# Minimum mass of a black hole

1. Dec 9, 2004

### DB

Knowing that the escape velocity of the Sun is 617 km/s and that it's mass is 1.989 e30 kg, and the speed of light: 299 792 km/s, tell me if this calculation is right. I'm trying to calculate the minimum mass of a black hole. And we know that light cannot escape a black hole so...
Comparing to the sun:

299 792/617 km/s = approx. 485.88, so now if a mass of 1.989 e30 kg has an escape velocity of 617 km/s then:

1.989 e30 kg x 485.88 = 9.66 e32 kg (a pretty much exact mass of an object that can barely emit light.)

So this would mean that a black hole can have a minimum mass of about
1 e33 kg???

100000000000000000000000000000000 kg is some heavy stuff lol.

So are my calculations right?

Thnx

Last edited: Dec 9, 2004
2. Dec 9, 2004

### krusty the clown

I was under the impression that black holes had to exceed a certain density, not mass. I am not sure of the exact reasons but I am sure someone else will have a better explination.

3. Dec 9, 2004

### The Bob

Balck holes are singularities. They are very dense and will not allow anything to esape them as their gravity it so large. I was under the impression (because of all of the talk about time travel) that it was simply like a hole in space-time or a continous curving of it.

About the mass though I am not sure if it actually has any or not. I believe it must do but if it did then it would get bigger and the universe would be getting emptier and emptier. I don't know.

4. Dec 9, 2004

### The Bob

I also don't know where the 485.88 has come from and also how you made your assumption on the minimum mass.

5. Dec 9, 2004

### DB

485.88 is how many times more the escape velocity of an object of a mass like the sun would have to be to have an escape velocity of the speed of light 299792 km/s

6. Dec 10, 2004

### Orion1

concept conceptualism...

DB, I believe that you are attempting to conceptualize a BH model based upon a quantum integer mass solar model:

$$n_1 = \frac{c}{v_{\odot}} = \sqrt{ \frac{r_{\odot}}{2GM_{\odot}}} c$$
$$n_1 = \sqrt{ \frac{r_{\odot}}{2GM_{\odot}}} c$$
$$n_1 = 485.436$$
$$v_{\odot} = 6.176*10^5 m*s^{-1}$$ - Solar escape velocity

Solving for M,n for 'constant' solar radius:
$$M_{BH} = n_1^2 M_{\odot} = \frac{r_{\odot}c^2}{2G}$$
$$\boxed{M_{BH} = 4.687*10^{35} kg}$$

Last edited: Dec 10, 2004
7. Dec 10, 2004

### Chronos

The current theoretical limit for minimum size of a black hole is the plank mass - about 21.77 micrograms. That yields a schwarzchild radius of a planck length. Why would you think minimum black hole mass has anything to do with a solar mass? In your calculations, you treat the sun as stationary. In the presence of a mass as imposing as that you propose, the sun would have about as much chance of winning that gravitational tug-o-war as a flea against a locomotive.

8. Apr 14, 2008

### Disgod

I think the proper question that needs to be asked is: What is the theoretical minimum size that a black hole can be that can sustain itself?

The problem is Hawking Radiation, the smaller the black hole the more Hawking Radiation it releases, and the quicker the black hole dissipates. So, at some point the black hole is going to be releasing so much energy out that it will collapse, explode, dance a merry jig (I'm not entirely sure about what will happen). I heard once that the theoretical minimum is somewhere around the mass of Mount Everest, but I could be wrong.

9. Apr 14, 2008

### Oberst Villa

I would think the minimum mass for which a black hole can sustain itself would depend on its environment.

if you consider a black hole in a theoretical universe that is really completely "empty" (no radiation, no matter), then it would continously loose energy (and therefore mass) by the Hawking Radiation you described. for a big black hole this might take a very long time, but in the end it would vanish.

on the other hand, in our "real" universe, we always have the cosmic background radiation (2.7K I think). so if the black hole is so big that the temperature of its Hawking Radiation is colder than 2.7K, the black hole would absorb more radiation than it emits, gaining energy/mass and growing bigger.

furthermore, our "real" universe is not completely empty of matter but contains some "dust". so the black hole might also survive if it could suck in this cosmic dust faster than it looses energy/mass due to Hawking Radiation, even if its so small that Hawking Radiation is hotter than 2.7 K

10. Apr 14, 2008

### Disgod

Oberst Villa

You are right about the Cosmic Background Radiation being too hot for most black holes to evaporate due to Hawking Radiation, but there might be black holes that are small enough that they have heated up beyond 2.7k, so called primordial black holes which were created right after the big bang that are small enough that they do emit more energy than they absorb and could evaporate within the current lifetime of the universe. They are theoretical at this point, but, even if they do not exist, eventually the CBR will drop to a point below the temperature of all black holes and they will all start evaporating. It will just take a really, really long time before any black hole will be small enough to worry about reaching the evaporation point.

11. Apr 14, 2008

### kamerling

This would only be valid for a black hole with a radius equal to that of the sun. If you would compress the sun, it's escape velocity would become bigger.

12. Apr 15, 2008

### Oberst Villa

when I was writing about the CBR temperature I thought of it as constant, but certainly on a cosmological timescale it will decrease - stupid me ! thanks for the clarification, Disgod.

13. Apr 15, 2008

### Oberst Villa

a principal problem with this approach is, that you try to understand a black hole in the framework of Newtonian Mechanics, treating light as an ordinary particle and comparing its speed to a bodies escape velocity.

but when we talk about velocities that approach the speed of light we have to use Relativistic calculations. in this case, because we are talking about a gravity field, those of the General Theory of Relativity. unfortunately, doing this calculations is way beyond my own (current) capabilities, but I just wanted to "warn" you.

