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Chapter 16: Problem 15

T/F: A 1.2- \(M_{\circ}\) white dwarf has a smaller radius than a \(1.0-M_{\circ}\) white dwarf.

Short Answer

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Step by step solution

01

Understanding White Dwarf Properties

White dwarfs are stellar remnants with no more fusion reactions occurring. Their properties depend on their mass due to degenerate electron pressure.
02

Chandrasekhar Limit

The Chandrasekhar limit (approximately 1.4 times the mass of the Sun) is the maximum mass a stable white dwarf can have before collapsing into a neutron star or black hole.
03

Inverse Mass-Radius Relationship

For white dwarfs, the radius decreases as the mass increases, due to the principles of degeneracy pressure in quantum mechanics. This relationship is contrary to most everyday objects where mass and size are directly proportional.
04

Comparing the Given Masses

Compare the given masses: 1.2 solar masses and 1.0 solar masses. The white dwarf with 1.2 solar masses will have a smaller radius compared to the one with 1.0 solar masses due to the inverse mass-radius relationship.
05

Conclusion

Based on the inverse mass-radius relationship, a 1.2-\(M_{\u2008\circ}\) white dwarf indeed has a smaller radius than a 1.0-\(M_{\u2008\circ}\) white dwarf.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chandrasekhar limit
White dwarfs have a critical mass called the Chandrasekhar limit. This limit is about 1.4 times the mass of our Sun. If a white dwarf's mass exceeds this limit, it can no longer support itself against gravitational collapse.
Chandrasekhar discovered that beyond this limit, the internal pressure required to counteract gravity would be insufficient. This would lead the white dwarf to collapse into a denser object, like a neutron star or black hole.
This limit is fundamental in understanding the lifecycle of stars and the behavior of stellar remnants.
degenerate electron pressure
White dwarfs are held up by a unique force: degenerate electron pressure. This is a quantum mechanical effect arising from the Pauli exclusion principle.
In simple terms, electrons are packed so closely that they create a pressure preventing the star from collapsing. Even without nuclear fusion, this pressure supports the white dwarf against gravity.
The degenerate electron pressure depends heavily on the density. As the star's mass increases, the electrons are packed more tightly, increasing this pressure even further.
inverse mass-radius relationship
One of the intriguing properties of white dwarfs is their inverse mass-radius relationship. Unlike regular objects where more mass usually means a larger size, white dwarfs behave differently.
Here, as the mass of a white dwarf increases, the radius decreases. This is due to the degenerate electron pressure that becomes more significant as density increases.
This relationship allows us to predict that a 1.2 solar mass white dwarf would indeed have a smaller radius compared to a 1.0 solar mass white dwarf. Understanding this concept is crucial in studying the characteristics and evolution of white dwarfs.

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Most popular questions from this chapter

What will the escape velocity be when the Sun becomes an AGB star with a radius 200 times greater and a mass only 0.7 times that of today? How will these changes in escape velocity affect mass loss from the surface of the Sun as an AGB star?

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T/F: All stars eventually die.
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