A sort of uber-world envisioned by Larry Niven. The concept is of a gigantic ring encircling a star, with the living space on the inside. It would have a radius of 1 AU, and be spinning at 770 mi/sec to give it 1 Earth gravity. This would cause approximately Earth-like conditions on the entire inner surface of the ring. Niven estimates that one could be built of this size out of roughly the mass of Jupiter. This would give us roughly three million times the usable surface area of Earth. Or, as he put it, by the time we need one we'll know how to build it.

Larry Niven has fleshed out this concept to an amazing degree. For example, you'd put shadow squares in an orbit closer to the star, which would allow for day-night cycles. You produce energy by lining the sunward side of the squares with solar collectors, and beam it back to recieving dishes on the ringworld as microwaves. A set of 1000-mile high walls on the edges hold the atmosphere in. Any oceans and mountains would be curved directly into the material the ringworld was made out of- you'd get a bas relief by looking at the other side. You could put a series of electromagnetic rings on the rim to create an em catapult for launching ships with. A series of Bussard ramjets would keep the Ringworld on-center. Oceans would have to have a plumbing system to keep all the soil from winding up at the sea bottom. A meteor defense system would be necessary (in the book this was accomplished by using the star as an astronomically sized gas laser).

In other words, it's a gigantic concept to get your head around. It's probably a more practical concept than the Dyson Sphere, but vastly more efficient than these lame ball-shaped worlds we use now.

StrawberryFrog's write-up is good, but is perhaps a little short-sighted about solving some of the technical issues involved in building one of these suckers- for example, spinning up the ringworld is an easy problem to solve. All you have to do is take the ramscoop motors mounted on the rim walls and point their exhausts tangentially- and wait. I suspect anyone who's building a ringworld can wait long enough for the world to spin itself up. If not, well, you could always use the flare devices in the mirror squares to increase the star's output.

As for finding the matter, that's a relatively simple enterprise (heh- sci-fi pun!) as well. The easy way to do is to find a really large star and mine it. If you spin up a star (left as an exercise to the reader), a bunch of the matter will escape, and all you have to do is catch it (and if you've already got ramscoop drives, that's obviously not difficult).
Would that Larry Niven could flesh out his character and plot development to the same degree as the technicalities of a ring shaped world. Okay, I admit, it is a very cool idea, and I loved that part of the book. But Jesus, has he ever met a real human? Does he have any concept of how they might talk or interact with each other? Does he really think women, even outer space women, are flawlessly beautiful and want to take off their clothes and skronk every guy who comes along?

I could buy the whole ring world thing fine. It was the dialogue that strained my suspension of disbelief.

It's hard to be a female science fiction fan sometimes.

One can visit the web site "Rendering Larry Niven's Ringworld" at http://www.rahul.net/rootbear/graphics/ringworld/ for a visual illustration of looking down on Ringworld. It has some amazing visual perspectives that give you a very good sense of what it would be like to approach a structure of this magnitude. In particular, it is quite instructive to view the ring against the background of space and stars. One is also struck by how hard it is to see the ring from interplanetary distances.

Incidentally, the mass of our sun is 1.989 x 10^30 kg which is only about 1,000 times the mass of the Ringworld.


or, an extra-large writeup for an extra-large concept

Ringworld is a science fiction novel by Larry Niven that features the eponymous idea in stellar construction, derived from the Dyson sphere. It is built structure larger than a planet: a ring around a star.

Ringworld is brilliant idea which, like most of them is obvious in retrospect: One problem with the science fiction version of Dyson sphere (i.e. a solid shell around the sun) is that there is no gravity inside, and thus no way to stick the air to the surface short of some as-yet-undiscovered gravity generator.

So spin the sphere, to make gravity by centrifugal force. Once it is spinning, no further input is needed to keep the gravity there - no gravity generator to invent, design, build, maintain and keep powered. But this only provides gravity around an equatorial band of the sphere. So only keep the equatorial part, and don't build the rest. Make it a flat ribbon around the sun, with walls at the edges to keep the air from spilling off the sides.

