1. Stars II Stellar Characteristics: Mass, Temperature, & Size

2. Attendance Quiz Are you here today? (a) yes (b) no
(c) see? I told you so!

3. Today’s Topics Stellar luminosities Stellar masses
Stellar temperatures and sizes
Laws of Thermal Radiation
Stefan-Boltzmann Law
Luminosity, Temperature and Size
Hertzsprung-Russell Diagram (intro)
Wein’s Law
Stellar Temperatures

4. Stellar Luminosities (from last time)
Stellar luminosities vary from L–1,000,000 L, ten orders of magnitude
Note that most of the stars in this image are at the same distance, so their relative apparent brightness is the same as their relative luminosities
Note that there are many more faint stars than bright stars, suggesting that less luminous stars are far more common

5. Stellar Masses Stellar masses are quite difficult to measure
However, about 2/3 of stars are part of a binary system
In those cases, we can use Kepler’s 3rd law to find masses
p2  a3
where the proportionality constant depends on the masses of the system
In general, for two objects orbiting their center-of-mass
For the Solar System M1+M2  M

6. Stellar Masses
For binary stars, M1 and M2 are more similar than in the solar system
However, if we can measure the relative speeds of the two stars as they orbit, these allow us to determine the ratio of the two masses, which together with Kepler’s 3rd Law, allows us to find the masses individually
There are three observable types of binary star systems
Visual binaries
Spectrocopic binaries
Eclipsing binaries

7. Binary Star Systems
In visual binaries, it is possible to measure a and p directly, and get v from the orbit
In spectroscopic binaries, the absorption lines shift back and forth as the stars orbit each other, due to the Doppler shift (Interactive Figure 15.8)
From the Doppler shifts, we can find the speeds (along the line of sight) of the orbits
Eclipsing binaries are the most important of the binary systems: the orbit is almost edge-on, and the stars move in front of each other, causing dips in their brightness (Interactive Figure 15.7)
We can find p from the time between eclipses; v from the Doppler shifts (which we know is LOS), and find a from p and v

8. Stellar Masses To summarize:
For a binary system, if we can find the period p, and the semi-major axis, a, of the orbit, then we can find the sum of the stellar masses using Kepler’s 3rd Law
If we can find the relative speeds of the two stars in their orbits (using either our knowledge of orbital mechanics or the Doppler shift), we can use this information to find the relative masses
Together, this information allow us to find the masses of the two stars
Stellar masses vary from 0.08M to 150M
Less massive stars are more common than more massive stars

9. Stellar Temperatures and Sizes
As we have seen, stars emit an absorption spectrum
The hot inner core emits thermal (continuous) radiation
The cooler atoms in the outer atmosphere of the star absorb light at the specific wavelengths corresponding to the transitions within those atoms
The continuous part of this spectrum can be used to find stellar temperatures
We can also use a combination of luminosity and temperature to determine stellar sizes (radii)

10. Laws of Thermal Radiation
A plot of intensity v. wavelength of a continous spectrum looks like the curves below
There are two rules that govern curves of thermal radiation
Stefan-Boltzmann Law - each square meter of a hotter object emits more light at all wavelengths than a cooler object (L/m2  T4)
Wein’s (“veen’s”) Law - hotter objects emit photons with a higher average energy (shorter wavelength) (max  1/T)
(Active Figure 5.19)

11. Stefan-Boltzmann Law
The Stefan-Boltzmann Law means that objects can be more luminous for two possible reasons
If an object is hotter, it will give off more total energy (L  T4)
Since the energy per square meter of surface is the same for all objects at the same temperature, an object which has a bigger surface area will give off more total energy, for a given temperature
For a sphere, surface area = 4πR2, so L  T4R2

12. Luminosity Quiz I
A lump of lead is heated to a high temperature. A lump of gold of
the same size as the lump of lead is also heated to the same high
temperature. Which lump of material is brighter?
The lump of lead is brighter.
The lump of gold is brighter.
Both lumps are equally bright.
You cannot tell which lump is brighter without knowing more about the chemistry of lead and gold.

