welcome let me remind you of what we have
been discussing distances in the context of
cosmology and let me start of by recapitulating
the things that we have discussed in the last
class so we started off by looking at the
cosmological line element
in terms of the conformal time eta and the
advantage of the conformal time is that the
propagation of light in comoving coordinates
looks like straight line if i draw it on a
space time diagram where i use eta instead
of usual cosmic time that is the big advantage
of using this conformal time and let me also
remind you that there are two different functions
over here which come into the line element
one is the r and the other is s
if the matric is specially flat then r and
s are same otherwise they are two distinctly
different functions if the space is curved
and the significance of this i had explained
in the last class r is the distance so let
us take a another point over here a galaxy
let us say over here r is the comoving coordinate
to the galaxy so if i consider of sphere of
radius r the surface area of that sphere is
given by 4 pi into this square because this
is the thing that multiples these solid angular
terms so the surface area of the sphere is
given by 4 pi s square in general and the
distance is r
so the ratio of the surface are to the radius
square is not a constant it is a function
of r that is the way the curvature of space
manifests itself and this ratio becomes a
constant when the radius is much smaller than
the radius of curvature comoving radius you
recovered the value 4 pi but if the space
is curved as you approach the length scale
the comoving curvature length scale you have
significant deviation in this ratio from the
value 4 pi
so there are two things over here this is
the radial distance this is the transverse
distance and the physical situation that we
considered let us move on to that
so were considering a physical situation where
there is a galaxy from which we are receiving
radiation and the radiation is red shifted
the observed red shift is z and we have seen
that this red shift will be the ratio of the
scale factors between the epoch when the light
was emitted and the light was observed and
the comoving distance to this galaxy is equal
to c into the difference in the conformal
time between these two events and we worked
out how to calculate this
so to calculate this you have to carry out
integral this integral allows us to determine
the comoving distance to any source whose
red shift is known usually in cosmology red
shift is the observed quantity everything
else has to be determined from the red shift
usually so if the red shift is known you can
use this formula to determine the comoving
distance also the difference in conformal
time
and the way to use this formula you have the
hubble parameter as a function of scale factor
here so one has to use the dynamics of the
universe which depends on the constituents
of the universe to determine this and we had
worked out one particular example the example
that we had worked out is where omega matter
is 1
there is only matter in the universe and for
this modal we had seen that the comoving distance
goes as 1-1/ square root 1+z into 2c/h0 let
me show you what this looks like we had drawn
this in the last class let me show you here
in the graph what this looks like so you see
that it is increases initially as expected
according to hubble’s law this we had checked
we have seen that when red shift is extremely
small the is the same as hubble’s law
as expected according to hubble’s law this
we had checked we have seen that when red
shift is extremely small this is the same
as hubble’s law
we had worked this out in the last class and
then what happens is that as we will go to
larger red shifts there will be deviations
from the linear relation because the dependence
is different and finally you find that at
very large red shift it saturates which is
a constant and we had also discussed the implication
of this this implies that red shift infinity
corresponds to a finite distance
that finite distance is the horizon that is
the furthest distance that we can be see observe
at present a photon which was emitted at the
big bang from this surface will reach us now
and we had also introduced a length scale
called the hubble radius c/h0 and in this
particular cosmological modal the horizon
is twice the hubble radius and we can make
an estimate of how much this is so let me
quickly do
this is a very important number let us just
quickly make an estimate so the hubble radius
c/h0 3 into 10 to power 8 meter per second
which we can write as 10 to power 5 kilometers
per second
that is the speed of light divided by 100h
kilometers per second per mpc so we see that
the hubble radius has a value which is 3000
here we have 100 here we have 10 to the power
5 so it is 3000 h inverse mpc and the horizon
in this modal is twice this 6000 megeparsec
h inverse megeparsec so the furthest distance
that you can see in this omega matter =1 cosmological
modal is 6000 h inverse megeparsec that is
the size of the horizon
another point to note which i have mentioned
earlier also that the entire distance scale
crucial hinges on this factor h inverse on
the hubble parameter if the hubble parameter
is smaller all the length scale are larges
hubble parameter is larger all the length
scale inversely related so this is just a
numerical estimate now let us go back to this
