welcome we have been discussing thermal radiation
and let me briefly recapitulate right at the
start what we had discussed in the last class
thermal radiation is radiation that is emitted
by material which is in thermal equilibrium
and if we have radiation that is enclosed
in a cavity like this and the whole enclosure
and the radiation inside comes to thermal
equilibrium at a temperature t then the radiation
that is inside this cavity is referred to
as blackbody radiation
and we had seen in the last class that this
radiation the spectrum of this radiation the
specific intensity is just a function of temperature
and this function is called the planck function
we had also seen that if we have some material
which is at a temperature t the material itself
is in equilibrium then the source function
for this material which is in thermal equilibrium
is the planck function
so these are two different things so this
is called a thermal radiation that originates
from such a material is called thermal equilibrium
is called thermal radiation it need not be
the radiation itself need not be in equilibrium
with the material so the specific intensity
of the radiation could in principle be anything
if the specific intensity of the radiation
comes to equilibrium with the material then
the specific intensity becomes the planck
function
otherwise just the source function is the
planck function this is called thermal radiation
this is called black body radiation okay then
having discussed this we went ahead and we
calculated the entropy how the entropy on
this depends on the temperature and the volume
and based on this we finally reached the conclusion
that the energy density of this blackbody
radiation is proportional to t to the power
4
and this constant of proportionality we called
it ab the stefan constant further we also
saw that the planck function this we had seen
earlier that the specific intensity is related
to this multiplied by c and divided by 4pi
you get the specific intensity so if you integrate
this over frequency the planck function over
frequency then that is equal to c into the
energy density by 4 pi
the flux the brightness this is the brightness
is c into the energy density by 4pi which
also scales as t to the power 4 and the flux
from any surface of a black body cavity we
saw is also scaled as t to the power 4 and
this constant is called the stefan boltzmann
constant sigma and it is related to this stefan
constant like this so it is c into the stefan
constant by 4 we also calculated the entropy
and the entropy we saw is proportional to
t cube one power of t less and it is equal
to 4 by 3 into the energy density by the temperature
into the volume of the system so the entropy
density is 4 by 3 energy density by t after
that i told you okay we also looked at what
happens when you have adiabatic expansion
of radiation and we saw that for radiation
which is made to expand or contract adiabatically
pv to the power 4 by 3 is a constant
so radiation behaves like an adiabatic medium
with adiabatic index 4 by 3 okay having done
all of this i told you that from my macroscopic
thermodynamic considerations you cannot proceed
much further to determine these constants
the stefans constant or the stefans boltzmann
constant you need to look at the microscopic
nature of this so we looked at that is what
we were looking at and in the last class at
the end of the last class this is where we
had reached
so we had calculated a relation 
between the specific intensity remember the
specific intensity inu is for raised in a
certain direction so this is a unit vector
along the direction of the rays let me draw
a picture so these are my rays that okay so
they along some unit vector n and they will
be in a spread of solid angles so the specific
intensity at the frequency nu in this particular
direction of the unit vector n is related
to the occupation number of the photons
this n over here is the photon occupation
number and photon occupation number of a particular
mode the mode being the vector wave vector
k how is this mode related to this frequency
and the direction in which the wave is propagating
the ray is propagating the wave vector k so
the direction of this it is obvious is going
to be same as the unit vector n that is the
direction in which the light ray is propagating
and the magnitude is going to be 2pi omega
and omega we know is so omega is equal to
c into k right so we can write this as 2 pi
c into k so omega 2pi omega and omega is c
into so we can so this is going to be right
omega so this is the modulus of k the modulus
of k is omega by c and omega is 2pi nu so
we can write this as 2pi nu by c right that
is a modulus of k the magnitude into the direction
of the vector
so we have a relation relating the specific
intensity in this direction to the occupation
number of a mode k with these factors outside
these factors one of them comes from the energy
of the photons and others come from just counting
considerations h nu comes from the energy
