Question

If the distance between Earth and Jupiter is \(810\times10^6\text{ km}\) and the angular diameter of Jupiter measured from Earth is \(36''\), then the diameter of Jupiter (in km) is nearly:

• A. \(2.4\times10^6\)
• B. \(8.2\times10^6\)
• C. \(4.9\times10^5\)
• D. \(1.4\times10^5\)

Hint:

For astronomical objects having very small angular sizes, use \(d=\theta D\) with \(\theta\) expressed in radians.

Answer 1

The Correct Option is D

Concept: For very small angles, the angular diameter formula is \[ \theta=\frac{d}{D} \] where \[ \theta = \text{angular diameter in radians} \] \[ d = \text{actual diameter} \] \[ D = \text{distance from observer} \] This approximation is widely used in astronomy.

Step 1: Convert angular diameter into radians. Given, \[ \theta=36'' \] Since \[ 1^\circ=3600'' \] therefore \[ 36''=\frac{36}{3600}^\circ \] \[ =0.01^\circ \] Converting into radians, \[ \theta=0.01\times\frac{\pi}{180} \] \[ =1.745\times10^{-4}\text{ rad} \]

Step 2: Apply the small-angle formula. Given distance, \[ D=810\times10^6\text{ km} \] Using \[ d=\theta D \] we obtain \[ d=(1.745\times10^{-4})(810\times10^6) \] \[ d\approx1.41\times10^5\text{ km} \]

Step 3: Compare with options. The calculated value is approximately \[ 1.4\times10^5\text{ km} \] which matches option (D).

Step 4: Final answer. \[ \boxed{1.4\times10^5\text{ km}} \]