Problem 19 - College Physics

Question

\(\bullet\) White dwarfs and neutron stars. Recall that density is mass divided by volume, and consult Chapter 0 and Appendix \(E\) as needed. (a) Calculate the average density of the earth in \(g / c m^{3},\) assuming our planet to be a perfect sphere. (b) In about 5 billion years, at the end of its lifetime, our sun will end up as a white dwarf, having about the same mass as it does now, but reduced to about \(15,000 \mathrm{km}\) in diameter. What will be its density at that stage? (c) A neutron star is the remnant left after certain supernovae (explosions of giant stars). Typically, neutron stars are about 20 \(\mathrm{km}\) in diameter and have around the same mass as our sun. What is a typical neutron star density in \(\mathrm{g} / \mathrm{cm}^{3} ?\)

Step-by-Step Solution

Answer

Earth's average density is about 5.5 g/cm^3. White dwarf's density will be much higher, around 1 x 10^6 g/cm^3, while a neutron star is even denser, about 1 x 10^14 g/cm^3.

Step 1: Calculate the Earth's Volume

Assume Earth is a perfect sphere. The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). The Earth's radius is approximately 6,371 km, which is 6,371,000 m or 6,371,000,000 cm. Plug this into the formula to find the volume: \[ V = \frac{4}{3} \pi (6,371,000,000)^3 \text{ cm}^3. \]

Step 2: Calculate the Earth's Mass

The mass of the Earth is approximately \(5.972 \times 10^{27} \text{ g} \).

Step 3: Calculate the Earth's Density

Density is mass divided by volume. Using the Earth's mass from Step 2 and the volume from Step 1, calculate the density: \[ \text{Density} = \frac{5.972 \times 10^{27} \text{ g}}{\frac{4}{3} \pi (6,371,000,000)^3 \text{ cm}^3}. \] Solve for the density.

Step 4: Calculate the White Dwarf's Volume

At the end of its lifetime, the sun is expected to become a white dwarf with a diameter of 15,000 km. The radius is half of that: 7,500 km or 7,500,000 cm. Using \( V = \frac{4}{3} \pi (7,500,000)^3 \text{ cm}^3 \), calculate the volume.

Step 5: Calculate the Sun's Mass as a White Dwarf

The mass of the sun is approximately \(1.989 \times 10^{33} \text{ g} \), which remains approximately the same as a white dwarf.

Step 6: Calculate the White Dwarf's Density

Using the white dwarf's mass from Step 5 and volume from Step 4, find the density: \[ \text{Density} = \frac{1.989 \times 10^{33} \text{ g}}{\frac{4}{3} \pi (7,500,000)^3 \text{ cm}^3}. \] Solve for the density.

Step 7: Calculate the Neutron Star's Volume

A neutron star has a typical diameter of 20 km, so its radius is 10 km or 1,000,000 cm. Calculate the volume with \( V = \frac{4}{3} \pi (1,000,000)^3 \text{ cm}^3 \).

Step 8: Calculate the Neutron Star's Density

Using the sun's mass from Step 5 (as neutron stars also have similar mass), calculate the neutron star's density: \[ \text{Density} = \frac{1.989 \times 10^{33} \text{ g}}{\frac{4}{3} \pi (1,000,000)^3 \text{ cm}^3}. \] Solve for the density.

Key Concepts

Understanding White Dwarfs

White dwarfs are fascinating stellar remnants left behind when stars, like our sun, exhaust their nuclear fuel. At the end of its life, a star like the sun will shed its outer layers, leaving a hot core behind. This core becomes a white dwarf. What makes white dwarfs particularly interesting is their dense nature. Despite having a mass similar to that of the sun, they are drastically smaller in size.
This is because the mass of the sun compacts into a volume as small as about 15,000 kilometers in diameter. To provide some perspective, this is nearly comparable to Earth's diameter! Due to this extreme reduction in size while maintaining mass, white dwarfs have exceptionally high densities. Calculating the density involves dividing the sun's unchanged mass by its new volume as a white dwarf. The result is an astonishingly high density, which poses interesting questions about the physics and internal structure of these remnants.

Neutron Stars: The Densest Stars

Neutron stars are even denser than white dwarfs. They form from the remnants of massive stars after a supernova explosion. Despite having a mass comparable to the sun, a neutron star's diameter shrinks to just about 20 kilometers! Imagine all that mass packed into such a small space. This incredible density arises due to the immense gravitational forces at play during the star's collapse after the supernova.
Within a neutron star, gravity is so strong that it compresses atoms tightly. Electrons and protons combine to form neutrons, leaving a dense core made almost entirely of neutrons, hence the name. The typical density of a neutron star can be calculated using the same approach as for white dwarfs: divide the sun's mass by the neutron star's remarkably small volume. This calculation gives rise to densities that are the highest known in the universe, illustrating the extreme conditions these stars embody.

Density Calculations Made Simple

Density calculations are crucial for understanding the properties of stellar remnants like white dwarfs and neutron stars. The basic principle is to divide the mass of the object by its volume using the formula:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
For spheres, which include stars and planets, the volume is determined via the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius. Each step in the density calculation process involves substituting the known mass and the calculated volume into the density formula. For example, to find the density of a white dwarf, one uses the sun's mass and its volume when it becomes a white dwarf.
The same procedure applies to neutron stars, where extremely small radii are used, leading to remarkably high densities. Understanding these calculations helps unlock the mysteries of stars and provides insights into their life cycles and remnants, turning abstract concepts into comprehensible numbers.