You can approximate a day-night cycle by placing plates of solid material between the Ringworld and the star. Wire them together and spin them in their own smaller ring. They will double as power stations by collecting solar energy.

You still have three million times the surface area of the earth. And all of that surface designed to your specifications, not subject to the vagarities of geography. No icy polar wastelands, no deserts, no great oceans, unless you want them.

Note that there is one drawback to a Ringworld over a Dysan sphere. To give it 1g gravity, it must spin very fast, at aound 1 239 000 m per second (770 miles per second). The journey around the sun that the Earth completes in 364 1/4 days, a point on the Ringworld makes in 8.78 earth days, and so it moves 41 times as fast as the Earth. This requires a really strong construction material.

"The Ringworld has a simplicity of its own: the math is that of a suspension bridge with no endpoints. The Ringworld's engineers used robust and simple technology -- the Soviet space program. Everything I have written about the Ringworld since that first tale has touched on four simple, basic ideas: 
One was to run a ribbon of strong material in an Earthlike orbit around a Sol-type sun, spin it for gravity, and terraform the inner surface. 
The second was to introduce Teela Brown, the enigmatic woman bred for luck. 
The third was to offer the notion that every symptom of aging in a human being is an aborted sign of a juvenile's attempt to reach its adult form: to become the Protector
The fourth was to depict the human urge to play God"
 - Larry Niven

"Spectroanalysis and frequency shifts give us a relative difference in velocities. The ring is clearly rotating at 770 miles per second, a velocity high enough to compensate for the pull of gravity from a primary, and to provide an additional centripetal acceleration of 9.94 meters per second. Consider the tensile strength needed to keep the structure from disintegrating under such a pull!

"Something new then," said Louis. "Chiron, how big is this ring? How massive is it?" 

It massed a little more than the planet Jupiter.

- Ringworld. The metric system has an uphill battle ahead if even advanced aliens won't use it when speaking to humans.

The books

Two sequels were written. To call them a trilogy is a slight misnomer, as most trilogies are planned as one long story, and written back to back. These are seperate stories, written decades apart.
Ringworld (Copyright 1970)
The Ringworld Engineers (Copyright 1980) The classic sequel to Ringworld, following Louis Wu and Teela Brown as they manipulate events on the Ringworld in an attempt to save it.
The Ringworld Throne (Copyright 1996) IMHO the weakest of the three books.

Ringworld parameters

From the appendix of The Ringworld Throne verbatim, except for the metric conversion:

30 hours = 1 Ringworld day 
1 turn = 7.5 Ringworld days = 1 Ringworld rotation 
1 falan = 10 turns = 75 Ringworld days 
mass = 2 * 10 30 grams
radius = 0.95 * 10 8 miles ( 1.52 * 10 11 m) 
Circumference = 5.97 * 10 9 miles (9.6 * 10 12 m) 
Width = 997 000 miles (1604 000 000 m) 
Surface area = 6 * 10 14 square miles = 3 * 10 6 times the surface area of Earth (approx) 
Surface gravity = 31.7 feet/second/second = .992 gee 
Rim walls rise inward, 1000 miles (1609km)
Repair center = forty miles high by .56 * 108 square miles area = 1.12 * 10 8 cubic miles 
Near great ocean = 600 * surface area of earth. 
Star G1 or G2, barely smaller and cooler than Sol.


A Ringworld designed from humans, around a sol-like star would have a radius equal to the earth's distance from the sun. (But perhaps you'd want a bit more to moderate the climate. The sun's rays will hit all of the ringworld perpendicularily, whereas on earth they do so only in the tropics. This is a major influence on climate. Thanks to LeoDV for pointing this out.) The width that is suggested is "as wide as the sun". I do not know if this width is optimal, but it seems that a narrower or wider Ringworld would work as well.

