bracketleftBigg
F
(
γ,α
)
−
E
(
γ,α
)
tan
β
sin
β
tan
γ
+
cos
β
E
(
γ,α
)
tan
3
γ
cos
2
α
−
cos
2
β
tan
2
γ
cos
2
α
bracketrightBigg
,
(3.140)
respectively, where
E
(
γ,α
)
=
integraldisplay
γ
0
(1
−
sin
2
α
sin
2
θ
)
1
/
2
d
θ,
(3.141)
F
(
γ,α
)
=
integraldisplay
γ
0
d
θ
(1
−
sin
2
α
sin
2
θ
)
1
/
2
,
(3.142)
are special functions known as
incomplete elliptic integrals
.
4
The integral
α
0
, defined in (3.107), transforms to
α
0
=
2 (cos
β
cos
γ
)
1
/
3
a
0
sin
γ
F
(
γ,α
)
.
(3.143)
Finally, making use of some of the analysis in the previous two sections, the normalized angular momentum, and
normalized mechanical energy, of the planet can be written
hatwide
L
=
=
√
6
10
1
+
cos
2
β
(cos
β
cos
γ
)
2
/
3
hatwide
ω,
(3.144)
hatwide
E
=
−
3
5
(cos
β
cos
γ
)
1
/
3
sin
γ
F
(
γ,α
)
+
3
20
1
+
cos
2
β
(cos
β
cos
γ
)
2
/
3
hatwide
ω
2
,
(3.145)
respectively.
Now, the constraint (3.139) is obviously satisfied in the limit
β
→
0, since this implies that
α
→
0 and
E
(
γ,α
),
F
(
γ,α
)
→
γ
. Of course, this limit corresponds to the axisymmetric Maclaurin spheroids discussed in the previous sec
tion. Jacobi, in 1834, was the first researcher to obtain the very surprising result that (3.139) also has nonaxisymmetric
ellipsoidal solutions characterized by
β >
0. These solutions are known as the
Jacobi ellipsoids
in his honor. The
properties of the Jacobi ellipsoids, as determined from Equations (3.139), (3.140), (3.144), and (3.145), are set out in
Table 3.2, and illustrated in Figures 3.5 and 3.6. It can be seen that the sequence of Jacobi ellipsoids
bifurcates
from the
sequence of Maclaurin spheroids when
e
13
=
0
.
81267. Moreover, there are no Jacobi ellipsoids with
e
13
<
0
.
81267.
However, as
e
13
increases above this critical value, the eccentricity,
e
12
, of the Jacobi ellipsoids in the
x
1

x
2
plane
grows rapidly, approaching unity as
e
13
approaches unity. Thus, in the limit
e
13
→
1, in which a Maclaurin spheroid
collapses to a
disk
in the
x
1

x
2
plane, a Jacobi ellipsoid collapses to a
line
running along the
x
1
axis. Note, from
Figures 3.5 and 3.6, that, at fixed
e
13
, the Jacobi ellipsoids have
lower
angular velocity and angular momentum than
Maclaurin spheroids (with the same mass and volume). Furthermore, as is the case for a Maclaurin spheroid, there is a
maximum angular velocity that a Jacobi ellipsoid can have (
i.e.
,
hatwide
ω
=
0
.
43257), but no maximum angular momentum.
Figure 3.7 shows the mechanical energy of the Maclaurin spheroids and Jacobi ellipsoids plotted as a function of
their angular momentum. It can be seen that the Jacobi ellipsoid with a given angular momentum has a
lower
energy
3
See
On Jacobi’s Figure of Equilibrium for a Rotating Mass of Fluid
, G.H. Darwin, Proc. Roy. Soc. London
41
, 319 (1886).
4
See
Handbook of Mathematical Functions
, M. Abramowitz, and I.A. Stegun (Dover, New York NY, 1965).
Hydrostatics
51
e
12
e
13
hatwide
ω
hatwide
L
−
hatwide
E
e
12
e
13
hatwide
ω
hatwide
L
−
hatwide
E
0.00
0.81267
0.43257
0.30375
0.50452
0.60
0.85585
0.42827
0.30984
0.50138
0.05
0.81293
0.43257
0.30375
0.50459
0.65
0.86480
0.42609
0.31296
0.49975
0.10
0.81372
0.43257
0.30375
0.50459
0.70
0.87510
0.42288
0.31760
0.49734
0.15
0.81504
0.43256
0.30377
0.50458
0.75
0.88705
0.41807
0.32462
0.49372
0.20
0.81691
0.43253
0.30380
0.50457
0.80
0.90102
0.41069
0.33562
0.48814
0.25
0.81934
0.43248
0.30388
0.50453
0.85
0.91761
0.39879
0.35390
0.47908
0.30
0.82237
0.43237
0.30402
0.50445
0.90
0.93778
0.37787
0.38783
0.46295
0.35
0.82603
0.43220
0.30427
0.50432
0.95
0.96340
0.33353
0.46860
0.42782
0.40
0.83037
0.43191
0.30468
0.50410
0.96
0.96950
0.31776
0.50078
0.41499
0.45
0.83544
0.43146
0.30532
0.50376
0.97
0.97605
0.29691
0.54672
0.39771
0.50
0.84131
0.43078
0.30628
0.50326
0.98
0.98317
0.26722
0.62003
0.37241
0.55
0.84808
0.42976
0.30772
0.50250
0.99
0.99101
0.21809
0.76872
0.32842
Table 3.2:
Properties of the Jacobi ellipsoids.
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 Fluid Dynamics, Fluid Mechanics, stress tensor, Fluid Motion