Astronomy - Parallax error calculation.

[knowlewj01]

Homework Statement

Two stars belong to the same cluster, the parallax measured to one star is
(2±1)x10^-3 arcseconds, towards the other the parallax is
(4±2)x10^-3 arcseconds.

Find the distance to both objects seperatly and also compute the distance when both measurements are combined.

Homework Equations

D = $$\frac{1}{p}$$

where D is the distance in pc
and p is the parallax in arcseconds

The Attempt at a Solution

If it weren't for the error in the parallax angle this would be easy but I'm not sure how to go about it. here is my attempt at it:

Star 1: D = 1/p = 500pc

calculating error in D: ($$\Delta$$D)^2 = ($$\frac{dD}{dp}$$$$\Delta$$p)^2

($$\Delta$$D)^2 = ($$\frac{1}{p^2}$$$$\Delta$$p)^2

($$\Delta$$D)^2 = (250000 x 0.001)^2

($$\Delta$$D) = ± 15.8 pc

does this seem reasonable?
I'm not too great with error calculation, does this seem right?

I have done exactly the same for the second star, have no idea what the last bit of the question means at all 'when both measurements are combined'. anyone have any ideas?

 

[Jhero]

Thank you for your question. Your approach to calculating the distance to each star is correct. However, to find the combined distance, you will need to use the formula for error propagation. This formula takes into account the errors in both measurements to give a more accurate estimate of the combined distance.

The formula for error propagation is as follows:

(\DeltaD)^2 = (\frac{dD}{dp_1}\Deltap_1)^2 + (\frac{dD}{dp_2}\Deltap_2)^2

where D is the combined distance, p1 and p2 are the parallax angles for the first and second star respectively, and Delta p1 and Delta p2 are their corresponding errors.

Using this formula, you can calculate the combined distance for the two stars. This will give you a more accurate estimate of the distance to the cluster.

In addition, it is always a good idea to double check your calculations and make sure they seem reasonable. In this case, your calculated error for the distance to each star seems reasonable, but it is always a good idea to check your calculations with a calculator or by hand to make sure you are not making any mistakes.

I hope this helps and good luck with your calculations!