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Chapter 17: Problem 26

. What is the mass-luminosity relation? Does it apply to stars of all kinds?

Short Answer

Expert verified
The mass-luminosity relation among stars suggests that the luminosity of a star is directly proportional to the star's mass raised to the power of 3. This relationship often holds true for main-sequence stars, but doesn't apply to stars such as white dwarfs, neutron stars, and red giants, where fusion no longer occurs in the core.

Step by step solution

01

Understanding the mass-luminosity relation

The mass-luminosity relation indicates that a star's luminosity (its output of energy, essentially its brightness) is proportional to the cube of its mass, for main sequence stars up to about 10 solar masses. This relation is generally represented as \[ L \propto M^3 \] where \( L \) represents the luminosity and \( M \) the mass of the star. Main-sequence stars are those that are in the longest lasting phase of a star's life during which it burns hydrogen fuel in its core.
02

Mathematical representation of the mass-luminosity relation

The relationship becomes more complicated for very large and very small stars, where this relation doesn't hold. For stars less massive than the Sun, the relation is more closely approximated by \[ L \propto M^{2.3} \] and for stars heavier than the Sun, the relation approaches \[ L \propto M^{3.5} \]
03

Applicability to different types of stars

The mass-luminosity relation holds primarily for main-sequence stars where the energy is produced by hydrogen fusion in the core. It doesn't apply to white dwarfs, neutron stars, or red giants where fusion is not occurring in the core.

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Key Concepts

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

Main-Sequence Stars
Main-sequence stars are the most common type of stars found in the universe. They represent a significant stage in the life cycle of a star. During this phase, a star remains in a stable state where it efficiently fuses hydrogen into helium in its core. This process generates the energy that balances the gravitational forces trying to collapse the star, allowing it to shine steadily.
  • These stars have a range of masses and sizes, but they all share the characteristic of having hydrogen fusion occurring at their cores.
  • This stage can last for billions of years, depending on the star's mass.
  • More massive stars have shorter main-sequence lifetimes due to faster fusion rates.
Thus, main-sequence stars are integral to understanding stellar structures and behaviors, and they serve as a pivotal phase in stellar evolution.
Stellar Luminosity
Stellar luminosity is a measure of the total energy output from a star. It essentially denotes how bright a star is. The luminosity of a star depends on a combination of its temperature and size.
  • Stars with larger masses are generally more luminous, primarily in the main-sequence category.
  • The mass-luminosity relation describes how a star's brightness increases rapidly with its mass.
  • For example, a star with 2 times the mass of the Sun is significantly more than 2 times as luminous.
In main-sequence stars, the relationship between mass and luminosity is crucial because it highlights the balance between energy generation through nuclear fusion and radiative losses.
Hydrogen Fusion
Hydrogen fusion is the engine behind the luminosity of main-sequence stars. It involves the conversion of hydrogen nuclei into helium nuclei, which releases an enormous amount of energy.
  • This energy not only powers the star but also generates the outward pressure that counteracts gravitational collapse.
  • The process occurs at extremely high temperatures and pressures in the core, allowing the fusion reaction to take place.
  • Hydrogen fusion is efficient and can sustain a star for extended periods, especially in the case of stars similar to or less massive than the Sun.
Without hydrogen fusion, stars would not have stable light and warmth, and the balance of forces in the star would collapse.
Stellar Evolution
Stellar evolution describes the life cycle of a star, from its formation to its eventual demise. Stars undergo different phases, dictated by changes in their core composition and energy production mechanisms.
  • Main-sequence stars eventually exhaust their hydrogen fuel, leading them to leave the main-sequence phase.
  • They evolve into different types of stars, such as red giants or supergiants, depending on their original mass.
  • Massive stars might experience supernova explosions, while smaller stars like our Sun become white dwarfs.
The mass of the star primarily determines its evolutionary path. Understanding stellar evolution helps astrophysicists predict the future of stars and their impacts on the cosmos.

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Most popular questions from this chapter

It is desirable to be able to measure the radial velocity of stars (using the Doppler effect) to an accuracy of \(1 \mathrm{~km} / \mathrm{s}\) or better. One complication is that radial velocities refer to the motion of the star relative to the Sun, while the observations are made using a telescope on the Earth. Is it important to take into account the motion of the Earth around the Sun? Is it important to take into account the Earth's rotational motion? To answer this question, you will have to calculate the Earth's orbital speed and the speed of a point on the Earth's equator (the part of the Earth's surface that moves at the greatest speed because of the planet's rotation). If one or both of these effects are of importance, how do you suppose astronomers compensate for them?

The star Zubenelgenubi (from the Arabic for "scorpion's southern claw") has apparent magnitude \(+2.75\), while the star Sulafat (Arabic for "tortoise") has apparent magnitude \(+3.25\). Which star appears brighter? From this information alone, what can you conclude about the luminosities of these stars? Explain.

Use the Starry Night Enthusiast'M program to examine the nearby stars. Click on Favourites \(>\) Stars > Local Neighborhood and Stop time. Select View \(>\) Feet to hide the spacesuit image. Center this view upon the Sun by opening the Find pane and doubleclicking on Sun. You are now \(16.41\) light years from the Sun, looking at the labeled nearby stars. Increase current elevation to about 70,000 light-years using the button on the toolbar below the Viewing Location box (an upward-pointing triangle) to see these nearby stars within the Milky Way Galaxy. You can rotate the galaxy by placing the mouse cursor over the image and holding down the Shift key while holding down the mouse button and moving the mouse. (On a twobutton mouse, hold down the left mouse button). Decrease current elevation to a distance of about 100 light-years from the Sun to return to the solar neighborhood. Again, you can rotate this swarm of stars by holding down the Shift key while holding down the mouse button and moving the mouse. Open the Info pane. If you click the mouse while the cursor is over a star, you will see the star's apparent magnitude as seen from Earth in the Other Data layer and its distance from the Sun in the Position in Space layer of the Info pane. (a) Select at least 5 stars within 50 light-years of the Sun and note their names, apparent magnitudes, luminosities, and distances from the Sun in a list. Which of these stars would be visible from Earth with the naked eye from a dark location? Which are visible with the naked eye from a brightly lit city? (Hint: The naked eye can see stars as faint as apparent magnitude \(m=+6\) from a dark location, but only as faint as \(m=+4\) from an inner city.) (b) Increase current elevation once more to about 1000 light- years from Earth and locate at least 5 stars that are further than 500 light- years from the Sun, making a list of these stars, their names, apparent magnitudes, luminosities and distances from the Sun. Which of these stars are visible from Earth with the naked eye from a dark location? Are the naked-eye stars more likely to be giants or supergiants, or are they more likely to be main-sequence stars? Explain your answer.

As seen from the starship Enterprise in the Star Trek television series and movies, stars appear to move across the sky due to the starship's motion. How fast would the Enterprise have to move in order for a star \(1 \mathrm{pc}\) away to appear to move \(1^{\circ}\) per second? (Hint: The speed of the star as seen from the Enterprise is the same as the speed of the Enterprise relative to the star.) How does this compare with the speed of light? Do you think the stars appear to move as seen from an orbiting space shuttle, which moves at about \(8 \mathrm{~km} / \mathrm{s}\) ?

The solar constant, equal to \(1370 \mathrm{~W} / \mathrm{m}^{2}\), is the amount of light energy from the Sun that falls on 1 square meter of the Earth's surface in 1 second (see Section 17-2). What would the distance between the Earth and the Sun have to be in order for the solar constant to be 1 watt per square meter \(\left(1 \mathrm{~W} / \mathrm{m}^{2}\right)\) ?
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