Identify two other characteristics of the CMB radiation that are predicted from the Hot Big Bang theory.

A spectral line in the hydrogen spectrum measured in the laboratory today has a wavelength of 21cm. Since the emission of the CMB radiation, the cosmic scale factor has changed by a factor of 1100. Determine the wavelength of the 21cm spectral line in the CMB radiation when it is observed today.

isotropic/appears the same from every viewing angle

homogenous/same throughout the universe

black-body radiation

23 100 «cm»
OR
231 «m»

State what is meant by a binary star system.

(i) Calculate $\frac{b_{\text{A}}}{b_{\text{B}}}=\frac{\text{apparent brightness of Alpha Centauri A}}{\text{apparent brightness of Alpha Centauri B}}$.

(ii) The luminosity of the Sun is 3.8 × 10²⁶ W. Calculate the radius of Alpha Centauri A.

Show, without calculation, that the radius of Alpha Centauri B is smaller than the radius of Alpha Centauri A.

Alpha Centauri A is in equilibrium at constant radius. Explain how this equilibrium is maintained.

A standard Hertzsprung–Russell (HR) diagram is shown.

Using the HR diagram, draw the present position of Alpha Centauri A and its expected evolutionary path.

two stars orbiting about a common centre «of mass/gravity»
Do not accept two stars orbiting each other.

i
stars are roughly at the same distance from Earth
OR
d is constant for binaries

$\frac{L_{\mathrm{A}}}{L_{\mathrm{B}}}=\frac{1.5}{0.5}=3.0$

Award [2] for a bald correct answer.

ii
$r=\sqrt{\frac{1.5\times 3.8\times {10}^{26}}{5.67\times {10}^{-8}\times 4\pi \times {5800}^4}}$

= 8.4 × 10⁸ «m»

Award [2] for a bald correct answer.

«A=$\frac{L}{\sigma T^4}$» B and A have similar temperatures

so areas are in ratio of luminosities

«so B radius is less than A»

radiation pressure/force outwards

gravitational pressure/force inwards

forces/pressures balance

Alpha Centauri A within allowable region

some indication of star moving right and up then left and down ending in white dwarf region as indicated

State two characteristics of the cosmic microwave background (CMB) radiation.

The present temperature of the CMB is 2.8 K. Calculate the peak wavelength of the CMB.

Describe how the CMB provides evidence for the Hot Big Bang model of the universe.

Determine the distance to this galaxy using a value for the Hubble constant of H₀ = 68 km s^–1$$Mpc^–1.

Estimate the size of the Universe relative to its present size when the light was emitted by the galaxy in (c).

black body radiation / 3 K

highly isotropic / uniform throughout
OR
filling the universe

Do not accept: CMB provides evidence for the Big Bang model.

[2 marks]

«$\lambda =\frac{2.9\times {10}^{-3}}{2.8}$» ≈ 1.0 «mm»

[1 mark]

the universe is expanding and so the wavelength of the CMB in the past was much smaller

indicating a very high temperature at the beginning

[2 marks]

«$z=\frac{v}{c}\Rightarrow$» v = 0.16 × 3 × 10⁵ «= 0.48 × 10⁵ km$$s⁻¹»

«$d=\frac{v}{H_0}\Rightarrow v=\frac{0.48\times {10}^5}{68}=706$» ≈ 710 «Mpc»

Award [1 max] for POT error.

[2 marks]

$z=\frac{R}{R_0}-1\Rightarrow \frac{R}{R_0}=1.16$

$\frac{R_0}{R}=0.86$

[2 marks]

State what is meant by a main sequence star.

Show that the mass of Theta 1 Orionis is about 40 solar masses.

The surface temperature of the Sun is about 6000 K. Estimate the surface temperature of Theta 1 Orionis.

Determine the distance of Theta 1 Orionis in AU.

Discuss how Theta 1 Orionis does not collapse under its own weight.

The Sun and Theta 1 Orionis will eventually leave the main sequence. Compare and contrast the different stages in the evolution of the two stars.

stars fusing hydrogen «into helium»

[1 mark]

$M=M_{\odot }{\left(4\times {10}^5\right)}^{\frac{1}{3.5}}=39.86M_{\odot }$

«$M\approx 40M_{\odot }$»

Accept reverse working.

[1 mark]

$4\times {10}^5={13}^2\times \frac{T^4}{{6000}^4}$

$T\approx 42000$ «K»

Accept use of substituted values into $L=\sigma$4$\pi$R²T⁴.

