Red Shift and Hubble

Light From A Stationary Source

Due to the distances involved, it is not possible to directly sample stars in order to find out their composition. However, scientists can use another method to identify the elements that make up a star, even millions of light-years away, by using the light they give out. By observing the light spectrum of the star, every element present can be found. There are two main types of spectra:-

1. Continuous spectra - a 'rainbow' (light of all Frequencies).

2. Line Spectra - a 'rainbow' cut down to several lines of colour (light of specific frequencies).

Rainbows and Continuous Spectra

When a ray of white light passes through a Triangular Prism, the ray is refracted twice. The overall effect of these refractions is to cause the white light to be dispersed. This causes a spectrum to be observed:-

This is the same process of how a rainbow is seen on a rainy day:-

When we observe the light dispersed by a raindrop we see a rainbow, we see an approximate continuous spectrum of light from red to violet. The diagram below shows a true continuous spectrum in visible light:-

In the above image there are no gaps in this spectrum, it runs continuously from violet to red.

Note - a rainbow is not a true continuous spectrum, but our eyes cannot resolve the gaps as they are so small, the light forms an absorption line spectrum (see below).


Line Spectra

The light emitted from the Sun starts out as a continuous spectrum, but when it passes through the Solar Photosphere (the outer layer of the Sun's atmosphere), some of the light is absorbed by the gases present. This means that the spectrum that reaches the Earth is incomplete, missing certain Frequencies. The diagram below shows the absorption line spectrum of the Sun:-

As can be seen above, the light from the Sun is littered with these gaps, and it is these lines of absorption that allow scientists to identify the elements present in the Sun.

By observing the spectra given off by a particular element in vapour-form, it can be shown that each individual element has a unique set of emission lines, which match exactly the gaps in the Solar spectrum. The diagram below shows the emission line spectra for several elements:-

By observing a star's spectrum, and by matching the absorption lines to the known pattern for each element, a star's composition can be found.


Light From a Moving Source

If the star emitting the spectrum is moving relative to a stationary observer, then due to the Doppler Effect, the Frequency of the observed light can change. The diagram below shows the effect of a moving light source on the spectra generated:-

As can be seen in the above absorption spectra, the relative motion of a light source causes the entire spectrum to be shifted, either towards the red or blue end of the spectrum:-

1. Source moving away from observer - Spectrum is Red-shifted.

2. Source moving towards the observer - Spectrum is Blue-shifted.

The same formula used to calculate the Doppler Shift for sound can also be used to calculate the Doppler shift in light, with the wave speed equal to the speed of light:-

Note - The above formula is only useful for nearby galaxies. Distant galaxies are moving at relativistic speeds, and due to this, the above formula cannot be used without taking relativity into account (which is beyond the scope of the Higher course).

Note - When discussing light, by convention, it is usual to discuss the Doppler Effect (and all other light phenomena) in terms of Wavelength. In order to convert the change in Frequency caused by the Doppler Effect to a change in Wavelength, the Wave equation (see Wave Equations) can be used.

Example 1 -

NGC 3718 is a galaxy in the Constellation Ursa Major. A particular emission line in its spectra is observed to be 1003 nm. In a lab, the same emission line is measured as 1000 nm.

Calculate:-

1. The direction the galaxy is travelling.

2. The Velocity of the galaxy relative to Earth.


As the Wavelength has increased, this means that the galaxy is moving away from the Earth.

λ0 = 1003 nm = 1003x10-9 m

λs = 1000 nm = 1000x10-9 m

v = 3x108 ms-1

vs = ?

V = f λ

f = V / λ

fs = 3x108 / 1000x10-9

fs = 3.000x1014 Hz

f0 = 3x108 / 1003x10-9

f0 = 2.991x1014 Hz

Vs = (V (fs / fo)) - V

Vs = (3x108 (3.000x1014 / 2.991x1014 ) - 3x108

Vs = 902,708 ms-1

Vs = 902 kms-1

Note - As can be seen from the above example, when dealing with light the Doppler Effect gives only very small changes in wavelength, even when the object is travelling very fast. This is why Redshift is only visually seen (the galaxies look "more red") on very distant, fast moving galaxies.

