Describe how some white dwarf stars become type Ia supernovae.

Hence, explain why a type Ia supernova is used as a standard candle.

Explain how the observation of type Ia supernovae led to the hypothesis that dark energy exists.

Calculate the rotation velocity of stars 4.0 kpc from the centre of the galaxy. The average density of the galaxy is 5.0 × 10^–21 kg m^–3.

Explain why the rotation curves are evidence for the existence of dark matter.

Outline, with reference to star formation, what is meant by the Jeans criterion.

In the proton–proton cycle, four hydrogen nuclei fuse to produce one nucleus of helium releasing a total of 4.3 × 10^–12 J of energy. The Sun will spend 10¹⁰ years on the main sequence. It may be assumed that during this time the Sun maintains a constant luminosity of 3.8 × 10²⁶ W.

Show that the total mass of hydrogen that is converted into helium while the Sun is on the main sequence is 2 × 10²⁹ kg.

Massive stars that have left the main sequence have a layered structure with different chemical elements in different layers. Discuss this structure by reference to the nuclear reactions taking place in such stars.

In 2017, two neutron stars were observed to merge, forming a black hole. The material released included chemical elements produced by the r process of neutron capture. Describe two characteristics of the elements produced by the r process.

Derive, using the concept of the cosmological origin of redshift, the relation

T $\propto \frac{1}{R}$

between the temperature T of the cosmic microwave background (CMB) radiation and the cosmic scale factor R.

The present temperature of the CMB is 2.8 K. This radiation was emitted when the universe was smaller by a factor of 1100. Estimate the temperature of the CMB at the time of its emission.

State how the anisotropies in the CMB distribution are interpreted.

The Sun is a second generation star. Outline, with reference to the Jeans criterion (M_J), how the Sun is likely to have been formed.

Suggest how fluctuations in the cosmic microwave background (CMB) radiation are linked to the observation that galaxies collide.

Show that the critical density of the universe is

$$\frac{3H^2}{8\pi G}$$

where H is the Hubble parameter and G is the gravitational constant.

The graph shows the variation with time t of the cosmic scale factor R in the flat model of the universe in which dark energy is ignored.

On the axes above draw a graph to show the variation of R with time, when dark energy is present.

The density of the observable matter in the universe is only 0.05 ρ_c. Suggest how the remaining 0.27 ρ_c is accounted for.

The density of dark energy is ρ_Λc² where ρ_Λ = ρ_c – ρ_m. Calculate the amount of dark energy in 1 m³ of space.

The mass of visible matter in the galaxy is M.

Show that for stars where r > R₀ the velocity of orbit is v = $\sqrt{\frac{GM}{r}}$.

Draw on the axes the observed variation with r of the orbital speed v of stars in a galaxy.

Explain, using the equation in (a) and the graphs, why the presence of visible matter alone cannot account for the velocity of stars when r > R₀.

The light from a distant galaxy shows that $z=0.11$.

Calculate the ratio $\frac{\mathrm{size} \mathrm{of} \mathrm{the} \mathrm{universe} \mathrm{when} \mathrm{the} \mathrm{light} \mathrm{was} \mathrm{emitted}}{\mathrm{size} \mathrm{of} \mathrm{the} \mathrm{universe} \mathrm{at} \mathrm{present}}$.

Outline how Hubble’s law is related to $z$.

Hubble originally linked galactic redshift to a Doppler effect arising from galactic recession. Hubble’s law is now regarded as being due to cosmological redshift, not the Doppler effect. Explain the observed galactic redshift in cosmological terms.

Show by calculation that Eta Aquilae A is not on the main sequence.

Estimate, in $\mathrm{pc}$, the distance to Eta Aquilae A using the parallax angle in the table.

Estimate, in $\text{pc}$, the distance to Eta Aquilae A using the luminosity in the table, given that $L_{\odot }=3.83\times {10}^{26} \mathrm{W}$.

Suggest why your answers to (b)(i) and (b)(ii) are different.

Eta Aquilae A is a Cepheid variable. Explain why the brightness of Eta Aquilae A varies.

Eta Aquilae A was on the main sequence before it became a variable star. Compare, without calculation, the time Eta Aquilae A spent on the main sequence to the total time the Sun is likely to spend on the main sequence.

State the nature of the anisotropies observed in the CMB radiation.

Identify two possible causes of the anisotropies in (a).

Explain the evidence that indicates the location of dark matter in galaxies.

Outline why a hypothesis of dark energy has been developed.

Distinguish between a constellation and an open cluster.

