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Imagine a world like Pellucidar or Skartaris. This world exists on the inside of a 'hollow Earth'. The laws of physics apply as normal, except as follows

  • There is a gravitational force that attracts all objects to the interior surface of the sphere. This force falls off as an inverse square law until in the very center of the hollow sphere there is no gravity. Gravity can be expressed as $g = g_o\left(r^2/a^2\right)$ where $g$ is gravity at a point, $g_0$ is gravity at the 'surface' (9.8 m/s), $r$ is distance from the center, and $a$ is radius of the hollow world.

  • The interior is entirely filled with breathable air. The air pressure is atmospheric (1 bar = 100 kPa) at the 'ground' level. Pressure may vary with gravity at height.

  • In that very center, there is a light and heat creating point source of radiation. Unlike other hollow Earths, this one goes on and off, so there is both night and day.

The question is this: What is the fastest way to get across the sphere in a balloon?

The base answer would be just to travel along the surface of the sphere, but there are no internal combustion engines, so you would either have to 'sail' your way there, pedal your way there, or tether your balloon to a donkey and walk there.

Is it possible to take a short cut? Can you use the balloon's buoyancy and the decreasing gravity with height to take some sort of shortcut across the center of the sphere?

Considerations

  • The interior surface has the same surface area as the Earth.

  • The light and heat felt from the 'sun' is the same as that felt at the equator on Earth. Feel free to work backwards to see how powerful the 'sun' has to be.

  • Across the sphere means a point 180 degrees away (in all directions); all the way across the sphere.

  • If you need to fly past or through the center, you can do it at night so as not to get burned.

  • There are no breathing apparatus, so make sure you don't suffocate by piloting into a low air pressure zone, if any exist.

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    $\begingroup$ This is a great question. Let's get all the energy on the plate, what's the wattage/sqmeter of our central "solar" source? Is three any volcanism? $\endgroup$ – JBH Jan 24 '18 at 3:06
  • $\begingroup$ @JBH The solar source generates the same energy on the surface as our sun as seen at the equator. I dunno about vulcanism, so lets default to no. $\endgroup$ – kingledion Jan 24 '18 at 3:25
  • $\begingroup$ Once you get near to the center of the sphere you will start to slow as the air friction increase relative to the buoyancy of the vehicle. Someone smarter than I can do the math, but intuition says that once you get near to the center you gravitate to the center and get stuck because of neutral buoyancy. $\endgroup$ – pojo-guy Jan 24 '18 at 3:36
  • $\begingroup$ If you have an inverse square drop-off of gravity as you describe, then the center of the hollow will not have a zero gravity. If you want to create a model gravity that's not as problematic (and that is) then use e.g.$g=g_0(r/a)$ where $g_0$ is gravity on the shell and $a$ is the radius of the shell and $r$ is the radius from the center. This is mathematically less problematic (although you've of course still completely rewritten physics anyway). $\endgroup$ – StephenG Jan 24 '18 at 4:45
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    $\begingroup$ @PyRulez I said I was going to try. $\endgroup$ – kingledion Jan 25 '18 at 13:35
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Let's consider a few issues.

  • There would be no atmospheric escape into outer space. This isn't much to begin with, but with the atmosphere being completely trapped, you have more to work with.

  • Next, The Earth's atmosphere is about 300 miles thick on the outside of a sphere. A sphere's volume is 4/3 pi*r3 The earth's radius is 6.37x106 meters for a volume of 1.08x1021 cubic meters. Add 300 miles (482,803 meters) and you get 1.35x1021 cubic meters for a atmospheric volume of 270x1018. If we process all that math to get the inner atmospheric thickness we get 359 miles.

  • Now, I've ignored the fact that the atmosphere thins as altitude increases, but this actually works in our favor as the thickness would be more then 359 miles due to the compression of density near the surface. Adding the effect of no atmospheric escape, and using an itch on my right elbow as my guide, I'm going to boldy declare that the thickness of your atmosphere is approximately 400 miles.

  • Finally, I'm going to note that, simplistically, the difference between having a sun that illuminates the entire world during the day and a sun that only illuminates half the world at any one time is that while on Earth you get winds, on your world you'll get a daily increase of barometric pressure and a nightly decrease. This will also have the effect of increasing the atmosphere thickness, which will reach its peak in the mid-afternoon. (I'm ignoring everything like seasonal conditions such as your sun switching on for longer periods during the "summer" and shorter periods during the "winter." Just one season for my answer, thanks!). How much the barometric pressure will increase (and therefore the atmospheric thickness) depends on how much the atmosphere heats and how much ocean you have to evaporate water into the air. But, I'm being outrageous, so let's claim that gets us to 450 miles of thickness at 3:00pm.

