Last week

Completed discussion of methods of measuring distances

Stellar masses using Kepler’s 3rd Law

This is confined to measurements of binary systems

relatively close to us.

However from these measurements we find a clear relationship between

Mass and luminosity.

We then began to look at the work of Edwin Hubble.

Mass – Luminosity Relationship for stars.

Putting many mass and luminosity measurements together show a clear relationship between the two.

This now allows us to infer stellar masses for stars beyond the reach of parallax measurements

We also conclude that these stars all belong to a common class of objects which we will come to see as Main Sequence Stars.

Hubble’s Law

Many redshifts measured by Vesto Slipher

Hubble and his assistant Milton Humason measured the corresponding distances - d.

v/c

Hubble put these results together to get

v = H0.d Hubble’s Law

where H0 is a constant. If v is in kms-1 and d is in Mpc then

H0 is in kms-1Mpc-1

A 1936 version of Hubble’s results published by him.

Hubble’s Law

Figure shows the distances and recessional velocities for a sample ofgalaxies.The error bars are for measurements of distances made using type 1a supernovae as Standard Candles.

From these resultsH0=65 kms-1Mpc-1

Value is hotlydisputed.We can take H0 as between40 and 100 kms-1Mpc-1

Hubble’s Law

It is Hubble’s Law which established the idea of an expanding Universe and led naturally to the idea of the Big Bang Model..

The Law immediately raises a number of questions in our minds.

1.Has the Universe always been expanding?

2.Will it continue to expand?

3.How can we explain this?

4.How fast is it expanding?

Cosmological principle

• Astronomers accepted quite a long time ago that we have no privileged place in the Universe -we have no special observing position

• Einstein’s Theory of Relativity emphasised this with its stress on relative motion.

• If we are typical observers then our view of the Universe will be the same no matter the point of observation-the Universe would be expanding away from us.

• COSMOLOGICAL PRINCIPLE:-If we are typical observers then our view of the Universe will be the same as at any other point, namely it will be expanding away from us according to Hubble’s Law.

• In other words over very large distances the Universe is homogeneous (every region is the same) and isotropic (same in every direction).

Olbers’ Paradox.-A Fundamental Question

1610:- Kepler asked why is it dark at night if the Universe is infinite with stars distributed at random throughout it?

No matter which direction you look your line of sight will end on a star. Although the light from one star dims by the inverse square law the farther away we look we will see more stars- this will compensate for the dimming.

18th Century:- Halley and Cheseaux asked the same question.

1826:- Olbers raised the question again and made it a popular question. Hence the paradox is named after him.

Solution:-We only become aware of the existence of another galaxy when its light reaches us. If the Universe is not infinitely old then there must be distant galaxies whose light has not reached us yet.

A Non-static Universe?

Another hint of a Universe that is not static comes from a considerationof Gravity. Newton, having developed his gravitational Law, wonderedif the Universe was collapsing under its own self-gravity.

Everything should get pulled together by gravity.

Newton did not pursue this further and the view persisted until the 20th Century that the Universe was static.

What does expansion mean here?

Hubble’s Law suggests that the Universe is expanding.It is often likened to an explosion but this is not correct.Instead it is better to think in terms of the expansion of the surface of a balloon.

A

BPoints A and B are marked on the surface of the balloon.As the balloon expands the two points move apart.The further apart they are away from each other to start with the more rapidly they move apart.This is true no matter where you are on the balloon(Cosmological Principle).In essence it is the entire surface which is expanding.

B

A

A

Surface of balloon has no centre. There is also noedge.Similarly it is pointless to ask “Where is thecentre of the Universe?”or”What is it expanding into?” or “Where is the edge?”These questions are meaningless in this model!!

Another way to think of it is that theframe of reference expands as theUniverse expands.

Remember = 1 +Z 0

Effectively Z is a measure of stretching of

In other words what is happening is that the wavelength gets longer because of the stretching of the whole fabric of space.

Remember = (1 +Z ) 0

Expanding Universe

Effectively Z is a measure of stretching of

So if we observe an object with Z = 6 the Universe has expanded by ( 1+Z ) in the

time it has taken for the light (radiation) to reach us.

Accordingly if there is a galaxy which is now 7 Mpc away from us then

at the time the light from the distant object was emitted this galaxy was

7 Mpc/ (1+ 6) = 1Mpc away

And the Universe was smaller in volume by ( 1+ Z )3 = 73 = 147

MW

MW

S

S

Time

Distance to milky Way

Location

of Milky way

At time

Of

Supernova

Other galaxy now

Expanding Universe – Meaning of Distance

photons

Supernova in another galaxy

Light travels to us at velocity c

If the light takes 400 million years to reach us we say it is 400 Mly away. but

are we talking about when the light was emitted or the distance we are from S now?

We have to think in terms of spacetime.

400 million years is the lookback time forthe supernova.

The expansion stretches out the wavelength of the photons. This is the Cosmological Redshift.

