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I understand the Moon could hold onto an Earthlike Atmosphere for a matter of centuries.

Titan has a higher escape velocity than our Moon, it you increased its temperature to that of out Moon, it would lose atmosphere much more quickly, but it would still be a matter of centuries to a millennium for it to lose its atmosphere.

Titan · Escape velocity

Titan · Escape velocity

8,658 feet/s (2,639 m/s)

Moon's Escape velocity (km/s)
2.38
The Moon's gravity is 0.165 of Earth.
Titan's is 0.14 of Earth.

Titan's Mass is
(1.3452±0.0002)×10^23 kg

Titan's orbit period is 15.945 days, in a 24 hour period Titan turns 360/15.945  = 22.578 degrees so in a 24 hour period an object which orbits over 360 degrees of its surface has to orbit a total of 360 + 22.578 = 382.578, that means its actual orbital period would be 24*360/382.578 = 22.584 Hours.

An orbit around Titan at a radius of 11455 km will produce an orbit period of 24 hours relative to Titan's surface, that means this artificial sun will rise and set and rise again as seen from a given location on Titan's surface in 24 hours. It is probably easiest to orbit this Sun over the equator and not have seasons, otherwise the seasonal cycle will last about 16 days without constant orbital correction. An artificial light source could instead vary its intensity instead to create seasons over the entire satellite. the radius of the Sun is 695,700 km, over a distance of 150,000,000 as seen from Earth.

11,455km/150,000,000 km = 7.6366666666666666666666666666667e-5 Multiply this by 695,700 km and we get a radius of 53.12829 km for our artificial Sun. The surface temperature of the Sun is Photosphere (effective): 5,772 K. the boiling point of Tungstein is Boiling point: 5,555 °C or 5,828K You know what that means? That means Tungstein remains liquid at a temperature equal to the surface temperature of the Sun. So an artificial sun can be a ball of glowing molten tungstein that is 53.12829 km in radius, as seen from the surface of Titan, such a glowing ball would look exactly like the Sun as seen from Earth.

The density of liquid Tungstein is 17.6 g/cm3 or 17.6 tons/meter^3
Volume of the sphere is 628143703822703 meters cubed times 17.6 tons = 11,055,329,187,279,572.8 tons of Tungstein or 1.1e16 tons.
Now the question is what's the best way to heat this molten ball of Tungstein to the temperature of the surface of the Sun, the great thing about liquids is they hold their volume, rather than expand as a gas, in a condition of microgravity, surface tension will keep it to a ball of white hot liquid. So what's the best way to heat it? Perhaps a laser, a very powerful solar powered x-ray laser, target this ball with it, but avoid hitting Titan and this ball of molten metal and illuminate one hemisphere of Titan heating up its surface to Earth like temperatures. This leads to another problem. Much of the surface is ice, not only water ice, by dry ice, its nitrogen atmosphere will probably get a lot of carbon-dioxide as the dry ice on its surface sublimates, the water ice will eventually melt and form an ocean. The good news is there are a lot of carbon compounds in Titan's atmosphere and crust, through chemical rearrangement, they can be formed into artificial rock, to form a shell in the shape of the topography of this Moon.

Usually you do that if you want a avoid a filter for using foul language. You misspell the foul word so it doesn't get removed.

That is third person, bet you don't know what happens first person when you die, no one does! Knowledge and belief are not the same thing.

In total, at the distance of 15,000 km, it would have to be 338.24 kilometers in diameter, this will take into account the fact that the mirror will have to be angled at 45 degrees to reflect incident light rays from the Sun, so this reduces its apparent diameter to 0.7 of its actual diameter. At a distance of 15,000 kilometers the orbital period is 8 hours and change, if we want, we could increase the orbit to 24 hours, but then we'd have to increase the size of the mirrors too. the size of the mirror would have to be proportional to the mirror's distance from the Earth. If we increase to orbit to 35,870 km we would get a 24-hour orbit, if we did this however...
35,870/15,000 km = 2.39133.
2.39133 * 338.24 = 808.84 km
We'd have to increase the mirror diameter to 808.84 kilometers the Sun would then rise on a 24-hour schedule, it could be synchronized with the appearance of the real Sun to produce twice as much sunlight, the mirror doesn't have to be 100% reflective of course, and the real Sun when it appears is at a low angle, heavily filtered by the atmosphere, so it won't be quite twice as much as we see the Sun at high noon in the New York area for example. You probably do want to warm it up more to compensate for the cooling effect of the surrounding glaciers.

