Measuring Stellar
Oscillations
Capturing the Sound of
Stars
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Stellar oscillations, like those observed in
the Sun, manifest themselves by periodic variations in the surface
temperature, radius and overall brightness of the star.
In a star like the Sun, the amplitude of these
variations is very small. In temperature, the variations are a tiny
fraction of a degree and for light intensity (brightness), the
variations are a few parts per million. Such tiny changes are
exceedingly difficult to measure from the Earth, mainly because of
disturbances from the Earth's atmosphere.
You can
read and see more about the disturbances from the Earth's atmosphere
here.
However, when observing
with a satellite such as Kepler, we are no longer observing through
the atmosphere, which makes extremely precise measurements possible.
Measurements that are precise enough to observe solar-like
oscillations in stars other than the Sun.
For other types of
pulsating stars, such as those described in the previous chapter,
the amplitudes are generally higher and can be measured from Earth.
However, many of these stars oscillate with a few dominant
frequencies (earlier referred to as "tones") with high oscillation
amplitudes, and then in a much larger number of frequencies having
lower amplitude. The nature of the mathematical technique used to
analyse pulsating stars (this technique will be described below) is
that the more data that are available, the smaller amplitudes can be
detected.
Since performing long-duration, continuous
observations from the Earth is very demanding, extensive datasets
only exist for a very small number of stars. One of these stars is
the Delta Scuti star FG Vir, where astronomers have collected almost
2000 hours of time-series data so far. This huge effort has involved
many astronomers using dozens of telescopes all across the globe,
over a time span of more than a decade. The more data that is
collected on this star, the more low-amplitude oscillation
frequencies are being detected.
This means, that even for the
"classical" variables (such as Delta Scuti, Beta Cep, SPB stars
etc.), which can be observed from ground, Kepler will provide
fantastic datasets, simply because of the high-precision space
measurements, and because of the continuity of observations over
several years. The Kepler observations will allow detection of many
more oscillation frequencies compared to what can be obtained from
ground. And, with more frequencies, stricter bonds on the
theoretical stellar models are obtained.
Analysis
of Time-Series Data
Kepler collects time-series
data, which simply means that the brightness of the individual stars
is measured every minute for several years (there are some
modifications to this - follow the link to the Kepler homepage at
the end of these pages for more details). This long duration of the
observations is the first of the major strengths of Kepler, as is
illustrated by the two next examples.
The figure below shows
the light curve of a star with two oscillation frequencies (of
periods 36 and 37 hours, respectively) with high amplitudes. The
combined effect of the two frequencies is to cause the brightness of
the star to vary sometimes with high amplitudes, sometimes with low.
This effect of a slow change between constructive and destructive
interference is called "beating" between the two frequencies, and is
easily seen in the figure.
 As the small segment above the light
curve shows, we have no chance of determining both oscillation
frequencies if we only observe the star for a few days - it will
actually look as if the star only has a single oscillation
frequency. Only by observing the star for a very long time can we
determine the true nature of the star's variability, and detect both
frequencies.
To understand the next example, we first need
to describe "how" oscillation frequencies are actually measured -
this is illustrated in the figure below. Top left is a light curve
of a pulsating star with a single period, the period of oscillation
is easily found by measuring the times between two wave tops. Top
right is a corresponding, so-called amplitude spectrum. On the
x-axis is frequency (here in cycles per day; a frequency of 24
cycles per day is the same as a period of 1 hour - the oscillation
have time to go through 24 cycles in one day), on the y-axis is
amplitude.
Such a figure is also referred to as a Fourier
spectrum, after the mathematician who first developed the technique.
 The amplitude spectrum on the top
right is obtained by testing how well frequencies in some interval -
here from 1 to 19 cycles per day - matches the observed light curve
shown on the left. This is done by mathematical fits (using the
computer, naturally) of sine-waves to the light curve, starting with
a sine of, for instance, 1.00 c/d, then 1.01 and so on, up to 19
c/d.
For frequencies far from the oscillation frequency (10
c/d), this fit will be poor, resulting in low amplitude in the
amplitude spectrum. But at the frequency actually present in the
light curve, the fit will be very good - a sine wave with a
frequency of 10 c/d matches the light curve well - resulting in a
peak at that frequency in the amplitude spectrum.
This
technique becomes very powerful if more than one oscillation
frequency is present in the light curve, as shown in the bottom
plots of the figure. Now we have a star oscillating in five
frequencies, resulting in a very complicated light curve. But, using
the amplitude spectrum we can tell that the 5 frequencies are 5, 7,
8.5, 9 and 11 c/d, something we would never have been able to do
from the light curve alone. This is also the technique used for
extracting the millions of oscillation frequencies in the Sun,
mentioned in the previous chapter.
Noise and
Disturbances
So, the amplitude spectrum allows us
actually to find the oscillation frequencies, even in very
complicated-looking light curves. But it has another interesting
feature too. Measurements of starlight, even from space, are
subjected to noise: there is an unavoidable, natural and fundamental
noise called photon-noise, or counting statistics - a true measure
of some object of nature such as a star will always have some degree
of uncertainty to it.
Then there is instrument noise, noise
from scattered Sun/Earth/moon-light into the telescope, noise from
the data reduction procedures, and so on. In short, even the best
observations will have some noise. And if this noise is too high,
much higher than the amplitudes of oscillation for some star, then
these oscillations cannot be measured.
Or, at least, they
cannot be "seen" in the light curve. This is shown in the figure
below. The upper panel shows how observations of a distant,
solar-like star will look. This plot just shows two days worth of
data, out of a much longer time series. It looks just like noise.
But the amplitude spectrum in the lower panel (30 days of data were
used in the calculation and the result is shown as power, which is
just the amplitude squared), clearly reveals the presence of
oscillation frequencies. These were hidden in the noise in the time
series but can be extracted using the Fourier spectrum, because here
the noise level drops as more data are becoming available (there is
simply more data to fit to).
So, even low-amplitude
frequencies embedded in noise in the time series can be detected
using the Fourier technique. The long duration of the Kepler
observations is very important indeed.

The need for
continuous observations
There is another important
aspect of Kepler that makes it superior to observations from ground:
the observations are continuous, not affected by the day/night
rhythm or bad weather.
Imagine we are using a telescope here
on Earth to observe the star from above, oscillating with just a
single period. During daytime we can of course not observe. This
introduces gaps in the time series as is illustrated in this figure:

Here, the Fourier technique runs
into trouble.
We don't know the light curve of the star in
the daytime (because we don't measure it) and this lack of data
introduces false peaks in the amplitude spectrum - these are called
aliases and occur simply due to the absence of information in the
daytime, as is described here.
They
make it very difficult to determine which peaks are due to true
oscillation frequencies in the star, and which are caused by the
gaps.
The SONG Project
So the
continuous observations of Kepler is a very strong advantage
compared to ground-based observations.
This is in fact also the
main argument for building SONG, a network of ground-based, automatic
telescopes, which will do time series observations of stars much
brighter than the ones which will be observed with Kepler. SONG
will use the technique of spectroscopy, which is less disturbed by
the atmosphere, but which is limited to bright stars.
Although the science goals of Kepler and SONG are similar,
the two projects are not in competition, but are complementing each
other. We also note that although planets will be detected with
SONG, these will only be planets in close orbit around its parent
star, an earth-like planet in an earth-like orbit cannot be detected
from ground with any present-day technique. It is simply impossible
because of the level of precision required to obtain the detection.
But such a planet can be detected by intensity measurements from
space - it can be detected by Kepler.
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