Question

Choose the best answer to each of the following. Explain your reasoning with one or more complete sentences. Which of these things has the smallest radius? (a) a \(1.2 M_{\text {Sun }}\) white dwarf (b) the event horizon of a \(3.0 M_{\text {Sun }}\) black hole (c) the event horizon of a \(10 M_{\text {Sun }}\) black hole

Short answer

The event horizon of the \(3.0 M_{\text{Sun}}\) black hole has the smallest radius.

Step-by-step solution

Understanding Stellar Objects

A white dwarf is a dense stellar remnant formed after a star has exhausted most of its nuclear fuel. Its size is comparable to Earth, characterized by high density but relatively smaller compared to stars. A black hole is a region in space where the gravitational pull is so strong that nothing can escape from it, not even light.

Evaluating White Dwarf Radius

Typically, the radius of a white dwarf is roughly comparable to that of Earth, even though its mass could be similar to the Sun's. Therefore, a white dwarf with a mass of \(1.2 M_{\text{Sun}}\) would have a radius approximately 7,500 km or slightly less, considering its increased mass.

Calculating Black Hole Event Horizon

The radius of the event horizon of a black hole, known as the Schwarzschild radius, can be calculated using the formula: \(R_s = \frac{2GM}{c^2}\). For a black hole with \(3.0 M_{\text{Sun}}\), the event horizon radius \(R_s\) is approximately 9 km. For a black hole with \(10 M_{\text{Sun}}\), \(R_s\) is about 30 km.

Comparing Radii

Now we compare all the radii: the white dwarf's radius is approximately 7,500 km; the event horizon of a \(3.0 M_{\text{Sun}}\) black hole is 9 km; and the event horizon of a \(10 M_{\text{Sun}}\) black hole is 30 km. Clearly, the event horizon of the \(3.0 M_{\text{Sun}}\) black hole has the smallest radius.

Conclusion

Based on the calculations and comparisons, the object with the smallest radius is the event horizon of the \(3.0 M_{\text{Sun}}\) black hole.

Key concepts

White Dwarf

A white dwarf is the remnant core of a star that has ceased fusing nutrients in its core. White dwarfs represent the final evolutionary state of stars whose mass is not enough to become a neutron star or black hole. These stellar remnants are incredibly dense. To give you an idea, a white dwarf's mass can be similar to that of our Sun, yet its size is only comparable to Earth. This means that a teaspoon of white dwarf material would weigh tons!

White dwarfs are sustained by electron degeneracy pressure, which opposes gravity and keeps the star from collapsing further. Despite their small size, white dwarfs still shine brightly, thanks to residual thermal energy left over from their previous nuclear fusion. However, over billions of years, they gradually cool and fade away into what’s theoretically referred to as a "black dwarf," a star that no longer emits significant heat or light.

Black Hole

Black holes are fascinating cosmic objects. They occur when a massive star collapses under its own gravity at the end of its life cycle. The gravitational pull of a black hole is so intense that even light cannot escape from it, which is why it's "black."

Structurally, we can think of a black hole as having a core known as a singularity, where mass is concentrated into an infinitely small point. Surrounding this singularity is the event horizon. Once something crosses this boundary, it cannot escape the black hole's gravitational grip and is effectively lost to the universe. Black holes can vary greatly in mass—from a few times the mass of the Sun to those millions or even billions of times solar mass, known as supermassive black holes, which often reside at the centers of galaxies.

Event Horizon

The event horizon is the "point of no return" for a black hole. It's an invisible boundary that surrounds the black hole. Anything that crosses this boundary, whether it's light, information, or matter, will be drawn inexorably inward towards the singularity.

The size of the event horizon is determined by the mass of the black hole, which is defined by the Schwarzschild radius. The more massive a black hole, the larger its event horizon will be. The event horizon represents the surface area or the limit within which the escape velocity equals the speed of light, rendering escape impossible. Therefore, trying to observe anything within the event horizon is impossible with current technology, since no information can travel out of it.

Schwarzschild Radius

The Schwarzschild radius is a critical concept in understanding black holes. This term refers to the radius of the boundary surrounding a non-rotating black hole, up to which objects are inevitably drawn toward the black hole once passed. It is named after Karl Schwarzschild, who first provided solutions to Einstein's field equations of general relativity, describing the gravitational field outside a spherical mass.

Mathematically, the Schwarzschild radius \(R_s\) can be found using the formula \[ R_s = \frac{2GM}{c^2} \],where \(G\) is the gravitational constant, \(M\) is the mass of the object, and \(c\) is the speed of light. This radius determines the size of the event horizon for a black hole. For a given mass, the Schwarzschild radius sets the scale of how tightly packed the mass must be to form a black hole. As noted, for a black hole with a mass three times that of the Sun, the Schwarzschild radius is approximately 9 kilometers.