This is to cover an area on the ground that is 50 km in radius, beyond that, you get a reflected partial image of the sun, and a transition zone to a normal polar climate. This zone will be quite isolated by these ice sheets, you won't get penguins or seals or walruses, or krill that are disturbed by this, and very little life exists in the interior of Antarctica, except for that which we place there.

If we can use material from the Moon, it might prove cheaper to do this, that to build a 50 km radius floating platform on the Ocean, and less environmental impact as well. Reflectors in space don't have to withstand the rigors of a churning ocean after all.

I thought about this too, then I thought, do I want this "Sun" to be up all the time? If we were to turn off this "Sun" so the people in Antarctica can have some night, this statite would fall towards the Earth, you see what keeps it up is radiation pressure, and it not only has to reflect light towards a particular area on Earth, it also has to levitate itself with that same light, the direction the light has to be reflected in in order to levitate itself might not be in the same direction we want to reflect light in in order to warm a patch of Earth. It might be better to use a thicker, heavier mirror, still a solar sail, but one which can't levitate itself against the force of gravity, instead it uses light pressure to constantly change it orbit, when it is not over Antarctica, the orbit change is so that it is orbiting at right angles to the direction of the Sun, we call this a Solar Synchronous Orbit, it orbits in such a way that the Sun is always  visible, it orbits over the poles, and is angled at 45 degrees to the incident light rays, and as it passes over that particular patch, it changes its orientation, using light pressure, to reflect light on that particular patch of ground. From the point of view of those on the Ground, the Sun will appear to rapidly rise from the South, go directly overhead, and then set in the opposite horizon, at times where will appear to be two Suns in the sky at the same time, one, low above he horizon, and one rising to its zeneith almost directly over head and then setting in about 4 hours. During winter, the transantarctic mountains will experience a series of 8-hour days with about 4 hours of sunlight and 4 hours of darkness.

The surrounding areas are ice sheets, one particular effect I think would be and artificial hurricane, you see warm air would rise in the center pulling it surrounding colder air. I think it would to a large extent be a "snow Hurricane" as a lot of precipitation from it will be snow. Snow will tend to pile up on top of the glaciers, the glaciers would tend to move inward toward the melt water lake and then be melted, and thus we continue the water cycle. If the area is small, say about 50 kilometers in radius, the rest of the continent wouldn't be affected much at all. All that ice surrounding it would isolate the ecology around the transantarctic patch from the more environmentally sentivie coastal areas of Antarctica, so the penguins and seals would hardly be affected at all, as the warmed up areas would be deep into the interior of the continent. And this would be good practice for terraforming Mars.

Because that's what were talking about! How many people fly in airplanes? Millions. Why is the idea about millions of people traveling in space such an incredible idea to you, just because we're not doing it now? Million of people fly in airplanes, you for you it seems space travel is mostly for audiences watching live pictures on their televisions. For you an astronaut is a celebrity figure who signs autographs, that has got to change, what's called The Right Stuff has go to change. If you need to be a super genius at the pinnacle of your career and be hired by the government to go into space, this is not a real and true space age. Suppose President Thomas Jefferson wanted to create an Aeronautical Agency to support balloon flights and taxpayer expense. Thomas Jefferson had seen balloon flights in France, wouldn't that be comparable to what NASA is doing today? We are in the hot air balloon age of space flight. Most people in the early 19th century didn't fly at all. though a few did, the Wright Brothers weren't the first.

