Ingram & van der Klis (2013)

Transcription

1 An exact analytic treatment of propagating mass accretion rate fluctuations in X-ray binaries Adam Ingram Chris Done, Michiel van der Klis, P Chris Fragile Ingram & van der Klis (2013) 4 th September

2 Energy Flux GRO J Soft State Energy Hard State Done, Gierlinski & Kubota (2007) Truncated disk model

3 Truncated disk model Normalized flux Soft State Time (seconds) XTE J Hard State Ingram (2012)

4 Truncated disk model Frequency Power Frequency GX Frequency Power Frequency Ingram (2012) Soft State Hard State

5 Truncated disk model Frequency Power Frequency 1/ t visc (r o )

6 Truncated disk model Frequency Power Frequency 1/ t visc (r o )

7 RMS-flux relation L (arbitrary units) t (s) Uttley & McHardy (2001); Uttley, McHardy & Vaughan (2005)

8 RMS-flux relation L (arbitrary units) t (s) Uttley & McHardy (2001); Uttley, McHardy & Vaughan (2005)

9 RMS-flux relation Uttley & McHardy (2001); Uttley, McHardy & Vaughan (2005)

10 Frequency Lynden-Bell & Pringle (1974) Lyubarskii (1997) Propagating fluctuations r 1 MRI

11 Frequency Lynden-Bell & Pringle (1974) Lyubarskii (1997) Propagating fluctuations r 1 M (ν) 2 1/ t visc (r 1 )

12 Frequency Lynden-Bell & Pringle (1974) Lyubarskii (1997) Propagating fluctuations r 1 r 2 M (ν) 2 1/ t visc (r 1 ) 1/ t visc (r 2 )

13 Lynden-Bell & Pringle (1974) Lyubarskii (1997) Propagating fluctuations r 1 r 2 r 2 t visc (r) dr r 1 r

14 Propagating fluctuations Lyubarskii (1997); Arevalo & Uttley (2006), Kotov et al (2001)

15 Propagating fluctuations Flux Frequency x Power Time (s) Frequency (Hz) Arevalo & Uttley (2006) Ingram & Done (2011, 2012a)

16 Propagating fluctuations Flux Time (s) RMS (absolute) Flux Arevalo & Uttley (2006) Ingram & Done (2011, 2012a)

17 Propagating fluctuations Ingram & van der Klis (2013) 0.01 < A j 2 > χ ν2 =0.985 Power χ ν2 =1.037 < B j 2 > Mdot/mean 0 5 a k Frequency (Hz) Mdot/mean 0 5 b k Time (seconds) Time (seconds)

18 Propagating fluctuations Ingram & van der Klis (2013) 0.01 < A j 2 > χ ν2 =0.985 Power χ ν2 =1.037 < B j 2 > Multiply the two time series Mdot/mean 0 5 x k Frequency (Hz) Time (seconds)

19 Propagating fluctuations Ingram & van der Klis (2013) 0.01 < A j 2 > χ ν2 =0.985 < X j 2 > Power χ ν2 =1.037 < B j 2 > x k Frequency (Hz) Multiply the two time series Mdot/mean 0 5 Simulate10,000 realisations SLOW!!!!!!! Time (seconds)

20 Analytic propfluc Ingram & van der Klis (2013) 0.01 < A j 2 > χ ν2 =0.985 < X j 2 > Power χ ν2 =1.037 < B j 2 > Frequency (Hz) Convolve the two power spectra! One analytic calculation! FAST!!!!!!!

21 ies Analytic propfluc generated via the TK95 method, gethertogetx k = a k b k.thedft X j = N/2 k= N/2+1 A j k B k. Mdot/mean 0 5 a k Mdot/mean 0 5 x k Mdot/mean 0 5 b k (5). Since A j k is periodic on the in Time (seconds) Time (seconds) Time (seconds)

22 ies 1994) < X j 2 >= N/2 Analytic propfluc < s jk 2 >. (6) generated via the TK95 method, k= N/2+1 gethertogetx Substituting definition for k = s jk aand k busing k.thedft the fact that A j k B k 2 = A j k 2 B k 2,wefind < X j 2 >= N/2 k= N/2+1 < A j k 2 >< B k 2 >. (7) + cross terms. values Since of the analytic A functions j k is A(ν) periodic 2 and B(ν) on 2! the in- Since the averaging here is over infinite realisations, we can write this in terms of power spectra a x X(ν k j k ) 2 = A(ν j ) 2 B(ν j ) 2, (8) Mdot/mean 0 5 Mdot/mean 0 5 where denotes a convolution. We thus have an expression to obtain N/2+1values of the analytic function X(ν) 2 from N/ Time (seconds) Time (seconds) Time (seconds) simulations. We take our two input power spectra to be Lorentzians Mdot/mean 0 5 (5) As a demonstration, we compare our analytic calculation to b k Figu b k a lines simu lines

23 ies 1994) < X j 2 >= N/2 Analytic propfluc < s jk 2 >. (6) generated via the TK95 method, k= N/2+1 gethertogetx Substituting definition for k = s jk aand k busing k.thedft the fact that A j k B k 2 = A j k 2 B k 2,wefind < X j 2 >= N/2 k= N/2+1 < A j k 2 >< B k 2 >. (7) + <cross terms>. values Since of the analytic A functions j k is A(ν) periodic 2 and B(ν) on 2! the in- Since the averaging here is over infinite realisations, we can write this in terms of power spectra a x X(ν k j k ) 2 = A(ν j ) 2 B(ν j ) 2, (8) Mdot/mean 0 5 Mdot/mean 0 5 where denotes a convolution. We thus have an expression to obtain N/2+1values of the analytic function X(ν) 2 from N/ Time (seconds) Time (seconds) Time (seconds) simulations. We take our two input power spectra to be Lorentzians Mdot/mean 0 5 (5) As a demonstration, we compare our analytic calculation to b k Figu b k a lines simu lines

24 Analytic propfluc x k = a k b k. < X j 2 >= N/2 k= N/2+1 < A j k 2 >< B k 2 > 0.01 χ ν2 =0.985 < X j 2 > < A j 2 > Power χ ν2 =1.037 < B j 2 > Ingram & van der Klis (2013) Frequency (Hz)

25 Analytic propfluc Ingram & van der Klis (2013)

26 Analytic propfluc Ingram & van der Klis (2013)

27 Analytic propfluc Ingram & van der Klis (2013)

28 Analytic propfluc Frequency x Power ([σ/µ] 2 ) rings: χ ν2 = rings: χ ν2 = rings: χ ν2 = rings rings: χ ν2 =1.014 Frequency (Hz) Ingram & van der Klis (2013)

29 Analytic propfluc Fragile et al (2007) Ingram, Done & Fragile (2009) Ingram et al (in prep)

30 Analytic propfluc Frequency x Power ([σ/µ] 2 ) Multiplicative QPO χ ν2 =106.46/88 XTE J Δχ Frequency (Hz) Ingram & van der Klis (2013)

31 Conclusions The propagating mass accretion rate fluctuations model can explain the data Using Monte Carlo simulations to treat the model are very expensive and fitting lots of data isn t feasible We can calculate the effect of propagation analytically, making fitting data feasible! This is a general result - whatever the model for the variability process in the flow, our formulae can be used to calculate the effect of propagation We will release this as an XSPEC model

32 LOFT Ingram et al (in prep)

33 LOFT ratio Model Energy (kev) Ingram & Done 2012b

34 LOFT Hard State Intermediate State r o = 60; i = 60 r o = 7; i = 60 Rise / Fall Rise / Fall E (kev) E (kev)