2 (a) Suppose you are observing two stars, Star A and Star B. Star A is 3 magnitudes fainter than Star B. How much longer do you need to observe Star A to collect the same amount of energy in your detector as you do for Star B?
(b) Stars have both an apparent magnitude, m, which is how bright they appear from the Earth. They also have an absolute magnitude, M, which is the apparent magnitude a star would have at d = 10 pc. How does the apparent magnitude, m, of a star with absolute magnitude M, depend on its distance, d away from you?
(c) What is the star’s parallax in terms of its apparent and absolute magnitudes?
(b) Stars have both an apparent magnitude, m, which is how bright they appear from the Earth. They also have an absolute magnitude, M, which is the apparent magnitude a star would have at d = 10 pc. How does the apparent magnitude, m, of a star with absolute magnitude M, depend on its distance, d away from you?
(c) What is the star’s parallax in terms of its apparent and absolute magnitudes?
(a) Suppose you are observing two stars, Star A and Star B. Star A is 3 magnitudes fainter than Star B. How much longer do you need to observe Star A to collect the same amount of energy in your detector as you do for Star B?
Let's use the recursive relation for magnitudes to solve this problem.
\[\frac{F_B}{F_A}=2.5^{(m_A-m_B)}=2.5^3=16\]
From this, we know that Star B is 16 times brighter than Star A.
Energy scales to \(Flux \times Time\), thus:
\[E=Ft\]
Let's let \(E_A\) be the energy received from Star A in the time \(t_A\). We want \(E_B=E_B\), so:
\[E_A=F_A t_A=E_B=F_B t_B\]
\[F_A t_A =F_B t_B\]
We already established that \(F_B=16F_A\), so:
\[F_A t_A =16F_A t_B\]
\[\boxed{t_B=\frac{T_A}{16}}\]
Thus, one should observe Star B for one sixteenth of the time that one observed Star A in order to get the same amount of energy capture.
(b) Stars have both an apparent magnitude, m, which is how bright they appear from the Earth. They also have an absolute magnitude, M, which is the apparent magnitude a star would have at d = 10 pc. How does the apparent magnitude, m, of a star with absolute magnitude M, depend on its distance, d away from you?
We have established in previous posts that Flux received from a star at a given distance is:
\[F=\frac{L}{4\pi d^2}\]
So, for our star at 10 pc with magnitude M, we have:
\[F_M=\frac{L}{4\pi (10pc)^2}\]
Comparatively, at another distance d, the same expression reads:
\[F_m=\frac{L}{4\pi d^2}\]
In this expressions, L is constant, so we can set the two equations equal to one another.
\[F_M 4\pi (10pc)^2 = F_m 4\pi d^2\]
With rearranging, we can get:
\[\frac{F_M}{F_m}=\frac{d^2}{(10pc)^2}\]
With arrangement, we can add another term: the magnitude relation.
\[\frac{F_M}{F_m}=\frac{d^2}{(10pc)^2}=2.5^{(m-M)}\]
We are trying to solver for magnitude in terms of distance, so let's discard the Flux comparison:
\[\frac{d^2}{(10pc)^2}=2.5^{(m-M)}\]
Next, let's solve for m in terms of d and M.
\[2log(\frac{d}{10pc})=(m-M)log(2.5)\]
\[m-M=\frac{2log(\frac{d}{10pc})}{log(2.5)}\]
\[m=\frac{2log(\frac{d}{10pc})}{log(2.5)}+M\]
\[m=5log(\frac{d}{10pc})+M\]
\[m=5(log(d)-log(10pc))+M\]
\[\boxed{m=5(log(d)-1)+M}\]
This is also known as the Distance Modulus.
(c) What is the star’s parallax in terms of its apparent and absolute magnitudes?
We know that:
\[p=\frac{1}{d}\]
If we solve the distance modes for distance, we get:
\[m=5(log(d)-1)+M\]
\[\frac{m-M}{5}+1=log(d)\]
\[d=10^{\frac{m-M}{5}+1}\]
Plugging this in, we get:
\[\boxed{p=10^{-(\frac{m-M}{5}+1)}}\]
I worked on this problem with B. Brzycki, G. Grell, and N. James.
Very nice job. 5
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