Cosmological redshift

When we see light from a galaxy and it is redshifted we have a choice. Either

a. The light is Doppler shifted because of the motion away from us

or

b. The wavelength of the photons is stretched – the idea of Cosmological redshift.

The latter is preferable because it becomes difficult for us to define precisely what

we mean by the speed of the galaxy at great distances.

Cosmological redshift tells us how much space has expanded during the lookback

time to a galaxy.

Cosmological Horizon

So we now think of space itself as expanding with all of us being carried along with it.

Cosmological redshift tells us how much Universe has expanded in lookback time.

Universe does not have an edge!!

However it does have a horizon – a place beyond which one cannot see.

Cosmological Horizon = limits of the observable Universe = boundary in time.

It exists because we cannot see back before the Universe began.

If Universe is 1.4 x 1010 years old then nothing can have a lookback time longer

than this time. The lookback time to the cosmological horizon = Age of Universe.

At any one moment we are looking back in space and time. We cannot look past the horizon since we cannot look back before the Universe began.

As we will see in a couple of lecture’s time we cannot even see that far.

Expanding Universe Big Bang

• Hubble’s Law leads naturally to the idea of the Big Bang. -At earlier times the galaxies would have been closer and earlier still they would have been even closer-

• Most distant galaxies move away fastest.Since c is a constant this means these distant galaxies were seen at an earlier time. Hence the highest velocities occurred earlier on in the Universe and this is characteristic of an “explosion”.

• So the basic idea is that if we go back far enough the Universe was in a highly condensed state initially-”A primeval atom” and expansion began at a specific time.

(George Lemaitre’s idea)

Age of the Universe

• Basic Idea-Universe began in a state of infinite density and expanded as if in an explosion.How long ago was this?

v

r

V = H0r

• If we run the Universe backwards to the Big Bang then v = H0r

If T0 is the time from the beginning of the Universe then

T0 = r = r = 1

assuming that v is constant.

v rH0 H0

• So T0 = 25 x 109 years if H0 = 40 km s-1Mpc-1 and = 12.5 x 109 years if H0 = 100 km s-1Mpc-1

• Former value assumes v = constant.This is unlikely since gravity should slow things down.

Big Bang Model

Cosmic singularity:- In the Big Bang model the Universe encompasses, at all times, all of space. “There is nothing beyond the edge.” At time zero the density of matter was infinite. Space and time had no clear meaning. Phrases such as “before the Big Bang” or “at the Big Bang” are meaningless.

Planck Time;- Prior to the Planck time( tp ) our physics does not work. Tp = [ Gh/c5 ]1/2 = 1.35 x 10-43 seconds

Models of the Universe

Must explain

- Hubble’s Law

- Olber’s paradox

- Why the Universe does not collapse under its own gravity.

- diversity of objects seen in our telescopes.

It must satisfy

-Cosmological Principle

and be -Homogeneous and isotropic on largest scale.

Steady State Universe and Continuous Creation

• Alternatives to the Big Bang Model were proposed-do not get the idea that it is the only possible model!! One such was the “Steady State Model” put forward by Bondi, Gold and Hoyle(1948,1949). In essence it extended the Cosmological Principle to say that the Universe is homogeneous,isotropic and appears the same at all times.

• Now Hubble’s Law gives us the characteristic time for the creation of matter rather than the age of the Universe. It only needs a few hundred atoms per cubic metre every 1010 years to keep the steady state.This is far too small to be measured. This counterbalances the mass moving away from us. How this happened was “ignored” in the model.• Creation could be at random points in space or it could be in centres of galaxies.This would be consistent with observed extra activity at centres of galaxies.• Overall individual galaxies form out of intergalactic matter, evolve and recede while new matter enters to keep density constant.

• Gamow et al tried to explain creation of all the elements in the Big Bang [see later] but this does not work. Instead Burbage, Burbage, Fowler and Hoyle showed that it happened in stars. This might seem to favour the Steady State Model.

Steady State Universe and Continuous Creation

• However another piece of evidence came along which could be explained by the Big Bang Model but not its competitors, namely the Cosmic Microwave Background (CMB)

Before we look at that we should remind ourselves of the propertiesof Black-body radiation which we studied at the beginning of this course.

Blackbody Radiation

• General question:-What is the spectrum of EM radiation emitted by an object of arbitrary temperature T in thermal equilibrum.

We assume that this “blackbody” reflects no radiation at any

•Max Planck showed that the spectrum is given by

ud = 8hc -5.d[exp(hc/kT) - 1]

where ud is the energy density =energy/unit volume

• Although no perfect blackbody exists solids and stars follow Planck’s Law very closely. [Note that picture is on log-log scale.]

Wien’s Displacement Law

• Doubling T increases P by 16 since PA = .T4

• Note that maximum wavelength max

shifts with .This can be quantified in Wien’s Displacement Law.

max.T = const.

= 2.9 x 10-3 m.K

• This quantifies the observation that an object changes colour with Temperature e.g.At room temp. spectrum peaks in infra-red.

