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Date	May 2018	Marks available	2	Reference code	18M.3.HL.TZ1.19
Level	Higher level	Paper	Paper 3	Time zone	1
Command term	Explain	Question number	19	Adapted from	N/A

Question

A galaxy can be modelled as a sphere of radius R₀. The distance of a star from the centre of the galaxy is r.

M18/4/PHYSI/HP3/ENG/TZ1/19

For this model the graph is a simplified representation of the variation with r of the mass of visible matter enclosed inside r.

The mass of visible matter in the galaxy is M.

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

[1]

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

[2]

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.

[2]

Markscheme

$\frac{{m{v^2}}}{r} = \frac{{GMm}}{{{r^2}}}$ and correct rearranging

[1 mark]

linear / rising until R₀

then «almost» constant

[2 marks]

for v to stay constant for r greater than R₀, M has to be proportional to r

but this contradicts the information from the M-r graph

if M is constant for r greater than R₀, then we would expect v $\propto {r^{\frac{{ - 1}}{2}}}$

but this contradicts the information from the v-r graph

[2 marks]

Examiners report

[N/A]

Syllabus sections

Option D: Astrophysics » Option D: Astrophysics (Additional higher level option topics) » D.5 – Further cosmology (HL only)

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18N.3.HL.TZ0.21a:
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17M.3.HL.TZ2.20c:
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18M.3.HL.TZ1.19a:
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Show that for stars where r > R₀ the velocity of orbit is v = $\sqrt {\frac{{GM}}{r}}$ .
17M.3.HL.TZ2.20a:
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17M.3.HL.TZ1.17b.i:
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18N.3.HL.TZ0.21b:
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Option D: Astrophysics » Option D: Astrophysics (Additional higher level option topics)

Option D: Astrophysics

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