A LevelPhysicsAQARevision Notes9. Astrophysics9.3 Cosmology9.3.4 Hubble's Law

Hubble's Law (AQA A Level Physics)

Revision Note

Katie M

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Katie M

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Hubble's Law

  • Hubble’s law states:

The recessional velocity of a galaxy is proportional to its distance from Earth

  • This can be expressed mathematically as:

v space equals space H d

  • Where:
    • v = recessional velocity of an object (km s−1)
    • H = Hubble constant (km s−1 Mpc−1)
    • d = distance between the object and the Earth (Mpc)
  • Hubble’s law shows that:
    • The further away a star is from the Earth, the faster it is moving away from us
    • The closer a star is to the Earth, the slower it is moving away from us

Graph of Hubble's Law

Hubbles Law Graph, downloadable AS & A Level Physics revision notes

A key aspect of Hubble’s law is that the furthest galaxies appear to move away the fastest

The Hubble Constant

  • The constant of proportionality H in Hubble’s law is known as the Hubble constant:

H space equals space v over d

  • The value for the Hubble constant has been estimated using data from thousands of galaxies, and other sources, such as standard candles
  • Our current best estimate of the Hubble constant, based on CMB observations by the Planck satellite, is:

H = 67.4 ± 0.5 km s−1 Mpc−1

  • Note: this value is constantly under review as more data is collected

Worked example

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

 5-12-5-we-q_ocr-al-physics

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

Answer:

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

  • Hubble’s Law: v space equals space H d
  • The gradient of the speed-distance graph = H

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

5-12-5-we-solution_ocr-al-physics

  • From the graph: v = 20 000 km s–1
  • From the graph: d = 305 MPc

Step 3: Calculate the gradient of the graph

H space equals space fraction numerator 20 space 000 space minus space 0 over denominator 305 space minus space 0 end fraction space equals space 65.6 space km space straight s to the power of negative 1 end exponent space Mpc to the power of negative 1 end exponent

Hubble constant:  H = 66 km s–1 Mpc–1

Examiner Tip

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

Estimating the Age of the Universe

  • Hubble’s Law is extremely important as it can give us an estimate the age of the Universe
  • It can be used to find the time since the expansion began, and hence the age of the Universe
  • We can calculate the time taken to reach a distant object from the Earth if we know 
    • How far away it is
    • Its recessional speed
  • This requires a couple of assumptions:
    • All points in the Universe were initially together
    • The recessional speed of a galaxy is and has always been, constant
  • Comparing the equation for speed, distance and time:

t i m e space equals space fraction numerator d i s t a n c e over denominator s p e e d end fraction space equals space d over v

  • With the Hubble equation:

v space equals space H d space space space space space rightwards double arrow space space space space space H space equals space v over d

  • It can be seen that:

t i m e space equals space 1 over H

  • If we consider that all matter was at the same point at the very start of the Big Bang (t = 0), then the time taken for the galaxy to expand to its current state must be equal to the age of the Universe
  • Using current estimations of the Hubble constant, astronomers believe that the universe has been expanding for around 13.7 billion years

Worked example

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 ≈ 3.1 × 1016 m
  • 1 year = 3.16 × 107 s

Step 2: Convert 67.4 km s−1 Mpc−1 to m s−1 Mpc−1

  • H = 67.4 km s−1 Mpc−1 = 67.4 × 1000 = 6.74 × 104 m s−1 Mpc−1

Step 3: Convert 1 Mpc to m

  • 1 Mpc = (3.1 × 1016) × (1 × 106) = 3.1 × 1022 m

Step 4: Convert m s−1 Mpc−1 to s−1 

  • H = 6.74 × 104 m s−1 Mpc−1 = fraction numerator 6.74 blank cross times blank 10 to the power of 4 over denominator 3.1 blank cross times blank 10 to the power of 22 end fraction = 2.17 × 10–18 s–1
  • Hence, H = 2.17 × 10–18 s–1

Step 5: Calculate the age of the Universe

  • Age of the Universe:  t space equals space 1 over H space equals space fraction numerator 1 over denominator 2.17 blank cross times blank 10 to the power of negative 18 end exponent end fraction = 4.60 × 1017 s  
  • Age of the Universe:  t space equals space fraction numerator 4.60 blank cross times blank 10 to the power of 17 over denominator 3.16 blank cross times blank 10 to the power of 7 end fraction = 1.46 × 1010 years = 14.6 billion years

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    Katie M

    Author: Katie M

    Expertise: Physics

    Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.