If the acceleration due to gravity experienced by a point mass at a height h above the surface of earth is same as that of the acceleration due to gravity at a depth $$\alpha h$$ ($$ h \ll R_E$$) from the earth surface. The value of $$\alpha$$ will be _____. (use $$R_E = 6400$$ km)

We need to find $$\alpha$$ such that the acceleration due to gravity at height $$h$$ above the Earth's surface equals the acceleration due to gravity at depth $$\alpha h$$ below the surface, given $$h \ll R_E$$.

The acceleration due to gravity at a height $$h$$ above the surface (for $$h \ll R_E$$) is approximately $$g_h = g\left(1 - \dfrac{2h}{R_E}\right)$$.

The acceleration due to gravity at a depth $$d$$ below the surface is $$g_d = g\left(1 - \dfrac{d}{R_E}\right)$$.

Setting these equal with $$d = \alpha h$$: $$g\left(1 - \dfrac{2h}{R_E}\right) = g\left(1 - \dfrac{\alpha h}{R_E}\right)$$.

Cancelling $$g$$ and simplifying: $$-\dfrac{2h}{R_E} = -\dfrac{\alpha h}{R_E}$$, which gives $$\alpha = 2$$.

Hence, the correct answer is 2.