If you orbit too close within the Schwarzschild radius, you fall in and if you’re very far away, the orbits are the same as they’d be around any normal astronomical body. But, what happens in between those extremes, when you’re orbiting near a black hole but not right at the Schwarzschild radius?
Effective Potential Energy
To understand this, we can use the effective potential energy. We write the total energy in the form 1/2 mvr-squared plus U-effective, where U-effective is the sum of the real potential energy, minus-GMm over r, plus a term that depends on angular momentum, L-squared over 2mr-squared.
That L-squared term is the kinetic energy associated with angular motion, but because angular momentum is conserved, we write it in a form that depends only on r, making it resemble a type of potential energy.
Changing Radial Coordinate
The radial coordinate changes with time, as a body whirls around an attractor. In general relativity, there is an extra term in the effective potential energy: minus L-squared times the Schwarzschild radius over 2mr-cubed.
When we re-plot U effective for a black hole, we need to add the contributions of the negative-1 over r term, the positive-1 over r-squared term, and the negative-1 over r-cubed term. As r gets smaller, the new 1 over r-cubed term grows the fastest of all, and it’s negative. So, the effective potential dives downward at small radius.
The vertical axis shows U-effective divided by little mc-squared, and the horizontal axis shows the radius in units of the Schwarzschild radius. We take a value of angular momentum equal to 2.1 little mc times the Schwarzschild radius.
Orbiting Away from the Schwarzschild Radius
What we can see is that below one Schwarzschild radius, there’s a pit in the potential. Remember, we can visualize what’s going to happen to the radial coordinate r by imagining we drop a marble onto the curve starting from an initial height E, given by the total energy.
If we’re far away from the Schwarzschild radius, the marble rolls back and forth. That means r oscillates back and forth, as was the case in Newtonian gravity; the particle follows an orbit with some minimum and maximum distance.
But if we let go of the marble inside the Schwarzschild radius, it falls into the pit, regardless of energy. If you’re inside the event horizon, you’re doomed. And even if we start the marble further out, in the bowl, if the energy is high enough, it won’t just roll back and forth, it will go over the inner wall of the bowl and fall into the pit.
What that means is that in general relativity, you can hit the origin even if you have some nonzero angular momentum. There’s no more barrier here that seals off the singularity, as there was in Newtonian gravity. It’s as though, when you get close enough, the hole reaches out and grabs you. It absorbs both your mass, and your angular momentum. That’s right, black holes can rotate.
This article comes directly from content in the video series Introduction to Astrophysics. Watch it now, on Wondrium.
We have drawn this curve for a specific value with angular momentum, and different values lead to different shapes, for the bowl. Now, let us see what happens as we lower the angular momentum. The bowl gets pounded down on the left side, until it’s not even a bowl any more. There’s no more minimum. That means there’s no place where the particle can sit still at a constant radius. A circular orbit is impossible.
This is a famous result in general relativity. It’s called the ISCO—the Innermost Stable Circular Orbit. The ISCO for a non-rotating black hole turns out to be 3 times the Schwarzschild radius. For a rotating black hole, the ISCO can be larger or smaller, depending on which way it’s rotating.
Let’s go back now to a case with more angular momentum, so the effective potential energy curve still looks like a bowl. If you give the particle a modest amount of energy, the radial coordinate oscillates back and forth, just as it does for a planet going around the Sun.
For a planet, we found the trajectory is an elliptical orbit, with a minimum distance of a times 1 minus e, and a maximum distance of a times 1 plus e. But, remember, we only get ellipses for the very specific case of a force that goes like 1 over r-squared. And in general relativity, the force law doesn’t go exactly as 1 over r-squared. There’s that extra term in the potential. So, we don’t get perfect ellipses.
Close to the black hole, the orbits are not even approximately ellipses. They’re rosettes, whipping around fast as they approach the black hole, then slowing down as they recede, and coming back again from a different angle.
If you’re relatively far away, the orbits are very nearly ellipses but not quite. What happens is that the orientation of the ellipse gradually wheels around in space. It’s an effect called apsidal precession.
Orbiting Near a Black Hole
So, the planets orbiting a black hole, are not quite obeying Kepler’s first law. Remember, once you’re outside a spherical mass distribution, the gravitational effects are the same, no matter whether you’re orbiting a planet, a star, or a black hole. That means Kepler’s first law must be at least a little bit wrong in the solar system, too. The planetary orbits should be precessing.
And they are. The effect is strongest for Mercury, the planet with the smallest orbital distance. In fact, the precession of Mercury’s orbit has been observed for centuries. But it’s precessing mainly for a different reason. The gravitational forces from all of the other planets pulling on Mercury also causes a departure from the idealized problem of one star and one planet, and that causes precession, too.
Common Questions about Orbiting Near a Black Hole
The Innermost Stable Circular Orbit or ISCO for a non-rotating black hole turns out to be 3 times the Schwarzschild radius. For a rotating black hole, the ISCO can be larger or smaller, depending on which way it’s rotating.
Close to the black hole, the orbits are rosettes, whipping around fast as they approach the black hole, then slowing down as they recede, and coming back again from a different angle.
If you orbit too close within the Schwarzschild radius, you fall in and if you’re very far away, the orbits are the same as they’d be around a star, or a planet, or any normal astronomical body.