Question

Suppose two white dwarf stars have the same surface temperature. Why is the more massive of the two white dwarfs the less luminous?

Short answer

The more massive white dwarf is less luminous because it has a smaller surface area due to its greater compactness.

Step-by-step solution

Understanding Luminosity

Luminosity refers to the total amount of energy a star emits per second. For white dwarfs, luminosity depends on both temperature and surface area.

Exploring Surface Temperature

Both stars have the same surface temperature, meaning they emit energy at the same rate per unit area as described by the Stefan-Boltzmann Law: \( L = 4 \, \pi \, R^2 \, \sigma \, T^4 \). Here, \( R \) is the radius, \( T \) is the temperature, and \( \sigma \) is the Stefan-Boltzmann constant.

Considering White Dwarf Structure

In white dwarfs, more massive stars are more compact due to stronger gravitational forces compressing the star. This implies that the radius \( R \) decreases as mass increases.

Relating Mass and Luminosity

Since luminosity depends on \( R^2 \), a smaller radius results in a smaller surface area, thus reducing luminosity. Despite having the same temperature, the more massive star has a smaller surface area, resulting in lower luminosity.

Key concepts

Luminosity

Luminosity is a measure of the total energy a star emits every second. You can think of it as the star's power output. For white dwarf stars, determining luminosity is crucial because it tells us how bright these stars are in the sky.

The formula for luminosity, when applied to stars, includes surface temperature and surface area:

• The surface temperature, which shows how hot the star's surface is.
• The surface area, often related to the size of the star.

This means that even if we have two stars with the same temperature, their brightness or luminosity could differ based on how big they are.

In summary, luminosity depends on two primary factors: how hot the star is and how extensive its surface is. In the case of white dwarfs, which are very dense, the mass has a significant impact on their size, thereby affecting luminosity.

Surface Temperature

Surface temperature is a fundamental characteristic of a star because it determines how much energy is emitted per unit area. For white dwarfs, this is especially interesting due to their very high temperatures even though they have relatively small sizes compared to other types of stars.

The relationship between temperature and emitted energy comes down to the physics of thermal radiation. This explains why the temperature is such a pivotal factor in understanding a star's luminosity.

Therefore, in scenarios where two white dwarfs have the same surface temperature, they share the same energy emission rate per unit area. However, their total emitted energy or luminosity still varies based on their total surface area, dictated largely by their radii.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a key principle for understanding how stars radiate energy. It provides a profound connection between a star's temperature and the amount of energy it emits. The law is expressed with the equation:\[ L = 4 \pi R^2 \sigma T^4 \]This formula shows that the energy output (luminosity, \( L \)) of a star depends on:

• \( R \): the radius of the star, which affects its surface area.
• \( T \): the surface temperature of the star, raised to the fourth power, illustrating its significant effect.
• \( \sigma \): the Stefan-Boltzmann constant, a fixed value in the equation.

The formula highlights how a small change in surface temperature can lead to a large change in luminosity because the temperature is to the fourth power. However, for stars like white dwarfs, which can have similar surface temperatures, differences in their radii (and thus in their surface areas) become the primary reason for differences in luminosity. This is particularly true as more massive white dwarfs are more compact, resulting in smaller radii and lower luminosities.