Question

Be sure to show all calculations clearly and state your final answers in complete sentences. Neutron Star Density. A typical neutron star has a mass of about \(1.5 M_{\text {Sun }}\) and a radius of 10 kilometers. a. Calculate the average density of a neutron star, in kilograms per cubic centimeter. b. Compare the mass of \(1 \mathrm{cm}^{3}\) of neutron star material to the mass of Mount Everest \(\left(=5 \times 10^{10} \mathrm{kg}\right)\).

Short answer

A neutron star is extremely dense, with 1 cm³ weighing more than Mount Everest.

Step-by-step solution

Calculate the Mass of the Neutron Star

The mass of a typical neutron star is given as approximately \(1.5 M_{\text{Sun}}\). We know the mass of the Sun \(M_{\text{Sun}}\approx 1.989 \times 10^{30}\) kg. So, the mass of the neutron star \(M = 1.5 \times M_{\text{Sun}} = 1.5 \times 1.989 \times 10^{30}\) kg.

Calculate the Volume of the Neutron Star

The volume \(V\) of a sphere is given by the formula \(V = \frac{4}{3}\pi r^3\). Here, the radius \(r = 10\) km, which is \(10,000\) meters or \(1 \times 10^6\) centimeters since 1 kilometer = 1,000 meters and 1 meter = 100 centimeters. Thus, the volume is \(\frac{4}{3}\pi (1 \times 10^6)^3\) cubic centimeters.

Calculate the Average Density of the Neutron Star

The density \(\rho\) is mass divided by volume. Substituting the mass and volume from Steps 1 and 2, we find \(\rho = \frac{1.5 \times 1.989 \times 10^{30}}{\frac{4}{3}\pi (1 \times 10^6)^3}\) kg/cm\(^3\). Compute this value to find the density.

Compare the Masses

Now consider \(1\) cm\(^3\) of neutron star material. Using the density calculated in Step 3, this volume of material would have a mass equal to the density value in kg. Compare this to the mass of Mount Everest, which is \(5 \times 10^{10}\) kg, to see how much heavier the neutron star material is.

Key concepts

Neutron Star Mass

Neutron stars are incredibly massive objects, remnants of massive stars after a supernova explosion. In the exercise, we are given a neutron star mass of approximately 1.5 times the mass of our Sun.

• The mass of the Sun, denoted as \(M_{\text{Sun}}\), is about \(1.989 \times 10^{30}\) kg.
• Thus, the mass of the neutron star \(M\) can be calculated as \(1.5 \times M_{\text{Sun}} = 1.5 \times 1.989 \times 10^{30}\) kg.
• This results in an astonishing mass for the neutron star, about \(2.9835 \times 10^{30}\) kg.

This massive amount, while concentrated into a small space, is what makes neutron stars so dense and fascinating.

Sphere Volume Calculation

To understand the density of a neutron star, we first need to determine its volume. Neutron stars are roughly spherical, and their volumes can be calculated using the formula for the volume of a sphere:\[V = \frac{4}{3}\pi r^3\]Given a radius of 10 kilometers for the neutron star, we need to convert units to centimeters because density calculations often require it in kg/cm³.

• 1 kilometer equals 1,000 meters, and 1 meter equals 100 centimeters.
• Thus, 10 kilometers equals \(10 \times 1,000 \times 100 = 1 \times 10^6 \) centimeters.

Now, plug in the radius:\[V = \frac{4}{3}\pi (1 \times 10^6)^3\]This large volume illustrates how vast the sphere is, despite the radius seeming relatively small compared to earthly measurements.

Density Comparison

The concept of density helps us understand how tightly packed the mass of an object is within a given volume. In this exercise, the density of a neutron star is defined as its mass divided by its volume. Calculate the density \(\rho\):\[\rho = \frac{\text{Mass}}{\text{Volume}} = \frac{1.5 \times 1.989 \times 10^{30}}{\frac{4}{3}\pi (1 \times 10^6)^3}\]This calculation results in an incredibly high density, typically much greater than what we observe on Earth.

• This density shows neutron stars as some of the densest objects in the universe.
• Understanding these values helps compare them to everyday objects, like the materials we encounter on Earth.

Mass Comparison

To grasp the enormity of neutron star density, comparing the mass of a small volume, say 1 cm³, of neutron star material to a colossal earthly object like Mount Everest, provides clarity.

• Previously calculated, the density of a neutron star tells us that 1 cm³ of its material would be exceedingly heavy.
• Mount Everest's mass is approximately \(5 \times 10^{10}\) kg.
• However, even 1 cm³ of neutron star material could be vastly heavier due to its extreme density.

By visualizing this comparison, you appreciate how neutron star material is not only dense but fundamentally different from most mass-to-volume ratios we typically reckon with.