# How do astronomers measure the distance of the planets from its star?

I know they have accurate means of measuring the distance of the planets in the Solar System from the Sun. I'm skeptical how can they use the same tools or techniques for other systems so far away as 22 light years.

According to this link the planet GJ667Cc resides in the "habitable zone". How are they able to measure the distance of something from something while observing it from a distance of 22 light years?

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– nico Oct 2 '12 at 6:01
Yes, possibly triangulation. But what I actually meant to ask is, do they really have the tools to accurately use triangulation as far as 22 light years? I mean, we can surely measure the distance of something using a ruler but if you're going to use it for long distances, the magnitude of error will surely increase (of course I'm just oversimplifying here but you get the idea). And for a distance of 22 light years I'm skeptical about its accuracy to be any meaningful at all. – supertonsky Oct 2 '12 at 6:09

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There's no triangulation, since indeed that would be unusably imprecise. John Rennie's answer explains it all. However, you should take claims of "habitable zone" with a grain of salt. Everyone is very eager to find the first Earth analogue, and so sometimes claims of this nature get blown out of proportion. – Chris White Oct 2 '12 at 8:04

## 1 Answer

The Doppler shift in the light from the star tells you the period of the planet's orbit and also the velocity the star moves. You need to know the mass of the star, but this can be estimated to good accuracy from the star brightness and type. Once you know the mass of the star you can calculate the distance of the planet from it's period using:

$$r^3 = \frac{GM}{4\pi^2}P^2$$

where $M$ is the mass of the star and $P$ is the period of the oscillation.

Not that it's directly relevant to your question, but from the velocity of the star's oscillation we can calculate the minimum mass of the planet, because the velocity of the stars displacement depends on the gravitational force between the two. We can only calculate a minimum planet mass because if the plane of the system it tilted relative to us the true mass is higher than the one we calculate.

Having said this, these days most extrasolar planets are discovered because they transit their star, and these systems are not tilted relative to us (otherwise they wouldn't transit!). That means we can calculate an accurate mass for the planet.

In practice we normally turn the calculation over to a computer model (called a Bayesian Kepler periodogram if you want to Google it) because there are usually several planets and the oscillation is not a simple sine wave. We use a numerical fit to work out how many planets there are and how far from the star they are.

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– supertonsky Oct 2 '12 at 9:06

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– John Rennie Oct 2 '12 at 9:33
I mentioned the planet mass only because estimating the planet mass is an important part of finding Earth-like planets. As you say, it doesn't affect the calculation of the star-planet distance. – John Rennie Oct 2 '12 at 9:34
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