Hubble's Law

Hubble's Law

Using Pogson's Law and the Doppler Effect, we can make the biggest and most profound statement in Cosmology.

Pogson's Law can determine the distance a galaxy is from Earth, and the Doppler Effect can determine the galactic radial velocity relative to Earth. Plotting the radial distance against radial velocity on a graph gives an interesting result.

Remarkable...

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The line of best fit on the graph is a straight line which implies that the recessional velocity is directly proportional to the distance of the galaxy. This is also known as Hubble's Law.

Since Vr is conventionally expressed in units of km/s and r in units of Mpc, the units of the constant (Hubble's Constant) are km.s(^-1)/Mpc.

From the graph, we can see that the furthest galaxies are moving away from us at a faster rate than closer galaxies. This is what would be expected if all galaxies were to move away from one another from a single point as if from a big explosion - commonly known as 'the Big Bang'.

Hubble's Constant also gives us some information about how much time has elapsed since the Big Bang event. As H=V/r, by assuming that each galaxy has traveled a distance of r at constant velocity V in time T we can show that V=r/T. Hence, T=1/H.

By cancelling the units of Hubble's Constant to 1/s, the reciprocal of that value is the age of the universe in seconds. The agreed value of the age of the Universe is 4.3×10¹⁷ seconds, which is equivalent to around 13.8 billion years.

The fact that mankind can determine the age of the universe so easily still amazes me...

Remarkable...

Doppler Effect

The Doppler Effect

The Doppler effect causes waves, such as sound and light to change wavelength depending on the velocity of the observer in relation to the velocity of the emitter of the wave.

 

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The Doppler effect is expressed with the relation: Δλ/λ(rest) = v/c

v is the radial velocity of the emitter when the observer perceives themselves to be at rest.
λ(rest) is the wavelength of light (or sound) that would be observed from the object if it were at rest relative to the observer.
  Δλ is the difference between the observed wavelength and λ(rest).
c is the speed of light.

In stars with similar surface temperatures as the Sun, the spectral absorption of ionised calcium feature prominent H and K lines. These lines can be studied in an Earth-bound laboratory quite easily to determine the value of λ(rest). The K line has a λ(rest) of 393.37nm and the H line has a λ(rest) of 396.85nm.

Because these lines are easily observable in distant galaxies, a spectrometer can be used to determine the value for λ for both lines. Due to the Doppler Effect, the values for λ and λ(rest) for each line will differ by the same Δλ. Using the equation Δλ/λ(rest) = v/c, the radial velocity of the galaxy can be experimentally determined.