Award [2] for a bald correct answer.

[2 marks]

$4\times {10}^{-11}=4\times {10}^5\times \frac{1\text{A}{\text{U}}^2}{d^2}$

$d=1\times {10}^8$ «AU»

Accept use of correct values into $b=\frac{L}{4\pi d^2}$.

[2 marks]

the gravitation «pressure» is balanced by radiation «pressure»

that is created by the production of energy due to fusion in the core / OWTTE

Award [1 max] if pressure and force is inappropriately mixed in the answer.

Award [1 max] for unexplained "hydrostatic equilibrium is reached".

[2 marks]

the Sun will evolve to become a red giant whereas Theta 1 Orionis will become a red super giant

the Sun will explode as a planetary nebula whereas Theta 1 Orionis will explode as a supernova

the Sun will end up as a white dwarf whereas Theta 1 Orionis as a neutron star/black hole

[3 marks]

Describe what is meant by the Big Bang model of the universe.

State two features of the cosmic microwave background (CMB) radiation which are consistent with the Big Bang model.

Determine the distance to the galaxy in Mpc.

Describe how type Ia supernovae could be used to measure the distance to this galaxy.

theory in which all space/time/energy/matter were created at a point/singularity

at enormous temperature

with the volume of the universe increasing ever since or the universe expanding

OWTTE

[2 marks]

CMB has a black-body spectrum

wavelength stretched by expansion

is highly isotropic/homogenous

but has minor anisotropies predicted by BB model

T «= 2.7 K» is close to predicted value

For MP4 and MP5 idea of “prediction” is needed

[2 marks]

$\frac{v}{c}=z\Rightarrow v=0.084\times 3\times {10}^5=2.52\times {10}^4$ «km$$s^–1»

$d=\frac{v}{H_0}=\frac{2.52\times {10}^4}{68}=370.6\approx 370$ «Mpc»

Allow ECF from MP1 to MP2.

[2 marks]

type Ia have a known luminosity/are standard candles

measure apparent brightness

determine distance from d = $\sqrt{\frac{L}{4\pi b}}$

Must refer to type Ia. Do not accept other methods (parallax, Cepheids)

[3 marks]

State the most abundant element in the core and the most abundant element in the outer layer.

The Hertzsprung–Russell (HR) diagram shows two main sequence stars X and Y and includes lines of constant radius. R is the radius of the Sun.

Using the mass–luminosity relation and information from the graph, determine the ratio $\frac{\text{density of star X}}{\text{density of star Y}}$.

On the HR diagram in (b), draw a line to indicate the evolutionary path of star X.

Outline why the neutron star that is left after the supernova stage does not collapse under the action of gravitation.

The radius of a typical neutron star is 20 km and its surface temperature is 10⁶ K. Determine the luminosity of this neutron star.

Determine the region of the electromagnetic spectrum in which the neutron star in (c)(iii) emits most of its energy.

core: helium

outer layer: hydrogen

Accept no other elements.

[2 marks]

ratio of masses is ${\left(\frac{{10}^4}{{10}^{-3}}\right)}^{\frac{1}{3.5}}={10}^2$

ratio of volumes is ${\left(\frac{10}{{10}^{-1}}\right)}^3={10}^6$

so ratio of densities is $\frac{{10}^2}{{10}^6}={10}^{-4}$

Allow ECF for MP3 from earlier MPs

[3 marks]

line to the right of X, possibly undulating, very roughly horizontal

Ignore any paths beyond this as the star disappears from diagram.

[1 mark]

gravitation is balanced by a pressure/force due to neutrons/neutron degeneracy/pauli exclusion principle

Do not accept electron degeneracy.

[1 mark]

L = $\sigma$AT ⁴ = 5.67 x 10^–8 x 4$\pi$ x (2.0 x 10⁴)² x (10⁶)⁴

L = 3 x 10²⁶ «W»
OR
L = 2.85 x 10²⁶ «W»

Allow ECF for [1 max] if $\pi$r ² used (gives 7 x 10²⁶ «W »)

Allow ECF for a POT error in MP1.

[2 marks]

$\lambda =\frac{2.9\times {10}^{-3}}{{10}^6}=2.9\times {10}^{-9}$ «m»

this is an X-ray wavelength

[2 marks]

Determine the distance from Earth to the Cepheid star in parsecs. The luminosity of the Sun is 3.8 × 10²⁶ W. The average apparent brightness of the Cepheid star is 1.1 × 10^–9 W m^–2.

