Blackbodies and Flux: Blog 4, Worksheet 2.1, Problem 3

You observe a star you measure its flux to be \(F_{\star} \). If the luminosity of the star is \(L_{\star} \) 
(a) Give an expression for how far away the star is. 

Flux is energy per time per area, so we know that at a distance, d, flux is: \[F_{\star}= \frac{L_{\star}}{4 \pi d^2 } \] Solving for d gives: \[d^2 = \frac{ L_{\star} }{4 \pi  F_{\star} } \]  \[d = \left( \frac{ L_{\star} }{4 \pi  F_{\star} } \right)^{ \frac{1}{2}} \]

(b) What is its parallax? 

In the first worksheet we found the relationship: \[ \theta = \frac{ 1 \: AU }{d} \] Substituting for d gives: \[ \theta =  \left( \frac{ 4 \pi  F_{\star} }{L_{\star} } \right)^{ \frac{1}{2}} \] with parsecs that would be \[ \theta =  \frac{1}{2 \times 10^5} \left( \frac{ 4 \pi  F_{\star} }{L_{\star} } \right)^{ \frac{1}{2}} \]

(c) If the peak wavelength of its emission is at λo, what is the star’s temperature? 

In question 1 we found: \[ \lambda_0 = \frac{hc}{4kT} \] solving for T gives: \[ T = \frac{hc}{4k \lambda_0} \]

(d) What is the star’s radius, \(R_{\star} \)?

For this, we can use the relationship: \[ L_{\star} = 4 \pi R^2 T^4 \] Solving for R gives \[ R = \left( \frac{L_{\star}}{4 \pi \sigma T^4 } \right)^{\frac{1}{2}} \] Substituting for T gives:   \[ R = \left( \frac{L_{\star}k^4 \lambda_0^4 }{64 \pi \sigma h^4 c^4 } \right)^{\frac{1}{2}}\]