5.5.3 Cosmology

So in this section we'll use different unit for distance (as apposed to the meter). This is because the distances are so vast. The new units are as follows:

• Astronomical unit (AU)
• This is the average distance from the Earth to the Sun. It is most often used to express the average distance between the Sun and other planets in our Solar System.
• It is equal to 1.50 x 10¹¹ m
• Light-year (ly)
• This is the distance travelled by light in a vacuum in a time of one year. It is often used when expressing distances to stars or other galaxies.
• It is equal to 3.00 x 10⁸ x (365 x 24 x 60 x 60) = 9.46 x 10¹⁵ m
• Parsec (pc)
• So for this one, we need to be aware that in cosmology we measure angles in arcseconds and arcminutes. One arcsecond is 1/3600th of a degree (1 arcminute is 1/60th of a degree).
• The parsec is defined as the distance at which a radius of one AU subtends an angle of one arcsecond.
• The value of 1pc can be determined by tan(1/3600) = 1AU/1pc. This means that 1pc = 1AU/tan(1/3600). This equals 3.1 x 10¹⁶ m.
• It is important to realise that at a distance of n parsecs, the angle subtended by a radius of 1AU = 1/n arcseconds.

There is a special technique used to determine the distance to stars that are relatively close to Earth (like, less than 100 pc). This is known as stellar parallax. Parallax is the shift in position of a relatively close star against the backdrop of much more distant stars as Earth orbits the Sun. If p (parallax angle) is measured in arcseconds, the distance to the nearby star in parsecs is given by the following equation:

d = 1/p

This technique can't be used to measure the distance between stars bigger than 100pc from the Earth because as d increases the parallax angle decreases. Eventually becoming too small to measure accurately, even with the most advanced astronomical techniques.

The Doppler effect
The Doppler effect is used to determine the speed of moving objects.

When a wave source moves relative to an observer, the frequency and wavelength of the waves received by the observer change compared with what would be observed without relative motion. Originally, two points equidistant from the source would receive waves at the same frequency and wavelength as they were emitted from the source. When the source moves closer to one point, the waves received by this point will be compressed. They have a shorter wavelength and a higher frequency (therefore, a shorter period).

How fast the wave source moves relative to the observer affects the size of the observed shift in wavelength and frequency. For electromagnetic waves we can use the Doppler equation. The equation shows that the faster a source moves, the greater the observed change in wavelength and frequency:

Δλ/λ ≈ Δf/f ≈ v/c

NOTE: The Doppler equation can only be used for galaxies with speed far less than the speed of light.

As we know already, one technique to analyse starlight involves looking at the absorption spectra from stars. The Doppler effect can be used to determine the relative velocity of a distant galaxy. Any difference in the observed wavelengths of the absorption lines must be caused by the relative motion between the galaxy and the Earth.

NOTE: if the galaxy is moving towards the Earth the absorption lines will be blue-shifted (they move toward the blue end of the spectrum because the wavelength appears shorter). If the galaxy is moving away from the Earth the absorption lines will be red-shifted as the wavelength appears stretched.

Using data from the absorption spectra of many distant galaxies, Hubble made two key observations:

1. A confirmation that earlier observations that the light from the vast majority of galaxies was red shifted (they had a relative velocities away from the Earth)
2. He found that in general the further away the galaxy was the greater the observed red shift and so the faster the galaxy was moving

From these observations Hubble formulated his law (Hubble's Law): The recessional speed (v) of a galaxy is almost directly proportional to its distance (d) from the Earth.

This means that a graph of recessional speed against distance for all galaxies will produce a straight line graph through the origin. The gradient is a constant of proportionality - the Hubble constant (Ho). It's (current) value is 2.2  × 10^-18 km s^-1 Mpc^-1. From Hubble's law it can be derived that:

v ≈ Ho x d

Hubble's law has been very useful in determining key evidence for the Big Bang theory and the model of the expanding Universe (following the Big Bang). This model is the accepted explanation of the observation that the light from nearly all the galaxies we can see is red-shifted. The further two points are apart the faster their relative motion (the more red-shifted their spectra).

The cosmological principle is the assumption that (when viewed on a large enough scale) the Universe is homogenous and isotropic and the laws of physics are universal:

• Homogenous means the matter is distributed uniformly across the Universe (ie the density of the universe is uniform
• Isotropic means that the Universe looks the same in all directions to every observer (ie there is no end to the Universe)
• The laws of physics can be applied across the Universe (meaning that theories/models tested on Earth can be applied to everything within the Universe).

The Big Bang
Hubble's law and the microwave background radiation are two key pieces of evidence for the Big Bang theory. Hubble's law shows that space is expanding as the galaxies are receding from each other because space itself is expanding in all dimensions.