Explain why Cephids are used as standard candles.

from first graph period=5.7 «days» ±0.3 «days»

from second graph $\frac{L}{L_{\text{SUN}}}=2300$ «$\pm \text{200}$»

d = «$\sqrt{\frac{2500\times 3.8\times {10}^{26}}{4\pi \times 1.1\times {10}^{-9}}}=8.3\times {10}^{18}\text{m}$» =250 «pc»

Accept answer from interval 240 to 270 pc If unit omitted, assume pc.
Watch for ECF from mp1

Cepheids have a definite/known «average» luminosity

which is determined from «measurement of» period
OR
determined from period-luminosity graph

Cepheids can be used to estimate the distance of galaxies

Do not accept brightness for luminosity.

Show that the apparent brightness $b\propto \frac{A{T}^4}{d^2}$, where $d$ is the distance of the object from Earth, $T$ is the surface temperature of the object and $A$ is the surface area of the object.

Two of the brightest objects in the night sky seen from Earth are the planet Venus and the star Sirius. Explain why the equation $b\propto \frac{A{T}^4}{d^2}$ is applicable to Sirius but not to Venus.

substitution of $L=\sigma A{T}^4$ into $b=\frac{L}{4{\mathrm{\pi d}}^2}$ giving $b=\frac{\sigma A{T}^4}{4{\mathrm{\pi d}}^2}$

Removal of constants $\sigma$ and $4\mathrm{\pi }$ is optional

equation applies to Sirius/stars that are luminous/emit light «from fusion» ✓

but Venus reflects the Sun’s light/does not emit light «from fusion» ✓

OWTTE

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

Estimate, in $\text{pc}$, 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.

$«\frac{L}{L_{\odot }}=\frac{M^{ 3.5}}{{M_{\odot }}^{3.5}}=5.{70}^{3.5}=» 442$ ✓

the luminosity of Eta (2630$L_{\odot }$) is very different «so it is not main sequence» ✓

Allow calculation of $L^{\frac{1}{3.5}}$ to give $M=\mathit{9}\mathit{.}\mathit{5}\mathit{ }{M}_{\odot }$ so not main sequence

OWTTE

$d«=\frac{1}{2.36\times {10}^{-3}}»=424 «\mathrm{pc}»$ ✓

Use of $d=\sqrt{\frac{L}{4\mathrm{\pi b}}}$ ✓

$=\sqrt{\frac{2630\times 3.83\times {10}^{26}}{4\mathrm{\pi }\times 7.20\times {10}^{-10}}}$ ✓

$«=\frac{1.055\times {10}^{19}}{3.26\times 9.46\times {10}^{15}}»=342«\mathrm{pc}»$ ✓

Award [3] marks for a bald correct answer between $\mathit{340}$ and $\mathit{344}\mathit{ }«\mathrm{pc}»$

parallax angle in milliarc seconds/very small/at the limits of measurement ✓

uncertainties/error in measuring $L$ οr $b$ or $\theta$ ✓

values same order of magnitude, so not significantly different ✓

Accept answers where MP1 and MP2 both refer to parallax angle

OWTTE

reference to change in size ✓
reference to change in temperature ✓
reference to periodicity of the process ✓
reference to transparency / opaqueness ✓

Outline one reason for the difference in wavelength.

Determine the velocity of the galaxy relative to Earth.

galaxies are moving away

OR

space «between galaxies» is expanding

Do not accept just red-shift

«$\frac{\mathrm{\Delta }\lambda }{\lambda }=$» $\frac{1.04}{115}=\frac{v}{c}$

0.009c

Accept 2.7×10⁶ «m s^–1»

Award [0] if 116 is used for $\lambda$

Distinguish between the solar system and a galaxy.

Distinguish between a planet and a comet. 

a galaxy is much larger in size than a solar system

a galaxy contains more than one star system / solar system

a galaxy is more luminous

Any other valid statement.

[1 mark]

a comet is a small icy body whereas a planet is mostly made of rock or gas

a comet is often accompanied by a tail/coma whereas a planet is not

comets (generally) have larger orbits than planets

a planet must have cleared other objects out of the way in its orbital neighbourhood

[1 mark]

Main sequence stars are in equilibrium under the action of forces. Outline how this equilibrium is achieved.

A main sequence star P, is 1.3 times the mass of the Sun. Calculate the luminosity of P relative to the Sun.

