Light Curves, Power Spectra, &
Periodicities
a guided study of the x-ray
binary Centaurus X-3
by John Kolena
november 2003
Cen X-3, discovered nearly 40 years ago, beautifully
illustrates the
process of astronomical discovery.
Using the x-ray data, we can
reconstruct the contents, properties, and behavior of the entire system. We will find that Cen X-3 is a binary system
that contains a tiny neutron
star and a much larger and more massive companion. We will
determine the rotation period of the neutron star, the orbit size of
the neutron
star, the size and mass of the
companion
star, the luminosity of the source (about 100,000 times brighter than
the Sun!), and much more. The key is discovering that the system displays periodic
changes in brightness (two different periods, in fact). The
measurement of such periodicities can then be used to determine the
nature of the objects that in turn produced those periodicities.
This set of activities will guide you through
the process of analyzing an object that varies in brightness with
time. You will learn how to discover if the light varies with a
regular period, how to determine the value of that period
accurately, and how to determine something about the masses and sizes
of the
objects involved and/or the scale of the system.
Part 1: acquiring the image
Start DS9, connect to one of the Virtual
Observatory Chandra servers, and click on the link that says: Load the
Cen X-3
image. Under the "Color" menu
, select "he."
The image you see (figure at right)
consists of a
black
spot surrounded by bright filled circle of light, along with streaks of
light
also going off on either side. These
streaks
occur because this observation
uses the Chandra gratings, which act like prisms to break up the x-ray
light
into their component x-ray "colors," much as a rainbow breaks up
sunlight into
visible light colors. (The green circle and long skinny
rectangles are "regions" used to analyze the data. Don't
delete these green regions!)
The central black
spot results because Cen X-3 is so bright that
the
satellite collects more photons (or counts) at these pixels than the
detectors can comfortably handle. This phenomenon is called
pile-up and, if necessary, the "correct" image can be
reconstructed
from the original data.
But we will not need to do that here.
Part 2: making a light curve
To detect a periodicity in the brightness of an object, we need to have
a plot of the object's brightness as a function of time (aka a "light
curve").
And to create a light curve, you must first selection a
"region" of the image to study. For this image of Cen X-3,
however, the regions have
already been selected (the green circle and the long skinny
rectangles). So we'll go on to the next step. Go to the
Analysis
drop-down menu, and select FTOOLS/Light
Curve. Click OK
in the pop-up box that
appears. After a short time, the light
curve will
appear! It will look something like
What is being plotted here? Brightness (in
counts/sec) on the vertical axis (except it's labeled "Rate") is
plotted versus time
(in seconds) on the horizontal axis. The plot looks like a
solid mass of black, because the light variation happens so rapidly
compared to the
length of the observation (nearly 50,000 seconds or14 hours). You
can see that the brighntess of the
source varies
dramatically from just under 25
counts/sec to nearly 150 counts/sec.
How can we determine the period of something varying so rapidly?
Part 3: fine
tuning the light curve
Because the light variation is so rapid, let's zoom in on the
data and
see what's happening over a small
time interval. Place your cursor
near the base of the
light curve, just above the tick mark at 30000, and left click. While
holding down the button, drag the cursor upward and over into a skinny
tall box about a
quarter
of an inch wide. (Note: it is NOT
important exactly where
you start and end the data selection; you just want to isolate a small
portion of
the
curve.) After you have selected your
box, left click again.
A new light curve will appear that zooms
in on the box you just selected. Repeat this
procedure again, and after one or two more tries, you will have zoomed
in on a portion of the curve where individual pulses can be
easily seen (a horizontal range
of about 100 seconds in time is good). By the way, If you make a mistake, or don't
like the selection you made, or just want to try again, all you have to
do is right
click on the plot to go
back to the previous graph.
Eventually your light curve should look something like
You can now
easily see the x-ray light from Cen X-3 varying up and down with time. What would you estimate the period of the
variation to be? ("Period" is the time from one peak to the next
peak. )
It looks like Cen X-3 is pulsing in brightness about
every 5
seconds (or just under), right? The frequency of the light
variation (the
reciprocal of period) is therefore approximately 0.2 s-1 .
Making a rough estimate of the period of light
variation is easy in
this case (as long as you can see the individual peaks and troughs of
brightness), but what if the period isn't very obvious?
And how can you determine the period in the most
accurate way?
And ultimately we would like to know why the
brightness is varying so quickly, yet so regularly.
Part 4: the power spectrum:
how to find an accurate period of
variation
Ideally, of course, we would like to analyze ALL
of the data for this source at once (as opposed ot the small portion of
the data that we looked at above). In order to find the most
accurate period
possible that represents all of the data in the observation, we use a
technique called a power spectrum
analysis. A power spectrum analysis
determines what period or periods best fit the entire data set.
Go to the Analysis menu and
click on FTOOLS/Power Spectrum. Click OK in
the pop-up box
that appears.
After a short time, a power spectrum will appear.
It will look like
The power spectrum plots the likelihood that a frequencey is present in
the data (represented by "power" on the vertical axis) as a function of
frequency (on
the horizontal axis). Notice that the plot consists of what
appears to be a few sharp spikes, indicating that only a few
frequencies
(or periods) are present
in the data. The tallest spike is at
about 0.2 Hz. This means that the most
likely frequency (the reciprocal of period) in the light variation is
0.2
cycles per
second. This corresponds to 1 cycle or period about every 5
seconds. This matches the period
we estimated by looking at the
data by eye above.
