Burning Questions
Here is a conversation I recently had with noted physicist Casey "Morrison" Davis of UC Davis, known to undergrads of all departments as "the druid" and to the livejournasphere as barnabas-truman; rumored to be a wizard whose bushy brown beard is considered an omen of good fortune, whose very presence is believed to cause bystanders to understand math and physics better; whose pennywhistle music on the quad can tame wild beasts and even Calpirgers. for the sake of imagining this conversation, instead of picturing us at our computers talking on facebook, as actually happened, why don't you go ahead and picture us wearing bowler hats sitting in a victorian parlor. I do in fact currently have a handlebar mustache, which will fit in with this setting nicely.
Me: Casey dear chap, can light itself orbit a black hole?
Casey J Morrison: Hm. I think it could technically be possible at the event horizon.
(Event horizon being a specific radial distance outwards from the center of the black hole itself)
Are you familiar with the concept of escape velocity?
Me: yes. the speed needed to break out of a gravity well yes?
Casey J Morrison: Yes.
Specifically the speed needed to just keep on *coasting* outwards forever without ever falling back "down," without any need for further thrust.
Escape velocity depends on both the mass of the star (or planet or whatever) *and* how far you are from it; e.g. the escape velocity from the top of a mountain is going to be slightly less than the escape velocity at sea level.
Likewise if you're already in high orbit, escape velocity's not going to be very large.
If a star shrinks in size while remaining the same mass, its *surface* escape velocity increases, because its surface is now closer to its center of mass.
A black hole is a star that has shrunk enough that its surface escape velocity is greater than the speed of light, so even light can't escape it.
Casey J Morrison: The distance from the center out to the location where escape velocity is exactly the speed of light is called the "Schwarzchild radius"; this radius depends only on the mass of the star, and a black hole can be defined simply as an object that fits entirely inside its own Schwarzchild radius.
The region of space exactly one Schwarzchild radius away from the black hole's center is a thin spherical shell known as the event horizon.
Anything inside the event horizon *will* fall into the center, even if it's going as fast as light.
Anything outside the event horizon could escape if it's going fast enough.
On the other hand, you don't have to be going at escape velocity to *orbit* a planet or star or whatever.
In fact it's entirely possible for, say, a planet to orbit a black hole. If the sun collapsed into a black hole right now, Earth and etc. would continue orbiting exactly as they do now, because the sun's mass hasn't changed, and the planets are all still as far away from it as they were before.
I suppose light could potentially orbit *anything*; the only relevant property of light in this case is its speed.
For a given mass of the object you're orbiting (star or planet or whatever), your speed and your orbital radius are related to each other, so each orbital speed would be tied to a specific orbital radius. If we use the speed of light in that equation, we should be able to calculate at what radius light should orbit.
I think I happen to have a spreadsheet for calculating this very thing that I was using for Kerbal a while back...
Me: so light can orbit something outside the schwarzchildren radius?
and/or a planet can be rotating a black hole within the radius and life could go on on it?
Casey J Morrison:What I'd have to do is compare the formula for Schwarzchild radius and the formula for how fast you must be going to orbit at a given radius.
Me: I see I see
Casey J Morrison:Okay, found the formulas.
Here we go:
The Schwarschild radius is
R = 2 * G * M / c^2
where G is the universal gravitational constant, M is the mass of the star, and c is the speed of light
Me: this is a fascinating idea though, if a planet can continue to orbit within a black hole and things can remain relatively normal on the surface
Casey J Morrison: The speed you have to be going in order to orbit something (say, a star) in a circular orbit is
v = √ ( G * M / r )
where G is the universal gravitational constant, M is the mass of the thing you're orbiting, and r is the orbital radius (i.e. distance from center of star to your location)
::the butler comes in with a plate of scones and stares aghast at Casey, who is drawing a diagram on the wall with a piece of cheese::
So I think what we can do is substitute the Schwarzchild radius R in place of r in the second equation.
