Q. 7.54 - An Introduction to Thermal Physics

Question

The sun is the only star whose size we can easily measure directly; astronomers therefore estimate the sizes of other stars using Stefan's law.

(a) The spectrum of Sirius A, plotted as a function of energy, peaks at a photon energy of$2.4\mathrm{eV}$, while Sirius A is approximately $24$ times as luminous as the sun. How does the radius of Sirius A compare to the sun's radius?

(b) Sirius B, the companion of Sirius A (see Figure 7.12), is only $3\%$ as luminous as the sun. Its spectrum, plotted as a function of energy, peaks at about$7\mathrm{eV}$. How does its radius compare to that of the sun?

(c) The spectrum of the star Betelgeuse, plotted as a function of energy, peaks at a photon energy of $0.8\mathrm{eV}$, while Betelgeuse is approximately$10,000$times as luminous as the sun. How does the radius of Betelgeuse compare to the sun's radius? Why is Betelgeuse called a "red supergiant"?

Step-by-Step Solution

Answer

(a) . The radius of Sirius A is $1.69$$\text{ (In units of suns radius) }$.

(b) . $\text{ The radius of Sirius B is 0.007}$.

(c) . $\text{ The radius of Betelgeuse is 310.}$

Step 1. Given information

The formula used is $R=\sqrt{\left(\frac{L}{T^4}\right)}$ to calculate the desired result.

Step 2. Calculating the radius of Sirius A

$\text{ The surface temperature of Sirius }A\text{ is given by }$

$\left(\frac{2.4\mathrm{eV}}{1.41\mathrm{eV}}\right)=1.702$

$\text{ So the radius of Sirius A should is}$

$=\sqrt{\left[\frac{24}{(1.70{)}^4}\right]}$

$=1.69\text{ (In units of suns radius) }$

The radius of Sirius A is $1.69$.

Step 3. Calculating the radius of Sirius B

$\text{ The surface temperature of Sirius B is given by }$

$\left(\frac{7\mathrm{eV}}{1.41\mathrm{eV}}\right)=4.96$

$\text{ Which is nearly five times the suns temperature, so the radius of Sirius B should be }$

$R=\sqrt{\left(\frac{L}{T^4}\right)}$

$=\sqrt{\left(\frac{0.03}{(4.96{)}^4}\right)}$

$=0.007$

As it is less than $1\%$ of suns radius and just slightly smaller than the earth's radius. This result is in rough agreement with that of $7.23\left(d\right)$ where we calculated that, one solar - mass white decay should have a radius first slightly larger than earth's.

Step 4. Calculating the radius of  Betelgeuse 

$\text{ The surface temperature of Betalgeuse is given by }$

$\frac{(0.8\mathrm{eV})}{(1.41\mathrm{eV})}=0.57$

$\text{ the radius of Betelgeuse will be }$

$R=\sqrt{\left(\frac{L}{T^4}\right)}$

$=\sqrt{\left[\frac{10,000}{(0.57{)}^4}\right]}$

$=310$

As the radius is  larger than the radius of earths orbit, and nearly as large as the orbit of mass. Super giant is certainly an approximate term. As for "red" the spectrum of Betelgeuse is certainly redder than the sun, due to its lower temperature of $3300 \mathrm{K}$ which makes its spectrum peak well into the infrared and fall of considerably at the blue end of the visible range. But this temperature is still slightly hotter than the filament of an incandescent bulb, so the color of Betelgeuse shouldn't be any redder than that of incandescent light, "yellow-orange' would be a more accurate description.