• Very important since it allows us to obtain a measure of the SURFACE TEMPERATURE of a star from max.For the Sun max is in blue but with a lot of radiation in red so it looks yellow.For stars with T = 3000k max is in infrared but significant amount in red. Red Giants are at this T.

Cosmic Microwave Background(CMB)

• Essentially only three pieces of evidence underpin the Big Bang Model and the CMB is one of them.

• It was discovered by Penzias and Wilson in 1965.

They are shown here with thehorn antenna they had builtfor telecommunications viasatellites.They found a signal from alldirections at = 7.35 cm.Theonly explanation was that itis real.Work quickly began tomeasure the background at other wavelengths.

Cosmic Microwave Background

With the addition of more and more points it became clear that it was

a thermal curve at a temp.of 2.7K.Later more precise measurements tell us that itfits a black-body spectrumat T = 2.726 +/- 0.005K perfectly.

Note:- In this particular graph the wavelength increases`to the left.

Cosmic Microwave Background

Cosmic Microwave Background Explorer( COBE), a satellite launched by NASA in 1989 was the first to show that there are very smallfluctuations in the CMB at the level of one part in 105.

Summary of CMB Measurements

Summary of CMB Measurements

Summary-CMB Measurements

1.CMB is seen in all directions as a perfect BB at T = 2.726+/-0.005K

2.At sensitivity levels of 1 in 103 there is a large scale anisotropy in the CMB due to the motion of the Earth through the frame of reference in which the CMB is uniform.

Earth is moving at about 350 km s-1 w.r.tthis frame.We are moving towards LEO andaway from Aquarius.[Galaxy is moving insame direction at 600 km s-1 ]This motion has to be subtracted from the measured spectrumin order to see the real spectrum.

Cosmic Microwave Background

Cosmic Microwave Background Explorer( COBE), a satellite launched by NASA in 1989 was the first to show that there are very smallfluctuations in the CMB at the level of one part in 105.

Origin of CMB

Universe starts as a singularity. Very high density and Temperature.

As it expands it cools and becomes less dense.

Initially the temperature is so high that any particle (and its anti-particle) that can

exist, does exist. We get a soup of quarks , gluons, electrons, positrons, neutrinos,

anti-neutrinos ---------

When temperature falls enough the quarks become locked into nucleons (protons

and neutrons) for ever.

At next stage elements begin to form but process halts quickly.

This is because no stable isotopes of any element with masses 5 or 8 exist.

The Universe is 75% H and 25% He.

It is also flooded with electromagnetic radiation. Any H atom that forms is

immediately broken up in collisions with photons.

Origin of the CMBWhen Universe was younger it wasnot only denser but hotter. In early Universe it consisted of 75% H:25% He and it was full of radiation.As a result all the H wasIonised[See a) on left].If an atom formed it soon disintegrated in collision with a photon. Universe was opaque.There was a strong coupling between Matter and Radiation.

At some point( 3 x 105 years) T dropped to 4000K.Now the no.of photonswith sufficient energy is too small to stop atoms forming.

There is an equilibrium. As fast as atoms form they are broken up.

Origin of the CMBNow H atoms form, photons no longer interact strongly with Matterand the Universe becomes transparent. The photons can nowspread freely throughout the Universewith a BB spectrum initially at 4000Kbut with time it has been shifted to 2.726K.

Since radiation was continuouslyscattered up to this point it is known asthe Last Scattering Surface.We cannot see anything prior to thistime because the Universe was opaque.

Tiny ripples represent non-uniformity at that time.Universe was 103 times smaller.

Uniformity was remarkable but present non-uniformity comes from it.

This clinched the Big Bang Model since no other could explain it.

Expansion of the Universe

As the Universe expands the volume increases proportionally tothe cube of the scale factor.

Evolution of the Universe in Big Bang Model

1 Assume adiabatic process = No heat into or out of the system[Universe]2.Assume Universe is an idealised, smooth fluid.A good assumption in the early stages.3.Constituents are all in thermal eqbm. E = kT4.First Law of Thermodynamics dU = dQ + dW In an adiabatic system dU = dW = pdV. For a non-viscous fluid p = 1/3 where = energy density.5.As Universe expands V and must change. We introduce a scale factor R(t) with dimensions of length. So V = 4/3.. R(t)3

6.Now take a very small volume with only one photon in it. R = h/V = hc/V 1/V 1/ R(t)3 1/R(t)4

where depends on redshift and must scale with R(t).7.Now E = R.V R(t)3 . 1/R(t)4 1/R(t)

8.Hence since E = kT we have T 1/R(t)

Thus as Space expands [R(t) increases], T decreases.

The table gives a history of theUniverse as seen in terms of the Big Bang Model.

The History can be divided intoepochs characterised by the mainprocesses at work in each epoch.

For each epoch the time from theBig Bang and the temperature areindicated.

Some critical points

An Inflationary Universe