That should tell us how fast you have to be going in order to orbit along the event horizon itself. If you want to orbit *within* the event horizon you'd have to go faster.
So!
v = √ ( G * M / R )
becomes
v = √ ( G * M / ( 2 * G * M / c^2 ) )
The G's and M's cancel out, leaving us with
v = √ ( 1 / (2/c^2) )
This simplifies to
v = c * √ (1/2)
or about 0.707c
In other words, if you're *at* the Schwarzchild radius, you should be able to orbit if you can travel at 71% of the speed of light.
In order to orbit while *inside* the Schwarschild radius, you'd have to go even faster.
We could also find a formula for an even smaller radius where the necessary orbital velocity would be c; inside that radius, light not only couldn't escape--it couldn't even orbit.
Me: interesting
I had been under the vague impression one would get crushed if one got anywhere near the event horizon but if one could somehow slingshot oneself off to the side and go into orbit one might theoretically survive into orbit (assuming one had the other necessary elements of space survival and such)?
Casey J Morrison: The "crushing" part is actually due to something different called tidal forces. It's not just a property of a black hole; it will occur to a large enough object that is very close to any very massive planet/star/whatever, but since black holes are so compact the effect is much more dramatic there.
Consider, for instance, Jupiter's moon Io.
Jupiter is very massive, and Io orbits pretty close to it.
Usually we just treat stuff like moons and planets as single points, because the radius of each object is very small compared to the distance between the objects, but in cases like this the distance between the objects is small enough that the radius of the object becomes relevant.
Me: is Io's size in relation to jupiter greater than the moon in relation to us? or perhaps its just much closer?
this is a fascinating conversation, I may have to post it on livejournal ;D
Casey J Morrison: Io's distance from the center of Jupiter is pretty similar to our moon's distance from the center of Earth, but Jupiter is so much more massive that the effects are significantly amplified.
Io is also pretty close in size to our own moon.
So really the only difference here is that Jupiter is way more massive than Earth, and thus Io is experience much stronger gravity than Luna is.
Me: ::nodnod::
Casey J Morrison: Specifically about 318 times stronger. Yikes.
Anyway, the thing is, treating Io as a single point is a reasonable approximation, but isn't entirely accurate.
The critical difference is that the side of Io facing *towards* Jupiter is experiencing a stronger gravitational pull than the side of Io facing *away* from Jupiter, because it's closer!
(This is also true of Earth's moon, of course; Io just experiences a more dramatic difference in forces because Jupiter is 318 times more massive.)
Me: are you about to use the phrase "hill sphere" ? ::asks excitedly::
Casey J Morrison: Er, wasn't planning to?
Oh, is this like the sphere of influence of the body?
Me: this is the hill sphere of which I speak: http://en.wikipedia.org/wiki/Hill_sphere
"An astronomical body's Hill sphere is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sphere of its own. Any object within that distance would tend to…"
Also, is the reason most moons rotate with one face facing their mother planet related to these tidal forces of which you speak?
Casey J Morrison: Related, yes.
That's known as "tidal locking" because it's caused by exactly the same thing.
Anyway, this imbalance in gravitational forces leads to a continual strain on Io's interior. Analogously, imagine two people pulling a box in the same direction, but one of them is pulling much harder than the other. The box gets a lot of strain.
In the case of Io, the tidal strain leads to enormous amounts of internal heat and pressure, which is suspected to be the reason why Io's surface is the most volcanic of any moon or planet we know of.
Me: I see I see ::twists left end of mustache thoughtfully::
Casey J Morrison: The reason it's all called "tidal" whatever is that it's exactly what causes ocean tides here on Earth. Water on the side of the Earth currently facing the moon experiences a slightly stronger pull from the moon's gravity; water on the side facing away from the moon experiences slightly weaker pull.
On any individual small object, this difference is negligible, but for something as massive and sizeable as the ocean, it's significant.
Me: hmmm. But it seems to me it just pulls the other object towards it, same as gravity normally works, so in the case of a black hole it would pull you towards the black hole and therefore not necessary crush you as long as you're in the very act of falling towards it at full velocity?
