> Dear Dr. Schneertz, > > I had a question about black holes. If you were to compress the mass of a > human into a singularity, it would have a Schwartzchild radius, albeit > really small, right? What would happen if something, like a finger, came > near it's event horizon? > > || Bill Wendling wendling@ncsa.uiuc.edu Dear Mr. Wendling, Interesting question ... Hmmm.... Here's what I would envision: Schwarzschild radius of a human being (say 10^5 g) ~ 1.5 x 10^-23 cm (!) This is much smaller than the dimensions of an atom, even the dimensions of a nucleus. Can the gravity of such a black hole rip apart a typical nucleus before the nucleus enters the black hole? Typical binding energies per nucleon (CNO group elements) are ~8 MeV/nucleon (see Fig. 10.9), which implies nuclear forces of order F = 8 MeV/fermi = 1.3 x 10^8 dynes. This is the (classical) strength of the gravitational field of a human black hole at a distance of r = (GM/F)^(1/2) = 7 x 10^-6 cm, which is actually ~10^2 *atomic* radii. So, yes, it could rip apart nearby atoms and even nuclei. Bad news for the finger! The human black hole begins gobbling up this matter. The matter is heated by compression as it flows in toward the black hole, and some fraction of its rest energy is radiated away. Estimates I'm familiar with for non-rotating black holes put this fraction at ~8%. The energy radiated away will be a mixture of photons and other particles created in matter-antimatter pairs. (The antimatter particles quickly annihilate with the matter particles, however, to create more photons.) The rate at which the human black hole eats fingers will be limited by the radiation pressure of the outflowing photons. This is the Eddington limit argument. Classically, that limit (see p. 463) is (eq. 12.19) L_Ed = 1.3 x 10^5 ergs/g (for CNO composition), meaning our human black hole would radiate L_Ed = 1.3 x 10^10 ergs/s = 1300 kW of power, while consuming the finger at a rate dM/dt = L_Ed/(0.08c^2) = 1.8 x 10^-10 g/s. Actually, this is probably a serious underestimate of L_Ed and dM/dt, because the black hole is so small that the Planck peak of the photon distribution lies in the ~50 GeV range -- very hard gamma-rays, and at these energies the absorption/scattering cross-sections of electrons are much much smaller than in the classical (Thomson-scattering) limit. So the bottom line, as I see it, is this: The black hole shreds the tip of the finger, and, in eating just a tiny fraction of it, becomes such a powerful gamma-ray source that it (a) vaporizes the tip of the finger, (b) cooks the rest of it, and (c) kills the person to whom the finger belongs through radiation poisoning! Have a nice day! Dr. Schneertz |
Dear Mr. Wendling, It occurred to me on the way home Friday that I had made another error in my response to your "human black hole" question. (Well, nobody's perfect!) In comparing nuclear with gravitational forces, one should really compare the nuclear force with the *difference* in gravitational forces over a nuclear distance. Furthermore, I inadvertently compared gravitational acceleration to nuclear force. Starting over, then, one has F_nuc ~ E_bind/del r = 1.3 x 10^8 dynes as estimated before, but del F_grav = 2 (GMm/r^3) del r , where M in the mass of the black hole, m the mass of a nucleon, r the distance to the black hole, and del r the nuclear dimension (1 fermi = 10^-13 cm). Then the distance at which gravity shreds nuclei is r_shred (nuclei) ~ (2GMm del r/F_nuc)^(1/3) = (2GMm (del r)^2/E_bind)^(1/3) = 2.6 x 10^-16 cm. This is obviously a lot smaller than my previous estimate -- much smaller than typical nuclear dimensions. This means that nuclei will NOT be shredded, but instead the black hole will "infiltrate" individual nuclei, and eat them out from the inside. If I repeat the above exercise, but for electrons orbiting in atoms, then m is the electron mass (factor of 2000 smaller), del r the typical atomic dimension (~1 angstrom = 10^-10 cm = 10^3 fermi), and E_bind is smaller by a factor of order 10^6. This makes the shredding radius r_shred (atoms) ~ 2 x 10^-13 cm, that is, smaller than atomic dimensions. (All of this confirms that gravity is a much weaker force than nuclear forces or electromagnetism.) The black hole is therefore unable to shred atoms either. All of this changes the complexion of the problem substantially.... If the black hole is unable to shred atoms, much less nuclei, then I see events unfolding differently: The black hole gravity pulls in the nearest nucleus (nothing prevents this), and begins eating nucleons once it's inside the nucleus. It can eat neutrons at will; each time it eats a proton, it acquires a positive charge, and so quickly eats an electron for dessert! However, since the gravitational force is not strong enough to mash nearby atoms, the net rate at which the black hole eats matter is limited by the rate at which the void it created by eating an atom can be refilled from the surrounding sea of atoms. I'll have to return to this later, for lack of time at present, but the rough estimates I can make quickly (if correct) indicate that this process of refilling the void is going to "starve" the black hole -- i.e., feed it mass at a rate much smaller than the Eddington-limited rate I estimated last time. If that's the case, the power output from the black hole is going to be much smaller than previously estimated. Dr. Schneertz |
Dear Mr. Wendling, Back to your "human black hole"... It occurred to me getting ready for today's lecture that I should have checked out the rate of Hawking radiation from this hypothetical black hole. It turns out that this completely dominates everthing else: The black hole radiates a luminosity L = (h c^6)/(30720 pi^2 G^2 M^2) at effective temperature T = (h c^3)/(16 pi^2 G M k) giving an evaporation time scale [with L = -(dM/dt)c^2) ] t = (M c^2)/(3 L). Plugging in numbers for a 10^5 g black hole, one finds: L = 3.56 x 10^35 ergs/s (about 100 L_sun!) T = 1.227 x 10^21 K and t = 8.41 x 10^-11 s . In short, such a black hole would be the end of civilization as we know it, never mind poking it with your finger! Third time's a charm... Have a nice day. Dr. Schneertz |