Worksheet 5.1, Problem 2: Solar Flux and Luminosity

Consider the amount of energy produced by the Sun per unit time, also known as the bolometric luminosity, L$_{\odot}$. That same amount of energy per time is present at the surface of all spheres centered on the Sun at distances r > R$_{\odot}$. However, the flux at a given patch on the surface of these spheres depends on r.

The first part of this problem asked how flux, F, is dependent on luminosity, L, and distance, r.

An object's luminosity diffuses according to the inverse square law, $L=4\pi r^2F$, which says that luminosity is equal to the flux at a given distance, r, times the surface area of the sphere with radius r.
Given the inverse square law, one can see that the relationship between flux, luminosity, and distance is:
$F\propto \frac{L}{r^2}$

The second part of the problem asked us to find the effective temperature of the sun given the following information:

• The flux felt on the Earth from the sun, which I will call the Earth Flux, F$_{\oplus}$, is $1.4\times10^6 \frac{ergs}{s\cdot cm^2} $ 
• The angular diameter of the sun is $\theta =.57^{_o}$
• $L_{\odot}(r=R_{\odot})=L_{\odot}(r=1 AU)$
• The Stefan-Boltzman constant, $\sigma$, is $5.7\times10^-5 \frac{ergs}{s\cdot cm^2\cdot K^4} $

Using the third bullet point, we set up the following equality

$4\pi R_{\odot}^2F{_\odot}=4\pi d^2F_{\oplus}$

where d is the distance between the Earth and the sun, 1.5x10$^{_13}$ cm.  As we derived on last week's worksheet (work can be seen in the post entitled Flux and Luminosity), flux is related to temperature by $F=\sigma T^4$.  This gives us the following equation:

$R_{\odot}^2\sigma T_{\odot}^4=d^2F_{\oplus}$

$T_{\odot}=\left ( \frac{d^2F_{\oplus}}{\sigma R_{\odot}^2} \right )^{1/4}$

In order to find the radius of the sun, R$_{\odot}$, we had to use the provided angular diameter.  Because the angle is so small, we were able to use small angle approximation, which says $\sin \theta =\tan \theta =\theta $.  According to this super advanced picture,

where $\alpha=\frac{\theta}{2}$, 

$\tan \alpha =\frac{R_{\odot}}{d}$

$R_{\odot}=\frac{d\alpha }{2}$

Once alpha has been converted from degrees to radians, the final equation for the temperature of the sun can be read as 

$T_{\odot}=\left ( \frac{d^2F_{\oplus}}{\sigma \left ( \frac{d\theta }{2} \right )^2} \right )^{1/4}$

After substituting all of the given values for their variables, we found that the temperature of the sun is about 10$^4$ K.  Any error in this result is due to the many times we rounded during our calculations at the end because we didn't have a calculator.  A quick google search let me know that the real temperature of the sun is closer to 1.6x10$^{7}$, so the rounding error is really significant.