Question
During a total solar eclipse the moon almost entirely covers the sphere of the sun. Write the relation between the distances and sizes of the sun and moon.
✓
Answer
Key point: In geometry, a solid angle (symbol: $\Omega$ or w) is the twodimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large the object appears to an observer looking from that point. In the International System of Units (SI), a solid angle is expressed in a dimensionless unit called a steradian (symbol: sr).
A small object nearby may subtend the solid angle as a larger object farther away. For example, although the Moon is much smaller than the Sun, it is also much closer to Earth. Diagram given below shows that moon almost entirely covers the sphere of the sun. $R_{me}$ = Distance of moon from earth $R_{se}$ = Distance of sun from earth Let the solid angle made by sun and moon is $\text{d}\Omega,$ we can write
$\text{d}\Omega=\frac{\text{A}_\text{sun}}{\text{R}^2_\text{se}}=\frac{\text{A}_\text{moon}}{\text{R}^2_\text{me}}$
Here, $A_{sun}$ = Area of the sun $A_{moon}$ = Area of the moon$\Rightarrow\theta=\frac{\pi\text{R}_\text{s}^2}{\text{R}_\text{se}^2}=\frac{\pi\text{R}_\text{m}^2}{\text{R}_\text{me}^2}$
$\Rightarrow\Big(\frac{\text{R}_\text{s}}{\text{R}_\text{se}}\Big)^2=\Big(\frac{\text{R}_\text{m}}{\text{R}_\text{me}}\Big)^2$
$\Rightarrow\frac{\text{R}_\text{s}}{\text{R}_\text{se}}=\frac{\text{R}_\text{m}}{\text{R}_\text{me}}\ \text{or}\ \frac{\text{R}_\text{s}}{\text{R}_\text{m}}=\frac{\text{R}_\text{se}}{\text{R}_\text{me}}$
(Here, radius of sun and moon represent their sizes respectively)
A small object nearby may subtend the solid angle as a larger object farther away. For example, although the Moon is much smaller than the Sun, it is also much closer to Earth. Diagram given below shows that moon almost entirely covers the sphere of the sun. $R_{me}$ = Distance of moon from earth $R_{se}$ = Distance of sun from earth Let the solid angle made by sun and moon is $\text{d}\Omega,$ we can write
$\text{d}\Omega=\frac{\text{A}_\text{sun}}{\text{R}^2_\text{se}}=\frac{\text{A}_\text{moon}}{\text{R}^2_\text{me}}$
Here, $A_{sun}$ = Area of the sun $A_{moon}$ = Area of the moon$\Rightarrow\theta=\frac{\pi\text{R}_\text{s}^2}{\text{R}_\text{se}^2}=\frac{\pi\text{R}_\text{m}^2}{\text{R}_\text{me}^2}$
$\Rightarrow\Big(\frac{\text{R}_\text{s}}{\text{R}_\text{se}}\Big)^2=\Big(\frac{\text{R}_\text{m}}{\text{R}_\text{me}}\Big)^2$
$\Rightarrow\frac{\text{R}_\text{s}}{\text{R}_\text{se}}=\frac{\text{R}_\text{m}}{\text{R}_\text{me}}\ \text{or}\ \frac{\text{R}_\text{s}}{\text{R}_\text{m}}=\frac{\text{R}_\text{se}}{\text{R}_\text{me}}$
(Here, radius of sun and moon represent their sizes respectively)
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