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The Astrophysical Journal, 848:109 (9pp), 2017 October 20
© 2017. The American Astronomical Society. All rights reserved.
Multipoles and Force on External Points for a Two-layered Spheroidal
Liquid Mass Rotating Differentialy
Joel U. Cisneros-Parra
, Francisco J. Martinez-Herrera2, and J. Daniel Montalvo-Castro2
Facultad de Ciencias, UASLP, Zona Universitaria, San Luis Potosi, S,L,P, 78290, Mexico
Instituto de Fisica, UASLP, Zona Universitaria, San Luis Potosi, S,L,P, 78290, Mexico
Received 2017 May 24; revised 2017 August 25; accepted 2017 September 11; published 2017 October 20
We recently reported on a series of equilibrium figures for a self-gravitating heterogeneous liquid body, consisting
of two concentric distorted spheroids, “nucleus” and “atmosphere,” each endowed with its own internal motion of
differential rotation. In our current work, we calculate the body’s force at external points and obtain a multipolar
expansion of the potential. We also give an account of figures with prolate nuclei, which remained unnoticed by us
in our former paper.
Key words: gravitation – hydrodynamics – planets and satellites: general – stars: rotation
The homogeneous series is relatively easy to systematize;
though, the heterogeneous ones are not because they depend on
six independent parameters.
In the current paper, we calculate the body’s force at
external points and expand on the potential in multipoles; we
also report on figures with prolate nuclei, which remained
unnoticed to us in our former work. Because the heterogeneous series are multiparametric, a wide variety of
equilibrium figures can be obtained that, after establishing
their stability (in process), will supply a large body of data of
possible usefulness for those researchers who model, for
example, planets and stars. The current paper belongs to the
field of those like Kong et al. (2013) and Hubbard (2013), for
the study of rotating fluid planets, or that of Schubert et al.
(2011) on rotating two-layer Maclaurin spheroids for
modeling planets and satellites. Another related paper is that
of Marchenko (1979), who calculates quadrupoles to
compare them with those of planets.
Aside from practical applications, which are not now in our
line of work, the present paper is related to those of the cited
authors in the sense that our model is stratified in an “onionlike” fashion, and matter is incompressible. We do not follow
their mathematical methods because, for example, we do not
dispose of a starting exact analytical relation between angular
velocity and eccentricity e (or the parameter l = e 1 - e 2 ),
as Kong et al. (2013) have. In our approach, the (variable)
angular velocity is related numerically to model parameters
only. Hence, if we adjust the model to a particular instance, we
must proceed correspondingly (see Section 5).
1. Introduction
In a past paper (Montalvo et al. 1983), we obtained
equilibrium figures for a self-gravitating heterogeneous liquid
mass consisting of two confocal rotating spheroids, “nucleus”
and “atmosphere,” the nucleus being denser, flatter, and
rotating faster than the atmosphere:a Maclaurin-like spheroid
(the core) can host an envelope (the atmosphere); the body’s
relative density was denoted by e (=(rn - ra ) ra ). When the
rotation is effected with common angular velocity, and for
confocal geometry, neither spheroids nor ellipsoids stratified in
multi-layers (Lyttleton 1951; Chandrasekhar 1969) are possible
(Hamy’s theorem Hamy 1889). In a more general case of a
heterogeneous liquid built up as thin concentric ellipsoidal
shells, Chambat (1994) demonstrates that equilibrium is
Because of Hamy’s theorem, clearly, insofar as heterogeneous ellipsoids are concerned, we have to take distance
from a quadratic equation. On this account, we departed from a
distorted surface, vaguely recalling Jeans’ work (Jeans 1914)
on a Jacobi ellipsoid, with the difference that in ours the degree
of distortion is not necessarily slight.
To simplify matters, our study was appointed, first, to a
homogeneous distorted spheroid. It comes out in natural
fashion that the equilibrium of this mass is sustained by an
internal motion of differential vorticity (heretofore called
differential rotation); grounded on Bernoulli’s equation, a
series of figures was obtained, which, following Jeans, we call
“spheroidal”; none of our results refers to “ellipsoidal” figures.
Next, our interest was aimed at a heterogeneous mass
consisting of two concentric spheroidals, each endowed with
its own internal motion of differential rotation, from which a
more elaborated series resulted. In this last analysis, the
parameter ε was swept from moderate (»1) values, to high
(»25), the ensuing series consisting exclusively of oblate
figures for both nucleus and atmosphere; we refer to these
series as “oblate-oblate” (for nucleus-atmosphere). Furthermore, it is possible for there to be figures in which the nucleus
and atmosphere are proper spheroids (Maclaurin-like), or a
combination of spheroid-spheroidal and spheroidal-spheroid,
moreover, for the case in which the envelope is denser than the
core (i.e., for e < 0).