Cosmological Distances

As was seen within the National 5 course, measuring distance in metres was impractical, it was more useful to measure distance in light-years:-

1 Lightyear (ly) = 9.5x1015 m

This is a useful distance measurement when working within our own galaxy but even this large distance becomes too small on the cosmological scale.

When dealing with the largest cosmological distances a more useful unit to use is the Parsec (as well as kPc and MPc).

The word Parsec is shortened form of "Parallax of one Arcsecond". Parallax is the apparent shift in position of a nearby object against a distant background as the viewpoint changes. This can be seen simply be holding a pen close to your face and observing it through different eyes:-

A Parsec is defined as the distance of an object that experiences a parallax angle of one second when the Earth's position changes by 1 A.U.:-

As can be seen in the above diagram, the change in position of 1 astronomical unit occurs simply by observing the star 6 months apart. The position of the star is measured accurately against the background starfield and by using trigonometry, the distance to the star can be calculated.

1 Parsec (Pc) = 3.26 Lightyears (ly)


Redshift

For non-relativistic celestial objects, astronomers can use the following Redshift ratio to calculate the relative Velocity of that object :-

Z = (λ0 - λs) / λs

and

Z = Vs / C

Where :-

Z = Redshift (ratio - no units)

λ0 = Wavelength observed (m)

λs = Wavelength emitted by source (m)

Vs = Relative Velocity of source (ms-1)

C = Speed of light (3x108 ms-1)

Note - By convention, Blueshift (object moving towards an observer) is generally referred to as negative Redshift and as such has a negative value for "Z".

Example 2 -

NGC 3718 is a galaxy in the constellation Ursa Major. A particular emission line in its spectra is observed to be 1003 nm. In a lab, the same emission line is measured as 1000 nm.

Calculate:-

1. The direction the galaxy is travelling.

2. The Velocity of the galaxy relative to Earth.


As the Wavelength has increased, this means that the galaxy is moving away from the Earth.


Z = ?

λ0 = 1003 nm = 1003x10-9 m

λs = 1000 nm = 1000x10-9 m

Vs = ?

C = 3x108 ms-1


Z = (λ0 - λs) / λs

Z = (1003x10-9 - 1000x10-9) / 1000x10-9

Z = 3x10-3


Z = Vs / C

Vs = ZxC

Vs = 3x10-3 x 3x108

Vs = 900,000 ms-1

Vs = 900 kms-1


As can be seen above, using either the Doppler formula or Redshift formula will give the same answer for a star's velocity.

Note - The Doppler formula will not apply to any object moving with a Velocity 0.1C or greater. Any faster than 0.1C and relativistic effects must be taken into account (which is beyond the scope of the Higher Course).

Hubble's Law

In 1929, the American astronomer Edwin Hubble published his own work on the Doppler Effect on light from distant galaxies. Hubble had been studying a class of star called a Cepheid Variable Star, which was a type of star that that changes size and brightness following a very fixed period of time. By observing the the period and apparent brightness of distant Cepheid Variables, Hubble was able to show that galaxies were much further away than previously thought.

Hubble also noticed that the further away the Cepheid Variable star was from Earth, the greater the Redshift of the star. This meant that the further from Earth a star was, the faster it was receding. In fact, Hubble found that if he plotted a graph of distance against Recessional Velocity, the two were directly proportional:-

From this work, Hubble derived the following formula:-

Where :-

V = The Recessional Velocity (ms-1)

H0 = Hubble's Constant (2.3x10-18 s-1)

d = Distance to object (m)

Example 3 -

The Black Eye galaxy (Messier object 64) is approximately 24 Mly from Earth. What is the galaxy's Recessional Velocity?

V = ?

H0 = 2.3x10-18 s-1

d = 24 Mly = 24x106 ly = (24x106 x 9.5x1015) = 2.28x1023 m

V = (2.3x10-18) x (2.28x1023) = 524000 ms-1

V 524 kms-1

The video below shows an explanation of Hubble's law by Brian Cox:-