Life on the Main-Sequence
242-249
A coherent picture of how stars form and produce energy is starting to emerge. In this section you will explore the differences among the main-sequence stars and also be able to answer an age-old question "How old are the Stars?"
It's All About Mass (mostly)
The most significant factor that determines the structure of a main-sequence star is its mass. Radius, central pressure, temperature and density as well as surface temperature can all be related to mass. Use the following applet to investigate this.
| Applet "StarMaker" illustrates how the gross physical properties of a main-sequence star are determined by mass. |
Example 9.8 Use the applet "StarMaker" to answer the following: Which main sequence stars have the highest core temperatures and which have the highest central densities?
Solution: The most massive stars will have the highest central temperatures. A 30 Mo star will have a central temperature of about 35 million K while a 0.1 Mostar has a central temperature of about 4.4 million K. On the other hand, the low mass star is much denser with a core density of over 300 g/cm3 . The 30 Mostar has a central density of only a few g/cm3.
Comparing Main-Sequence Stars
Let's consider some of the key differences between main-sequence stars:
Low Mass Stars (0.1 Mo- 0.5 Mo)
Stars with masses less than 0.5 solar mass will have cool atmospheres (less than 4000 K) and relatively cool cores (4 - 5 million K). This means that such stars must rely upon the PP-cycle to provide energy. Indeed, for the lowest masses the PP-cycle is barely able to work. If the mass of the star drops below 0.08 Mo then fusion will not occur and the star will be a brown dwarf.
Figure 9.25 indicates that convection is the dominant energy transport method for low mass stars. Even though their core temperatures are the lowest of the main-seqence it is the temperature gradient that is important in determining whether convective or radiative energy transport occurs. Because the temperature drops from about 5 million K to only a few thousand degrees in a span of little more than 0.1 solar radii the temperature gradient is very large. |
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| The fact that convection is operating throughout the star ensures that very good mixing occurs between the core and the outer layers. .As you saw in Chapter 8 (Figure 8.18) these stars are the most common in the solar neihbourhood and yet, because of their very low luminosity, they remain invisible to the un-aided eye. | |
Figure 9.25 A fully convective low-mass star.Middle-Mass Stars (0.5 Mo- 1.0 Mo)
Our sun is an example of a typical "middle-mass" star. Core temperature is in excess of 10 million K and the inner region transports heat primarily through radiation. In the outer envelope convection takes over and becomes the dominant energy transport mechanism. Figure 9.26 illustrates this. Surface temperatures are now in the range of 4500 K - 6000 K. |
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| Figure 9.26 Middle mass star with radiative core and convective outer envelope. |
High Mass Stars (mass > 1Mo)
| For stars more massive than the sun the way in which energy is transported makes a significant change. Because the weight of the overlying layers is so large the core is compressed to very high temperatures. The CNO-cycle becomes the dominant fusion reaction and the core becomes convective. Now the outer envelope becomes radiative - essentially the opposite of the middle-mass stars. This is shown in Figure 9.27. |  |
| Figure 9.27 Massive star with convective core and radiative outer envelope. |
So - What Makes a Star a "Main-Sequence Star"?
Fundamentally a main-sequence star in one for which Hydrogen fusion in the core provides most of the star's energy. As you will see in the next chapter, some massive stars can, as they age, begin to fuse heavier elements, but all stars have Hydrogen fusion in common. We can therefore define a main-sequence star as one for which Hydrogen fusion (and only Hydrogen fusion) is occuring in the core.
Example 9.9 Do stars "burn"?
Solution: Although we often use the term "burning" when we talk about stars it is important to understand that this not the same as the burning that occurs when you light a match or sit around a campfire. A burning match is releasing "chemical energy" which is the energy released by the breaking of bonds in a chemical reaction. This kind of burning could be called chemical burning. Stars derive energy from the conversion of mass into energy via nuclear fusion. This could more accurately should be called nuclear burning.
How Long Do Stars "Live"?
If you took a large enough sample of stars (and avoided the slection effect of preferentially picking the brightest ones) you would discover that 90% of those stars are either on or very close to the main-sequence when plotted on an H-R diagram. You can iterpret this to imply that a star spends 90% of its lifetime as a main-sequence star. This begs the question - How long can a star be on (or close to) the main-sequence? To answer this, consider the following example:
Example 9.10 How much more energy does a 5 solar mass mid B-type star emit than our sun?
Solution: Recall from the last chapter that luminosity is the measure of how much energy a star emits each second and that luminosity depends on mass as expressed by the Mass-Luminosity relation . This means that the 5 Mo star emits (5)3.5 = 280 times as much energy as the sun each second.
We can infer the main sequence lifetime of a star relative to the lifetime of our sun by simply considering the rate at which the star emits energy (which depends on its mass) to the mass available for nuclear burning - that is - the mass of the star itself. Expressed as a formula this becomes:
The units for this are "solar lifetimes". As you saw in a previous chapter, the sun can continue to fuse Hydrogen in its core for about 10 billion years. If we consider theA 5 solar mass star from Example 9.10, it has 5 times as much mass as the sun but burns its fuel 280 times fatser! Compared to the sun its lifetime is:
This means that the main-sequence lifetime of a 5 solar mass star is less than 2% as long as the sun. To put this in units of years the main-sequence lifetime of a 5 Mo star is (0.018)(10 billion years) = 180 million years.
The following applet provides an interactive graph that allows you to determine the main sequence lifetime of a star.
| Applet "AgeCalc" provides you with a simple interactive graph to determine main-sequence lifetime of a star. Units are "mega-years". |
Practice
- Compare the main-sequence lifetimes of M-type and O-type stars. Which lives longer and by how much? (You will need to make resonable estimates for the masses of the stars).
- A "galactic year" is the time required for the sun to orbit our galaxy - about 250 million years. During this time stars can drift appreciably from the regions in which they form. True or false (and give your reasons!): Massive stars are always seen very close to where they formed.
- Use Stellarium (or any other planetarium software) to investigate the Pleiades cluster (otherwise known as M45). Find the spectral classification for the bright Pleiads - Atlas, Pleione, Alcyone, Asterope,Maia and Electra and from this estimate the age of the Pleiades.
- The bright red star Betelgeuse (the upper left corner of Orion) has a spectral type of M2Ib. Explain why you could not use the lifetime calculation developed in this section to estimate Betelgeuse' lifetime.
- Explain clearly when both the mass-luminosity and lifetime formulae can be used.
To understand the evolution of main-sequence stars
Chp 12.1
The fact that stellar "personality" is primarily a function of mass and composition is sometimes referred to as the "Russell-Vogt Theorem" in honour of Henry Norris Russell and Heinrich Vogt.
Temperature gradient is the rate at which temperature changes over distance.