2. Theoretical Background
In this section, we recall the theoretical proceedings on
which our study is based. The surface equation of our mass is
x 2 + y2
= 1,
(1 )
which has cylindrical and tri-planar symmetry, and where
-1 4  d  ¥.3 We take e1 as the figure’s largest axis; e3 is
A negative d (>-1 4 ) makes a figure more bloated, relatively to the
spheroid, on the poles (zM > e3), while a positive d makes it more
flattened (zM < e3).
The Astrophysical Journal, 848:109 (9pp), 2017 October 20
Cisneros-Parra, Martinez-Herrera, & Montalvo-Castro
not the smallest one, but zM is
z M = e3
4d + 1 - 1
where f is an arbitrary function. In other words, the angular
velocity distribution has cylindrical symmetry. Substituting k of
Equation (13) into Bernoulli’s Equation (5), we get
(2 )
- f (R 2 ) - V (R 2 , z 2 ) +
In these papers, Bernoulli’s equation (Landau & Lifshitz 1987)
and the equation of continuity were used for obtaining steadystate spheroidal figures. The time-independent equations of
motion for a self-gravitating incompressible fluid are
(v · grad) v = grad V -
grad p ,
div v = 0,
div v = 0.
- f (R 2 ) - V (R 2 , z 2 ) = 0
(3 )
(4 )
w 2 = -2
(5 )
x 2 + y2
= 1,
(7 )
x 2 + y2 +
(8 )
(9 )
In a cylindrical coordinate’s system (R, j, z ), the velocity field
(tangent to circles) has a j-component alone:
v = w (0, R , 0).
Equation (6) will have two terms only:
⎛ ¶w
= w ⎜R 2 2 + w ⎟ ,
⎝ ¶R
= R2w 2 .
¶z 2
2 e dVN
w 2n = +
w 2a,
1 + e dr
w 2 = 2 f ¢ (R 2 ) ,
1 2 2
R w + f (R 2 ) ,
w 2a = - 2
From here we deduce that
k = R 2 f ¢ (R 2 ) + f (R 2 ) =
= 1,
with no geometrical restriction between the axes being
assumed. The figures’ axes are normalized relative to the
atmosphere’s greatest axis, so that e1 < 1. Since the atmosphere
is more extended than the nucleus, we must have zMn  zMa
(see Equation (2)). In Cisneros et al. (2016), we defined the
“contact figure” as that model with zMn = zMa , i.e., with the
poles of atmosphere and core being one on top of another.
As boundary conditions, we took (i) pa=0 on the envelope
surface, and (ii) pa=pn on the core surface, where, in addition,
the potential is assumed to be continuous. This all leads to the
knowledge of core’s and envelope’s angular velocities
(Cisneros et al. 2016):
Assuming, additionally, that the velocity field is symmetric
about the z-axis, we can express it as
w = w (x 2 + y 2 , z 2 ).
and the atmosphere, with equation
thus, the angular velocity must be a function of the kind
w = w (x 2 + y 2 , z ).
As described in Cisneros et al. (2016), our heterogeneous
model consists of two layers:4 the nucleus, with surface
equation (see Equation (1))
The continuity equation div v = 0 leads to
¶y 2
dR 2
w 2 = -2
Let us now suppose that each fluid point rotates around the zaxis with nonconstant angular velocity ω. Hence, the velocity
field is given by
v = w ( - y , x , 0 ).
f (R 2) = - V (R 2 , z 2) , (15)
which is the general angular velocity distribution law for any
axial-symmetric, incompressible, self-gravitating fluid with
p=0 on its surface.
Using the variable r (=x 2 + y 2 = R2 ), Equation (16) can be
written as
k being constant along the streamline. Generally, k changes
from one streamline to another, it being an overall constant
only when rot v = 0. The changes in k are dictated by
grad k = v ´ (rot v).
where z is a function of R for a figure with cylindrical
symmetry. Equation (15) allows us to determine the function f
(=-V ), and thus ω is established:
For a streamline (tangent to v), we get
1 2
v - V + p = k,
Since function f depends on R but not on z (i.e., it is constant on
cylinders), we can determine it using only the surface equation,
on which p=0:
where v is the velocity field, V is the potential, and p is the
pressure. These equations can also be written as
1 ⎞
grad ⎜ v2 - V + p ⎟ = v ´ (rot v) ,
r ⎠
p = 0.
This is a particular instance of the Bizyaev et al. (2015) multi-shell model of
stratified confocal spheroids, which was worked out recently. With the twoshell model, we intend to imitate rotating masses with the central part being
denser than the outer ones.
