Lecture Notes

Arny Chapter 12

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The relationships between stellar (star) properties like luminosity, color, size, etc. is summarized in the Stellar Properties Chart.

Chapter 12 Measuring the Properties of Stars
Our Sun is a star, the stars are all Suns.
Are all stars like our Sun?
Or are there many different types of stars?

Section 12.1 - Measuring a Star's Distance
How do we determine how far away a star is?

This is a fundamental problem in astronomy.
How do you determine the distance to something that you can't reach physically?
We will spend a lot of time the rest of the quarter discussing methods of determining distances to stars, galaxies, and the like.

We can determine distances using triangulation or parallax.
Parallax: The shift in the direction to an object as the observer moves.
[See figures 12.1 and 12.2 on pages 351 and 352.]

Baseline: The distance moved by the observer.
Parallax angle = half of the angular shift.

Stars, being very distant, show very little parallax.
Using a larger baseline gives larger parallax angles and hence more accurate distance measurements.
The most convenient large baseline
available is the diameter of the
Earth's orbit = 2 AU.
In our textbook, the baseline is
considered to be half this, 1 AU.
This simplifies the calculations for the
text because it uses the parallax angle
rather than the angular shift in calculations.
[This is not as simple as it looks because the solar system is moving at the same time that the Earth is orbiting and the other star may be moving as well.]

Even with a baseline of 1 AU, no star shows a parallax angle larger than 1 arc-second ( = 1/60 of 1/60 of 1 degree).
One arc-second is the angular size of a quarter at 3.25 miles.

The general formula for determining distances using parallax is derived in Extending Our Reach: Measuring the Distance to Sirius on page 353 of the text.
The general formula is called the Parallax Formula:

d = 360 b / (2 pi p)
where
d is the distance to the star
b is the baseline (called AC in text rather than b), usually b = 1 AU
pi = 3.14
p = parallax angle in degrees of angle

Whatever units are used for b will be the units for d.
We will use the same formula many times in the Distances computer lab, although we will be using b as the 'full' baseline, and the angular shift rather than p.

Stellar (star) distance measurements using triangulation (parallax) are most conveniently calculated in units of "parsecs".
1 parsec = 1 pc = 3.26 light-years = 3.26 LY
A parsec is the distance at which an object viewed with a baseline of 1 AU will show a parallax of 1 arcsec.
"par-sec" = parallax of 1 arc-second.

Using the parsec unit gives a simple formula for computing distances measured using parallax.
When using a baseline of 2 AU (or "1 AU"), opposite sides of Earth's orbit.
Distance to star in parsecs = 1/(parallax angle in arcseconds)
d = 1/p
This formula only applies when using a baseline of 2 AU.
The formula was obtained from the above result by putting in 1 AU for b, changing units from AUs to pc, degrees to arc-seconds.

Section 12.2 Measuring the Properties of Stars from their Light
How do we determine the luminosity of a star?

The rate at which a star emits light (energy) is called its luminosity.
Rates of energy are given in units of watts (W).
The luminosity of the Sun is L = 4 x 10^26 W (that's a lot of light bulbs!).
This is the total rate at which light energy comes off the star's surface.

But not all of that light hits the Earth.
In fact, only a very small fraction hits the Earth, and even less enters your eye.
The brightness of the Sun is measured by giving the rate at which light energy will hit each square meter of area on the Earth from the Sun.
That value is 1400 W, or 1400 W/m^2.
This is how we measure brightness.

Some stars appear bright, others dim.
A bright star means we are receiving more light from it.
The brightness we see depends on two things:
1. The luminosity of the star
Luminosity = total light output.
2. The distance to the star
If the star is further away, it will appear less bright.

There is a formula for this (seems like there's a stupid formula for everything, huh?):

B = L / (4 pi d^2)
where
B = brightness (measuring, again, how bright it looks to us)
L = luminosity
pi = 3.14
d = distance away of star

Note that with luminosity in units of watts, and distance in meters, this gives brightness in units of W/m^2.