The luminosity of the Sun L${}_{\odot }$ is 3.85 × 10²⁶ W. Determine the luminosity of Gacrux relative to the Sun.

The distance to Gacrux can be determined using stellar parallax. Outline why this method is not suitable for all stars.

draw the main sequence.

plot the position, using the letter P, of the main sequence star P you calculated in (b).

plot the position, using the letter G, of Gacrux.

Discuss, with reference to its change in mass, the evolution of star P from the main sequence until its final stable phase.

photon/fusion/radiation force/pressure balances gravitational force/pressure

gives both directions correctly (outwards radiation, inwards gravity)

OWTTE

[2 marks]

«L $\propto$ M³⁵ for main sequence»

luminosity of P = 2.5 «luminosity of the Sun»

[1 mark]

L_Gacrux = 5.67 × 10^–8 × 4π × (58.5 × 10⁹)² × 3600⁴

L_Gacrux = 4.1 × 10^–29 «W»

$\frac{L_{Gacrux}}{L_{\odot }}$ «= $\frac{4.1\times {10}^{29}}{3.85\times {10}^{26}}$» = 1.1 × 10³

[3 marks]

if the star is too far then the parallax angle is too small to be measured

OR

stellar parallax is limited to closer stars

OWTTE

[1 mark]

line or area roughly inside shape shown – judge by eye

Accept straight line or straight area at roughly 45°

[1 mark]

P between $1L_{\odot }$ and ${10}^1{L}_{\odot }$ on main sequence drawn

[1 mark]

at ${10}^3{L}_{\odot }$, further to right than 5000 K and to the left of 2500 K (see shaded region)

[1 mark]

ALTERNATIVE 1

Main sequence to red giant

planetary nebula with mass reduction/loss

OR

planetary nebula with mention of remnant mass

white dwarf

ALTERNATIVE 2

Main sequence to red supergiant region

Supernova with mass reduction/loss

OR

Supernova with mention of remnant mass

neutron star

OR

Black hole

OWTTE for both alternatives

[3 marks]

The astronomical unit ($\mathrm{AU}$) and light year ($\mathrm{ly}$) are convenient measures of distance in astrophysics. Define each unit.

$\mathrm{AU}$:

$\mathrm{ly}$:

Comets develop a tail as they approach the Sun. Identify one other characteristic of comets.

Identify one object visible in the image that is outside our Solar System.

$AU$: «average» distance from the Earth to the Sun ✓

$ly$: distance light travels in one year ✓

made of ice «and dust» ✓

«highly» eccentric/elliptical orbit around the Sun ✓

formed in the Oort Cloud ✓

star / named star / stellar cluster/ galaxy/ constellation ✓

Answer may be indicated on the photograph.

Estimate, using the data, the age of the universe. Give your answer in seconds.

Identify the assumption that you made in your answer to (a).

On the graph, one galaxy is labelled A. Determine the size of the universe, relative to its present size, when light from the galaxy labelled A was emitted.

use of gradient or any coordinate pair to find H₀ «= $\frac{v}{d}$» or $\frac{1}{H_0}$ «= $\frac{d}{v}$»

convert Mpc to m and km to m «for example $\frac{82\times {10}^3}{{10}^6\times 3.26\times 9.46\times {10}^{15}}$»

age of universe «= $\frac{1}{H_0}$» = 3.8 × 10¹⁷ «s»

Allow final answers between

3.7 × 10¹⁷ and 3.9 × 10¹⁷ «s» or 4 × 10¹⁷ «s»

[3 marks]

non-accelerated/uniform rate of expansion

OR

H₀ constant over time

OWTTE

[1 mark]

z « = $\frac{v}{c}$» = $\frac{4.6\times {10}^4\times {10}^3}{3.00\times {10}^8}$ = 0.15

$\frac{R}{R_0}$ = «z + 1» = 1.15

$\frac{R_0}{R}$ = «$\frac{1}{1.15}$ =» 0.87

OR

87% of the present size

[3 marks]

Suggest, using the graphs, why star X is most likely to be a main sequence star.

Show that the temperature of star X is approximately 10 000 K.

Write down the luminosity of star X (L_X) in terms of the luminosity of the Sun (L_s).

Determine the radius of star X (R_X) in terms of the radius of the Sun (R_s).

Estimate the mass of star X (M_X) in terms of the mass of the Sun (M_s).

Star X is likely to evolve into a stable white dwarf star.