Modern flight began in 1783 when Joseph-Michael and Jacques-Ètienne Montgolfier engineered the first hot-air balloon flights. On Oct. 15, 1783, the Montgolfiers brothers launched a balloon on a tether with Jean-François Pilâtre de Rozier, a chemistry and physics teacher, as the passenger. In that era, nobody knew if a person could withstand the rigors of being up in the air, so a previous flight had included animals, to see if they survived. They did, as did de Rozier. Later that year, the first untethered flight was made.

An Island Three would by comparison have about 250 square miles of living space, the Shackleton Dome would have 78.5 square miles, still it is a great deal more than the O'Neill Bernal Sphere Island One. So if an Island Three O'Neill Cylinder can hold 10 million, with similar living space, the shackleton Dome could be home to as many as 3 million.

You know we don't have to work out of just one launch pad, if the demand is high enough, more launch pads can be built, we could launch multiple rockets at once, multiple rockets can fly in every launch window, just as we have multiple airplanes taking off and landing at multiple airports all the time. If the cost can be brought down, then the demand will go up. Each ship can carry 100 people, but if we launch 100 ships at once we can send 10,000 people to Mars on a convoy of spaceships. We don't do that now, because the cause is too high, but if we can bring down the cost, we can certainly justify building 100 launch pads to launch 100 rockets at once, there is nothing that says we couldn't do this, there is plenty of space to launch all of those rockets in a single launch window and not have them bump into each other. So if we could send 10,000 human beings every launch window that is how much in 70 years?, about 350,000. Okay, so if we built 300 launch pads and launched 300 ships at a time, we could have one million people on Mars in 70 years, not too bad. What sort of airplanes did we have in 1916?

I think they looked something like this. How do you go from this

to this in 70 years? What would be the early 22nd century equivalent to a 777 in space? Would it be those crude things we're talking about now?

There is no Terran Government to arrest you, instead there are lots of countries, land in the right country and they won't arrest you!

Interesting response Gary. The Mars Colonial Transporter is a reusable two-stage to orbit vehicle with two different second stages, one for carrying passengers and cargo and the other is a fuel tanker for refueling the passengers and cargo module. The ship relies on a chemical methane-oxygen combustion to not only achieve orbit but also to make the trans-Mars injection, no use of ion, plasma, or nuclear drives is mentioned. The crew-cargo module also has a fan-shaped deployable solar power array to power the ship during its cruise between planets. The passengers float around in zero-g during the cruise, the ship lands without a parachute and then produces its fuel for the return journey. I assume there is some sort of nuclear reactor of the type Zubrin mentioned to power the manufacture of return fuel, but perhaps extensive arrays of photovoltaic panels could do the same job if there are enough of them. Either the first explorers will be brought home, or we'll have to keep on resupplying them from Earth. We'll need long range rovers to do a thorough exploration of Mars around the landing site and to search for water, not the type of rover Mark Watney had to jury rig in the movie The Martian. We probably should preposition several long range rovers for the explorers to use, and plenty of spare parts to repair the same rovers. the rovers should be pressurized, because its hard to live in a spacesuit for those long journeys across Martian terrain, and the rovers should either have an airlock if a docking port for externally mounted spacesuits. I think an airlock is a better idea, as it allows the space suits to be cleaned and maintained within the pressurized rover.

What if the gold is still in the ground and unmined? It is there now. Why can't you buy and sell the Martian gold right now without mining it? I'm sure there is more gold in Mars right now than in the vaults of Fort Knox. I kind of suspect you have to do something with the gold other than leave it in the ground before you can buy and sell it. The difference between gold on Earth and gold on Mars is that you can't make jewelry out of gold that has been unmined and still on Mars. Or imagine this, a Hollywood star at the academy awards, wearing the latest fashion, and hanging on a string around her neck is a photograph of a necklace that was made out of gold and is still on Mars, and they Hollywood star insists that its hers to the reporters that ask. Gold on Mars is not the same as having gold on one's person.

How about building a hab next to a canyon wall of the Vallis Marineris, that should provide some radiation protection.