A redshift derivation question

[Harry2005]

Hello

I have been reading a textbook on cosmology and have come up against a self test question I just can't crack! Any help would be greatly appreciated:

"The Cosmological redshift of a quasar is normally deduced from its emission lines. Let this redshift be ze. Suppose the quasar has ejected a cloud of gas at velocity v with respect to the quasar. We detect the gas as absorption at redshift za. Show that the velocity of ejection is given by:

v/c = [ (1+ze)^2 - (1+za)^2 ] / [ (1+ze)^2 + (1+za)^2 ]

The "e" and "a" should be subscript, and the ^2 should be superscript (squared).

Thanks very much.

 

[Chronos]

Hi Harry2005!

Welcome to PF. The formula you cited is standard. Just plug in the Z values. It works because you subtract the difference between quasar emission and intervening absorption lines. The intervening gas clouds are nearer to us and have lower redshifts. This is referred to as the Lyman alpha forest [enormous, primordial gas clouds blocking our view creating a 'forest' of dark lines in quasar emission spectrums].

 

[Harry2005]

Hi

Thanks very much for your reply. The question in the textbook asks to derive the formula...I know the redshift and doppler shift formulas but just can't seem to get them to combine to that form... :grumpy:

Cheers

 

[cragwolf]

Neither can I. I get cross-terms which I can't get rid of. My idea was that the difference in redshifts between the gas cloud and the quasar is due solely to the (longitudinal) relativistic doppler effect.

 

[babbloo]

Hi Harry

I tried to find the equation of z from standard formula

1+z=(Lambda_obv)/(Lambda_emitted)

Now Observed wavelength will now be the difference of the wavelengths due to Cosmological Red Shift and the Doppler Blue Shift(Neutral Hydrogen atoms in the cloud).
Hence, 1+z=(Lambda_e - Lambda_a)/(Lambda_emitted)

But, i still could not get the equation which you have specified.
Anyone .. Any Idea! Can someone derive it?

 

[babbloo]

The doppler shift is actually due to Cloud of gas ejected from the quasar.
Also, This should either be red shifted or Blue shifted depending on the direction in which it is emitted.

 

[Chronos]

I disagree. The lyman absorption lines are not even close [in redshift] to the emission lines.

 

[babbloo]

Yes .. you are right ..
Lyman absorption lines are due to the neutral hydrogen atoms in the intervening space (Between Earth and Quasar)
But, there is a cloud of gas being ejected from the quasar. There should be a doppler shift when light waves from the quasar travel through this. This leads to absorption lines in the quasar spectra. This is not due to the Lyman alpha forest.

And it is either blueshifted or redshifted depending on the direction in which the cloud of gas is ejected from quasar.

Hence, the overall redshift component will be a combination of the Cosmological redshift due to the expansion of space and a doppler shift due to the cloud of gas ejected from the quasar.

Do you agree now?

 

[matt.o]

Generally, the redshift break down is given as follows:

(1+z_obs )= (1+z_cos)(1+z_pec)

where z_obs is the observed redshift of the object, z_cos the cosmological redshift contribution and z_pec is due to any peculiar velocity on top of the cosmological redshift.

Redshift contributions are in general not simply added. Hope this helps.

 

[babbloo]

Ztot=(1+Zcos)(1+ZDopp)

I also got a piece of reference material from Kembhavi and Narlikar’s book about the Quasar Redshift upon reading which I found the following:

• Zabs of the Quasar’ generally is due to Doppler and the cosmological effects. And, Zabs will be less than Zem in most of the cases.
• The relative red shift in terms of Zabs and Zem can be given as

Ztot = (Zem – Zabs) / (1+ Zabs) ------------------ > Eq. 1

Substituting Eq. 1 in the formula for the Quasar redshifts

v/c = [(Z+1)^2 - 1] / [(Z+1)^2 +1 ]

The final equation for v/c interms of Zem and Zabs is

v/c = ( [1+ Zem] ^2 - [1+ Zabs] ^2 ) / ([1+ Zem] ^2 + [1+ Zabs] ^2)

Hope this is what you guys looking for.