Outline why the radius of a white dwarf star reaches a stable value.

the wavelengths of the dips correspond to the wavelength in the emission spectrum

the absorption lines in the spectrum of star X suggest it contains predominantly hydrogen

OR

main sequence stars are rich in hydrogen

[2 marks]

peak wavelength: 290 ± 10 «nm»

T = $\frac{2.9\times {10}^{-3}}{290\times {10}^{-9}}$ = «10 000 ± 400 K»

Substitution in equation must be seen.

Allow ECF from MP1.

[2 marks][

35 ± 5L_s

[1 mark]

$\frac{L_{\text{X}}}{L_{\text{s}}}=\frac{R_{\text{X}}^2\times {\text{T}}_{\text{X}}^4}{R_{\text{s}}^2\times {\text{T}}_{\text{s}}^4}$

OR

$R_{\text{X}}=\sqrt{\frac{L_{\text{X}}{\text{T}}_{\text{s}}^4}{L_s{\text{T}}_{\text{X}}^4}}\times R_{\text{s}}$

$R_{\text{X}}=\sqrt{\frac{35\times {6000}^4}{10{000}^4}}\times R_{\text{s}}$ (mark for correct substitution)

R_X = 2.1R_s

Allow ECF from (b)(i).

Accept values in the range: 2.0 to 2.3R_s.

Allow TS in the range: 5500 K to 6500 K.

[3 marks]

M_X = $(35{)}^{\frac{1}{3.5}}$M_s

M_X = 2.8M_s

Allow ECF from (b)(i).

Do not accept M_X = (35)${}^{\frac{1}{3.5}}$ for first marking point.

Accept values in the range: 2.6 to 2.9M_s.

[2 marks]

the star «core» collapses until the «inward and outward» forces / pressures are balanced

the outward force / pressure is due to electron degeneracy pressure «not radiation pressure»

[2 marks]

State what is meant by a binary star.

The peak spectral line of Sirius B has a measured wavelength of 115 nm. Show that the surface temperature of Sirius B is about 25 000 K.

The mass of Sirius B is about the same mass as the Sun. The luminosity of Sirius B is 2.5 % of the luminosity of the Sun. Show, with a calculation, that Sirius B is not a main sequence star.

Determine the radius of Sirius B in terms of the radius of the Sun.

Identify the star type of Sirius B.

draw the approximate positions of Sirius A, labelled A and Sirius B, labelled B.

sketch the expected evolutionary path for Sirius A.

two stars orbiting a common centre «of mass»

Do not accept “stars which orbit each other”

«$\lambda$ x T = 2.9 x 10^–3»

T = $\frac{2.9\times {10}^{-3}}{115\times {10}^{-9}}$ = 25217 «K»

use of the mass-luminosity relationship or ${\left(\frac{M_{\text{Sirius}}}{M_{\text{Sun}}}\right)}^{3.5}$ = 1

if Sirius B is on the main sequence then $\left(\frac{L_{\text{Sirius}\text{B}}}{L{}_{\text{Sun}}}\right)$ = 1 «which it is not»

Conclusion is given, justification must be stated

Allow reverse argument beginning with luminosity

$\left(\frac{L_{\text{Sirius}\text{B}}}{L{}_{\text{Sun}}}\right)$ = 0.025

r _Sirius = «$\sqrt{0.025\times {\left(\frac{5800}{25000}\right)}^4}$ =» 0.0085 r _Sun

white dwarf

Sirius A on the main sequence above and to the left of the Sun AND Sirius B on white dwarf area as shown

Both positions must be labelled 

Allow the position anywhere within the limits shown.

arrow goes up and right and then loops to white dwarf area

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 $M_{\odot }$.

Describe how the chemical composition of a star may be determined.

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

λ = «$\frac{2.9\times {10}^{-3}}{4600}$ =» 630 «nm» ✔

black body curve shape ✔

peaked at a value from range 600 to 660 nm ✔

$\frac{L}{L_{\odot }}={\left(\frac{0.73R_{\odot }}{R_{\odot }}\right)}^2\times {\left(\frac{4600}{5800}\right)}^4$ ✔

L = 0.211 $L_{\odot }$ ✔

M = «${0.21}^{\frac{1}{3.5}}{M}_{\odot }$ =» 0.640 $M_{\odot }$ ✔

Obtain «line» spectrum of star ✔

Compare to «laboratory» spectra of elements ✔

red giant ✔

planetary nebula ✔

white dwarf ✔

A galaxy is 1.6 × 10⁸ ly from Earth. Show that its recessional speed as measured from Earth is about 3.5 × 10⁶ m s^-1.