picture so here i have drawn this function
for you so the function that i am showing
over here is essentially this
where we have worked out the integral which
allows us to calculate we have worked out
this integral for the particular cosmological
modal where omega matter is 1 in this picture
i also show you another –the units here
are megeparsec i have chosen hubble parameter
h 70 and models are both spatially flat i
have also shown the comoving distance r for
another cosmological modal which has only
cosmological constant and no matter
you see the difference so the way the comoving
distance varies with red shift is crucially
dependent on the cosmological modal that we
have through the dependence of the hubble
parameter h on the scale factor the scale
factor has different dependence on a for different
cosmological models the other thing that is
important which i should also show you now
is that you have another function which is
s
you have two functions coming into the picture
so we have one more function which is s let
me also show you how s behaves with red shift
because given the red shift you can determine
r and for the r you can determine s if the
modal is specially flat s and r are exactly
same but if the model has curvature then they
will be different
so the entire discussion subsequent discussion
is going to be in the context of these four
cosmological models we have given them names
so i will request you to keep this in mind
that there are four cosmological model in
the context of which we are going to have
the discussion the first one is the standard
cold dark matter cosmological modal and this
has omega matter not equal 1
so the entire universe is filled with dark
matter i have told you that there is evidence
for dark matter and this is a modal where
assume that the entire universe is filled
with this matter so that it is exactly equal
to the critical density we do not see so much
matter in the universe so let us assume that
it is dark and this is the modal this is another
modal where there is more density then the
critical density
and in addition to this the universe –if
there is more density then the critical density
the potential energy is going to be negative
which will make the universe collapse and
this modal is going to be specially curved
omega curvature is negative so it is going
to be specially curve so the sum total of
these two has to be one so this is the contribution
from curvature so this is a modal which i
am following kcdm curvature it has some curvature
which is positive
the third modal is open cdm this is cosmological
modal where omega matter the matter contribution
pressure less matter is 3 there are observations
which indicate that this is the matter contribution
at present is around 30 percent of the critical
density total matter contribution so the rest
of it we are assuming here is in curvature
and this is a fourth model where again omega
matter is 3 and the rest of it is in cosmological
constant lambda 7
this is called lambda cdm lcdm let me also
mention that this is the currently favoured
cosmological modal observation seem to favour
the last one so let us now see what is the
behavior of this function s
how does it behave as a function of red shift
for these four different cosmological models
this picture shows you exactly that so you
see in this modal in the curve one which is
positively curved the function s is basically
sign and for large red shift this will grow
slower then the linear sign function it grows
slower than the linear function as sin theta
for small theta it is linear and then it becomes
slower so you see this modal falls below all
the other models
the modal over here falls below all the other
models kcdm and then we have the standard
cdm which is completely better dominated has
only matter in it and here we have the open
cdm and the lambda cdm models the value is
zr comparable but again you see that at low
red shift the lambda cdm modal dominates whereas
at it is reversed at larger end shifts
and this shows you the same function plotted
over a large red shift range in a log scale
it is the same function but plotted over a
large red if you want to display some function
over a large red shift range obviously you
cannot use a linear scale you will only see
the behavior over a small part of it so this
shows you the same function over a large red
shift range all the way to red shift 1000
and this shows you the behavior of the different
functions
so you see if you look at around redshift
1000 the open cdm modal the s z functioning
increases much more is much higher than the
other three models remember this is a log
scale so a small increase this is an increase
of 10 times so this is more than double the
difference between this and this is more than
double similarly the difference between this
and this is roughly double 
so the point i am trying to make here is that
both are an s have depend for the red shift
dependence of both of these things is different
in different cosmological models
but then i told you that comoving distance
is no a physical quantity the first physical
quantity that you can define is the proper
distance which is actual physical distance
between two points but again you cannot measure
this so for the observational point of view
and first important thing that we discussed
is the angular diameter distance and let us
recapitulate how the angular diameter distance