This formula is called the Inverse Square Law.
For an object with luminosity L, the brightness B that we will see depends on distance d, we must divide by the distance squared.
Dividing by means multiplying by inverse, second power means square, hence the "inverse square" law.

Solution: B = ?
L = 4 x 10^26 W
d = 1 AU = 150 million km = 1.5 x 10^11 m
B = (4 x 10^26 W) / (4 x 3.14 x (1.5 x 10^11 m)^2) = 1415 W/m^2
agreeing (with some slight rounding errors) with the 1400 W/m^2 claimed before.

If we see a bright star, we don't know if it's got a high luminosity, if it's nearby, or both.
Planets appear very bright in our sky, not because they have a high luminosity but merely because they are extremely close.

How bright a star appears depends on both its luminosity and its distance away.
If we know the luminosity or distance, we can solve for the other.
This gives us a different way of finding star distances.
[Called the method of standard candles.]
We'll return to this idea in section 12.8.

Surprisingly, most of the bright stars you see at night appear so bright because they are highly luminous, not because they are the closest to us.
Of the 6000 stars that can be seen with the naked-eye, only 6 are less luminous than our Sun.
Why is luminosity more important than distance?

The luminosity of a star depends on its
(1) temperature and (2) size.
The hotter an object the more light it emits from each bit of surface area.
The bigger an object, the more total area from which it emits light and hence the larger the total of light emitted.

The Stefan-Boltzmann Law describes these relationships mathematically.

L = (4 pi R^2) s T^4
L = luminosity
R = radius
(4 pi R^2) = surface area of object
s = Greek letter sigma = Stefan-Boltzmann constant = I'm not going to tell you because we won't do any math with the formula anyway.
T = surface temperature (in Kelvin).

Bigger => more surface area
Hotter => each bit of surface emits more light
Stars can vary in luminosity tremendously, there are stars with luminosities more than a million times that of our Sun, and others 10,000 times less luminous.
Size matters more than temperature for stars because stars vary much more in size than they do in temperature.
Big stars are really, really, big.

The big stars have such huge luminosities that the distance doesn't have much affect on them.
Most stars are very dim, so we can barely see them even if they are nearby.
Of the 40 closest stars, only 3 are more luminous than our sun.

How can we determine the temperature of a star?
The surface temperature determines the star's overall color.
We talked about this in chapter 3, Wien's law.

Yes, stars come in different colors!
I know, they all look white.
That's mainly because humans can see dim things only in black and white, it has to do with the physiology of the eye.
With experience, you can notice the different (but subtle) colors of stars.
There are red stars and blue stars and yellow stars and more.

Of course, cameras can be used to take color pictures of stars.
You'll find lots of pictures in the text where you can see different star colors.
For instance, the pictures on pages 26 and 369 shows many colored stars.

Anyway, we can determine the surface temperatures of stars simply by looking at them, or more precisely measuring the wavelength of peak radiation and using the Wien's law formula.

How do we determine the size of stars?
Using the Stefan-Boltzmann law, L = (4 pi R^2) s T^4,
we can solve for the size (radius R) of stars when we know the other factors (luminosity L and surface temperature T).
There are also ways (not easy but they have been done) of directly viewing stars and measuring their size. Called interferometry.
Eclipsing binary stars (where two stars orbit each other and eclipse each other occassionally as seen from Earth), star sizes can be determined from the duration of eclipses. (Sec. 12.4)

Stars can be hundreds of times larger than our Sun (called giant or supergiant stars) or hundreds of times smaller than our Sun (called dwarfs).


The magnitude system.
Astronomers don't usually measure the brightness of stars in W/m^2.
Instead, they use a system that dates back to Hipparchus, called the magnitude system.
We will describe this in more detail in lab.
Note: Apparent magnitude, or just magnitude, is another way of expressing brightness.
Absolute magnitude is a different way of expressing luminosity.

