# Mass-Radius relation of a Neutron star

• Tuugii
In summary, the Mass-Radius relationship for a neutron star can be determined by combining the equations for neutron degeneracy pressure and hydrostatic equilibrium. The resulting expression is M(r) = (5/2)(h^2/Gm_n^4)(n^5r^2).
Tuugii

## Homework Statement

Hey all,

I need a help to determine the Mass-Radius relationship for a neutron star. I've done it for a white dwarf, but for a neutron star I need to know the Neutron degeneracy pressure expression, can anyone please help me to solve it?

I am thinking that if I have the n.deg.pressure expression then I can use the hydrostatic equilibrium, and assume the masses of proton and neutron to be exactly equal;

I am not sure, but I might also need the density ratio? is it correct? for instance for a white dwarf, I have [ro_c]/[ro_mean] = 5.99, I don't know the value for a neutron star.

thanks,
T

eddyHomework Equations Neutron degeneracy pressure = p_deg = (2/5) h^2 n^5/m_n^4Hydrostatic equilibrium equation: P(r) = -Gm(r)/r^2The Attempt at a Solution The Neutron degeneracy pressure expression is given by: p_deg = (2/5) h^2 n^5/m_n^4, where h is the Planck's constant, n is the number density of neutrons and m_n is the neutron mass. The hydrostatic equilibrium equation can be used to determine the Mass-Radius relationship for a neutron star. The equation is given by: P(r) = -Gm(r)/r^2, where G is the gravitational constant and r is the radius of the neutron star. Assuming that the masses of protons and neutrons are equal, we can combine the two equations to obtain the Mass-Radius relationship for a neutron star. The expression for the Mass-Radius relationship is given by: M(r) = (5/2)(h^2/Gm_n^4)(n^5r^2). You do not need to know the density ratio for a neutron star, since this value is not relevant for the Mass-Radius relationship.

## 1. What is a neutron star?

A neutron star is a highly dense, compact object that is formed after a massive star undergoes a supernova explosion. It is composed primarily of neutrons and has a radius of about 10 kilometers.

## 2. How is the mass of a neutron star related to its radius?

The mass of a neutron star is directly proportional to its radius, meaning that the larger the mass, the larger the radius will be. This relationship is governed by the neutron star's equation of state, which describes the relationship between its density, pressure, and temperature.

## 3. Why is the mass-radius relation of a neutron star important?

The mass-radius relation of a neutron star is important because it helps us understand the physical properties of these incredibly dense objects. It also allows us to make predictions about the structure and behavior of neutron stars, which can help us study the physics of extreme environments.

## 4. What is the maximum mass a neutron star can have?

The maximum mass a neutron star can have is called the Tolman-Oppenheimer-Volkoff (TOV) limit, which is estimated to be around 2-3 times the mass of the Sun. Beyond this limit, the gravitational forces would be too strong and the neutron star would collapse into a black hole.

## 5. How is the mass-radius relation of a neutron star determined?

The mass-radius relation of a neutron star is determined through observations and theoretical models. Astronomers measure the mass of a neutron star by observing its gravitational influence on a nearby companion star, while the radius is estimated through observations of the neutron star's X-ray emission. Theoretical models use the neutron star's equation of state to predict the relationship between mass and radius.

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