Question

(i) The earth-moon distance is about 60 times the earth radius. What will be the diameter of earth (approximately in degrees) as seen from the moon.

(ii) Moon is seen to be of ${\left(\frac{1}{2}\right)}^{\circ }$ diameter from the earth. What must be the relative size compared to the earth?

(iii) From parallax measurement, the sun is found to be at a distance of about 400 times the earth-moon distance. Estimate the ratio of sun-earth diameters.

Answer 1

(i) Step 1: Draw a rough sketch of the given situation.


Step 2: Convert the angle in radian.

As we know that
$\theta =\frac{arc}{radius}=\frac{R_E}{60R_E}=\frac{1}{60}rad$


Step 3: Find the diameter of earth.

It is clear from the figure that the diameter of earth (in degree) as seen from the moon will be the angle from the moon to the diameter of earth.
i.e., $2\theta =2\times \frac{1}{60}\times \frac{180}{\pi }=\frac{6^{\circ }}{\pi }$
Substituting the value of $\pi$ we get –
$\frac{6}{3.14}\cong 2^{\circ }$

Final answer: $\theta =2^{\circ }$

(ii) Given, moon is seen from earth
(diametrically)
angle = ${\frac{1}{2}}^{\circ }$

As we seen if earth is seen from moon, the angle = $2^{\circ }$
Relative size of Moon = $\frac{\text{size of moon}}{\text{size(diameter) of earth}}$
= $\frac{{\left(\frac{1}{2}\right)}^{\circ }}{2^{\circ }}=\frac{1}{4}$

Size of moon is $\frac{1}{4}$ the size (diameter) of earth .

Final answer: $\frac{D_{earth}}{D_{moon}}$ = 4

(iii) From parallax measurement, given that sun is at a distance of about 400 times the earth-moon distance.

Hence it can be written as,
$\frac{r_{sun}}{r_{moon}}=400;$ r = distance
Sun and moon both appear to be of the same angular diameter as seen from the earth.
(D = Diameter)
$\frac{D_{sun}}{r_{sun}}=\frac{D_{moon}}{r_{moon}}=\Rightarrow \frac{D_{sun}}{D_{moon}}=400$
$\frac{D_{earth}}{D_{moon}}=4\Rightarrow \frac{D_{sun}}{D_{earth}}=100$

Final answer: $\frac{D_{sun}}{D_{earth}}=100$