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

Scott Stobbe scott.j.stobbe at gmail.com
Sat Jul 30 18:16:31 EDT 2016

```Thanks Tom, I would agree LIGOs efforts are beyond heroic, I will try to
find some of their phase noise plots.

Regarding Q of the earth, I would agree one could compute an UNloaded Q for
the earth as if it were a mass element in some form of a mechanical
oscillator. The first sticky point is which Q, a loaded Q (QL) would assume
it's oscillating (which as many have outlined) it's not, an unloaded Q
(1/DF, or X/R) would be a reasonable value to commute for some specific
frequency. It is interesting (although expected) I have arrived to similar
Q as you, though through different reasoning, but similar assumptions.

Taking inspiration from the many brilliant controls and analog Engineers
from the analog computing days, we can create a circuit equivalent model
for the rotational dynamics of the earth.

Some physical parameters:

omega_e = 7.3E-5 rad/s         (angular rate of the earth)
alpha_e = -6.3E-22 rad/s^2     (angular deceleration of the earth)
J_e = 8E37 kg m^2                (Inertia of the earth)

Mapping the torque-(angular displacement) space to the volt-coulomb space

Capacitor - Torsional Spring
Q = C V,  tau = k theta

Resistor - Damper / Friction
Qdot = V/R,    tau = B thetadot

Inductor - Mass
Qdotdot = V/L,    tau = J thetadotdot

In this circuit equivalent model the inertia of the earth would be
represented by an inductor of inductance
L = 8E37 kg m^2

An approximate rotational friction coefficient (tidal friction, doubt it's
a first order relation, but for the sake of Q, assume it is, and other
losses) can be found from the net angular deceleration.
B = J * |alpha_e|/omega_e = 6.9E20 Nm s

Solving as an equivalent resistance yields,
Rs = 6.9E20 Nm s

Finally the unloaded Q at 11 and change uHz,
Q = XL/Rs = (7.3E-5)(8E37)/(6.9E20) = 8.5E12

8.5 Trillion, not bad for an inductor at 11 uHz... Now if you really wanted
an 11 uHz oscillator you could ram a torsional spring up the earth's south
pole.

>From a circuit perspective, the earth's rotation looks like a monster near
superconducting inductor that at some point and somehow was precharged to a
current Io, and then had its terminals crowbarred. Our solar time is like
watching a reference electron run round and round a coil.

Björn and Dave, thanks for the gyro reference I will take a look.

On Fri, Jul 29, 2016 at 1:19 PM, Tom Van Baak <tvb at leapsecond.com> wrote:

> Scott Stobbe wrote:
> > I believe a phase noise plot deep into the uHz or lower would apply to
> the
> > rotation rate of the earth.
>
> Yup. You'll see lots of uHz to Hz noise plots by people working with
> seismic noise, for example. My introduction to the subject were the many
> plots and papers that describe the heroic effort LIGO goes through to
> measure tidal and seismic noise in order to keep their gravity wave servos
> locked. Short-term, the surface of the earth is a very noisy place. But if
> you can model or measure it before it hits the mirrors, you can mostly back
> it out.
>
> One can use PN and ADEV statistics on earth rotation, just like any other
> clock or oscillator. And it seems we can also compute Q for the earth, as
> if it were a mechanical oscillator.
>
> The remaining question in this thread is if earth Q measurement has actual
> meaning, that is, if the concept of Q is valid for a slowly decaying
> rotating object, as it is for a slowly decaying simple harmonic oscillator.
> And that's were get into the history and definition(s) and applicability of
> Q to non harmonic oscillators, such as coils, capacitors, atomic clocks,
> planets, pulsars, etc.
>
> /tvb
>
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```