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:   $$R = \left( \frac{L_{\star}k^4 \lambda_0^4 }{64 \pi \sigma h^4 c^4 } \right)^{\frac{1}{2}}$$