Required Preparation for Section 9, but it is suggested for all sections plus whatever your section instructor requires.
Lab Preparation:
- Read Lab 8
- Read a sufficient amount of the articles linked to the following terms etc. so that you can define and/or understand the terms etc. at the level of our class: Doppler effect, electromagnetic spectrum, Mercury, Mercury's spin-orbit resonance, radar, radial velocity.
- Write out the definitions required in Part A of Lab 8.
Suggested Supplementary Preparation. The items are often alternatives to the required preparation.
- Bennett (2008 edition): p. 149--157 and 163--174 on light and p. 215 on Mercury's spin-orbit resonance, or the corresponding pages in similar books.
- IAL 12: The Moon and Mercury: Spin-Orbit Resonance.
- Background Notes
Quiz Preparation:
There is no end to the studying you can do, but it is only a short quiz.
It's worth moderate amount (8 %) of a 1-credit-hour course.
One to two hours prep should suffice.
There will be 10 or so questions and the time will be 10 or so minutes.
The questions will range from quite easy to challenging.
After the quiz, the solutions are posted at
The Doppler Effect---The Rotation of Mercury:
Quiz Solutions.
The students will learn a little about
electromagnetic radiation (EMR),
frequency,
the Doppler effect,
and radar astronomy.
The geometry in the lab
may be a bit more than they can absorb in the lab period.
The Startup Presentation should usually be no more than about 10 minutes.
Students lose patience and want to get at the work---and that is right---labs are
active learning, not passive learning.
For this lab a few bullets in point form written on the board with drawn diagrams
and handwaving
might be best.
You will have to give up presenting when you run out of time (10 minutes or so)---and leave the rest of
the things you could say to TA mode.
The other settings are computed for you.
Then lauch a radar pulse to
Mercury.
It will take 13 minutes---real-time minutes---to get back.
So now I will give a presentation which will die in 10 minutes or so no matter where
it gets too.
In the limiting case, that you run away at the
vacuum light speed c
the frequency will go to zero.
You can't reach c, of course, but nothing forbids you from
coming arbitrarily close, in principle.
A derivation of this formula is given in the Background Notes: Doppler Effect.
But you-all don't need to know that---but it would be good for your soul.
Well Mercury
"observes" the
radar pulse
with frequency shift given by
We assume for simplicity that its rotation axis is perpendicular to the
Mercury-Earth
line.
Because it is an extended rotating object,
there is not one signal with one shifted
frequency coming back from
Mercury, but
a continuum range of signals with
a continuum range of shifted frequencies.
And this range varies with time since different parts of
Mercury
are different distances away---so we get a time range of returning signals.
The returning first signal that comes back comes from sub-radar point---the
surface point of Mercury
directly on the
Mercury-Earth line.
This reflection region expands into an annulus that grows from the
sub-radar point until it reaches
Mercury's
limb (the circumference
region of Mercury as seen on the sky).
After the limb signal returns,
the radar pulse cannot reflect anymore and
the returning signals die off abruptly.
Do NOT worry about the derivation of this formula.
We can solve for v_rot:
You feed in df_r and df_b into the appropriate fields in the
Excel spreadsheet
and the other required data,
and the Excel spreadsheet will calculate
Mercury's
rotation period P:
This rotation period is relative to the
fixed stars which define
an inertial frame.
So it is a sidereal time interval
and NOT the Mercurian solar day P_sol
which is actually twice the
Merurian orbital period.
Mercury's
rotation period P
and
orbital period P_orb
are locked in a
Mercury's 3:2 spin-orbit resonance:
P_orb:P=3:2.
Gravitational effects have driven
Mercury
to this
Mercury's 3:2 spin-orbit resonance
and drive it back toward it
if perturbed off the exact 3:2 ratio.
So we actually have the ratio P_sol:P_orb:P=4:3:2.
Where the signal spectrum seems to dive steeply at the
red and blue ends before it disappears into background noise.
Waves are
an oscillatory motion of a continuum of something extending in space.
Here we will just consider
traveling waves
in one dimension like waves on a string.
A simple
traveling wave motion
has spatial cycle with a length called the
wavelength.
If you move along in space beside the
traveling wave motion,
the wave shape repeats every wavelength.
