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I have an extremely silly problem I am working on, having to do with celestial mechanics under ridiculous situations, and I have some questions about whether I'm doing it all right, if anyone is willing to spare a few minutes and half their sanity. (I am probably going to post my final results on my blog, but since nobody reads my blog I am basically only seeking these answers for my own amusement.) Below first is the work I've done so far, and my questions are at the end...
So for the last week or so I've been playing a video game called "Super Mario Galaxy". SMG is the new 3D Mario game, and its gimmick is that where previous Mario games had you jumping between platforms floating in the air in the Mushroom Kingdom, the new one has you jumping between little bitty planetoids floating in space. Mario Galaxy makes absolutely no pretenses at realism (for one thing, Mario does not need to breathe in the vacuum-- but he does run out of air underwater; also, Mario's shadow is always cast in the current direction of gravity. Also there's the sentient mushrooms...) but, I thought it would be fun to try to sit down and work out whether some of the weird things that happen in Mario Galaxy are actually possible given the physics of the actual universe.
Here's the main thing I'm wondering about: Most of Mario Galaxy is spent running around on the surfaces of these little asteroids. The asteroids vary in size and shape, although most are roughly spherical and on average each asteroid is about the size of a Starbucks franchise. At least some of these are revealed to be in fact hollow spheres with black holes at the center, like the one in the first 30 seconds of this video:
http://youtube.com/watch?v=xTrgJ95BMsw
The trick is: every single one of these asteroids has normal Earth gravity. So here is what I am trying to figure out: would it be physically possible, in reality, to actually construct little asteroids like this, with compact volume but Earth gravity, assuming an absurdly high degree of technology and resources at one's disposal but no new physics?
So here's about as far as I've got.
The planetoid at the start of that video looks like it's about as wide as a 747 is long. Let's consider just that one planetoid. Wikipedia says a 747 is 230 feet long.
Newton's formula for gravity is m_1*a = G * m_1 * m_2 / r^2, where I take m_1 as the mass of Mario and m_2 is the mass of the planetoid. Solving for m_2, I get:
m_2 = a * r^2 / G
r is 115 feet (half a 747), and a is Earth gravity, 9.8 m/s/s.
Plugging into google calculator, I get:
((9.8 m (s^(-2))) * ((115 ft)^2)) / G = 1.8043906 × 10^14 kilograms
So, if you want to get earth-like gravity on the surface of a 230-foot-diameter sphere, you're going to need about 1.8 * 10^14 kg of mass. This is not that much! Checking Wikipedia I find even the smallest moons and dwarf planets in the solar system get up to about 10^20 kg. in fact, 2 * 10^14 kg is just about exactly the mass of Halley's Comet. This sounds attainable; other characters are shown hijacking comets for various purposes elsewhere in Mario Galaxy, so no one will notice a few missing.
Meanwhile, Wikipedia's formula for escape velocity for a planet ( sqrt( 2GM / r ) ) tells us that escape velocity from this planetoid will only be about:
sqrt((2 * G * (1.8043906 × (10^14) kilograms)) / (115 feet)) = 26.2110511 m / s
Which is about 60 MPH. So the cannon stars Mario uses in the video to move from planet to planet should work just fine.
So at this point the only question is: If one were to attempt to fit Halley's Comet into a 230-foot-diameter sphere, would this turn out to be impossible for any reason? I've got some numbers crunched on this one, but this is where I may have to start asking some questions. I'm going to consider two cases: One where the mass is in a black hole at the center of the sphere; and one where the mass is distributed through the sphere evenly. In both I'll try to see if anything really bad happens.
So, first, the black hole case. Looking http://curious.astro.cornell.edu/question.php?number=160 I'm told that the event horizon of a mature black hole should be equal to G * M / c^2, and Wikipedia tells me that the Schwarzschild radius is about twice that (is that actually right??). So plugging this in, I find that the radius of this black hole at the center of the sphere is going to be:
(G * (1.8043906 × (10^14) kilograms)) / (c^2) = 1.33970838 × 10^-13 meters
... uh oh! Now we seem to be running into trouble. If someone wanted to construct a black hole with the mass of Haley's comet, they'd somehow have to pack the mass of the whole thing into... well, google claims the diameter of a gold atom is 0.288 nanometers, so... about one-hundredth of the diameter of a gold atom?! That doesn't sound very feasible.
On the other hand, considering the case where the sphere is solid, one finds that the required density is going to be about:
(1.8043906 × (10^14) * kilograms) / ((4 / 3) * pi * ((115 feet)^3)) = 1.00023843 × 10^9 kg / m^3
As remarkable coincidence would have it, 10^9 kg/m^3 is, according to wikipedia, exactly the density of the crust of a neutron star! So this is sounding WAY easier than the black hole plan: All you have to do is go in and chip off 180,000 cubic feet of the crust of a neutron star, and you've got your planetoid right there. Although I suspect maybe neutronium wouldn't be very pleasant to walk on?
