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Solar convex core

Our galaxy is full of stars, the Sun being one of them. Around the Sun, they are distributed at a rate of roughly one per 250 cubic light years or so.^[1] With their random distribution within space, this did make me wonder: if you could form a shape placing vertices on other stars, what is the largest convex shape you could form without any stars except the Sun? In other words, what is the solar convex core?

Algorithm

The algorithm to find the solar convex core (SCC) obviously requires a list of stars with their position S. I sourced the list from Wikipedia's List of nearest stars. We then create a list of "filtered" stars F, which initially contains every star in S. Every star a in F is then checked, for every star b in F, if star a is farther from the Sun than star b, a is removed from F. After the process, the vertices of solar convex core are the stars left in F. I have written an implementation of this algorithm in C, found in solarconvexcore.c.

Results

The solar convex core is composed of nine stars:

The edges of the polyhedron mapped out in the sky. Made with Stellarium.
Source

Here are some properties of these stars:

• All stars are the closest of their respective constellation;
• The only star visible by the naked human eye (app. mag. < 6.5) is Sirius, being the brightest star in the night sky;
• If substellar objects are included, WISE 0855−0714 would be placed between Barnard's Star and Wolf 359;
• The largest distance between two vertices is between Lacaille 9352 and Lalande 21185 at 19.0259 ly;
• In the resulting convex polyhedron, Ross 248 and Lacaille 9352 have pentagonal vertex figures, Barnard's Star and Sirius have hexagonal vertex figures, and the remaining have quadrilateral vertex figures;
• The polyhedron has 14 triangular faces. As it is virtually impossible for more than three stars to lie in a single plane, the faces must trivially be triangles;
• The polyhedron has 21 edges;
• The volume of the polyhedron is 885.29 ly³ = 25.516 pc³ = 7.4965×10⁵⁰ m³.

Edges

Sorting the stars 0–8 by distance, 0 being Proxima Centauri, raw edge data is as follows:

An approximation of this in Stellarium, only using HIP stars, shown in the above image is:

Faces

Using the same numbering system as above:

This allows us to build a 3D model of the polyhedron:

• OFF model: made by Qhulling the vertices and converting them in OFFBuilder v2.
• STL model: simply converted the OFF model to an STL model in Image To STL.