A line in the hydrogen spectrum when measured on Earth has a wavelength of 486 nm. Calculate, in nm, the wavelength of the same hydrogen line when observed in the galaxy’s emission spectrum.

Outline how observations of spectra from distant galaxies provide evidence that the universe is expanding.

d = «1.6 × 10⁸ × 9.46 × 10¹⁵ =» 1.51 × 10²⁴ «m»✔

v = «H₀d = 2.3 × 10⁻¹⁸ ×1.5 × 10²⁴ =» 3.48 × 10⁶ «m s^–1» ✔

Answer given, correct working required or at least 3sf needed for MP2.

$\mathrm{\Delta }\lambda =«\frac{{\lambda }_{0}v}{c}=\frac{4.86\times {10}^{-7}\times 3.48\times {10}^6}{3\times {10}^8}=$» 5.64«nm»  ✔

observed λ = «486 + 5.64 =» 492 «nm»✔

all distant galaxies exhibit red-shift ✔

OWTTE

This very simple application of Hubble’s law was answered correctly by the vast majority of candidates.

Many candidates subtracted the change in wavelength and obtained a blue shift. Others were unsure which wavelength λo is in the data book equation. But correct answers were common.

Nearly all candidates were able to mention redshift as the evidence for galaxy recession and the universe expansion.

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.

In cluster, stars are gravitationally bound OR constellation not ✔

In cluster, stars are the same/similar age OR in constellation not ✔

Stars in cluster are close in space/the same distance OR in constellation not ✔

Cluster stars appear closer in night sky than constellation ✔

Clusters originate from same gas cloud OR constellation does not ✔

d = 275 «pc» ✔

because of the difficulty of measuring very small angles ✔

Outline the processes that produce the change of luminosity with time of Cepheid variables.

Explain how Cepheid variables are used to determine distances.

Determine the peak apparent brightness of δ-Cephei as observed from Earth.

Calculate the peak surface temperature of δ-Cephei.

Astronomers claim to know the properties of distant stars. Outline how astronomers can be certain that their measurement methods yield correct information.

Cepheid variables expand and contract

OR

Radius increases and decreases

OR

Surface area increases and decreases ✔

Surface temperature decreases then increases✔

Surface becomes transparent then opaque ✔

OWTTE

Do not reward ‘change in luminosity/brightness’ as this is given in the question.

Accept changes in reverse order

the «peak» luminosity/actual brightness depends on the period

OR

More luminous Cepheid variables have greater period✔

measurements of apparent brightness allow distance determination

OR

Mention of ✔

OWTTE

$d=«273\times 3.26\times 9.46\times {10}^{15}=»8.42\times {10}^{18}$«m»  ✔

$b=«\frac{L}{4\pi d^2}=\frac{7.70\times {10}^{29}}{4\pi {(8.42\times {10}^{18})}^2}=»8.6\times {10}^{-10}$«Wm^–2»  ✔

«$T=\frac{2.9\times {10}^{-3}}{4.29\times {10}^{-7}}$»

=6800«K»  ✔

Data subject to peer review/checks by others ✔

Compare light from stars with Earth based light sources ✔

measurements are corroborated by different instruments/methods from different teams ✔

OWTTE

The expansion and contraction of Cepheid stars was commonly mentioned. Changes in surface temperature and opacity were less commonly mentioned. A common misconception seems to be that the variation of luminosity is due to a change of the rate of fusion. A few candidates left this question unanswered.

Many candidates knew that if the luminosity of the Cepheid is known then the absolute brightness can be used to determine distance. But far fewer candidates could link luminosity with the period of the Cepheid star. Many seemed to think that the luminosity of all Cepheids is the same.

Calculating the brightness of a star from its luminosity was an easy question for most candidates. But quite a few did not convert parsecs into metres especially at SL.

This simple calculation using Wien’s law was very well answered.

Many candidates correctly stated that astronomers can use peer review or different methods in checking that the information obtained from stars is correct.

Outline how Hubble measured the recessional velocities of galaxies.

Using the graph, determine in s, the age of the universe.

measured redshift «z» of star ✔

use of Doppler formula OR z∼v/c OR v = $\frac{c\mathrm{\Delta }\lambda }{\lambda }$ to find v ✔

use of gradient or any point on the line to obtain any expression for either $\text{H}=\frac{\text{v}}{\text{d}}$ or $\text{t}=\frac{\text{d}}{\text{v}}$ ✔

correct conversion of d to m and v to m/s ✔

= 4.6 × 10¹⁷ «s» ✔

Distinguish between a constellation and a stellar cluster.

