This lecture is about Cepheid variables and how they're used to find distances in space.
A Cepheid variable is a type of extremely luminous star that undergoes periodic changes
in luminosity, so its luminosity varies, so it's a variable star. They're so luminous that we're able
to observe individual Cepheid variables in other galaxies. The properties of Cepheid variables can
be used to determine distances in space. Cepheid variables vary their luminosity periodically,
which means they repeat over equal amounts of time. The graph of the Cepheid variable's
luminosity is not a sine graph however. The time it takes to regain luminosity is less than the
time it takes to lose luminosity. We can use luminosity-time graphs like the one below to
identify the period of a Cepheid variable. So here I can see the period the time it takes the Cepheid
to repeat one full cycle of becoming brighter and dimmer is 5 days, so the period of a Cepheid
variable can be found on a luminosity time graph by just looking at the time between two points of
equal luminosity. Cepheid variables have an outer layer that expands and contracts. When the outer
layer is larger and less dense, more light from the core of the star can pass through,
and the star is brighter. When the outer layer is smaller and more dense, less light from the core
can get through, and the star is dimmer, so more luminosity means the star is in its larger phase
and less luminosity means it's in its smaller phase. So a very common Cepheid question is to be
given a graph like this luminosity-time graph and to be asked where is the star largest and where
is its smallest? So it's going to be largest at its highest luminosity and smallest at its lowest
luminosity. Variable stars are called Cepheid variables because the first variable star to be
observed was delta sephi in 1784 by John Goodrick. The reason why we care about Cepheid variables so
much is that they're what allowed humans to find the exact distance to objects in space
that were too far away to apply the parallax method. So a big part of this lecture is going
to be about the history of astronomy and just how important Cepheid variables were to that history.
Before 1908, our only method for measuring distant objects in space was parallax,
and parallax only works for angles we can clearly measure. When the angles become too small,
our atmosphere begins to interfere with the measurements, and so the farthest we could
measure with parallax was about 30 parsecs, or about 100 light years. So we could not
measure the distance to any object beyond that length of 100 light years in space, and that's
an incredibly small distance to be able to measure in space compared to just how much we can observe.
Just as an example, the radius of the Milky Way, if I were to draw the Milky Way as a circle,
that radius is 50,000 light years, and so if we could only observe 100 light years out, this blue
circle represents how far we could measure in any direction before Cepheids and before 1908, and the
circle is not drawn to scale. If I were to compare this blue circle to the galaxy circle on the left,
this tiny little dot at the very end of that arrow is the total range of distance we could measure
before 1908, and before Cepheid variables, so you can see that's incredibly small compared to even
just the Milky Way galaxy, which is just one small part of the observable universe. Cepheid variables
allowed us to jump from a limit of 100 light years to a limit of 20 million light years after their
discovery, so just to compare how far we could measure after we discovered Cepheid variables: if
I were to place this red circle to scale inside of the purple circle, it would appear as this dot at
the very end of this arrow. So Cepheid variables alone completely changed our understanding of the
universe and how far away different objects were, which also allowed us to understand the shape of
our galaxy and the shape of other galaxies and eventually prove that the universe was expanding,
and this gigantic step was made possible by a pattern that was observed by an astronomer in 1908
astronomers had been aware of variable stars since the late 18th century, but in 1908 the astronomer
Henrietta Leavitt observed a relationship between the period and maximum luminosity
of variable stars that would allow scientists to find the exact distance of any variable star
she could observe, even stars that were in other galaxies. Before we talk about how she did this
we need to observe a problem with measuring distances in space. When we were working with
the brightness equation, which relates the brightness that we observe on earth to the
luminosity of a star and the distance from earth to the star, you may have wondered why we can't
just use this equation to understand exactly how far away any star is based on its brightness,
and the answer is that we don't know how luminous the star is. As an example, in the picture on
the bottom left we could imagine that star A appears brighter than star B in the night sky.
We could conclude from that that "Oh maybe star A is closer to us than star B and that's why it
appears brighter" but that wouldn't make sense because it could just as easily be the case
that star A is just more luminous than star B. It gives off more energy per second than star B
and star a could be farther away and still appear brighter in the sky. So what I'm
trying to say is that a star's brightness alone cannot tell us how far away it is.
