It depends on how realistic solar eclipse do you want. Solar eclipses have two important characteristics:
- Naked eye visibility of the solar corona.
- Darkness.
To have darkness like the real thing you would need a shadow that's as large as the Moon's umbra during a total ecipse, that's hundreds of kilometers, that would need to be a huge object. That's isn't practical I think.
The naked eye visibility of the corona probably easier. In order to see the corona somehow we need to get rid of the forward scattered light. The moon does it by simply have a large shadow so the whole atmosphere is under the umbra so there is no forward scattered light.
I think the same idea can be done in small. Just have an umbra large enough to provide just a little view around the sun without forward scattered light, in this case we have a geometry like this:
There $\alpha$ is the angular radius of the sun, which is roughly $0.25$ degrees.
The $\beta$ is the desired angular radius of the area of dakness where we would like to observe the corona. Let's say it's $0.5$ degrees.
The $h$ is the height of the atmosphere where scattering becomes negligible so the sky is totally dark. I don't know exact numbers here. I only have an educated guess.
One is from the many pictures or videos taken from high altitude balloons that reach 30km altitude regularly an example of such view is:
But I'm not sure if it's really that dark or just the white balance.
Another guess comes from making a calculation based on twilight brightness. After sunset, higher and higher layers of the atmosphere scatter the light until there is completely dark. From wikipedia:
Evening nautical twilight is defined to begin at sunset and end when
the center of the sun is 12 degrees below the horizon. In general,
nautical twilight ends when navigation via the horizon at sea is no
longer possible.
So we need to calculate $h$ when we put 12 degrees in into $\alpha$ at the center of Earth.
I did the math and got roughly 35km. ($\sqrt{R^2 + (R \mathrm{tan}(\alpha/2))^2} - R$)
So I think setting $h$ to 30km is reasonable.
But in order avoid the shading object itself blocking the view of the corona we need to send it even higher, so it appears smaller. The highest altitude a balloon ever reached is above 50km. So we can fix it at 50km altitude for example.
Now let's see the numbers.
At 30km a circular disk that have 0.5° apparent radius from the ground have 262m radius (30*tan(0.5°)). Considering the angular size of sun, at 50km high we would need a sphere that has 349m radius (0.262+(50-30)*tan(0.25°)) to cast 262m radius shadow at the 30km level. So a really large balloon would probably do it.
A sperical balloon with 349m radius would have 178 millions of cubic meter capacity, which is 200 times more that was used in Felix Baumgartner's jump.
It would have an umbra on the ground with a radius of 131m (0.349-50*tan(0.25°)). And a penumbra with a radius of 567m (0.349+50*tan(0.25°)).
It's apparent size would be a circle with an angular radius of 0.4°. So it would still block most of the view, but the dark patch will always extend beyond the apparent view so at near the end of the totality we may have a better view of the corona.
Although there are still some problems:
Scattered light from the environment. The shadow tube might not be dark enough after all. When the sun sets at west the shadow of earth rises at the east. It's dark but it's not completely black even if it's a shadow going through the entire atmosphere. But it will probably be dark enough to reveal the corona.
The balloon must be made of a very opaque or a very reflective material, so it appears truly dark for the shadow side. Or must be simply thick enough.
My calculations are for the case when the sun is at the zenith. When it's not then we would need an even larger ballon.
So apart from the enormous ballon we would need it seems to entirely possible to make naked eye visibility of the corona from the ground with a shading object.
For scientific purposes it's much easier to bring a coronagraph satellite like SOHO into the space, or using polarizers and image processing in a ground based coronagraph.