The peak wavelength of radiation from Eta Cassiopeiae A is 490 nm. Show that the surface temperature of Eta Cassiopeiae A is about 6000 K.

The surface temperature of Eta Cassiopeiae B is 4100 K. Determine the ratio $\frac{\mathrm{radius} \mathrm{of} \mathrm{the} \mathrm{Eta} \mathrm{Cassiopeiae} \mathrm{A}}{\mathrm{radius} \mathrm{of} \mathrm{the} \mathrm{Eta} \mathrm{Cassiopeiae} \mathrm{B}}$.

The distance of the Eta Cassiopeiae system from the Earth is 1.8 × 10¹⁷m. Calculate, in terms of $L_{\odot }$, the luminosity of Eta Cassiopeiae A.

On the HR diagram, draw the present position of Eta Cassiopeiae A.

State the star type of Eta Cassiopeiae A.

Calculate the ratio $\frac{\mathrm{mass} \mathrm{of} \mathrm{Eta} \mathrm{Cassiopeiae} \mathrm{A}}{\mathrm{mass} \mathrm{of} \mathrm{the} \mathrm{Sun}}$.

Deduce the final evolutionary state of Eta Cassiopeiae A.

stars in a cluster are gravitationally bound OR in constellation are not ✔

stars in a cluster are the same/similar age OR in constellation are not ✔

stars in a cluster are close in space/the same distance away OR in constellation are not ✔

stars in a cluster originate from same gas cloud OR in constellation do not ✔

stars in a cluster appear much closer in night sky than in a constellation ✔

Notes: Take care to reward only 1 comment from a given marking point for MP1 to MP5.

«$T=\frac{2.9\times {10}^{-3}}{490\times {10}^{-9}}$»

5900 K ✔

NOTE: Answer 6000 K is given in the question.
Answer must be to at least 2 s.f. OR correct working.

«from b∝ L∝ R² T⁴»

realization that $R^2\propto \frac{b}{T^4}$ «for binary stars which are same distance away» ✔

$\frac{R_{\mathrm{A}}}{R_{\mathrm{B}}}=\sqrt{\frac{\left({\displaystyle \frac{1.1\times {10}^{-9}}{5.4\times {10}^{-11}}}\right)}{{\left({\displaystyle \frac{5900}{4100}}\right)}^4}}$ ✔

$\frac{R_{\mathrm{A}}}{R_{\mathrm{B}}}=2.2$ ✔

NOTE: Award [2] for answer 0.46 from inverted ratio.

«use of $L=4\mathrm{\pi }{d}^{2}b$»

$L=4\mathrm{\pi }\times {\left(1.8\times {10}^{17}\right)}^2\times 1.1\times {10}^{-9} «=4.48\times {10}^{26} \mathrm{W}»$✔

$L=1.2 L_{\odot }$ ✔

approximately correct position on the main sequence as shown, within highlighted region ✔

main sequence star
OR
type F or G star ✔

$\frac{M}{M_{\odot }}=1.2^{\frac{1}{3.5}}=1.05$ ✔

mass of the «remnant» star $<1.4M_{\odot }$ OR Chandrasekhar limit
OR
mass OR luminosity similar to the Sun ✔

the final stage is white dwarf ✔

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.

Estimate the age of the universe in seconds using the Hubble constant H₀ = 70 km s^–1Mpc^–1.

Outline why the estimate made in (b)(i) is unlikely to be the actual age of the universe.

spectra of galaxies are redshifted «compared to spectra on Earth» ✔

redshift/longer wavelength implies galaxies recede/ move away from us
OR
redshift is interpreted as cosmological expansion of space ✔

«hence universe expands»

NOTE: Universe expansion is given, so no mark for repeating this.
Do not accept answers based on CMB radiation.

ALTERNATIVE 1

$z=\frac{392-122}{122}=2.21$ ✔

$\frac{R}{R_0}=«2.21+1=» 3.21$ ✔

ALTERNATIVE 2

$\frac{R}{R_0}=\frac{{\displaystyle 392}}{122}$ ✔

=3.21 ✔

$H=«\frac{70\times {10}^3}{\left({10}^6\times 3.26\times 9.46\times {10}^{15}\right)}=»2.27\times {10}^{-18} «{\mathrm{s}}^{-1}»$ ✔

$T=«\frac{1}{2.27\times {10}^{-18}}=» 4.4\times {10}^{17} \mathrm{s}$ ✔

because estimate assumes the «present» constant rate of expansion ✔
it is unlikely that the expansion rate of the universe was ever constant ✔
there is uncertainty in the value of H₀ ✔

NOTE: OWTTE

Identify, on the HR diagram, the position of the Sun. Label the position S.