Casey J Morrison: See, with black holes, the problem is that it's *so* massive that the tidal strain is hugely amplified.
Right here on Earth, if you're simply standing upright on the surface, your feet are experiencing a stronger gravitational pull than your head is, because they're closer to the center of the planet.
But the percentage difference between [distance from Earth's center to your feet] and [distance from Earth's center to your head] is very tiny, and Earth's gravity isn't very strong anyway.
Me: ah so the differential in pull between the end of you or your spaceship that's closer to the center of the black hole and the other end will pull it apart
Casey J Morrison: Exactly.
And that's entirely because the black hole's gravity is so strong.
Me: I see. So how do we overcome this Casey?
Casey J Morrison: Make the spaceship really really small.
Me: so this hypothetical planet orbiting within the event horizon would have been destroyed by tidal forces before the mother star ever went black hole, since its mass would be the same, yes?
Casey J Morrison: The thing is, if the star isn't a black hole yet, the planet can't be orbiting within the star's event horizon, because the event horizon is inside the star itself.
If the event horizon is outside the star, then the star fits inside its own Schwarzchild radius, and it's already a black hole.
Me: ah yes. so this hypothetical planet couldn't be rotating within the event horizon because before it was a black hole that space was taken up by the star itself.
what causes a star to collapse anyway?
Casey J Morrison: For instance, the Schwarzchild radius of our own sun is about 3 kilometers.
Me: ::gasp:: we're orbiting a black hole!!
::runs out of the house screaming::
Casey J Morrison: The simple version is "stars collapse when they use up their fuel."
Me: yeah but what does that mean. the atoms have fissioned into things that will no longer combust? but does that reduce the volume? it seems like it would increase it?
Casey J Morrison: The more complicated version is that the size of any star is due to the equilibrium of gravity (trying to pull the matter inwards) versus pressure (trying to push the matter outwards).
A typical star like our own sun, for instance, is made mostly of hydrogen.
This really massive clump of hydrogen is pulled inwards by its own gravity.
But the smaller it gets, the higher the pressure becomes, and pressure pushes outwards.
If these were the only factors involved, this colossal hydrogen gas cloud would find a stable equilibrium at the volume where gravity's pull and pressure's push balance each other out.
However, if the hydrogen cloud is massive enough (e.g. the sun), the interior hydrogen atoms are mashed so close together that they can undergo fusion--hydrogen atoms smash together and form helium.
This releases some energy, because one helium atom has a lower total energy level than two hydrogen atoms.
As this happens over and over again, the released energy raises the temperature of the star, which increases the pressure.
Thus the actual size of the star is the equilibrium volume that allows balance between gravity pulling inwards and pressure (including extra pressure due to fusion) to balance each other.
Any questions so far?
Me: ah no I think I follow. so the fusion is a necessary part of maintaining the current size
and without that fusion it would just be a massive gas giant planet yes?
Casey J Morrison: Exactly. That's the deal with Jupiter; it's mostly hydrogen and if it were much more massive it would have had enough gravity to initiate fusion and become a star.
Of course a star doesn't have an unlimited amount of hydrogen.
When a significant percentage (not sure how much, but a lot) of the hydrogen has been "used up"--that is, converted to helium--there is no longer enough fusion going on to generate as much heat as there was before.
Without the extra heat, we're now looking at just gravity and pressure, so the new equilibrium volume would be smaller. The star therefore collapses to its new "right size."
However, by collapsing even smaller, the pressure at the core is now stronger than it was before, and in fact may be strong enough to start mashing the helium atoms close enough together that *they* start fusion.
So the star continues to shine, though I think it would be emitting less heat and maybe different wavelengths of light, because helium fusion is going to involve different energy levels than hydrogen fusion.
This sort of collapse would presumably then happen again when it runs out of helium.
Me: and it keeps getting reduced to heavier elements?
Casey J Morrison: Yup.
