To solve this problem, we need to calculate the radius of a neutron star with a mass 10 times that of the Sun, given that the density of a neutron star is comparable to the density of an atomic nucleus.

Step 1: Determine the mass and density of the neutron star

• The mass of the Sun, $M_{\odot}$, is: $M_{\odot} = 1.99 \times 10^{30} \, \text{kg}$
• The mass of the neutron star, $M_{\text{NS}}$, is 10 times the mass of the Sun: $M_{\text{NS}} = 10 \times M_{\odot} = 10 \times 1.99 \times 10^{30} \, \text{kg} = 1.99 \times 10^{31} \, \text{kg}$

The density of the neutron star is approximately the same as the density of an atomic nucleus, which is around: $\rho_{\text{NS}} = 10^{17} \, \text{kg/m}^3$

Step 2: Use the formula for the volume of a sphere to find the radius

The volume $V$ of a sphere is related to its radius $R$ by the formula: $V = \frac{4}{3} \pi R^3$ The mass $M$ of the neutron star is related to its density $\rho$ and volume $V$ by: $M = \rho \times V$ Substitute the expression for the volume of a sphere: $M = \rho \times \frac{4}{3} \pi R^3$ Solve for $R$: $R^3 = \frac{3M}{4\pi\rho}$ $R = \left( \frac{3M}{4\pi\rho} \right)^{1/3}$

Step 3: Plug in the values for mass and density

Substitute $M = 1.99 \times 10^{31} \, \text{kg}$ and $\rho = 10^{17} \, \text{kg/m}^3$: $R = \left( \frac{3 \times 1.99 \times 10^{31}}{4 \pi \times 10^{17}} \right)^{1/3}$

Let’s calculate this next.

The radius of the neutron star would be approximately $36.2 \, \text{km}$.

Would you like any additional details or further explanations on this?

Here are five related questions you might find interesting:

1. How does the density of a neutron star compare to that of other stellar remnants like white dwarfs?
2. What physical processes occur during the formation of a neutron star?
3. What role do neutron stars play in the study of gravitational waves?
4. How do scientists measure the mass and radius of neutron stars?
5. Can a neutron star form from a supernova in any type of star?

Tip: The immense density of neutron stars is such that a single cubic centimeter of neutron-star material would weigh about the same as a large mountain on Earth!