A Treatise on the Dyson Sphere

Originally by:
Anders Sandberg with ideas and additions from:
Richard Treitel
Stefan E. Jones
Dani Eder
Cheradenine Zakalwe
Steve Linton
David Lorenzo Duffy
Erik Max Francis
Frank Palmer
Christopher P. Winter
Steve Willner
HTML Conversion and E2 Adaptation by Dialogue

Notes from Dialogue:
This paper was originally published in a journal I occasionally read. I got in touch with the author many years ago (almost 6, now that I think about it). I offered to make it into an HTML page and host it on my web site. That site has been down for almost two years now, and just recently I ran across this while reviewing my old html materiel. I reserved the right to republish it on any web site I was involved in. Therefore, I now share it with you fine folks.
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1.What is a Dyson Sphere?

The Dyson sphere (or Dyson shell) was originally proposed in 1959 by the astronomer Freeman Dyson in "Search for Artificial Stellar Sources of Infrared Radiation" in Science as a way for an advanced civilization to utilize all of the energy radiated by their sun. It is an artificial sphere the size of an planetary orbit. The sphere would consist of a shell of solar collectors or habitats around the star, so that all (or at least a significant amount) energy will hit a receiving surface where it can be used. This would create a huge living space and gather enormous amounts of energy.

A Dyson sphere in the solar system, with a radius of one AU would have a surface area of at least 2.72e17 km^2, around 600 million times the surface area of the Earth. The sun has a energy output of around 4e26 W, of which most would be available to do useful work.

The original proposal simply assumed there would be enough solar collectors around the star to absorb the starlight, not that they would form a continuous shell. Rather, the shell would consist of independently orbiting structures, around a million kilometers thick and containing more than 1e5 objects. But various science fiction authors seem to have misinterpreted the concept to mean a solid shell enclosing the star, usually having an inhabitable surface on the inside, and this idea was so compelling that it has been the main use of the term in science fiction. The earliest appearance of this version seems to be Robert Silverberg's novel Across a Billion Years.

A third kind of shell would be very thin and non-rotating, held up by the radiation pressure of the sun. It would consist of statites (see below, in the section about stability). Essentially it is a "Dyson bubble", where reflecting sails reflect light onto collectors for use in external habitats. Its mass would be very small, on the order of a small moon or large asteroid.

In the following I will call solid Dyson spheres Type II or dyson shells and independently orbiting spheres Type I.

2.Who is Dyson?
Freeman Dyson, professor emeritus, Schppö pf Natural Sciences, Institute for Advanced Study, Princeton, N.J. President of the SSI (Space Studies Institute)

Books by Dyson:

  • Infinite in all Directions
  • Disturbing the Universe
  • Imagined Worlds
    Major papers: Time Without End: Physics and Biology in an Open Universe

    3.Was Dyson First?
    No, he admitted himself that his original inspiration came from The Star Maker by Olaf Stapledon, written in 1937.

    As the aeons advanced, hundreds of thousands of worldlets were constructed, all of this type, but gradually increasing in size and complexity. Many a star without natural planets came to be surrounded by concentric rings of artificial worlds. In some cases the inner rings contained scores, the outer rings thousands of globes adapted to life at some particular distance from the sun. Great diversity, both physical and mental, would distinguish worlds even of the same ring.

    Stapledon, in turn, may have got the idea from J. D. Bernal, who also influenced Dyson directly. Bernal describes in The World, the Flesh, and the Devil spherical space colonies:

    Imagine a spherical shell ten miles or so in diameter, made of the lightest materials and mostly hollow; for this purpose the new molecular materials would be admirably suited. Owing to the absence of gravitation its construction would not be an engineering feat of any magnitude. The source of the material out of which this would be made would only be in small part drawn from the earth; for the great bulk of the structure would be made out of the substance of one or more smaller asteroids, rings of Saturn or other planetary detritus. The initial stages of construction are the most difficult to imagine. They will probably consist of attaching an asteroid of some hundred yards or so diameter to a space vessel, hollowing it out and using the removed material to build the first protective shell. Afterwards the shell could be re-worked, bit by bit, using elaborated and more suitable substances and at the same time increasing its size by diminishing its thickness. The globe would fulfill all the functions by which our earth manages to support life. In default of a gravitational field it has, perforce, to keep its atmosphere and the greater portion of its life inside; but as all its nourishment comes in the form of energy through its outer surface it would be forced to resemble on the whole an enormously complicated single-celled plant.