Me: I see I see ::twists the right end of mustache thoughtfully::
Casey J Morrison: Up until iron, anyway; once you get past iron, fusion would be losing energy instead of gaining energy.
I forget where elements heavier than iron come from; I think it has something to do with the even higher compression in supernovas or something like that.
But this is why there's so much iron everywhere--very light elements tend to undergo fusion to become more stable heavier elements; very heavy elements tend to undergo fission to become more stable lighter elements; iron is the most stable equilibrium in the middle.
Me: which now has me wondering how it is that some planets like Jupiter get away with being mostly hydrogen
Casey J Morrison: Fortunately for us, iron is pretty useful for building stuff!
Well, the universe seems to have started out as all hydrogen (or maybe some helium? I forget), so even with all the fusion that's happened in stars, there's still plenty of hydrogen left.
Me: why didn't the iron and hydrogen and everything else in our system get more equally distributed
and are you saying the solid planets have all been chewed up and spit out by stars?
Casey J Morrison: Pretty much, or at least they're made from material that has been chewed up and spit out by stars.
The general picture I've got is something like this: old star goes through the fusion -> collapse -> less efficient fusion -> more collapse -> etc. cycle for a while, building up a mixture of hydrogen, helium, and also heavier elements, and eventually explodes when it's spinning too fast... oh wait, I didn't tell you about spinning, did I?
Me: noe?
Casey J Morrison: Suppose you're on skates or whatever and spinning, and holding heavy weights out at arm's length. What happens if you pull the weights in close to your chest?
Me: hmmm you become more likely to lose your balance and go sprawling?
Casey J Morrison Har har yeah.
Me: so the whole mess accelerates in rotation as it collapses?
Casey J Morrison: Exactly.
Moment of inertia (a description of both mass and how it's arranged) decreases, so rotational velocity must increase to conserve rotational momentum (rotational velocity times moment of inertia). ::Casey makes a diagram in the air with his finger, it hangs there glowing warmly::
As it spins faster and faster, the heavier elements are forced to the outside and eventually flung out.
Me: also so there could be some really massive conglomerations of iron that have star like mass but lack the ability to conduct fusion?
Casey J Morrison: I think if it gets that massive the gravity at the core is so strong that the atoms don't just fuse, they shatter, and you're left with just a soup of protons, electrons, and neutrons.
Electrical forces push out the protons and electrons, and you're left with just an extremely dense core of pure neutrons--a neutron star.
Me: oh interesting. So it's flinging iron out as it goes. Could this cause the whole thing to come apart (lose the mass necessary to conduct fusion) before it ever does anything dramatic like become a black hole?
Casey J Morrison: I'm not entirely sure of the details, but I think what happens (or at least one possible result) is that the heavier elements get flung outward but maybe there's enough lighter stuff (hydrogen and helium) still remaining at the center that the core can reach a new stable equilibrium and begin hydrogen fusion again.
The result: a standard sun-like star in the middle, and a disc of the shattered heavier elements orbiting around it.
If there happens to be a clump of heavier elements (iron or whatever) somewhere in this disc that's denser than its surroundings, its gravity will start pulling the surrounding material towards it, and over long periods of time this causes most of the disc's material to form into planets orbiting the central star.
This is how solar systems form.
Me: ::gasp::
Casey J Morrison: You can see a really nice simulation of this here:
http://www.nowykurier.com/toys/gravity/gravity.html
(press "generate proto disk" in the lower left corner)
Me: hmmm my Internet is down I'll have to try creating a solar system later
Casey J Morrison: When you can access it, try it out; it's a pretty fun physics simulation toy.
Me: will do. in the mean time, good night! thanks for the extensive explanations!
Casey J Morrison: You're welcome! I enjoy discussing such matters; it's one of my favorite topics and a good way to make sure I understand as much of it as I think I do.
Goodnight!
And with that that Casey disappeared in a puff of smoke, leaving me sipping tea and gazing out the window thoughtfully.
And here is a picture that is only related inasmuch as it's of the sun, since I don't have any pictures of any black holes at the moment. (: