Orbits and Tidal Accelerations

Newtonian version

Suppose that a small body, such as a space station — too small to have a significant gravitational field of its own — is orbiting a planet in a perfectly circular orbit. Suppose that the space station has become “tidally locked” so that it always keeps the same face towards the body it’s orbiting, just as the moon keeps the same face towards the Earth.

The centre of mass of the space station will move in a circle at a constant rate; anyone floating there will feel weightless, and will remain fixed relative to the walls around them. Now, suppose they place test particles a short distance away from the centre in various directions. If those particles are allowed to move freely, starting from rest (relative to the walls of the station), how will their motion look from within the station?

The easiest case to consider is when the test particles have been shifted away from the centre either in the direction in which the station is orbiting, or in the opposite direction. Although the orbit is curved, our space station is so small compared to the orbit that there is essentially no difference between displacement along a straight line tangent to the orbit, and displacement along the curve of the orbit itself. Now, if the test particles not only lie on the orbit, but are also moving in such a manner that they will stay on it, the result will be very simple: in a reference frame locked to the space station, they won’t be seen to move at all, but will remain exactly where they were placed.

Test particles displaced along orbit of centre of massTest particles displaced along orbit of centre of mass

That turns out to be exactly what happens. Because the space station keeps one face locked towards the body it’s orbiting, being at rest in the station anywhere along the line of the orbit amounts to the same thing as moving along the orbit at exactly the same rate as the station itself. So, as shown in the animation above, the displaced test particles will follow the same orbit as the station’s centre of mass — a little further along, or lagging slightly behind it — and in the station’s reference frame they will simply stay put.

Next, suppose we place a test particle a short distance away from the station’s centre of mass in a direction that points out of the plane of the orbit. The particle will still be almost the same distance from the planet as the station’s centre of mass, and it will be moving with the same speed and in the same direction, but the plane that contains the particle, the centre of the planet, and the particle’s velocity vector, will be inclined at a small angle to the plane of the station’s orbit.

Test particle displaced above orbit of centre of mass

So, the test particle’s orbit will be just like the station’s orbit, but inclined at an angle to it. Seen from within the station, the particle will oscillate up and down as its orbit takes it above and below the plane of the station’s orbit.

That oscillating motion describes what happens if we allow the particle to move freely. What if we consider an astronaut supported by some structure within the space station, a short distance above or below the station’s centre of mass? Because the structure is preventing the astronaut from moving in towards the centre, she will feel it exerting a force on her body. In other words, she will feel as if she possesses weight, with the apparent gravitational pull pointing towards the centre of the station. The strength of this pull will be proportional to the distance from the centre; the astronaut’s weight will grow linearly with distance.

Net force on astronaut at a fixed distance above or below plane of orbit

The weight that the astronaut feels, or the acceleration that she measures for a free-falling particle, is known as tidal acceleration. Another way to think about this is to note that the planet the station is orbiting is tugging on the astronaut with a gravitational pull at a slight angle to the plane of the station’s orbit (see image on the right), whereas if the astronaut remains a fixed distance above or below the plane the centrifugal force associated with her orbital motion will lie precisely parallel to the plane. The centrifugal and gravitational forces that balance perfectly at the station’s centre of mass will not balance for the astronaut, and she will feel the difference between them as a weight directed towards the station’s centre. But in some ways it’s simpler just to note that the astronaut, by failing to follow the inclined orbit of the free-falling test particle, must be subject to a force in the direction required to resist that natural motion.

Dust cloud free-falls towards Earth

Now the tidal acceleration we’ve seen in this case, drawing things closer to the centre of the station, raises a small puzzle. Imagine a cloud of dust free-falling straight towards the Earth from space (see animation on the left). Because all the dust is moving towards a single point at the centre of the Earth, the cloud will be squeezed in both horizontal directions, with the trajectories of all its particles lying on a cone that draws ever narrower as it approaches the Earth. If gravity squeezes a cloud like that in the two horizontal directions, shouldn’t there be a tidal force experienced by astronauts in our space station, pulling them towards the centre, not only when they are above or below the plane of the orbit, but also when they are displaced from the centre in the direction along the orbit, like the test particles of our first example?

