5.46 Equations for Cosmology

• Recall the Doppler effect is defined as:

The apparent change in wavelength or frequency of the radiation from a source due to its relative motion away from or toward the observer

• • On Earth, the Doppler effect is most easily observed when something which is both fast moving and emitting a loud sound (such as an ambulance or fast car) moves past an observer, producing a characteristic change in the pitch of the sound that is heard
• In space, the Doppler effect of light can be observed when spectra of distant stars and galaxies are observed, this is known as:
  • Redshift if the object is moving away from the Earth, or
  • Blueshift if the object is moving towards the Earth

• Redshift is defined as:

The fractional increase in wavelength (or decrease in frequency) due to the source and observer receding from each other

• For non-relativistic galaxies, Doppler redshift can be calculated using:

• Where:
  • Δλ = shift in wavelength (m)
  • λ = wavelength emitted from the source (m)
  • Δf = shift in frequency (Hz)
  • f = frequency emitted from the source (Hz)
  • v = speed of recession (m s^-1)
  • c = speed of light in a vacuum (m s^-1)

Part (a)

Step 1: Write down the known quantities

• • Emitted frequency, f = 4.570 × 10¹⁴ Hz
  • Shift in frequency, Δf = (4.547 – 4.570) × 10¹⁴ = –2.3 × 10¹² Hz
  • Speed of light, c = 3.0 × 10⁸ m s^–1

Step 2: Write down the Doppler redshift equation

Step 3: Rearrange for speed v, and calculate

Part (b)

Step 1: Compare the frequencies and answer the question

• • The observed frequency is less than the emitted frequency (the light from a laboratory source),
  • The source is receding (moving away from) from the Earth

• In 1929, the astronomer Edwin Hubble showed that the universe was expanding
  • He did this by observing that the absorption line spectra produced from the light of distant galaxies was shifted towards the red end of the spectrum
  • This doppler shift in the wavelength of the light is evidence that distant galaxies are moving away from the Earth
• Hubble also observed that light from more distant galaxies was shifted further towards the red end of the spectrum compared to closer galaxies
  • From this observation he concluded that galaxies or stars which are further away from the Earth are moving faster than galaxies which are closer

•  Hubble’s law states:

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

 

• Hubble’s law can be expressed as an equation:

v ≈ H₀d

• Where:
  • v = recessional velocity of an object, the velocity of an object moving away from an observer (km s^-1)
  • H₀ = Hubble constant, this will be provided in your examination along with the correct units (km s^-1 Mpc^-1)
  • d = distance between the object and the Earth (Mpc) 

• Alternatively, if the velocity of the receding object was measured in km s^–1 and the distance from the earth to the object was measured in km, then then unit for the Hubble constant would be s^–1

• Hubble’s law shows:
  • The further away a star is from the Earth, the faster it is moving away from us

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

Step 1: List the known quantities:

• • d = 20 light years
  • H_o = 2.2 x 10^-18 s^-1

Step 2: Convert 20 light-years to m:

• • From the data booklet: 1 ly ≈ 9.5 x 10¹⁵ m
  • So, 20 ly = 20 x (9.5 x 10¹⁵) = 1.9 x 10¹⁷ m

Step 3: Substitute values into Hubble's Law: 

• • From the data booklet: v ≈ H₀d 
  • So, v ≈ (2.2 x 10^-18 ) x (1.9 x 10¹⁷) = 0.418 m s^-1

Step 4: Confirm your answer:

• • The velocity of the galaxy as it moves away from Earth 0.42 m s^-1