is measured
the angular diameter distance is defined so
that if i have an object whose proper size
is known then the angle it subtends is the
angular diameter the ratio of the proper size
to the angular diameter distance or what is
more useful is that if i know the angular
diameter distance and if i can measure the
angular subtends i can get the proper size
and usually there are certain things called
standard rulers
for which we have some idea of what the proper
size is and the angle can be measure this
is where the angular diameter distance is
particularly relevant and the angular diameter
distance is defined as the ratio of sz/1+z
and if its specially flat then it is r/1+z
so this picture shows you the angular diameter
distance for different cosmological modals
the picture on the screen shows you the angular
diameter distance how it varies with red shift
for different cosmological modals
so you see that for all the modals it has
a maxima somewhere around between 1 and 2
and then it falls off and again you see that
the angular diameter distance in this red
shift range between 1 to 0 to 6 lcdm is much
larger then all of them lcdm and ocdm are
actually quite close nearby whereas the ones
which have significant amounts of matter omega
matter is close to 1 the distance are much
smaller considerably smaller
and if you look at the behavior over a large
red shift range this is what it looks like
and at large red shifts the open cdm modal
has a larger angular diameter distance compared
to lcdm an lcdm has larger than the standard
cdm or curve one with the positive curvature
so this is the angular diameter distance it
basically refers to angles the next thing
that we were discussing in the last class
is the luminosity distance
and i had told you what the luminosity distance
is it is based on the relation between luminosity
and the flux so now let us derive the luminosity
distance r l so let us consider a source whose
luminosity the amount of radiation it emits
in the frequency range l nu emitted delta
nu emitted
this is the energy emitted 
in the frequency intervals so energy emitted
in the time interval per time so l nu is the
energy emitted in the frequency interval delta
nu e basically so there is a source over here
which is emitting radiation and this is the
luminosity in a particular frequency interval
so if i multiply with frequency interval i
get the amount of energy it radiates per unit
time
and we want to find what will be the flux
that will be measured by an observer over
here 
how is the derivation done the derivation
is done that you consider the photons that
come out of here they will be distributed
over a sphere like this and you divide that
by the area of the sphere you will get the
flux the time over which it is emitted and
the time over which it is received is the
same usually not in the usual context
so you have a 100 watt bulb and you want to
do it but here you see the time interval also
gets changed because of the expansion of the
universe so let us put in the time interval
and the frequency also get changed because
of red shift so let us now calculate the correct
relation between the luminosity and the flux
taking into account all of these so to do
this let us calculate first the number of
photons that were emitted the number of photos
will not get changed
so the number of photons that were emitted
delta n emitted is equal to --in a time interval
so this is the number of photons that were
emitted in a time interval delta t e in the
frequency interval delta nu e is given by
lnu e delta nu e delta t e/ h nu e this is
the number of photons that were emitted in
the frequency interval delta nu emitted in
the time interval delta t emitted this is
the energy that was emitted in this frequency
interval
and time interval divided by the energy of
each photon i will get the number of photos
now the same number of photons that were emitted
will propagate out and they will pass through
a surface of some radius at some later time
so the same number of photons will be received
over here and to determine the flux at the
observer what i have to do is i have to divided
delta nu delta n emitted by i have to multiply
this with the frequency of each photons
so the frequency of the photons would have
changed by the time it comes from here to
here it will be h into nu observed this is
the total energy that is crossing this surface
and it will be received by an observer over
here over a time interval delta t observed
over a frequency interval delta nu observed
this will be the flux in the frequency interval
observed over here now the question what is
the ratio of these emitted frequency and observed
frequency
we have already worked this out sorry divided
by the area of the sphere and the area of
the sphere is going to be 4 pi into s the
area of the sphere is going to 4 pi into s
square that is the area it is not going to
be c delta t it is going to 4 pi s square
that is how you calculate area in cosmology
the distance is r the distance from here to
here is r so the area is 4 pi/s square not
4 pi/r square so this is the flux and we know
how to calculate s
so the situation that we are considering the
observer is at the red shift zero the source
is at the red shift z so we know what this
r is we know what s is s is a known function
and we want the physical area here this is
the comoving area you have to divide it by