The existence of microwave background radiation is the second piece of evidence for the Big Bang. Microwave background radiation can only be explained by the Big Bang and the expansion of space. It's existence can be explained in two ways:

• When the Universe was young and extremely hot space was saturated with high-energy gamma photons. The expansion meant that space itself was stretched over time. The expansion stretched the wavelength of these high-energy photons so we now observe this primordial electromagnetic radiation as microwaves.
• The Universe was extremely hot and dense when it was young. Expansions over billions of years reduced it's temperature to about 2.7 K. The Universe can be treated as a black-body radiator. At this temperature the peak wavelength would correspond to about 1mm (in the microwave region of the spectrum).

We can estimate the age of the Universe by assuming that it has expanded uniformly over time since the Big Bang. This actually isn't the case lol. Results from recent observations show that the expansion of the Universe is accelerating. Nonetheless, this assumption will give a crude indication the Universe's age.

So, Hubble's law shows galaxies are receding from each other. If a galaxy at a distance 'd' is moving away at a constant speed 'v' then a time (d/v) must have elapsed since it was next to our galaxy. This time is roughly the age of the Universe. The ratio d/v is equal to 1/Ho meaning that:

age of the Universe 't' ≈ 1/Ho

NOTE: This gives the age of the Universe to be 4.5 x 10¹⁷ s (≈14 billion years).

As I mentioned above, it is now known that the Universe appears to be expanding at an increasing rate. The most widely accepted theory includes the concept of dark energy. It is suggested that this hypothetical form of energy fills all of space and tends to accelerate the expansion of the Universe. The two most significant discoveries that have changed our understanding of the Universe is the discovery of dark energy and dark matter.

We need energy to accelerate things. The term 'dark energy' was coined to describe a hypothetical form of energy that permeates all space. It is estimated that dark energy makes up around 68% of our Universe.

In the 1970s astronomers studying the Doppler shift in light from galaxies found that the velocity of the stars in the galaxies did not behave as predicted. It was expected that their velocity would decrease as the distance from the centre of the galaxy increases. This effect is observed in other gravitational systems where most of the mass is in the centre (e.g the moons of Jupiter). The observations can be explained if the mass of the galaxy is not concentrated in the centre. We currently think that there must be another type of matter which we cannot see. This dark matter is spread throughout the galaxy, explaining the observations. The Universe must be made up of 27% of this matter (according to calculations).

All we know about dark matter is we know it cannot be seen directly with telescopes and it neither emits not absorbs light.

The rest of the universe is made up of a small percentage of ordinary matter.

5.5.2 Electromagnetic radiation from stars

When electrons are bound to their atoms in a gas they can only exist in one of a discrete set of energies - the energy levels of an electron:

• An electron cannot have a quantity of energy between two levels
• The energy levels are negative because external energy is required to remove an electron from the atom.
• The energy leve with the most negative value is the ground state/ground level
• An electron with zero energy is free from the atom

An atom is said to be excited when an electron moves from a lower to a higher energy level within an atom in a gas. Raising an electron into a higher energy level requires external energy - for example when photons of specific energy are absorbed by the atom. Similarly, when an electron moves from a higher energy level to a lower energy level it loses energy. As we know, energy is conserved. This means that as the electron makes a transition between levels a photon is emitted from the atom (this can be known as de-excitation). In order for an electron to make a transition from -3eV to -6.8eV (for example) it must lose 3.8eV. It emits this in the form of a photon with energy 3.8eV. The energy of a photon emitted in an electron transition from a higher to lower energy level is given by the equation:

E = hf

NOTE: it is important to realise that each element has its own unique set of energy levels.

Different atoms have different spectral lines - the spectra from starlight can be used to identify the elements within stars without a direct sample (as a ample of a star is pretty hard to obtain lol). There are three kinds of spectra:

• Emission line spectra - each element produces a unique emission line spectrum because of its unique set of energies
• Continuous spectra - all visible frequencies/wavelengths are present. The atoms of a heated solid metal (e.g. a filament lamp) will produce this type of spectrum
• Absorption line spectra - this type of spectrum has a series of dark spectral lines against a continuous spectrum. The dark lines have the same wavelengths as the bright emission spectral lines for the same gas atoms.

If the atoms are excited then when the electrons drop back into the lower energy levels they emit photons with a set of discrete frequencies specific to the element. This produces a characteristic emission line spectrum and each spectral line corresponds to photons with a specific wavelength. These spectra can be observed in a laboratory from heated gases. Each coloured line represents a unique wavelength/frequency of photon emitted when an electron moves between two specific energy levels.

