Determining Parallax Accuracy for Stellar Distances

[~Sam~]

Homework Statement

Out to what distance in parsecs can we find the distances of stars to 10% accuracy or better; i.e., at what distance would the 1 mas accuracy of the measurements result in a 10% uncertainty in the distance? (HINT: To reasonable accuracy, the percentage uncertainty in the inverse of a quantity equals the percentage uncertainty in the quantity itself.)

Homework Equations

Tan(θ)= opp/adj

The Attempt at a Solution

My teacher tells me that this is all the information I need, this question mentioned the Hipparcos catalogue, but I was told I didn't need it. Any ideas? I know how to calculate the distance of a celestial object using parallactic angle..but how are you suppose to know the uncertainity?

 

[tiny-tim]

Hi ~Sam~!

With what accuracy can you measure angles?

 

[~Sam~]

Hi ~Sam~!

With what accuracy can you measure angles?

I'm not quite sure I understand. Do you mean the usage of the small angle approximation? And the relative accuracy for small angles, thus distance objects?

 

[tiny-tim]

To measure distance by parallax, you have to measure angles (at two different times).

The accuracy of the distance depends on the accuracy of the angles.

So how accurate are the angles?

 

[~Sam~]

To measure distance by parallax, you have to measure angles (at two different times).

The accuracy of the distance depends on the accuracy of the angles.

So how accurate are the angles?

Sorry, I'm just not getting it, and I don't have data or anything..So I'm not sure how to determine the accuracy of the angle.

 

[~Sam~]

Ok wait, so I can accurately measure the angle to the milliarsecond. So with that we can get 1000 parsecs relatively accurate. But where does the 10 percent come into play? How would I figure that out?

 

[tiny-tim]

Ok wait, so I can accurately measure the angle to the milliarsecond.

Good.

Now what is the formula connecting the parallax angle with the distance?

 

[~Sam~]

Good.

Now what is the formula connecting the parallax angle with the distance?

tan(θ)= opp/adj, so opp=1parsec and adj=our distance.
But more importantly there is p('')= 1/ r(pc) so r(parsecs)=1/ p(arsecs)..I could divide and get 1000pc.?

Still, I have difficulty understanding how to incorporate the 10%..or is it already incorporated with the answer being 1000pc?

 

[tiny-tim]

tan(θ)= opp/adj, so opp=1parsec and adj=our distance.

No, adj is our distance from the star, and opp is our distance from the sun (or is it twice that? ).

So θ ~ tanθ = sun-distance/star-distance.

Assuming we know sun-distance extremely accurately, what is the relation between uncertainty in star-distance and uncertainty in angle?

 

[~Sam~]

No, adj is our distance from the star, and opp is our distance from the sun (or is it twice that? ).

So θ ~ tanθ = sun-distance/star-distance.

Assuming we know sun-distance extremely accurately, what is the relation between uncertainty in star-distance and uncertainty in angle?

Since θ =sun-distance/star-distance. Then star distance= sun distance/θ. I'm not sure what you mean by uncertainty in star distance and angle (do you mean inverse proportional?). Thanks for the help so far.