Timing analysis
What is timing, and why is it important in astronomy?
This figure shows timelapse images of the
Geminga pulsar (bottom) and how they translate into a light curve
(top).
Timing analysis is one of the main tools an astronomer uses to
conduct scientific investigations of astrophysical systems using the
electromagnetic radiation that they produce. Where imaging gives
information on the spacial structure and spectroscopy gives information
on the elemental composition, ionization state and temperatures of
astronomical objects, timing analysis allows astronomers to study the
dynamic properties of an object. The targets of timing studies include
accretion
flows, oscillations, and accretion disk
instabilities, as well as magnetic field
configurations and instabilities in compact and noncompact stellar
systems and active galaxies.
Anyone who has played a musical instrument knows how important it is
to tune the instrument. Occasionally, a tuning fork is used to do that.
A typical tuning fork used by musicians is for a pure A tone, of 440 Hz.
But how does that apply in this situation? One Hertz (Hz) is one wave, or cycle, per second. A pure tone like an A
travels just like a wave through the air, received by the eardrum as a
signal at 440 waves per second. Now, imagine a musical chord. A Cmajor
chord is made up of the notes C, E and G. The chord can be decomposed
into three waves, each having a different frequency in Hz. In fact, any sound can be
decomposed into a number of frequencies — some stronger, some
weaker.
Sound is not the only type of data that can be decomposed into a
number of frequencies. We can also think of time series (a series of
data point values of a variable taken at successive intervals) in this
way. Astronomers use a number of tools that allow them to take time
series data and study it in terms of the frequencies present in the
data. This is useful to astronomers because it allows them to determine
periodic components of the data, which can then be related to physical
properties of the system, like rotation, binary period, and so on.
Methods of timing analysis
Fourier transform
A Fourier transform is a mathematical operation that changes data
from time domain to frequency domain, which allows the data to be
represented as the sum of a series of sines and cosines at various
amplitudes, that occur at specific points in time, rather than a
continuous function.
Astronomers use Fourier transform to visualize the data as a large
number of periodic components, each with a different period and a
different strength. Very strong components may appear in data where
there is a periodicity present. Astronomers then must estimate the
significance of this feature (which is another way of saying that they
must determine how much confidence they have that it is a true
periodicity in the data). When astronomers are looking for new
phenomena, such as pulsars, they are very conservative in what they
take to be a real signal. A frequency has to be many times stronger than
it would be in a random data set to be taken seriously. A Fourier
transform is based on a mathematical theorem that any signal can be
decomposed into an infinite number of sine waves. This means that
Fourier analysis works best when the signal we are looking for is very
similar to a sine wave.
Fourier analysis of the time series on the
left results in the Fourier power spectra on the right. It is clear
that the analysis does identify the periodic behavior in the
data.
Epoch folding
Another method that is useful for a signal with arbitrary shape is
epoch folding. This is done by chosing a range of periods, and "folding"
the data at those periods.
As an example, say an astronomer has one reading per second for 500
seconds of an object. She suspects that the object has a period of 25
seconds, so whe will "fold the data on a period of 25 seconds." To do
this, she would start with the first 25 points, and then add the second
25 points (points 2650) to the first 25. Now add the third set of 25
points, and so on, until she reaches the end of the data set. If the
period of the source is not close to 25 seconds, then times where the
signal is high will cancel out with times where the signal is low in
each of the 25 "bins", and the resulting epoch folded light curve will look sort of flat
and boring. If, however, 25 seconds is very close to a period that is
actually in the data, then bins where the signal is high will add
together, and bins where it is low will add together, and the result
will be a very nice looking epoch folded light curve. At this point,
the astronomer must again assess how significant the resulting light
curve is. This is generally done by looking at the spread of values, or
errors, in the typical bin, and comparing how much higher the high bins
are than the standard error.
Light curve for the Xray binary system
Circinus X1 taken by the AllSky Monitor aboard the RXTE
satellite.
Wavelet analysis
Fourier transforms are great at finding systems that have a
consistent periodicity, like pulsars. However, many sources of interest
have periodic signals that vary over time and in their amplitude; other
sources have many frequencies present in their data. For sources like
this, wavelet analysis is used. Wavelet analysis also decomposes a
signal into time and frequency space simultaneously.
Wavelet analysis is used primarily on signals where there are many
frequencies present in a data set, that may change with time. Both the
epoch folding and the Fourier techniques tell you only about what
frequencies are present in an entire data set, but nothing about how the
strengths or values of these frequencies may change with time. Wavelet
analysis is designed to capture timevarying frequencies and display
them. Since more information is available than was gained from Fourier
or epoch folding techniques, we need to represent the results of wavelet
analysis differently. This is usually done in the form of an image that shows what frequencies
were present at what times in the data set. While wavelet techniques are
very powerful in helping astronomers uncover complex nonstationary
behavior in data sets, they must be used with caution. The extra
information that can be gained comes at the cost of increased difficulty
in evaluating whether the result is significant. However, the fact that
many of the most intriguing astronomical data sets appear to contain
time varying frequency components demands that astronomers use every
tool in their time series analysis toolbox, including wavelets.
Use Hera to try your hand at timing
analysis
with real astronomy data.
Updated: June 2011
