Hello. Last time, we saw how we can use, various
relationships to measure distances to stars, or star clusters.
Chief among them, relations between period, and luminosity for pulsating
variables like Cepheids, but also fitting of the main sequence for the star
clusters. These are examples of a really more
general idea of Distance indicator relations.
So let's just recap what that really means.
In general, if we can find a correlation between some distance independent
quantity, like color or temperature or stars or period was for pulsating
variables, this correlates well with something that Does depend on distance
like, stellar luminosity. And we can calibrate that relation, then
we can see how stars in, other clusters, of different distances, fit on that
relation. So now when we compare, these
correlations inapparent quantities, say apparent magnitudes instead of absolute
magnitudes, clusters of different distances Will systematically shift from
each other. From magnitude of shift, we can find out
what the relative distance is between them.
So thus, we can use these distance to indicate the relations as a measure of
relative distance. If we can actually calibrate them, in the
absolute sense, then of course we can get the optimal distances as well.
In this is how Cepheids work. So now the question is, can we do the
same for galaxies, and the answer is yes. The first relation, that we'll consider,
is so called, surface brightness fluctuations.
And, conceptually, it's fairly simply. Suppose we are looking at some galaxies
with a detector that has square pixels, like, typically they do, and each of
these pixels would cover, a certain number of stars.
Now if we, remove the galaxy twice as far, then there'll be, four times as many
stars in each pixel, and that will look smoother.
The mathematical reason be, behind this is, fairly simply.
For elliptical galaxies and bulges, most of the light comes from luminous red
giants. And it's the fluctuations in the numbers
of the red giants that will determine what's going on here.
So if we have, say, stars, that contribute most of the light and the
average flux is f and the number of them in a pixel is, and the total amount of
flux from the pixel will be n*f. And the variance of that will be due,
given by the square root of the number of stars.
It's simple statistics. So, if we move galaxy further, the number
will, go as the distance squared, but flux will decline as the distance square.
And, therefore, the variance will scale with the square.
Of the distance, universally. And square root of that, which is rms or
sigma as the first power. This is why a galaxy that's twice as far
would appear twice as smooth. So now, if we actually knew what real
luminosity is for an average star that corresponds to that.
Average f, then we can determine the distance.
Roughly speaking, the absolute mag, mean the absolute magnitude M corresponding to
that main flux F, is the one at the top of the Red Giant branch, and we can try
to calibrate that using Andromeda Galaxy to which we measure distance using other
techniques. Using that gives the following
calibrating relationship here. Note that this a purely empirically
calibrated relationship using a galaxy to which we know the distance.
However, there is likely to be some dependence on the actual coefficients of
this equation, or I should say correlation, that will depend on things
like the mean metallicity of stars in a given population, and so on.
We also have to take care of things like removing the effects of foreground,
extinction, presence of foreground stars in our galaxy, background galaxies behind
the galaxy we're studying, and so on. Nevertheless, this is a very powerful
method, and can be used to measure distances up to 100 megaparsecs using
Hubble-Space telelscope. Here is a simulation of how that works.
The 2 columns of pictures correspond to 2 galaxies, fictitious galaxies that 1 of
which is twice as far as the other. Top panels schematically show where the
stars would be relative to the pixel grid.
Now if we average the fluxes we can see in the grey scale images the second row
what say. Picture might look like.
Now let's add more pixels or alternatively we can move galaxies
further out and so we see those grainy structure which is indeed what is
observed. Finally, we convolve this with
atmospheric turbulence thus seeing because we are not observing in a vacuum.
Don't have to do this last step if observing with a space telescope.
And the last row shows selected images of 2 different galaxies, of 2 different
distances from us. Obviously the one that's further away is
much smoother. 100 megaparsecs is a good distance but
we'd like to push even further into what's called pure Hubble.
So remember, the definition of Hubble's Law is that distance is proportional to
the recession of velocity. That's due to the expansion of the
universe. However, in reality, galaxies do move.
Since the time they form, they acquired so called peculiar velocities because of
the mutual gravitational attraction. You can imagine, that if there is a large
concentration of mass, in some direction from a galaxy, it will be accelerated
towards it. And over time it will develop certain
velocity. So the actual observed radio velocities
of galaxies have these two components, the components that's due to the
expansion of the universe, the Hubble flow, and the other one which has to do
with dynamics of large scale structure. Now, typically Today these peculiar
velocities of galaxies are a few hundred kilometers per second.
For example Milky Way moves 600 kilometers per second relative to the
cosmic micro background which is a good physical embodiment of the combo and
coordinate grid, and galaxies new cluster.
So, in our case, we are a member of the local group, and that is part of a larger
concentration, the local super cluster. And our entire local group is falling to
it with a speed of a few hundred kilometers per second.