Now zoom in
(using the same click-and-drag technique described above)
on the tallest peak in the power spectrum, until
you get a plot that spreads the spike across most of the graph. Your new power spectrum plot should look like
The power spectrum now shows frequencies ranging from about
.20783 Hz to about .2083 Hz. This is
equivalent to a period range from about 4.812 s to about 4.801 s.
But why is it that
frequencies covering a range are present in the data? Why isn't there just one frequency (or period)
present?
Part 5: interpreting the
power spectrum: how fast is the object moving?
Other observations furnish
additional data. Every 2.1 days the
x-rays disappear for about 12 hours, and it seems most likely that the
disappearance is caused by the XRS being eclipsed by a larger
companion. When
the x-rays reappear, the x-ray pulses reach us at higher frequencies
(eventually reaching a frequency of 0.2083 pulses per second); 1.05
days (half the orbit period) later, the pulses start reaching us at
lower frequencies (eventually dropping to a frequency of 0.20783 pulses
per second). This oscillation in the pulse frequency continues,
back and forth, every 2.1 days. What we
are apparently seeing is the x-ray
source moving towards us (giving us a higher frequency Doppler shift)
and then
moving away from us, on the other side of the orbit (giving us lower
frequencies). Because the variation of the
frequency with time is sinusoidal, we can conclude that the orbits of
the two stars are circular. How
fast is the XRS moving?
We can use the Doppler shift formula to find out:
(change in
frequency)/(average frequency) [Df/f] = (orbital
speed)/(speed of light) [ v/c ]
Df = observed change in
frequency = (0.2083 s-1
- 0.20783 s-1)/2
vorbital/c
= 0.000235 s-1/0.208065
s-1 = .00113
vorbital =
339 km/s
Part 6: the orbit size of the
XRS (x-ray source)
Because we know the orbital period of the objects in the binary
system (because of the periodic eclipses of the XRS by
its companion) and its orbital speed, we can find the orbit size of the
XRS:
circumference [C] = 2
* pi* orbital radius [ = 2 p
rorbital]
= (orbit speed) (orbit period) = [vorbital Porbital]
rorbital = vorbital Porbital/(2 p)
rorbital = (339 km/s) (2.1 days)
(86400 s/day)/(2 * 3.14)
rorbital = 9.8 x 106 km
compare this to the size
of the Earth's orbit around the sun!
Part 7: the luminosity of the XRS
We can now determine the luminosity of the pulsar by using the inverse
square law along with the known flux of and the distance to the pulsar.
You may want to remind yourself how to determine the flux of a source
from the Chandra observations.
Part 8: how do we know what type of
objects are in this system?
A
simple set of
observations has told us much about the nature of the Cen X-3 binary
star
system. But how did astronomers figure out what type of object
the XRS is? How are the x-rays produced?
What is the nature of the XRS's companion? A key piece of the
puzzle fell into place in
1968 when the first neutron stars were discovered.
For some ideas on how
astronomers determined that one of the objects in this system is a a
neutron star, see the astrophysics
of short-period light variations.
A neutron star is a tiny object (typically 10 km in radius)
which has
an intense magnetic field (trillions of times more powerful than
that
of the
Earth). A young single, isolated neutron star often appears as a
"pulsar" which produces a pulse of luminosity once per rotation
period. The neutron star is surrounded by a plasma of electrons
and positrons accelerated by high voltages and magentic fields.
Current models favor a lighthouse-like effect in which the bright
magnetic poles of the neutron star flash
in our sight line once for every rotation of the star.
The case of Cen X-3 is a bit more complicated because the neutron
star's binary companion is leaking material that eventually ends up in
a spinning disk (an "accretion disk") around the neutron star. As
the charged particles streaming from the giant companion are
decelerated in the process of impacting the accretion disk, they
release their kinetic energy as x-rays.
[This is somewhat similar to what
happens when
aurorae (the "northern lights") are produced on Earth.
The sun ejects particles from solar
storms, and the earth intercepts some of them. The earth's
magnetic
field
lines funnels them towards the terrestrial poles, where they produce
the light of an
aurora.]
So the Cen X-3 picture is
this: there are two stars in the
system, a tiny dense one (imagine all the water in Lake Erie in your
bathtub) that is responsible for the x-rays we see, and
a
companion, which in this case turns out to be a giant (spectral class
O6). They revolve around
their common center of mass every 2.1 days. The
giant companion provides the matter that feeds the neutron star. The
neutron star's magnetic field funnels some of this accreting material
onto the magnetic
poles
of the neutron star. As these poles
come into view periodically, due to the rotation of the neutron star,
the intense x-rays emitted from the hot material are
detected as
"pulses" of radiation every 4.8 seconds.
Part 9: are we done?
Even though we have
uncovered much about the stars in the Cen X-3 system from the
observations, more questions remain: Does the rotation period of
the neutron star remain constant? Or is the rotation
slowing as the neutron star uses some of its kinetic energy to
power its x-ray luminosity? Or might the rotation be
speeding up, continually kicked up a notch by infalling matter from the
giant star? (OK, we actually now know the answer to this
question, because astronomers have been monitoring the pulsar period
over the past few decades.) Neutron stars
are generally the result of the explosion of a massive star in the
supernova process. How did the binary partnership survive that
tremendous explosion?
things to try on your own
Now that you know how to obtain a light curve and analyze it for
periodicity, use the GK Per
observations to determine as much as you can about this object and its
environment.
acknowledgments
This page is based on an activity originally written by Dr.
Terry Matilsky of Rutgers University. The original ideas,
analysis outline, and many of the words are his. Any inaccuracies
in the screen shots, calculations, etc. can fairly be blamed on me