Niven suggests that the impression of a Ringworld inhabitant would be that they are standing on a flat earth, with an arch rising overhead, and the sun supported beneath the apex of the arch. The apex of the arch would have an apparent width of 1/2 as wide as the sun, it being of the same width but twice as far away. The horizon would not curve significantly, and the actual curve would be upward, so you would be able to see very far (to the limits imposed by atmospheric haze). If you got up on a mountain you would be above most of that.

The visualisations at http://www.rahul.net/rootbear/graphics/ringworld/ give a sense of scale: in a view where the Ringworld is a wide ribbon, the earth would be a pale blue dot. Thus any feature that you can see on the Ringworld would be really macroscopic. Even continents and oceans on the same scale as on the earth would be lost. The rim walls, being only 1600 kilometers high, are invisible at this scale.

Evolution of the concept

Niven's original Ringworld is in some ways industrial era engineering taken to its limits. The structure is impossibly large, but it is described as inert and homogenous. Adjustments made via a few large mechanisms controlled by small number of human (or equivalent biological sentient) operators at a central point.

Niven's Ringworld has cables in the scrith (ringworld construction material) set thousands of kilometres apart. IMHO, all parts of a structure this important would have detail, design, complexity and variation on every scale from the microscopic to the largest.

For instance, the floor of the Ringworld would likely not just be a bed of inert scrith, but will also have conduits and access points to distribute the electricity collected by the shadow squares. Communications network cables and points (I doubt it would be a token ring - it takes light 52 minutes to travel all the way around). Plumbing for electricity, data, water, air and anything else. Sensors to monitor the load and integrity of the system. Repair mechanisms. Even subterranean transport systems. Niven describes the scrith on the floor of the ring as being slippery, almost frictionless. Later on, when the characters are at a spaceport on top of the rimwall, he mentions that the scrith there is ridged for traction. I don't think that any part of the scrith would be smooth and featureless.

As Iain Banks's 90's riffs on these themes suggest, any structure this complex and important is likely to literally have a mind of it's own (if not more than one), with it's own self-preservation instinct. It would interpret damage as injury to itself, and be wired to seek a happy state of health and future, and to promote civilised inhabitants who care for it.

Given that making the Ringworld would require a few aeons of progress in physics and large-scale construction techniques, we can hope for some advances in AI and smart materials as well in this time.

The mind of the Ringworld would be far beyond human in age, memory, size and the extent of it's worldly powers.

As Iain Banks notes, transport is much easier if, instead of climbing out of the gravity well, you can simply drop a few 100 meters down into the vacuum of space below you. Transport around the ring is easier with no air resistance, so transport systems will be found here.


What if, 50 000 years later, inhabitants of the Ringworld have a bad nuclear war and puncture the ring?

What if the meteorite defences fail?

A planet is stable in its orbit - it stays there. A solid Dyson sphere is neutrally stable - if it drifts off center to its star, it stays offcenter. But a ringworld is unstable, like a boulder balanced on the top of a hill. Any drift offcenter will be amplified, until it impacts the sun with catastrophic results. Therefore it needs active stabilistaion (e.g. rocket motors) to keep it in place, like a Segway. What if they fail? These topic provide the plot impetus for The Ringworld Engineers.

Having airlocks built into the floor of the Ringworld is an invitation for trouble later on. How do you fix one if it blows?

It may be preferable to build bulkheads across the Ringworld - at right angles to the rim walls. This will contain the consequences of a catastrophic air loss (e.g., a meteorite strike or nuclear war) to one section. It is an open question as to if it is best to make a few sections (e.g. to divide the Ringworld into quarters) or many. Bear in mind the monstrous scale - dividing the Ringworld into a million segments separated by rim wall-like bulkheads leaves each segment still three times as large in area as planet Earth.

If these walls were put at 45 degrees rather than right angles to the rim walls, and criss-crossed, they would contribute to the structural integrity of the ring.

This segmentation would also give the opportunity for different ecologies, different planetary environments in different segments, thus allowing different species to share the cost and benefits of the Ringworld.

