How far, in parsecs, is an object that has a parallax p of 0.010 arc-second? How far is it, in light-years?

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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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Hello!

Without some information about object's size the answer is "It Is unknown". If two objects has the same parallax but the first is twice larger than the second, then the first is also twice farther.

Let the size of our object be about one astronomical unit.

By definition, one parsec is the distance from which one astronomical unit has parallax of 1 arc-second. Astronomical unit is about the distance from the Earth to the Sun.

Please look at the picture attached.

For such small angles (parallaxes) the triangles can be treated as right. Also, small angle is almost equal to its tangent, `tan(p) = p` (when p is in radians). For p in arc-seconds `tan(p)=C*p` where `C` is a constant.

So in the first triangle

`(A.U.)/(1pc)=tan(1arc-sec)=C*1=C,`

in the second triangle

`(A.U.)/d=tan(0.01 arc-sec)=0.01*C.`

Therefore

d/1pc = 1/0.01=100, or d = 100 pc.

Now for light-years. One parsec is about 3.26 light-years, so d=3.26*100=326 (light-years).