    A star is essentially an immense reservoir of energy, which is being dissipated as rapidly as its bulk will allow. It may be that, in the future, man will have no use for energy and be indifferent to stars except as spectacles, but if (and this seems more probable) energy is still needed, the stars cannot be allowed to continue to in their old way, but will be turned into efficient heat engines. The second law of thermodynamics, as Jeans delights in pointing out to us, will ultimately bring this universe to an inglorious close, may perhaps always remain the final factor. But by intelligent organization the life of the universe could probably be prolonged to many millions of millions of times what it would be without organization. Besides, we are still too close to the birth of the universe to be certain about its death.

    According to Stefan E. Jones <stefanj@io.com<, Raymond Z. Gallun, an American SF author may have come up with a similar concept independently.

    4.Why build a Dyson sphere?

    Energy and space. As described above, the amount of collected energy would be immense, and the living space simply unimaginable. Dyson pointed out that so far the energy usage of mankind has increased exponentially for at least a couple of thousand years, and if this continues we will soon consume more energy than the Earth receives from the sun, so the natural step is to build artificial habitats around the sun so that all energy can be used. The same goes for population in the long run (it should be noted that this is not a solution, just a logical result of growth). It is also possible that the Dyson sphere simply stores the energy for future use, for example in the form of antimatter.

    Even if cheap and efficient fusion power can be developed, eventually the waste heat has to be radiated away by a Dyson sphere-like cooling system.

    Other proposed uses have been for security (although it is hard to hide the infrared emissions; energy could be radiated away in certain directions, but thermodynamics places some limits on it), or just for the fun of it (if you have a sufficiently advanced technology megaengineering could become a hobby activity; after all, ordinary people today perform engineering or crafting feats far beyond the imagination of previous eras).

    5.What would a Dyson Sphere look like from the outside?

    A Type I Dyson sphere would probably not cover the star perfectly, so occasional glimpses of its surface would be seen as the habitats orbited. A type II Dyson sphere would be totally opaque (unless it had openings). The spheres would hence be invisible from a distance, just a black disk on the sky. But they would shine powerfully in the infrared, as the waste heat from the internal processes radiate away. The apparent temperature would be

    T = (E / (4 pi r^2 eta sigma))^1/4

    where E is the energy output of the sun, r the radius of the sphere, eta the emissivity and sigma the constant of Stefan-Boltzman's law.

    This would correspond to an infrared wavelength of lambda = 2.8978e-3 / T m (assuming a blackbody sphere) which for reasonable sizes lies in the infrared. Dyson predicted the peak of the radiation at ten micrometers.

    6.What would a Dyson Sphere look like from the inside?

    The curvature of the "ground" would be even less than on Earth, so to an observer close to it would look perfectly flat. In a solid dyson sphere with atmosphere, the atmosphere would limit the range of sight due to its opacity, and the horizon would be slightly misty.

    The sky would be filled with the surface of the sphere, giving the impression of a huge bowl over a flat earth, covered with clouds, continents and oceans although for a real Dyson shell these would have to be immense to be noticeable. The angular size of an object at a distance d and diameter l is 2arctan(l/2d). For an object of diameter 10,000 km (like the Earth) at a distance of a 100 million km (around 120 degrees away from the observer on the shell), the angular size would be around 10^-4 rad or 0.005 degrees, roughly the size of a pea 100 meters away.

    It should be noted (as Richard Treitel has pointed out) that even a very dark surface will shine intensely, making the sky much brighter than on Earth. The albedo of Earth is around 0.37, so an interior with an earthlike environment would have a sky where each patch reflects a noticeable fraction of the sunlight.

    In a type I dyson sphere roughly the same things would be seen: a plane wall of orbital habitats, solar collectors and whatnot stretching away into what looks like infinity (although here the curvature may become noticeable for observant viewers) and a hemispherical bowl covering the rest of the sky, centered around the sun. Solar collectors would have a very low albedo, but it is still likely that the interior will be very bright.

    7.Is a Dyson sphere stable?