All rim walls should have ramps up them at regular intervals, so that all you need to get to the top (and perhaps over the other side) is a vehicle, a spacesuit and a lot of determination. If the ramp is 100m wide, and underneath it slopes slightly back towards the rim wall at 89 degrees, it will merge with the rim wall's usual thickness after 8.9 km, which is nothing considering that the wall is 1609Km high.

A ramp at 30 degrees incline would be about 3200 km (2000 miles) long to get to the top.

The long term

"Once they lost the secret of transmutation, they would never recover."
"Oh? Why not?"
"Look about you, Louis. What do you see?"
Louis did. He saw a distant lightning-storm developing ahead; he saw hills, valleys, a distant city, twin mountain peaks, tipped with the dirty translucency of raw Ring flooring.
"Land anywhere on the Ringworld, and dig. What do you find?"
"Dirt" said Louis. "So?"
"And then?"
"More dirt. Bedrock. Ring floor material," said Louis. And as he said these words the landscape seemed to alter. Storm clouds, mountains, the city to spinward and the city dwindling behind, the edge of brilliance far away on the infinity-horizon, that might be a sea or a sunflower invasion . Now the landscape showed as the shell it was. The difference between an honest planet and this was the difference between a human face and an empty rubber mask.
"Dig on any world, " the puppeteer was saying, and eventually you will find some kind of metal ore. Here you will find forty feet of soil, and then the Ring foundation. That material cannot be worked. If it could be pierced, the miner would strike vacuum - a harsh reward for his labour."
"Give the Ring a civilisation capable of building the Ring, and it must necessarily have cheap transmutation. Let them lose the technology of transmutation - no matter how - and what would be left? Surely they would not stockpile raw metals. There are no ores. The metal of the Ring would all be in tools and in rust. Even interplanetary capability would not help them, for there is nothing to be mined around this star. Civilisation would fall, and never rise."
- Ringworld

Spaceship earth drives just fine with no one at the wheel. Not so the Ringworld.

The Ringworld is a conundrum - compared to a planet, it is a rather fragile and ephemeral construction, but that which supports an immense number of intelligent beings. So many, that there is not much opportunity to evacuate them, or indeed any other place that could accommodate them, should things go wrong. So the Ringworld must be designed to attempt to last and to be repaired until the sun goes out.

Larry Niven is quite right in that significant evolution of the inhabitants will most likely take place during the lifespan of the Ringworld. The Ringworld is on too large a scale to be considered merely species-defining.

So it must be designed to outlast species, not to mention their civilisations. Yet it cannot persist without continued intelligent intervention, and access to tools and knowledge on roughly the same scale as was used to construct it. 

But there is enough room for civilisations to be rising, falling, flourishing and lying fallow on the ring all at the same time, 

If the construction is a awesome task, so is the social design needed to make sure that it stays in place for millions of years.

It is possible that, among the many billions of inhabitants of the Ringworld, the honour of being recruited by the World maintenance engineers would be a coveted honour. Or perhaps the world and its subordinate machine intelligences would do all the repair work..

Would the entire Ringworld be vulnerable, for instance, to a small group of fanatics who had hijacked an interstellar craft? Intuition suggests that it might, but intuition does not always work when scaled up so large.


"Sure, it takes near-magical technology: the tensile strength of the scrith floor is about that of an atomic nucleus" - Larry Niven

Construction of the Ringworld is challenging on a perhaps insurmountable scale. You need as much mass as Jupiter to build it, If you find it - how do you tow it to the right solar system? I would guess that you pick a star with the materials on hand. But when the Ringworld is finished, it is advisable to clear out all asteroids and other bodies that might impact the ring, so a system without planets is preferable.

If this matter is all in one lump, how do you break it up? How do you demolish a planet larger than Jupiter, and keep all the bits that come off? Can you imagine the energy needed to pull all of the mass of Jupiter out of it's own gravity well?

Large planets, "gas giants" - are often mostly hydrogen gas - and that's not a very good building material.