The Astrophysical Journal, 848:109 (9pp), 2017 October 20
Cisneros-Parra, Martinez-Herrera, & Montalvo-Castro
where VN = e Vnn + Vna and VA = e Van + Vaa are the total
potentials at points of nucleus and atmosphere surfaces; since ω
has cylindrical symmetry, it is known for all points on the
body. Vna is the potential in nucleus points (first index) due to
the atmosphere (second index), and so on. The body’s relative
density is
rn - ra
our model in the chosen direction (y = 0, z = 0 ) are given by
where ta , tn , and τ are volumes of nucleus, atmosphere, and the
whole body, respectively; Q and O are the quadrupole and
octupole moments. Strictly speaking, Q (and O) is a
combination of quadrupole moment components Qij for the
given direction. Indeed, taking into account the model
symmetry, we can show that cq for any direction ϑ, measured
from the rotation axis to the equator, is given by
cq = sin (J)((2Q11 - 5Q33) cos2 (J) - 3Q11 sin2 (J)) ,
for the FR force component, whereas for the Fz force
component we have
cq =
where V is the potential (in the unit Gra ) at the point (R, j, z ).
We will restrict our calculations, first, to points on the
equator plane alone, so that ¶V ¶z = 0 , i.e., the force is
radial and depends on R only. All of this, with the aim of
getting an idea regarding the force’s nature. The coordinate’s
origin is placed on the mass center, and the radius vector r is
thought to be on the xz-plane (since our figures have axial
According to the geometry normalization of our models,
the equator radius is R = r = 1, it being the maximal
dimension. Since V is not known analytically, the force can
only be obtained numerically. As an example, the force for
some models was calculated at the following fixed external
cos (J)(5 (Q11 - Q33) cos (2J) - 5Q11 - Q33) , (26)
Q11 =
Q33 =
(2 x 2 - y 2 - z 2 ) d t ,
(2 z 2 - x 2 - y 2 ) d t .
òt +t
òt +t
Generally, Equation (23) represents the force with reasonable precision (10−3), the c constants being interpreted as
moments of three multipoles in the origin that describe the
body’s force at external points in an equivalent form. However,
as the model flattening increases, cq gets smaller until it
vanishes and becomes negative (precision does not diminish
dramatically), although the Q sign does not change. In such
cases, we may take
z = 0.
+ 4 + o6
We wish to abbreviate these 13 results with a short approximate
formula inspired in a truncated series expansion of the exact
+ 4 + o6 .
Models such as ours can be used to understand the rotation
of isolated bodies, like detached stars or nebulae, or for
studying the interaction with matter located in the vicinity or
far away, in which case the force exerted by the body at
external points may be needed (supposing that matter does not
noticeably deform the model). Let us examine this possibility.
In cylindrical coordinates, the force is given by
(2 x 2 - y 2 - z 2 ) d t
3. Multipoles and Force at the Body’s External Points
r = 1.00, 1.25, 1.50, 1.75, 2.00, 2.25,¼,4.00,
(2 x 2 - y 2 - z 2 ) d t
(2 x 2 - y 2 - z 2 ) d t
tn 2
(8x 4 - 24x 2 ( y 2 + z 2) + 3 ( y 2 + z 2)2) dt
t 8
(8x 4 - 24x 2 ( y 2 + z 2) + 3 ( y 2 + z 2)2) dt ,
tn 8
The six independent parameters fixing the heterogeneous series
are ε; e1 (the relative size of nucleus and atmosphere in the
equatorial direction); d n, da; and en , ea (or, alternatively, zMn, zMa ).
⎛ ¶V
¶V ⎞
, 0,
⎝ ¶R
¶z ⎠
òt +t
as an approximate force, where Q is the fixed true moment
(24), and only two constants cm , co need to be established by a
fitting procedure. The precision increases somewhat, and the cq
sign change is avoided.
Certainly, at our disposal is also the exact infinite series
expansion of the potential at external points, that in cartesian
coordinates is written as (mass center at origin)
We fix constants cm , cq , co by the best fit of Equation (23) to the
obtained values. The first term in Equation (23) corresponds to
a force of a monopole on a unit mass, the second to that of a
quadrupole, and the last to that of an octupole. Certainly, these
multipoles do not necessarily agree exactly with those of our
model, since expansion (23) is approximate and not the exact
infinite series coming from the potential. The true moments for
+ 5
å å Qij xi xj
i=1 j=1
å å å å Qijkl xi xj xk xl + ....
i= 1 j= 1 k= 1 l= 1
The Astrophysical Journal, 848:109 (9pp), 2017 October 20
Cisneros-Parra, Martinez-Herrera, & Montalvo-Castro
M cos (J)
3 (5 cos (2J) - 1) cos (J) Q11
2r 4
5 (29 - 28 cos (2J) + 63 cos (4J)) cos (J) Q1111
24r 6
The Qʼs represent the multipoles of several orders. The used
coordinate system is uncommon in astrophysics, but
Equation (28) can be easily changed to any other system. For
example, in spherical coordinates, we use the known
transformations in Equation (28) from which relations between
Q and J can be established (see Equation (33)):
J2 = 2Q11,
J4 =
fz = -
Where total mass M is given by
Q1111 ,....
d z5 ⎞
M = 2p ⎜z Ma - Ma2 - a Ma
5ea4 ⎠
d z5 ⎞
+ 2pe12 ⎜z Mn - Mn2 - n Mn
⎟ e.
