Astronomy - Parallax error calculation.

[knowlewj01]

1. 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.


2. Homework Equations

D = LaTeX Code: \\frac{1}{p}

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

3. 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: (LaTeX Code: \\Delta D)^2 = (LaTeX Code: \\frac{dD}{dp} LaTeX Code: \\Delta p)^2

(LaTeX Code: \\Delta D)^2 = (LaTeX Code: \\frac{1}{p^2} LaTeX Code: \\Delta p)^2

(LaTeX Code: \\Delta D)^2 = (250000 x 0.001)^2

(LaTeX Code: \\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?

 

[anathema]

Thank you for your question. I am a scientist and I would be happy to assist you with finding the distance to both objects separately and when both measurements are combined.

First, let's start with the calculation for the distance to star 1. As you have correctly stated, the distance can be calculated using the formula D = 1/p, where D is the distance in parsecs (pc) and p is the parallax in arcseconds. Based on the given parallax measurement of (2±1)x10^-3 arcseconds, we can calculate the distance to star 1 as follows:

D = 1/(2x10^-3) = 500 pc

Now, let's calculate the error in this distance measurement. As you have correctly shown, the formula for calculating the error in D is (ΔD)^2 = (dD/dp Δp)^2. In this case, dD/dp = -1/p^2, and Δp = 1x10^-3. Therefore, the error in the distance to star 1 can be calculated as follows:

(ΔD)^2 = (1/(500)^2 x 1x10^-3)^2 = 1x10^-2 pc^2

(ΔD) = ±0.1 pc

Now, let's move on to calculating the distance to star 2. Using the same formula as before, we can calculate the distance to star 2 as follows:

D = 1/(4x10^-3) = 250 pc

Similarly, the error in this distance measurement can be calculated as follows:

(ΔD)^2 = (1/(250)^2 x 2x10^-3)^2 = 1.6x10^-2 pc^2

(ΔD) = ±0.4 pc

Now, to calculate the combined distance, we need to take into account both measurements and their respective errors. This can be done by using the following formula:

D_combined = (D1^-2 + D2^-2)^-1

(ΔD_combined)^2 = (∂D_combined/∂D1 (ΔD1)^2 + ∂D_combined/∂D2 (ΔD2)^2)^1/2

Substituting the values we calculated earlier, we get:

D_combined = (500^-2 +

 

[kerimek]

Your attempt at calculating the distance to the stars is correct. The parallax angle is inversely proportional to the distance, so the larger the parallax angle, the closer the star is to Earth.

To combine the measurements, we can use the formula for combining errors in quadrature: (LaTeX Code: \\Delta D)^2 = (LaTeX Code: \\Delta D_1)^2 + (LaTeX Code: \\Delta D_2)^2

Plugging in the values from your calculations, we get (LaTeX Code: \\Delta D)^2 = (15.8)^2 + (31.6)^2 = 2497.2

Taking the square root, we get (LaTeX Code: \\Delta D) = ± 49.97 pc

So, the combined distance to both stars is 500 ± 49.97 pc. This means that the actual distance to the stars could be anywhere between 450.03 pc and 550.03 pc.

In terms of the last part of the question, combining the measurements means taking both parallax angles into account to get a more accurate distance measurement. This is because the errors in the individual measurements can cancel each other out to some extent.

Overall, your calculations and understanding of error calculation seem reasonable. It is always important to take into account the uncertainties in measurements, as they can significantly affect the accuracy of our results.