Section 12.3 Spectra of Stars
How do we determine the composition of stars?
From the spectra of stars, the colors missing in their spectrums.
In the early 1900's, the spectra for thousands and thousands of stars were classified by Edward Pickering and Annie Cannon.
Most of these stars had an absorption spectrum (a full rainbow but with a few colors missing due to absorption of some colors by the atmosphere of the stars).
The spectra of most stars included lines due to hydrogen.

The initial classification of stars was made based on how clearly the hydrogen lines could be seen in the spectrum.
Star types A, B, C, D, E, F, ...
The detailed early history of this classification is discussed in the text.

This does not mean that some stars have lots of hydrogen and others little.
All stars are composed of about 3/4 H and 1/4 He, plus a small fraction of "metals".
The variations in the spectra of stars is due to different surface temperatures.
Because of temperature variations, the hydrogen, helium, and metals can be ionized or excited in different ways.

It wasn't until later that the relationship between star temperatures and spectra were fully understood.
The modern stellar classification system ranks stars in order of temperature:
O B A F G K M
high temp to low temp

Note some classification letters were absorbed or dropped.
Each category is broken down into 10 sub-divisions,
G -> G0, G1, ..., G8, G9
G0 is hotter than G3 hotter than G9, F9 is almost the same as G0.

How are we supposed to remember OBAFGKM?
Mnemonics handout. (Oh Be A Fine Girl/Guy, Kiss Me)

These letters (the stellar classification system) rank the stars in order of temperature.
They also rank the stars in terms of color.
Why? Because of Wien's Law.

Spectral Class	Star Color	Surface Temperature
O	violet-blue	40,000 K
B	blue	16,000 K
A	blue-white	8500 K
F	white	6500 K
G	yellow-white	5500 K
K	orange	4200 K
M	red	3000 K

Astronomers have recently added spectral classes L and T to the end of this list, they are even cooler than M-type stars.

Our Sun is a G2 type star (5800 K), close to F type.

As we shall see, spectral class also correlates with mass, luminosity, size, and lifetime of stars.
But there are some exceptions.

How can we determine the motion of stars?
We can use the Doppler shift (chapter 3).
In most stars we see the familiar spectrum of hydrogen.
But sometimes all the lines are shifted towards blue, from the Doppler shift we understand this to be a star moving towards us.
From the amount of the shift we can determine the speed of the star.
Shifts to the red indicate stars moving away.
[See figure 12.12 on page 366.]

A similar procedure can also give information about how fast a star is spinning or the rate at which stars orbit each other or orbit in galaxies.

Section 12.4 Binary Systems
How do we determine the masses of stars?
Whenever we see objects orbiting each other, we can use Newton's version of Kepler's third law to determine the total mass.
And since over half of all stars are in "binary systems", we can determine the mass of most stars!

Newton's version of Kepler's third law (in the simplest form we worked out in chapter 2) is

M + m = a^3 / P^2
where
M and m = masses of stars in units of solar masses
one solar mass = mass of our Sun, symbol = Mo or Ms
A star with a mass of 4.4 Mo has a mass 4.4 times more than our Sun.
a = average distance between stars in AUs
P = period of time for orbit in years.

Imagine you're watching a pair
of stars orbiting each other (a
"binary" system, see figure 12.13
on page 368) in a big telescope.
The period P is easily determined.
Just time how long it takes for the
stars to return to their original locations.
(It may take hours, days, months,
years, or even decades.)

It may look just as easy to determine the separation distance a.
But:
(1) We are seeing the angular separation.
To get the true separation we must know the distance to the pair.
And distances can be tough to get (parallax only if close enough).
(2) We might be seeing the binary system more edge-on, rather than face-on.
In the extreme edge-on case, we don't know how far the stars are moving in or out (of our line-of-sight) and hence we don't know the true separation.

We can use the Doppler shift to help determine this.
From the shifting red and blue shifts as the stars orbit (see figure 12.13 on page 374) we can determine their speeds which helps determine distances.
Sometimes, the only way we know we are looking at a binary star system is that we see two spectra, these are called spectroscopic binaries.