the physical area the physical area the proper
area this will have to be multiplied with
the scale factor square this will give me
the comoving area but the scale factor at
present has a value unity
so actually there should be a a0 square here
also like this is the area of the sphere at
the instance when the observation is done
but a0 we have assumed to be 1 because the
observer is at present so this is 1 this will
give us the flux in the interval delta nu
so now you see we have to determine what is
the ratio observed frequency to emitted frequency
let me put this here so what we will have
is that the flux is equal to the luminosity
and then i will have here the nu observed
by nu emitted coming from this and this i
will have delta nu 
observed by delta nu sorry delta nu emitted/
delta nu observed delta t emitted/ delta t
observed into 1pi/ 4 pi s square that is what
we have so i have just put this delta n over
here so i will have the ratio delta t emitted/
delta t observed nu observed/ nu emitted and
delta t and delta nu ratio is okay so this
is what i will get fine
now we also know that nu observed by nu emitted
is equal to 1+z which is also = t emitted/
t observed we have worked this out how delta
t emitted how time intervals change because
it is light that is propagating so we have
worked this out right in the beginning so
you see that this ratio will cancel out with
this ratio and this ratio will give me a factor
of 1+z so delta t emitted/ delta t observed
this will be the inverse of this so delta
t emitted/ delta t observed
so what we have here is that the flux 
no this will be other way round sorry the
frequency of light as it propagates what happens
does it increase or decrease how much distance
source will the frequency be higher or smaller
"professor - student conversation starts”
smaller it will be smaller right so the observed
frequency will be less than the emitted frequency
observed wavelength is more so it will be
1/1+z and the time interval is opposite is
the frequency
so there will be a factor of 1+z-1 here so
what i will get is 1+z sz square compare this
with the definition of the luminosity distance
this is 4 pi dl square z so what we get from
here is that the luminosity distance dl is
equal to square root of 1+z into sz
this is if you are dealing with luminosity
and the flux per frequency interval now quite
often you do not deal with this what you deal
with is the bolometric luminosity 
so what is the bolometric luminosity the bolometric
luminosity is the total luminosity integrated
over all frequencies so this is the bolometric
luminosity which is l nu integrated over d
nu similarly the bolometric flux is going
to be f nu integrated over d nu
now the point to remember is that here the
integral will be over the emitted frequency
and here the integral will be over the observed
frequency after all this is what you observe
you see the light that you observe here this
is the observed frequency this is emitted
frequency they are not the same you are actually
what this gives you is the relation between
the luminosity at a certain emitted frequency
and the flux at another observed frequency
some observed frequency which is different
and these two are related the frequencies
are related through the red shift that red
shift formula here this formula so if you
take this expression and integrate it let
us say we integrate it over the observed frequency
so we integrate this over the observed frequency
then what we get is that the flux the bolometric
flux is equal to --so i will have 1/ 4 pi
1+z s square
that is what i will have here and then i have
the integral the luminosity d nu observed
because i have integrated this over the observed
frequency and this i have to write in terms
of the emitted frequency because i know the
luminosity the bolometric luminosity is the
integral of this over all emitted frequencies
so i have to then multiply rather divide this
no what i have to do so 
the flux can be written as 1/4pi 1+z square
s square z lnu emitted dnu emitted
because i have to multiply this with the factor
of 1+z this becomes the d nu emitted and this
thing is the bolometric luminosity l so we
see that if you use the bolometric flux and
the bolometric luminosity then the luminosity
distance is equal to 1+z into the --and throughout
our discussion we shall we using this definition
so this is if you use the bolometric luminosity
and the bolometric flux 
so let me show you how this varies with red
shift so this picture it is graph over here
the graph on the computer screen shows you
how the luminosity distance varies with red
shift for the different cosmological models
that we were discussing so you see again the
ones which have large amounts of matter dominated
both these models
the behavior is quite different from the modal
where omega matter is 3 and 
you see again here there are differences between
the open cdm and lcdm model this shows you
the same thing how the luminosity distance
varies with red shift over a large red shift
range so we see that the distances which are
important in observational cosmology are essentially
the luminosity distance and the angular dimeter
distance these are the two things which are
important in observational cosmology
instances that we have been discussing till
now the luminosity distance the angular diameter
distance comoving distance etcetera all of
them have the property that at low red shifts