of the photon and others come from just counting
considerations and factor 2 comes here because
light has two polarizations that we are assuming
that the light is unpolarized
so both are equally present okay so this is
a very important relation and it has got several
applications and we shall see these as we
go along this course right now it is a very
general relation between the specific intensity
and the occupation number the modes of the
photon okay right now we are interested in
a situation where the photons in this cavity
are in thermal equilibrium with the walls
of the cavity
so they are in thermal equilibrium at a temperature
t now a photons we know follow the there are
two statistics fundamental statistics one
is the bose statistics the other is the fermi
statistics now photons are what are photons
they have spin one so they are bosons right
so photons follow the bose statistics so the
occupation number n so if the photons are
in thermal equilibrium that is what we are
assuming right
if they are not in thermal equilibrium the
occupation number could be anything but if
they are in thermal equilibrium then the occupation
number is given by the bose statistics and
there are two parameters here one is the temperature
of the system and the other is the chemical
potential mu which so there is a system which
is bosons in equilibrium so to completely
specify the system you need to tell the temperature
and the chemical potential right
the bose distribution has two unknown parameters
the temperature and the chemical distribution
chemical potential now photons are not conserved
that is something i should have mentioned
right in the beginning also so you can use
these thermodynamic considerations to you
to calculate the energy and then everything
else because photons can be destroyed and
created okay
so the number of photons essentially adjusts
itself to the temperature depending on the
temperature the number of photons will go
up or down okay so for things for particles
bosons that are not conserved whose number
is not conserved okay there is no conserved
quantity associated with a photon there is
no charge there is no mass mass is not conserved
any way but there is no charge associated
with the photon
so these are massless uncharged particles
so the chemical potential they have no charge
whatsoever no kind of charge so the chemical
potential is zero and then you can straight
away write down the occupation number in thermal
equilibrium this is isotropic and it is one
by the bose distribution and with zero chemical
potentials it is one by exponential their
energy by kt and the energy we know is h nu
by kt minus 1 that is the bose distribution
so the wave number is converted to frequency
we have just seen how to do that and you put
it in here and this tells you how many photons
there are in this mode okay so once you assume
that it is in thermal equilibrium it is you
can straight away just plug it in here and
you get the planck function okay so just plug
it in here if so these photons are in thermal
equilibrium with at temperature t with some
material right
photons do not interact with each other so
they cannot come to thermal equilibrium by
themselves you required the cavity or some
material which will bring it to thermal equilibrium
once they come to thermal equilibrium their
occupation number is given by this bose distribution
and we can straight away write down the planck
function 
which is 2 h nu cube by c square into the
bose distribution was occupation number
so we have to 2h h here is the planck constant
okay bear this in mind and k we shall use
kb here to distinguish it from the wave number
so kb is the boltzmann constant h is the planck
constant and kb is the boltzmann constant
so the planck function is 2 hnu cube by c
square and here we have 1 by exponential h
nu by kt minus 1 okay so that is the planck
function that is the specific intensity of
blackbody radiation this is kb of blackbody
radiation 
okay
so let us now look at the behaviour of the
planck function both as a function of frequency
and as a function of temperature so let us
now spend little time just looking at this
so the spectrum of blackbody radiation how
it depends as a function of frequency at a
given temperature or if you fix the frequency
how does it vary with temperature let us look
at this okay
so first we will consider a certain limit
the limit being the situation where h nu by
kb the boltzmann constant into t is much less
than one so this is called the rayleigh jeans
limit okay so we will first study the planck
spectrum in the rayleigh jeans limit what
we mean is that hnu by kbt is much less than
1 so the temperature of your system define
the frequency and that frequency is basically
kbt by the planck constant
if the frequency that you are looking at is
much more than that is much smaller than that
okay so the frequency that you are looking
at is much smaller than that let us see what
happens so if h nu by k t is much smaller
