The Doppler Effect and Hubble's Law

SPEC 3.9.3.1 & 3.9.3.2 • COSMOLOGY

Red shift, recession velocity, age of the universe, and evidence for the Big Bang

Learning Objectives

By the end of this lesson, you will be able to:

Explain the Doppler effect and its application to light from distant galaxies

Use the equations

z = \frac{v}{c} = \frac{Δ f}{f} = \frac{Δ λ}{λ}

Apply Hubble's law

v = H_{0} d

to calculate distances to galaxies

Derive and use the age of the universe as

t \approx \frac{1}{H _{0}}

Calculate the critical density using

ρ = \frac{3 H _{0}^{2}}{8 π G}

Describe evidence for the Big Bang : CMBR, H:He abundance ratio

Why This Matters

The Doppler effect allows astronomers to measure how fast distant galaxies are moving away from us, leading to one of the most profound discoveries in science: the universe is expanding from a common origin.

The Doppler Effect

Definition

The Doppler effect is the apparent change in frequency and wavelength of waves when a source and observer move relative to each other.

As a wave source moves, the wavelengths in front are compressed (shorter wavelength, higher frequency) and those behind are spread out (longer wavelength, lower frequency).

Everyday Example

You hear this with ambulance sirens: higher pitch as it approaches (compressed waves), lower pitch as it recedes (stretched waves).

Red Shift and Blue Shift

When the Doppler effect is applied to light from stars and galaxies:

MotionEffect on SpectrumNamez Value

Moving away from EarthSpectral lines shift to longer wavelengths (towards red)Red shiftz > 0 (positive)

Moving towards EarthSpectral lines shift to shorter wavelengths (towards blue)Blue shiftz < 0 (negative)

Key Observation

Almost all distant galaxies show red shift, not blue shift. This is evidence that the universe is expanding — distant galaxies are moving away from us.

The more distant the galaxy, the greater its red shift.

The Red Shift Formula

The red shift (z) of an object can be calculated using:

Formula

z = \frac{v}{c} = \frac{Δ f}{f} = \frac{Δ λ}{λ}

SymbolMeaningUnit

zRed shift (dimensionless ratio)—

vRecession velocity of objectm s⁻¹

cSpeed of light (3 × 10⁸ m s⁻¹)m s⁻¹

Δf, fChange in frequency, original frequencyHz

Δλ, λChange in wavelength, original wavelengthm

Important

This formula is only valid when v ≪ c (velocity much less than the speed of light). For objects moving close to c, relativistic corrections are needed.

Worked Example 1

Calculating Red Shift01

A galaxy is moving away from Earth at 26,000 km s⁻¹. Calculate the wavelength and frequency change of a 650 nm line in its spectrum. Take c = 3 × 10⁸ m s⁻¹.

1

Calculate the wavelength change:

Δ λ = \frac{λ v}{c} = \frac{650 \times 1 0 ^{- 9} \times 26000 \times 1 0 ^{3}}{3 \times 1 0 ^{8}}

Δ λ = 56.3 nm

2

Calculate original frequency:

f = \frac{c}{λ} = \frac{3 \times 1 0 ^{8}}{650 \times 1 0 ^{- 9}} = 4.6 \times 1 0^{14} Hz

3

Calculate frequency change:

Δ f = \frac{f v}{c} = \frac{4.6 \times 1 0 ^{14} \times 26000 \times 1 0 ^{3}}{3 \times 1 0 ^{8}}

Δ f = 4.0 \times 1 0^{13} Hz

AnswerΔλ = 56.3 nm, Δf = 4.0 × 10¹³ Hz

Worked Example 2

Finding Recession Velocity02

A galaxy is moving relative to Earth. A 600 nm spectral line is found to be Doppler shifted by 30 nm towards the red.

(a) Is the galaxy receding or approaching?

(b) Calculate its speed relative to Earth.

(a)

Receding — the shift towards red means longer wavelength, so the galaxy is moving away from Earth.

(b)

Use the red shift formula:

v = \frac{c Δ λ}{λ} = \frac{3 \times 1 0 ^{8} \times 30 \times 1 0 ^{- 9}}{600 \times 1 0 ^{- 9}}

v = 1.5 \times 1 0^{7} m s^{- 1} = 15000 km s^{- 1}

Answer(a) Receding (b) v = 1.5 × 10⁷ m s⁻¹

Applications: Binary Star Systems

The Doppler effect can be used to identify binary star systems, where two stars orbit a common centre of mass.

Spectroscopic Binaries

Binary systems where the stars are too close to be resolved by a telescope. They are identified using Doppler shifts in their spectral lines.

As the stars orbit:

When one star moves towards us (blue shift), the other moves away (red shift)

Each spectral line splits into two

When both stars move perpendicular to our line of sight, there is no Doppler shift

Eclipsing Binaries

Definition

Eclipsing binaries are binary systems where the orbital plane is in line with Earth, so the stars periodically cross in front of each other.

