Blog #4, Worksheet #2.1, Black Bodies and Flux

3. You observe a star and 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. 

Luminosity is a measured constant dependent on the specific star. Flux is the amount of power that is emitted from a certain area of a black body. The two are related with the simple equation \[F = \frac{L}{A}\] where A is the area of a sphere through which the flux is being measured. The area of a sphere is defined as \[A = 4 \pi d^2\] where d is the distance away from the star. Thus to solve for distance we simply rearrange and solve for d and get \[d = (\frac{L_\star}{4\pi F_\star})^{\frac{1}{2}}\]

b) What is its parallax?

From the first worksheet we derived that the parallax is \[P = \frac{1}{D[pc]}\] It is important to note that the distance, D, here is in parsecs. To use our equation from part A we must convert parsecs to a metric unit, centimeters, by multiplying the denominator \[1pc = 3.08 x 10^{18} cm\]We can then use our equation for distance from part A to solve for the parallax of the star.

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

We use the Wein Displacement Law to relate peak wavelength to temperature with \[\lambda_0 = \frac{b}{T}\] where b is a constant of proportionality equal to \[b = 2.89 x10^{-3} m \cdot K\] Rearranging to solve for T, we get \[T = \frac{b}{\lambda_0}\]

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

Luminosity is directly related to both radius and temperature through the equation \[L_\star = 4\pi {R_\star}^2 \sigma T^4\] where \[\sigma = 5.67x10^{-8} W \cdot m^{-2} \cdot K^{-4}\] and is known as the Steffan-Boltzmann constant. We then rearrange to solve for \(R_\star\) and get \[R_\star = (\frac{L_\star}{4\pi \sigma T^4})^{\frac{1}{2}}\]And for a final step we can put in our answer from part C for the temperature and get \[R_\star = (\frac{L_\star {\lambda_0}^4}{4\pi \sigma b^4})^{\frac{1}{2}}\]