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DoctorOfSpace
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13 Mar 2017 06:09

Well now that the math is out of the way someone should begin building an infinite improbability drive. 
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midtskogen
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13 Mar 2017 14:25

Of course.  The probability is obviously 2^267,791 to 1.
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13 Mar 2017 16:10

That's weird, a vase of petunias just crashed on my floor from nowhere.
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13 Mar 2017 16:44

Mosfet, by any chance, was there a whale too?
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13 Mar 2017 18:07

No, but I heard the firefighters are nearby.
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Watsisname
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13 Mar 2017 19:31

I wasn't expecting anyone to try to calculate it, but was curious to see how small people would take it with guesses, and if it would be small enough.  FFT had the right idea to introduce recursive scientific notation, as it turns out the actual answer is so small that it can't be written without it!  It is about 10^-10^52.  That's one chance in 1 with 10^52 zeros after it.  I... uh... can't think of anything that even remotely describes how small that is in everyday terms.  If you do happen to try it with the Heart of Gold, beware of summoning space whales.

Gnargenox:  Nice job with the method for figuring it out.  Basically we need to map out the gravitational potential energy between Earth and Saturn (most of it is due to the Sun, and then there's a quick downturn very close to Saturn).  Then apply the Schrödinger equation.

Image
That's a pretty freaking huge hill to cross, and we're trying to tackle it with... about half a kilojoule.  The Little Engine That Could.  But the wave function does goes through, amplitude dying exponentially as it does.  

It's difficult to solve the Schrödinger equation with this function for potential, but it helps to make a simplifying approximation that it is a rectangle, with an average height of about 660 Megajoules per kilogram, and width of ~9AU.  Then using the wide barrier approximation, the tunneling probability is

Image

All the stuff in front of the exponential part is pretty much irrelevant.  We could multiply it by a googol and it wouldn't change the answer.  The exponential is huge(ly small), about e^-10^52.
 
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Gnargenox
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15 Mar 2017 00:25

Thank you, I love your cleaned up formula! I can even see where the 16 comes from lol. I assume U means constant velocity, or acceleration independent of time? When you say things like 10^10^52, I only think of fertilization rates for lawn care lol. Luckily I had this book by Cook laying around on the bookshelf. Sounds like the chances of quantum tunneling to Saturn are the same as picking a random atom in the entire universe and you picking the same one.
Green - 01.jpg

Anyway, I was zooming around the universe earlier today and wished I could find more Green Stars... I think I understand why I won't see any, even though our own sun is considered to emit mostly green wave lengths of visible light. Basically since all stars emit light in all wave lengths the color green is always drowned out by the other ends of the spectrum creating a "white" looking star. But then I wondered how could I see other objects that appear green to me without them also being washed out and appearing white? Since color is determined by temperature and brightness I guess the other objects in space that are green must be so dim that our eyes are not overwhelmed by all the other wavelengths of light. Here are a few snap shots of Green stars I have seen due to optical illusions, some nebula that are portrayed as green and some naturally appearing green objects in space.
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Watsisname
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15 Mar 2017 01:42

Gnargenox wrote:
Source of the post I assume U means constant velocity, or acceleration independent of time?

Nah, U is the potential energy of the barrier (660MJ in this case... and don't ask me why physicists use U for potential energy).  It's also the same thing as the V0 in the formulae you showed -- you can think of it as V for voltage, since usually in the context of electron tunneling it's with an electrostatic barrier.

The velocity of the particle is considered in the "E", for kinetic energy.  In this case it ends up being pretty much irrelevant -- it's the height (U), width (L), and your mass (m) that make the probability so small for tunneling to Saturn.  (I hope I didn't just call you fat.)

Comparing to the probability of picking the same atom out of all atoms in the observable universe is fun.  There are about 10^80 of them.  If I pick one at random, the odds of you picking the same atom is 1 in 10^80.  Ok, that's not so impressive after all.  The first picker hasn't changed the odds.  But what if more people pick atoms at random?  

If three people do it then the odds that all three pick the same atom are 1 in (10^80)^2.  The extra exponent is simply the number of people who pick, minus 1.  

What if every person on Earth picks an atom out of the universe at random?  Then the odds that every person picks the same atom is still better than the odds of tunneling to Saturn!  (It's 1 in (10^80)^(7.5x10^9), which is about 10^-10^12).  

Long story short, the probability for tunneling to another planet is as ridiculous as the idea.   And now I've talked enough about quantum mechanics; someone else should talk about green things. :)
 
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Gnargenox
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15 Mar 2017 02:48

oh, duh, (U = potential energy). Yes, picking the same atom certainly puts tunneling into perspective!
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16 Mar 2017 07:31

How large would a black hole need to be to eat Earth if it was at one meter above the surface of Earth?
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Hornblower
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16 Mar 2017 10:04

Xoran, well how long to you want it to take? Ignoring Hawking radiation, a black hole of any size can be used.
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Xoran
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16 Mar 2017 10:14

Hornblower wrote:
Source of the post Xoran, well how long to you want it to take? Ignoring Hawking radiation, a black hole of any size can be used.

The question includes hawking radiation :)
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Watsisname
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18 Mar 2017 02:52

I've been thinking about this and working some stuff out, and hopefully should have a solution before too long. :)
 
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Watsisname
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19 Mar 2017 21:56

Okay, here's what I've come up with.  Probably the best answer for how large a black hole would need to be (if it formed 1m above the surface) to eat the Earth is a bit smaller than the size of an atom.  Perhaps this doesn't seem too surprising, but the reasons why this is the best answer may not be what you expect.  

So I'll detail here my approach, and the insight that came out of it.


Part I: Making it to Ground

Wherein a black hole is born and takes up skydiving... screaming the whole way.

► Show Spoiler


Part II: Overcoming the Eddington Limit

Wherein the black hole learns to eat properly, and thus began the age of terror.

► Show Spoiler



Part III: To Swallow a Planet

"Now I am become death, destroyer of worlds."

► Show Spoiler
 
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DoctorOfSpace
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19 Mar 2017 22:01

Watsisname wrote:
Source of the post Part II: Overcoming the Eddington Limit

This was the bit I knew about, but since you seem to be the resident physicist it seems best to just let you run away with these answers.
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