Minimal mass of a neutron star.

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Discussion Overview

The discussion revolves around the minimal mass of a neutron star, particularly in relation to the semi-empirical mass formula and its applicability to neutron stars. Participants explore theoretical implications, stability concerns, and calculations related to neutron star mass.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question the applicability of the semi-empirical mass formula to neutron stars, noting that it is traditionally used for nuclei containing protons.
  • One participant suggests that the Coulomb term would cancel out, leading to a positive binding energy, and proposes a formula for calculating the minimal mass based on various constants and parameters.
  • There is a discussion about why "one neutron" cannot be considered the minimal mass, with some arguing that a single neutron is unstable on relevant timescales.
  • Another participant points out that a neutron star should not be conceptualized as a large nucleus but rather as a collection of nucleons held together by gravity.
  • Participants express uncertainty about the assumptions underlying the semi-empirical mass formula's application to neutron stars and the implications of gravitational binding energy calculations.

Areas of Agreement / Disagreement

Participants generally agree that a neutron star is a collection of nucleons held together by gravity, but there is no consensus on the minimal mass or the validity of applying the semi-empirical mass formula in this context. Multiple competing views remain regarding the stability of a single neutron and the calculations presented.

Contextual Notes

Limitations include the dependence on the assumptions made in applying the semi-empirical mass formula to neutron stars and the unresolved nature of the gravitational binding energy calculations, which assume uniform mass distribution.

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What is the minimal mass of a neutron star, if the semi empirical mass formula is used?
 
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If you mean the formula for the mass of a nucleus, you probably know that it doesn't apply where there are no protons.
 
clem said:
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.
 
Why isn't the answer "one neutron"?
 
Vanadium 50 said:
Why isn't the answer "one neutron"?

:biggrin: :approve:
 
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.
 
Vanadium 50 said:
Why isn't the answer "one neutron"?
Because "one neutron" is unstable ?
 
humanino said:
Because "one neutron" is unstable ?

an ordinary star is also unstable
 
malawi_glenn said:
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.
 
  • #10
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'
 
  • #11
malawi_glenn said:
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.
 
  • #12
humanino said:
Yes, and I think we all agree on that.

well the OP seemed to be confused :-)
 

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