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The Question
(Submitted July, 21, 2010)
I've read something about how astronomers discover exoplanets. I
understand the basic concept - the star is bright and the planets
are dim so you cannot see them directly. But you know that if
there are planets, the star will wobble a bit because the planets
have a gravitational effect on it. OK, this is fairly clear. However,
you cannot know how many planets orbit the star. Would the wobble
not be highly erratic if there were, say, 10 planets, possibly with
irregular orbits, pulling the star to and fro? Do you have tools to
separate the impacts of individual planets? Or does the current
technology only permit to find a planet if it is the only planet
circling its star?
The Answer
This is an excellent question! The answer is that yes, the motion of
the star becomes quite complicated when there are multiple planets to
consider However, we have very good mathematical tools for separating
out the effects of each planet.
Consider the simple case of a planet in a perfectly circular orbit.
The motion it imparts to the star will then have the shape of a sine
wave, v = A * sin( omega * t ), where A is the amplitude of the
effect and omega is the rate at which the planet orbits the star. This
is illustrated in the example on the page:
http://en.wikipedia.org/wiki/Doppler_spectroscopy
Now, if there is another planet orbiting the star, it will be at a
different distance from the star, and hence omega will be different.
So now you have the sum of two sine waves at different frequencies,
and so on for further planets in the system. Each additional planet
is one additional frequency that will show up in the signal (all at
different amplitudes as well, but that doesn't help us separate them).
Fortunately, it is easy to separate out multiple sine waves at various
frequencies. The mathematical tool used for this is the Fourier Transform,
which gives you all the frequencies and amplitudes in one operation.
It works very much like your ear does when you hear multiple tones at
once.
There are two complications that make reality a little more work, however
First is that the Fourier Transform is not actually the best tool for most
accurately determining A and omega, so somewhat more sophisticated tools
are actually used. These basically involve calculating the shape of the
expected sine wave as a function of frequency, and then trying various
values of amplitude and frequency until you find the best match to the actual
signal. You may recall Newton's Method of finding a solution to an equation;
that is basically what is done here, but with two variables.
The second complication is that the planets may not have circular orbits.
In fact, they are always at least somewhat eccentric (i.e. elliptical).
This means that each planet's signature is not quite a sine wave, and thus
the Fourier Transform doesn't work perfectly. For orbits that are only
slightly eccentric, the Fourier Transform still works pretty well, but for
very eccentric orbits it becomes less sensitive. In those cases it's really
required to use the search algorithm described above, but with eccentricity
included as a third value that is varied while searching for the best fit to
the data.
There are also some other methods of finding exoplanets besides mapping out
the wobble of the star by doppler spectroscopy. Until now that has been the
most successful method, but with the recent launch of the Kepler mission,
the transit method is being used. There is a nice article about all the
methods of finding exoplanets at:
http://en.wikipedia.org/wiki/Methods_of_detecting_extrasolar_planets
We hope that helps.
-Kevin and Ira,
for "Ask an Astrophysicist"
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