    In a type I Dyson Sphere all the structures orbit independently around the star, and their orbits are normal keplerian elliptic or circular orbits. Since the mass of the shell is negligible compared to the sun, the self-gravity can be ignored (it will merely cause some precession of elliptical orbits). And if two orbits intersect, they can be adjusted by using solar sails, ion engines, magsails or similar low-energy devices.

    Another version would be based on statites (this is probably due to Robert L. Forward): each solar collector will also be a solar sail, and hover without orbiting above the sun, held up by light pressure. By adjusting the sail area statites can move in and out, and by adjusting their angle they can move away if needed. Traffic control may be a problem, but can likely be handled in various ways, for example by local flight control centers or automatic systems based on flocking behavior.

    The force on a statite would be F = L/(4 pi c r^2) - GMm/r^2, where L is the total luminosity of the sun (3.9e26 W), M is the mass of the sun, m is the density of the statite, r the distance to the sun and c is the speed of light. To remain in balance, the statite will have to have the density

    m=E/(4 pi c G M)

    (this assumes a 100% reflective statite). Note that this is independent of distance to the sun, closer to the sun the gravitational pull is greater, but the radiation pressure is stronger. The density depends only on the mass/luminosity of the sun. For a statite in the solar system, the density would be around 0.78 g/m^2

    A rigid dyson sphere is not stable, since there is no net attraction between a spherical shell and a point mass inside. If the shell is pushed slightly, for example by a meteor hit, the shell will gradually drift off and eventually hit the star. This is a classic problem in elementary mechanics and is usually solved in introductory textbooks.

    Gauss Law

    One easy way to derive it is from Gauss Law: the integral of the force across an arbitrary closed surface is proportional to the amount of mass inside it. If the surface is a sphere surrounding the dyson sphere, there is obviously an inward force on the surface of the sphere since there is a mass inside it. But if the sphere is inside the dyson sphere (the sun is ignored in this calculation, as we are only interested in the gravity of the dyson sphere), there is no mass inside it and hence the integral must be zero, which means that there is no gravitational field inside the sphere.

    Elementary Proof

    It can also be proven using only elementary (brute force) calculus. This treatment is from Kleppner & Kolenkow, An Introduction to Mechanics (p. 101) and deals with the force between a point of mass m at radius r on the x-axis from a spherical shell centered at the origin:

    Divide the shell into narrow rings. Let R be the radius of the shell, t its thickness (t << R). The ring at angle theta, which subtends an angle dtheta, has a circumference 2 pi R sin theta, width R dtheta an thickness t, which gives it a volume of

    dV=2 pi R^2 t sin theta d theta

    and a mass of (M/2) sin theta dtheta where rho is the density of the shell.

    Each part of the ring is the same distance r´ from m, and by symmetry the force from the ring is directed along the axis with no transversal component. Since the angle alpha between the force vector and the line of centers is the same for all sections of the ring, the force components along the line of centers add to give

    dF=G m rho dV cos alpha / r´^2

    for the whole ring. This is then integrated: F = int (G m rho dV/r´2) cos alpha. By expressing cos alpha as a function of polar angle we get:

    F = [GMm/2] int_0^pi ( (r - R cos theta) sin theta dtheta)/(r^2 + R^2 - 2 r R cos theta)^2/3

    (where int_0^pi is the integral from 0 to pi). Through the substitution u=r-Rcos(theta), du=Rsin theta dtheta we get:

    F = [GMm/2R] int_{r-R}^{r+R} (u du) / (R^2 - r^2 +2ru)^(3/2)

    which is a standard integral resulting in:

    F = (GMm/2R)(1/2r^2)[sqrt(R^2-r^2+2ru)-(r^2-R^2)/sqrt(R^2- r^2+2ru)]_{r-R}^{r+R}

    For r<R we get:

    F=(GMm/4Rr^2){(R+r)-(R-r)-(r^2-R^2)(1/(R+r)-1/(R-r))} = 0

    8.How strong does a rigid Dyson shell need to be?

    Very strong. According to Frank Palmer:
    Any sphere about a gravitating body can be analyzed into two hemispheres joined at a seam. The contribution of a small section To the force on the seam is  g(ravity)*d(ensity)*t(hickness)*A(rea)*cos(angle).  The integral of A*cos(angle) is (pi)*R^2.