than 1 then we can expand this in a taylor
series and this will become 1 plus h nu by
k t minus 1 so 1 cancels out so the occupation
number in this limit nk let us just look at
the occupation number that also is interesting
is kbt by hnu which is much greater than 1
because that is the inverse of this so each
mode is occupied by a large number of photons
that is the limit that we are looking at each
mode is occupied by a large number of photons
it is very densely populated okay that is
the limit that we are looking at and in this
limit the planck function 
so this whole factor is now kbt by hnu so
one factor of h nu cancels out and what we
have is 2 nu square by c square the boltzmann
constant into the temperature of the radiation
which can also be written in a form that is
quite easy to remember 2 kbt by lambda square
okay
so in the rayleigh jeans limit we see that
the planck function takes a relatively simple
form if at a fixed temperature you look at
the specific intensity then the specific intensity
or the amount of energy increases proportional
to the square of the frequency that is the
first thing or it goes down as 1 by lambda
square so the higher the frequency the more
the energy contained in this radiation
this works as long as you are in this regime
okay and it is linearly proportional to the
temperature so if you increase the temperature
make it twice the energy contained will also
just double okay and the other interesting
feature to note is that the planck constant
does not appear anywhere over here it is purely
classical in that you do not need any quantum
mechanics to understand this okay which is
again can be understood intuitively from the
fact that you have many photons in each mode
so the photon nature the discrete nature does
not manifest itself you have a large number
of photons in each mode and you can think
of the electromagnetic wave is actually a
wave the photon nature does not manifest itself
okay and much of radio astronomy works in
this regime in the rayleigh jeans regime so
this is very useful if you are doing radio
astronomy for example radio astronomers always
use this
they do not have to bother about in most situations
they did not bother about the entire planck
spectrum this is adequate now there is the
other limit that is the wien limit or the
wiens spectrum and here it is assumed that
h nu by kbt is much greater than 1 so if h
nu by kbt is much greater than 1 the occupation
number is essentially exponential minus h
nu by kbt because this number is a very large
number much greater than 1 okay
so the occupation number 
and you see that this is obviously much less
than 1 so the photons in each mode there are
very few photons much less than 1 okay that
this is an average occupation number so you
expect to find typically less than 1 photon
in each mode okay and in this regime the particle
nature of the electromagnetic radiation actually
starts to manifests itself it is in this regime
okay
and in this regime the planck spectrum now
becomes 2h nu cube by c square exponential
h nu by kbt minus so there is an exponential
cut off this function falls off rapidly because
of this exponential so at a fixed temperature
if you increase the frequency this function
falls off rapidly with temperature okay so
let me show you a graph showing the these
are the two limiting situations
let me show you a curve this picture over
here shows you the behaviour of the planck
function this shows you the planck function
for different values of the temperature so
let us just focus on any one of them okay
for first so let us just consider anyone of
them let us consider say this one over here
okay this is for 1 kelvin 10 kelvin 100 1000
all the way to a million so if you look at
any one of them you see at low frequencies
this is a log minus log scale
so both of these are intervals are logarithmically
spaced so this is proportional to nu s quare
and then it increases and then there is a
maximum somewhere and beyond which it falls
off exponentially the exponential cuts it
off okay so that is the first feature that
we see of the planck spectrum so at low frequencies
it increases as the frequency square and then
there is a maximum somewhere and then beyond
that it falls off
this is a characteristic frequency and obviously
the characteristic frequency is divided decided
by kb into t it depends on the temperature
the higher the temperature the larger the
characteristic frequency okay so this is something
that okay before we come to this let us ask
the question what happens if i fix the temperature
so if i fix the temperature and look at a
fixed frequency
so at sorry if i fix nu and look at b nu as
a function of t okay at a fixed frequency
let us say i have look at this point over
here which is 10 to the power 10 hertz and
vary the temperature increase the temperature
now it turns out that del b 
you can do this and check for yourself it
is straight forward i am not going through
the exercise you just differentiate it with