5en4 ⎠
In practice, we can use a somewhat less accurate approximation for the force (Equation (23)) by taking the first three
terms of a Taylor’s series, i.e., monopole, quadrupole, and
octupole, instead of using a computer program like ours. To
this end, one needs to establish Qij and Qijkl by integrals like
(24), without factors 3 and 5. Fortunately, the integrals can be
evaluated in a closed form, and the moment components are
expressed concisely by means of functions
q (e , e1, d , z) =
Core and envelope volumes, tn and ta , can be expressed also
in terms of a figure’s geometry:
d z5 ⎞
tn = 2pe12 ⎜z Mn - Mn2 - n Mn
5en4 ⎠
d z5 ⎞
ta = 2p ⎜z Ma - Ma2 - a Ma
⎟ - tn.
5ea4 ⎠
pe12 z
[35d 2e12 z 8
+ 90d (2e2 + e12 ) e2z 6
- 2d ) e12 +
210 (2e2 + e12 ) e6z 2] ,
63e 4z 4 ((1
4e 2 )
315e12 e8
4. Prolate-oblate Series
In one of our papers (Cisneros et al. 2015), a series of
homogeneous oblate spheroidal figures were obtained, the
prolate form not being allowed.5 In a subsequent paper
(Cisneros et al. 2016), we obtained a multiparametric series
for a heterogeneous mass consisting of two layers. Heretofore,
the oblate shape was present in all of our figures, which lead us
to believe that this form was actually a requisite for equilibrium
to hold. However, a detailed examination of the low ε series,
disclosed that they encompass brief regions where the nucleus
is prolate (no prolate atmospheres were found). As usual, the
series begins with figures having an oblate nucleus, which
becomes prolate for values greater than a certain en, and so for
the rest of the series, we refer to this last portion as the “prolateoblate” series. In general, the series is continuous, but some of
them can have a brief region of forbidden figures. We now
discuss the prolate-oblate series, distinguishing two cases:
e > 0 and -1 < e < 0 (i.e., rn < ra ), thereby avoiding
negative densities.
pe12 z
o (e, e1, d , z ) =
[3465d 3e14 z12
+ 12285d 2 (4e 2 + e12 ) e12 e 2z10
+ 5005de 4z 8 (3 (1 - d ) e14
+ 8e 4 + 24e12 e 2) - 9009e 8z 4 (3 (1 - d ) e14
+ 8e 4 + 24e12 e 2) + 6435e 6z 6 (12 (1 - 2d ) e12 e 2
+ (1 - 6d ) e14 + 8e 4) - 45045e14 e12
+ 45045 (4e 2 + e12 ) e12 e10z 2].
Indeed, Q11 and Q1111 depend on the equilibrium model
parameters through relations
Q11 = q (ea , 1, da, z Ma) + eq (en , e1, dn, z Mn) ,
Q1111 = o (ea , 1, da, z Ma) + eo (en , e1, dn, z Mn).
One can show that the other nonzero components are given by
Q33 = - 2Q11,
Q1133 = - 8Q1111,
4.1. Case e > 0
Q3333 = Q1111. (33)
In this section, various prolate-oblate series for e > 0 are
established, along with their various limits.
Hence, the approximate force components fR , fz for any
direction ϑ taking up to octupoles is given by
Cassini (Cassini 1718; Todhunter 1873) believed, based on Descartes’
vortex theory, that the Earth’s shape was that of an elongated (prolate)
spheroid, it being confirmed by his measures of one-degree arc length at
different places of an Earth meridian; however, his measures had errors, so that
Newton’s conclusions, inspired by Cassini’s observations of the Jupiter’s
oblateness, and grounded on gravitational theory had to be accepted:the
rotating Earth necessarily was an oblate spheroid. At that time (Hooke–)
Newton gravitational theory was not universally accepted, hence the truth
about Earth’s shape models built with vortex or gravitational theories was
decided by observation (through the famous Peru and arctic expeditions).
M sin (J)
3 sin (J)(3 + 5 cos (2J)) Q11
2r 4
5 sin (J)(15 + 28 cos (2J) + 21 cos (4J)) Q1111
8r 6
fR = -
The Astrophysical Journal, 848:109 (9pp), 2017 October 20
Cisneros-Parra, Martinez-Herrera, & Montalvo-Castro
Table 1
Limits for Series with Prolate Nuclei, for Increasing ε
First Prolate
Last Prolate
Note. e1 = 0.5, dn = da = -1 8.
Table 2
Some Members of the Prolate Oblate Series
for e = 0.1, e1 = 1 2, dn = -1 8, da = -1 8
Figure 1. Final model with prolate nucleus.
