# B Black Holes and Hawking radiation

#### wolram

Gold Member
Is there a limit as to how much a black hole gathers mass and how much it losses mass via hawking radiation so that the black hole becomes in equilibrium, neither gaining mass or loosing mass
How long would it take a hypothetical isolated black hole to loose one solar mass due to Hawking radiation.

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#### russ_watters

Mentor
There wouldn't be a stable size because Hawking Radiation decreases with increasing mass. Counter-intuitive.

#### wolram

Gold Member
There wouldn't be a stable size because Hawking Radiation decreases with increasing mass. Counter-intuitive.
Can you please give a reference for this Russ.

#### stevebd1

Gold Member
Is there a limit as to how much a black hole gathers mass and how much it losses mass via hawking radiation so that the black hole becomes in equilibrium, neither gaining mass or loosing mass
In an overly simplistic situation where you have just the black hole and the cosmic microwave background radiation, then you can calculate the mass a static black hole would need to be, to be in equilibrium with the CMB radiation that was falling into the BH and the Hawking radiation the BH was giving off. Using the following equation-

$$T=\frac{1}{M}\cdot\frac{\hbar c^3}{8k\pi G}$$

rearranged so that-

$$M=\frac{1}{T}\cdot\frac{\hbar c^3}{8k\pi G}$$

with T set at 2.76 Kelvin, you get a mass of about 4.446e+22 kg (the moon is 7.35e+22 and Europa is 4.8e+22 kg) and a Schwarzschild radius of 6.602e-05 m.

In reality, you would need to take into account the light of distance stars also (which would bump up the T of 2.76 K), any random debris that would fall into the BH (hydrogen, other interstellar matter), not to mention the possibility of an accretion disc which would slowly add mass to the BH. Dark energy and dark matter would also play a part. It's also worth mentioning that spin and charge reduce any HR output a black hole might have (see the bottom part of What is Hawking radiation).

#### Chronos

Gold Member
A black hole could, in principle, reach thermal equilibrium with the CMB and cease to shrink via Hawking radiation, which is currently the case for even small [subsolar] mass black holes. However, the CMB temperature itself also decreases over time due to expansion., meaning a black hole currently in thermal equilibrium will fall out of equilibrium in the distant future and shed energy via the Hawking process. The calculations border on science fiction, predicting a lifetime of a googol years or more for a SMBH, but, nevertheless everything dies. Just some things are permitted by physics to become unimaginably ancient.

#### russ_watters

Mentor
Can you please give a reference for this Russ.

This is described in the wiki on the subject, though somewhat buried or not explicit:
A black hole of one solar mass (M) has a temperature of only 60 nanokelvin (60 billionths of a kelvin); in fact, such a black hole would absorb far more cosmic microwave background radiation than it emits. A black hole of 4.5 × 1022 kg (about the mass of the Moon, or about 13 micrometers across) would be in equilibrium at 2.7 kelvin, absorbing as much radiation as it emits. Yet smaller primordial black holes would emit more than they absorb and thereby lose mass...

Unlike most objects, a black hole's temperature increases as it radiates away mass. The rate of temperature increase is exponential, with the most likely endpoint being the dissolution of the black hole in a violent burst of gamma rays.