You then need to process the matter, probably using nuclear transmutation on an ungodly scale.

but no chemical element, compound or alloy is strong enough to support the weight, and expecially the rotation of the Ringworld. This may be the single biggest hurdle. Unlike a space elevator, which has been calculated to be doable, the Ringworld will almost certainly remain science fiction forever.

And then you have to spin it:

"Do you know how much energy it takes to spin the mass of the Ringworld up to seven hundred and seventy miles per second?"
"Why do you ask?"
"It takes a lot. Thousands of times the yearly energy output of this kind of sun. Where would the Ringworld engineers get all that energy? What they had to do was disassemble a dozen Jupiters, or a superjovian planet a dozen times Jupiter's mass. All mostly hydrogen, remember. They'd use some hydrogen in fusion for the energy to run that project, and reserve more of it in magnet bottles. After they made the Ringworld from the solid residues, they'd have fuel for fusion rockets to spin it up to speed."
- Ringworld engineers
This page calculates that the energy involved is "1.6*1039 joules of kinetic energy. This is equal to the entire energy output of a Sunlike star for 130,000 years"


In one sense, it is not possible to tell "the story" of the Ringworld. It is just too large. Even the epic journey in the first Ringworld book doesn't even cover the whole distance the short way across the Ringworld. It would be 3 million times harder than to tell the story of the Ringworld than the impossible "story of earth and everyone on it". the Ringworld is a place when uncountably many stories will happen.

But in another sense, unlike a planet, it has a human (sapient?) story of how, why and when it was made, and what happened to it after that.

Even Niven's Ringworld odysseys cannot give more than a brief taste - there is simply to much ground to cover.


It is likely that a highly advanced civilisation will not set out to build a Ringworld. However they may have many habitats, solar-cell farms, automated factories, captured asteroids, etc. etc orbiting around their star.

Later on, they may move them into synchronous orbit and link them up with cables, and spin the resulting ring for gravity.

Once this is in place, it would be possible to build onto this ring with "Plates" (CF Iain Banks) that support domed or walled 1-gravity habitats that simulate a planet's surface, as a city grows building by building.

The resulting structure will have much in common with a Ringworld, but will not have been build all in one shot, but added onto and accreted over many thousands of years, inhabited and viable at every step of the way.

It would resemble Niven's Ringworld in much the same way that a rainforest resembles a concrete continent painted green.

On the other hand, any species that has been space-bourne for so long may no longer be interested in or able to live under gravity any more, and thus may not be bothered to build a Ringworld at all. Here Dyson's original concept of the sphere (and not the Science-Fiction interpretation) begins to seem more feasable: no solid sphere, just so many orbiting solar panels and habitats that little or no sunlight escapes.

Ringworld book plot summary:

NB: major spoilers ahead. Skip the rest of the writeup if you haven't read the books.


Though the book is written in the third person, the point of view is clearly that of Louis Wu. We are privy to his thoughts and observations. And he gets twice as much sex as any of the other characters.

The alien race known as puppeteers, who by human standards are congenital cowards, are fleeing the galaxy. In their migration, they catch sight of the artefact called the Ringworld. A four-member crew is assembled to investigate Nessus: a puppeteer. Nessus is by puppeteer standards mad. This makes him suitable as an intermediary between the puppeteers and other species, and capable of adventure. His behaviour however is erratic. He exhibits a manic-depressive (or rather foolhardy-extrovert to catatonic cowardly withdrawal cycle) cycle. He recruits:
Louis Wu: protagonist, 200 year old adventurer, lover, xenophile and all-round capable guy.
Teela Brown: 20-year old nice but naïve human female possessed of extraordinary luck (descended from 6 generations of lottery winners) who has never experienced any significant hardship, loss or deprivation. Louis' lover.
Speaker to animals: A Kzin warrior. The title shows how important diplomacy is to the Kzin.

In payment the puppeteer offers the others a more advanced spaceship hyperdrive. He in turn receives payment from his more reticent countrybeings by them allowing him to breed.