Since these large scale non-uniformities extend out to scales of maybe 100
megaparsecs or so. Up until that scales, or thereabouts,
there, there are going to be peculiar velocity components.
Going to larger scales, things pretty much average out, and if the peculiar
velocity is few hundred kilometers per second, few thousand kilometers per
second hubble flow would make that 10% if we go to.
Tens of thousands of kilometers per second of recession velocity.
Then the peculiar velocities will only matter on a percent level.
This is why I would like to measure Hubble constant to distances that are
considerably larger than 100 megaparsecs. And for that, we need some very luminous
objects. Galaxies and supernovae are such luminous
objects and both have been used to push measurement of the Hubble constant all
the way out to the Hubble flow. This is where galaxy scaling relations
come in. Remember we explain how it works at the
beginning of this clip and if we can find a quantity for galaxies that correlates
with another one What the first one, with being distance dependant, such as
luminosity. The outer one being distance independent.
Then we can, actually do the same trick, as we did with cephates.
There are 2 examples, of such, distant indicator relations for galaxies.
1 for spiral galaxies, called the Tully-Fisher relation, and 1 for
elliptical galaxies, called the fundamental plane.
Because these distance indicator relations, provide a relative Distance of
galaxies from each other, they do have to be calibrated locally.
So, for example, the Tully-Fisher relation can be calibrated with distances
to spirals that are now unsafe from Cepheids.
Or as the Fundamental Plane relation can be calibrated from the distances using
surface brightness fluctuations. Each of which, of course, has been
calibrated with the lower ranks of the distance ladder.
Now one thing to note here is that we have some idea where these relations come
from, but their exact physical standing for evolution are not yet well
understood. So we have to beware.
We're using a tool whose origins and the right functions are not perfectly well
understood. The Tully-Fisher relation, for spirals,
is a very useful one, it, connects the luminosity of a galaxy, with its maximum
rotational speed. When we talk about, properties of
galaxies, later on in the class, we will address these relations in much greater
detail, but, for now, just take it for granted, that this is indeed the case,
and I'll show you the plot. The luminosity of spiral galaxies are
well protected with rotational speeds, and the relation has the power of slope
of 4, namely luminosity is proportional to the fourth power of the circular
velocity. Roughly speaking, the power varies
depending on the filter that is used and so on.
We can also express that in terms of magnitudes, because log of the luminosity
* -2.5 gives the absolute magnitude, will manifest itself as the width of the
observed radio line of neutral hydrogen. Because as the hydrogen atoms move in the
galaxy their Doppler shifted this way or that and faster the rotation speed.
More of the Doppler Browniung rays. So we can establish this, this
correlation using spiral galaxies with well measured distances nearby and then
try to apply it to much more distant ones.
In practice the scatter of the Tully-Fisher relations is typically 10 to
20%. The very best that has been done is about
9% and that's including all the measurement errors.
It's actually a very good correlation when done well.
There are some problems. The light from spiral galaxies is
affected by the extinction in them. There is lots of dust that absorbs the
blue light, or absorbs light at all wavelengths, but more so in the blue.
And, the fluctuation in star formation can affect the luminosity of galaxy on
short time scales. So here is the actual Tally-Fisher
relation for a set of galaxies nearby, in 3 different filters.
There is a gradual change in slope. But that's okay.
Whatever we do, we can just calibrate to that slope.
But also note that the scatter improves, the further in the red we go, for the
reasons I just mentioned. There is less effect of the extinction,
and less susceptibility fluctuations due to the very luminous young stars.
The other important correlation is so called fundamental plane.
This is actually a set of bivariate correlations, which connect one property
of elliptical galaxies with the combination of 2 others.
But it's usually shown in this form, where a distance dependent quantity.
Radius determining a certain consistent fashion is correlated against a
combination of velocity dispersion. Which, again, is the Doppler broadening
of spectroscopic lines in galaxy spectrum.
And it means surface brightness. Surface brightness not depending on the
distance, at least in Euclidean. Space.
The observed scatter of the fundamental plane visible bands say a is about 10%.
So it is as good as the Tully-Fisher. It may be even slightly lower and
actually why is it so small is an interesting problem enough itself, which
will come back to when we talk about properties of elliptical galaxies.
Its zero point nowadays, tends to be calibrated with surface brightness
fluctuations. The fundamental plane comes in another
flavor, so called DN sigma relation, where DN is the diameter of a galaxy at
which its surface brightness reaches certain value.
This turns out to be a slightly oblique projection of the fundamental plane, but
it basically works in the same way. Next time we will talk about use of
supernova as a standard candle, a very popular method these days.