This is as far as I got so far. So:
QUESTIONS
1. Above, I assume I don't even need to think about relativity. Do any relativistic effects come into play when you start talking about 10^14ish kg cramped into the volume of a 747?
2. The black hole solution, again, requires one to fit 10^14ish kg into a space about a hundredth of the diameter of a gold atom. This sounds pretty obviously silly in practice, but is there in principle any reason why this would be impossible? Assuming one (due to some unknown technology) has the ability to exert enormous amounts of pressure on that 10^14 kg, is there anything realistically stopping you from making a black hole? Say, would degeneracy pressure get in the way, or would that just mean that much more force to overcome?
3. In the solid-sphere solution I half-seriously suggest grabbing 10^14 kg of degenerate matter from the surface of a neutron star. Assuming one were to somehow do this, and just dump the 747's worth of neturon star out in outer space-- would the degenerate matter do anything inconvenient, like expand or explode? I assume the surface of a neutron star is requiring some pretty intense gravitational pressure to hold it to the star. What happens if you suddenly take that gravitational pressure away? Let's say, just to complete the silliness, I wanted to build a shell to keep my little chunk of neutron star from exploding. How would I go about calculating the amount of pressure per square inch that shell would have to be able to take from the inside?
4. In the solid-sphere solution I assume the sphere can be treated like a single point mass. Is this valid, considering the sphere is not that large compared to the person walking on it?
5. Although I only consider the case of a spherical planetoid here, a lot of the planets in Mario Galaxy have really weird shapes-- shaped like hamburgers, say, or cubes, or avocados. I'm going to just ignore these shapes because they're clearly ridiculous, but there's one that I'm curious about: A torus. At several points in Galaxy, Mario is suddenly found walking on a torus, and at all times, wherever he walks on the torus (even on the inside "rim"), gravity holds him to its surface. Is there ANY basis, however bizarre, by which this could actually work? What would happen if you got something around the amount of mass for the planetoid proposed above, squeezed it evenly distributed into a 747-scale torus, and tried to walk on it? Would gravity hold you to the torus's surface, or would you be pulled toward the center of the torus? And what would happen if you walked into the inside "rim"? Would the gravity still "work"? How does one go about solving problems like this, concerning the gravitational effects of irregular bodies?
Any help would be appreciated, thanks! :)
So for the last week or so I've been playing a video game called "Super Mario Galaxy". SMG is the new 3D Mario game, and its gimmick is that where previous Mario games had you jumping between platforms floating in the air in the Mushroom Kingdom, the new one has you jumping between little bitty planetoids floating in space. Mario Galaxy makes absolutely no pretenses at realism (for one thing, Mario does not need to breathe in the vacuum-- but he does run out of air underwater; also, Mario's shadow is always cast in the current direction of gravity. Also there's the sentient mushrooms...) but, I thought it would be fun to try to sit down and work out whether some of the weird things that happen in Mario Galaxy are actually possible given the physics of the actual universe.
Here's the main thing I'm wondering about: Most of Mario Galaxy is spent running around on the surfaces of these little asteroids. The asteroids vary in size and shape, although most are roughly spherical and on average each asteroid is about the size of a Starbucks franchise. At least some of these are revealed to be in fact hollow spheres with black holes at the center, like the one in the first 30 seconds of this video:
http://youtube.com/watch?v=xTrgJ95BMsw
The trick is: every single one of these asteroids has normal Earth gravity. So here is what I am trying to figure out: would it be physically possible, in reality, to actually construct little asteroids like this, with compact volume but Earth gravity, assuming an absurdly high degree of technology and resources at one's disposal but no new physics?
So here's about as far as I've got.
The planetoid at the start of that video looks like it's about as wide as a 747 is long. Let's consider just that one planetoid. Wikipedia says a 747 is 230 feet long.
Newton's formula for gravity is m_1*a = G * m_1 * m_2 / r^2, where I take m_1 as the mass of Mario and m_2 is the mass of the planetoid. Solving for m_2, I get:
m_2 = a * r^2 / G
r is 115 feet (half a 747), and a is Earth gravity, 9.8 m/s/s.
Plugging into google calculator, I get:
((9.8 m (s^(-2))) * ((115 ft)^2)) / G = 1.8043906 × 10^14 kilograms
So, if you want to get earth-like gravity on the surface of a 230-foot-diameter sphere, you're going to need about 1.8 * 10^14 kg of mass. This is not that much! Checking Wikipedia I find even the smallest moons and dwarf planets in the solar system get up to about 10^20 kg. in fact, 2 * 10^14 kg is just about exactly the mass of Halley's Comet. This sounds attainable; other characters are shown hijacking comets for various purposes elsewhere in Mario Galaxy, so no one will notice a few missing.
