newtonian mechanics - Why is the total energy of an orbiting system negative?

Assume it's an circular orbit. Object A orbits around object B. Take object B as frame of reference.

.$E=KE_a + GPE$

.$E=\frac 12m_av_a^2 +(-\frac {GM_bm_A}r)$

.$E=\frac 12m_a(GM_br)+(-\frac {GM_bm_a}r)$

.$E=-\frac {GMm}{2r} < 0$

What does negative total energy at any instant of time mean?

Answer

Negative energies are totally fine, because you had to pick a zero-point for energy. In your calculation you picked it to be at infinity. You could have chosen the zero-point for potential energy in such a way that your system had zero energy, or whatever. Only changes in energy are meaningful, in general.

Consider this: what happens if you add energy to this system? It gets closer to zero, and zero for us is the point where the particle is at rest, but is infinitely far away from the other particle. So negative energy represents the fact that to "free" the particle from the central potential requires you to add energy. This comes up a lot in quantum mechanics--the ground state energy of the hydrogen atom is -13.6 eV.