14. May 7, 2008

### Orion1

$$T_s = \frac{\hbar c^3}{8 \pi G M_s k_b}$$
$$k_b$$ - Boltzmann constant

Cosmic microwave background radiation black body spectrum temperature:
$$T_u = 2.725 \; \text{K}$$ - Universe temperature

Black Hole black-body radiation temperature equivalent to Cosmic microwave background radiation temperature:
$$\boxed{T_s = T_u}$$

Minimum mass black hole that can existentially exist naturally in known Universe in absentia of quantum black holes:
$$\boxed{M_s = \frac{\hbar c^3}{8 \pi G k_b T_{u}}}$$

Theoretical minimum black hole mass:
$$\boxed{M_s = 4.502 \cdot 10^{22} \; \text{kg}}$$

$$r_s = \frac{2G M_s}{c^2}$$
$$\boxed{r_s = 6.687 \cdot 10^{-5} \; \text{m}}$$

Reference:
Cosmic microwave background radiation - Wikipedia
Boltzmann constant - Wikipedia

Last edited: May 7, 2008
15. May 7, 2008

### Orion1

Integration by substitution:
$$M_s = \frac{r_s c^2}{2G} = \frac{\hbar c^3}{8 \pi G k_b T_u}$$

$$\boxed{r_s = \frac{\hbar c}{4 \pi k_b T_u}}$$

$$\boxed{r_s = 6.687 \cdot 10^{-5} \; \text{m}}$$

Reference:

Last edited: May 7, 2008
16. May 9, 2008

### rubecuber

i think that Krusty the clown isright, but id also look up on google: chandransaker limit. That qwill give you your answer!

17. May 10, 2008

### Orion1

Chandrasekhar limit maximum mass:
$$M_c = \frac{\omega_3^0 \sqrt{3\pi}}{2}\left ( \frac{\hbar c}{G}\right )^{3/2}\frac{1}{m_H^2}\; \; \; \; \; \; \mu_e = 1$$

Black Hole minimum mass:
$$\boxed{M_c = 1.1401 \cdot 10^{31} \; \text{kg}}$$

Black Hole minimum mass:
$$\boxed{M_c = 5.7318 \cdot M_{\odot}}$$

$$r_s = \frac{2G M_s}{c^2}$$

Schwarzschild mass is equivalent to Chandrasekhar mass:
$$\boxed{M_s = M_c}$$

Integration by substitution:
$$M_s = \frac{r_s c^2}{2G} = \frac{\omega_3^0 \sqrt{3\pi}}{2} \left( \frac{\hbar c}{G}\right )^{3/2}\frac{1}{m_H^2}$$

$$r_s = \frac{\omega_3^0 \sqrt{3\pi}}{2} \left( \frac{2G}{c^2} \right) \left( \frac{\hbar c}{G}\right )^{3/2}\frac{1}{m_H^2}$$

$$\boxed{r_s = 16.933 \; \text{km}}$$

key:
$$m_H$$ - Hydrogen mass
$$\omega_3^0$$ - Lane-Emden equation solution constant

The Lane-Emden equation solution for the $$\omega_3^0$$ constant proof, has not yet been published on Wikipedia.

Reference:
Lane-Emden_equation - Wikipedia
Chandrasekhar limit - Wikipedia

Last edited: May 10, 2008
18. May 10, 2008

### Orion1

$$r_s = \frac{\omega_3^0 \sqrt{3\pi}}{2} \left( \frac{2G}{c^2} \right) \left( \frac{\hbar c}{G}\right )^{3/2} \frac{1}{m_H^2} = \frac{\omega_3^0 \hbar}{c} \sqrt{ \frac{3 \pi \hbar c}{G} } \frac{1}{m_H^2}$$

$$\boxed{r_s = \frac{\omega_3^0 \hbar}{c} \sqrt{ \frac{3 \pi \hbar c}{G} } \frac{1}{m_H^2}}$$
$$\boxed{r_s = 16.933 \; \text{km}}$$

Last edited: May 10, 2008
19. May 10, 2008

### Orion1

Theoretical quantum mass limit: (minimum black hole mass)
$$M_q = \frac{m_p^3}{m_H^2}$$

Planck mass:
$$m_p = \sqrt{ \frac{\hbar c}{G}}$$

$$\boxed{M_q = \left( \frac{\hbar c}{G} \right)^{\frac{3}{2}} \frac{1}{m_H^2}}$$

$$\boxed{M_q = 3.6801 \cdot 10^{30} \; \text{kg}}$$

$$\boxed{M_q = 1.8501 \cdot M_{\odot}}$$

$$r_s = \frac{2G M_s}{c^2}$$

Schwarzschild mass equivalent to quantum mass:
$$\boxed{M_s = M_q}$$

Integration by substitution:
$$M_s = \frac{r_s c^2}{2 G} = \left( \frac{\hbar c}{G} \right)^{\frac{3}{2}} \frac{1}{m_H^2}$$

Identity:
$$r_s = \left( \frac{2 G}{c^2} \right) \left( \frac{\hbar c}{G} \right)^{\frac{3}{2}} \frac{1}{m_H^2} = \frac{2 \hbar}{c} \sqrt{\frac{\hbar c}{G}} \frac{1}{m_H^2}}}$$

$$\boxed{r_s = \frac{2 \hbar}{c} \sqrt{\frac{\hbar c}{G}} \frac{1}{m_H^2}}}$$
$$\boxed{r_s = 5.4659 \; \text{km}}$$