In fact there is such a tidal force, but it’s cancelled exactly by the centrifugal force of the station’s spin! In order to keep one face locked towards the body it’s orbiting, the station is rotating, and that means that anything away from the axis of spin — which is perpendicular to the plane of the orbit — will need some force acting upon it to keep it from flying away. For the motionless particles of our first example, that role is filled by the gravitational tidal force that squeezes everything that’s displaced “horizontally” (that is, in any direction perpendicular to the direction of the centre of the planet). Of course, this doesn’t negate our earlier observation that test particles strung along the station’s orbit will appear motionless from within the station simply because they’re following the same orbit. Both perspectives make sense, and lead to the same result.

Test particles displaced inwards and outwards from orbit of centre of mass

What happens when we place test particles a little closer to, or a little further from, the planet than the station’s centre of mass? For a falling cloud of dust, being further from the Earth than the cloud’s centre means the gravitational pull is a little weaker, so particles there lag ever further behind; being closer to the Earth than the centre means a stronger gravitational pull, so particles there race further ahead. In other words, the cloud is stretched out vertically.

Something similar happens to test particles in our orbiting station. A particle starting out a little further from the planet than the station’s centre of mass will lie on a larger orbit, but it won’t be a circular orbit. If the particle starts at rest in the station, it will have precisely the same angular velocity in its motion around the planet as the station itself. But because it’s slightly further out, the gravitational pull on it will be slightly weaker and the orbital centrifugal force slightly stronger (see second image on the right), so instead of the perfect balance between gravity and centrifugal force that allows the station to maintain a circular orbit, centrifugal force will dominate, and the particle will move outwards. As a secondary effect, its orbital angular velocity will then diminish (conserving angular momentum), causing it to lag behind the station’s slightly faster orbit. So as well as moving outwards from the station’s centre the particle will appear to swerve backwards in the counter-orbital direction (see animation below). But its initial acceleration will be outwards, and an astronaut in the same initial location who is held in place by some supporting structure will feel a weight directed precisely outwards.

For a test particle displaced in the opposite direction, everything is reversed. The gravitational pull on it will dominate over its orbital centrifugal force, so it will accelerate inwards, its orbit carrying it closer to the planet. At the same time, as a secondary effect its orbital angular velocity will increase, so from the point of view of someone in the station it will swerve forwards in the orbital direction. But an astronaut in a fixed position displaced inwards from the centre of the station will feel a weight pointing precisely inwards.

Test particles displaced inwards and outwards from orbit of centre of mass

What is the role of the station’s spin in all this? The acceleration found from looking at these test particle orbits is greater than the usual tidal stretching by an amount which can be attributed, equally well, either to a difference in orbital centrifugal force at different radii, or to centrifugal force associated with the spin of the station around its axis. For our dust cloud falling towards the Earth, the vertical stretching is double the horizontal squeezing, at an equal distance away from the cloud’s centre. For astronauts in our station, the radial stretching is three times the squeezing towards the orbital plane. In other words, an astronaut ten metres from the space station’s centre in a radial direction will feel a weight three times greater than that of an astronaut who is ten metres above or below the orbital plane. (See the Mathematical Details page for more on quantifying these effects.)

A second role for the station’s spin is in making sense of the way the test particles swerve sideways once they’re in motion. We’ve explained this in terms of the orbital angular velocity changing as the particles move away from, or closer to, the planet they’re orbiting. But from the perspective of someone in the station, this motion can be ascribed to the Coriolis force. This is the same force that makes air initially travelling due south in the Earth’s northern hemisphere swerve to the west: the Earth’s rotation was initially carrying the air eastwards with the planet itself, but at lower latitudes the ground is moving more rapidly, so the air can’t keep up. Similarly, anything moving away from the centre of the station (which has a counter-clockwise spin in the figure above) will swerve in a clockwise direction, because its tangential momentum is not enough to keep up with the station’s greater speed further from the centre.

There’s one more simple experiment worth considering: what happens if we take a test particle that lies on the orbital line, and give it a small radial push? Given that a particle displaced a small distance in a radial direction accelerates away rapidly, you might think this particle would do the same, but it doesn’t. Instead, it cycles around inside the station along a bounded elliptical trajectory.

