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Date	May 2019	Marks available	2	Reference code	19M.3.HL.TZ2.22
Level	Higher level	Paper	Paper 3	Time zone	2
Command term	Justify	Question number	22	Adapted from	N/A

Question

The homogeneous model of the universe predicts that it may be considered as a spherical cloud of matter of radius r and uniform density ρ. Consider a particle of mass m at the edge of the universe moving with velocity v and obeying Hubble’s law.

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

[2]

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

[2]

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

[1]

Markscheme

total energy=kinetic energy+potential energy

total energy= $\frac{1}{2}m{v^2} - \frac{{GMm}}{r}$ $\frac{1}{2}m{v^2} - \frac{{GMm}}{r}$ ✔

substitution of M = $\frac{4}{3}\pi {r^3}\rho$ $\frac{4}{3}\pi {r^3}\rho$ ✔

«Hence answer given»

Answer given so for MP2 look for clear evidence that M_Universe $\left( {\frac{4}{3}\pi {r^3}\rho } \right)$ $\left( {\frac{4}{3}\pi {r^3}\rho } \right)$ is stated and substituted.

substitutes H₀r for v ✔

«total energy = 0»

$\frac{1}{2}mH_0^2{r^2} = \frac{4}{3}\pi G\rho {r^2}m$ $\frac{1}{2}mH_0^2{r^2} = \frac{4}{3}\pi G\rho {r^2}m$ ✔

«hence ρ_c= $\frac{{3H_0^2}}{{8\pi G}}$ $\frac{{3H_0^2}}{{8\pi G}}$ »

Answer given, check working carefully.

9.5 × 10⁻²⁷ « kgm^–3» ✔

Examiners report

The vast majority of the candidates could state that the total energy is equal to the sum of the kinetic and potential energies but quite a few did not use the correct formula for the gravitational potential energy. The formula for the mass of the sun was usually correctly substituted.

This was a relatively easy demonstration given the equation in 22a. However many candidates did not show the process followed in a coherent manner that could be understood by examiners.

The question was well answered by many candidates.

Syllabus sections

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

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Option D: Astrophysics » Option D: Astrophysics (Additional higher level option topics)

Option D: Astrophysics

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