Problem 15 - Astronomy: At Play in the Cosmos

Question

A star has a mass of \(0.7 M_{\text {suat }} .\) Which statement about it is true? a. It will live less than 10 billion years. b. Its temperature is greater than 16 million \(\mathrm{K}\). c. It may be using the CNO cycle for nuclear fusion. d. It is a brown dwarf. e. It is using the \(P\) -P process of nuclear fusion.

Step-by-Step Solution

Answer

The star is using the P-P process for nuclear fusion (option e).

Step 1: Understand the Mass of the Star

The mass of the star is given as \(0.7 M_{\text{suat}}\), which is 0.7 times the mass of the Sun. In astrophysical terms, this is a star with less solar mass.

Step 2: Analyze the Lifespan of a Star

Generally, stars less massive than the Sun live longer than 10 billion years because they burn their nuclear fuel more slowly. Therefore, option (a) stating that it will live less than 10 billion years is not true for a star with \(0.7 M_{\text{suat}}\).

Step 3: Evaluate the Surface Temperature

Stars with \(0.7 M_{\text{suat}}\) tend to have cooler temperatures because they are less massive and don't generate as much energy. A temperature greater than 16 million \(\mathrm{K}\) is typical for more massive stars, making option (b) incorrect.

Step 4: Check the Nuclear Fusion Process

Stars less massive than the Sun, like this \(0.7 M_{\text{suat}}\) star, primarily use the proton-proton (P-P) chain reaction, not the CNO cycle, due to the lower core temperature. This makes option (c) incorrect and option (e) correct.

Step 5: Determine if the Star is a Brown Dwarf

Brown dwarfs are substellar objects with an insufficient mass (less than \(0.08 M_{\odot}\)) to sustain hydrogen fusion. Since the star in question has \(0.7 M_{\text{suat}}\), it is not a brown dwarf, eliminating option (d).

Key Concepts

Main Sequence Stars

Main sequence stars are those stars that are in the longest-lasting stage of stellar evolution, where they fuse hydrogen into helium in their cores. This stage can last for billions of years and contributes to the star's stability.

• Lifespan: The lifetime of a star on the main sequence largely depends on its mass. Less massive stars, like the one with a mass of 0.7 times that of the Sun, typically have longer lifespans, often exceeding 10 billion years.
• Temperature: Main sequence stars range in temperature, but less massive stars usually display cooler surface temperatures than heavier stars.

These stars are the most common type visible in our universe, and their characteristics are largely influenced by their position on the Hertzsprung-Russell diagram, which plots stars based on their brightness and temperature.

Proton-Proton Chain

The proton-proton chain is a series of nuclear reactions that are the primary source of energy generation within stars like our Sun and those with lower masses. In this process, hydrogen nuclei (protons) combine through several steps to form helium, releasing energy.

• Low Temperature Fusion: The proton-proton chain is efficient in stars with lower core temperatures compared to the more massive stars, which conduct the CNO cycle.
• Energy Production: Despite its relative complexity, this chain is crucial as it allows stars to shine by converting mass into energy according to Einstein's famous equation, \(E=mc^2\).

Stars like the one discussed, with 0.7 times the solar mass, primarily utilize the proton-proton chain due to their cooler interiors, making it a fundamental concept in understanding stellar energy production.

Mass-Luminosity Relation

The mass-luminosity relation is a crucial principle in stellar astrophysics, which states that the luminosity of a star—how much energy it emits—depends on its mass. Generally, the more massive a star, the brighter it is.

• Proportionality: The relation is approximately proportional to the fourth power of the star's mass. This means if a star's mass doubles, its luminosity could increase by a factor of 16.
• Implications for Lifespan: More luminous stars burn through their fuel faster, leading to shorter lifespans. Conversely, stars with lower mass and luminosity, such as those with 0.7 solar masses, have longer, more stable main sequence phases.

Understanding the mass-luminosity relation is essential when analyzing how different stars age and evolve within the main sequence of stellar life cycles.