How Big Are the Stars?

173-183

No matter how big a telescope we use we really can't see the surfaces of stars. They are just too far away. This means that in order to answer the question "How big are the stars?" we must resort to a number of indirect methods. Indeed, it is quite remarkable that we can do this at all! As you now appreciate, the distances between us and the nearest stars are enormous. How could we then determine the size of star?

Luminosity, Radius and Temperature - the Critical Link

In an earlier section we introduced the oddly-named "black body" to describe how hot objects emit light energy. You learned that the energy emitted by a blackbody (i.e. Flux) depends on the "4th power" of the temperature. This is a very strong relationship. For example, if you double the temperature of an object then the flux of energy radiating from the object increases by a factor of 2⁴ or 2x2x2x2 =16 times! As you have just seen, the luminosity of a star also depends on its surface area. So, combining these ideas tells you that knowing the intrinsic brightness of a star also tells you about its size and temperature. This is illustrated for you in the following applet.

Applet Stellar Luminosity

Example 8.11 Use the following data to compare the size of the bright star Capella and our sun. Do this without using any formulas but remember that a magnitude difference of 5 is the same as a brightness factor of 100.

Solution: Since Capella and our sun have essentially the same temperature any difference in intrinsic brightness must be due to a difference in their surface areas. The difference in the absolute magnitudes of the two stars is nearly 5, so Capella must be about 100 times brighter. Since we just concluded that this is due to a difference in surface area we see that Capella must have about 100 times the surface area. Surface area depends on the square of radius so Capella must be approximately 10 times the radius of the sun.

Example 8.12 The bright star Arcturus has an apparent magnitude of -0.29. Its distinctly orange colour is consistent with a surface temperature of 4300 K. The parallax for this star is 0.089". Use this information to estimate the size of Arcturus. (Hint: use some of the applets given in this and previous sections to assist you.)

Solution: This is a multi-part problem which combines ideas from the previous 3 sections. Use the following strategy:

1. First determine the distance to Arcturus.
2. Given the apparent magnitude you will be able to find the absolute magnitude of the star.
3. Next, using the surface temperature and the temp-radius applet you can find an appropriate radius to give an appropriate intrinsic brightness.

• Distance: Use the distance - parallax relation () to determine that Arcturus is 1/0.089 = 11.2 pc away.
• Use the distance-modulus applet to conclude that at a distance of 11.2 pc the distance modulus is m - M = 0.25
• Therefore the absolute magnitude of Arcturus is M = m - 0.25 = -0.29 - 0.25 = -0.54
• Using the temp-radius applet suggests that Arcturus has a radius of 22.3 times the radius of the sun or 22.3R_o

The Hertzsprung-Russell Diagram

All of what you have learned in the last 3 sections can be combined with what you learned earlier about stellar spectral classification to construct one of the most significant relationships in astronomy - the Hertzsprung-Russell or H-R diagram. The H-R diagram shows a relationship between the intrinsic brightness of a star (expressed either as absolute magnitude M_v or as luminosity) versus the star's temperature (or spectral type).

Figure 8.3 Rollover figure of the Herztsprung-Russell diagram showing temperature or spectral class on horizontal axis.

The H-R diagram is remarkable. When stars are plotted according to their intrinsic brightness and temperature a number of very distinct patterns emerge. This is shown in the following diagram produced by data from the Hipparchus satellite and represents one of the most complete and accurate H-R diagrams produced to date.

One of the first things to note in the H-R diagram is the grouping of stars. Roughly 90 percent of the stars fall along a band that stretches diagonally from upper left o lower right on the diagram. This is called the main-sequence. The main-sequence and other "groupings" are more correctly called luminosity classes and represent specific epochs in the evolutionary life-cycle of a star. The luminosity classes are summarized in table 8.2 below. Letter Designation Luminosity Class VII: White dwarfs; very faint, extremely small but at times very hot VI Subdwarfs; similar to main sequence but somewhat smaller and less luminous V Main sequence; diagonal band stretching across the HR diagram. IV Subgiants; brighter than the main sequence, somewhat larger stars III Giants; very large and luminous stars II Bright giants; somewhat larger and brighter giants Ib, Ia, IO Supergiant stars; exceedingly bright and massive stars.

Figure 8.4 Hertzsprung-Russel diagram generated from Hipparchus space satellite data.