The parallax angle of Mintaka measured from Earth is 3.64 × 10^–3 arc-second. Calculate, in parsec, the approximate distance of Mintaka from Earth.

State why there is a maximum distance that astronomers can measure using stellar parallax.

The Great Nebula is located in Orion. Describe, using the Jeans criterion, the necessary condition for a nebula to form a star.

Determine the peak wavelength of the radiation emitted by Epsilon Indi.

Using the axis, draw the variation with wavelength of the intensity of the radiation emitted by Epsilon Indi.

The following data are available for the Sun.

Surface temperature  = 5800 K

Luminosity                  = $L_{\odot }$

Mass                          = $M_{\odot }$

Radius                       = $R_{\odot }$

Epsilon Indi has a radius of 0.73 $R_{\odot }$. Show that the luminosity of Epsilon Indi is 0.2 $L_{\odot }$.

Epsilon Indi is a main sequence star. Show that the mass of Epsilon Indi is 0.64 $M_{\odot }$.

The Sun will spend about nine billion years on the main sequence. Calculate how long Epsilon Indi will spend on the main sequence.

Describe the stages in the evolution of Epsilon Indi from the point when it leaves the main sequence until its final stable state.

Outline how Hubble measured the recessional velocities of galaxies.

Use the graph to determine the age of the universe in s.

Outline what is meant by dark energy.

State two candidates for dark matter.

Light from distant galaxies is redshifted. Explain the cosmological origin of this redshift.

Draw, on the axes, a graph to show the variation with time of the cosmic scale factor R for the flat model of the universe with dark energy.

Compare and contrast, the variation with time of the temperature of the cosmic background (CMB) radiation, for the two models from the present time onward.

Describe what is meant by dark matter.

The distribution of mass in a spherical system is such that the density ρ varies with distance r from the centre as

ρ = $\frac{k}{r^2}$

where k is a constant.

Show that the rotation curve of this system is described by

v = constant.

Curve A shows the actual rotation curve of a nearby galaxy. Curve B shows the predicted rotation curve based on the visible stars in the galaxy.

Explain how curve A provides evidence for dark matter.

Outline how the light spectra of distant galaxies are used to confirm hypotheses about the expansion of the universe.

Light from a hydrogen source in a laboratory on Earth contains a spectral line of wavelength 122 nm. Light from the same spectral line reaching Earth from a distant galaxy has a wavelength of 392 nm. Determine the ratio of the present size of the universe to the size of the universe when the light was emitted by the galaxy.

State what is meant by the critical density.

Calculate the density of matter in the universe, using the Hubble constant 70 km s^–1Mpc^–1.

It is estimated that less than 20 % of the matter in the universe is observable. Discuss how scientists use galactic rotation curves to explain this.

Describe the formation of a type Ia supernova.

Show that the distance to the supernova is approximately 3.1 × 10¹⁸ m.

State one assumption made in your calculation.

Justify that the total energy of this particle is $E=\frac{1}{2}m{v}^2-\frac{4}{3}\pi G\text{r}{r}^{2}m$.

At critical density there is zero total energy. Show that the critical density of the universe is: ${\text{r}}_c=\frac{3H_0^2}{8\pi G}$.

The accepted value for the Hubble constant is 2.3 × 10⁻¹⁸ s⁻¹. Estimate the critical density of the universe.

Describe the mechanism of formation of type I a supernovae.

Describe the mechanism of formation of type II supernovae.

Suggest why type I a supernovae were used in the study that led to the conclusion that the expansion of the universe is accelerating.

Proxima Centauri is a main sequence star with a mass of 0.12 solar masses.

Estimate $\frac{\mathrm{lifetime} \mathrm{on} \mathrm{main} \mathrm{sequence} \mathrm{of} \mathrm{Proxima} \mathrm{Centauri}}{\mathrm{lifetime} \mathrm{on} \mathrm{main} \mathrm{sequence} \mathrm{of} \mathrm{Sun}}$.

Describe why iron is the heaviest element that can be produced by nuclear fusion processes inside stars.

Discuss one process by which elements heavier than iron are formed in stars.

Show that the temperature of the universe is inversely proportional to the cosmic scale factor.

The present temperature of the cosmic microwave background (CMB) radiation is 3 K. Estimate the size of the universe relative to the present size of the universe when the temperature of the CMB was 300 K.

Outline, with reference to the Jeans criterion, why a cold dense gas cloud is more likely to form new stars than a hot diffuse gas cloud.

Explain how neutron capture can produce elements with an atomic number greater than iron.

Explain the formation of a type I a supernova which enables the star to be used as a standard candle.

Describe the r process which occurs during type II supernovae nucleosynthesis.