# [time-nuts] Q/noise of Earth as an oscillator

Sun Jul 31 18:50:02 EDT 2016

```I agree with those who repudiate the use of "Q" with respect to the
length of an Earth day, and I feel that the statement about the Earth
near the end of "The Story of Q" is incorrect when viewed from a modern
perspective. I base my arguments on:

(1) The existing use of Q to describe the tidal dissipation in a body
for over 50 years. This has nothing to do directly with the
stability of the rotation rate. See "Q in the Solar System" for a
discussion of how Q refers to tidal dissipation:
http://www.es.ucsc.edu/~pkoch/EART_206/09-0127/Goldreich%20&%20Soter%2066%20Icarus%205-375.pdf

(2) The fact that the Earth is not a harmonic oscillator with a periodic
forcing function or natural resonance. The length of a day is an
accident of history.

Let's consider the rotation of a body in a near-vacuum. First note that
there is no sinusoidal physical quality of the rotating body which
stands out. For harmonic oscillators we are familiar with periodic
position, velocity, voltage, current, electric field, and magnetic field
values which have a natural period based on physical characteristics.
Harmonic oscillators transfer energy every cycle between fields. But if
you correct for the spacetime distortion due to the mass of the Earth
and the electromagnetic forces holding the surface from the center of
our planet, an object on the surface of the Earth is moving each 24
hours in a manner constrained by geometry and similar to the free motion
of a body in space far from gravitational fields. In other words, the
motion of an object on the surface is similar to that of a satellite in
free-fall. The motion does not involve any periodic change of kinetic
and potential energy as in a pendulum, flexing quartz crystal, L-C
resonant circuit, cavity, or the similar quantum mechanical constraints
which cause resonance lines in the exchange of energy between
electromagnetic fields and electrons in atoms.

I would describe the rotation rate of the Earth in a similar manner to a
rigid object (which is not spinning) in inertial movement in free space
(away from gravitation and other fields) with respect to an arbitrary
distance scale. For example, consider a point on Earth which is 100
meters from the north axis, and an object in free space moving
inertially at a rate of 628 meters per 24 hours (described in atomic
clock seconds, of course). Either the point on the Earth or the object
in free space can't be described as in motion without reference to some
external scale (due to relativity). The point on Earth (and the object
in free space) can be described in motion at a rate of 628 meters per 24
hours with respect to our Sun or to the distant stars (cosmic microwave
radiation, as in Mach's principle). Both the Earth and the object in
free space have inertia due to their mass, but in neither case does the
size, mass, composition, or other characteristics of the body cause a
preferred velocity. The velocity of that point 100 meters from the north
axis (assuming the axis hadn't moved over time) was quite different
several million years ago (due to tidal effects), and there is no
preferred velocity for that point on the Earth, just as there is no
preferred velocity for the object in free space. If a comet strikes
either one the velocity will change to any arbitrary value imparted by
the kinetic energy of the impacting object.

A point on the Earth's surface is constrained by the geometry of the
spinning body to be in motion at a certain rate relative to other points
on the body, and due to the body being reasonably rigid we consider
there to be a "rotation rate" for the Earth. Similarly, different points
on the body in free space are constrained by geometry and the rigid
nature of that body so that they are moving at the same rate with
respect to the Sun or the distant stars. In either case, if there are no
outside influences (gravity, tidal friction, radiation pressure, solar
or interstellar wind pressure, etc.) the body will remain in constant
motion. That's Newton's First Law. The periodicity of the rotation of a
spherical object is a geometric illusion, not a material property. The
only reason some think of the Earth as a periodic oscillator is that
geometry and gravity causes the motion of a point on the surface to be a
helix in space-time. This causes an apparent periodicity constrained by
geometry which is unrelated to the exchange in energy between fields
which characterizes harmonic oscillators.

Consider an artificial satellite which is rotating. Just as with the
Earth, the rotation rate can take any value when kinetic energy is added
from external interactions (constrained by relativistic effects and
breakup of the body).  There is no natural resonant frequency related to
rotation (or linear motion of the body I earlier described moving in
free space). The one Earth rotation per 24 hours rate is just a
constraint of geometry and the accidental rotational inertia at a
specific moment in our history, and is not controlled by physical
properties (density, Young's modulus, mass, etc.).

The situation is completely different for parameters of objects which
exhibit Q. The Earth has a resonant vibration frequency due to material
properties, and this causes a characteristic Q due to material
properties (as described in the paper referenced above). But this has
only a tiny effect on the rotation of the Earth, which can speed up or
slow down largely irrespective of the Q. A harmonic oscillator is
constrained to a specific resonant frequency, which changes only
slightly with material properties sensitive to temperature and other
effect. If you add energy to a harmonic oscillator the amplitude of
oscillation increases but not the frequency (which is constrained by the
Q to be centered on a natural resonance frequency).

My final argument is that the rotation frequency of the Earth is
affected by tidal friction, but the amplitude of the motion of that
point 100 meters from the axis is unaffected. The amplitude of a
harmonic oscillator is directly affected by friction or other losses,
but the effect on resonant frequency is tiny. So loss effects frequency
in one situation and amplitude in the other. How can Q relate to both
situations?
