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Date	November 2019	Marks available	2	Reference code	19N.3.SL.TZ0.10
Level	Standard level	Paper	Paper 3	Time zone	0 - no time zone
Command term	Show that	Question number	10	Adapted from	N/A

Question

Eta Cassiopeiae A and B is a binary star system located in the constellation Cassiopeia.

The following data are available.

Apparent brightness of Eta Cassiopeiae A = 1.1 × 10^–9 Wm^–2
Apparent brightness of Eta Cassiopeiae B = 5.4 × 10^–11 Wm^–2
Luminosity of the Sun, $L_{⊙}$ = 3.8 × 10²⁶ W

A Hertzsprung–Russell (HR) diagram is shown.

Distinguish between a constellation and a stellar cluster.

[2]

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.

[2]

b(i).

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

[3]

b(ii).

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

[2]

b(iii).

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

[1]

c(i).

State the star type of Eta Cassiopeiae A.

[1]

c(ii).

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

[1]

c(iii).

Deduce the final evolutionary state of Eta Cassiopeiae A.

[2]

c(iv).

Markscheme

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.

b(i).

«from b∝ L∝ R² T⁴»

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

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

$\frac{R_{A}}{R_{B}} = 2.2$ ✔

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

b(ii).

«use of $L = 4 π d^{2} b$ »

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

$L = 1.2 L_{⊙}$ ✔

b(iii).

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

c(i).

main sequence star
OR
type F or G star ✔

c(ii).

$\frac{M}{M_{⊙}} = 1 . 2^{\frac{1}{3.5}} = 1.05$ ✔

c(iii).

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

the final stage is white dwarf ✔

c(iv).

Examiners report

[N/A]

b(i).

[N/A]

b(ii).

[N/A]

b(iii).

[N/A]

c(i).

[N/A]

c(ii).

[N/A]

c(iii).

[N/A]

c(iv).

Syllabus sections

Option D: Astrophysics » Option D: Astrophysics (Core topics) » D.2 – Stellar characteristics and stellar evolution

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17N.3.SL.TZ0.12e.ii: sketch the expected evolutionary path for Sirius A.
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Write down the luminosity of star X (L_X) in terms of the luminosity of the Sun (L_s).
17N.3.SL.TZ0.12a: State what is meant by a binary star.
16N.3.SL.TZ0.16b: Explain why Cephids are used as standard candles.
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Show, without calculation, that the radius of Alpha Centauri B is smaller than the radius of Alpha Centauri A.
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Show that the mass of Theta 1 Orionis is about 40 solar masses.
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Determine the peak wavelength of the radiation emitted by Epsilon Indi.
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Outline why the neutron star that is left after the supernova stage does not collapse under the action of gravitation.
19M.3.SL.TZ2.15c:
Outline why the Sun will maintain a constant radius after it becomes a white dwarf.
17N.3.SL.TZ0.12b:
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.
19M.3.SL.TZ2.15b: Suggest the conditions that will cause the Sun to become a red giant.
18N.3.HL.TZ0.18b:
Epsilon Indi is a main sequence star. Show that the mass of Epsilon Indi is 0.64 ${M_ \odot }$ .
16N.3.SL.TZ0.15e:
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.
18M.3.SL.TZ1.11c:
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.
19N.3.SL.TZ0.10c(iii):
Calculate the ratio $\frac{mass of Eta Cassiopeiae A}{mass of the Sun}$ .
20N.3.HL.TZ0.22a:
Show by calculation that Eta Aquilae A is not on the main sequence.
18M.3.SL.TZ1.11a.i:
Suggest, using the graphs, why star X is most likely to be a main sequence star.
17M.3.SL.TZ2.11b:
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 $\frac{{{\text{density of star X}}}}{{{\text{density of star Y}}}}$ .
19N.3.SL.TZ0.10c(i): On the HR diagram, draw the present position of Eta Cassiopeiae A.
19N.3.SL.TZ0.10c(ii): State the star type of Eta Cassiopeiae A.
19N.3.SL.TZ0.10c(iv):
Deduce the final evolutionary state of Eta Cassiopeiae A.
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Determine the radius of star X (R_X) in terms of the radius of the Sun (R_s).
19M.3.SL.TZ2.13bii:
Calculate the peak surface temperature of δ-Cephei.
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plot the position, using the letter P, of the main sequence star P you calculated in (b).
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On the HR diagram in (b), draw a line to indicate the evolutionary path of star X.
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plot the position, using the letter G, of Gacrux.
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The present temperature of the CMB is 2.8 K. Calculate the peak wavelength of the CMB.
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Estimate the mass of star X (M_X) in terms of the mass of the Sun (M_s).
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Discuss, with reference to its change in mass, the evolution of star P from the main sequence until its final stable phase.
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Show by calculation that Eta Aquilae A is not on the main sequence.
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draw the main sequence.
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Show that the temperature of star X is approximately 10 000 K.
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Option D: Astrophysics » Option D: Astrophysics (Core topics)

Option D: Astrophysics

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Markscheme

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