Minimal mass of a neutron star.

[hhhmortal]

What is the minimal mass of a neutron star, if the semi empirical mass formula is used?

 

[Meir Achuz]

If you mean the formula for the mass of a nucleus, you probably know that it doesn't apply where there are no protons.

 

[hhhmortal]

If you mean the formula for the mass of a nucleus, you probably know that it doesn't apply where there are no protons.

I thought the coulomb term would cancel and you would have to add a 0.6GM^2 / R and some how the binding energy would be positive.

 

[Vanadium 50]

Why isn't the answer "one neutron"?

 

[malawi_glenn]

Why isn't the answer "one neutron"?

 

[hamster143]

It's a straightforward substitution. When you discard negligible factors, you get simply this
$$E_{binding} = (a_V-a_A) A - a_s A^{2/3} + 0.6 G m_n^2 A^2 / (r_0 A^{1/3}) = 0.6 G m_n^2 A^{5/3} / r_0 - (a_A-a_V) A$$

$$A_{min} = (5/3 (a_A-a_V) r_0 / (G m_n^2))^{1.5}$$

$$a_A-a_V \approx 8 MeV$$
$$G \approx 6.674 * 10^{-11} m^3 kg^{-1} s^{-2}$$
$$r_0 \approx 1.25 * 10^{-15} m$$
$$m_n \approx 940 MeV \approx 1.67*10^{-27} kg$$

and we may need to insert a few powers of c to get the dimensions right. one way to do it is this

$$A_{min} \approx (5/3 * 8/940 * (3*10^8 m/s)^2 * (1.25 * 10^{-15} m / (6.674 * 10^{-11} m^3 kg^{-1} s^{-2} * 1.67*10^{-27} kg))^{1.5} = (5/3 * 8/940 * 9 * 1.25 / (6.674*1.67) 10^{39})^{1.5} \approx 5.4 * 10^{55}$$

$$M_{min} = 9*10^{28} kg = 0.05 M_{sun}$$

and of course we should be happy to get the right order of magnitude, considering how crude our model is ... for example, the formula for gravitational binding energy assumes uniform mass distribution, which is not the case in a neutron star. And applicability of the semi-empirical mass formula to neutron star is a very big and rather unfounded assumption.

 

[humanino]

Why isn't the answer "one neutron"?

Because "one neutron" is unstable ?

 

[malawi_glenn]

Because "one neutron" is unstable ?

an ordinary star is also unstable

 

[humanino]

an ordinary star is also unstable

A neutron is [huge]VERY[/huge] stable for strong interaction, but unstable for time scales relevant to stars, and the gravitational interaction.

 

[malawi_glenn]

The thing Vanadium is pointing out is perhaps that a neutron star is an object of nucleons which is held together by gravity, thus one should not imagine a neutron star as a huge 'nuclei'

 

[humanino]

The thing Vanadium is pointing out is perhaps that a neutron star is an object of nucleons which is held together by gravity, thus one should not imagine a neutron star as a huge 'nuclei'

Yes, and I think we all agree on that.

 

[malawi_glenn]

Yes, and I think we all agree on that.

well the OP seemed to be confused :-)