Very Very Basic Numeracy Question - I'm Having a Dumb Moment?

[Physics Enemy]

Hi,

Basically the question is about doppler shift. The formula is:
(f2 - f1)/f1 = -kT ; f1 = Freq in object rest frame, f2 = Freq in lab frame

The Q: Often, frequencies are compared to 1 part in 10^15. In order to make use of this level of accuracy, to what extent can T fluctuate?

I think I'm having a serious dumb moment. Does this mean (f2 - f1) is at least 1/10^15? i.e.) We can't discern less than that? But then what do I do; are they expecting a % error?

Thanks.

 

[Redbelly98]

Hi,

Basically the question is about doppler shift. The formula is:
(f2 - f1)/f1 = -kT ; f1 = Freq in object rest frame, f2 = Freq in lab frame

There is a problem with your equation, since the units do not match. The left side is unitless, while kT has units of energy.

The Q: Often, frequencies are compared to 1 part in 10^15. In order to make use of this level of accuracy, to what extent can T fluctuate?

I think I'm having a serious dumb moment. Does this mean (f2 - f1) is at least 1/10^15? i.e.) We can't discern less than that? But then what do I do; are they expecting a % error?

It means |f2-f1| is comparable to f1/10¹⁵, or f2/10¹⁵.

 

[Physics Enemy]

There is a problem with your equation, since the units do not match. The left side is unitless, while kT has units of energy.

Sorry - The k is just a constant I put in, it's actually various quantities bundled together. So the dimensions do turn out to be correct. I just wanted to simplify and focus on the Q.

It means |f2-f1| is comparable to f1/10¹⁵, or f2/10¹⁵.

This is the key point and thanks for your answer. Could you perhaps expand on this? At the moment it just seems a little random to me and I don't understand it in laymans terms. And why is it one or the other (f1/10^15 'or' f2/10^15) ?

Thanks!

 

[Redbelly98]

Sorry - The k is just a constant I put in, it's actually various quantities bundled together. So the dimensions do turn out to be correct. I just wanted to simplify and focus on the Q.

Okay, that makes more sense. (The letter k is also used for Boltzmann's constant, which often appears in equations with temperature in the form kT, so I wrongly assumed you meant that.)

This is the key point and thanks for your answer. Could you perhaps expand on this? At the moment it just seems a little random to me and I don't understand it in laymans terms. And why is it one or the other (f1/10^15 'or' f2/10^15) ?

Thanks!

Well, in layman's terms, it means the difference |f2-f1| is a lot smaller than f1. How much less? Well, it's 1/10¹⁵ times smaller than f1. We could also say that f2 is 0.999999999999999*f1, or maybe it's 1.000000000000001*f1 -- depending on whether f1 or f2 is larger than the other. Either way, it doesn't matter whether |f2-f1| is being compared to f1 or f2, because f1 and f2 are so very nearly equal to each other.

Or, put another way, the ratio

|f2 - f1| / f1 = 1/10¹⁵

 

[Physics Enemy]

Okay, that makes more sense. (The letter k is also used for Boltzmann's constant, which often appears in equations with temperature in the form kT, so I wrongly assumed you meant that.)

Yes you're right, it was sloppy of me. :)

Well, in layman's terms, it means the difference |f2-f1| is a lot smaller than f1. How much less? Well, it's 1/10¹⁵ times smaller than f1. We could also say that f2 is 0.999999999999999*f1, or maybe it's 1.000000000000001*f1 -- depending on whether f1 or f2 is larger than the other. Either way, it doesn't matter whether |f2-f1| is being compared to f1 or f2, because f1 and f2 are so very nearly equal to each other.

Or, put another way, the ratio

|f2 - f1| / f1 = 1/10¹⁵

Thanks, I'm happy with this. So therefore:

1/10^15 = Constant * (Variation in T)
Variation in T = Some Number

So it's not a %, just +/- this number. Correct?

 

[Redbelly98]

Yes, correct.

 

[Physics Enemy]

Thank you RedBelly. :-)