Question

The angular diameter of the sun is measured to be $1920''$. The distance or gap of sun from the earth is $1.496 \times {10^{11}}m$. Find the diameter of the sun.

Answer 1

The given angular diameter is a measure of angle between ends of sun and it’s measured from some point on earth. As we have given an angular diameter which is equal to the diameter of sun divided by distance between the sun and earth. Diameter of sun is arc length of circle with radius equal to distance between sun and earth and angle of arc is equal to angular diameter of earth.

Complete step by step answer:

Given the angular diameter of the sun is $1920''$ and distance R between sun and earth is $1.496 \times {10^{11}}m$.
Angular diameter in radian is $\theta = \dfrac{{1920}}{{3600}} \times \dfrac{\pi }{{180}}rad$ ( ${1^o} = 60'$ and $1' = 60''$).
$\theta = 0.0093rad$
As shown in figure, diameter of sun D is arc length of circle with radius equal to distance between sun and earth and angle for arc is $1920''$.
Then $D = \theta \times R$
$D = 0.0093 \times 1.496 \times {10^{11}} = 1.39 \times {10^9}m$
Hence the diameter of the sun is $1.39 \times {10^9}m$.

Note: Here angle $\theta$ is too small that we consider arc as line that why it gives us the more accurate diameter of sun. Here we use $D = \theta \times R$ because for too small an angle in radian, angle is equal to ratio of arc length and radius of circle. As we know that earth revolves around the sun elliptical orbit which means the distance of the earth changes with time but given reading are such that angle is measured when earth is at given distance. If we measure angle at different time distance change and angle also changes such our answer remains the same for each case.