Hubble's Law (AQA A Level Physics): Revision Note

The graph shows how the recessional velocity $v$ of a group of galaxies varies with their distance $d$ from the Earth.

Use the graph to determine a value for the Hubble constant and state its unit.

Answer:

Step 1: Recall Hubble's Law and the Hubble constant

• Hubble’s Law: $v=Hd$

• The gradient of the speed-distance graph = $H$

Step 2: Read values of v and d from the graph

• From the graph: v = 20 000 km s^–1

• From the graph: d = 305 MPc

Step 3: Calculate the gradient of the graph

$H=\frac{20000-0}{305-0}=65.6\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$

Hubble constant:  $H$ = 66 km s^–1 Mpc^–1

Measurements of type 1a supernovae are used to find a value for the Hubble constant. The distance from Earth is known for many type 1a supernovae.

Describe how these values of distance are used, with other data, to find the Hubble constant.

Your answer should include:

• the other data needed and how these data are used

• the graph plotted, including appropriate units for the axes

• how the Hubble constant is obtained and any limitations on the result.

Answer:

Step 1: Describe the data needed for the determination of the Hubble constant and how it will be used

• The Hubble constant can be described by the ratio of recession velocity and distance

$H=\frac{v}{d}$

• Therefore, as well as measurements of distance $d$ to the Type 1a supernovae, measurements of red shift $z$ are also needed

• The recession velocities $v$ of the Type 1a supernovae can be calculated from red shift using:

$v=zc$

• Values of red shift $z$ can be obtained by measuring the wavelength of spectral lines from the Type 1a supernovae and comparing them to the same spectral lines from a source in a laboratory

Step 2: Describe how to plot the graph to determine the Hubble constant

• Once the measurements of recession velocity $v$ and distance $d$ are obtained, they can be plotted on a graph with

  • velocity $v$, in km s^-1, on the y-axis

  • distance $d$, in Mpc, on the x-axis

• Then, the gradient of the graph is the Hubble constant $H$, in km s^-1 Mpc^-1

Step 3: Discuss any limitations in the obtained value of the Hubble constant

• Values of distance are determined using the distance modulus equation:

$m-M=5\log \left(\frac{d}{10}\right)$

• For Type 1a supernovae, the absolute magnitude $M$ is known, but measurements of apparent magnitude $m$ are required, which may be affected by light passing through gas and dust in space

• A large amount of data is needed to reduce uncertainty, however, Type 1a supernovae are rare events, and their occurrence is difficult to predict

Other potential limitations:

• There is a large variation in the data due to variations between galaxies or random errors in measurement

• Type 1a supernovae are typically observed at large distances, where data suggests that the expansion of the Universe is accelerating, so this leads to discrepancies in values of $H$ compared to nearer sources

The units for the quantities in Hubble's law and the Hubble constant can change depending on the situation. Make sure you convert them to appropriate units and express your final answer correctly. 

The Hubble constant is given in the data booklet as $H=65\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$

To convert this into SI units:

• Multiply by 1 km

• Divide by 1 Mpc, where 1 parsec = 3.08 × 10¹⁶ m (from the data booklet)

$H=\frac{65\times {10}^3}{\left(3.08\times {10}^{16}\right)\left({10}^6\right)}=2.1\times {10}^{-18}{\mathrm{s}}^{-1}$

In 2020, the best estimate for the Hubble constant was 67.4 km s⁻¹ Mpc⁻¹.

Use this value to calculate the age of the Universe.

Answer:

Step 1: List the known quantities

• Hubble constant, $H$ = 67.4 km s⁻¹ Mpc⁻¹

• 1 parsec = 3.08 × 10¹⁶ m (from the data booklet)

• 1 year = (60 × 60 × 24 × 365) = 3.15 × 10⁷ s

Step 2: Convert the Hubble constant into SI units

• First, multiply by 1 km, or 10³ m

$H=67.4\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}=67.4\times {10}^3\mathrm{m}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$

• Then, divide by 1 Mpc, or 3.08 × 10²² m

$H=\frac{67.4\times {10}^3\mathrm{m}{\mathrm{s}}^{-1}}{3.08\times {10}^{22}\mathrm{m}}=2.19\times {10}^{-18}{\mathrm{s}}^{-1}$

Step 3: Calculate the age of the Universe

• The age of the Universe is equivalent to the reciprocal of the Hubble constant:

$t=\frac{1}{H}=\frac{1}{2.19\times {10}^{-18}}=4.57\times {10}^{17}\mathrm{s}$ 

$t=\frac{4.57\times {10}^{17}}{3.15\times {10}^7}$ = 1.45 × 10¹⁰ years = 14.5 billion years