A bit more on absorption line spectra: This is formed when light from a source that produces a continuous spectrum passes through a cooler gas. As the photons pass through the fas some are absorbed by the gas atoms, raising electrons up into higher energy levels and so exciting the atoms. Only photons with an energy exactly equal to the difference between the different energy levels are absorbed (meaning that only a specific wavelength are absorbed) - this creates dark lines in the spectrum. These lines show which photons have been absorbed. When the electron drops back down to a lower energy level the photons are re-emitted in all directions so the intensity in the original direction is reduced. 

We need to know about how we detect which elements are present on stars (without a sample). Basically, when the light from a star is analysed it is found to be an absorption line spectrum. Some wavelengths of light are missing - these are the photons that have been absorbed by atoms of cooler gas in the outer layer of the star. If we know the line spectrum of a particular element we can check whether the element is present in the star (if a particular element is present its characteristic pattern of spectral lines will appear as dark lines in the absorption line spectrum).

A diffraction grating is an optical component with regularly spaced slits/lines that diffract and split light into beams of different colours travelling in different directions. These beams can be analysed to determine the wavelengths of spectral lines in the laboratory/from starlight. It is slightly different to the double slit (Youngs Double Slit experiment) in which it consists of a large number of lines ruled on a glass/plastic slide and each line diffracts like a slit producing a clearer and brighter interference pattern than the double slit. The direction of the beams produced depends on the spacing of the lines/slits of the grating and the wavelength. 

Like in the double slit, maxima and minima are still formed. The interference pattern is the result of the superposition of the diffracted waves in the space beyond the grating.  The formation of maxima at a particular point depends on the path difference and the phase difference of the waves from all the slits.

The zero-order maxima (n=0) is formed when the path difference is zero, that is at an angle θ=0. The angle θ is measured relative to the normal to the grating/to the direction of incident light. For the nth order maxima the path difference QY at an angle θ will be equal to nλ. Also, the distance PQ is the separation between adjacent lines/slits on the grating. 

We can use the following equation to determine any of the above features:

sinθ = QY/QP = nλ/d

dsinθ = nλ

NOTE: n must be an integer value. 

Thus us known as the grating equation. It can be used to accurately determine the wavelength of monochromatic light.


At any given temperature (above absolute zero) an object emits electromagnetic radiation of different wavelengths and different intensities. We can model a hot object as a black body. This is an idealised object that absorbs all the electromagnetic radiation that shines onto it and (when in thermal equilibrium) emit a characteristic distribution of wavelengths at a specific temperature.

Wein's displacement law relates the absolute temperature (T) of a black body to the peak wavelength (λmax) at which intensity is a maximum. It's a bit confusing because λmax isn't the maximum λ, it's actually the most abundant λ. The law states that λmax is inversely proportional to T:

λmax ∝ 1/T

Therefore, for any black body emitter λmax T = constant. the constant is Wein's constant - 2.90x10^-3 mK.

Modelling objects as approximate black bodies helps scientists to determine temperatures of objects simply by analysing the electromagnetic radiation they emit. It is important to realise that, as the temperature of an object changes, the distribution of the emitted wavelengths changes. As temperature increases, the peak wavelength reduces and the peak of the intensity-wavelength graph becomes sharper.

Stephan's law: The total power radiated by a star is called luminosity. Stephan's law states that the total power radiated per unit surface area of a black body is directly proportional to the fourth power of the absolute temperature of the black body. Luminosity can be found using the equation below:

L = 4πr²σT⁴

NOTE: σ is known as the Stephan constant (5.67 x 10^-8 WmT^-2K^-4).

Stephan's law shows that the the luminosity (L) of a star is directly proportional to:

• it's radius² (L ∝ r²)
• it's surface area (L ∝ 4πr²)
• it's surface absolute temperature⁴ (L ∝ T⁴)

We can use both Wein's displacement law and Stephan's law to estimate the radius of a distant star. Once the radius is known we can calculate it's density and mass using Newton's law of gravitation.

5.5.1 Stars

Okay so we need to know some definitions:

• Planets - An object in orbit around a star with three important characters:
• A mass large enough for its own gravity to give it a round shape
• It has no fusion reactions
• It has cleared its orbit to most other objects
• Planetary satellites - A body in orbit around a planet
• Comets - Small irregular bodies made up of ice, dust, and small pieces of rock. All orbit the sun. Can develop tails as they near the sun. Range from a few hundred metres to tens of kilometers across
• Solar systems - Contains the sun and all objects that orbit the sun.
• Galaxies - A collection of stars, and interstellar dust and gas. On average it will contain 100 billion stars.
• Universe - A collection of all the galaxies

Nebulae are gigantic clouds of dust and gas. They are formed over millions of years as the tiny gravitational attraction between particles of dust and gas pulls the particles towards each other forming vast clouds. The gravitational collapse accelerates as the dust/gas gets closer together and denser regions form which pull in more dust and gas, gaining mass and getting denser. They also get hotter as gravitational energy is eventually transferred to thermal energy. In one part of the cloud a prostar forms - this is a very hot, very dense, sphere of dust and gas but is not yet a star.

