Propagating fluctuations in the mass accretion rate of a precessing flow as a power spectral model for black hole binaries

This model and description are due to Adam Ingram.

The power spectrum of black hole binaries (BHBs) in the rise to outburst generally consists of a quasi-periodic oscillation (QPO) and additional band limited noise. This can be phenomenologically modelled using a number of broad Lorentzians for the band limited noise and a narrow Lorentzian for each harmonic of the QPO. Here, we assume a physical origin for this variability in a truncated disc / hot inner flow geometry. Mass accretion rate fluctuations are generated everywhere in the flow (primarily) at the local viscous frequency and propagate inwards towards the black hole. We also assume that the entire flow is precessing due to frame-dragging (Lense-Thirring precession), which gives rise to a QPO. The key assumption is the surface density profile as this sets the precession (and therefore QPO) frequency but also, by mass conservation, sets the viscous frequency as a function of radius. Further details of the model, propfluc, are included in Ingram & Done (2011), particularly in the appendix.

We assume that the surface density is given by a smoothly broken power law, consistent with the results of general relativistic magneto hydrodynamic (GRMHD) simulations. The break occurs at the bending wave radius, rbw, with the power law dependence on radius parametrised by zeta for r>>rbw and by lambda for r<<rbw. The sharpness of the break is given by another parameter, kappa. The QPO frequency is calculated self-consistently and the power spectrum of the QPO is represented by Lorentzians centred at f_QPO, 2f_QPO, 3f_QPO and 1/2f_QPO to represent the 1st, 2nd, 3rd and sub harmonics respectively.

As this is a model for the power spectrum rather than the spectral energy distribution, the process for loading the data into XSPEC is slightly different. First of all, a power spectrum can easily be created from a light curve using powspec from the XRONOS package (for example). The power spectrum will then typically be written in the form
f, df, P, dP
where f is the frequency, P the power and df and dP denote the corresponding error. XSPEC, however, expects a .pha file in the form
Emin, Emax, F(Emax-Emin), dF(Emax-Emin)
where Emin and Emax are the lower and upper bands of each energy bin and F is the flux. It is therefore necessary to create a data file with the inputs
f-df, f+df, 2Pdf, 2dPdf.
This data file can then be converted into a .pha file using flx2xsp which will also generate a diagonal response (.rsp) file. The data can then be read into XSPEC in the usual way, using the command data 1:1 'filename'.pha. The command ip euf will then present the data and model in terms of fP plotted against f (even though, by default the axes will be labelled in the units of EF vs E). More information about this procedure can be found in appendix A in Ingram & Done (2011).

Parameters in propfluc:

1. Sigma0: normalisation of the surface density in dimensionless units (see Ingram & Done 2011).
2. rbw: radius at which the surface density begins to drop-off.
3. kappa: governs the sharpness of the break.
4. lambda: power law dependence at small radius.
5. zeta: power law dependence at large radius.
6. Fvar: governs level of intrinsic variability generated at each annulus.
7. fbmin: the viscous frequency at the truncation radius.
8. ri: inner radius of the flow.
9. sig_qpo: width of the QPO - higher harmonics are tied to have the same quality factor.
10. sig_subh: width of the sub-harmonic - this can have a different quality factor to the other harmonics.
11. n_qpo: normalisation of QPO (fundamental).
12. n_2h: normalisation of 2nd harmonic.
13. n_3h: normalisation of 3rd harmonic.
14. n_subh: normalisation of sub-harmonic.
15. em_in: emissivity index.
16. dL: fractional Gaussian error introduced to the model light curve in order to give white noise. Usually best to set this to zero and fit the model to a white noise subtracted power spectrum.
17. nn: sets the length of the generated light curve. Duration = 2^nn dt. Must be an integer as the model uses fast Fourier transforms. We recommend nn=22 for fitting but model runs faster for nn=20.
18. Ndec: sets the number of annuli the flow is split up into. This is number of rings per decade in viscous frequency. We recommend Ndec=15.
19. Normalisation included by XSPEC. Fix to unity.

Download the tar file propfluc.tar. If you are using this without any other models in xspec12 then untar this in a clean directory, fire up xspec12 and type @load. The model should then be set up. Otherwise, build it as normal with other local models.

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