    So the total force is g*d*t*(pi)*R^2. Which is independent of distance, neatly enough.

    The area resisting the force is 2*(pi)*R*t.

    Thus, the pressure is g*d*R/2; this can be translated into a cylindrical tower of a given height on Earth. If that tower built of that material could stand, then the compression strain is not too great.

    At 1 AU, that comes to 2*([pi]*AU/YR)^2, or -- by my calculations -- in the neighborhood of 80 to 90 THOUSAND kilometers high.

    The tendency to buckle, moreover, is another problem.

    9.What about gravity on a rigid Dyson shell?

    A nonrotating dyson shell would have just two sources of gravity: the shell itself and the star. As mentioned above, on the inside only the gravity of the star would be felt and everything would fall down into it, while on the outside there would be weak gravity (for a 1 AU sphere centered around the sun, the gravity would be 6e-3 m/s^2).

    The only ways to make a rigid Dyson shell habitable on the inside would be either to provide it with some sort of antigravity (which is unlikely) or to rotate it, which would make only the equatorial band habitable unless the interior was terraced. A rotating Dyson sphere would be under immense strains; see the section about the ringworld for a simple calculation. Niven pointed out that if you want to spin a Dyson sphere, it is better to build it like a film canister for reasons of structural strength, and then you have a Ringworld.

    It has been suggested that one could live on the outside of the sphere, especially if the interior star is rather cool; it appears that a terrestrial environment is possible around M stars just at the end of the main sequence. Erik Max Francis gives the following derivation of this kind of sphere:

    First, know the luminosity-mass relation for main sequence stars:

    L = k M^nu,

    where k is a constant of proportionality and nu is between 3.5 and 4.0. (k depends on the choice of nu, obviously.) You can find the constant k, given nu, based on the fact that the Sun has a luminosity of 3.83 x 10^26 W and a mass of 1.99 x 10^30 kg.

    Second, know the gravitational acceleration:

    g = G M/R^2.

    Third, the blackbody power law (we're approximating the star as a blackbody, which isn't too bad of an approximation):

    L = e sigma A T^4.

    Knowing these factors, you can combine them to get an equation which relates the mass of the star to the desired temperature and gravity of the sphere:

    k M^(nu - 1) = 4 pi e sigma G T^4/g.

    Substituting ideal conditions (g = 9.81 m/s^2, T = 300 K), you find that M must be between 0.054 and 0.079 masses solar (the variance is dependent on the variance in the exponent in the mass-luminosity relation). The end of the main sequence is at about 0.08 masses solar, for comparison.

    This would produce spheres with a radius of 0.0057-0.0069 AU (852,720 - 1,032,240 km).

    It might also be possible to have a biosphere between two dyson spheres (this is used in Baxter's The Time Ships).

    10.Would the solar wind be a problem?

    If an earthlike ecology was built inside a large rigid dyson shell, there would be an influx of ions (mostly hydrogen) from the solar wind. The solar wind has a density of around 5 ions/cm^3, moving at around 500 km/s; that would lead to an influx of 2.5e12 ions/m^2/s. This might appear large, but is actually a tiny amount, just 4e-12 mol (one gram of hydrogen is approximately one mol). Since the hydrogen could not naturally escape from the atmosphere it would gradually become more and more hydrogen rich, but it would take trillions of years before the effects became significant. The net force from the solar wind and the light pressure (which is larger than the solar wind pressure) is also minor compared to the attraction of the sun and the internal strains of a rotating dyson shell. In a type I dyson sphere the light pressure could be used to keep satellites hanging in space.

    It should be noted that there would be no auroras in a Dyson shell, since there is no magnetic field. This also would also mean that more radiation would reach the ground from the sun since it cannot naturally be deflected (although one could imagine megaengineering systems to provide an artificial magnetic field).

    11.Can a Dyson sphere be built using realistic technology?

    A type I Dyson sphere can be built gradually, without any supertechnology or supermaterials, just the long-term deployment of more solar collectors and habitats. This work could start today (and one might argue that our satellites are the first step). Using self-replicating machinery the asteroid belt and minor moons could be converted into habitats in a few years, while disassembly of larger planets would take 10-1000 times longer (depending on how much energy and violence was used).

