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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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Most popular questions from this chapter

In the spectrum of a particular star, the Balmer line \(\mathrm{H}_{\alpha}\) has a wavelength of \(656.15 \mathrm{~nm}\). The laboratory value for the wavelength of \(\mathrm{H}_{\alpha}\) is \(656.28 \mathrm{~nm}\). (a) Find the star's radial velocity. (b) Is this star approaching us or moving away? Explain. (c) Find the wavelength at which you would expect to find \(\mathrm{H}_{\alpha}\) in the spectrum of this star, given that the laboratory wavelength of \(\mathrm{H}_{\alpha}\) is \(486.13 \mathrm{~nm}\). (d) Do your answers depend on the distance from the Sun to this star? Why or why not?

Sketch a Hertzsprung-Russell diagram. Indicate the regions on your diagram occupied by (a) main-sequence stars, (b) red giants, (c) supergiants, (d) white dwarfs, and (e) the Sun.

From its orbit around the Earth, the Hipparcos satellite could measure stellar parallax angles with acceptable accuracy only if the angles were larger than about \(0.002\) arcsec. Discuss the advantages or disadvantages of making parallax measurements from a satellite in a large solar orbit, say at the distance of Jupiter from the Sun. If this satellite can also measure parallax angles of \(0.002 \mathrm{arcsec}\), what is the distance of the most remote stars that can be accurately determined? How much bigger a volume of space would be covered compared to the Earth-based observations? How many more stars would you expect to be contained in that volume?

Sketch the light curve of an eclipsing binary consisting of two identical stars in highly elongated orbits oriented so that (a) their major axes are pointed toward the Earth and (b) their major axes are perpendicular to our line of sight.

The star HD 3651 shown in Figure 17-13 has a mass of \(0.79 \mathrm{M}_{\odot}\). Its brown dwarf companion, HD \(3651 \mathrm{~B}\), has about 40 times the mass of Jupiter. The average distance between the two stars is about \(480 \mathrm{AU}\). How long does it take the two stars to complete one orbit around each other?
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