Elliptical Orbits and Making a Torus

There is currently no easy way to make elliptical orbits for solar sails but using some techniques it is still possible. The method is only practical in creative mode. The first part of this guide will explain how to make elliptical orbits and some of the derivation involved. The second part of this guide will provide an example use of elliptical orbits and provide the general steps and information I used to make a torus. For this guide, the x axis is the axis at 0 degrees longitude.

When a node is deleted, it releases free orbit solar sails. If the node was placed on the equator of the star, the sails would continue to orbit in a perfect circle, eventually spreading out and forming a ring. However, if the node was placed away from the equator, its velocity would be lower than the required velocity to orbit in a perfect circle at that distance. The sails will then orbit the star in an elliptical orbit.

With the position vector r and velocity vector v of the sails, we can use the formula

where h = r x v is the specific relative angular momentum, µ is the standard gravitational parameter, and e is the eccentricity vector. The magnitude of the vector will be the eccentricity while the direction will point towards the periapsis. If the node was placed on the equator, we know that r = (x, 0, 0)ᵀ and v = (0, y, 0)ᵀ, where x and y are any positive real numbers. Now

Therefore

Calculating the magnitude shows

Since the orbit is circular the eccentricity will equal 0, hence
In the case where the node was placed on the latitude θ, we know that r = (x * cos(θ), 0, x * sin(θ))ᵀ and v = (0, y * cos(θ), 0)ᵀ. Now

Therefore

The magnitude of this vector will be the eccentricity of this orbit while the direction will point towards the periapsis. Placing a node above the equator then deleting it will release free sails. The sails will then spread out and orbit around the star in an ellipse similar to figure 1.1.

Figure 1.1 - Result after deleting a dyson node on the 45° latitude line. To specify the inclination and longitude of the orbit, a series of matrix transformations can be applied. For a rotation along the x axis, the matrix

can be applied. For a rotation along the y axis, the matrix

can be applied. For a rotation along the z axis, the matrix

can be applied. If the vector (0,0,1)ᵀ represents the north pole of the sphere, the result of applying the transformations to the vector (0,0,1)ᵀ will represent the location of the north pole after transformation. With this, the longitude θ and inclination ϕ of the sphere can be calculated as shown in figure 1.2.

Figure 1.2 - Calculating longitude and inclination.

From figure 1.1, it can be seen that the periapsis has an angle below the xy plane of θ. To move the periapsis onto the xy plane, a rotation of θ along the y axis can be applied. To give the orbit an inclination, a rotation of ϕ along x axis can be applied.

Now an orbit can be created by placing any number of nodes on the appropriate latitude. Position the camera on the x axis and delete the nodes as they pass over the star as shown in figure 1.3. The result should look similar to figure 1.4.

Figure 1.3 - Inclination and longitude of the shell is set to create an orbit with an inclination of 29°.
Figure 1.4 - Final result after deleting all nodes.

One way to create a torus is to have orbits with some eccentricity and inclination. For the following example, the torus is made using 10000m radius, 0.5 eccentricity and inclination.

At a latitude of θ = 45°,

To make the periapsis on the xy plane, a rotation of 45° along the y axis is needed. To give the orbit an inclination, a rotation of 0.5 radians = 29° along the x axis is needed. Hence the transformation needed is

Applying this to the vector we get

From the vector, the longitude θ and inclination ϕ of the shell needs to be

Place 24 dyson nodes along the 45° latitude of the shell. While facing the star from 0° longitude, delete the nodes as they pass over the star as seen in figure 2.1. This will create one orbit of the torus. Shift the longitude of the shell and the camera by 15° as shown in figure 2.2. Repeat the process until a full rotation is made. The final result should look similar to Figure 2.3.

Figure 2.1 - Creating the first ring of the torus.
Figure 2.2 - Creating the second ring of the torus.
Figure 2.3 - Final result.

The aim of this guide was to provide a method for people who may want to incorporate elliptical orbits into their dyson swarm designs. The method does take some time as placing and deleting the nodes is a tedious task. I am unware of any methods to change the colour of the sails after deleting the nodes. To perform matrix multiplication done in section 2.1, any online calculator would do the job. If there are any questions or confusion regarding the guide, let me know and I would be happy to answer them.

Below are some pictures taken outsied the editor.