Suggest the conditions that will cause the Sun to become a red giant.

Outline why the Sun will maintain a constant radius after it becomes a white dwarf.

During its evolution, the Sun is likely to be a red giant of surface temperature 3000 K and luminosity 10⁴ L_☉. Later it is likely to be a white dwarf of surface temperature 10 000 K and luminosity 10-4 L_☉. Calculate the $\frac{\text{radius of the Sun as a white dwarf}}{\text{radius of the Sun as a red giant}}$.

the letter S should be in the region of the shaded area ✔

the fusion of hydrogen in the core eventually stops

OR

core contracts ✔

the hydrogen in a layer around the core will begin to fuse ✔

Sun expands AND the surface cools ✔

helium fusion begins in the core ✔

Sun becomes more luminous/brighter✔

Ignore any mention of the evolution past the red giant stage

electron degeneracy <<prevents further compression>> ✔

Ignore mention of the Chandrasekhar limit.

Award [0] for answer mentioning radiation pressure or fusion reactions.

Locating the Sun’s position on the HR diagram was correctly done by most candidates, although a few were unsure of the surface temperature of the Sun.

The evolution of a main sequence star to the red giant region is reasonably well understood. However many struggled to find three different facts to describe the changes. Answers were often too vague, when writing about a change in temperature or size of a star, the candidates are expected to mention whether they are referring to the core or the surface/outer layer. A surprising number of candidates wrote that the Sun must be less than eight solar masses.

The mention of electron degeneracy pressure was fairly common, but incorrect answers were even more common at SL.

Calculating the ratio of the radius of a white dwarf to a red giant star was done quite well by most candidates. However quite a few candidates made POT errors or forgot to take the final square root.

Explain how international collaboration has helped to refine this value.

Estimate, in Mpc, the distance between the galaxy and the Earth.

Determine, in years, the approximate age of the universe at the instant when the detected light from the distant galaxy was emitted.

experiments and collecting data are extremely costly

data from many projects around the world can be collated

OWTTE

[1 mark]

v = «zc = 0.19 × 3 × 10⁸ =» 5.7 × 10⁷ «ms^–1»

d = «$\frac{v}{H_0}=\frac{5.7\times {10}^4}{70}$» = 810Mpc     OR     8.1× 10⁸ pc

Correct units must be present for MP2 to be awarded.

Award [2] for BCA.

[2 marks]

ALTERNATIVE 1

$\frac{R_{\text{now}}}{R_{\text{then}}}$ = 1 + z = 1.19

so (assuming constant expansion rate) $\frac{t_{\text{now}}}{t}$ = 1.19

t = $\frac{14}{1.19}$ = 11.7By = 12«By (billion years)»

ALTERNATIVE 2

light has travelled a distance: (810 × 10⁶ × 3.26 =) 2.6 × 10⁹ly

so light was emitted: 2.6 billion years ago

so the universe was 11.4 billion years old

MP1 can be awarded if MP2 clearly seen.

Accept 2.5 × 10²⁵ m for mp1.

MP1 can be awarded if MP2 clearly seen.

[3 marks]

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 $z$.

« $\frac{R_0}{R}=$ »

$\frac{1}{1.11}$  OR  $0.90$  OR  $90\%$ ✓

«Hubble’s » measure of v/recessional speed uses redshift which is $z$
OR
redshift ($z$) of galaxies is proportional to distance «from earth»
OR
combines $v=Hd$ AND $z=\frac{v}{c}$ into one expression, e.g. $z=\frac{Hd}{c}$. ✓ 

OWTTE

Distinguish between a constellation and a stellar cluster.

stars in a cluster are gravitationally bound OR in constellation are not ✔
stars in a cluster are the same/similar age OR in constellation are not ✔
stars in a cluster are close in space/the same distance away OR in constellation are not ✔
stars in a cluster originate from same gas cloud OR in constellation do not ✔
stars in a cluster appear much closer in night sky than in a constellation ✔

NOTE: Take care to reward only 1 comment from a given marking point for MP1 to MP5.