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The flux released by the accretion process is also observed to vary in time and this provides further information on the mechanisms involved. There are various ways in which information about the nature of the variability may be extracted from the raw lightcurves. These methods are commonly divided into time-domain and Fourier-domain methods. In time-domain methods, statistics are calculated directly from the flux at each time, while in Fourier-domain methods, a Fourier transform is first applied to the lightcurves, separating fast and slow (high and low frequency) changes in the lightcurves.

1.5.1

Time domain

The simplest variability property is the amount of variability. The observed variance (S2) will have a contribution from measurement uncertainties (σerr) as well as intrinsic variability. An estimator of the intrinsic variability is the excess variance (Vaughan et al., 2003a),

σ_XS2 = S2− ¯σ_err2 (1.3)

which may also be viewed as its fractional amplitude (Edelson et al., 1990)

F_Var= s S2_{− ¯σ}2 err ¯ x2 (1.4)

Where two lightcurves (A(t), B(t), for example in different energy bands) have been measured, relations between them may also be investigated. They can be compared by testing the extent to which they vary together in the same way that any pair of sets of values may vary together, or be correlated,

r2= Cov(A(t), B(t)) 2

Var(A(t))Var(B(t)) (1.5)

Additionally, time series are located in time; this provides further information on how the series relate, for example if changes in one series are reflected some time later in another. This is expressed as the cross-correlation function (CCF, or auto-correlation function, ACF, if A = B),

CCF(τ) = Cov(A(t), B(t + τ))

pVar(A(t))Var(B(t)) (1.6)

Typically, astronomical lightcurves do not consist of equispaced measurements due to the constraints of observatory operations, Sun avoidance etc. (and where equispaced data are available, Fourier methods may be more powerful). Therefore, some adjustment is required to account for the different spacings in the data, which may also differ between the two lightcurves which are to be compared.

If both lightcurves are generally well-covered, it is reasonable to simply interpolate each lightcurve onto a regular grid of times and calculate the CCF as before (producing the ICCF). Where the sparsity or irregularity of each lightcurve is more significant, interpolation will not give a reliable estimate of the true lightcurve so methods relying on the measured values

should be used. This is typically done with the discrete cross-correlation function (DCF, Edelson et al., 1990)

DCF(τ) = ⟨(Ai− ¯A)(Bj− ¯B)⟩|ti−tj−τ|<ε

pVar(A(t))Var(B(t)) (1.7)

Here, the average across all points of a given offset, τ, is replaced by the average across offsets within a given range around τ.

Any of these variants of the cross-correlation function may be used to show the relation between the processes responsible for producing the flux in each lightcurve: a peak at a given lag, τ, shows the difference in time between variations in the two lightcurves. This time difference can be attributed to the time taken for the signal which determines the changes (e.g. a photon flux) to propagate between the two regions (e.g. the light travel time). Hence distances between the two regions can be inferred.

The measured DCF depends on how the times of observations correspond to the variability timescales of the lightcurve; once observations are sparse and irregular, it is hard to determine analytically how each observation contributes to the DCF. Therefore, errorbars on quantities derived from the DCF (such as time lags) are usually calculated by bootstrapping; details of the method implemented are given at the point in the text where this is used.

1.5.2

Fourier domain

The variability in a process can also occur rapdily or slowly; a convenient way to express this (since it has many convenient mathematical properties) is through the Fourier transform:

˜ A( f ) =

Z ∞ −∞A(t)e

−i2π f t_{dt (1.8)}

or in discretised form, as applicable to observations of a lightcurve: ˜

A( f ) =

_∑

A(t)e−i2π f t (1.9)

where T is the set of measured times in the lightcurve.

While this calculation can be made for any f and T , it is usual to use equispaced T = {0,t, 2t, ..., (N − 1)t} and f = {−1/2t + 1/Nt, −1/2t + 2/Nt, ..., 1/2t} since this is invertible and the resulting ˜A( f ) are statistically independent. Further, for real A(t) (as is the case for lightcurves), ˜A(− f ) = ˜A∗( f ), so the full information is carried in the positive frequency components.

Since a set of complex numbers is no easier to interpret than a lightcurve, various products are produced from combinations of Fourier transforms of one or more lightcurves. These typically separate the amount from the relative time of any variability. Each complex ˜A( f ) may be considered in its polar form, a( f )eiφ ( f ), with a and φ real. a is then the amplitude (amount) of variability at a given frequency and φ is the phase (delay in time). This phase is usually with respect to the (arbitrary) start of the measured lightcurve, so only phase differences have physical meaning.

The power spectrum P( f ) = ˜A( f ) ˜A∗( f ) = a2( f ) (where∗denotes the complex conjugate) isolates the magnitude of the variability at each frequency. This can be smooth if there are no specific timescales on which variability occurs most strongly, or have peaks if variability occurs at particular frequencies. These can be very sharp (e.g. spin period of a neutron star, resonant frequency of a pipe) or somewhat broadened (e.g. precession of a gas disc, resonant frequency of a soft cavity).

There is significant scatter in a power spectrum taken from a single lightcurve. Using a longer lightcurve does not improve this as it generates a power spectrum with more points over a wider range of frequencies. Instead, the scatter is reduced by splitting a lightcurve into many sections and averaging over at least each section and sometimes also adjacent frequencies. The power spectrum is then

P( fi) = 1 MN N

∑

∑

Provided MN is large enough, the errors on such power spectra (or other products) may be approximated as Gaussian; typically, 50 is considered sufficient (Vaughan et al., 2003a). Another common product of Fourier analysis is lags between different lightcurves. The phase lag ∆φ ( f ) = φA− φBis taken from the Fourier transforms (e.g. Nowak et al., 1999) and converted to a time lag as

∆t( f ) = ∆φ ( f )

2π f (1.11)

Higher order products may also be constructed; for example the bispectrum,

which can be used to study connections between different components of variability (e.g. Arur and Maccarone, 2019; Maccarone and Coppi, 2002; Maccarone and Schnittman, 2005).