 # Grapefruit sized black hole as planet 9? sonhouse Science 11 Jul '20 01:44
1. 14 Jul '20 02:41
This is a back of an envelope calculation and is out by about 3 orders of magnitude, according to the estimate given in the introduction of  which is based on a far more sophisticated calculation.

Given the mass of the neutron, proton and electron, and assuming that the mass of neutrinos is zero (which is fine for this) the amount of energy liberated by the decay of a free neutron into a proton, electron and anti-neutrino is 0.782326 MeV (the error caused by the mass of the neutrino at most affects the last digit given). The gravitational binding energy of a neutron at the edge of a neutron star, in the Newtonian approximation is:

E = -GMmₙ/r = -GAmₙ²/r

Where G is Newton's constant, A is the number of neutrons, mₙ the mass of a neutron and r the radius of the neutron star.

A = 4/3 π n r³

n is the number density of nuclear matter which we take to be 1.60E+44 . Substituting this into the gravitational energy equation we have:

E = D M ^ (2/3)

where D = G cuberoot(4/3 π n mₙ⁴) and M is the mass of the neutron star (= A mₙ). This gives an estimate for the mass required to supress beta decay gravitationally. Assuming I haven't goofed up on the spreadsheet the figure is 0.000325 solar masses or 108 times the mass of the Earth. Reference  gives the figure of 0.189 solar masses, so I'm out by a barn mile, but it's not a very sophisticated calculation.

What this means is that the object refered to in the OP cannot be made of neutron matter as it can't have a mass much in excess of 5 solar masses.

 https://arxiv.org/pdf/astro-ph/9707230.pdf
 https://en.wikipedia.org/wiki/Nuclear_density
2. 14 Jul '20 06:00
@DeepThought
What do you think about my idea that a grapefruit sized BH would be selective in the wavelengths it could absorb since it would just be too small, if you thought of it as a 4 cm wide dipole antenna, that antenna would have an extreme VSWR to radiation say at 1 Ghz as compared to the roughly 9 or so Ghz as a 4 cm dipole.
So TV signals passing by this BH would just maybe have some phase shift of the wave but not much lost to the BH itself.
It would have above the event horizon a bit of a gravitational lens though but the frequencies would have to be above 10 or so Ghz to be focused by this BH.
3. 14 Jul '20 14:39
@sonhouse said
@DeepThought
What do you think about my idea that a grapefruit sized BH would be selective in the wavelengths it could absorb since it would just be too small, if you thought of it as a 4 cm wide dipole antenna, that antenna would have an extreme VSWR to radiation say at 1 Ghz as compared to the roughly 9 or so Ghz as a 4 cm dipole.
So TV signals passing by this BH would j ...[text shortened]... tional lens though but the frequencies would have to be above 10 or so Ghz to be focused by this BH.
Well, it's not that massive or very big so you'd expect some gravitational lensing effects but not masses of it.
4. 14 Jul '20 15:06
@deepthought said
This is a back of an envelope calculation and is out by about 3 orders of magnitude, according to the estimate given in the introduction of  which is based on a far more sophisticated calculation.

Given the mass of the neutron, proton and electron, and assuming that the mass of neutrinos is zero (which is fine for this) the amount of energy liberated by the decay of ...[text shortened]...

 https://arxiv.org/pdf/astro-ph/9707230.pdf
 https://en.wikipedia.org/wiki/Nuclear_density
The problem with this is that the particle that needs to remain bound is an electron. So the energy = GM mₑ/r, where mₑ is the mass of an electron. This changes the constant D:

D = Gmₑ (4/3 π n mₙ)

This changes the mass estimate to 44 solar masses, which is too high. It implies that D should be about 32 times larger. I've assumed the electron gets all the energy and that relativity can be neglected, which I'd expect to change things by a factor of 4. So possibly the density's higher in an object that big.
5. 14 Jul '20 23:06
@DeepThought
I wasn't thinking of the lensing, just the absorbing of RF and that would have a lower frequency limit I would think since say a 1 meter (300Mhz) would just be too big physically to be collected.
6. 15 Jul '20 06:55
@deepthought said
The problem with this is that the particle that needs to remain bound is an electron. So the energy = GM mₑ/r, where mₑ is the mass of an electron. This changes the constant D:

D = Gmₑ (4/3 π n mₙ)

This changes the mass estimate to 44 solar masses, which is too high. It implies that D should be about 32 times larger. I've assumed the electron gets all the energy ...[text shortened]... d expect to change things by a factor of 4. So possibly the density's higher in an object that big.