Two stars are 10 light years away from the earth. They are seen through a telescope of objective diameter 30 cm. The wavelength of light is 600 nm. To see the stars just resolved by the telescope, the minimum distance between them should be (1 light year = $$9.46 \times 10^{15}$$ m) of the order of: 

The two stars will appear as two distinct objects to the eye only if the telescope can just resolve them. For a telescope with a circular aperture, the condition for just-resolution is given by Rayleigh’s criterion, which states:

$$\theta_{\text{min}} \;=\;1.22 \,\frac{\lambda}{D}$$

Here $$\theta_{\text{min}}$$ is the smallest angular separation (in radians) that the telescope can resolve, $$\lambda$$ is the wavelength of the light being observed, and $$D$$ is the diameter of the objective.

We have the numerical values

$$\lambda = 600 \text{ nm} = 600 \times 10^{-9}\,\text{m},$$

$$D = 30 \text{ cm} = 0.30\,\text{m}.$$

Substituting these into Rayleigh’s formula, we obtain

$$ \theta_{\text{min}} =1.22 \,\frac{600 \times 10^{-9}}{0.30} =1.22 \times \left( \frac{600}{0.30}\right)\times 10^{-9} =1.22 \times 2000 \times 10^{-9} =2440 \times 10^{-9} =2.44 \times 10^{-6}\,\text{rad}. $$

The linear distance $$s$$ between the two stars that corresponds to this angular separation is related by the simple geometry formula

$$s = \theta_{\text{min}}\; L,$$

where $$L$$ is the distance of the stars from the earth.

Each star is 10 light-years away, so

$$ L = 10 \times (1\;\text{light year}) = 10 \times 9.46 \times 10^{15}\,\text{m} = 9.46 \times 10^{16}\,\text{m}. $$

Now we multiply:

$$ s = (2.44 \times 10^{-6}) \times (9.46 \times 10^{16}) = (2.44 \times 9.46) \times 10^{\,(-6 + 16)} = 23.0824 \times 10^{10}\,\text{m}. $$

Shifting the decimal place to get a single significant figure before the power of ten

$$ s = 2.30824 \times 10^{11}\,\text{m}. $$

To express this in kilometres, we divide by $$1000$$:

$$ s = \frac{2.30824 \times 10^{11}}{10^{3}} = 2.30824 \times 10^{8}\,\text{km}. $$

This is of the order of $$10^{8}\,\text{km}.$$

Hence, the correct answer is Option A.