Great. So now we have a catalog, a growing
catalog of stars to which we know the distance.
We have pulled those stars. We, some of them stars nearby,
down off of the celestial sphere. We know the distance to them, we live, they live
in our three-dimensional universe. And what does that tell us about them?
Well, tells us a lot. Let's start by seeing what we can tell
about a star just by looking at it, that's all we can do with stars.
But, assuming that someone tells us because we measure carefully the distance
to it. Well, one thing we can do is we can do a
photometric measurement. We can measure the brightness of a star,
literally aim a telescope with a known aperture area at the star and measure the
rate at which the telescope gets energy from the star.
You correct for dark sky. There's a process,
it's not that difficult. But if we know the brightness, then we
can remember, remind ourself that brightness is related to luminosity of
the star by this expression. Right?
And this is luminosity divided by four pi D^2, if we know the brightness because we
measured it. We know the distance because we measured
it. We can extract an expression by solving
this for the luminosity. So we know the actual luminosity of the
star. Now we can tell you, is this some very
huge, very luminous star, or some very tiny star?
this is norm way to write the equation, a more convenient form in which to write
it is to scale it to something known. In other words, this same equation holds
for any star in particular, it holds for the sum.
And the advantage of writing it in this way is now whether this suggests is if I
divide the left-hand sides and set that equal to the ratio of the right-hand
sides, first of all the, four pis and whatever constants are sitting around
cancel, but more usefully I now get a product of dimensionless quantities.
I get an expression that says that this divided by that is this divided by that
or this expression here, where I have plugged in that the value of the distance
of the sun from earth is one astronomical unit and the sun's brightness is the
solar constant. So we have rate of dimensionless object.
the luminosity of the star in units of solar luminosity written as the product
of two others. Whenever possible, we'll write it that
way. Okay.
Now we know the luminosity of the star. What else can I get just by looking at
it? Well, if I'm careful I can measure the
color, star's color. And the star's color as we know gives us
information about its temperature. If I find the maximum of its black body
spectrum, I can get an estimate from a temperature from Wien's law.
So there is Wien's law. It relates the surface temperature of a
star to the wavelength at which it emits its maximum emission,
this is written in Kelvin. And now, if I know the temperature and I
know the luminosity, I can remind myself that I have yet another expression for
the luminosity of any star. L was equal to the sigma T^4, given by V
Stefan-Boltzmann law for the rate at which each square meter of the star
surface radiates times the surface area of the star, which in terms of the star's
radius is given by four pi R^2, where R is the star's radius.
And now, if I know T and I know L, I can extract an expression for R.
you find that R^2 is L over four pi sigma T^4 or if I just want R, I simply take
the square root of both sides. And I can rewrite that as the square root
of L over four pi sigma times the square root of one over T^4 is T minus two, one
over T^2. And in this form, again I can compare to
the sun by dividing this expression for the star by the same expression for the
sun. The radius of the sun is luminosity of
the sun square root divided [INAUDIBLE] one over the temperature, solar
temperatures square, and this gives me another one of those
nice scaling expressions. But then, bottom line is if I know the
temperature and luminosity, for which I needed observations and the distance, I
can get the radius of the star. Now,
now some stars, a few tens have actually been imaged and this is a recent
development. There's a few tens of stars for which the
radius has actually been measured particularly optically by imaging them
and measuring the angular distance between the left and right side of the
star. This is a new result, but most stars,
when we say we know their radius, we know it from this.
This is brilliant, we can know a lot from a star, about a star if we know it's
distance. the problematic measure here is the
temperature Remember we said the black body spectrum is very broad.
If you have a very broad mountain, identifying exactly where the summit is
can be a problem, in the same way, finding the precise maximum of a very
broad black body spectrum is a difficult and imprecise task.
We want a better thermometer, since measuring the stellar temperature is so
important. And, it turns out that we have a better
thermometer. it's also less subject to distortion,
for example, if there is a cloud of gas or dust between us and the star, we know
that we'll preferentially scatter blue light star viewed through dust or gas
will appear redder than it is and that might mislead us into thinking it is
cooler than it is. In fact, if you looked at the sun at
sunset and tried to devise it's temperature from it's color or even in
the middle of the day, you would conclude that it's yellow rather than green.
Now something that is not distorted by interception intervening
media is the star's absorption line spectrum.