The repeated pattern is the wave cycle.
The traditional symbol for wavelength
is the Greek letter lambda---which
in HTML, we have to spell out in Roman letters.
The velocity at which a wave cycle moves along is the
phase velocity.
For light in vacuum,
the phase velocity
is the vacuum light speed
always symbolized by c.
The time for one wave cycle to pass a point in space is the wave period:
Because of the formula fL=v, the frequency
and wavelength contain essentially
the same information for waves
with constant phase velocity
like light in vacuum.
Thus, people characterize such by either or both of
frequency
and wavelength
as suits their convencience.
Electromagnetic radiation (EMR)
(AKA light) is
a self-propagating
electromagnetic wave.
For EMR,
there are no in principle limits on
frequency
and wavelength
between 0 and infinity.
Actual existing processes that produce
EMR
probably set some limits on
frequency
and wavelength, but
I don't think we know what those limits are.
There may be frequencies
and wavelengths
smaller or greater than any we have ever measured.
The entire continuum of
frequency
and wavelength
is called the
electromagnetic spectrum.
Below is figure illustring the
electromagnetic spectrum
in multiple ways.
Caption: A diagram
illustrating the
electromagnetic spectrum
in multiple ways.
Note
Credit/Permission:
© User:Inductiveload,
2007 /
Creative Commons
CC BY-SA 3.0.
Image linked to Wikipedia.
The Doppler effect
is the dependence
frequency
of a wave motion
on the velocity of the observer (who could be either source or receiver of
the wave motion).
The most common everyday
manifestation of the
Doppler effect
is the change in frequency
of sirens
depending on the motion of the source.
The Doppler effect
is rather different for the cases of
mechanical waves
(e.g., sound waves)
and
light.
However, conveniently the 1st order formulae for low velocities
for the two cases are the same.
What is a low velocity depends on the case, in fact.
Here we will derive the
Doppler effect formula
for light
in the low velocity limit: i.e., in the non-relatativistic limit where
(v/c)<<1.
Consider the system below with a traveling wave cycle and frames
that can only move in the direction of the motion of the
light beams.
Now say we take A as the source and A' as the receiver.
Since the light been is going to the right, A' can only recieve when it is to the
right of A.
With this assumption, we see that v>0, gives df<0 and v<0 gives df>0.
In astro jargon, df<0 is called a
redshift
and df>0 is called blueshift.
This is jargon is just because in
visible light
red is a low frequency
and blue is high frequency.
The jargon is used for all light,
not just visible light.
Violet is the highest frequency visible light.
Probably, because violetshift doesn't trip of the tongue.
Notice that our derivation is for
frames that move only in the direction of light beam
Our formulae df/f=-v/c and dL/L=v/c are valid in other cases, but
v is only the component of velocity along the line of sight between the
source and receiver.
In the astro context, this component of velocity is called the
radial velocity.
The velocity component perpendicular to the line of sight (i.e., the transverse velocity) causes
1st order Doppler effect shift.
There is
transverse Doppler effect
shift, but it is 2nd order in v/c, and so is usually negligibly small.
The derivation of the
relativistic Doppler effect
for light is just
the one given 1st order
Doppler effect for
light in section
Doppler Effect
with a little extra bit.
So it is actually cinchy.
Our derivation of the
Doppler effect for
light could be tweaked
to work for sound too.
But doing that obscures important difference
between the cases of
light and
sound.
In any case, the clear-cut derivation of the
sound Doppler effect is cinchy.
In preparing for a quiz on the Doppler effect
and Mercury,
you should read over your report on lab 8
and go over the required lab preparation (see above).
df/f=v/c ,
where df is the change frequency,
f is the initial or final frequency
(it doesn't matter which to our level of error),
v is the velocity of the "observer" along the beam direction relative to whatever the
initial frame of rest was.
v>0 for an "observer" plowing into the waves
v<0 for an "observer" running away from the waves.
This formula is only valid for df/f<<1, and that is why
can be either the initial or final frequency.
If df>0, the frequency shift is
called a blueshift.
If df<0, the frequency shift is
called a redshift.
df_1/f=v/c , where v is Mercury's
velocity in the direction of the Earth
or as we call the
radial velocity
of Mercury
from the Earth's
perspective.
