6 High-Frequency Variability in Neutron-Star Low-Mass X-ray Binaries
269
significantly to the variability. Instead of the width, some authors use the quality
factor, Q, (sometimes also called the coherence, but we will reserve the name
coherence for another property of the QPO signal), defined as the ratio of the
centroid frequency and the FWHM of the QPO, Q = ν 0 /Δ, to characterise the
width of the Lorentzian. A narrow Lorentzian would then have a high quality factor.
The width or, equivalently, the quality factor, can provide information about the
lifetime of the process that produces the QPO, or how much the frequency of the
QPO changes over the time interval that was used to produce the PDS. On the
other hand, an initially very narrow QPO could be broadened if the oscillations
are damped in an intervening medium between the source and the observer, e.g., an
X-ray corona very close to the accreting object. Finally, the normalisation, N, equal
to the integral of the Lorentzian from −∞ to ∞, measures the total power of that
variability component. As mentioned earlier, the integral of the power density over a
certain frequency range gives the power contributed by, in this case, the Lorentzian
component that represents the QPO and, because of Parseval’s theorem, this is the
part of the variance in the light curve that is produced by the QPO. The amplitude
of the QPO is the square root of N, and is usually expressed as the rms variability of
the signal that produces the QPO divided by the average intensity of the source (and
normally given in percent), the so-called rms fractional amplitude, or rms amplitude
for short [18, 118]. When it is not normalised by the average intensity, this amplitude
is called the absolute rms variability [158]. Both the fractional and the absolute rms
amplitudes provide a measure of the variability of the light curve of the source over
the range of frequencies (or, equivalently, times scales) where the QPO dominates
the power spectrum. The rms amplitude as a function of energy provides information
about the radiative process that produces the QPO.
A narrow component, with a Q factor larger than 2, is usually called a QPO. The
definition is a bit vague (should a component with a Q factor just a bit smaller
or bigger than 2 be also called a QPO?), but it has been useful, and hence it
sticked. Components that have Q < 2 are usually called bumps and, if the central
frequency of this component is at ν 0 = 0, they are called zero-centred Lorentzians.
In general, all components that produce power over a broad frequency range are
called broad-band noise components. (Notice that, in this case, the word noise
refers to variability from the source.) Sometimes a broad-band noise component
can be fitted by a combination of several, relatively broad and weak, Lorentzians.
Since a Lorentzian is the Fourier transform of a sine (or cosine) function whose
amplitude drops exponentially with time, a so-called shot, there have been many
attempts to understand the variability in these sources in terms of a combination of
shot noise components, with different periods, amplitudes and decay times, that add
up together to produce the observed light curve. (The decay time of a shot in the light
curve is inversely proportional to the FWHM of the Lorentzian in the Fourier PDS.)
In recent years, however, it has been shown that the variability in these sources is
inconsistent with additive shots, but it is rather a multiplicative process [159]. This
raises the question of whether the bumps and broad-band noise components in the
PDS of these sources are in reality a combination of several narrow Lorentzians.
This standpoint is attractive because a broad-band noise component is complex, and
269
significantly to the variability. Instead of the width, some authors use the quality
factor, Q, (sometimes also called the coherence, but we will reserve the name
coherence for another property of the QPO signal), defined as the ratio of the
centroid frequency and the FWHM of the QPO, Q = ν 0 /Δ, to characterise the
width of the Lorentzian. A narrow Lorentzian would then have a high quality factor.
The width or, equivalently, the quality factor, can provide information about the
lifetime of the process that produces the QPO, or how much the frequency of the
QPO changes over the time interval that was used to produce the PDS. On the
other hand, an initially very narrow QPO could be broadened if the oscillations
are damped in an intervening medium between the source and the observer, e.g., an
X-ray corona very close to the accreting object. Finally, the normalisation, N, equal
to the integral of the Lorentzian from −∞ to ∞, measures the total power of that
variability component. As mentioned earlier, the integral of the power density over a
certain frequency range gives the power contributed by, in this case, the Lorentzian
component that represents the QPO and, because of Parseval’s theorem, this is the
part of the variance in the light curve that is produced by the QPO. The amplitude
of the QPO is the square root of N, and is usually expressed as the rms variability of
the signal that produces the QPO divided by the average intensity of the source (and
normally given in percent), the so-called rms fractional amplitude, or rms amplitude
for short [18, 118]. When it is not normalised by the average intensity, this amplitude
is called the absolute rms variability [158]. Both the fractional and the absolute rms
amplitudes provide a measure of the variability of the light curve of the source over
the range of frequencies (or, equivalently, times scales) where the QPO dominates
the power spectrum. The rms amplitude as a function of energy provides information
about the radiative process that produces the QPO.
A narrow component, with a Q factor larger than 2, is usually called a QPO. The
definition is a bit vague (should a component with a Q factor just a bit smaller
or bigger than 2 be also called a QPO?), but it has been useful, and hence it
sticked. Components that have Q < 2 are usually called bumps and, if the central
frequency of this component is at ν 0 = 0, they are called zero-centred Lorentzians.
In general, all components that produce power over a broad frequency range are
called broad-band noise components. (Notice that, in this case, the word noise
refers to variability from the source.) Sometimes a broad-band noise component
can be fitted by a combination of several, relatively broad and weak, Lorentzians.
Since a Lorentzian is the Fourier transform of a sine (or cosine) function whose
amplitude drops exponentially with time, a so-called shot, there have been many
attempts to understand the variability in these sources in terms of a combination of
shot noise components, with different periods, amplitudes and decay times, that add
up together to produce the observed light curve. (The decay time of a shot in the light
curve is inversely proportional to the FWHM of the Lorentzian in the Fourier PDS.)
In recent years, however, it has been shown that the variability in these sources is
inconsistent with additive shots, but it is rather a multiplicative process [159]. This
raises the question of whether the bumps and broad-band noise components in the
PDS of these sources are in reality a combination of several narrow Lorentzians.
This standpoint is attractive because a broad-band noise component is complex, and