Question

Choose the best answer to each of the following. Explain your reasoning with one or more complete sentences. Which of these objects has the smallest radius? (a) a $1.2M_{\mathrm{Sun}}$ white dwarf (b) a $0.6M_{\mathrm{Sun}}$ white dwarf (c) Jupiter.

Short answer

A 1.2 solar mass white dwarf has the smallest radius.

Step-by-step solution

Understand White Dwarfs

White dwarfs are the remnants of stars and have comparable masses to our Sun but a much smaller size. According to the white dwarf mass-radius relationship, more massive white dwarfs are actually smaller in size due to degeneracy pressure.

Compare Masses of White Dwarfs

Given the choices are two white dwarfs with masses of 1.2 and 0.6 times the mass of the Sun. The mass-radius relation tells us that the 1.2 solar mass white dwarf would be smaller in radius than the 0.6 solar mass white dwarf due to higher degeneracy pressure.

Compare Size of White Dwarfs and Jupiter

Compared to white dwarfs, even planets like Jupiter are much larger in radius. A white dwarf, even though very massive, is around the size of Earth, whereas Jupiter is much larger than Earth.

Determine the Smallest Radius

Now that we've established the relationship between mass and radius in white dwarfs, and having compared the sizes of white dwarfs with Jupiter, we can conclude that the 1.2 solar mass white dwarf is the object with the smallest radius among the given options.

Key concepts

Mass-Radius Relationship

In the intriguing world of white dwarfs, the mass-radius relationship presents a fascinating paradox. Contrary to what we might expect, as the mass of a white dwarf increases, its radius actually decreases. This seemingly counterintuitive concept is a key characteristic of white dwarfs.

This behavior can be attributed to the balance of forces within the star. As a white dwarf accretes more mass, the gravitational force pulls the star's matter more tightly towards the center. However, the white dwarf does not collapse, thanks to the opposing pressure created by the electrons within its core—a phenomenon known as degeneracy pressure. Due to this unique scenario, a more massive white dwarf, such as the one with a mass of 1.2 times that of the Sun, exhibits a smaller radius compared to a less massive one with a mass of 0.6 times that of the Sun.

Degeneracy Pressure

Degeneracy pressure is a fundamental concept in understanding white dwarfs. It arises from the principles of quantum mechanics. In white dwarfs, the atoms are under immense pressure. They are so tightly packed that typical atomic forces need help to resist gravity.

Here, degeneracy pressure comes into play. It originates from the Pauli Exclusion Principle, which states that two fermions, such as electrons, cannot occupy the same quantum state. This principle creates a powerful resistance against further compression.

• Degeneracy pressure is different from thermal pressure, which is related to temperature.
• This pressure depends primarily on the density of particles rather than their temperature.

It remains constant even when the white dwarf cools down, which is why more massive white dwarfs are smaller, despite their greater gravitational pull. This unique pressure is what keeps white dwarfs stable and prevents them from collapsing further.

Astronomical Objects Comparison

When we compare white dwarfs to other astronomical objects like planets, differences become starkly evident. Let’s consider white dwarfs and Jupiter. White dwarfs, often having masses comparable to the Sun, are tremendously dense and compact. Despite their high mass, they achieve a size similar to that of Earth.

Jupiter, on the other hand, is one of the largest planets in our solar system, vastly more enormous than Earth—and consequently white dwarfs in terms of volume. This means, even a smaller mass white dwarf, like one with 0.6 times the mass of the Sun, has a much smaller radius than Jupiter’s. This contrast highlights the density and compact nature of stars like white dwarfs compared to planets.

• White dwarfs are stellar remnants, far denser than planets.
• Planets like Jupiter are composed mostly of lighter elements and gases.

These differences in composition and structure lead to white dwarfs having significantly smaller radii despite their considerable mass.