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1. Gravity of a torus vs gravity of a sphere.

Have you ever wondered how gravity vector of a toroidal planet would look like? I have.

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This plot presents gravity of a sphere (yellow line) and gravity of a torus (orange/red line). Two hatched circles are intersections of our torus and sphere with the plane of our plot.

On the X-axis is distance from the center. Both torus and sphere are centered at the same point. On the y-axis - gravity vector. Positive value means that it is directed to the left.

We can see, that as we go further away from both bodies, their gravity looks more and more similar. We also note that inside the sphere gravity changes linearly with distance.

2. Equations

First we find a formula for a little chunk of mass of our torus:

$$dm=\rho rR\, dr\, d\alpha\, d\theta$$
$$dm=k rR\, dr\, d\alpha\, d\theta$$

Acceleration is G times mass divided by squared distance. Distance from the center is:

$$d_{o}=\sqrt{(R+r\cos\alpha)^{2}+(r\sin\alpha)^{2}}$$

Distance from some point at distance d from the center:

$$d_{d}=\sqrt{(d_{o}\cos\Theta-d)^{2}+(d_{o}sin\Theta)^{2}}$$

After subsitution this gives us:

$$d_{d}=\sqrt{\left(\sqrt{(R+r\cos\alpha)^{2}+(r\sin\alpha)^{2}}\cos\Theta-d\right)^{2}+(R+r\cos\alpha)^{2}(\sin\Theta)^{2}+(r\sin\alpha)^{2}(\sin\Theta)^{2}}$$

Magnitude of \(d\vec{a}\) vector is:

$$\left|d\vec{a}\right|=\frac{G\rho rR\, dr\, d\alpha\, d\theta}{\left(\sqrt{(R+r\cos\alpha)^{2}+(r\sin\alpha)^{2}}\cos\Theta-d\right)^{2}+(R+r\cos\alpha)^{2}(\sin\Theta)^{2}+(r\sin\alpha)^{2}(\sin\Theta)^{2}}$$

We are interested only in radial component (because it will not cancel out). Our radial component is:

$$da_{x}=(R-d)\left(\frac{1}{\left(\sqrt{(R+r\cos\alpha)^{2}+(r\sin\alpha)^{2}}\cos\Theta-d\right)^{2}+(R+r\cos\alpha)^{2}(\sin\Theta)^{2}+(r\sin\alpha)^{2}(\sin\Theta)^{2}}\right)^{\frac{3}{2}}$$

3. Some results

For r=1, R=2:

d torus a sphere a
000.0000
0.2500-0.65161.2337
0.5000-1.41462.4674
0.7500-2.36033.7011
0.8750-2.94444.3180
1.0000-3.44944.9348
1.1250-1.75025.5517
1.2500-0.26556.1685
1.50001.89907.4022
1.75003.36798.6359
2.00004.41069.8696
2.50005.74586.3165
3.00006.30484.3865
3.25005.13033.7376
3.50004.19923.2227
3.75003.51272.8074
4.00002.99072.4674
4.50002.25661.9496
5.00001.77161.5791
6.00001.18361.0966
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