Table 3
Limits for Prolate-oblate Series, for Increasing ε
The lowest en value for the occurrence of the prolate
(zMn = e1) form is given by the relation
en =
4d n + 1 - 1
2d n
Prolate Begin
Series Ceiling
0.5274 = ea
0.5274 = ea
0.5274 = ea
Last Prolate
0.8048 = ea
0.5908 = ea
0.5307, ea=0.8010
0.5274, ea=0.8348
Note. e1 = 0.5, dn = da = 1 8.
Going on with the series taken as basic in our paper
(Cisneros et al. 2015), namely, for e1 = 1 2 (so that the
atmosphere is twice as large as the nucleus), and d n = -1 8,
we have that the core is prolate from en=0.4619 upward. If
we choose da = -1 8 and the same en, then the series also
starts at ea=0.4619 (the floor) and ends at a specific ea value
(the ceiling); further series can be obtained increasing en.
Repeating this procedure for higher ε values, we come to emax
(=0.5286, for this instance), beyond which the prolate form no
longer occurs. In this way, the complete set of series for
these particular parameters are obtained. The corresponding
limits are given in Table 1, whose rearmost row is reserved for
emax . This frontier series begins with a contact figure
(en = ea = 0.4619), has a certain ceiling (in this case, =en ),
and a final one-member series, which is a contact model
(en = ea = 0.4619); in other words, the case emax = 0.5286
consists of only one figure, it being a contact model. For lower
ε values, the series are more populated. For example, for
e = 0.1, there is a continuum of series, with the initial series
having a “first” prolate nucleus (en=0.4619), which is in
contact with its envelope, and with its ceiling at ea=0.7356.
The ending (contact) prolate figure occurs at en = ea = 0.6328
or zMn = zMa = 0.6850 . For the same ε, taking higher en values
(cores become more prominently prolate), new series come out,
each beginning with a stronger prolate nucleus in contact with
its atmosphere, and ending with a figure having the same core
and an extended envelope. For even higher en, the series
become narrower, i.e., their ceilings lower, until ea equals en;
that is, the final en series consists of an isolated contact model
with a maximal prolate nucleus (en=0.6328, or “last
prolate”). In Table 1, we also see that the lower ε is, the last
figure has a nucleus with the prolate shape at its top. Table 2 is
built for the series corresponding to en=0.5500.
Series for e > 0 begin normally with a contact figure
(zMn = zMa ), and end with a figure having an identical nucleus
but an extended envelope. The set finishes with a one-member
series (Figure 1) that can be, either a contact figure, or a
Table 4
Series Limits with Prolate Nucleus for Increasing ε
Prolate Begin
Series Ceiling
Last Prolate
en ea
0.5274, ea = 0.4619
0.5274, ea = 0.4619
Note. e1 = 0.5, dn = 1 8, da = -1 8.
Table 5
Series Limits with Prolate Nucleus for Increasing ε
Prolate Begin
Series Ceiling
Last Prolate
en ea
0.4619, ea = 0.5274
0.4619, ea = 0.5274
0.4619, ea = 0.5274
Note. e1 = 0.5, dn = -1 8, da = 1 8.
noncontact one, depending on d n, da . Tables 1–5 comprise
limits for some prolate-oblate series.
Table 3 is worked out for d n = da = 1 8, and the results are
somewhat similar to those of Table 1:series with a prolate
nucleus are possible for e  0.5225; there is a series
continuum, each beginning with a contact model and a higher
ceiling than in the former case. Moreover, for a given ε, the
ending one-member series with prolate core is not always a
contact figure:at about e  0.5, the last prolate nucleus has an
extended envelope.
The Astrophysical Journal, 848:109 (9pp), 2017 October 20
Cisneros-Parra, Martinez-Herrera, & Montalvo-Castro
Table 6
e1 Against emax for Prolate Figures
d n = da = - 1 8
d n = da = 1 8
Now, taking opposite signs for dn and da, we build Tables 4
and 5. This time, a contact figure (zMn = zMa ) has en ¹ ea . For
example, for the -+ case, the contact figure from the start of a
prolate region has en = 0.4619, ea = 0.5274, with emax =
0.6525; for the +- case, we have en = 0.5274, ea = 0.4619,
and emax = 0.3913. Again, Tables 4 and 5 simply contain limit
parameters. For a given ε value (emax ), the initial series has
what we call an oblate-prolate nucleus (zMn = e1), and a
specific ceiling, the last series consisting of only one member.
For example, in the -+ case, with e = 0.1, the first series has
its ceiling at ea=0.7759, and the terminal one has only the
member en = 0.6300, ea = 0.6481, with a well-formed prolate
nucleus and an extended atmosphere; between these two
extremes, there is a continuum of en series, a particular occurrence
of which is the en=0.6 series, with a certain ea ceiling.
The outcomes of Table 5 may be described in a similar way
as those of Table 4, with the difference that last one-member
series for emax is a contact model here.