Table 8.2 Stellar Luminosity Classes

Exploring the H-R Diagram

Use the following applet (HRexplorer) to explore the wealth of information contained within the H-R diagram. To assist you, complete the following tasks:

1. Investigate how the position of a star changes if you vary the temperature of the star but do not change the luminosity. Explain why the position and radius changes as they do.
2. Repeat the steps from number 1 above but this time vary the luminosity and not the temperature. Again - explain why the position and radius changes as they do.
3. By varying both luminosity and temperature "move" along the main-sequence.

Just What are Luminosity Classes?

The division of the H-R diagram into luminosity classes is not "accidental". Rather it points to the underlying physics of how stars create energy and how they evolve. Roughly 90% of stars are found in the main-sequence section of the diagram because this is where a stars spends 90% of its lifetime. As a star ages important changes occur to the star's surface temperature and size. This means a specific star's position on the H-R diagram changes as the star evolves. Thus, knowing which luminosity class a star is in tells us something about the age of that star.

So, how do you know what luminosity class a star belongs to? This is not simple but ultimately rests on looking at the spectrum of light coming from the star. The following figure shows two spectra for two very different stars. Each of the stars is an A-type stars. One, however, has much narrower spectral lines than the other. The narrow line tells the astronomer that the star producing this absorption line ha a very rarefied atmosphere and is a super-giant star. In the other spectrum, the lines are broader suggesting a much denser atmosphere.

Spectrum of an A3 I supergiant star
Spectrum of A3 V main-sequence star
Figure 8.5 Comparison of spectral line widths: Supergiant star and a main-sequence star.

Using the HR Diagram - Spectroscopic Parallax

• estimating stellar distances measuring stellar age
• estimating stellar lifetimes

• spectral-luminosity type relating to temperature
• distance modulus

Example 8.13 Through the telescope you find a faint blue star. Spectroscopic examination reveals that it is a B5 V star and your CCD photometer tells you that it has an apparent magnitude of 6.7. How far away is the star?

Solution: Use the applets HRexplorer and distanceModulus to help answer this.

The first thing to do is to look at the HRexplorer . Since a B5 V star has a surface temperature of 15 500 K we can judge that its absolute magnitude is -1.1. We now know the distance modulus:

If you use distanceModulus you can conclude that the star must be about 360 pcs away or about 1200 ly away. 

Spectroscopic Parallax for Math Geeks! (or people who just like equations...)

There is a relatively straight-forward mathematical connection between distance modulus and distance that all starts with the brightness-magnitude relation that you learned at the beginning of the course. If we let 'r' stand for the distance (in pcs), m the apparent magnitude and M the absolute magnitude then we get two important formulae:

Example 8.14 Use the formulae given above to confirm that the star in Example 8.13 is 360 pcs away.

In example 5.13 you concluded (by the method of spectroscopic parallax) that the distance modulus of the star is 7.8. Put this into to get

The formulae gives a more exact answer of 363.1 pcs.

Practice

1. Use the applet StellarLuminosity and the data from Example 8.11 to compare the size of the bright star Capella and our sun.
2. Use Stellarium (or any other web based data source) and the HRexplorer applet to compare the radius and temperature as well as luminosity class for the following stars:

3. How can we be certain that white dwarfs are very small?
4. How does an H-R diagram make it clear that giant stars are larger than main-sequence stars of the same temperature?
5. If a star is 8 times the radius of the sun but one-half the temperature how will its luminosity compare to that of the sun? Please express your answer quantitatively.

To determine the size of the stars

Chapter 9.3,9.4

Altair is one of the very few stars that we have been able to resolve. The image shown below is a direct image of Altair produced by the CHARA array at Mount Wilson. Altair is not spherical! Its high rate of rotation squashes into this oblong shape.

The bright star Capella appears low in the north during Canadian summers and high over head during the winter months. The star is actually a binary system consisting of two giant G-types stars.

The H-R diagram is a fundamental diagram in astrophysics that relates stellar luminosity to temperature.

Eijnar Hertzsprung (1873-1967) was a Danish astronomer who was also trained as a chemical engineer. Other achievements include the recognition that the thin shape of spectral lines in supergiant stars indicate the very low density of the atmosphere of these stars.

Henry Norris Russel (1877-1957) was one of the most influential American astronomers of the last centaury. He made fundamental contributions in many parts of astrophysics as well as atomic physics.