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Chapter 29: Problem 3

A neutron star is a star that has collapsed into a collection of tightly packed neutrons. Thus, it is something like a giant nucleus; but since it is electrically neutral, there is no Coulomb repulsion to break it up. The force holding it together is gravity. Suppose the Sun were to collapse into a neutron star. What would its radius be? Assume that the density is about the same as for a nucleus. Express your answer in kilometers.

Short Answer

Expert verified
If the Sun were to collapse into a neutron star with a density approximately the same as a nucleus (approximately \(2.3 \times 10^{17}\, kg \cdot m^{-3}\)), its radius would be approximately 12.19 kilometers.

Step by step solution

01

Find the mass of the Sun

To find the mass of the sun, we can simply use the known value of its mass, which is \(1.989 \times 10^{30}\) kg. We will denote this as \(M_{sun}\): $$ M_{sun} = 1.989 \times 10^{30} \, kg. $$
02

Find the volume of the Sun as a neutron star

To find the volume, we will use the mass-density relation, which is: $$ Volume = \frac{Mass}{Density}. $$ We are given the density of a nucleus(\(\rho\)), which is approximately \(2.3 \times 10^{17} \, kg \cdot m^{-3}\). We can now compute the volume of the Sun when it becomes a neutron star(\(V_{ns}\)) using the formula: $$ V_{ns} = \frac{M_{sun}}{\rho} = \frac{1.989 \times 10^{30} \, kg}{2.3 \times 10^{17} \, kg \cdot m^{-3}} \approx 8.65 \times 10^{12} \, m^3. $$
03

Find the radius of the Sun as a neutron star

To find the radius of the neutron star, we need to find the radius of a sphere that has the volume we just computed. The formula for the volume of a sphere is: $$ V = \frac{4}{3} \pi r^3. $$ We can now solve for the radius (\(r\)) by rearranging the formula and plugging in the volume of the neutron star: $$ r = \sqrt[3]{\frac{3 V_{ns}}{4 \pi}} \approx \sqrt[3]{\frac{3 (8.65 \times 10^{12} \, m^3)}{4 \pi}} \approx 1.219 \times 10^4 \, m. $$
04

Convert the radius to kilometers

Finally, we need to convert the radius from meters to kilometers. To do this, we simply divide by 1000: $$ r_{km} = \frac{1.219 \times 10^4 \, m}{1000} \approx 12.19 \, km. $$ Thus, the radius of the Sun if it were to collapse into a neutron star would be approximately 12.19 kilometers.

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