfunction solution for g satisfying the BC’s is given by Laguerre polynomials with associated eigenvalues • The energy functionh satisfies the differential equation with λ = λn

  43. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars Transport Equation Formalism (Becker & Wolff, ApJ, January 1, 2007): • The solution for h is: Whittaker functions where • The Green’s function can be evaluated using the expansion: where the expansion coefficients Cn are computed using the orthogonality of the Laguerre polynomials

  44. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars Transport Equation Formalism (Becker & Wolff, ApJ, January 1, 2007): • Integration of the PDE with respect to energy yields the derivative jump condition • Substituting the expansion for fG yields

  45. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars Transport Equation Formalism (Becker & Wolff, ApJ, January 1, 2007): • Substituting in the solution for h gives: where the Wronskian is defined by • Hence we obtain • Next we utilize the orthogonality relation

  46. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars Transport Equation Formalism (Becker & Wolff, ApJ, January 1, 2007): • The result obtained for the expansion coefficient Cn is: • The corresponding exact solution for the Green’s function is given by: • The Green’s function for the escaping photon spectrum is:

  47. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars Transport Equation Formalism (Becker & Wolff, ApJ, January 1, 2007): • The Green’s function for the escaping photon spectrum is: kTe= 0.86 keV where = 1.15  = 0.20, 0.79 0= 0.1 • In terms of the escaping Green’s function becomes:

  48. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars Transport Equation Formalism (Becker & Wolff, ApJ, January 1, 2007): • We can’t resolve the vertical structure of the emission column • Hence the emission we see is the vertically integrated spectrum • The vertically-integrated Green’s function for the escaping photon spectrum is calculated using: kTe= 0.86 keV = 1.15  = 0.20, 0.79 • The exact solution is given by:

  49. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bulk vs. Thermal Comptonization Photon Energization: • The totalphoton energization rate is: total rate recoil losses thermal Compton bulk Compton • This can be rewritten as: • Total energization rate vanishes at the critical energy: Low-temperature limit

  50. Bulk and Thermal Comptonization inAccretion-Powered X-Ray Pulsars Theory: Bright Pulsars Theory Parameters: • The spectrum should approach the bulk Comptonization result in the limit Te  0 (  ) Column-integrated Green’s function for escaping photon number spectrum for different plasma electron temperatures =0.79 Becker & Wolff (2005) pure-bulk Comptonization spectrum based on exact velocity profile ε=εbulk 3.95 39.5 ∞ • The value of determines the importance of the Wien bump

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