Blog Post 14, Worksheet 5.1, Problem 2

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?

We know from Problem 1 that the relationship between Star A and Star B's flux and magnitude is given by the following equation.
\[\frac{F_A}{F_B} = 10^{0.4(m_B-m_B)},\] which can also be written as \[\frac{F_A}{F_B} \approx 2.5^{(m_B-m_A)}\]

We know that Star A is 3 magnitudes fainter than Star B, so the quantity \(m_B-m_A\) is equal to 3. We get: \[ \frac{F_A}{F_B} = 2.5^3\]
\[ F_A = 15.625\: F_B\] In order to collect the same amount of energy for Star A as for Star B, you have to observe Star A for 15.625 times the amount of time you would need to observe 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?

We know that the relationship between the fluxes and magnitudes of two stars at a distance 10 kpc and a distance d away is given by the following equation:
\[\frac{F_{10 \text{ pc}}}{F_d} = 10^{0.4(m-M)}\] We also know that we can calculate the flux for each star using the following formula: \[F = \frac{L}{4πd^2}\]. From this, we have: \[F_d = \frac{L}{4πd^2}\] \[F_{10 \text{ pc}} = \frac{L}{4π(10 \text{ pc})^2}\] We can plug these values for each star's flux into the previous equation to relate the two: \[\frac{F_{10 \text{ pc}}}{F_d} = 10^{0.4(m-M)}\] \[\frac{\frac{L}{4π(10 \text{ pc})^2}}{\frac{L}{4πd^2}} = 10^{0.4(m-M)}\] \[\frac{d^2}{100} = 10^{0.4(m-M)}\] \[d^2 = 10^{2+0.4(m-M)}\]If we take the logarithm (base 10) of each side of the equation, we get: \[2 \text{log}(d) = 2 + 0.4(m-M)\] \[\frac{2(\text{log}(d) - 1)}{0.4} = m - M\] \[m = 5(\text{log}(d) - 1) + M\] This equation relates the apparent magnitude, m, of a star with absolute value M to its distance, d, away from the observer.

(c) What is the star's parallax in terms of its apparent and absolute magnitudes?

From a previous worksheet, we know that the relationship between parallax and distance of an object is given by the equation: \[\text{parallax [arcseconds]} = \frac{1}{\text{distance[pc]}}\] From part (b), we know: \[d^2 = 10^{2+0.4(m-M)}\] \[d = (10^{2+0.4(m-M)})^{1/2}\] We can plug this value for d into the relationship between parallax and distance to get: \[p = (10^{2+0.4(m-M)})^{-1/2}\]