These produce characteristic light curves with periodic dips:

Primary minimum: Brighter star eclipsed — deepest dip

Secondary minimum: Dimmer star eclipsed — shallower dip

Hubble's Law

In 1929, Edwin Hubble discovered that the recession velocity of galaxies is directly proportional to their distance from Earth.

Hubble's Law

v = H_{0} d

SymbolMeaningUnit

vRecession velocity of galaxykm s⁻¹

H₀Hubble constantkm s⁻¹ Mpc⁻¹

dDistance to galaxyMpc (megaparsec)

Hubble Constant

Current value: H₀ ≈ 71.5 km s⁻¹ Mpc⁻¹

This means a galaxy 1 Mpc away recedes at ≈ 71.5 km s⁻¹, a galaxy 10 Mpc away at ≈ 715 km s⁻¹, etc.

Hubble's Law: The Evidence

Hubble plotted recession velocity against distance for many galaxies and found a linear relationship:

Interpretation

The gradient of this graph gives the Hubble constant H₀. The linear relationship implies all galaxies started from a common point — evidence for the Big Bang.

Worked Example 3

Hubble's Law03

Find the distance to a galaxy with a recession velocity of 1200 km s⁻¹ if H₀ = 71.5 km s⁻¹ Mpc⁻¹. Give your answer in Mpc and light-years.

(1 Mpc = 3.26 × 10⁶ ly, 1 pc = 3.086 × 10¹⁶ m)

1

Apply Hubble's Law:

d = \frac{v}{H _{0}} = \frac{1200}{71.5} = 16.78 Mpc

2

Convert to metres:

d = 16.78 \times 1 0^{6} \times 3.086 \times 1 0^{16} = 5.19 \times 1 0^{23} m

3

Convert to light-years:

d = 16.78 \times 3.26 \times 1 0^{6} = 5.5 \times 1 0^{7} ly

Answerd = 16.8 Mpc = 55 million light-years

Age of the Universe

Hubble's Law can be used to estimate the age of the universe. If we assume:

The universe has been expanding at a constant rate since the Big Bang

All galaxies started from the same point

Then using

v = H_{0} d

and

d = v t

:

Age of Universe

t_{0} = \frac{d}{v} = \frac{1}{H _{0}}

Converting H₀ to SI units:

1

H₀ = 71.5 km s⁻¹ Mpc⁻¹ = 71.5 × 10³ m s⁻¹ Mpc⁻¹

2

1 Mpc = 3.086 × 10²² m

3

H₀ = (71.5 × 10³)/(3.086 × 10²²) = 2.32 × 10⁻¹⁸ s⁻¹

Result

t_{0} = \frac{1}{H _{0}} = \frac{1}{2.32 \times 1 0 ^{- 18}} = 4.32 \times 1 0^{17} s \approx

13.7 billion years

Critical Density of the Universe

The critical density determines the ultimate fate of the universe. It's the density needed for the universe to just stop expanding (but not contract).

Derivation: The recession velocity of a galaxy can be considered as its escape velocity from the rest of the universe:

Critical Density Formula

ρ_{c} = \frac{3 H _{0}^{2}}{8 π G}

Where G = 6.67 × 10⁻¹¹ N m² kg⁻² (gravitational constant)

Typical Value

Using H₀ = 2.32 × 10⁻¹⁸ s⁻¹:

ρ_{c} = \frac{3 \times ( 2.32 \times 1 0 ^{- 18} ) ^{2}}{8 π \times 6.67 \times 1 0 ^{- 11}} \approx 9.6 \times 1 0^{- 27} kg m^{- 3}

This is equivalent to about 6 protons per cubic metre!

The Fate of the Universe

The ratio of actual density to critical density is called Ω (omega):

Density Parameter

Ω = \frac{actual density}{critical density} = \frac{ρ}{ρ _{c}}

ConditionGeometryFate

Ω < 1Open (negative curvature)Runaway expansion forever

Ω = 1FlatExpansion slows but never stops

Ω > 1Closed (positive curvature)Expansion stops, then Big Crunch

Current Evidence

Observations suggest Ω ≈ 1 but the expansion is accelerating, driven by mysterious dark energy.

Worked Example 4

Critical Density04

Calculate the critical density of the universe if H₀ = 70 km s⁻¹ Mpc⁻¹.

(G = 6.67 × 10⁻¹¹ N m² kg⁻², 1 Mpc = 3.086 × 10²² m)

1

Convert H₀ to SI units:

H_{0} = \frac{70 \times 1 0 ^{3}}{3.086 \times 1 0 ^{22}} = 2.27 \times 1 0^{- 18} s^{- 1}

2

Apply the critical density formula:

ρ_{c} = \frac{3 H _{0}^{2}}{8 π G} = \frac{3 \times ( 2.27 \times 1 0 ^{- 18} ) ^{2}}{8 π \times 6.67 \times 1 0 ^{- 11}}

3

Calculate:

ρ_{c} = \frac{3 \times 5.15 \times 1 0 ^{- 36}}{1.68 \times 1 0 ^{- 9}} = 9.2 \times 1 0^{- 27} kg m^{- 3}

Answerρ_c ≈ 9.2 × 10⁻²⁷ kg m⁻³

The Big Bang Theory

The Big Bang

The universe began from a singularity — an infinitely small, infinitely hot, infinitely dense point — approximately 13.8 billion years ago. Time and space both originated at this moment.

