I was reading through this post today and was very impressed by the response that was given. However, what would have to happen to the velocity in order to collide with the Earth?
Velocity of satellites greater than required velocity
I was think of setting up an equation as follows. If the orbit changes from a circular orbit at some height $h$ with velocity $v$, then an elliptical orbit will occur if the velocity decreases to $\lambda v$, for some $\lambda \in (0,1)$.
From the post made earlier, we know that the original velocity is given by $$v_0^2 = \frac{GM}{R_E+h}$$ and the new velocity is given by $$\lambda^2 v_n^2 = \lambda^2 \Bigg ( GM \Bigg ( \frac{2}{R_E+h} - \frac{1}{a} \Bigg ) \Bigg ).$$ Therefore, solving $$\lambda^2 v_n^2 \leq \frac{GM}{R_E}.$$ Should yield a viable restriction on $\lambda^2$.
But this doesn't give me what I want. A satellite should crash into the earth if it breaks through the atmosphere, i.e when $h < R_E + R_A$, where $R_A$ is the atmospheric height.
How do I determine this $R_A$ from the general theory?
I'm aware that the escape velocity is given by $V_E = \sqrt{\frac{GM}{R_E+h}}$.