Question

Use the information provided below about three main-sequence stars \((\mathrm{A}, \mathrm{B},\) and \(\mathrm{C})\) to complete the following and explain your reasoning. \- Star A has a main-sequence life span of 5 billion years. \- Star \(\mathrm{B}\) has the same luminosity (absolute magnitude) as the Sun. \- Star C has a surface temperature of \(5,000 \mathrm{~K}\). a. Rank the mass of these stars from greatest to least. b. Rank the energy output of these stars from greatest to least. c. Rank the main-sequence life span of these stars from longest to shortest.

Short answer

a. A > B > C; b. A > B > C; c. C > B > A.

Step-by-step solution

Determine the Mass of Star A

Main-sequence stars follow the mass-luminosity relation and their life span is roughly inversely correlated with their masses. Given that Star A has a main-sequence life span of 5 billion years, we can compare this with the Sun, which has a life span of about 10 billion years. Star A should have a mass greater than the Sun, as stars with higher mass have shorter life spans.

Identify the Characteristics of Star B

Star B has the same luminosity as the Sun. According to the mass-luminosity relation, if the luminosity is similar, the mass should also be approximately the same as the Sun. Hence, Star B is likely to have a mass similar to the Sun.

Analyze Surface Temperature of Star C

Star C has a surface temperature of 5000 K. The Sun's surface temperature is around 5778 K. Since star mass affects temperature and Sun-like stars typically have temperatures around 5778 K, a cooler star like Star C would typically have a lower mass. Therefore, Star C likely has the smallest mass among the three.

Ranking the Mass from Greatest to Least

From the analysis above, we can rank the mass of the stars: Star A (greatest mass due to shorter lifespan), Star B (similar mass as the Sun), and Star C (lowest mass due to lower temperature). So, the order is A > B > C.

Determine Energy Output Based on Luminosity

Star B, having the same luminosity as the Sun, provides a baseline. Star A likely has a greater energy output because typically, higher mass leads to higher energy and luminosity. Star C, with lower mass and temperature, is likely to have the lowest energy output. Thus, the order of energy output is A > B > C.

Ranking the Life Span from Longest to Shortest

Stars with greater mass usually have shorter life spans. Therefore, since Star C likely has the lowest mass, it will have the longest life span; Star B, with a Sun-like mass, comes next; and finally, Star A with the greatest mass has the shortest life span. Thus, the ranking is C > B > A.

Key concepts

Mass-Luminosity Relation

The mass-luminosity relation is a crucial concept in understanding the lifecycles of main-sequence stars. This relation indicates that a star's luminosity (brightness) is strongly dependent on its mass. Specifically, for main-sequence stars, the luminosity increases as the mass increases, and this relationship can be approximated by the formula: \[ L \propto M^{3.5} \] Here, \(L\) represents luminosity and \(M\) mass. This implies that even a slight increase in mass leads to a significant increase in luminosity. The mass-luminosity relation helps astronomers determine a star's mass just by knowing how bright it is.
For example, if Star A is more luminous than Star B, Star A is likely more massive. Conversely, a star like Star C, with lower luminosity compared to Star B, would be less massive. Understanding this relationship aids in predicting other stellar properties, such as lifespan and energy output.

Stellar Lifespan

A star's lifespan is inversely connected to its mass. Massive stars consume their nuclear fuel much faster than their lower-mass counterparts. As a result, they have shorter lifespans. For example, the Sun is expected to live about 10 billion years due to its moderate mass. A larger star, such as Star A, with a lifespan of 5 billion years, indicates it is more massive than the Sun. Meanwhile, smaller stars like Star C, although cooler and less massive, would have substantially longer lifespans.
The reason behind this is that massive stars have a higher core pressure and temperature, leading to faster nuclear fusion rates. This rapid fusion exhausts their fuel supply more quickly, cutting their lives short. Conversely, smaller stars burn their fuel at a slower pace and can live for tens to hundreds of billions of years, making them the longest-lasting stars in the galaxy.

Star Luminosity

Luminosity is a measure of the total energy a star emits per second. It is an intrinsic property, independent of a star's distance from us. Within the universe, stars differ widely in luminosity due to variations in mass. Stars like B, with Sun-like luminosities, indicate a balance that is neither too high nor too low, suggesting a moderate level of mass. In contrast, luminous stars like A are massive and radiate more energy.
The immense outward pressure from intense fusion at their cores drives this energy release. Conversely, Star C, with its reduced luminosity, emits less energy, correlating with its lower mass. This diversity in brightness is what brings the night sky to life, with some stars shining brilliantly and others casting a gentle glow. Luminosity not only reveals mass but also gives insights into temperature and age.

Surface Temperature in Stars

The surface temperature of a star offers valuable information about its mass and luminosity. Typically expressed in Kelvin (K), temperature is a direct reflection of the nuclear reactions happening at the star's core.
For example, our Sun has a surface temperature of about 5778 K. Star C, with a cooler temperature of 5000 K, suggests a star with a lesser mass compared to the Sun. Stars with higher masses are typically hotter and exhibit higher surface temperatures due to the intensified nuclear fusion occurring in their cores.
We can infer a star's type, size, and even stage in its stellar evolution by examining its temperature. Cooler stars, like Star C, often belong to the K or M spectral classes and glow with a reddish hue, while hotter stars tend to occur in the O, B, or A spectral categories, emitting a blue or white light. This broad spectrum of temperatures across stars explains their varying colors seen from Earth.