Key evidence for the Big Bang:

1

Red shift of galaxies — almost all galaxies are moving away from us, indicating expansion from a common origin

2

Hubble's Law — the linear v-d relationship supports expansion from a single point

3

Cosmic Microwave Background Radiation (CMBR)

4

Primordial element abundances — the H:He ratio

Common Misconception

The Big Bang was not an explosion in space — it was the rapid expansion of space itself. There is no "centre" of the universe; every point is moving away from every other point.

Cosmic Microwave Background Radiation

In 1965, Arno Penzias and Robert Wilson discovered a uniform background radiation coming from all directions in space.

CMBR Properties

Temperature: approximately 2.7 K

Peak wavelength: approximately 2 mm (microwave region)

Uniform in all directions (isotropic)

Why CMBR supports the Big Bang:

After the Big Bang, the universe was filled with high-energy radiation

As the universe expanded and cooled, this radiation was red-shifted

What was once gamma rays is now microwave radiation

The CMBR is the "afterglow" of the Big Bang — predicted by theory before discovery

Nobel Prize

Penzias and Wilson received the 1978 Nobel Prize in Physics for their discovery of CMBR.

Primordial Element Abundances

During the first few minutes after the Big Bang, the universe was hot enough for nuclear fusion to occur:

Hydrogen nuclei (protons) fused to form helium

This "Big Bang nucleosynthesis" lasted only a few minutes before the universe cooled too much

Approximately ¼ of hydrogen was converted to helium

Predicted vs Observed

ElementBig Bang PredictionObservedHydrogen~75%~73%Helium~25%~25%Everything else~0%~2%

Why This Matters

The close match between predicted and observed abundances is strong evidence for the Big Bang. The H:He ratio of approximately 3:1 by mass cannot be explained by stellar fusion alone.

Quasars

Definition

Quasars (quasi-stellar objects) are extremely luminous objects with very large red shifts, indicating they are very far away.

Key properties:

Red shifts up to z ≈ 7 (meaning v ≈ 0.97c using relativistic formula)

Distances of billions of light-years

Despite their distance, they appear very bright

Using the inverse square law, their power output equals that of an entire galaxy

What Are They?

Quasars are thought to be active galactic nuclei — supermassive black holes at the centres of distant galaxies, emitting huge amounts of energy as matter falls into them.

Because they are so distant, we see quasars as they were billions of years ago. They provide evidence that the early universe was different from today.

Timeline After the Big Bang

Time After Big BangEventTemperature

10⁻⁴³ sPlanck era — physics unknown—

10⁻¹⁰ sParticle soup dominates10¹⁵ K

1 sNeutrons and protons formed10¹⁰ K

3 minutesHelium nuclei formed (nucleosynthesis)10⁹ K

300,000 yearsAtoms form, universe becomes transparent, CMBR released6000 K

1 billion yearsFirst stars and galaxies form18 K

13.8 billion yearsPresent day2.7 K

Practice

A distant galaxy has a red shift z = 0.15.

(a) Calculate the recession velocity of this galaxy. (c = 3 × 10⁸ m s⁻¹)

(b) Using H₀ = 71.5 km s⁻¹ Mpc⁻¹, calculate the distance to the galaxy in Mpc.

(c) A hydrogen spectral line has a laboratory wavelength of 656 nm. What wavelength would be observed from this galaxy?

(a)

v = zc = 0.15 \times 3 \times 1 0^{8} = 4.5 \times 1 0^{7} m s^{- 1} = 45000 km s^{- 1}

(b)

d = \frac{v}{H _{0}} = \frac{45000}{71.5} = 629 Mpc

(c)

Δ λ = z λ = 0.15 \times 656 = 98.4 nm

Observed wavelength = 656 + 98.4 = 754 nm

Summary

Key Formulas

Red shift:

z = \frac{v}{c} = \frac{Δ f}{f} = \frac{Δ λ}{λ}

Hubble's Law:

v = H_{0} d

where H₀ ≈ 71.5 km s⁻¹ Mpc⁻¹

Age of universe:

t_{0} = \frac{1}{H _{0}} \approx 13.7

billion years

Critical density:

ρ_{c} = \frac{3 H _{0}^{2}}{8 π G} \approx 1 0^{- 26} kg m^{- 3}

Key Concepts

Doppler effect: Waves compressed/stretched by relative motion

Red shift: Light shifted to longer wavelengths (galaxy receding)

Big Bang evidence: CMBR (2.7 K), H:He ratio (3:1), galaxy red shifts

Fate of universe: Depends on Ω (open/flat/closed)

Quasars: Distant, luminous objects with very high red shifts