So Mercury observes
frequency
f+df_1
and reflects some signal back to Earth.
Now Earth
is plowing into the return signal at velocity v relative to
Mercury, and so
there is another shift given by
df_2/(f+df)= approximately df_2/f=v/c .
The frequency
measured at Earth is thus
f+df_1+df_2 .
The total shift is just
df=df_1+df_2 and is given by
df/f=2v/c .
So that is where the mysterious 2 comes from in this lab.
If Mercury is moving
toward Earth, v>0 and df>0.
Otherwise, v<0 and df<0.
df_r/f=v_r/c , where v_r=v-v_rot*sin(theta)
to
df_b/f=v_b/c , where v_b=v+v_rot*sin(theta)
where r stands for reddest and b for bluest,
theta is the angle of the annulus reflection region from the sub-radar point
with the angle subtended at Mercury's center,
and v_rot is Mercury's
equatorial rotation velocity.
v_rot=(v_b-v_r)/[2*sin(theta)]=(c/f)*(df_b-df_r)/[2*sin(theta)] ,
which is undefined as theta goes to zero, but it doesn't go
to infinity formally since df_b-df_r goes to zero likewise
as theta goes to zero.
Actual observed spectra have noise and are averages over
some range of time.
So one does not see a sharp spike in
frequency
as theta goes to zero, but a sort of broadened peak for which
df_b and df_r not really definable.
P=2*pi*R_Mercury/v_rot , where R_Mercury is
Mercury's radius.
P=L/v ,
where P is period,
L is wavelength,
and v is phase velocity.
Say N wave cycles pass a point.
The frequency f or
number of cycles per unit time is
f=N/(NP)=1/P .
So frequency is just the
inverse of the period.
Note
v=L/P=fL
or, as one usually see it,
fL=v .
For light in vacuum,
fL=c .
wavelength = c / frequency
where c is the vacuum light speed.
Thus,
wavelength and
frequency
contain the same information in a different form.
Time 1: The wave cycle is just starting to pass point A.
__________
| |
|__________|
A defines a frame A
A' defines a frame A' moving with respect to A at velocity v
Time 2: The wave cycle is just past point A and is still passing over moving point A'
__________
| |
|__________|
A
A'
The wave period in frame A is P=L/c,
where L is wavelength and c is wave speed as measured in frame A.
Time 3: The wave cycle is just past moving point A'
__________
| |
|__________|
A A'
The wave period in frame A' is P'=(L+vP')/c,
where vP' is the distanc point A' has moved and L+vP' is the distance the
end of the wave cycle has moved in getting to A'
Actually, here is where we make the low velocity approximation.
Time actually flows differently in different
inertial frame
according to
special relativity.
The correction for this effect requires our P' to be multiplied by a correction
factor to give the true P'.
The correction factor is sqrt[1-(v/c)**2].
If (v/c)<<1, then the correction factor is 1 to 1st order in (v/c).
So we do not need the correction for a 1st order correct formula.
Now for some algebra:
P'=(L+vP')/c=L/c+(v/c)P'=P+(v/c)P'
P'[1-(v/c)]=P
P'=P/(1-v/c)
This formula relates A and A' frame periods.
Converting to frequency,
we obtain the 1st order
Doppler effect formula
for frequency
f'=f(1-v/c)
or
(f'-f)/f=-v/c
or
df/f=-v/c
where df is the change in frequency.
Note v is the relative velocities between the two frames.
In the Startup Presentation, there
was no minus sign in df/f=-v/c because we defined the velocity to
be positive for the direction opposite to the beam direction.
Here we have defined it to be positive in the beam direction.
Both choices have their advantages and both are used.
Why not violetshift instead of blueshift?
The 1st order
Doppler effect formula
for wavelength is derived
as follows:
f'=f(1-v/c) , and so L'=c/f'=c/[f(1-v/c)]=L/(1-v/c) .
For v/c<<1, 1/(1-v/c) = approximately 1+v/c, and thus
L'=L(1+v/c)
or
dL/L=v/c ,
where dL=L'-L is the change in wavelength.
Measurements of df or dL allow one to solve for v:
v=c(df/f)=c(dL/L) .
Such measurements are very important in astronomy.