Doppler Effect Example: Chapter 5 #43
Let's do the first part of the problem the long way, then we'll do the
rest without any real calculations other than adding and subtracting.
In this question, we are told the rest wavelength of the level 2 to 1
transition in H is 121.6 nm. This is the "true" or "laboratory"
wavelength. This is what you'd measure if there were no motions.
In Star A the observed wavelength is given as 120.5 nm.
So we know the wavelengths and of course we know the speed of light "c",
so the only thing we don't know is the velocity. So let's compute the
velocity.
Write out Doppler's Law: (lambda is the wavelength)
v / c = (lambdaobserved - lambdatrue) /
lambdatrue
Now let's plug in the values one by one. And always keep the units:
v / c = (120.5 nm - 121.6 nm) / 121.6 nm
v / c = (-1.1 nm) / 121.6 nm
v / c = -0.009046
Notice how the units of length (the nanometer) in the numerator and
denomination (top and bottom of the fraction) cancel out. So what remains
is a pure number on the right hand side. This is correct, because the
left hand side is the ratio of a velocity over a velocity, and this is
also a pure number with no units.
Also notice that the number is negative - this means the object is moving
towards us. (Remember that negative velocity is motion towards us and
causes a blueshift; positive velocity is motion away and causes a
redshift).
Ok, back to the formula:
v / c = -0.009046
We want v, not v/c, so bring the "c" over to the right hand side. Then we
have:
v = c x (-0.009046)
So now lets replace "c" with its actual value of 300,000 km/s.
v = 300,000 km/s x (-0.009046)
And now, finally:
v = -2713.8 km/s , moving towards the Earth.
See how the units of velocity (km/s) show up correctly.
Okay, now let's answer the question without any caculations.
- Which stars are moving toward us?
If a star is coming towards us, it will have a negative velocity. This
means that the measured wavelength minus the rest wavelength has to be
negative. For this to be true, the measured
wavelength has to be LESS than the rest wavelength. So let's see:
Star A: 120.5 nm is less than 121.6 nm, so it is moving towards us
Star B: 121.2 nm is less than 121.6 nm, so it is moving towards us
Star C: 121.9 nm is more than 121.6 nm, so it is moving away.
Star D: 122.9 nm is more than 121.6 nm, so it is moving away.
- Which star is moving the fastest?
The largest speed is the case where the difference between the measured
and the true (rest) wavelength is the largest.
Star A: 120.5 nm - 121.6 nm = -1.1 nm
Star B: 121.2 nm - 121.6 nm = -0.4 nm
Star C: 121.9 nm - 121.6 nm = +0.3 nm
Star D: 122.9 nm - 121.6 nm = +1.3 nm
So which one of these number is largest in absolute value? The answer is
Star D.
Doppler Effect Example: Chapter 5 #52
Once you know the question involves the Doppler effect, start off by
writing down Doppler's law. Then identify each of the knowns and
unknowns.
{(lambdaobserved - lambdatrue) /
lambdatrue} = V / c
We are given the observed wavelength (121.9 nm for Star C) and the
rest wavelength (lambdatrue = 121.6 nm), and we know the speed
of light "c". We want to solve for the radial velocity.
Let's start by re-arranging the Doppler formula to solve for the velocity
v:
c x {(lambdaobserved - lambdatrue) /
lambdatrue} = V
So then put in the numbers:
3x105 km/s x {(121.9 nm - 121.6 nm) / 121.6 nm} = V
Now just punch it into a calculator step by step and you've got the
answer:
3x105 km/s x {(0.3 nm) / 121.6 nm} = V
3x105 km/s x { 0.002467 } = V
740 km/s = V
The answer is positive, so Star C is moving away (redshifted).
The answer for Star D is: moving away at 3207 km/s.
Did you notice that Questions 51 and 52 are the continuation of question
43?