The Astrophysical Journal, 848:109 (9pp), 2017 October 20 https://doi.org/10.3847/1538-4357/aa8d20 © 2017. The American Astronomical Society. All rights reserved. Multipoles and Force on External Points for a Two-layered Spheroidal Liquid Mass Rotating Differentialy 1 Joel U. Cisneros-Parra 1 , Francisco J. Martinez-Herrera2, and J. Daniel Montalvo-Castro2 Facultad de Ciencias, UASLP, Zona Universitaria, San Luis Potosi, S,L,P, 78290, Mexico 2 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 Abstract 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 impossible. 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 z2 z4 + + d = 1, e12 e32 e34 (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 3 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). 1 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 . 2d 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 - 1 grad p , r div v = 0, div v = 0. - f (R 2 ) - V (R 2 , z 2 ) = 0 (3 ) (4 ) w 2 = -2 (5 ) (6) x 2 + y2 z2 z4 + + d = 1, n e12 en2 en4 (7 ) x 2 + y2 + (8 ) (9 ) (10) 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). (11) Equation (6) will have two terms only: ⎛ ¶w ⎞ ¶k = w ⎜R 2 2 + w ⎟ , 2 ⎝ ¶R ⎠ ¶R ¶k ¶w = R2w 2 . ¶z 2 ¶z dVA , dr 2 e dVN 1 w 2n = + w 2a, 1 + e dr 1+e w 2 = 2 f ¢ (R 2 ) , 1 2 2 R w + f (R 2 ) , 2 (19) w 2a = - 2 (12) From here we deduce that k = R 2 f ¢ (R 2 ) + f (R 2 ) = z2 z4 + d = 1, a ea2 ea4 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 ). (18) and the atmosphere, with equation thus, the angular velocity must be a function of the kind w = w (x 2 + y 2 , z ). (16) dV . (17) dr 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 ¶w ¶w , = 2 ¶x ¶y 2 dV , 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). or 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 1 v - V + p = k, r 2 (14) 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 1 ⎞ grad ⎜ v2 - V + p ⎟ = v ´ (rot v) , ⎝2 r ⎠ 1 p = 0. r (20) 4 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. (13) 2 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 e= rn - ra . ra our model in the chosen direction (y = 0, z = 0 ) are given by Q= a ò 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)) , (25) for the FR force component, whereas for the Fz force component we have (22) 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 symmetry). 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 points: 1 cos (J)(5 (Q11 - Q33) cos (2J) - 5Q11 - Q33) , (26) 2 where Q11 = Q33 = 1 (2 x 2 - y 2 - z 2 ) d t , 2 1 (2 z 2 - x 2 - y 2 ) d t . 2 òt +t a n òt +t a n 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. cm Q c + 4 + o6 2 r r r F= We wish to abbreviate these 13 results with a short approximate formula inspired in a truncated series expansion of the exact force: cq cm c + 4 + o6 . 2 r r r òt ò 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 F= 3 (2 x 2 - y 2 - z 2 ) d t 2 ò (21) 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, n 3 (2 x 2 - y 2 - z 2 ) d t 2 3 +e (2 x 2 - y 2 - z 2 ) d t tn 2 5 O= (8x 4 - 24x 2 ( y 2 + z 2) + 3 ( y 2 + z 2)2) dt t 8 5 +e (8x 4 - 24x 2 ( y 2 + z 2) + 3 ( y 2 + z 2)2) dt , tn 8 (24) = 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 ⎞ ⎟, F=⎜ , 0, ⎝ ¶R ¶z ⎠ òt +t (27) 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) (23) 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 V= 1 Q0 + 5 r r + 3 1 r9 3 3 3 å å Qij xi xj i=1 j=1 3 3 3 å å å å Qijkl xi xj xk xl + .... i= 1 j= 1 k= 1 l= 1 (28) 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 + r2 2r 4 5 (29 - 28 cos (2J) + 63 cos (4J)) cos (J) Q1111 . 24r 6 (35) 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 8 Q1111 ,.... 3 ⎛ z3 d z5 ⎞ M = 2p ⎜z Ma - Ma2 - a Ma ⎟ 3ea 5ea4 ⎠ ⎝ ⎛ z3 d z5 ⎞ + 2pe12 ⎜z Mn - Mn2 - n Mn ⎟ e. 3en 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: ⎛ z3 d z5 ⎞ tn = 2pe12 ⎜z Mn - Mn2 - n Mn ⎟, 3en 5en4 ⎠ ⎝ ⎛ z3 d z5 ⎞ ta = 2p ⎜z Ma - Ma2 - a Ma ⎟ - tn. 3ea 5ea4 ⎠ ⎝ pe12 z [35d 2e12 z 8 1260e8 + 90d (2e2 + e12 ) e2z 6 + - - 2d ) e12 + 210 (2e2 + e12 ) e6z 2] , 63e 4z 4 ((1 4e 2 ) + (36) 315e12 e8 (37) (29) 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 480480e12 + 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]. (30) 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) , (31) Q1111 = o (ea , 1, da, z Ma) + eo (en , e1, dn, z Mn). (32) One can show that the other nonzero components are given by Q33 = - 2Q11, Q1133 = - 8Q1111, 4.1. Case e > 0 8 Q3333 = Q1111. (33) 3 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 5 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 + 2 r 2r 4 5 sin (J)(15 + 28 cos (2J) + 21 cos (4J)) Q1111 , 8r 6 (34) fR = - 4 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 ε First Prolate en=ea Ceiling ea Last Prolate en=ea 0.4619 0.4619 0.4619 0.4619 0.7356 0.5802 0.4693 0.4619 0.6328 0.4973 0.4665 0.4619 0.1 0.3 0.5 0.5286 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 en ea 0.55 0.55 0.55 0.60 0.55 0.65 Figure 1. Final model with prolate nucleus. 