respect to time the expression that we have
with respect to temperature
and you will find that this is positive okay
so it means that if i increase the temperature
the value of the specific intensity is at
a fixed frequency the value of the specific
intensity is going to increase okay so that
is what we have here so if i fix the frequency
and look at the curves for different temperature
they all lie on top of one another okay so
as the temperature increases the value of
the specific intensity goes up okay
so these curves never intersect right because
at a higher temperature the curve is always
above the one at a lower temperature so these
curves never intersect so if i know if i observe
some radiation and i can measure if i know
it is a black body and if i can measure just
anyone point on this curve i know that the
temperature immediately it is unique okay
so if from if i know that it is a blackbody
spectrum and if i can measure just one frequency
then the temperature is uniquely determined
because the curves do not intersect okay so
if i am just one measurement over here so
i measure if i can determine the entire temperature
provided i have some a priori information
or if i assume that it is a blackbody radiation
okay this is the first thing second thing
is you can determine where the maxima occur
okay 
in frequency at a fixed temperature fix t
and ask the question del b where is this zero
this will give you a the value of the maxima
okay and we see that if you do this if you
do this exercise or if you look at the curve
over here so if you look at the curves over
here you will find that nu max by t and this
is a constant okay and the value of this constant
it is 588 into 10 so hertz by kelvin okay
so this tells us that which you can determine
this again i am not going through the algebra
but you can determine this it is quite straight
forward and exercise you have to just differentiate
the planck spectrum with respect to frequency
and then find set it equal to zero so you
have to just differentiate this with respect
to nu so there is nu here and nu here and
then set it equal to zero if you do this exercise
what you find is that the frequency where
you have maxima scales proportional to the
temperature and the constant is given over
here
this is called wien displacement law 
okay now here we have been discussing the
specific intensity which is the energy contained
in a ray per frequency interval now people
also can often discuss another quantity which
is the frequency the energy contained per
wavelength interval okay which is b lambda
the planck function as a function of lambda
okay it is not just the planck function as
a function of lambda
so b lambda d lambda is equal to b nu t nu
the same energy is contained in the corresponding
energy wavelength interval and we know that
so we know that we can write this as the mod
of d nu d lambda d lambda so which essentially
tells us that b lambda is equal to b nu and
this if you differentiate nu by lambda you
will get c by lambda square okay so these
are two different quantities not only do you
write it as a function of lambda
but it also tells you the energy arriving
in that interval of wave length in an interval
of the unit wave length interval whereas the
specific intensity that we have been dealing
with tells us the energy per unit frequency
interval now these curves let me just show
you a picture of these curves and these curves
obviously look different so this picture shows
you b lambda okay effectively forget about
the scale over here this shows you the behaviour
of b lambda
here again you have a maxima and as you increase
the temperature the maxima shifts to a smaller
wavelength okay so you can differentiate this
also with respect to lambda and set it equal
to zero and you have then you will get a relation
that lambda max that is the wavelength where
b lambda is a maximum into the temperature
t this is equal to 029 centimetre kelvin okay
this is also known as this also is known as
the wien displacement law
so you can use either of them either of them
both of them refer to wien displacement law
and this tells us where b lambda has a maximum
and the one that we considered earlier tells
us where the b nu has a maxima now you cannot
convert lambda max and lambda you cannot relate
them lambda and nu max by just a factor of
c because they are the maxima of two different
functions okay they are slightly different
okay
so the key point here is that as you increase
the temperature as you increase the temperature
the maxima shifts to higher and higher frequencies
which ever you look at high shifts to higher
and higher frequencies or smaller and smaller
wavelengths okay and just as an example look
at these curves look at these curves then
we see that at around 1 kelvin or 10 kelvin
somewhere over here the maxima are all in
the radio microwave region okay
whereas for a few thousand over here it is
in the optical visual range or ultraviolet
visual infrared and if you are looking at
