Question

from parallax measurement, the sun is found to be at a distance about 400 times the earth- moon distance. estimate the ratio of sun-earth diameters

Answer 1

Given,
$$\begin{gathered} \frac{\mathrm{Distance}\mathrm{of}\mathrm{sun}\mathrm{from}\mathrm{earth}}{\mathrm{Distance}\mathrm{of}\mathrm{moon}\mathrm{from}\mathrm{earth}}=\frac{{\mathrm{r}}_{\mathrm{sun}}}{{\mathrm{r}}_{\mathrm{moon}}}=400 \\ \mathrm{and} \\ \frac{\mathrm{Diameter}\mathrm{of}\mathrm{sun}}{\mathrm{Diameter}\mathrm{of}\mathrm{earth}}=\frac{{\mathrm{D}}_{\mathrm{sun}}}{{\mathrm{D}}_{\mathrm{earth}}}=? \\ \mathrm{As}\mathrm{seen}\mathrm{from}\mathrm{the}\mathrm{earth},\mathrm{both}\mathrm{sun}\mathrm{and}\mathrm{moon}\mathrm{appear}\mathrm{to}\mathrm{be}\mathrm{of}\mathrm{same}\mathrm{angular}\mathrm{diameter}. \\ \mathrm{So}, \\ \mathrm{\theta }=\frac{{\mathrm{D}}_{\mathrm{sun}}}{{\mathrm{r}}_{\mathrm{sun}}}=\frac{{\mathrm{D}}_{\mathrm{moon}}}{{\mathrm{r}}_{\mathrm{moon}}} \\ \frac{{\mathrm{D}}_{\mathrm{sun}}}{{\mathrm{D}}_{\mathrm{moon}}}=\frac{{\mathrm{r}}_{\mathrm{sun}}}{{\mathrm{r}}_{\mathrm{moon}}}=400----\left(\mathrm{i}\right) \\ \mathrm{So},\frac{{\mathrm{D}}_{\mathrm{earth}}}{{\mathrm{D}}_{\mathrm{moon}}}=4---\left(\mathrm{ii}\right) \\ (\mathrm{Since}\frac{{\mathrm{R}}_{\mathrm{earth}}}{{\mathrm{R}}_{\mathrm{moon}}}=4) \\ \mathrm{Dividing}\left(\mathrm{i}\right)\mathrm{by}\left(\mathrm{ii}\right),\mathrm{we}\mathrm{get} \\ \frac{{\mathrm{D}}_{\mathrm{sun}}}{{\mathrm{D}}_{\mathrm{earth}}}=\frac{400}{4}=100 \\ \mathrm{Or}{\mathrm{D}}_{\mathrm{sun}}:{\mathrm{D}}_{\mathrm{earth}}=100:1 \end{gathered}$$