0.55 0.6714 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 = e1 4d n + 1 - 1 2d n . (38) ε Prolate Begin en Series Ceiling ea 0.1 0.3 0.5 0.5225 0.5274 = ea 0.5274 = ea 0.5274 = ea L 0.9707 0.9726 0.9339 L Last Prolate en 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 ε 0.1 0.3 0.3915 Prolate Begin en Series Ceiling ea Last Prolate en ea 0.5274, ea = 0.4619 0.5274, ea = 0.4619 L 0.7759 0.7176 L 0.6300/0.6481 0.5632/0.4934 0.5277/0.5060 Note. e1 = 0.5, dn = 1 8, da = -1 8. Table 5 Series Limits with Prolate Nucleus for Increasing ε ε 0.1 0.3 0.5 0.5225 Prolate Begin en Series Ceiling ea Last Prolate en ea 0.4619, ea = 0.5274 0.4619, ea = 0.5274 0.4619, ea = 0.5274 L 0.9705 0.9721 0.6141 L 0.7337/0.8375 0.5550/0.6336 0.4896/0.5589 0.4619/0.5274 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. 5 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 e1 emax 0.845 0 0.8 0.1992 0.7 0.4695 0.6 0.5612 0.5 0.5286 0.4 0.4288 0.3 0.3125 0.2300 0.2340 0.2 0.2276 0.1 0.2275 0.5 0.4753 0.4 0.4912 0.3 0.5125 0.2 0.5248 0.1 0.5290 d n = da = 1 8 e1 emax 0.9196 0 0.9 0.3832 0.8 0.7065 0.7 0.6757 0.6 0.5790 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 (39) e1 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 cylinder. surface (Cisneros et al. 2015, Equation (44)). Here, we call pan the atmosphere pressure at nucleus points, which is given by pan = VA + fa (R 2) = VN - VA (R 2) , (40) ra 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 Re1 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 pan * + Vaa * ), = eVnn + Van - (eVna (41) ra 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 s= 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. 0.5 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 6 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 ε −0.5 −0.6 −0.7 −0.8 −0.9 −0.95 −0.99 en ea w 2n pan 0.5459 0.5219 0.5007 0.4920 0.4650 0.4572 0.4512 0.6850 0.6875 0.6900 0.6900 0.6900 0.6900 0.6900 3.2...4.2 5.2...4.0 5.0...6.7 7.7...10.0 14.0...18.0 27.0...35.0 132...167 0.43...0.84 0.46...0.80 0.48...0.77 0.53...0.73 0.49...0.70 0.49...0.68 0.49...0.66 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 ε −0.6 −0.7 −0.8 −0.9 −0.95 −0.99 −0.999 4.4. Spheroids with Prolate Nucleus rn w 2n pan 0.4ra 0.3ra 0.2ra 0.1ra 0.05ra 0.01ra 0.001ra 1.6...2.0 2.1...2.7 3.1...4.2 6.4...9.1 13.3...19.1 68.7...99.5 692...1005 0.16...0.24 0.14...0.22 0.12...0.22 0.11...0.21 0.11...0.21 0.10...0.20 0.10...0.20 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. 7 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 8 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). ORCID iDs Joel U. Cisneros-Parra 8785-6527 https://orcid.org/0000-0002- References Bizyaev, I. A., Borisov, A. V., & Mamaev, I. S. 2015, CeMDA, 122, 1 Cassini, J. 1718, De la Grandeur et de la Figure de la Terre (Suite des Memoires de la Academie Royale des Sciences) Chambat, F. A. 1994, A&A, 292, 76 Chandrasekhar, S. 1969, Ellipsoidal Figures Of Equilibrium (New Haven, CT: Yale Univ. Press) Cisneros, J. U., Martínez, F. J., & Montalvo, J. D. 2015, RMxAA, 51, 119 Cisneros, J. U., Martínez, F. J., & Montalvo, J. D. 2016, RMxAA, 52, 375 Hamy, M. 1889, in Annales de l’Observatoire de Paris, Memoires, Vol. 19 (Paris: Gauthier-Villars) Hubbard, W. B. 2013, ApJ, 768, 43 Jeans, J. H. 1914, Phil. Trans., 215, 27 Kong, D., Zhang, K., & Schubert, G. 2013, ApJ, 764, 67 Landau, L. D., & Lifshitz, E. M. 1987, Course of Theoretical Physics: Fluid Mechanics (New York: Pergamon) Lyttleton, R. A. 1951, The Stability of Rotating Liquid Masses (Cambridge: Cambridge Univ. Press) Marchenko, A. N. 1979, SvAL, 5, 106 Montalvo, J. D., Martínez, F. J., & Cisneros, J. U. 1983, RMxAA, 5, 293 Schubert, G., Anderson, J. 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