we recovered the hubble’s law so this is
a very important thing so all the laws at
low red shifts recovered hubble’s law and
you can all the distances you can check that
from the formula and these if you have observations
that low red shift then these observations
do not depend on the cosmological modal they
only depend on the value of h0 which sets
the ch0 overall length scale so the basic
idea that we are going to discuss now is how
to determine h0 and observation that low red
shift can be used to do this so if you can
measure the distances to different galaxies
independently and the red shift independently
if you can measure red shift and distances
independently you can then determine the value
of the hubble parameter i have also told you
several classes ago that determining distances
is the most difficult thing in cosmology you
can infer the distance from the red shift
that is easy but independently determining
the distances is extremely difficult
so let us now take a look at this so this
is what hubble has done to start with hubble
has measured distances using cepheid variables
distances to galaxies and he has measured
the red shifts on the spectra and this is
a picture of edwin hubble and he has measured
these two quantities and he found that a straight
line that they have a linear dependence hubble’s
observation unfortunately were based on very
low red shifts
so he had only a few galaxies located very
close to us and the slope that he got was
far too high he got a slope of around 400
which brings down the entire distance scale
of the universe
so let us now take a closer look at the future
development the subsequent developments since
hubble the basic problem in determining the
hubble parameter likes in the fact that you
have peculiar velocities and the motion of
a peculiar velocity something new so let me
send a little time discussing this so in the
picture that we have learnt the observer here
so in the picture that we have learnt the
observer here and this observer will see that
all the galaxies are moving away with the
velocity which is proportional to the distance
and this velocity in the new by universe and
this velocity can be interpreted as the red
shift that is in the new by universe if you
look at the entire picture then it is better
to interpret it in terms of the scale factor
but in the nearby universe this is fine
now this is the component of velocity due
to hubble expansion you can think of it like
that now in reality in addition to this expansion
the velocity due to expansion of the universe
the galaxies have other components of velocities
also and this other component is called the
peculiar velocity so peculiar velocity is
by definition the actual velocity of a galaxy
minus what you expect from the hubble law
this definition is in the nearby universe
if further away it gets slightly modified
but it is essentially is this it is the deviation
from the hubble expansion why are there deviations
from hubble expansion from the motion predicted
by the expansion of the universe these deviations
arise because the universe is not exactly
homogenous and isotropic there are deviations
and some parts of the universe have more matter
so this is more matter some parts have less
matter so i will put a different color and
the places which have more matter exert an
extra force on this galaxy for example or
a place where there is less matter will exert
an extra repulsive force so the analysis till
now has assumed that the universe is homogeneous
and isotropic but if there are deviations
these will cause deviation in the velocities
also and these are what are called peculiar
velocities
for example andromeda galaxy is not moving
away from us it is moving towards us why because
in the neighbor between our galaxy and andromeda
galaxy in this region there is excess matter
compared to the average density of the universe
which is causing the andromeda and our galaxy
to come together so there are peculiar velocities
in the universe this is an observed fact and
the peculiar velocities typically have speeds
of the order of few 100 kilometer per second
so you could take a typical speed to be 300
kilometers per second and if you convert this
to red shift using z is equal to v/c on the
order of v/c then you get that this corresponds
to red shift of 0001 i have chosen 300 so
that this comes out to be the division as
easy so peculiar velocities will contribute
to the red shift and the contribution will
be of the order of 10 to the power -3 so the
red shifts that you see have a contribution
due to the hubble expansion
in addition they have a contribution due to
peculiar velocities let us call it the peculiar
red shift peculiar velocity/c and this is
of the order of 001 that is what we saw this
introduces errors in the red shift the red
shift that you measure will not be just due
to expansion it will have a contribution due
to peculiar velocities so this will be reflected
in an uncertainty in the hubble parameter
because what you will do is you measure the
distance we will measure the red shift and
look at the ratio if red shift has some other
contribution not due to the expansion of universe
that will be reflected in an uncertainty to
the hubble parameter and we would like the
uncertainty in the hubble parameter in this
we would like the error due to this peculiar
velocity to be less than 10 percent let us
put an restriction like this
