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

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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. This version of the model is calculated analytically using the formalism of Ingram & van der Klis (2013) and is consequently a lot faster than the original model.

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.

There is some extra functionality to this version of the model. Parameters 15 and 16 are respectively `hard' and `soft' band emissivity indices (gamma_h and gamma_s). Parameter 20 (mode) specifies whether the output is the hard band power spectrum (mode=1), soft band power spectrum (mode=2) or the time lag between the two energy bands (mode=3). If gamma_h > gamma_s, the model predicts hard lags. Note, if mode=1, gamma_s should be fixed and if mode=2, gamma_h should be fixed (to anything, the fit is insensitive to the parameter). Parameter 21 (conv) specifies whether the QPO signal is added to the broad band noise signal (conv=0) or multiplied with it (conv=1). The latter is more physically motivated (Ingram & van der Klis 2013). Handy tips for working with the model are provided in Rapisarda, Ingram & van der Klis (2014).

As this is a model for the power spectrum (or lag spectrum if mode=3) 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 ftflx2xsp 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). A similar process can be used for converting a lag spectrum into a format which can be read in by XSPEC.

Parameters in propfluc:

1.	Sigma0	normalisation of surface density profile.
2.	rbw	bending wave radius.
3.	kappa	surface density profile parameter.
4.	lambda	surface density profile parameter.
5.	zeta	surface density profile parameter.
6.	Fvar	fractional variability per decade in radius.
7.	ro	truncation radius.
8.	ri	inner radius of the flow.
9.	Q	quality factor, Q=centroid/FWHM, of the QPO harmonics.
10.	Qsub	quality factor of the subharmonic (can be different from Q).
11.	n_qpo	normalisation of fundamental.
12.	n_2qpo	normalisation of 2nd harmonic.
13.	n_3qpo	normalisation of 3rd harmonic.
14.	n_05qpo	normalisation of sub-harmonic.
15.	gamma_h	hard band emissivity index.
16.	gamma_s	soft band emissivity indenx.
17.	M	BH mass in Solar masses.
18.	a	dimensionless spin parameter.
19.	Ndec	number of rings per decade in radius.
20.	mode	1 = hard band PSD, 2 = soft band PSD, 3 = lag spectrum.
21.	conv	0 = add QPO, 1 = multiply QPO.
22.	Norm	included by XSPEC - fix to unity.

The final four parameters are all just settings, not free parameters.

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.