Gravity of a Torus

What is the gravitational field of a torus? In which direction would the net force of gravity point at various locations near a ring of matter? People sometimes think that, perhaps for reasons of symmetry, an object in the interior of a ring of matter would be drawn toward the center, but this is not the case – at least not for objects in the plane of the ring. To see why, consider a very thin ring of mass treated as a circle of radius R in the plane, and a particle inside this ring at a distance r from the center. Construct an arbitrary line passing through this particle, striking the ring in two opposite directions at distances L₁ and L₂. If we rotate this line about the particle through an incremental angle dθ, it will sweep out sections of the ring proportional to L₁ cos(α)dθ  and L₂ cos(α)dθ, where α is the angle the chord makes with the normals to the circle at the points of intersection. The net gravitational force exerted by these two opposing sections of the ring is proportional to the masses in these small sections divided by the squares of the distances, i.e., the force is proportional to dθ cos(α) (1/L₁ – 1/L₂) in the direction of the L₁ intersection point. Hence the net force is in the direction of the closest point on the ring, directly away from the center.

Of course if the particle was inside a spherical shell instead of a circular ring, the conical region surrounding each ray through the particle would sweep out a section of the shell proportional to the square of the distances L₁ and L₂, which would then cancel out with the inverse squares in the force expressions, leaving a net zero force on the particle. However, in the case of a ring of matter, the distances L₁ and L₂ appear only to the first power in the mass terms, so they cancel only one of the powers in the inverse squares. This is why the net force along every ray is positive in the direction of the closer point on the ring, which implies that a particle in the interior plane of the ring would move outward toward the ring.  There would be an equilibrium point at the center of the ring, but this point would not be stable, i.e., it would be a local maximum of the potential energy.

To quantitatively determine the Newtonian gravitational force for a massive ring of radius R, we can note first that the field has cylindrical symmetry, so all that matters is the radial distance (from the ring’s center) of a test particle and its elevation above the plane of the ring. It’s convenient to use cylindrical coordinates, r,θ,z, where r is the radial distance from the center of the ring, θ is the angle in the plane of the ring, and z is the elevation above (or below) the plane of the ring. Without loss of generality we can focus on a test particle at a distance r from the origin and at an angle θ = 0. Letting ρ denote the mass per unit length along the circumference of the ring, each increment of the ring contributes the incremental mass dm = ρRdθ.  This mass is at a distance s from the test particle, given by

The corresponding incremental potential is –dm/s, so the total gravitational potential is given by the integral

The gravitational force in the r and z directions is given by the derivatives dψ/dr and dψ/dz. There is no closed-form expression for the above integral, but we can bring the differentiations inside the integrals, giving the following expressions for the radial and axial forces

Using these equations, with R = ρ = 1, the radial gravitational force near a ring of matter at various elevations above the plane of the ring is plotted in the figure below.

The curve labeled “z = 0.0” is for points in the plane of the ring. As discussed previously, for points away from the center the net gravitational force is positive, i.e., outward, directly away from the center. However, at an elevation of 0.8R above the plane of the ring, the net radial force is negative as we move away from the central axis, meaning the radial force tends to push a test particle back toward the central axis. Even for elevations of just 0.1R, there is a point inside the ring’s radius where the radial force passes through zero and then becomes negative. For each elevation (other than zero) there is a radial position of stable equilibrium. Of course, these locations are not equilibrium positions in the axial direction, as shown by the plot below, which gives the axial force as a function of axial distance (from the plane of the ring) for various radial positions.

This plot shows that there is always a net axial force toward the plane of the ring. Hence a test particle is always driven toward the z = 0 plane, on which it is always driven toward the r = R radial position, i.e., it is propelled toward the ring. The r = 0 curve in this plot represents the axial force as a function of axial position for a test particle on the central axis. For each radial position there is a unique elevation of maximum force, pulling the test particle back toward the plane of the ring. A similar plot for radial positions outside the radius of the ring is shown below.

We can also express the gravitational forces as power series. If r is less than R it’s convenient to write the radial force equation (with z = 0) in the form

where μ = r/R. Expanding this in powers of μ and dividing out the geometric series gives

Thus at r = 0 the radial force is zero, but at any non-zero value of r there will be a net outward force, initially proportional to r, and then increasing sharply as r approaches R. (The ring is assumed to have infinite density so the outward force goes to infinity at r = R.) On the other hand, if r is greater than R, it’s convenient to write the radial force equation (again with z = 0) in the form

where ν = R/r. Expanding this gives the series

So this starts out infinite on the outer surface (where ν = 1), drops off steeply in the near field, and then approach an inverse-square relation in the far field.

It’s interesting to consider how closely the gravitational field of a general torus is to that of the same mass distributed on a thin ring. Obviously if we arrange a set of spheres in a circle, their field is the same as if their masses were located at points on the ring. We can also arrange a set of overlapping spheres with most of the overlap removed by internal spheres, all of which are centered on the ring, so in this way we can approximate very closely a general torus with just positive and negative spheres that can be treated as points on the central ring.

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