Tables 1–5 are built supposing e1=0.5. However, taking
other e1 values, we arrive at different emax figures. Working out
the d n = da = -1 8 and d n = da = 1 8 possibilities only, the
results of Table 6 describe a relation between emax and e1.
We see in Table 6 that, for the−−case, upper limit for
obtaining prolate cores is at e1=0.845, which corresponds to the
homogeneous model. As e1 decreases, emax increases first to a
maximum value at about e1=0.6, diminishing then until it
remains constant from e1 » 0.23 on. In the emax » const. range,
we found that the limiting one-member series is not a contact
model (as in Tables 3 and 4), whereas outside this range it is
(Tables 1 and 5). Instance ++ behaves qualitatively similar:there
is an upper e1 limit at 0.9196, a emax maximum at about 0.7, and,
for low e1, emax has a constant value of approximately 0.529.
Depending on the values of e1, dn, and da, we can have a prolate
nucleus even for emax > 1; for example, if e1=0.75, d n =
da = -0.01, we have emax = 2.4.
We may define the degree σ of prolateness of the nucleus by
z - e1
s = Mn
Figure 2. To compute the pressure on the nucleus surface, we use the
atmosphere polar caps with a circular base at the top and bottom of the inside
surface (Cisneros et al. 2015, Equation (44)). Here, we call pan the
atmosphere pressure at nucleus points, which is given by
= VA + fa (R 2) = VN - VA (R 2) ,
since V is continuous on the nucleus surface and fa = -VA; we
remark that currently the rotation axis distance R refers to core
points, even in fa (and in wa ) function. For establishing wa
(=-fa¢), which has cylindrical symmetry, we need only to take
points on the surface of the atmosphere (as was done in paper
Cisneros et al. 2015); hence, we build now VA (R2 ) at envelope
surface points with R„e1 alone, e1 being the nucleus equator
* and Vaa
* the core and envelope
radius (see Figure 2). Call Vna
potentials at envelope surface points above (below) the circle’s
surface R2 = x 2 + y 2 = e12 (polar cap of Figure 2). Consequently, the pressure at nucleus points is given by
* + Vaa
* ),
= eVnn + Van - (eVna
where V is the potential of one body’s part (first subindex) on
the other one (second subindex).
which measures the relative difference between the rotation
axis and the equator radius. For example, if e = 0.01, e1=0.5,
d n = da = 0.001 (en » 0.97), we have
4.3. Case e < 0
From the definition of ε, we conclude that rn > ra if e > 0 , a
common case in stars where central parts are denser than outer
ones. On the other hand, for negative ε, we have e = 0 and
e = -1 as limits. In the first case rn = ra , and so the whole
body is homogeneous. The second instance means that rn = 0,
i.e., the core is empty. This last case (e < 0 ) accepts also
equilibrium configurations, i.e., it is physically possible. Here,
however, we do not ask about stability (Goldreich–Schubert–
Fricke criterion, for example) of equilibrium, and take the case
as a curiosity for the time being. A stability study is left for a
forthcoming paper.
0.97 - 0.5
= 0.94.
4.2. Pressure on Nucleus Surface
For a better understanding of the proceedings of Section 4.3,
we now determine the pressure at core’s surface points.
One of the boundary conditions used in Cisneros et al. (2015)
was the equality of envelope and core pressures on the core
The Astrophysical Journal, 848:109 (9pp), 2017 October 20
Cisneros-Parra, Martinez-Herrera, & Montalvo-Castro
Table 7
en , ea for the Last Series Figure, as ε Decreases
w 2n
Note. w 2n and pan approximate variation ranges are given. dn = da =
-1 8, e1 = 1 2 .
Figure 3. Radial force FR on the equator plane as a function of distance R from
the center. R=1 corresponds to a surface point. The force due to the spherical
mass (in red) is also given.
Table 8
Equilibrium Figures, as ε Decreases
4.4. Spheroids with Prolate Nucleus
w 2n
Although the current paper was supposed to deal only with
spheroidal heterogeneous figures, we will, nevertheless, pay
attention to the special case when d n = da = 0, i.e., when the
nucleus and the atmosphere are properly spheroids. Still, this
can be considered a generalization of our 1983 model, since we
are abandoning the confocal restriction and allow any
geometrical configuration for core and envelope, including
similarity of surfaces. Here, the angular velocities need not
necessarily to be constant; instead, they obey the general
relation w = w (R2 ), that is, the models, in general, have
differential rotation; we will be interested only in figures with a
prolate nucleus. In Sections 4.1 and 4.3, we determined the
prolate core region by establishing its beginning with that
figure for which zMn = e1, that is, rotation axis equates to
equator radius (so that the nucleus is just beginning to prolate).
