I will either sleep as sound as a really, really heavy rock or hit the ceiling jumping out of a startling dream tonight!
Thank you!! AMAZING stuff every time.....
Now I'm starting to think Dark Matter really isn't made up of primordial multi-ton microscopic black holes.

So, when I said I'll leave the question of the time it takes for the trillion-ton, atom-sized black hole to consume the Earth for another time...?  

Yeah, it's about a billion years.  Lol, I thought it would take a long time, but not THAT long.  The Eddington Limit really ...er... limits it, as it can only start out eating about a kilogram per second, and it pretty much stays that way until near the end.

The actual solution for how the black hole grows with time is an exponential of the form

where  

For this case, plotting the Earth's remaining mass over time gives a curve like this, where the y-axis is in kilograms and the x-axis is time in millions of years.



Making the black hole more massive doesn't change the timescale a whole lot.  Even if we made the black hole the same mass as Earth, it still takes about 30 million years to truly consume Earth, though at least it has a more steady effect:

Now I'm starting to think Dark Matter really isn't made up of primordial multi-ton microscopic black holes.

There could be some primordial microscopic black holes out there, and actually there are some researchers following the idea that it could explain much of the dark matter.  If so they must have a particular distribution of masses to be consistent with observations (or lack thereof). I think more astronomers are in the "new fundamental particle" mode, but either way the models can only benefit from more data at this point.

Even if we made the black hole the same mass as Earth, it still takes about 30 million years to truly consume Earth

At what point does Earth stop being capable of supporting life?  That seems to be the more interesting aspect of this and probably the hardest one to answer.

At what point does Earth stop being capable of supporting life?  That seems to be the more interesting aspect of this and probably the hardest one to answer.

That depends on what you mean with "life", because it will be uninhabitable to humans long before it is uninhabitable to bacteria.

How big would our black hole need to be to consume Earth in 10 minutes or less? And/or one minute or less?

How big would our black hole need to be to consume Earth in 10 minutes or less? And/or one minute or less?

Not exactly what you asked but :

 [youtube]8nHBGFKLHZQ[/youtube]

I wonder how people would react to a black hole suddenly appearing and beginning to slowly devour Earth. Hopefully nobody would try to jump into it.

I wish I had more time to read this, but I was doing a quick search of Stars made out of Water, and stumbled across this study of a protostar in the constellation of Perseus inside the molecular cloud NGC 1333. Apparently the young star is squirting out atomic oxygen and creating a disc of water around it. Simply mind boggling! I'm sure Space Engine will have sprinkler system stars in a future version.

by Lars Kristensen
http://www.aanda.org/articles/aa/full_h ... 44-13.html

Another one is TW Hydrae (fitting name hehe)
http://www.aanda.org/articles/aa/abs/20 ... 72-16.html

something related to NGC 1333
http://iopscience.iop.org/article/10.10 ... /1244/meta

You might find that interesting...

I wonder how people would react to a black hole suddenly appearing and beginning to slowly devour Earth.

Panic. Lots and lots of panic.

I wonder how people would react to a black hole suddenly appearing and beginning to slowly devour Earth. Hopefully nobody would try to jump into it.

I can quite easily imagine small fanatical black-hole-worshipping suicide cults. Heck, I still remember hearing about that cult suicide when Hale-Bopp came through in the mid-90s.

At what point does Earth stop being capable of supporting life?  That seems to be the more interesting aspect of this and probably the hardest one to answer.

I think it would be very fast -- as in a timescale of minutes to hours (for the Earth-mass black hole).

While it would be tempting to look at the gravitational effects (which are impressive... for example 1000g of gravity at 200km), the bigger problem would be the brightness of the accretion disk.  It would be gobbling up several million tons of material every second, and the the intensity of the radiation coming off of it would be comparable to a solar flare... going off continuously right here on and within the planet.  As a number, we're looking at about 10²⁵ watts, which is enough power to melt the entire Earth in less than a day.

How big would our black hole need to be to consume Earth in 10 minutes or less? And/or one minute or less?

This one is surprisingly harder to answer.  The main problem is getting around having the material form an accretion disk.  Because the Earth rotates, material falling into the hole will have some angular momentum, and this is enough to reach orbital velocity before plunging through the event horizon if the hole is not big enough.  (If that happens the material will still be drawn into the hole, but at a much slower rate governed by the Eddington Limit.)  By my calculations I figure a black hole of at least a few tens or hundreds of meters across should be big enough such that everything can plunge straight in.  This corresponds to a black hole with a few percent the mass of the Sun!

However, simply falling straight in isn't good enough.  Everything is getting rapidly compressed to a common and very small center, so the density and pressure will increase tremendously.  Things will heat up to incredible temperatures before they can fall through the horizon, even if they lack the angular momentum to form a disk, and this will almost certainly limit the hole's growth rate.  So the black hole needs to be even bigger to get around that limitation.  Exactly how much bigger is hard to say, and at this point I'm basically just guessing. Certainly it does not need to be a significant fraction of the size of Earth, but perhaps approaching a few percent of it, which would correspond to ~100 solar masses.

I think it's worth showing just how dramatically the Eddington Limit affects a black hole's growth rate using this Earth-devouring example.  If the material forms an accretion disk, then the size of black hole needed to eat one Earth mass in one second is truly remarkable.  When we look at distant quasars, we're seeing black holes that may weigh up to a billion solar masses, with accretion disks shining at around 10⁴⁰ watts (brighter than the entire rest of the galaxy they reside in)!  Yet the actual growth rate of the black hole is only a few solar masses per year, which is less than one Earth mass per second.  Very slow growth considering their immense size.

which is enough power to melt the entire Earth in less than a day.

Well that is depressing and awesome

Well that is depressing and awesome

Mostly awesome

I think I found an exoplanet! I need some help with calculating the orbital radius and mass of the planet. Here's the data:
Size: 2.6 Earth Radii
Orbital Period: 71.22 days
Star: KIC 7801296
I realize it might not be possible to calculate the mass, but atleast a range of values for the orbital radius would be great! Thanks in advance, 
(Also, let me know if you need to know anything else about the planet, discovered by transit)

I think I found an exoplanet! I need some help with calculating the orbital radius and mass of the planet. Here's the data:
Size: 2.6 Earth Radii
Orbital Period: 71.22 days
Star: KIC 7801296
I realize it might not be possible to calculate the mass, but atleast a range of values for the orbital radius would be great! Thanks in advance, 
(Also, let me know if you need to know anything else about the planet, discovered by transit)

How did you manage that? Backyard telescope or something bigger / more sophisticated?