A New Analytic Model for the Production of X-ray Time Lags in Radio Loud AGN and X-Ray Binaries John J. Kroon Peter A. Becker George Mason University MARLAM 2 10 October, 2014

What is a Time Lag? Van der Klis et al. (1987) Take FT of hard and soft channel time series:  H(ν f ) and S(ν f ) Compute Complex Cross Spectrum, C, given by: C=S*(ν f ) H(ν f ) Time Lag: Angle Arg(C) is phase lag complex conjugate Time lags are commonly observed from accreting black holes!

Cyg X-1 from Miyamoto et al How are observed time lags produced? Time lags computed from upscattering of monoenergetic input photons Compton? Cyg X-1 D~1900 pc L~3-4*10 5 L sol M~15 M sol Dependence of time lag profile on injection spectrum!

Hua, Kazanas, Cui (1999) Monte Carlo Simulation n e ~1/r n e ~1/r 3/2 Homogeneous Coronae

Main Ideas HKC found quasi-monochromatic injection in an inhomogeneous cloud can reproduce the time lags. Time lag results depend sensitively on the density profile. What about the injection spectrum? Can we solve the time lag problem exactly for a variety of injection scenarios (injection spectrum AND density profile)?

Main Ideas Develop new computational model for thermal Comptonization as a mechanism for time lags. Obtain analytical solution for the fundamental Green’s function for the scattering process. Green’s function allows us to explore a variety of injection scenarios.

: Electron Blue: HARD X-ray Red: SOFT X-ray Monochromatic Injection 2 keV and 11 keV FRED curves Miyamoto Scenario Monochromatic injection Fast Rise and Exponential Decay due to spatial diffusion and Compton energization More time required to reach high energies Time Lag is a weak function of Fourier frequency ε 0 =0.1 keV time (s) Compton energy exchange Corona Detector Initial photon energy counts

Corona Broadband Injection time (s) 2 keV and 11 keV FRED Curves Our Model Broadband injection Fast Rise in both channels is due to prompt escape  Rapid time variation  Highest Fourier Frequency Prompt escape is coherent, therefore no Time Lag at high Fourier frequency (short timescale) Exponential Decay is due to gradual escape from cloud Yields constant Time Lag at low Fourier frequency (long timescale) Compton energy exchange Detector : Electron Blue: HARD X-ray Red: SOFT X-ray counts

Mathematical Model The transport equation is as follows: Here we focus on the inhomogeneous Corona, with the definition Next, we collect like terms by multiplying through by z α and let α=1 Where we have introduced the dimensionless parameter,

Mathematical Model Define the Fourier Pair Make all the dimensionless variable substitutions and simplify The Fourier Transform of the INHOMOGENEOUS dimensionless Transport Equation is given by

A Word on the Quiescent Spectrum Quiescent spectrum derived from time- independent transport equation. Spectrum produced from Comptonization of continually injected monochromatic photons. Conversely, time lags are produced from sudden localized flash of bremsstrahlung seed photons.

Bremsstrahlung Need a particular solution for localized broadband injection. Bremsstrahlung energy convolution! Particular Solution Green’s Function where,, x 0 is injection energy

Theoretical Fourier transform gives time lags using Van der Klis et al. (1987) formalism with,

Model Parameters η=sets the minimum number density Θ=kT e /m e c 2 ; dimensionless electron temperature ε abs =bremsstrahlung self-absorption low cut-off energy R=physical radius of the corona z 0 =injection radius z in =r in /R; location of inner free-streaming radius (maximum electron number density) There are 6 free parameters  Integrated model  Comptonization creates both the Quiescent X-ray Spectrum and the Time Lags  Three parameters are tied down by fitting the Quiescent X ray Spectrum: (z in,η,Θ)  The other parameters (z 0,R,ε abs ) are set by comparing the Time Lags with the data

Integrated Model Results for GX Set by quiescent spectrum fit T e =3.56x10 8 K, Θ=0.06  * =5 SAME set of parameters for both Quiescent Spectrum and X-Ray Time Lags for GX

ε abs =1.6 keV  0 =  * =2.5 R=30,000 km Black Curve Homogeneous Cloud, Localized Bremsstrahlung Injection (Kroon & Becker 2014) Inhomogeneous (and Homogeneous) Cloud, Quasi-Monochromatic Localized Injection (HKC 1999) Cyg X-1 in Different States Here Cyg X-1 in Same State Here

Comparison for Cyg X-1 Kroon & Becker 2014/KB 2015 Analytic diffusion model Optically thick ( τ ~ ) Small cloud (R ~ 3-5x10 4 km) Cool (T e ~3-7x10 8 K ) Cyg X-1 Spectrum is fit equally well with homogeneous and inhomogeneous model. Compton y-parameter must be similar in each of these models Moderate cloud heating Monte Carlo simulation Optically thin (τ ~ 1) Large cloud (R ≥300,000 km ) Warmer (T e ~1.2x10 9 K ) Cyg X-1 Spectrum is fit equally well with homogeneous and inhomogeneous model. Compton y-parameter must be similar in each of these models Problematic cloud heating HKC 1999

Conclusions Broadband injection fits the time-lag data better than monochromatic injection. Injection at outer edge of cloud works best! Both homogeneous and inhomogeneous number density profiles CAN reproduce time lags IF injection is broadband. Corona Density Profile Broadband Injection(Quasi-)Monochromatic Injection Inhomogeneous (1/r) Kroon & Becker 2015 (in progress) HKC (1999) Homogeneous Kroon & Becker (2014)  Miyamoto (1988), Kroon & Becker (2014)

Thank you!

Appendix: Inhomogeneous Model !!! Constant parameters with subscript are evaluated at outermost radius. Constant parameters with subscript ‘0’ are evaluated at injection radius. !!!

The outer and inner boundary conditions are… and The series solution for the Fourier Transformed Green’s function is…

Derivation of Quiescent Spectrum from Distributed Monochromatic Injection