For spheroids, this means that the nucleus is a sphere. In
general, the sphere will rotate differentially, retaining its shape
aided by the atmosphere’s pressure and the gravitational and
centripetal forces (having no spherical symmetry). Not
surprisingly, we find models. For given e1 and emax values,
the prolate region begins again with a spherical core. This
model is isolated (a one-member series) and can be a contact
figure, or else can have an extended atmosphere, in much the
same manner as it occurred in Section 4.1. Taking a particular ε
(<emax ), we obtain series (labeled by en > e1) with prolate core.
As en increases, the series become narrower, finally reducing to
a one-member figure with the prolate feature at its highest:it is
what we call “last prolate.” Even lower ε values (but same e1)
lead to richer series sets, ending at a model whose nucleus has
the prolate shape at its highest (last prolate). For example, for
e1=0.5 we have emax = 0.8357, and there is only one model
with en = 0.5, ea » 1, i.e., a whole, nonrotating sphere. Taking
now e = 0.4 < 0.8357, say, and en > 0.5, one gets a
continuous series set. Selecting some series of the set, say,
that with the prolate nucleus en=0.54, one finds that it begins
at ea=0.54 (contact figure), and ends at ea=0.7294, i.e., all
models have the same prolate core and envelopes, which
extend more and more, still remaining spheroids. For the
extreme case in which e1=0.95, i.e., when the atmosphere
is 5% greater than the nucleus, we have emax = 1500 , but
the corresponding series consists of only one figure,
with zMn = 0.95.
Note. w 2n and pan approximate variation ranges are given. en = 0.2,
ea = 0.3, dn = da = -1 8, e1 = 1 2 .
Series are again obtained from e < 0 to e > -1. Frequently,
the figures have oblate cores, but sometimes the nucleus can be
prolate. To see this, we worked out ε values from −0.5 to
−0.99, for the fixed parameters d n = da = -1 8, e1 = 1 2.
Series beginning with a contact figure are found here too (for
e > -0.3636 ). They end up with a one-member series that is
not a contact figure; prolate cores are possible for about
e > -0.9. In Table 7, we exhibit the characteristics of these
limiting figures, namely e, en , ea and approximate variation
ranges for w 2n and pan distributions.
One sees from Table 7 that prolate cores (en > 0.4619)
appear for about e > -0.9. Moreover, as ε approaches −1, the
angular velocity grows rapidly (to ¥, Cisneros et al. 2015
Equation (45)) to support the nucleus’ surface pressure
distribution that remains practically fixed (changing from
0.49 to 0.66). That is, as e  -1, the nucleus becomes
slighter and slighter and rotates quicker and quicker; in the
limit e = -1, we have a hole (a nothing) rotating with infinite
angular velocity, a physically unacceptable situation. We can
approach the limit without actually reaching it.
For a better comparison and as a reinforcement of the
above results, we take a constant geometry en = 0.2, ea =
0.3, d n = da = -1 8 and let ε vary from −0.6 to −0.999,
getting thus Table 8.
According to Table 8, the pressure distribution in the core
remains practically the same for all ε values. Once more, for
lower rn , the angular velocity grows rapidly (approaching ¥);
this is necessary for sustaining the approximate invariable
atmospheric pressure maintaining the body’s equilibrium.
The Astrophysical Journal, 848:109 (9pp), 2017 October 20
Cisneros-Parra, Martinez-Herrera, & Montalvo-Castro
5. Discussion
The terms “nucleus” and “atmosphere” used here to
designate the two parts of our mass should not be interpreted
literally, since we do not pretend to exclusively model a star or
planet, but rather a generic body whose central part can be
clearly distinguishable from the external one, for instance, a
cloud. We have devoted an important portion of the current
article to discuss the figures with prolate nuclei since, in the
absence of a secondary mass, they have not been reported
previously. As for the analysis of the force, it is practically
indistinct whether the nucleus is oblate or prolate. In an attempt
to investigate if the nucleus shape has a noticeable impact
on the external force, we established it at several points for
models of the series of spheroids (d n = da = 0) with
e = 0.3, ea = 0.8, e1 = 0.5, varying en from 0.2 (oblate) to
0.55 (prolate). The differences were negligible:the force when
the nucleus is prolate is but slightly higher than when it is
oblate. A typical aspect of the force is shown in Figure 3; for
comparison, the force from a spherical body of the same mass
is also given. The model’s force is stronger, especially near the
surface, than that of the spherical shape:for R=1, they are
−4.94 and −4.45, respectively.
The e < 0 case, core slighter than the envelope, is stunning
but nonetheless, a physical possibility, because it satisfies the
basic equilibrium equations. Its stability will be considered
later. Still, there is no reason to reject it in the case of being
unstable, since it might represent the initial stages of mass
transfer from regions of low to high density, as it apparently
occurs in the Crab Nebula.
