We all love to prove things [spectral broadening/relativity]

[fasterthanjoao]

Well, hopefully this is going to be my last help-thread for a while, i'd like to start repaying the members. You may have noticed I've posted a few threads asking for help tonight, and I thank the members who have read my threads very much. I've narrowed down points of my course that I remain unsure about (after hours, believe me) of reading my textbooks and any others I can find and I *think* this is the last of it.

I have attached two documents (yeah I know, attached) of scanned in questions that require prooving. Unfortunately, I haven't been able to prove these myself, yet, and I have spent a lot of time trying to do so. I also apologize for the lack of latex, I spent ages trying to 'tex these questions up, but kept getting error messages to be honest I need to take a break, so I'd appreciate it a lot if some of you guys would look at my problems.

Note: these are not homework problems, I'm just trying to understand my course - any help with any of these proofs is greatly appreciated - be it completed proofs or guidelines. I promise that when I'm back on after a nap, I'll give latex another bash and respond to any queries you may have about these questions.

They require pre-honours level astronomy knowledge (or good thermal/bit o' quantum physics) - The first ones are on spectral broadening, and there's a quick relativity related proof (I think these ones are the easiest).

Thank you, and yes I'm aware that several members don't like handwritten documents so i'll try and get it sorted. thanks,
$$\delta$$

ftj.

 

[SpaceTiger]

Alright, let's start from the beginning and treat these like homework problems. It won't help much if you don't work through them.

1a. What is the wavelength shift produced by the doppler effect if an object is moving at a velocity, v? Then, imagine you're viewing the disk edge-on and you're looking only at material located at a radius, R. Which material will exhibit the strongest doppler shift? How quickly is it moving relative to you?

b. What is the rms speed of a gas at temperature, T, composed of particles, each of mass, m? What is the doppler shift produced by particles moving at this speed? How does this relate to the width of the associated absorption/emission line?

 

[fasterthanjoao]

1a. What is the wavelength shift produced by the doppler effect if an object is moving at a velocity, v? Then, imagine you're viewing the disk edge-on and you're looking only at material located at a radius, R. Which material will exhibit the strongest doppler shift? How quickly is it moving relative to you?

The Doppler shift of material at a radius R is? with omega as the angular velocity?
$$\Delta \nu = \frac{\nu_{0}}{c}[\Omega(R) - \Omega(R_{0})]R_{0}$$

Though, I think if I take that the Doppler shift is in one direction (in my direction?) I can take the shift as: $$\Delta \nu = \frac{\Omega_{x}}{c}$$

Can I then change from frequency to wavelength and substitute the Keplerian speed in the process? Is it wrong to say that radio of a change in wavelength to the initial wavelength is the ratio of change in frequency to the initial frequency?

b. What is the rms speed of a gas at temperature, T, composed of particles, each of mass, m? What is the doppler shift produced by particles moving at this speed? How does this relate to the width of the associated absorption/emission line?

$$\frac{1}{2}m<v_{x}>^2 = \frac{1}{2}kT \\ v_{x}=\sqrt{\frac{kT}{m}}$$ - I think I see that this is the most probable speed rather than the rms, and I'm reading that the rms is:$$v_{rms}=\sqrt{\frac{8kT}{\pi m}}$$

I can go from this to say that since, $$\frac{\Delta \nu}{\nu_{0}} = \frac{\nu_{x}}{c}}$$ then:
$$\frac{\Delta \nu}{\nu_{0}} = \sqrt{\frac{8kT}{\pi m c^2}}$$.

 

[SpaceTiger]

The Doppler shift of material at a radius R is? with omega as the angular velocity?
$$\Delta \nu = \frac{\nu_{0}}{c}[\Omega(R) - \Omega(R_{0})]R_{0}$$

This is the maximum Doppler shift from material at radius, R, and yes, omega is the angular velocity. How is it expressed in terms of ordinary velocity?

Though, I think if I take that the Doppler shift is in one direction (in my direction?) I can take the shift as: $$\Delta \nu = \frac{\Omega_{x}}{c}$$

I'm not sure how you're getting this formula. What happened to $\nu_0$ and R?

Can I then change from frequency to wavelength and substitute the Keplerian speed in the process? Is it wrong to say that radio of a change in wavelength to the initial wavelength is the ratio of change in frequency to the initial frequency?

Yes to the first question. You can answer the second question yourself. What is the relationship between wavelength and frequency. What happens when I differentiate this formula (i.e., take small intervals $\Delta \lambda$ and $\Delta \nu$)?

and I'm reading that the rms is:$$v_{rms}=\sqrt{\frac{8kT}{\pi m}}$$

Where are you reading that? In a Maxwell-Boltzmann distribution, that looks like the mean speed.

I can go from this to say that since, $$\frac{\Delta \nu}{\nu_{0}} = \frac{\nu_{x}}{c}}$$ then:

What is $\nu_x$?