This scenario is quite problematic for two main reasons: evaporation and peak wavelength.

The black hole's lifetime is too short

We can make a rough estimate of the properties of the Hawking radiation coming from the black hole. First, let's start with the luminosity. Since $L\propto M^{-2}$, where $L$ is luminosity and $M$ is the mass of the black hole, it turns out that $$L=9.01\times {10}^{-29}{\left(\frac{M}{M_{\odot }}\right)}^{-2}\text{ Watts}=2.34\times {10}^{-55}{\left(\frac{M}{M_{\odot }}\right)}^{-2}{L}_{\odot }$$ where $M_{\odot }$ and $L_{\odot }$ are the mass and luminosity of the Sun. You need a very low-mass black hole to produce a significant amount of light. In fact, for a black hole to produce one solar luminosity worth of power, its mass must be about 960 kg. The big problem? Such a tiny black hole would evaporate in about 75 nanoseconds (and even during that time, the amount of optical light is' producing will be small - see below). You can prolong its lifetime by increasing its mass - the evaporation timescale is $\tau \propto M^3$ - but this will in turn decrease its luminosity, and so for the flux to be enough to make a planet habitable, you need to have you planet be closer to the black hole, which could be dangerous if the black hole is actively accreting matter.

Lots of gamma rays, no visible light

The other major issue is that the peak wavelength of the radiation won't be in the visible band. A black hole's temperature is inversely proportional to its mass, and its peak wavelength ${\lambda }_p$ is inversely proportional to its temperature. We then have the relation $${\lambda }_p=5.87\times {10}^{12}\left(\frac{M}{M_{\odot }}\right)\text{ nm}$$ and for our tiny, 960-kg black hole, the peak would be far, far, into the gamma ray portion of the spectrum - not great for life. For comparison, visible light has a wavelength of about 300-700 nm, and you'd need a black hole about 1% the mass of the Moon to produce optical Hawking radiation.

How about accretion?

Others have talked about the possibility of energy from infalling matter in the black hole's accretion disk. Let's think about this a bit. There's a relationship between the maximum allowed luminosity - the Eddington limit - and the black hole's mass: $$L_{\text{Edd}}=1.26\times {10}^{31}\left(\frac{M}{M_{\odot }}\right){\text{ Watts}}^{-1}=3.37\times {10}^4\left(\frac{M}{M_{\odot }}\right)L_{\odot }$$ Is this significant? Well, yes, definitely. But there are problems:

• A $1M_{\odot }$ black hole accreting at an efficiency of $ϵ=0.1$ (fairly typical) would accrete a $1M_{\odot }$ accretion disk in about 45 million years, not enough time for life to evolve.
• That accretion disk would be hot, producing more high-energy radiation.

posted over 5 years ago

HDE 226868‭

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