    A rigid dyson shell would require superstrong materials, and its construction is complicated since half a shell is unstable. One could conceive of some dramatic capping process, where a number of previously freely orbiting structural components at the same time moved inwards to lock together into a shell (for example twenty spherical triangles). This would require tremendous precision, but since supertechnology is already assumed for building a rigid shell, it seems almost trivial. As somebody put it, if you can build a dyson shell you don't need it.

    12.Is there enough matter in the solar system to build a Dyson shell?

    Dyson originally calculated that there is enough matter in the solar system to create a shell at least three meters thick, but this might be an overestimate since most matter in the solar system is hydrogen and helium, which isn't usable as building materials (as far as we know today). They could presumably be fusioned into heavier elements, but if you can fusion elements on that scale, why bother with a dyson sphere?

    If one assumes that all elements heavier than helium are usable (a slight exaggeration), then the inner planets are completely usable, as is the asteroid belt.

      Mass (1e24 kg)

    Mercury:          0.33022
    Venus:            4.8690
    Earth:            5.8742
    Moon:            0.0735
    Mars:            0.64191
    Asteroids:      ~0.002

    Sum:            11.78733e24 kg

    It is a bit more uncertain how much of the outer planets is usable. Jupiter and Saturn mainly consist of hydrogen and helium, with around 0.1% of other material. Jupiter is assumed to have a rock core massing around 10-15 times the Earth, and Saturn probably contains a smaller core massing around 3 times the Earth. Uranus and Neptune's seem to be mainly rock and ice, with around 15% hydrogen, so a rough estimate would be around 50-70% usable mass. Pluto seems to be around 80% usable.

                    Mass (1e24 kg)      Usable Mass (rough estimate)

    Jupiter:          1898.8            ~58
    Saturn:            568.41          ~17
    Uranus:              86.967          ~43
    Neptune:            102.85          ~51
    Pluto:                0.0129          ~0.01
    Kuiper belt objects: ~0.02            ~0.016

    Sum:              2657.06  Usable: ~170

    (this is based on the assumption that the size distribution of the Kuiper belt mirrors the asteroid belt)

    (these tables based on information from Physics and Chemistry of the Solar System by John S. Lewis and The Nine Planets by Bill Arnett)

    The inner system contains enough usable material for a dyson sphere. If one assumes a 1 AU radius, there will be around 42 kg/m^2 of the sphere. This is probably far too little to build a massive type II dyson sphere, but probably enough to build a type I dyson sphere where mass is concentrated into habitats and most of the surface is solar sails and receivers, which can presumably be made quite thin.

    With the extra material from the outer system, we get around 600 kg/m^2, which is enough for a quite heavy sphere (if it was all iron, it would be around 8 centimeters thick, and if it was all diamond around 20 centimeters).

    A Type III shell, a "dyson bubble", would have a very low mass. Since its density is independent of the radius (see the stability section), its mass would scale as r^2. For an 1 AU bubble, the total mass needed would be around 2.17e20 kg, around the mass of Pallas.

    13.Wouldn't a Dyson sphere heat up?

    Even if the civilization living in the Dyson sphere did its best to store available energy, thermodynamics eventually wins and the sphere begins to radiate away energy until equilibrium is reached. Its temperature becomes

    T=[E/(4 pi eta sigma r^2)]^(1/4)

    where eta is the emissivity (=1 for a blackbody), sigma the constant of Stefan-Bolzman's law (5.67032e-8 Wm^2K^-4)and E the total energy output of the star measured in watts.

    In theory, if eta is very low the interior of the sphere could become as hot as desired, but this is unlikely since the material of the sphere would start to melt or evaporate if the temperature moved above 2000-3000K or so. And if the surface of the star became hot enough, the outer parts of the star would expand and a new thermal equilibrium set in with less internal energy production. If the sphere was a perfect energy container the star would eventually expand until its fusion processes ended; if the temperature was lowered (by energy use) fusion would resume until an equilibrium was reached - a bottled star.

    It should be noted that at 1 AU, the energy flux is around 1.4e3 W/m^2, which calculates as around 395 K, or 122 degrees C if the sphere is a blackbody. This is a bit too hot for an earthlike biosphere (Earth is cooled by its rotation, which effectively halves the energy flux, and its spherical shape, that lowers it further), and a dyson shell need some rather impressive cooling to work.

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