1 million or so it is in the x ray okay that
is how it keeps as you increase the temperature
the spectrum the peak of the spectrum shifts
and the bulk of the radiation is also shifting
to at higher and higher frequency range okay
so that is a brief discussion of the nature
of the shape of the planck spectrum
now let us ask slightly different question
what is the total energy density in the planck
spectrum right so our aim was after all to
calculate that constant ab or sigma from these
microscopic considerations okay so let us
ask the question what is the total energy
density in this blackbody radiation and to
do this what are the things that we have to
do so the total energy density we have seen
we have to take the planck spectrum let me
go through this
so you have to take the planck spectrum 
and you have to so we know that the energy
contained in a particular set of rays in a
particular direction this 
in a particular direction this is just this
quantity into rather this quantity divided
by c and if you want the total energy density
then you have to also integrate over solid
angle and you have to also integrate over
frequency okay
so this size the frequency interval we shall
do next okay so this gives us the total energy
density of the blackbody radiation of ray
contained in rays in all direction and this
d omega integral can just be taken outside
so this is equal to how much is this let me
write down so this is equal to h pi sorry
no no 8 4pi this will give us 4 pi so i have
8pi i am just multiplying this with 4pi and
here i have h nu cube by c cube into 1 by
exponential h nu by kt minus 1
so this is the energy density at a frequency
nu in the blackbody spectrum and to calculate
the energy total energy density integrated
over all frequencies i have to integrate this
over all frequencies okay to do that it is
convenient to introduce a new variable let
us call that variable x 
such that x is equal to h nu by the boltzmann
constant into t right which makes this whole
thing dimensionless so let us do that
so to do that if you do that then this integral
this becomes the quantity that is integrated
is x cube dx e to the power x minus 1 right
that is what is it to be integrated and the
rest are just constants outside so let me
write down those constants i have 8 pi and
we shall have kb so we have to multiply with
kb to the power 4 and t to the power 4 right
so that we have because we have four xes over
here we had the n nues nu cube and d nu over
here
so i want to convert them into x so i will
have this factor k after multiply this with
kbt h to the power 4 so i have this and in
the denominator i will have c cube and i will
have h to the power of 3 1 power of h is cancelled
out with this okay and this integral we know
this integral has a value pi to the power
by 4 by 15 okay this integral has a value
pi to the power 4 by 15 okay so now you see
the energy density is some constants into
t to the power 4
and these constants we had given the name
the stefan constant so we can straight away
identify what the stefan constant is the stefan
constant ab is equal to the stefan constant
ab is equal to let us identify what it is
so i have 8 and i have in the numerator 8
pi to the power 5 kb to the power 4 by 15
c cube and h cube that is the stefan constant
and this has a value now you put in all of
these are all fundamental constants
this is the boltzmann constant this is a planck
constant this is the speed of light and pi
so you put in all the values and what you
get that this is equal to 756 into 10 to the
power minus 16 joules per meter cube 
by kelvin to the power 4 right so we have
worked out the stefan constant and we had
seen earlier that the stefan boltzmann constant
sigma the stefan boltzmann constant is c times
c by 4 times the stefan constant okay
and this has a value which is equal to 567
into 10 to the power 
minus 8 watts per meter square that is the
flux per kelvin to the power 4 okay so we
have calculated the two constants that we
had arrived at just from purely thermodynamic
and macroscopic considerations so this tells
us the total energy density in blackbody radiation
this tells us the flux that is emitted from
the surface of a blackbody it emits for this
watts per meter square kelvin to the power
is multiplied by the temperature okay
so you will get it in watts per meter squared
that is the flux okay so i hope all of this
is clear so this kind of brings to an end
our discussion of blackbody radiation okay
now in astrophysics we rarely we do encounter
blackbody radiation but the blackbody radiation
also serves as a very useful tool in quantifying
as a kind of model which we use to quantify
any other radiation okay
so there are various kinds of effective temperatures
so we it is often convenient say we receive
the radiation from some source from some astrophysical
source now we would like to associate a temperature
with that source based on the radiation that
we receive after all the radiation itself
we would like to use it to infer something
