Well let us start climbing the distance ladder.
First, we'll talk about, methods that we can find distances to nearby stars, or
star clusters. The Trigonometric Parallax is the most
basic measurement of distance in astronomy.
And, hopefully all of you are, well famililar with it.
It is pure geometry, and there is nothing uncertain about it.
So by measuring the annual apparent slooshing on the sky of a star, we can
figure out how far it is, if we know the distance between earth and the sun, which
we do know, with a great percision. So the method by itself is safe, the
problem is that, these are very small angles, and the current state of the art
is that we can measure distances using parallaxes To about one kiloparsec out,
more or less, and that's well within our own galaxy, never mind external galaxies.
The next one is so-called moving cluster method.
This is a statistical method. Stars do come in clusters, clusters move
relative to the solar system, and stars have internal motions.
Within a cluster itself. Since, ostensibly, all cluster stars are
moving roughly in the same direction. If we look from afar, we will see them
converging toward some distant point. Which is direction which they are going.
By measuring the spread of those angles, we can figure out, what's the angle of
their actual velocity vector to the line of sight? We also measure proper motions
of stars in angular seconds, per unit time on Sky, as well as their radial
velocities. We can assume that random motions within
the cluster are same both in radial and tangential direction.
So by knowing what the, what the angle is between The radial and tangential
components, we can figure out what is the parallax to the cluster.
Note that this makes some assumptions about the internal motions of the
cluster, and it is basically a statistical technique.
So the more stars you have the better, but cluster's have finite number of
stars. By measuring distances to a number of
stars using either one of two techniques, we can calibrate the Hertzsprung-Russell
or polar magnitude diagram for stars. Hopefully this is something you know
about very well. It is a lot of stellar luminosity versus
temperature. Temperature is sometimes measured through
color. The important, thing to note here, is
that stellar luminosities are distance dependent according to the inverse square
law from given measurements of apparent brightness.
Temperature, measured from spectra, or from colors, is not.
Star will have exact same temperature no matter how hot it is.
So, once we calibrate The main sequence of this Hipparcos's diagram.
If we can measure stellar colors or temperatures, then we can read off their
absolute magnitudes. From absolute and apparent magnitudes, we
can figure out how far they are. In a given cluster you may have thousands
of stars, and therefore you can determine the distance very precisely, the distance
to the cluster itself. Now this works fairly well for young
stellar clusters in the galactic disc. However, no globular cluster is as yet
close enough to measure parallaxes to it. And so something else will have to be
done about those. There what we do is we measure, we use
Field stars of the same type as those that live in globular clusters, the
population 2 stars. HR diagram measurement is a collective
measurement. But not all stars are created equal.
Between the temperature luminosity plane, there is a strip within which stars are
unstable to pulsation. So called instability strip.
On the main sequence, those will be the Cepheids, or delta Cepheid stars.
Among the globular clusters, horizontal branch, those will be RR Lyrae stars.
There are many different kinds of pulsating stars.
But those are the principal ones. And indeed, physics of Cepheids and RR
Lyrae is probably the best understood of all It turns out that there are
correlations between observed period, which again does not depend on distance,
and luminosities of these stars, which do.
Those are empirical relations and they can be calibrated if you had distances to
a number of these pulsating stars using one of the previous techniques.
At first, people did not know that there are different kinds of pulsating stars,
they all thought it was one kind. And the first star that Hubble figured
out in Andromeda was Cepheid, pulsating star that produced the first distance to
another galaxy. But people are confusing RR Lyrae which
are much dimmer than surface with surface themselves and that confusion were
through a an error of factor of 2 in the distance scale.
Once Walter Baade understood that there really are two different kinds of period
luminosity relations. That error was corrected.
Let's talk about cepheids in more detail. Because they remain among the most
important distance indicators altogether. They're young luminous stars.
Therefore, they'll be found in star forming discs.
And star forming regions, whereas Delta Ceti itself is relatively bright one
within our own galaxy. It was Henrietta Leavitt working with
Harold Shapley who recognized that there is a correlation between period and
luminosity, as they were studying stars in the Magellanic Clouds, about 50
kiloparsecs away. Since all of them were roughly the same
distance. Apparent magnitude would correlate with
period and then they understood that comparing that with nearby Cepheids they
can find out how far the Magellanic Couds are.
Surface are important because they are bright and so we can see them far away,
we can find them in galaxies up to maybe 25 mega parsecs or so.
So we can calibrate distances to a number of nearby galaxies using Cepheids, which
is not easy but it's possible. And then we can use distances to those
galaxies to calibrate some other relations.
It isn't all perfectly safe. The position of stars must depend on
their internal composition and opacity, and therefore metallicity. the exact
effects of metallicity are not firmly established as here.
Moreover, there are external problems such as extinction.
Cepheids are found in star-forming regions and they also tend to be dusty.
So one has to make a, a correction for it.
In very distant galaxies, they may be blended by other stars giving us wrong
luminosity. Cepheids remain keystones of the distant
scale and that also applies for the measurements with the Hubble space
telescope. Here are some examples of Cepheid period
luminosity relations in the modulaic clouds in different filters.
The scatter is biggest in the blue band and the smallest in the near infra red.
