Geology Reference
In-Depth Information
Hence,
dE
dt
=−
2
τ
E
,
(4.67)
and
dE
dt
T
0
2
τ
Δ
E
=
=−
E
/
f
0
,
(4.68)
where
T
0
1/
f
0
is the period of the motion and
f
0
is the resonant fre-
quency. Thus, the quality factor is
=
2π/ω
0
=
Q
=
π
f
0
τ,
(4.69)
and
i
Q
f
0
i
Q
f
0
.
σ
0
=
ω
0
+
=
2π
f
0
+
(4.70)
Now consider forced motion. The right-hand side of the polar motion equation
(4.58) no longer vanishes. If the excitation is of unit amplitude and angular fre-
quency ω, the right-hand side of equation (4.58) becomes
e
i
ω
t
. This will excite a
motion
re
i
ω
t
with
1
=
r
i
/
Q
)
.
(4.71)
i
ω
−
i
π
f
0
(2
+
The response at frequency
f
=
ω/2π is then
Q
r
=
π
f
0
1
f
0
)/
f
0
.
(4.72)
+
i
2
Q
(
f
−
The power of the response, such as might be observed in a spectral density estimate,
is then found to be
a
2
2
|
r
|
=
f
0
)/
f
0
2
,
(4.73)
4
Q
2
(
f
1
+
−
with
a
2
2
f
0
. This is just the form (4.15) fitted to the maximum entropy
spectrum of the VLBI polar motion path.
=
Q
2
/π
4.3.3 Relating theory to observations
A number of fundamental physical constants enter the foregoing theoretical discus-
sion. Among these are the precessional constant,
H
10
−
3
,
=
(
C
−
A
)/
C
=
3.27379
×
kgm
2
10
37
and the equatorial moment of inertia,
A
=
8.0100
×
(Stacey, 1992,
10
−
11
m
3
p. 409). As well, the Newtonian constant of gravitation
G
=
6.67428
×
kg
−
1
s
−
2
(Mohr
et al
., 2008, pp. 688-691), the mean rotation rate of the Earth
10
−
5
rad s
−
1
Ω=
7.292115
×
(Hofmann-Wellenhof and Moritz, 2006, p. 88), and
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