about the source so we use the radiation that
we receive to infer certain temperature of
the source okay
and this is called an effective temperature
for the source the real temperature of the
source we do not know but this gives us some
handle on what we call the effective temperature
of the source and these effective temperatures
okay these are related to the blackbody 
spectrum 
and there are different effective temperature
select which are used in astrophysics let
me go through this so one of them is called
the brightness temperature
now what is the brightness temperature let
us just see so we are receiving suppose we
have measured the brightness of the radiation
from a source by brightness we quantify the
brightness of the radiation using specific
intensity right so suppose we have measured
the specific intensity of a source of the
radiation from a source and now suppose we
assume that the source is blackbody if we
assume that then we can say that if it were
a blackbody then this should be equal to the
planck function at some temperature t
and i have measured the specific intensity
at some frequency from some source okay so
i have measured just one value of nu i have
measured the specific intensity at a particular
frequency if i assume that it is a blackbody
then that direction tells me the temperature
of the blackbody this is called this temperature
which i associated with the radiation is called
the brightness temperature of that radiation
okay it quantifies the brightness of the radiation
and in the rayleigh jeans limit this is rather
simple so if i take in the rayleigh jeans
which is very useful in radio astronomy this
is quite simple so in the rayleigh jeans limit
and this is going to be you take the specific
intensity and just multiply it with c square
and divided by 2nu square it will give you
the brightness temperature right so if it
were a planked spectrum the temperature would
have been this much
in the rayleigh jeans limit the temperature
would have been this okay so this is another
way of just quantifying the specific intensity
and it tells you the brightness of the radiation
this is called it associates a temperature
which quite often is easier to interpret we
have a better feeling for it okay so it comes
out in the units of kelvin the specific intensity
is in units of joules per meter square per
hertz per second first irradiance this quantity
is something easier to interpret
it comes out in the units of kelvin and it
is a measure of the brightness of the radiation
okay it is called the brightness temperature
and it is extremely simple the relation is
extremely simple in the rayleigh jeans limit
otherwise you have to put in the planck formula
and determine this okay and it often gives
you a good idea of what is going on in the
radio for in radio astronomy where this is
extensively used
now the radiative transfer equation in the
rayleigh jeans limit the radiative transfer
equation also takes on a very simple form
so in the rayleigh jeans limit we can instead
of using specific intensity we can just use
a brightness temperature so the radiative
transfer equation becomes d by d tau of the
brightness temperature is equal to the specific
intensity is now replaced by brightness temperature
there are all these multiplicative factors
which will appear throughout this is equal
to minus tb plus the temperature of the source
the source function gets just replaced by
the temperature of the material through which
the radiation is passing okay so the radiative
transfer equation itself becomes much simpler
in the if you use the brightness temperature
in the rayleigh jeans limit okay
and what happens now so if the medium is optically
thick we know what happens the specific intensity
becomes equal to the source function right
that is what we have seen if the material
is optically thick whatever specific intensity
whatever brightness comes on it what comes
out is basically the source function if the
material is optically thick so here what happens
whatever be the temperature brightness temperature
of the incident radiation the brightness temperature
of the radiation that comes out is essentially
the temperature of the material
it is a blackbody with that temperature blackbody
spectrum okay it is a temperature of the material
okay so this is one of the temperatures it
is called the brightness temperature then
we have another one which is called the colour
temperature 
okay now just before finishing this brightness
temperature see the brightness temperature
is used in for any arbitrary radiation i just
equated to the planck spectrum at that value
and find the temperature okay
so let us consider a hypothetical situation
where i measure i nu from some source which
looks like this 
okay this is obviously not a blackbody spectrum
so if i calculate the brightness temperature
at this point what will i do i will find which
blackbody spectrum goes through this and i
will associate that temperature if i find
