Hubble's law

In a nutshell

Edwin Hubble was an astronomer, known for his work using the doppler shift from absorption spectra to prove the universe is expanding. This gives evidence for the Big Bang model. This was used to formulate Hubble's law, which shows that the recessional speed of a galaxy is almost directly proportional to its distance from the Earth. 

Equations

Constants

Variables

Doppler shift

Whenever a wave source moves away from a receiver, the frequency and wavelength observed by the receiver change. The doppler effect is most commonly explained using sound waves, such as when an ambulance is moving away from a stationary observer, its wavelength becomes shorter and frequency higher. However, the doppler effect can occur with all different types of waves. 

Doppler shift in starlight

The doppler effect can be used to determine the relative velocity of a distant galaxy. The absorption spectrum of a stationary object is compared against the absorption spectrum of a distant galaxy, and the shift in wavelength calculated. This shift is caused by the movement of a galaxy relative to the Earth, and we can use it to then calculate the speed at which the galaxy is moving using the doppler equation:

​$$\dfrac{\Delta\lambda}{\lambda} \approx \dfrac{\Delta f}{f} \approx \dfrac{v}{c}$$​​

where $$\lambda$$ is the wavelength emitted from source and $$\Delta\lambda$$ is the shift in wavelength, $$f$$ is the frequency and $$\Delta f$$ is the change in frequency, $$v$$ is the velocity at which the source is travelling and $$c$$ is the speed of light. The doppler equation can only be used for galaxies travelling at a speed much less than the speed of light.

Example:

A particular emission line of hydrogen is emitted with a wavelength of $$667.1\,nm$$​ from a gas cloud. From a telescope, the wavelength is observed as $$667.4\,nm$$​, how fast is this gas cloud moving relative to the Earth?

State variables:

​

 $$\lambda = 667.1\, nm \newline \\[0.1in] \Delta\lambda = 0.3\,nm\newline \\[0.1in] c = 3\times 10^8 \, ms^{-1}$$

State equations:

​$$\dfrac{\Delta\lambda}{\lambda} \approx \dfrac{v}{c}$$​

Rearrange for $$v$$​ and solve:

​$$\dfrac{\Delta\lambda c}{\lambda} \approx v\newline \\[0.1in] v = \dfrac{\Delta\lambda c}{\lambda} \newline \\[0.1in] v = \dfrac{0.3 \times 10^{-9} \times 3 \times 10^8}{667.1 \times 10^{-9}}\newline \\[0.1in] v = 1.3 \times 10^5\,ms^{-1}$$

The gas cloud is moving at $$\underline{v = 1.3 \times 10^{5}\,ms^{-1}}$$.​

Hubble's law

Hubble's law states that the recessional velocity of a galaxy $$v$$ is almost directly proportional to its distance from the Earth $$d$$​. Due to these being proportional, we must introduce a constant of proportionality, called the Hubble constant $$H_0$$. The Hubble's constant has a value of $$65 \,km s^{–1} Mpc^{–1}$$ or $$2.2 \times 10^{-18}s^{-1}$$. Hubble's law can be described mathematically as:

​$$v= H_0d$$

where $$v$$ is the galaxy's recessional velocity, $$d$$ is the distance between the galaxy and the Earth and $$H_0$$ is the Hubble constant.​

The recessional velocity and distance can be plotted on a graph, where the Hubble constant is the gradient. 

Example:

A galaxy is found to be moving away with a speed of $$1.3 \times 10^{7} ms^{-1}$$ calculate its distance from the Earth. 

State variables:

$$v = 1.3 \times 10^{7} \, ms^{-1} \newline \\[0.05in] H_0 = 2.2 \times 10^{-8}s^{-1}$$

​

State equation:

​$$v= H_0d$$

Rearrange and substitute in:

$$d= \dfrac{v}{H_0} \newline \\[0.08in] d= \dfrac{1.3 \times 10^{7}}{2.2 \times 10^{-8}} \newline \\[0.08in] d= 5.9 \times 10^{14} \, m$$

The distance to the galaxy is $$\underline{5.9 \times 10^{14}\,m}$$.​​

The expanding universe

Hubble's law is evidence that the universe is expanding, as galaxies are moving away from each other in all aspects of 3D space at an increasing rate. This leads us to believe that at one point, all galaxies were together at one point in space, giving evidence for the Big Bang theory.

The cosmological principle

The cosmological principle states that the universe is isotropic and homogeneous when looked at on a large enough scale. Isotropic means it is the same across all directions and homogeneous that matter is evenly distributed across it.

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