Although this is no place to discuss practical applications, we
will outline a proceeding for dealing with this point. We have
already pointed out that it is not possible to follow the selfconsistent technics of D. Kong or W. Hubbard, since they start
with an exact relation between geometry and angular velocity
(Maclaurin spheroid), improving then the level surface for
describing the planet. We could proceed in a similar fashion with
our homogeneous model—yet on numerical grounds—employing
a fixed level surface (Equation (1)). As an input, we would have
the axes relation zM. Distortion parameter d and angular velocity
distribution will then be predicted using a best-fit procedure of our
numerical established field to the observed one. The best fit will
be laborious, since for each model, we must determine the
potential at several inner and outer points. The weakness of using
a fixed equation type against a self-consistent level surface is
somewhat compensated by adjusting the distortion parameter d
(that is, a measure of the deviation from the spheroid). Since the
Hubbard level surface does not differ very much from the
spheroid, we expect that our prediction of the differential rotation
would not be too far from reality.
Regarding our two-layer model, the level surface shapes are
again fixed by Equations (18) and (19), though distortion
parameters dn and da are not; the symbolic form is not modified.
In treating to find a model for a particular body, we could proceed
in a similar way as above. Still, the procedure would be even more
arduous, since only one input parameter is at our disposal:axes
relation zMa. The best fit will give the five independent parameters
zMn, d n, da, e1, and ε, that provide information on the inner
structure of the mass (excluding da). zMn, e1 describe the relative
nucleus size, dn its distortion, and ε matter concentration.
However, we would probably be confronted not by one fit, but
a number of them; we cannot anticipate it. Our models have
versatility. For example, there is no fixed relation between the
Figure 4. Angular velocity distribution of series d=4. Each one is labeled
according to the e3-value.
Figure 5. Mean angular velocity of series d = 4, 1, 0.125, 0 in dependence of
rotation semiaxes e3.
semiaxes of nucleus and atmosphere, as it happens when
similarity of surfaces is demanded (which, by the way, is
contained in our general solution), thus opening a wide variety of
geometrical configurations.
We have outlined a proceeding on application matters, but
the concrete method might differ somewhat.
Surface Equation (1) is our proposal for treating to
understand rotating celestial bodies. It is a model, and like
any model, it represents reality approximately. How good this
approximation is, we cannot say yet. We presume that this
shape can be approximately adjusted to the real one of a body,
for example, but not exclusively, a planet. The reason for
adopting this geometry is for making one step further from
Maclaurin figures (occasionally used in astrophysical applications, and retrieved when d = 0), and generalize the results.
However, if such were not the case, a new and important fact
merges:differential rotation is a property of any axialsymmetric configuration, not only of our model (1), a matter
not known within the context of Maclaurin spheroids. Clearly,
a fixed geometry, not present on the work of the cited authors,
can be a limitation.
A differential rotation profile is the result of a given
geometry (e1, e3, d ). Conversely, is there a geometry that
reproduces a certain profile? In general, there probably is not.
Nonetheless, there is at our disposal a huge number of profiles
for a wide variety of geometries. For example, in one series
(e3 = 0.1, ¼, 0.6, d = 4), the angular velocity distributions in
Figure 4 are obtained. If the given profile is of that kind, then it
is possible to find the corresponding geometry. This refers to
the homogeneous model. For the two-shell model, we get a
richer variety of angular velocity profiles, a sample of which is
depicted in Figure 2 of Cisneros et al. (2016). Depending on
the densities and the geometry, cases are obtained with angular
The Astrophysical Journal, 848:109 (9pp), 2017 October 20
Cisneros-Parra, Martinez-Herrera, & Montalvo-Castro
velocity profiles ranging from almost constant to rapidly
variable. There are also instances in which angular velocity
increases in a certain r-range and decreases in another one.
From Figure 4, one sees that one part of the series has nearly
constant angular velocity. In the other, it clearly varies. The series
members have high distortion (d = 4). However, when d is
moderate or small (d » 1 8), more members have, in better
approximation, constant angular velocities. In Figure 5 is plotted
the mean angular velocity for the series d = 4, 1, 1 8, 0, last
corresponding to the Maclaurin spheroid. Clearly, series d = 1 8
approaches Maclaurin’s better than the others. We believe that our
model could be applied using constant (mean) angular velocity, if
the distortion is moderate, without affecting the results dramatically. In this case, the Ω variability complicates the numerical
procedures. Since, given a geometry and a mass distribution, the
gravitational field is completely known (see Equations (29)–(37)).
Obviously, our model is not general enough to describe exactly
real flows in celestial bodies, since it does not take into account
more refined magneto-hydrodynamic, thermodynamical, and so
on, processes. It is a modest one; however, it can be improved.
Besides these drawbacks, the fluid dynamics is physically
acceptable since the basic equations are fulfilled, for the given
conditions. We cannot claim that the model will explain any
known physical process taking place in astrophysical objects, not
even a particular one, since it has not yet been applied to some
specific cases (this will be done in a coming paper).
Joel U. Cisneros-Parra
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