Meanwhile, Wikipedia's formula for escape velocity for a planet ( sqrt( 2GM / r ) ) tells us that escape velocity from this planetoid will only be about:
sqrt((2 * G * (1.8043906 × (10^14) kilograms)) / (115 feet)) = 26.2110511 m / s
Which is about 60 MPH. So the cannon stars Mario uses in the video to move from planet to planet should work just fine.
So at this point the only question is: If one were to attempt to fit Halley's Comet into a 230-foot-diameter sphere, would this turn out to be impossible for any reason? I've got some numbers crunched on this one, but this is where I may have to start asking some questions. I'm going to consider two cases: One where the mass is in a black hole at the center of the sphere; and one where the mass is distributed through the sphere evenly. In both I'll try to see if anything really bad happens.
So, first, the black hole case. Looking http://curious.astro.cornell.edu/question.php?number=160 I'm told that the event horizon of a mature black hole should be equal to G * M / c^2, and Wikipedia tells me that the Schwarzschild radius is about twice that (is that actually right??). So plugging this in, I find that the radius of this black hole at the center of the sphere is going to be:
(G * (1.8043906 × (10^14) kilograms)) / (c^2) = 1.33970838 × 10^-13 meters
... uh oh! Now we seem to be running into trouble. If someone wanted to construct a black hole with the mass of Haley's comet, they'd somehow have to pack the mass of the whole thing into... well, google claims the diameter of a gold atom is 0.288 nanometers, so... about one-hundredth of the diameter of a gold atom?! That doesn't sound very feasible.
On the other hand, considering the case where the sphere is solid, one finds that the required density is going to be about:
(1.8043906 × (10^14) * kilograms) / ((4 / 3) * pi * ((115 feet)^3)) = 1.00023843 × 10^9 kg / m^3
As remarkable coincidence would have it, 10^9 kg/m^3 is, according to wikipedia, exactly the density of the crust of a neutron star! So this is sounding WAY easier than the black hole plan: All you have to do is go in and chip off 180,000 cubic feet of the crust of a neutron star, and you've got your planetoid right there. Although I suspect maybe neutronium wouldn't be very pleasant to walk on?
This is as far as I got so far. So:
QUESTIONS
1. Above, I assume I don't even need to think about relativity. Do any relativistic effects come into play when you start talking about 10^14ish kg cramped into the volume of a 747?
2. The black hole solution, again, requires one to fit 10^14ish kg into a space about a hundredth of the diameter of a gold atom. This sounds pretty obviously silly in practice, but is there in principle any reason why this would be impossible? Assuming one (due to some unknown technology) has the ability to exert enormous amounts of pressure on that 10^14 kg, is there anything realistically stopping you from making a black hole? Say, would degeneracy pressure get in the way, or would that just mean that much more force to overcome?
3. In the solid-sphere solution I half-seriously suggest grabbing 10^14 kg of degenerate matter from the surface of a neutron star. Assuming one were to somehow do this, and just dump the 747's worth of neturon star out in outer space-- would the degenerate matter do anything inconvenient, like expand or explode? I assume the surface of a neutron star is requiring some pretty intense gravitational pressure to hold it to the star. What happens if you suddenly take that gravitational pressure away? Let's say, just to complete the silliness, I wanted to build a shell to keep my little chunk of neutron star from exploding. How would I go about calculating the amount of pressure per square inch that shell would have to be able to take from the inside?
4. In the solid-sphere solution I assume the sphere can be treated like a single point mass. Is this valid, considering the sphere is not that large compared to the person walking on it?
5. Although I only consider the case of a spherical planetoid here, a lot of the planets in Mario Galaxy have really weird shapes-- shaped like hamburgers, say, or cubes, or avocados. I'm going to just ignore these shapes because they're clearly ridiculous, but there's one that I'm curious about: A torus. At several points in Galaxy, Mario is suddenly found walking on a torus, and at all times, wherever he walks on the torus (even on the inside "rim"), gravity holds him to its surface. Is there ANY basis, however bizarre, by which this could actually work? What would happen if you got something around the amount of mass for the planetoid proposed above, squeezed it evenly distributed into a 747-scale torus, and tried to walk on it? Would gravity hold you to the torus's surface, or would you be pulled toward the center of the torus? And what would happen if you walked into the inside "rim"? Would the gravity still "work"? How does one go about solving problems like this, concerning the gravitational effects of irregular bodies?
Any help would be appreciated, thanks! :)
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