if you want this condition to be satisfied
then you have to look at a red shift of 001
if you consider a galaxy at a red shift of
001 then the effect on the peculiar velocity
we know is going to be within 10 percent and
red shift of 001 you have to look at galaxies
at a distance which is more than 001 into
c/h0 which comes out to be around 30 megaparsec
so for a reliable determination of the hubble
parameter
you have to look at this relation at distances
beyond 30 megaparsec only then will the contribution
of the uncertainty due to the peculiar velocity
will be less than 10 percent so you have to
look pretty far away you cannot look just
a nearby galaxies to determine the hubble
parameter that is the first thing that we
have to realize now once you look at galaxies
at 30 megaparsec you can no longer identify
cepheid variables at that distance
you can no longer identify the starts so a
galaxy at that distance you cannot identify
individual starts it looks like a fuzzy thing
in the sky that is all you will not be able
to make out the stars so one has to look at
other things other quantities to determine
the distance
other important thing is that if you can push
this exercise to larger red shifts red shifts
of order 1 at red shifts of order 1 we have
seen that the different distances for example
the angular diameter distance this the way
it varies with red shift depends on the cosmological
modal so if you can measure the distance as
a function of red shift you can determine
which cosmological model we are in so you
can essentially determine the values of these
omega density parameters
because the distance will be different for
different density parameter you will get one
value so you know which one we live in that
is the basic idea
so this brings us back to the issue of the
cosmological distance ladder and we have already
discussed the cosmological distance ladder
to some extent we have learnt that you can
use cepheid variables to determine distances
out to galaxies which are nearer so the globular
cluster and nearby galaxies were you can make
out stars you can use cepheid variables to
determine distances now let us look at the
cosmological ladder again
so we have to now determine distances beyond
30 megaparsec that is basic problem and it
is not possible to directly jump to 30 megaparsec
because you cannot see features there so you
do not know how to do it so you have to again
extend your ladder put various steps in between
which overlap so let us go through this so
we will now start the discussion from the
largest scales other way round
and work back to the place where he can use
cepheid variables so the basic idea is that
at large distances like 30 megaparsec you
can use properties of galaxies that is the
only thing that you can use by and large properties
of galaxies now question is that if you just
take the luminosity of a galaxy if you assume
that all the galaxies have the same luminosity
and use them as standard candles
so let us suppose we assume that i know the
luminosity of all the galaxies they are same
and then if i take the flux and interpret
that due to the variation in the distance
then i can get the luminosity distance if
i can measure the flux then i can get the
luminosity distance as a function of red shift
this will not yield correct results because
the galaxy luminosities are large scattered
galaxies do not have a fixed luminosity
the galaxy luminosity have a large scatter
they come in a large range so our galaxy for
example has a luminosity of around 10 to the
power 11 times the luminosity of the sun that
is the number of stars in our galaxy there
are galaxies which are considerably fainter
and considerably brighter so the variation
in the flux cannot be related to the distance
so one has to look for other means of determining
the luminosity of the galaxy
and the basic idea is that one has to look
at the correlation between the luminosity
and the property other measurable properties
of the galaxy so you cannot measure the luminosity
but there are other quantities which you can
directly measure or you cannot measure the
length of a galaxy but there are other quantities
which you can measure and if you can correlated
the length to those then you can use this
to determine the get standard candles and
rulers
so let us first start off with elliptical
galaxies so we have already learnt that the
distribution of specific intensity in an elliptical
galaxy looks like an ellipse and these are
believed to be ellipsoids three dimensional
ellipsoids now the specific intensity variation
inside the galaxy away from the center so
specific intensity obviously fall off as you
go away from the center this fall of the specific
intensity away from the center is well described
by what is called de vacouler’s r to the
power one fourth law
so the way the specific intensity surface
brightness falls off as you go away from the
center is well described by this law i0 exponential
–r /r0 to the power one fourth this is called
de vacouler’s r /r0 to the power one fourth
law at the specific intensity is something
that can be measured because the specific
intensity does not changes the light propagates
well that is not precisely true in cosmology
in cosmology the specific intensity will change
because of the red shift but that we know
the red shift you have measured the red shift
of the galaxy so you can predict what the
specific intensity is when the light is emitted
