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QUESTION: Suppose we discover a planet orbiting a nearby star. The distance to the star is 3 pc. We observe the angular radius of the planet’s orbit to be 0.1 arc sec. How many AU from the star is the planet?

**CONCERN: I am not sure what "The distance to the star is 3 pc" means. The earth's distance to the star? or the sun's distance to the star? or even better, the planet's distance to the star??!!

Any explanation of the question would be greatly appreciated!

My initial approach was some fixed position (say sun/earth's distance) to the star is 3pc, so I will have 0.1 arc sec = 3pc/distance, and solve for distance as the answer. (I reckon this is a very simple question for astrophysics, but I need to get basics right)

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  • $\begingroup$ On interstellar scales, the variation in the distance to the star as the earth orbits is negligible, so the distance of the Earth to the star, is equal to the distance of the Earth to the star. $\endgroup$ – James K Apr 8 '16 at 21:17
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Use this (http://www.ast.cam.ac.uk/~mjp/calc_parallax.html) but substitute the exoplanet for the Earth, you don't need to involve the Sun in your calculations, just assume the star is 3pc (parsecs) from Earth.

From Figure 2, the distance between the Sun and the star is : d = r / tan P If P is 1 second of arc: d = 150 000 000 / tan 1" = 30 million million km This distance is called one parsec and is a basic unit for measuring astronomical distances. Distance in parsecs = 1 / P in seconds of arc

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  • $\begingroup$ Aaaand, what happens if your link breaks? $\endgroup$ – SE - stop firing the good guys Feb 6 '16 at 18:18
  • $\begingroup$ That is better. $\endgroup$ – SE - stop firing the good guys Feb 6 '16 at 18:20
  • $\begingroup$ The link works for me, I will include the full link though as well as including the figure. $\endgroup$ – Dean Feb 6 '16 at 18:21
  • $\begingroup$ Links are not a problem by them selves, in fact, citing your claims is encouraged. But be sure that your answer is useful future visitors. $\endgroup$ – SE - stop firing the good guys Feb 6 '16 at 18:24
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I think this is suggesting that the star is 3 pc from us (the observer), and you need to find the distance of the planet from it's star.

If the 3 pc was the distance of the planet from the star, then the question is just asking you to convert parsecs to AU, which is very easy and would make the information about the angular radius useless.

So this is the information you have:

  • Distance form observer to star: 3pc
  • Observed radius of orbit: 0.1 arcsec

You want to use this to find the distance from the planet to the star, using trigonometry and converting to the correct units

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3 parsecs is the distance from us (Earth, Sun etc) to the star.

At a distance of 1 parsec, the apparent movement of a star wrt the background, as the Earth orbits on its 1 AU orbit is 1 arc second. So, from 1 parsec distant, the earth would move by 1 arcsec each 6 months. It woukd have an apparevt angular radiusof 0.5 arc sec.

Now this star is 3 parsecs distant, so an orbit with radius 1au would appear to have a angular radius 3 times smaller (here I am approximating, it is valid because the angle is small) or 1/6 arc sec. The planet in the question has a smaller orbit, 6/10 of the Earth, or 0.6 au

The convenient units, and tge the small angles mean that we can avoid trionometry in the solution.

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