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Planet Hunting Toolkit II: Astrometry March 5, 2009

Posted by CosmicThespian in Toolkit.

In my previous post, Why It’s Hard, I discussed the difficulties inherent in finding planets around other stars by directly imaging them.  Basically, the stars are much brighter than their planets and the planets appear very close to their host stars.  If we want to have any hope of finding other worlds, we’re going to need to do something different than just taking pretty pictures and hoping we see a planet.

Fortunately, nature has provided a way!

While we generally can’t directly see these other planets, we can detect how the presence of an orbiting planet affects the motion of its host star!

The trick to understanding this lies in knowing something about the mechanics of orbiting bodies.  Most people, if asked, would say that the Earth orbits the Sun or that the Moon orbits the Earth.  But, here’s a secret: they’re wrong!

Now, before you run off to some other blog to find someone who is a little more qualified, let me explain.  The notion that the Earth orbits the Sun is only approximately true, but it’s not the whole story.  In truth, the Sun and the Earth orbit around a common center, called the barycenter of the system.  Where this barycenter is located depends predominately on the relative masses of the two bodies.

Let’s take a simpler example:  a see-saw.  Imagine two twin brothers want to balance themselves on a see-saw.  Where should they sit?  It probably doesn’t come as a great surprise that the brothers should sit at equal distances from the center.  Any other arrangement, and the see-saw will tip one way or the other.  When the brothers have got the see-saw balanced, we can say that their center of mass is over the pivot of the see-saw.

Now lets take a 200 lb uncle and his 50 lb niece.  How should they sit?  Without boring you with the math, the uncle, being four times heavier, needs to sit four times closer to the center than his niece in order to achieve balance.  To use some real numbers, if the niece is 6 feet away from the center, the uncle needs to sit a foot and a half away on the other side.  Another way of looking at this is to say that the center of mass of the uncle and niece is four times closer to the uncle than his niece.

What about two objects where one is 1000 times heavier than the other?  That’s right, the center of mass would be 1000 times closer to the heavier object than the lighter one.  Hopefully, you see a trend here.  However many times a heavier object is than another, that’s how many times closer the center of mass is to that object (or, to think in see-saw terms, that’s how many times closer that object needs to be to the center to make the whole thing balance).

By now you’re probably wondering what this has to do with planets.  Well, this barycenter I mentioned is located at the center of mass of the two objects!

If you have two stars in orbit around one another, both with equal mass, they will be orbiting around a point exactly halfway between them.  If one star is four times more massive than the other, they will orbit around a point that is four times closer to the more massive star.  If one is 1000 times heavier…..well, you get the point.

However, when it comes to planets and stars, the difference in masses is enormous!  The Sun is over 330,000 times more massive than the Earth!  Which puts the center of mass of the Earth-Sun system 330,000 times closer to the center of the Sun.  Given the 93 million miles between the two, that means the barycenter of the Earth-Sun orbit is only about 280 miles from the center of the Sun.  But, wait!  The Sun has a radius of nearly 700,000 miles! That means the point around which the Sun and Earth orbit is deep in the interior of the Sun – only 4 hundredths of a percent of the distance from the center of the Sun to its surface!

So you can see that saying the Earth orbits the Sun is a really good approximation.  But, as we’ll see, that slight difference will make all the difference for helping us out.

Let’s make things simple and ignore the other planets for now; we’ll pretend that the Earth is the only planet orbiting the Sun.  In that situation, what we just calculated indicates that the Sun is orbiting around a point in its own interior, wobbling 280 miles to either side (for 560 miles of total wobble).  If a planet can subtly pull on a star, perhaps we can detect this motion rather than look for the planet itself.

The science of measuring the positions of stars on the celestial sphere is called astrometry.  In principle, you could use astrometry to look for changes in those positions that indicate they are wobbling in the sky.  In practice, this is pretty difficult.  The reason is that the wobbles are generally really small.  If an astronomer on a hypothetical planet orbiting the closest star to our Sun, about 4 light years away, were to attempt to measure the wobble of our Sun resulting from the movement of the Earth, she would be trying to measure a wobble that subtended only 4 millionths of an arcsecond!  Yikes!  That would be like trying to measure the thickness of a dime located a quarter of a million miles away – roughly the distance to the Moon 175,000 miles away – roughly 3/4 the distance to the Moon!!

And, remember, this is from a vantage point of only 4 light years distant, basically next door.  Most of the stars we’d want to measure are hundreds or even thousands of light years from Earth.

Unfortunately, no current telescope can come close to doing that.  It gets easier if instead of trying to find Earth-like planets, you instead try to look for more massive Jupiter-like worlds with larger orbits.  Astrometry favors finding massive planets on wide orbits around stars which are relatively close to Earth.  But it’s still very difficult with the current generation of telescopes.; future space missions will improve on that.

At this point, things may seem hopeless.  Planets don’t lend themselves to imaging, and the wobbles they impart on their host stars are just too small to measure.  The next article in this series will show how we can make use of this wobble without worrying about measuring the minute changes in position…

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