Blog #5: WS2.1, #3

3. You observe a star and measure its flux to be \(F_\bigstar\). If the luminosity of the star is \(L_\bigstar\), 
a) Give an expression for how far away the star is. 

\(F_\bigstar = \frac{L_\bigstar}{4\pi r^2}\)
\(r^2 = \frac{L_\bigstar}{4\pi F_\bigstar}\)
\(r=(\frac{L_\bigstar}{4\pi F_\bigstar})^{1/2}\)

b) What is its parallax? 

\(\theta = \frac{L}{D}\)
\(\theta = \frac{1AU}{(\frac{L_\bigstar}{4\pi F_\bigstar})^{1/2}}=1AU(\frac{4\pi F_\bigstar}{L_\bigstar})^{1/2}\)

c) If the peak wavelength of its emission is at \(\lambda_0\), what is the star's temperature? 

\(\lambda_0 = \frac{b}{T}\)
\(T = \frac{b}{\lambda_0}\)
\(b = 2.9*10^{-3}mK\) (Wien's displacement constant)

d) What is the star's radius, \(R_\bigstar\)? 

\(L_\bigstar = \sigma AT^4\)
\(A = \frac{L_\bigstar}{\sigma T^4} = 4\pi R_\bigstar ^2\)
\(R_\bigstar = (\frac{L_\bigstar}{4\pi \sigma T^4})^{1/2}\)