Initially a satellite of 100 kg is in a circular orbit of radius $$1.5R_{E}$$ This satellite can be moved to a circular orbit of radius $$3R_{E}$$ by supplying $$\alpha\times10^{6}J$$ of energy The value of $$\alpha$$ is ____. (Take Radius of Earth $$R_{E}=6\times10^{6}m\text{ and }g=10m/s^{2}$$)

A satellite of mass 100 kg is moved from a circular orbit of radius $$1.5R_E$$ to $$3R_E$$. We need to find the energy required in the form $$\alpha \times 10^6$$ J.

Recall that the total energy of a satellite in circular orbit is $$ E = -\frac{GMm}{2r} = -\frac{mgR_E^2}{2r} $$ using $$GM = gR_E^2$$.

At radius $$r_1 = 1.5R_E$$ the energy is $$ E_1 = -\frac{mgR_E^2}{2 \times 1.5R_E} = -\frac{mgR_E}{3} $$.

At radius $$r_2 = 3R_E$$ the energy is $$ E_2 = -\frac{mgR_E^2}{2 \times 3R_E} = -\frac{mgR_E}{6} $$.

Therefore the required energy is $$ \Delta E = E_2 - E_1 = -\frac{mgR_E}{6} + \frac{mgR_E}{3} = mgR_E\left(\frac{1}{3} - \frac{1}{6}\right) = \frac{mgR_E}{6} $$.

Substituting values gives $$ \Delta E = \frac{100 \times 10 \times 6 \times 10^6}{6} = \frac{6 \times 10^9}{6} = 10^9 \text{ J} = 1000 \times 10^6 \text{ J} $$.

Hence $$\alpha = 1000$$ and the correct answer is Option (2): 1000.