OK, you're trying to get to the diametric opposite of where you began. As Pojo-Guy points out, bouyancy will only get you so much altitude, but rather than worry about what kind of gas you're using and the weight of your dirigible, I'm going to estimate a best-case scenario.

As the crow flies, if a crow didn't need to breathe and its feathers were entirely heat resistant, we're talking about an 8,000 mile trip. But you can't reach an altitude of 4,000 miles.

What you can do is float straight up 450 miles in the afternoon, then proceed in a sinusoidal path (400 miles to 450 miles altitude depending on what time of day it is) until you're over your landing point, then you pull the rope and drop like a rock.

Thus, your fastest time would be the time needed to rise 425 miles (on average), (Tascend)) + the time needed to fall 425 miles (on average), (Tdescend) + the time to traverse the surface of a sphere 425 miles (on average) less than the surface the cows are standing on (and I'm assuming it's flat rather than worrying about whether or not we're starting/ending in a valley or on a mountain) (Tsphere), which is pi*r miles or 11,231 miles.

So...

  • You need to rise 425 miles.
  • You need to fall 425 miles.
  • You need to cross 11,231 miles.

If we assume the cruising speed of the Hindenburg, 76mph, and half that to ascend/descend, then it will take 22.37 hours to rise and fall, and 147.8 hours in transit for a hair over 7 days to make the entire trip.

Unless there's weather... Remember that nightly decrease in barometric pressure? Yup, rain. Lots of rain. Maybe even hail... the weather on your world will be interesting... and you'd hate one of those Far Side moments when your brother shuffles up next to you and touches your skin with a small hydrogen leak nearby....

And, lest we forget, you can't actually get that high. So the trip is probably a whole lot closer to 2 weeks.

You'll notice I'm using a dirigible, not a balloon. I'm not convinced you'll have significant winds in your hollow world. There's nothing pushing the air independent of the spin of the world other than the comparatively gentle push of evaporating and condensing water vapor. Methinks a balloon would be a curiosity as it can only ascend and descend, but not travel.

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  • $\begingroup$ This is exactly the kind of response I was looking for, just great +. $\endgroup$ – kingledion Jan 24 '18 at 11:48
  • $\begingroup$ Weather inside this hollow earth will be similar to the weather on the surface of our Earth. At least, it will be caused by the same physical factors generating meteorological conditions. Winds, storms, rainfall, and what-not caused by thermal differences due to geographical variation. Landmasses, seas and oceans, et cetera. While this answer assumes no variation in atmospheric density with altitude, this seems to be the least likely plausible assumption in these circumstances. $\endgroup$ – a4android Jan 24 '18 at 11:52
  • $\begingroup$ Reviewing the OP's question, I note one of its assumptions is that the interior volume is filled with breathable air. This seems contrary to the internal gravitational profile. However, if there is a point source of heat and light at its centre there should be a massive heat transference. To the atmosphere, at least. This suggests interesting thermal conditions within. If gravity is rewritten in this world, quite likely other physical laws are too. $\endgroup$ – a4android Jan 24 '18 at 12:00
  • $\begingroup$ @a4android So I did the math out on both your and JBH's proposals. Turns out JBH has to be correct. Given the gravity profile of my interior sphere, if the atmosphere at surface density extended up to 100 km, air pressure at the surface would be 100 bars. So the atmosphere has to thin with altitude, as it does on Earth, and as JBH assumes, although breathability would be limited to more like 10km as it is on Earth. Maybe some magical assistance can up this. Also, this thins the atmosphere so no weird effects from the 'sun' in the middle, which I hadn't thought of. Thanks for the comments! $\endgroup$ – kingledion Jan 24 '18 at 14:17
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    $\begingroup$ @kingledion Actually JBH "ignored the fact that the atmosphere thins as altitude increases", while I believed, I hadn't done the calculation so it was pure intuition, that the atmosphere had to thin with altitude. This was an improbable assumption, however, the question postulated breathable air everywhere. I am relieved that this hollow earth's atmosphere is similar to a normal planet's. I may not have made that clear in my comments. $\endgroup$ – a4android Jan 25 '18 at 0:42

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