--
Bill Byrom N5BB

On Wed, Jul 27, 2016, at 08:42 AM, Tom Van Baak wrote:
> Hi Michael,
>
> like to dig
> deeper for the truth on this.
>
> Clearly both the earth and a pendulum (and many other periodic
> systems)
> exhibit a decay of energy, when you remove the periodic
> restoring force.
> And if you take the classic definition Q = 2 pi * total energy
> / energy
> lost per cycle then it would seem earth has a Q factor.
>
> In fact, if you use real energy numbers you get:
>
> - total rotational energy of earth is 2.14e29 J
> - energy lost per cycle (day) is 2.7e17 J
> - so Q = 2pi * 2.14e29 / 2.7e17 = 5e12, the same 5 trillion as my
>   earlier
> calculation.
>
> But your point about resonance is a good one and it has always
> intrigued
> me. Is this one difference between a pendulum and the earth as
> timekeepers?
>
> On the other hand, if you swept the earth with an external powerful
> frequency in the range well below to well above 1.16e-5 Hz (1/86164 s)
> would you not see a resonance peak right at the center? Given
> the mass of
> the planet and its pre-existing rotational energy, it seems like
> there is
> a "resonance", a preference to remain at its current frequency. Plus
> it
> has a slow decay due to internal friction. This sounds like any other
> timing system with Q to me.
>
> Or imagine a planet the same size as earth made from a Mylar balloon.
> Much less mass. Give it the same rotational speed. Much easier to
> increase or decrease its energy by applying external force. Far lower
> Q
> than earth, yes?
>
> It might also be useful at this point, to:
>
> read the history Q and its definition:
> http://www.collinsaudio.com/Prosound_Workshop/The_story_of_Q.pdf
>
> and read the patent in which Q first appeared:
> http://leapsecond.com/pages/Q/1927-US1628983.pdf
>
> or view the first paragraph in which Q appeared:
> http://leapsecond.com/pages/Q/1927-Q-patent-600x300.gif
>
> /tvb
>
> ----- Original Message -----
> From: "Michael Wouters" <michaeljwouters at gmail.com>
> To: "Discussion of precise time and frequency measurement"
> <time-nuts at febo.com>; <attila at kinali.ch>
> Sent: Wednesday, July 27, 2016 5:43 AM
> Subject: Re: [time-nuts] Q/noise of Earth as an oscillator
>
>
>> On Wed, Jul 27, 2016 at 8:08 AM, Attila Kinali
>> <attila at kinali.ch> wrote:
>>
>> "I am not sure you can apply this definition of Q onto earth."
>>
>> It  doesn't make sense to me either.
>>
>> If you mark a point on the surface of a sphere then you can observe
>> that point as the sphere
>> rotates and count rotations to make a clock. If you think of just a
>> circle, then a point on it viewed in a rectilinear coordinate system
>> executes simple harmonic motion so the motion of that point
>> looks like
>> an oscillator, so that much is OK.
>>
>> But unlike the LCR circuit, the pendulum and quartz crystal, the
>> sphere's rotational motion does not have a
>> resonant frequency. Another way of characterizing the Q of an
>> oscillator, the relative width of the resonance, makes
>> no sense in this context.
>>
>> It seems to me that the model of the earth as an oscillator is
>> misapplied and that the 'Q' is not a meaningful number.
>> I think the confusion arises here because of a conflation of a
>> rotation of the sphere (which marks out a time interval) with an
>> oscillation. Both can be used to define an energy lost per unit time
>> but the former doesn't have anything to do with the properties of an
>> oscillator.
>>
>> Something else that indicates that the model is suspect is that the
>> apparently high 'Q' implies a stability which the earth does
>> not have,
>> as Tom observes. Viewed another way, this suggests that the model is
>> inappropriate because it leads to an incorrect conclusion.
>>
>>
>> Cheers
>> Michael
>>
>> On Wed, Jul 27, 2016 at 8:08 AM, Attila Kinali
>> <attila at kinali.ch> wrote:
>>> Hoi Tom,
>>>
>>> On Tue, 26 Jul 2016 12:36:37 -0700
>>> "Tom Van Baak" <tvb at LeapSecond.com> wrote:
>>>
>>>> Among other things, the quality-factor, or Q is a measure of how
>>>> slowly a
>>>> free-running oscillator runs down. There are lots of examples of
>>>> periodic or
>>>> damped oscillatory motion that have Q -- RC or LC circuit, tuning
>>>> fork,
>>>> pendulum, vibrating quartz; yes, even a rotating planet in space.
>>>
>>> I am not sure you can apply this definition of Q onto earth. Q is
>>> defined
>>> for harmonic oscillators (or oscillators that can be
>>> approximated by an
>>> harmonic oscillator) but the earth isn't oscillating, it's rotating.
>>> While, for time keeping purposes, similar in nature, the physical
>>> description of both are different.
>>>
>>> Attila Kinali
>>>
>>> --
>>> Malek's Law:
>>> Any simple idea will be worded in the most complicated way.
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