So the brightness equation seems like it could be a useful equations for finding the distances
to stars, but we would need to know their exact luminosity (how much power they're giving off)
in order to use the equation to calculate that actual distance and before 1908 scientists did
not know the luminosity of stars farther away than 100 light years. Because we can always observe the
brightness on Earth, lowercase b in this equation just represents the observed brightness on Earth.
If we have the luminosity of a star we can observe, we can always calculate its distance, and
if we have the distance to a star we can always calculate its luminosity,
but we need one in order to calculate the other, and before Cepheid variables came along
we had no way of understanding the luminosity of stars farther away than 100 light years,
so we couldn't use this equation to calculate distance. Henrietta Leavitt was a computer
which at the time just meant a data analyst at Harvard who made observations about more than
2500 variable stars. And just as a heads up: When I say that she made observations, I mean that she
observed the data recorded by other scientists, because even though she was the one to make this
groundbreaking discovery, she did it without ever using a telescope herself, because women
were not allowed to use the Harvard telescopes in the early 1900s. Leavitt knew that every variable
star had an exact period where its luminosity would get larger and smaller and repeat,
and she wanted to know if there was a relationship between the period of the star and the maximum
luminosity that it reached. She didn't know how luminous variable stars were compared to
each other. She had the same problem that we just had looking at that brightness equation
where we can't understand the true luminosity of stars if we don't know the distance to the star.
She couldn't compare how luminous Cepheid variables were compared to each other,
because she could only observe their brightness, and a brighter star could just be closer rather
than having more luminosity. There were variable stars within the distance that could be measured
by parallax, so she could find the distance of those stars and calculate their luminosity using
the brightness equation, but there were not enough variable stars within that distance to get a clear
pattern for what she was looking for. She came up with a really cool solution to this problem which
was to observe many different variable stars in a far-away cluster of stars: the Magellanic Clouds.
She reasoned that because these stars seemed to exist in a cluster separate from other stars,
they were probably all roughly the same distance from Earth as each other. We now
know that the Magellanic Clouds are actually two dwarf galaxies that orbit the Milky Way.
If the stars that she observed were all roughly the same distance as each other, then if star A
were brighter than star B, it meant that star A was definitely more luminous than star B.
So this gave her a way of understanding which star was more luminous than another star. You can see
based on the picture that I've drawn that if we're observing two different stars from earth and we
have a good reason to think that they're the same distance away, if one star appears brighter in
the sky than the other, that has to mean that that star is more luminous than the other star, because
we know distance can't be affecting the brightness because the distance for both is the same.
She didn't know the actual luminosity of the stars she was observing, but she could still compare
their relative luminosity to their periods, so what that would mean is she could observe the
periods of different Cepheids: just how much time it takes for them to flash from brighter to dimmer
and back, and recording their relative luminosity would mean to say something like: okay this star
appears to be twice as luminous as this other star, this other star is 10 times as luminous as
this other star, so even though I don't have an exact number value for that luminosity,
I can still compare how luminous they are in comparison to each other, relative to each other.
After observing many stars in the cluster, she found a relationship between the period
of the variable star and its luminosity. If she graphed both values logarithmically, a linear
graph resulted with a clear predictable line. So this graph on the right is the actual graph that
she published in her paper on Cepheid variables. The y-axis is a measurement of negative powers of
10 of the brightness observed on Earth, and the x-axis is the logarithm of the period in days of
the star and these two lines are the maximum brightness and the minimum brightness of the
Cepheid variables, because remember they oscillate back and forth between being bright and being dim.
This graph is a little difficult to read for students, so I'm going to change it
into a more understandable graph here. Because she was graphing their relative luminosity compared to
each other, we can say that the y-axis is made up of numbers all multiplied by some constant
that we don't know. This allows us to show a proportion without knowing the actual values.
