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

Tom Van Baak tvb at LeapSecond.com
Wed Jul 27 09:42:00 EDT 2016

```Hi Michael,

I sympathize with both your and Attila's comments and would 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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```