13. Hertzsprung-Russell (HR) Diagram
In the early 20th century, two astronomers independently had the idea of plotting stars on a temperature-luminosity plot
This diagram is named in their honor a Hertsprung-Russell diagram (HR diagram for short)
Note that the x-axis has temperature increasing to the left (backwards)
This is because HR actually plotted the stars using a measure of color (spectral type) from blue to red
This diagram (which will discuss in a great deal of detail) is the key to unlocking the secrets of how stars differ both in their properties and their evolution

14. Stefan-Boltzmann Law (summary)
The Stefan-Boltzmann Law means that objects can be more luminous for two possible reasons
If an object is hotter it will be more luminous
If an object has a bigger surface area it will give off more total energy, for a given temperature
A star can be luminous either because it is hot or because it is big

15. Lecture Tutorial: Luminosity, Temperature and Size, pp. 53-56
Work with one or more partners - not alone!
Get right to work - you have 20 minutes
For question 1, each of the four pairs of burners requires an answer to the question: which burner will cook the spaghetti more quickly?
For each pair, make sure to consider all the options:
the left-hand burner cooks faster,
the right-hand burner cooks faster, and
there is not enough information to tell
If you find yourself unsure of how to answer for any of the pairs, just put a question mark and move on

16. Luminosity Quiz II
In question 1D in the Luminosity, Temperature, and Size
LT, which hot plate cooks the spaghetti faster?
The smaller, hotter (left-hand) plate
The larger, cooler (right-hand) plate
There is not enough information to tell

17. Luminosity Quiz III
The stars Antares and Mimosa each have the same luminosity.
Antares is cooler than Mimosa. Which star is larger?
Antares
Mimosa
They are the same size
There is not enough information to tell

18. Luminosity Quiz IV
You observe two stars with the same luminosity and determine that
one is larger than the other. Which star has the greater temperature?
the smaller star
the larger star
The temperatures are the same
There is not enough information to tell

19. Luminosity Quiz V
Imagine you are observing two stars. One star is hot and small and
the other star is cooler and larger. Which star is more luminous?
the hotter star
the larger star
They have the same luminosity
There is not enough information to tell

20. Luminosity Quiz VI
Rigel is much more luminous than Sirius B. Rigel and
Sirius B have the same temperature. Which star has the
greater surface area?
Rigel
Sirius B
They have the same surface area
There is not enough information to tell

21. Stellar Temperatures (Colors)
As we have seen, a hot, dense object gives off a thermal spectrum
Wein’s Law states that the peak wavelength varies inversely with the temperature of the object (Interactive Figure 5.19)
Thus, bluer = hotter and redder = cooler
Distance does not affect color

22. Stellar Temperatures (Colors)
We can see the effect of Wein’s Law with stars
Note that many of the stars are blue, but many more are red
This is because there are many more cool stars than hot stars (similar to what we have seen with mass and luminosity)
To determine T it is not necessary to measure the entire spectrum
Simply measuring the relative amount of blue and yellow (or blue and red) light, it is possible to get a fairly accurate surface temperature for a star

23. Lecture Tutorial: Blackbody Radiation, pp. 57-60
Work with one or more partners - not alone!
Get right to work - you have 15 minutes
Read the instructions and questions carefully.
Discuss the concepts and your answers with one another. Take time to understand it now!!!!
Come to a consensus answer you all agree on.
Write clear explanations for your answers.
If you get stuck or are not sure of your answer, ask another group.
If you get really stuck or don’t understand what the Lecture Tutorial is asking, ask me for help.

24. Stellar Temperature Quiz I
For Question 15 (based on Figure 2c) of the Blackbody Radiation Lecture Tutorial, which star is larger?
Star E is larger
Star D is larger
They are the same size
There is not enough information to tell

25. Stellar Temperature Quiz II
Use the graph at right to determine which of the following best
describes how Star A would appear as compared with Star B
Star A would appear more red
than Star B
Both stars would appear more
red than blue
blue than red
Star A would appear more blue
None of the above

26. Stellar Temperature Quiz III
Use the graph at right to determine which of the two stars (A or B)
emits light with the longer wavelength peak
Star A
Star B
Both stars’ peak emissions are
at the same wavelength
None of the above are possible

27. Stellar Temperature Quiz IV
The graph at right shows the blackbody spectra for three
different stars. Which of the stars is at the highest
temperature?
Star A
Star B
Star C