The Doppler Effect

[Physics of Binary Stars] [Astronomy]  [Topics]

(1) What is the Doppler Effect ?

  The Doppler effect is the apparent change in the frequency of a wave motion when there is relative motion between the source of the waves and the observer.Well known examples include the change in pitch of an ambulance or police siren, as it approaches and then recedes from you, or the same effect from a passing train. The Doppler effect is observed for ALL wave motions (from g to radio), where there is relative motion. It is extensively used in astronomy to deduce the component of velocity in the line-of-sight of an approaching or receding planet/star/galaxy etc. Some examples are discussed below. Details of the Doppler effect may be found in any A-level (or equivalent) textbook, including a derivation of the general formulae for the Doppler effect:

 
Dl /l = - v/c and
D f/f = v/c

Note that these equations use v as the velocity of the source and that v is assumed positive when the source is approaching the Earth (ie. if the source moves towards the Earth,)
 
 

Dl is -ve (l decreases: is "blue-shifted") and
Df is +ve (f increases)).

These equations are also non-relativistic: they are only true
when v << c.



(2) Examples to consider:
 We will examine the use of the Doppler effect in the following situations:

(i) Red-shifts and the Expansion of the Universe:
We can observe features in the spectra obtained from distant galaxies and compare
their wavelengths/frequencies with those measured in the laboratory. We find that,apart from galaxies within our Local Group (such as the Andromeda spiral), the light from distant galaxies has been "red-shifted" - the wavelengths of spectral lines are longer than those measured on Earth (ie. Dl is +ve) SO THESE GALAXIES ARE MOVING AWAY FROM US. This is a consequence of the general expansion of the Universe: all clusters of galaxies are moving away from each other.

(ii) Hubble's Law and the Age of the Universe:

See also Special Relativity Problems1
SWe can use Cepheid variable stars to deduce the distances to relatively nearby galaxies, but other methods are needed to allow distance-measurement beyond the Local Group. These include: using the brightest supergiant stars as "standard candles"/assuming typical sizes for HII regions and comparing those in other galaxies with those in our own/comparing the brightest member of a cluster of galaxies with bright members in nearby clusters whose distances can be more easily measured. A "pyramid" of distance measurements thus allows an estimate (the most distant galaxies measured may have an error of 100% in their assigned distances !) to be made for d, a galaxy's distance, and an interesting fact is discovered. We find that:  
THE MORE DISTANT A GALAXY THE FASTER IT IS MOVING AWAY FROM US

and, even more useful:

VELOCITY OF GALAXY µ DISTANCE OF GALAXY FROM EARTH

This relation is expressed in Hubble's Law:
 

v = H d
.
 where H is Hubble's Constant. H is very difficult to measure, because of the difficulty of obtaining accurate values for d. A typical value is H = 5 x 104ms-1Mpc-1.

The most distant galaxies observed through the world's largest telescopes have recession velocities approaching that of light. Setting v = c in Hubble's equation gives a distance to the edge of the observable universe of d = c/H = 6000Mpc, around about 20000 Mly.
We can also use Hubble's Constant (H) to estimate the age of the universe if we assume that the universe has always expanded at its present rate. Of course, this is probably not true, but at least it gives us an order of magnitude for how long the universe has existed. The idea is as follows:

Suppose we measure the distance to galaxy "Z" by some means other than the Doppler effect, and find that Z is 50Mpc away. The value of H = 50000ms-1 Mpc-1 , giving the recession velocity of this galaxy (ie. its speed away from us) as v = Hd = 2.5 x 106 ms-1 . So now calculate how long it has taken (T), at this speed, to move 50Mpc away from us - this will be the time since our galaxy and galaxy Z were together in the Big Bang.

1pc = 3.26ly = 206265AU = 3.09 x 1016 m, so 50Mpc = 1.55 x 1024 m

T = Distance gone/speed = 2 x 1010y

Note that we must use a value for the distance to Z that has been obtained independently of v = Hd, otherwise we are simply proving that x = x !


(iii) Binary Stars:
[Physics of Binary Stars]
.
The majority of the stars in the galaxy exist as binary or multiple star systems. In a binary system, 2 stars revolve around their common centre of mass (or barycentre).For simplicity, we only need to consider the simplest possible system we could observe: 2 stars moving around each other so that we see them edge-on.The diagram below shows a top view and a side view of a binary system containing stars S1 and S2. The barycentre of the system is marked as B. Suppose we can observe a spectral line, due to, say,Hydrogen, in the spectrum of each star. The diagram also shows what happens to this spectral line as the stars rotate around B.
.

 At position X, star S1 is approaching the Earth and its line is blue-shifted, while S2 is receding from the Earth and its line is red-shifted.
               At position Y, both stars have no motion away or towards the
Earth.So there is no Doppler shift, and the observed
lines correspond to those measured in the lab.
At position Z, S1 is now receding from Earth and its line is red-shifted, while S2 approaches and its line suffers a blue-shift. The value of Dl will be different for both stars. Although they have the same Period of rotation (P), since they must always remain on opposite sides of B, in general, the radius of their orbits will be different and they therefore move with different velocities. P can be measured by observing the relative motion of S1 and S2 over a period of time, and hence the radii of their orbits can be deduced if their orbital velocities are known.

(iv) Rotation of Planets:
We see the planets by the light that they reflect from the Sun. If we observe this reflected light in more detail, we see that the spectrum of light from the planet shows:

(a) A continuous spectrum plus some absorption lines: these are the features produced
by the Sun itself, just being reflected to us,
(b) Additional absorption lines due to certain wavelengths of the solar radiation having been absorbed by materials in the atmosphere of the planets. These extra absorption features allow us to deduce the chemical composition of the atmospheres of Jupiter, Saturn etc.
(c) Since the planet is rotating, the light that it reflects is Doppler shifted. Light from the edge of the planet approaching us is blue-shifted, and vice versa.

(v) Doppler Broadening of spectral lines:

The atoms/molecules/ions that are emitting light within a star are not at rest: they are moving around in random directions, at high speed - the speed is determined by the temperature of the hot gas. The radiation emitted therefore suffers a Doppler shift, depending on the motion of the atom when it was emitted. Since the atom could be directly away from us, directly towards or somewhere in between, the radiation produced is not observed at one specific wavelength, but a slight spread in l occurs. This effect is called Doppler Broadening.

 

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