Question

The sun emits energy in the form of electromagnetic waves at a rate of . This energy is produced by nuclear reactions deep in the sun’s interior. (a) Find the intensity of electromagnetic radiation and the radiation pressure on an absorbing object at the surface of the sun (radius r = R/2) and at r = R/2, in the sun’s interior. Ignore any scattering of the waves as they move radially outward from the centre of the sun. Compare to the values given in Section 32.4 for sunlight just before it enters the earth’s atmosphere. (b) The gas pressure at the sun’s surface is about $\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}\mathbf{Pa}$; at r = R/2, the gas pressure is calculated from solar models to be $\mathbf{4}\mathbf{.}\mathbf{7}\mathbf{\times }{\mathbf{10}}^{\mathbf{13}}\mathbf{Pa}$. Comparing with your results in part (a), would you expect that radiation pressure is an important factor in determining the structure of the sun? Why or why not?

Short answer

A) The intensity of the electromagnetic radiation is at sun's surface $\mathit{6}\mathit{.}\mathit{4}\mathit{\times }{\mathit{10}}^{\mathit{7}}W\mathit{/}{m}^{\mathit{2}}$and the radiation pressure at sun's surface 0.21 Pa and the intensity of the electromagnetic radiation in sun's interior $2.6\times {10}^{8}W/m^2$and the radiation pressure in sun's interior 0.86 Pa

B) The radiation pressure is an important factor for determine the structure

Step-by-step solution

Concept of the the intensity of light and the radiation pressure

The intensity of light is expressed by the equation$\mathbf{l}\mathbf{=}\frac{\mathbf{P}}{\mathbf{A}}\mathbf{=}\frac{\mathbf{P}}{\mathbf{4}{\mathbf{\tau \tau r}}^{\mathbf{2}}}$where A is the cross-sectional area of the beam, P is the rate of energy emission r is the sun radius. The radiation pressure at sun surface is given as ${\mathbf{P}}_{\mathbf{rad}}\mathbf{=}\frac{\mathbf{l}}{\mathbf{c}}$l is the intensity at the sun's surface, c is the speed of the light

Calculate the intensity at sun surface and its interior

The rate of energy emission is$3.9\times {10}^{26}\mathrm{W}$, radius of sun is$6.96\times {10}^5\mathrm{km}$. The intensity at sun surface, is$\mathrm{l}=\frac{\mathrm{P}}{\mathrm{A}}$ Substitute the values we have,

role="math" localid="1663995886324" $$\begin{gathered} \mathrm{l}=\frac{3.9\times {10}^{26}\mathrm{W}}{4\mathrm{\pi }{\left(6.96\times {10}^5\mathrm{km}\right)}^2} \\ =6.4\times {10}^{7}W/m^2 \end{gathered}$$

If the sun's radius is r = R/2 then,

The intensity at sun's interior r = R/2 is

$$\begin{gathered} \mathrm{l}=\frac{3.9\times {10}^{26}\mathrm{W}}{4\mathrm{\pi }{\left(3.48\times {10}^5\mathrm{km}\right)}^2} \\ =2.6\times {10}^8\mathrm{W}/{\mathrm{m}}^2 \end{gathered}$$

Calculate the radiation pressure at sun surface and its interior                        

The radiation pressure at sun surface is${\mathrm{P}}_{\mathrm{rad}}=\frac{\mathrm{l}}{\mathrm{c}}$.Substitute the values we have,

$$\begin{gathered} {\mathrm{P}}_{\mathrm{rad}}=\frac{6.4\times {10}^7\mathrm{W}/{\mathrm{m}}^2}{3\times {10}^8} \\ =0.21\mathrm{Pa} \end{gathered}$$

The radiation pressure at sun's interior, is${\mathrm{P}}_{\mathrm{rad}}=\frac{\mathrm{I}}{\mathrm{c}}$ Substitute the values we have

$$\begin{gathered} {\mathrm{P}}_{\mathrm{rad}}=\frac{2.6\times {10}^8\mathrm{W}/{\mathrm{m}}^2}{3\times {10}^8} \\ =0.86\mathrm{Pa} \end{gathered}$$

Thus, the intensity of the electromagnetic radiation is at sun's surface $6.4\times {10}^7\mathrm{W}/{\mathrm{m}}^2$and the radiation pressure at sun's surface 0.21 Pa and the intensity of the electromagnetic radiation in sun's interior $2.6\times {10}^8\mathrm{W}/{\mathrm{m}}^2$and the radiation pressure in sun's interior 0.86 Pa

Concept that radiation pressure is an important

The gas pressure at sun's surface is $1.0\times {10}^4\mathrm{Pa}$, in the sun's interior r = R/2 the gas pressure is $4.7\times {10}^{13}\mathrm{Pa}$. Compare the given values with the results of part (a), at sun's surface the gas pressure ($1.0\times {10}^4\mathrm{Pa}$) is 50000 greater than the radiation pressure at sun's surface (0.21 Pa x 50000 =$1.0\times {10}^4\mathrm{Pa}$) and for sun's interior r = R/2 the gas pressure $4.7\times {10}^{13}\mathrm{Pa}$is $6\times {10}^{13}\mathrm{Pa}$ times greater than the radiation pressure. Therefore, radiation pressure is an important factor for determine the structure