Hawking Radiation From Black Holes:

[Lamdbaenergy]

If you only knew the temperature of the black hole, like, if for example, the temperature of a 4 solar mass black hole being around 1.5e-8 kelvin, how could you possibly be able to calculate what wavelengths of radiation the black hole would give off? Would a black hole like this really only give off only low energy radio waves? And how could one find out how much energy that black hole would release upon explosion?

 

[enorbet]

Under what conditions do you expect a black hole to explode?

 

[Lamdbaenergy]

Under what conditions do you expect a black hole to explode?

Under evaporation, sorry.

 

[Ken G]

The problem is that the only kinds of black holes we have already seen good evidence of would take way too long to evaporate, and would be so dim we'd never see their Hawking radiation. We'd need much much smaller black holes, left over from the Big Bang, which could be just evaporating now. If those don't exist, how we'll ever see Hawking radiation I do not know.

 

[Lamdbaenergy]

The problem is that the only kinds of black holes we have already seen good evidence of would take way too long to evaporate, and would be so dim we'd never see their Hawking radiation. We'd need much much smaller black holes, left over from the Big Bang, which could be just evaporating now. If those don't exist, how we'll ever see Hawking radiation I do not know.

Yeah, I'm familiar with that fact already. But I read somewhere from a physics article that a solar mass black hole would emit radio waves at a wavelength of over 100 kilometers and a cycle of 1800 per second. I also want to know what kinds of mathematical equations govern how these black holes would emit electromagnetic waves and what kinds of particles they would produce around them.

 

[Ken G]

Hawking radiation should be thermal radiation, so the peak frequency is proportional to temperature. Thus one can scale to the Sun-- it peaks at some 4000 A for 6000 K, so if we have a T that is about two trillion times lower, we are talking about a wavelength that is two trillion times larger-- or almost a thousand kilometers. One problem that appears is that this is much larger than the size of the event horizon of the black hole, and I would imagine that a black hole would have difficulty radiating at wavelengths larger than its event horizon, but it's tricky with all those gravitational effects, so I really don't know what wavelength you should expect the spectrum to peak. Still, we can agree it would be way into the radio.

 

[Lamdbaenergy]

Hawking radiation should be thermal radiation, so the peak frequency is proportional to temperature. Thus one can scale to the Sun-- it peaks at some 4000 A for 6000 K, so if we have a T that is about two trillion times lower, we are talking about a wavelength that is two trillion times larger-- or almost a thousand kilometers. One problem that appears is that this is much larger than the size of the event horizon of the black hole, and I would imagine that a black hole would have difficulty radiating at wavelengths larger than its event horizon, but it's tricky with all those gravitational effects, so I really don't know what wavelength you should expect the spectrum to peak. Still, we can agree it would be way into the radio.

One way I calculate it myself, even though I have no idea if it's correct, is to do 2.8977721e-3 divided by the temperature I get from the black hole using the following formula:

http://qph.is.quoracdn.net/main-qimg-d2b24b5a0f11f6b2649c16e321eecae9?convert_to_webp=true

Now in this case, I just did a calculation for a 1 solar mass black hole, getting about 6.1724676e-8 kelvin. That gave me a supposed wavelength of 48 kilometers and a frequency of 6400 cycles per second? What is being done differently here?

 

[Ken G]

If one scales to the Sun, the wavelength you are talking about now should be 100 billion times longer than the solar peak at about 5000 Angstroms. So that's about 50 km, you are fine. I must have made an error before.

 

[Matterwave]

One should also perhaps consider the gravitational red shift this Hawking radiation will have to go through...although it is possible that the Hawking radiation equation already accounts for this effect.

 

[Lamdbaenergy]

One should also perhaps consider the gravitational red shift this Hawking radiation will have to go through...although it is possible that the Hawking radiation equation already accounts for this effect.

Maybe the luminosity formula also has something to do with it, too?

I guess it probably already does account for the gravitational waveshifting, anyway.

And as for energy left from when the black hole evaporates, is it just a typical E = mc^2, or is the math different? A 1 solar mass black hole would live for about 2 x 10(67) years, and when it evaporated, it would release nearly 2e47 joules of energy, I'm guessing. Or am I wrong on that part?

 

[Lamdbaenergy]

I'm not sure about this, but I think I might be on to something. I used an equation for gravitational red shifting and multiplied the original wavelength I got at first by the parameter at the end. I got pretty close to 160,000 meters, somewhere at 159,600 actually. I don't know if this could just be a coincidence, but isn't that weird?