How close to a black hole can an object orbit elliptically?

Question

How close to a black hole can an object orbit elliptically? I know circular orbits are no longer stable at distance less than 3 times the Schwarzschild radius. But what about elliptical orbits? Can an object have a semi-major axis or perihelion at distance of less than 3 times the Schwarzschild radius?

Answer 1

A bound elliptical orbit around a Schwarzschild black hole must have $r > 2 r_s$ at all times (where $r_s = 2 M$ is the Schwarzschild radius). Deriving this result is a good exercise for students learning about the Schwarzschild geometry, so I won't go through all the details, but the basic sketch of the proof is as follows:

• Recall that a massive particle moving in a Schwarzschild geometry is equivalent to a particle moving in a classical "effective potential" given by $$ V_\text{eff}(r) = - \frac{M}{r} + \frac{\ell^2}{2 r^2} - \frac{M\ell^2}{r^3}, $$ where $M$ is the mass of the black hole and $\ell$ is the specific angular momentum of the particle.
• Note that for a bound orbit, we must have $V_\text{eff}(r) < 0$ at all times.
• Find the points at which $V_\text{eff}(r) = 0$ for a given value of $\ell$. This will be the closest possible value of perihelion for a bound orbit for a particular value of $\ell$.
• Find the value of $\ell$ that allows for the closest perihelion. It turns out to be $\ell = 4M$, and for that value of the angular momentum you must have $r > 2 r_s$ to satisfy $V_\text{eff}(r) < 0$.

Answer 2

Yes, highly eccentric orbits can go deeper. (Kostic 2012) derives analytic solutions in terms of elliptic functions, noting (in footnote 4) that the closest approach would be 2 Schwarzschild radii out for a $l=2$ orbit. Such orbits may not be very elliptic-looking since there can be multiple turns per perimelasma approach (OK, periapse is the more common term).

Answer 3

I reality, in the frame of general relativity, the real elliptic orbit does not exist and all bodies will end-up falling in the black hole. The nearer you get from the event horizon, the faster you will fall inside which means the "limit you are looking for does not exist. It's more like, the farther you are, the longer you will keep orbiting.

I addition, since orbits can be more or less eccentric it is delicate to just give one distance.