QUESTION: How much mass would it take to build a Dyson Sphere around the Sun? Is Jupiter enough?

The original question was posed (and the answer suggested) in this Star Talk clip with Neil Degrasse Tyson.

Premises

1. The sphere is of the shell variety, not the swarm variety.
2. The sphere will be constructed at a distance equal to that of Mercury’s orbit, as this distance is sufficiently far to prevent collector materials from melting.
3. The sphere material will be treated as uniform density.
4. The material strength of the sphere is ignored, as are tidal forces.
5. The efficiency of the collector is assumed to be 100%.
6. Jupiter’s average density will be used as the material density.

Setup

To compute the thickness of a Dyson Sphere made of a Jupiter’s worth of material, first calculate Jupiter’s volume. Then calculate the surface area of a spherical shell around the sun at Mercury’s average orbital distance (roughly, since its orbit is highly elliptic). Divide Jupiter’s volume by the shell’s area to get the thickness of the shell.

import math

jupiter_mass = 1.898E27 # kg
jupiter_vol = 1.43E18 # m^3
jupiter_density = jupiter_mass / jupiter_vol # 1.327 g/cm^3

sphere_radius = 5.0E10 # m
sphere_area = 4 * math.pi * (sphere_radius ** 2) # 3.142E22 m^2
sphere_depth = jupiter_vol / sphere_area # 4.552E-5 m or 0.04 mm

The amount of incident energy per unit area is equal to the Sun’s surface emissions, taken to be 6.294E7 J/m^2s ¹, divided by the ratio of the Sun’s surface area (6.09E12 km^2) to the sphere’s, since we assume the collector to be 100% efficient.

sun_surface_emission = 6.294E7 # J/m^2s
sun_area = 6.09E18 # m^2

sphere_incidence = sun_surface_emission * (sun_area / sphere_area) # 1.220E4 J/m^2s

Analysis

A Dyson Sphere made up of a Jupiter’s worth of material and constructed at Mercury’s orbital distance would only contain enough material to be 0.04mm thick, or around half the thickness of an average human hair.

Such a structure would have to be many orders of magnitude stronger that any known material, and would have to resist tremendous thermal stresses. Capturing 100% of incident solar energy in that thin a skin would deposit tremendous energy, which would need to be stored or transported away, maybe by laser.

Conclusion

It’s probably not feasible to construct an actual Dyson Sphere around any star; capturing all of its energy would require either more material than the star’s system contains, or exotic materials not yet known but which would necessarily be far to rare to be usable.

Instead, Dyson Swarms capturing some fraction of the energy would be a lot more feasible and maintainable for any civilization willing to build such a megastructure.

ANSWER: Nope. Needs more matter than exists in the solar system, given properties of known materials.