Error propagation of distance modulus and parallax

[muse19]

Hi!

Here is my problem: there is a star, for which we know the distance, d=21.2 pc,
the measurement error is delta_d=1.8 pc. The question is that how far should we put this star,
so that the following equation would be true: d/delta_d = 3?
The teacher told me to use two formulas: m-M=-5+5log(d) and pi=1/d (pi=parallax),
and then do some error propagation calculations.
Of course, I have already tried it using the equations above,
but unfortunately I can not solve the problem.
Does anybody have any idea?
Actually, it would be OK as well, if anybody could do it using
any other equations.
Thank you, and sorry for my english.

Cheers, Joe

 

[lippyka]

The first step would be to calculate the parallax (pi) of the star using pi = 1/d. Plugging in the given value of d, we get pi = 0.047 radians or 2.75 arcseconds. To solve the problem, we have to find the value of d that would satisfy the equation d/delta_d = 3. Rearranging the equation, we get delta_d = d/3. We can then substitute the value for delta_d into the equation m-M = -5 + 5 log(d). Solving this equation for d, we get d = 10^[(m-M + 5)/5]. Substituting our calculated value of delta_d = d/3 into this equation, we get d = 10^[(m-M + 5)/15]. Solving this equation, we get d = 21.2 pc. Thus, the star should be placed at a distance of 21.2 pc in order for the equation d/delta_d = 3 to be true.