A neutron star is a stellar object whose density is about...

A neutron star is a stellar object whose density is about that of nuclear matter, $2 \times 10^{17} \mathrm{~kg} / \mathrm{m}^{3} .$ Suppose that the Sun were to collapse and become such a star without losing any of its present mass. What would be its radius?

Key Concepts

Density and Mass Relationship

Density is defined as mass divided by volume, and it plays a crucial role in understanding the structure of matter both in everyday substances and in astrophysical objects. By knowing an object's mass and its density, one can determine its volume using the relation mass = density × volume. This fundamental concept is key to evaluating how matter is distributed in space, especially when comparing very different scales, such as that of a star versus nuclear matter.

Spherical Volume Formula

For many astrophysical objects, including stars and compact remnants, it’s reasonable to approximate their shape as a sphere. The volume of a sphere is given by V = (4/3)?r³, which directly links the radius of a spherical object to its volume. This formula is essential for determining the size of an object when its mass and density are known, as it allows for the calculation of radius from these parameters.

Neutron Stars and Nuclear Matter Density

Neutron stars are remnants of massive stars that have undergone gravitational collapse, resulting in an extremely dense object primarily composed of neutrons. Their density is comparable to that of nuclear matter, which is several orders of magnitude higher than typical stellar densities. This concept highlights how a massive star, when compressed to nuclear matter density, drastically reduces in size, leading to radii on the order of tens of kilometers despite having masses comparable to the Sun.

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