Even when we know P and a, that enables us to compute just (M + m), the total mass not the individual masses.
The individual masses can be determined only by learning additional details about the orbits.
Basically, the lower mass object moves faster and in a bigger orbit (like the Earth compared to the Sun in the solar system).
How much bigger and faster determines how much less mass.

Solution: (m + M) = a^3 / P^2
= (4)^3 / (2)^2 = 64 / 4 = 16
The combined mass of the two stars is 16 solar masses.

Section 12.5 Summary of Stellar Properties
Nearly all stars are 71% H, 27% He.
They have surface temperatures from 30,000 K to 3000 K.
The have masses from 100 to 0.1 solar masses.
Table 12.4 (page 372) summarizes how astronomers determine properties of stars.
This is an excellent table, you should know it all!

Section 12.6 The H-R Diagram
H-R is short for Hertzsprung-Russell.
Usually just called an H-R Diagram, sometimes called a color-magnitude diagram.

Imagine for every star, we plot the luminosity of the star (total energy output) against the temperature of the star (or stellar class).
The result is an H-R diagram:
[See figure 12.17 on page 373.]

[Full Page Diagram]

Our Sun is a G2 star on the main sequence.
The Main Sequence is the long diagonal across the diagram.
Most stars are main-sequence stars.
Red Dwarfs are main-sequence stars.
White Dwarfs, Red Giants, and Supergiants are groups of stars off the main-sequence.
[Figure 12.17 in the text]

O- and B-type stars are the least common.
But O-type stars (and non-white dwarf B stars) have huge luminosities.
Hence they stand out in our skies and appear to be more common.
Same with supergiants and red giants.
Generally, the stars with the greatest luminosity are those that are physically largest, so bigger stars are those higher in the diagram (see figure 12.19).

White dwarfs have high surface temperatures but very low luminosities.
They must have very little surface area!
The typical white dwarf is hotter than our Sun but only about as big as the Earth.
Despite their low luminosity, a lot of white dwarfs have been discovered, they must be quite common.
We'll have much more to say about white dwarfs later.

Red giants have low surface temperature but high luminosities.
These stars must be huge!
These stars can be as big as the orbit of the Earth!

Our Sun will one day turn into a red giant.
It will swell up in size and may swallow up the Earth.
Will happen maybe 5 billion years from now.
The Sun will be getting bigger, but won't be increasing in mass, it will have a much lower density (like cotton-candy).


From all the studies of binary systems, an important result has been obtained.
Called the mass-luminosity relation.
[See figure 12.20A on page 376.]
The mass-luminosity relation is a simple idea,
stars with greater mass are usually much more luminous.
It can be used to determine the mass from the luminosity or the luminosity from the mass (at least for main-sequence stars).

Another way of classifying stars is called luminosity class.
The details aren't important.
Figure 12.22 and Table 12.5 summarize this adequately, you will use luminosity class some in computer lab.

Section 12.7 Variable Stars
All stars vary in brightness (luminosity) slightly.
So you could say that all stars are variable stars.
Our Sun is variable, mainly due to the 11-year sunspot cycle.
Some stars vary in brightness a lot, and usually the term "variable star" only refers to those.
Reasons why stars are variable is discussed in the text, we'll mention some of that later in the quarter.

Section 12.8 Finding a Star's Distance by the Method of Standard Candles
A "standard candle" is any object for which we accurately know the luminosity.
Remember we directly see a star's brightness, not its luminosity.
The luminosity could be worked out using the mass-luminosity relationship, the H-R diagram, luminosity class, or some combination.

If we know the luminosity L, we can use its brightness B (which we can always easily measure) to calculate the distance d.
B = L / (4 pi d^2) (inverse-square law)
This is an extremely important idea that comes up lots more later in the quarter.


Chapter Summary
Table 12.4 makes a better summary than that on page 380 although figure 12.25 is nice.

End chapter 12 lecture.