Nuclear fission needs to occur for a prostar to become a star. Fusion reactions produce kinetic energy. Extremely high pressures and temperatures inside the core are needed in order to overcome the electrostatic repulsion between hydrogen nuclei in order to fuse them together to form helium nuclei. Sometimes, as more and more mass is added to the prostar, it grows so large and the core becomes so hot that the kinetic energy of hydrogen nuclei overcomes the electrostatic repulsion and hydrogen nuclei are forced together to make helium nuclei - it is here that a star forms. 

The star now remains in a stable equilibrium with almost a constant size since gravitational forces compress the star but the radiation pressure (from the photons emitted during fusion) and the gas pressure (from the nuclei in the core) push outward. The forces balance so an equilibrium is maintained. This is known as the stars main sequence. How long a star remains stable depends on the size and mass of it's core. The cores of large, massive supergiant stars are hotter than those of small stars so they release more power and convert the available hydrogen to helium in a shorter time meaning they are only stable for a few million years. Smaller stars (e.g. our sun) are stable for tens of billions of years. As stars run out of hydrogen in their core (because it's all fused to form hydrogen), the star begins to move off it's main sequence. 


Stars with a mass between 0.5 - 10 times our sun (M_☉) 
These stars eventually evolve into red giants. At the start of this phase the reduction in energy released by the fusion in the core means the gravitational force is greater than the radiation/gas pressure force and the core shrinks. Pressure increases enough to start fusion in the shell around the core. Red giant start have inert cores (fusion no longer takes place). This is because very little hydrogen remains and also the temperature is not high enough for the helium nuclei to overcome the electrostatic repulsion between them. Hydrogen fuses to helium in the shell around the core. This causes the periphery of the star to expand as the layers slowly move away from each other. These layers cool as thy expand making the star go red. 

Most of the layers of the red giant will drift away as planetary nebula. The core then becomes a very dense hot (30,000K) core as a white dwarf. No fusion reactions occur here, it only emits energy because it leaks photons created in its earlier evolution. According to the Pauli exclusion principle, no two electrons can exist in the same energy state. When the core of a star begins to collapse under the force of gravity, the electrons are squeezed together. This creates a pressure that prevents the core from further gravitational collapse. This pressure created by the electrons is known as the electron degeneracy pressure. 

It is important to realise that the electron degeneracy pressure is only sufficient to prevent gravitational collapse if the core has a mass less than 1.44M_☉. This is the Chandrasekhar limit. This limit is the maximum mass of a stable white dwarf.

Stars with a greater mass of 10M_☉

Since their mass is much greater their cores are much hotter meaning they use up their hydrogen supply in a shorter amount of time. As with smaller stars, when the hydrogen in the core runs low it begins to collapse under gravitational forces. As the cores of these larger stars are much hotter the helium nuclei (formed from fusion of hydrogen) are moving fast enough to overcome electrostatic repulsion and the helium nuclei fuse into heavier elements. These changes in the core cause the star to expand which forms a supergiant (sometimes known as a super red giant).Temperatures and pressures are high enough to fuse even massive nuclei forming a series of shells inside the star. This process continues until the star develops an iron core. Iron nuclei cannot fuse as the reaction would produce no energy. This makes the star very unstable and the star dies by supernova (type 2) - a catastrophic implosion of the layers that bounce off the solid core leading to a shockwave that ejects all the core material into space. Supernovae create all the heavy elements (everything above iron in the periodic table was created in a supernova (such events distribute these heavier elements throughout the universe.

After a supernova the remnant core is compressed into either a:

• Neutron star - if the mass of the core is greater than the Chandrasekhar limit, the gravitational collapse continues, forming a neutron star. These are almost entirely made up of neutrons and are very very dense.
• Black hole - if the core has a mass greater than 3M_☉ the gravitational collapse continues to compress the core resulting in a gravitational field so strong that an object must be travelling greater than the speed of light to escape it.

The last thing we need to know in this section of the spec is the Hertzsprung-Russell diagram. This is a graph of stars in our galaxy showing the relationship between luminosity and their average surface temperature. The luminosity of a star is the total radiant power output of the star. The luminosity and temperature of stars can both vary widely so the HR (Hertzsprung-Russell) diagram scales are logarithmic.

Lower mass stars evolve into red giants moving away from their red sequence. Then they lose their cooler outer layers and slowly move across the diagram crossing the main sequence line ending up as white dwarfs. Higher mass stars start at point X before rapidly consuming their fuel and swelling into red supergiants at Y before going supernova.