GG304 Lecture 4
1
LECTURE 4: TIDES AND ROTATION
The ocean surface is an equipotential surface that changes in elevation due to
the changing relative positions of the Earth, Sun, and Moon, creating the ocean
tides. The solid earth also deforms in response to changes in gravitational
potential, creating the earth tides.
The center of mass of the Earth-Moon system is the barycenter.
D
The distance d of the barycenter from the
center of the earth is:
ML
d=
rL = 4600 km
ME + ML
A
Barycenter
C
B
rL
d
ω
As the moon revolves around the Earth (about once per month), the Earth’s
€
center revolves around barycenter. Similarly, each point within the Earth traces
a circle of radius d.
At the center of the earth (point C), the centrifugal acceleration balances the
M
gravitational acceleration toward the moon:
aC = G 2L
rL
At the point closest to the Moon (point B), the centrifugal acceleration is the
same, but the gravitational pull toward the
€ moon is greater. The difference is:
2
"
%
"
%
"R %
1
1'
M L $ RE
E
$
aB − aC = GM L
−
=G 2 2
+ 3 $ ' + ...' directed toward the moon
$ r −R 2 r 2 '
$
'
r
r
# rL &
L &
L #
L
#( L
E)
&
On the point farthest from the Moon (pt. A), the centrifugal acceleration is the
same but the gravitational pull toward the moon is greater. The difference is:
2
"
%
"
%
"R %
1
1
M
R
L
E
E
$2
' away from the moon
' =G
aC − aA = GM L $ 2 −
−
3
+
...
$
'
2'
2 $
$r
'
r
r
r
#
&
r
+
R
L
L
L
L
(
)
#
&
L
E
#
&
On the ring of points 90° from the Earth-Moon line (e.g., point D), the tidal
acceleration is directed toward the center of the Earth.
Clint Conrad
4-1
University of Hawaii
GG304 Lecture 4
2
These changes in the gravitational acceleration elevate the equipotential surface
at A and B and decrease it at D. The magnitude of
ocean tide is several feet (and depends on
the seafloor shape). The magnitude of the
earth tide is several to tens of cm.
At the equator, there are two tides per day
To
Moon
(semi-diurnal tides). At high latitudes, there
is only one tide per day (diurnal tides), and
in between one tide is higher than the other
(diurnal inequality). The sun also produces a
tide that is up to 45% that of the moon.
The solar and lunar tides reinforce each other (full and new moons) during a
spring tide and they oppose each other for a neap tide.
Tidal variations produces gravity variations of ~0.3 mgal (larger at high tide).
The force holding a mass m on a (any) moon’s surface is:
FG = G
mML
RL2
When this force is balanced by the tidal force:
FT = G
mME " RL %
$2 '
rL2 # rL &
€
then the mass can become detached from the moon’s surface. This distance is
" M %1/ 3
" ρ %1/ 3
E
called the Roche limit:
rL = $ 2
' €RL = 1.26RL $ E '
M
#
# ρL &
L &
The tidal bulge lags the earth-moon€line by about 2.9°, or about 12 minutes.
The gravitational pull of the moon on this bulge increases the length of day by
about 2.4ms per 100 years.
Clint Conrad
4-2
University of Hawaii
GG304 Lecture 4
3
Conservation of angular momentum gives:
% M M (
2
IEωE + MEd 2ωL + ML rL − d ωL + ILωL ≈ IEωE + ' E L *rL2ωL = constant
& ME + ML )
(
)
The gravitational attraction of the moon is balanced by its centretpetal
"M %
" M
%
E
G$ 2E ' = ωL2 rL − d = ωL2rL $
'
€ acceleration:
# ME + ML &
# rL &
(
Combining gives: IEωE +
ME ML
ME + ML
)
GrL = constant
€
Thus, a decrease in the angular momentum of the Earth is compensated by an
increase in €the the earth-moon distance. The moon is currently moving away
from the Earth at about 3.7 cm/yr.
Euler Nutation and Chandler Wobble – polar motions about the rotation
axis due to differences between moments of inertia about the axes of the earth.
I
Euler Predicted: Teuler = A Tday = 305 days
IC − IA
Chandler Observed: Tchandler = 435 days (longer due to elastic yielding)
€
Precession and Nutation
are displacements of the rotation axis in space.
€
They are due to torques on the Earth caused by tidal attraction.
Retrograde precession of the 23.5° spin axis: Tprecession = 25,700 years
Nutations:
Tsolar = 0.5 years
Tlunar = 18.6 years
€
Milankovitch
induced by orbit/rotation changes:
€ cycles are climate variations
€
Orbital precession (21 kyr period)
Obliquity varies (21.9° - 24.3°) due to interactions with other planets (41 kyr)
Orbit eccentricity varies (e=0.005 to 0.058) (400 kyr and 100 kyr periods)
Clint Conrad
4-3
University of Hawaii