the brightness temperature at this point again
i will find which blackbody spectrum it will
be a different one that goes through this
and i will associate that temperature okay
so if i have something that is actually not
a blackbody then different parts of the spectrum
will have different brightness temperature
i hope that is clear if it is exactly a blackbody
i will get the same brightness temperature
throughout okay there is something called
the colour temperature let me now tell you
what the colour temperature is okay which
again is quite useful sometimes sometimes
i cannot measure the specific intensity
there are situations where i have no idea
of what the specific intensity is okay i can
only measure the flux 
okay there are situations let us consider
this situation suppose i have a source over
here and my telescope the instrument that
i am looking at has an angular resolution
which is larger than this okay now what is
the specific intensity the specific intensity
is the radiation coming per solid angle i
do not know the solid angle subtended by the
source
because the source is smaller than the smallest
angle i can measure so i can only measure
the flux the total radiation coming from this
to determine the specific intensity i have
to actually write this i have to divide by
the solid angle subtended by the source and
that will be some number okay but i cannot
do this because i cannot determine this it
is smaller than the smallest angle i can measure
with my telescope which happens
for example two stars i cannot determine the
diameter angular diameter of stars it is much
smaller than the resolution the smallest angle
that can be measured by most telescopes so
i can measure only the flux which is the integral
of the specific intensity which is the specific
intensity into the solid angle of the source
i do not know what this is so it will remain
there as an unknown let us assume that the
specific intensity is roughly constant over
the source okay
so i can have a measurement of the flux fnu
and i find that its shape is very similar
to a blackbody but i do not know what the
obviously cannot say it is equal to a blackbody
because i do not know the scale i cannot say
that it is equal to this b nu because it is
i do not know the scale that is the solid
angle involved okay so what i will do is i
will just take the shape of the flux as a
function of frequency and fit it to see which
blackbody spectrum the shape best matches
okay
and then determine use that to determine a
temperature called the colour temperature
okay so possibility is i could see that the
flux has a peak and we know that the position
of the peak is unique for a blackbody right
uniquely determined by the temperature so
if i can measure where at what frequency the
flux is maximum i would then get an idea of
the temperature and that would be one estimate
of the colour temperature okay
so this is what the colour temperature is
all about we have a third kind of thing which
is a third kind of definition effective temperature
which is also used and this is called the
effective temperature 
so suppose i have a source for which i can
measure the total flux that is being emitted
okay total flux that is being emitted so if
i can measure the total flux that is being
emitted by per solid from that surface of
the source integrated over all frequencies
okay
so this is the flux emitted per area of that
source okay and we know that if it is a blackbody
this will be sigma t to the power 4 so if
i can measure the flux that comes from per
unit area of that source and for example the
sun i know the area of the sun i know its
radius i know the so i can determine the area
and if i know how much energy is being emitted
per unit area of the sun how much flux so
then i can equate it to sigma t to the power
4
and this is what is called the effective temperature
this again is perfectly correct only for a
blackbody but in astrophysics we applied we
have to associate we would like to associate
a temperature with the radiation so if i apply
to some other source then this the temperature
that we estimate using this is going to be
called the effective temperature so if i tell
you that the effective temperature of the
sun has some value
basically we have estimated the flux that
is emitted by unit area of the sun and just
fitted it to this stefan boltzmann law put
it in the stefan boltzmann law now the temperature
the surface of the sun may not be that it
will be that if the sun were exactly a black
body if not it will be something different
but we have been able to we have used this
to characterize the radiation from the surface
of the sun okay
so let me bring todays discussion to a close
over here today we learned the nature of the
planck spectrum or the blackbody spectrum
how it is related to the bose distribution
and from there we derived the stefan minus
boltzmann constant and we then looked at various
effective temperatures that are defined it
used in often used in astronomy and physics