Approaching the Ringworld, in their ship, The Lying Bastard, there is no response from the inhabitants. They decide to first inspect the shadow squares to see what allows them to orbit faster than expected. They collide with a thin strong filament between the squares.

Their ship is damaged by an automated meteorite defence system and it crash-lands on the Ringworld, on the foothills of a gargantuan mountain. Their ship is capable of interstellar travel by hyper drive, but there is no way at hand to get it into space.

Teela Brown's luck is held to have failed.

They set out for a rim wall, in the hope of finding a spaceport.

The inhabitants are surprisingly closely related to Earthlings, but civilisation appears to have fallen (quite literally - they had floating cities), and the Ringworld is in some disrepair.

They find a floating tower, with a working map of the Ringworld inside.

They learn that the massive mountain is called Fist Of God by the natives.

They try to pass themselves off as Gods, with mixed success. The locals worship the "engineers" who made the world, and the world is believed to be flat, with an arch over it from which the sun is suspended on a wire.

They pass through a city upon which mile upon wire of fine wire (the wire that holds the shadow squares together) is falling.

There is tension, due to the revelation that the puppeteers have intervened (indirectly of course) in human & Kzin history, by ensuring that a wandering band of alien traders reached the Humans first, thus giving them a hyper drive that enabled them to win the first man-Kzin war & preventing the human race being enslaved by the Kzin. Both Teela and Speaker are upset to find that they are the results of puppeteers deliberately breeding for desired traits.

They eventually find a survivor of the fall of the Ringworld civilisation, a woman called Halrloprillalar (Pril). She is several thousand years old, but still young due to anti-aging drugs better than those available to Louis. In the process, Teela is missing, presumed killed. Pril becomes Louis' lover. Their transport is mostly destroyed, but Pril's floating building is pressed into service. Pril tells that her people's civilisation fell because a fungus appeared that ate their superconductors.

Louis hatches a plan, and the party retrace their path. Teela resurfaces, with her new boy, a local barbarian hero replete with sword, loincloth and big muscles in tow. He is called Seeker because he is travelling to the base of the arch. He is in fact several centuries old, posessing a "magical" potion of longevity.

Louis is upset that she abandoned him (I don't see why, he did it to her first). He begins to surmise that Teela's luck never failed, and that everything that has befallen them is part of the luck's plan to do what is necessary to develop her character.

They journey back, which takes months of story time, but only a few pages of book. Teela and Seeker disembark at the floating tower with the map. Pril elects to come back to earthly civilisation.

They seize the fallen wire, and carry to back to their ship. In the process, one of Nessus's two heads is cut off (Don't worry, it doesn't contain his brain), and he spends the rest of the story unconscious. There they tie Pril's building to the Lying Bastard using the wire, and drive the building up the slopes of the Fist of God.

Louis' plan is revealed: The Fist of God is recent feature, damage to the ring from the impact of a huge meteorite from underneath, where the meteor defences cannot hit it. The top of the mountain is open, and they dive through, flung out to deep space, dragging the Lying Bastard behind them, and are free.

See also The Ringworld Engineers and Ringworld Throne for the other plot summaries.


For the Ringworld as written by Niven, this is true. But there are ways to design around the problem.

First off, you could pick a smaller and/or less luminous star and build a smaller Ringworld. You could also reduce the amount of gravity. Earth-chauvinists will object to both of these solutions on aesthetic grounds, and I must say I agree that having a red sun would be kinda weird.

But we can still fix things if you keep the Ringworld around a Sun-like star and give it 1g of gravity. We just need to come up with some way of supplementing/replacing the compressional force currently provided by the scrith to balance the centripetal force provided by the Ringworld's rotation.

