Question

The hot glowing surfaces of stars emit energy in the form of electromagnetic radiation. It is a good approximation to assume that the emissivity e is equal to 1 for these surfaces.

Part A

Find the radius RRigel of the star Rigel, the bright blue star in the constellation Orion that radiates energy at a rate of 2.7×10^31 W and has a surface temperature of 11,000 K. Assume that the star is spherical.

Use σ=5.67×10^−8 W/m2⋅K4 for the Stefan-Boltzmann constant and express your answer numerically in meters to two significant figures.

Part B

Find the radius RProcyonB of the star Procyon B, which radiates energy at a rate of 2.1×10^23 W and has a surface temperature of 10,000 K. Assume that the star is spherical.

Use σ=5.67×10^−8 W/m2⋅K4 for the Stefan-Boltzmann constant and express your answer numerically in meters to two significant figures.

Answer 1

Part(A):-Rate of Energy emitted by Star Rigel

$$ \begin{array}{l} =e \sigma T^{4} \times \text { Area } \\ =e \sigma T^{4} \times 4 \pi R_{\text {Rigel }}^{2} \end{array} $$

put the values given in the problem in above equation we get-

\(2.7 \times 10^{31}=1 \times 5.67 \times 10^{-8} \times(11000)^{4} \times 4 \times 3.14 \times R_{\text {Rigel }}^{2}\)

\(\Rightarrow R_{\text {Rigel }}^{2}=\frac{2.7 \times 10^{31}}{1 \times 5.67 \times 10^{-8} \times(11000)^{4} \times 4 \times 3.14}\)

\(\Rightarrow R_{\text {Rigel }}^{2}=2.589 \times 10^{21}\)

\(\Rightarrow R_{\text {Rigel }}=5.088 \times 10^{10} \mathrm{~m}\)

\(\Rightarrow R_{\text {Rigel }}=5.1 \times 10^{10} \mathrm{~m}\)

(Up to two significant figures)

\(\operatorname{Part}(B):-\)

Rate of Energy emitted by Star Procyon \(\mathrm{B}\)

$$ \begin{array}{l} =e \sigma T^{4} \times A r e a \\ =e \sigma T^{4} \times 4 \pi R_{\text {ProcyonB }}^{2} \end{array} $$

put the values given in the problem in above equation we get-

\(2.1 \times 10^{23}=1 \times 5.67 \times 10^{-8} \times(10000)^{4} \times 4 \times 3.14 \times R_{\text {ProcyonB }}^{2}\)

\(\Rightarrow R_{\text {ProcyonB }}^{2}=\frac{2.1 \times 10^{23}}{1 \times 5.67 \times 10^{-8} \times(10000)^{4} \times 4 \times 3.14}\)

\(\Rightarrow R_{\text {ProcyonB }}^{2}=2.9488 \times 10^{13}\)

\(\Rightarrow R_{\text {ProcyonB }}=5.43 \times 10^{6} \mathrm{~m}\)

\(\Rightarrow R_{\text {ProcyonB }}=5.4 \times 10^{6} \mathrm{~m} \quad\) (Up to two significant figures)

Answer 2

Given :
.
Energy radiated (Q) = 2.7 * 1032 W
.
Temp (T) = 11000 K ;
.
Stefan-Boltzmann law states that the energy flux by radiationis proportional to the forth power of the temperature:
.
q = e ·s · T4
.
The total energy flux at a spherical surface of Radius R is :
.
Q =q·p·R² =e·s·T4 * p R²

Hence the radius is :
.
R = v( Q /(e·s·T^4·p) )
.
= v(2.7 * 1032 W / (1 * 5.67 *10-8 W/m²K4 *(1100K)4 · p) )
.
= 3.22 *1013 m
.
Similarly calculate for (b)
. Power ( P )= 2.1*10^23W
Temperature ( T ) = 10,000 K
emissivity ( e ) = 1
P = e s AT4
A = P / e sT4
But the area of the star ( A ) = 4 pr2
4 p r2 = P / es T4
= v [ P / ( es T4 ) * ( 4p ) ]
= -----