so we can measure the central specific intensity
for example of a galaxy that is not difficult
to do another quantity that you can measure
is the central velocity dispersion sigma v
what do we mean by velocity dispersion
so this galaxy is an ellipsoid like this let
us say and it is a collection is stars in
an elliptical galaxy the stars are moving
randomly in different directions like a gas
at some temperature and this random motion
that prevents the galaxy from collapsing otherwise
the galaxy would have collapsed under its
own weight so there are random motions which
prevent it from collapsing and the sigma v
what is sigma v
so we are observing from somewhere over here
large distance away and we can measure the
line of sight component of the velocity so
we will be able to measure only the component
of the velocity along our line of sight due
to adopt from doppler shifts so if you can
look at the spectrum of this galaxy in some
line then the line will be some broad thing
like this where the width is determine by
the velocity dispersion
you will not be able to make out individual
stars typically but you will be able to make
out you will get a broad line due to the random
motions of the galaxies the width of the line
will reflect the random motion of the galaxies
because each spectral line will not be at
the same place it will be shifted due to doppler
shifts and the rms variants standard deviations
of these velocities can be estimated from
the line width and that is what is sigma v
the velocity dispersion
so it is v-v mean the mean value of this square
that is how you calculate the mean square
rms value deviation from mean value so you
can measure the central velocity dispersion
so these things can be measured i0 and sigma
v
and it has been observed that there are some
relations which are referred to as the fundamental
plain so there is a relation which is referred
to as the fundamental plain so what is this
fundamental plain these are essentially co-relations
between different galaxy properties here the
three properties that we are looking at are
the total luminosity of the galaxy the specific
intensity at the center surface brightness
at the center of the galaxy and sigma v and
it is found purely empirically
that the luminosity is proportional to the
central specific intensity to the power x
and sigma v to the power y so this is a plain
in a 3 dimensional space which has a luminosity
central specific intensity and sigma v as
the axis it is called the fundamental plain
basically so galaxies lie in a plain in this
space of luminosity central specific intensity
and sigma v so if you can in principle if
you can measure the value of this central
specific intensity
and the velocity dispersion at the center
you can determine the luminosity because galaxies
all lie on a particular plain in the space
of these three parameters so this is one possible
thing that you can use another possibility
is what is called dn sigma relation dn is
the radius of the or radius diameter of the
galaxy and it is defined in such a way so
that the average specific intensity inside
this radius has a particular value
so you have an image of a galaxy you find
the diameter inside which the mean specific
intensity has a particular value it is found
that is related to the velocity dispersion
so if you can measure the velocity dispersion
of the galaxy then you can predict what the
length scale corresponding to this d you can
predict dn and from the image you can measure
the angle that this subtends so you know this
length you know the angle
so you can determine the angular diameter
distance well at low shift they do not differ
so we can determine the distance so essentially
if you can measure the sigma v you can predict
the diameter of the galaxy and you can measure
the angle from the image then you can use
this to determine the distance further there
is another possibility which is called the
faber jackson relation this relates the luminosity
to the central velocity dispersion
so these are all empirical things findings
some of these can be justified there is a
physical bases for some of these which depends
on the –you have physical arguments for
some of these relations based on the virial
theorem because we know that the galaxy is
in equilibrium so the kinetic energy should
and gravitational potential energy are related
one is half of the other from there you can
find that such relation should exist and these
are observationally found
so they allow us to determine either the luminosity
or the diameter of the galaxy from measured
quantities
for spiral galaxies so we have also learnt
that there is another type of galaxy called
a spiral galaxy in spiral galaxy we have a
disk and the disk is rotating spiral galaxy
it is found that again the rotational velocity
is related to the luminosity and this is called
the tully fisher relation so this can be done
either in optical or in infrared or 21 centimeter
neutral hydrogen which we have discussed
so here again if you can measure the rotational
velocity which is easy from doppler shift
you can predict the luminosity of the galaxy
so let me briefly summarize we have what we
have learnt in the later part of today’s
talk we have learnt about how we can use galaxies
as either standard rulers or standard candles
so let me bring today’s lectures to a close
over here we shall resume our discussion from
here in the next lecture