So, we can look and understand something about the chemical composition of the
star's atmosphere. This was known and there was a big
ongoing project late 1930, twentieth centure into the 20s to classify the
forms that stellar absorption spectra take.
It was clear that there was a lot of information there, but isn't clear until
the 20s exactly what it is that we are classifying when we classify stellar
spectrum. we now understand the classification.
Here is a sort of compilation of representative stellar
spectrum arranged in order of what is called spectral type.
And spectral type runs from A through B, A, F, G, K, and M.
And, within each category there are ten subdivisions,
so B0 is on one end and then it goes all the way up to B10,
and A0 up to A10 and so on. and,
if you notice there are sort of properties,
there are spectral lines that are very prominent, say in this region between the
Bs and the As, but almost absent in the O type stars and then disappear in the F
stars. This for example are the spectral lines
of hydrogen, these are [INAUDIBLE] lines of hydrogen and the interpretation that
is understood in the 20s is that this organization, and the reason B, B comes
before A is because there was previously another organizing principle, and this
is, this was understood later. this organization, this organization is
in direction of decreasing temperature. So O stars are the hottest stars and M
stars are the coolest stars. and for example if you look at the
hydrogen in the stellar atmosphere, then O stars are so hot that the hydrogen is
essentially ionized and in B stars much of the hydrogen is ionized.
An ionized hydrogen atom is just a free proton and electron, there is no
absorption line spectrum for a free proton.
So, hydrogen lines are almost completely absent from the spectrum of O and they're
really weak in the spectrum of B type stars.
in the cooler Bs and in the A type stars, hydrogen lines are very permanent because
hydrogen is now atomic. the ambient the, the light that the star
is emitting is at high frequencies so the and the temperature is high enough that
these hydrogen atoms get excited to their excited state in the observed
this language is associated to a hydrogen atom in it's first excited state the
hydrogen atom ground state excitations of the Hydrogen atom from the ground state
are ultraviolet and they don't appear in the visible light spectrum.
And indeed, as we proceed in lowering temperature, once we go from A to F type
stars, the hydrogen lines weaken and eventually disappear, because hydrogen in
the cooler atmosphere of F type stars is in the ground state.
And if it absorbs anything, it absorbs ultraviolet light,
very few of the atoms are in the first excited state where they can absorb
visible light protons, so hydrogen lines disappear.
What starts to appear is the spectral lines over here of ionized metals,
heavier elements. You see these forests with many, many,
many, many lines in them. and these are characteristics of more
complex atoms, so these are the spectral lines of ionized metals, and in cooler
stars, yeah, then cooler Gs and the Ks we find the spectrum of neutral metals.
So now, it's, the temperature is cool enough that heavier atoms do not become
ionized. And in the M types star the cooler M
types stars, we actually find these bands, these ray broadened bands of
absorption. This corresponds to the existence of
molecules with eternal degrees of freedom and so they can absorb light at various
wavelengths. And so, this progression is a progression
in temperature, and once you understand that, we know that we can use this as a
precise thermometer, here is the current list of spectral
types. they are listed according to the color,
the color the perception of stars is very subjective.
Here are the temperatures. Note that the temperatures at stellar
surfaces range from as hot as 50 or 60,000 Kelvin in hot O type stars to 2000
or 3000 Kelvin on the very cool, dark red M type stars.
And, these are the sort of description of the spectral features that characterize
them. This is the prevalence and I should be
careful, this is our understanding of the prevalence of this type of star among all
stars. I'll explain why this needs some
qualification. And note that it is precisely increasing
as you go to cooler stars. So, the coolest M type stars form three
quarters of all known stars, whereas the O type stars form a tiny, tiny fraction
and the cooler the star is the more likely it is to appear.
And here's some examples of stars of each spectral type.
And so dark red Betelgeuse and orange are Arcturus and Aldebaran are examples that
you are familiar with. The sun is a G type star.
In fact, G2 within the internal classification of G type stars and Sirius
and Vega are A type stars and so on. The bright stars of Orion's Belt are O
type stars. some of these stars are listed here in
not in black type and in the next clip, we'll understand what we now have is a
collection of data about stars. We now have a huge population of stars
about which we know their luminosity, their radius, their temperature,
and we can start trying to arrange this data as a sociologist would to do
population study and try to draw conclusions.
We'll figure out how to organize it in the next clip.