Test particle given a small radial push

What’s happening here? From the point of view of an observer inside the station, the Coriolis force causes the particle to swing around into a direction where the same force acts to oppose the tidal acceleration. The particle is never at rest in the station, so the Coriolis force is always in play, and it is strong enough to keep the particle executing a bounded, cyclic motion.

From the point of view of someone looking at the orbit of the particle, it is following an elliptical orbit that crosses the circular orbit of the station, and at its closest and furthest points from the planet lies inwards and outwards from the station’s orbit by the same amount. This means that the total width of the ellipse in its longest direction (its major axis) is equal to the diameter of the station’s orbit, and that in turn means that the two orbits have identical periods.

You can try out your own experiments, launching test particles from any location and with any velocity you like, using the Null Chamber applet.

Relativistic effects

The Newtonian analysis above is fine for a space station orbiting a planet, but what if it’s orbiting a black hole instead? And what if that black hole has a significant amount of spin, dragging the surrounding space-time around with it? In those situations, although we still expect the same kind of tidal accelerations to be present, some of the assumptions we’ve made for the Newtonian case are no longer valid. One result of this is that the ratio of three to one for the two kinds of tidal acceleration experienced in the station will not continue to hold. This means, surprisingly, that from the inside of a sealed space station it would sometimes be possible to tell that you were definitely not orbiting an ordinary planet, but had to be orbiting a black hole — and it would be a ratio of tidal forces, rather than their absolute strength, that revealed this. More details.

Precession of gyroscope orbiting Schwarzschild black hole

The first example from our Newtonian analysis does always hold up: test particles displaced from the station’s centre along the orbital direction will always stay fixed relative to the station, since they share exactly the same orbit. But the intuitively obvious notion that in order to remain tidally locked, the station should spin about its axis with exactly the same period as its orbital motion, is no longer quite correct. Although the station must end up facing the same direction relative to the distant stars after completing one orbit, if it carries a gyroscope as a reference direction with it for that orbit, the gyroscope will not end up pointing in the same direction relative to the stars (and hence it will also fail to return to its original orientation with respect to the structures of the station itself). This is a general property of curved spacetime: carrying a vector around a closed path, keeping it as close to parallel as possible all the way, will not restore it to its original direction (see image on the left). What’s more, as far as centrifugal and Coriolis forces are concerned, the rate of spin of the station that matters is that measured relative to the gyroscope, not relative to the distant stars.

Precession of periapsis of orbit around Schwarzschild black hole

Another assumption that doesn’t hold up in the relativistic case is the existence of elliptical orbits. In general relativity, non-circular orbits don’t form perfect closed ellipses; rather, the points of closest approach to the central body occur after the orbiting body has travelled more than one full rotation (see image on the right). This effect was famously confirmed when general relativity was shown to account for a missing contribution to the precession of Mercury’s perihelion: the gradual rotation around the sun of the point of the planet’s closest approach.

Precession of nodes of inclined orbit around extremal Kerr black hole

If our black hole is non-rotating, then the spherical symmetry of spacetime means that two circular orbits inclined at an angle to each other will still have identical periods, as in the Newtonian case, and the rate at which a test particle oscillates up and down relative to the plane of the station’s orbit will still agree with the orbital period. However, if the black hole is rotating, then that perfect spherical symmetry vanishes. We can still have the station in a circular orbit around the hole’s “equator”, but no perfect circular orbits exist at an inclination to the equator, and the locations along the orbit where the oscillating test particle crosses the orbital plane will not remain fixed (see image on the left); for an orbit in the same direction as the hole’s spin, these crossings will be spaced more than 180 degrees apart.

So while in the Newtonian case the orbital period is really all there is to know, in the relativistic case there are four separate periods: the orbital period, the period of rotation of a gyroscope, the precession of the periapsis (point of closest approach for a non-circular orbit), and the precession of the nodes (points where an inclined orbit crosses the equatorial plane). Measuring them, and untangling their meaning, is part of the struggle faced by the protagonists of Incandescence to understand their world.

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Incandescence / Orbits and Tidal Accelerations / created Tuesday, 1 January 2008 / revised Saturday, 12 January 2008
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