This graph is similar to what her graph showed. What this graph is telling us is that if the
period of one variable star is 10 days and the period of another variable star is only 1 day,
the luminosity of the 10 day period star will be about 10 times as big as the 1 day period star,
so even though she didn't know the true number values of the luminosity, she could get this
relative comparison of the luminosities of different stars. She then argued that finding
a single variable star's luminosity would change this graph from a graph of relative
luminosity to absolute luminosity. Just as an example, let's imagine that somehow we found the
absolute luminosity the real true luminosity of a variable star with a period of 1 day, and we found
that luminosity to be 5 watts. That would fill in the information about what that constant is,
and without making any more observations we could take that same constant and plug it into the other
values, because we know how much these stars compare to each other relatively, so we could
say "I know 100% based on her observations that if you multiply the period by 10 the luminosity
will also be multiplied by 10. So if I know the luminosity of a star with a period of 1 day is 5,
I definitely know that the luminosity of a star with a 10 day period is definitely 100%
50. So if you have an absolute luminosity graph, you can find the luminosity of a star
just using its period and not using its distance. So just to review what I've said so far: Leavitt
observes variable stars in the same cluster which all have the same distance to Earth,
which allows her to observe how luminous the stars are relative to each other,
but not their absolute number value, which allows her to make this graph and find this
pattern in the luminosity versus the period of the stars, which would tell her the true
luminosity of any variable star if she can find the luminosity of just one single variable star,
because if she found that she could fill in all the missing information in the chart
and that would allow her to find the absolute luminosity of any variable star
just given its period, which is easy to observe from Earth. So if you could find the
true luminosity of any variable star she would know the luminosity of every variable star. We
know based on the brightness equation that if we have the distance to a star we can find the
luminosity. Our only tool for measuring large distances in space at the time was parallax,
and that could only measure 100 light years out, so if variable stars could be found within
100 light years of Earth, she could measure the distance from the star to earth using parallax
and use that distance with the brightness observed on earth to calculate the star's luminosity,
and use that luminosity to complete her graph of all variable stars. So after she published
her paper astronomers did make observations of variable stars within 100 light years of earth
and used parallax to measure their exact distance, and then calculated their luminosity
using the distance and the brightness observed on Earth, and recorded the period of the star,
which allowed them to place that star on her graph. They did the same thing with several
variable stars where they would calculate the luminosity based on the brightness and
the distance they could observe, and the period of the star, and place that on her graph, and
do that one more time. So again they calculated the luminosity using the brightness and distance,
found the period and placed that on her graph, and once they had that absolute data filled in and
they had her graph with the predicted proportions of the period and the luminosity of any star,
they were able to fill in the entire graph with this information. A variable star with a period
of 1 day had a luminosity of about 300 times the luminosity of the sun, a variable star with a
period of 10 days had a luminosity of about 3,000 times the luminosity of the sun, and a period of
100 days had a luminosity of about 30,000 times the luminosity of the sun, and just for reference
the luminosity of the sun is 3.85 x10^26 watts. So now that they had this information they could
use this to find the distance to any variable star that they could observe, and the farthest variable
star that we can observe is about 20 million light years away. So as an example they could
look at a variable star in the far far distance, way beyond anything that parallax could observe,
and they could make an observation like this. They could observe "Oh,
this star seems to have a period of 100 days, and its brightness observed on earth is 4.57 x 10^-17
watts per meter squared, so how far away is the star? We don't need to use parallax because we
can now use the period and Leavitt's graph of luminosity to find what the luminosity of the
star is and plug that into the brightness equation." So if the period is 100 days,
I can see that the luminosity is 30,000 times the luminosity of the sun, which is equal to
1.16 x10^31 watts using the brightness equation. If I have the brightness and the luminosity,
I can rearrange the equation to find the distance to the star.
Plugging in my numbers gets me a distance of 1.42x10^23 meters.
Converting that to light years gets me 1.5x10^7, or 15 million light years. So I just measured the
distance to a very very distant star only using its period and its brightness observed on Earth,
and the observations made by Leavitt of the pattern between the period and the luminosity.
So this ability to measure the exact distance to any variable star allowed astronomers to
make a much more detailed map of the universe, because variable stars are clustered in galaxies,
because all stars are clustered in galaxies, and scientists were able to understand the shape of
the Milky Way, the shape of other galaxies, and the really vast distances between galaxies that
we'll talk about in another lecture video. But the pattern of Cepheid variables that
Leavitt observed is what allowed us to be able to do that. One very last note: variable stars
are in a group of space objects called "Standard Candles" which are sources of light that follow
some standard rule that allows us to calculate their luminosity, and therefore their distance,
using the brightness equation. Standard Candles are used to determine the distance to individual
objects in space that we can observe from earth, so if you're using a Standard Candle you're also
going to be using the brightness equation and the understood luminosity of the object.
That's everything that you need to understand about the nature and history of Cepheid variables
and how to use their properties to calculate their distance, even at incredibly large distances.