Any given area of Ringworld floor has a certain force pushing it outward, given by F=ma, with a in this case equaling ~9.7m/s^2 (one Ringworld gravity). If we look up the mass of the Ringworld, multiply by one Ringworld gravity to get its weight, and divide by the area, we can calculate the average force exerted over a unit area of the floor (the outward pressure exerted on the scrith). That is how much force must be balanced to keep the Rignworld from tearing apart, and all of that balancing is currently done by the tensile strength of the scrith.

Next, given the size of the Ringworld and the mass of the star, we can use a=GM/r^2 to calculate the acceleration due to the star's gravity at the Ringworld floor. If we take the outward force on the Ringworld floor, and plug that and our new value for acceleration into the F=ma equation, we can solve for mass with m=F/a.

If we position exactly that much mass outside the Ringworld with zero orbital velocity, the inward force of its weight will exactly cancel the outward force on the Ringworld. Less, and there will still be some net outward force for the scrith to handle, but it will still be reduced. More, and we'll have to start worrying about compressional strength.

The scrith is supposedly frictionless, or nearly so, so we could concievably rest all of that mass right on the bottom of the Ringworld. On the other hand, we're trying to do this with the minimum of magical materials, so let's say that the counter-ring is coupled to the main ring via electromagnets.

Once the counter-ring is in place, the tensile strength of the Ringworld floor material is irrelevant. If we really needed to, we could concievably construct the Ringworld out of such low-tech material as stainless steel, plus the superconductors to build the magnets to hold it all up. Note, however, that we'll need a whole heckuvalotavit, whatever material is used, as the counter-ring will be orders of magnitude more massive than the main ring, itself already the mass of Jupiter.

Time to run some numbers and see just how much a heckuvalotavit really is. In order to make things simple I'll use as my given area the total surface area of the Ringworld. Ringworld surface gravity is .992g or ~9.7m/s^2, and its total mass is 2E27 kilograms, so the outward force is


The star is noted as being just smaller than the Sun, which masses 1.989E30 kilograms, so I'll round that down to 1.95E30 as a rough estimate of the Ringworld star's mass. The Ringworld's radius is 1.52E11 meters, and at that distance from the star inward gravitational acceleration is

a=1.95E30/1.52E112=.028733726 m/s^2

We now know the force we need to balance and the acceleration we have available, and so can solve for the mass:

m=675287291317527006417476104560.891 kilograms
m=~6.753E29 kilograms

As expected, dividing a large number by a small number gives a very large number for the mass of the counter-ring, a little shy of 6 hundredths the mass of the star, or ~338 times more massive than the original main ring.

If we're willing to (almost) completely abandon Niven's vision, we can construct a Ringworld much more economically. First, lets flip it inside out and kill its rotation- now, the livable surface is on the outside, and the gravity is the star's own. In order to get it up to reasonable levels, you'll either need to make the star bigger or the Ringworld smaller. Rather than shadow squares on the inside, this version of the Ringworld would have mirror squares on the outside to create day/night cycles. Filters and phosphors could be used to color the sunlight however you want, so almost any star would be suitable, and power collectors could still be orbited inside the ring.

Now, the strength we need to hold it up is compressional, rather than tensional. If we look at this as a case of the counter-ring becoming the main ring, it's obvious that what we need is a counter-ring spinning on the inside to create outward centripetal force to hold the Ringworld up. And, in fact, just as the outer counter-ring described above is more massive than its associated inner main ring, the inner counter-ring in this case can be made as thin as desired, and simply spun faster to make up the difference. There are two ways to set this up.

First, a single counter-ring could span the entire underside of the main ring. The main ring would be flat, and would still have walls at the edges to hold in the atmosphere. This is the most obvious set up.

Second, you could have two counter-rotating counter-rings, one at each edge. This way, if you use the main ring as a reaction surface for spin-up, the net angular force imparted on it will by zero. If they are each half the width of the main ring, they'll still span the entire underside, and the main ring will still be flat. If they're thinner, however, only the edges would be supported directly, and the main ring would be curved into a catenary cross-section for strength. The curve will hold in the atmosphere, with pressure decreasing as you approach the edges, eliminating the need for heavy rimwalls.

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