However the amplitudes are biggest in blue and smallest in infra red.
It's a good idea to observe them in different bands so that the The effects
of extinction can be taken out. Until Hipparchus satellite flew, we did
not have parallax calibration of Cepheids.
Distances to Cepheids until then were based on the distances through clusters
in which they live. And distances through those clusters were
by in large use, determined using the cluster map.
However, with the Hipparcos, a handful of cepheids was within reach.
And these are the actual calibration relations for distances to cepheids.
As you can see, they're fairly noisy. But in any case, for the first time, they
gave us an absolute calibration of the period luminosity relations for cepheid.
This is going to get a lot better with the Gaia satellite, which is an astronomy
mission which will measure Cepheids to a much larger number of pulsating stars
with a much greater precision. The other important kind of pulsating
stars are RR Lyrae. Their also named after the prototype star
that was first recognized. Their at a population to stars, their not
on the main sequence, but on the horizontal branch, which is the helium
burning may sequence, and their found in old stellar populations, such as the
globular clusters. They do have an advantage that their
periods are short. So, it's much easier to observe full
periodic Pulsating curve for an RR Lyrae Star than it is for a cepheid.
Because they're dimmer, they can be really used, only within the local group
of galaxies. But that's still useful, and it provides
a welcome check, on the distances measure using Cepheids.
Now, let's take a closer look to what happens when a star is pulsating.
It's photosphere expands, but the temperature changes.
As well. So the radius changes, the temperature
changes therefore, luminosity must change.
If we observe stars spectroscopically, we can observe the velocity of the photo
sphere. Come storage us and go away from us.
So we can measure stellar temperatures using colors or spectroscopy.
We can measure velocity of the pulsating photosphere using spectroscopy and we can
measure the changes in the apparent brightness.
This forms the basis of so called Baade-Wesselink Method.
If the pulsating stars were perfect black bodies, this would be an excellent pure
physics based method to determine distances to them.
Unfortunately, real stars are not perfect black bodies, but they're not too far
either. So, at any given time, the flux from a
star will be it's luminosity, which is in itself, given by Stefan Boltzmann
formula. It's proportional to the temperature to
the fourth power. And to the surface area of the star,
which is proportional to the square of its radius.
And it's universally proportional to the square of the distance.
We can measure those quantities all throughout the pulsation period.
So temperatures are directly observable from photometry.
But so are the fluxes. And the only remaining question is, can
we find out the radius? We can, in a way, because if we integrate motion of the
photosphere as traced by the radial velocity, we can find out how much the
radius has been changing. So we have 3 equations in 3 unknowns and
we can solve for that. Therefore, we can obtain distances purely
from measurements. And assumptions about black body nature
of stellar photospheres. The problem is, the stars are not perfect
black bodies. And some modelling of stellar
photospheres has to be done in order to actually make the method to work.
So there is model dependence. And that's where the uncertainties come.
From. There are a couple more statistical
methods that are based on stellar indicators.
Globular star clusters themselves have a distribution of luminosities.
It turns out that their distribution function, the luminosity function of
globular clusters, seems to be universal among galaxies for reasons that are not
really well understood at all. Actually you can think of many reasons
why this shouldn't be the case. But empirically, the do seem to be very
similar. Thus, if we can calibrate the luminosity
function of globular clusters in the milky way using local distance
indicators, then we can apply it to luminosity functions of globular clusters
in other galaxies. A good thing about this is the globual
clusters are much brighter than most stars, and so they're easier to find and
easier to measure. Now one problem is that the number of
globular clusters vary widely among the galaxies.
Elliptical galaxies, early type spirals, have most, late type spirals.
Hardly have any and therefore there be a statistical uncertainty for those
galaxies. A similar method uses luminosity function
of planetary neubli. As you recall planetary nebuli
represents. Stellar envelopes that have been shed by
a star, following it's horizontal branch phase.
They are illuminated and iodized by the incandescent core that remains.
And most of their light emerges in recombination emission lines.
A very prominent line among those, is the line of ionized oxygen at 5007 angstroms.
We can measure luminositys of those lines alone and then we can form luminosity
function that is distribution of luminosities for that emission line alone
for planetary nebula. That too, turns out to be more or less
the same for the nearby galaxies. Ostensibly, that reflects the way in
which stars evolve. But there isn't solid strong physical
basis. This is an empirical relation.
And again It, it is statistical. It can work up to the distance of the
vertigo cluster, which is not so bad, but not beyond.
And finally, there is, the tip of the red giant branch.
The stars cannot get more luminous than certain amount.
This is related to the, so called Eddington Luminosity, which, hopefully,
you have heard about, and which we can address later.
So, empirically, they don't seem to get brighter than a certain limit.
And, if we can observe stars in other galaxies, nearby galaxies like Andromeda,
and find out what are the most luminous ones.
Where does luminosity stop? Then that threshold can be used as standard candle.
The advantage of this is, of course, that these stars are bright.
The disadvantage is that it is not a terribly well-defined indicator.
There aren't very many of those stars so the numerical fluctuations can affect the
result. That is it about the stellar.
Distance indicator. Next time we will talk about so called
Distance Indicator Relations for galaxies.