The shape of a rapidly rotating polytrope with index unity

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by  Jerzy Knopik

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed23January2017 (MNLATEXstylefilev2.2)
The shape of a rapidly rotating polytrope with index unity
Jerzy Knopik, Patryk Mach, Andrzej Odrzywo(cid:32)lek
Instytut Fizyki im. Mariana Smoluchowskiego, Uniwersytet Jagiellon´ski, L(cid:32)ojasiewicza 11, 30-348 Krako´w, Poland
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23January2017
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n ABSTRACT
a We show that the solutions obtained in the paper ‘An exact solution for arbitrarily
J
rotatinggaseouspolytropeswithindexunity’byKong,Zhang,andSchubertrepresent

0 only approximate solutions of the free-boundary Euler–Poisson system of equations
2 describinguniformlyrotating,self-gravitatingpolytropeswithindexunity.Wediscuss

the quality of such solutions as approximations to the rigidly rotating equilibrium
]
R polytropic configurations.
S Key words: planets and satellites: gaseous planets – planets and satellites: interiors
h. – stars: rotation.
p
-
o
r
t 1 INTRODUCTION Inthefollowingsectionswediscussnumericalsolutions
s
a of rigidly rotating n = 1 polytropes, testing the ellipsoidal
InarecentpaperKongetal.(2015)attackedalong-standing
[ approximation. As expected, the deviation from the ellip-
problem of obtaining analytic solutions representing rigidly
soidalshapeissignificantforrapidlyrotatingconfigurations.
1 rotating stationary polytropes with the polytropic index
We also pay special attention to the model that was com-
v n = 1 (polytropic exponent γ = 1+1/n = 2). The quest
puted in Kong et al. (2015) assuming observational param-
2 for such solutions started with the works of Chandrasekhar
8 (1933a)1andKopal(1937,1939),butthemajorityofpapers eters of α Eri.
7
onthissubjecthasbeenpublishedduringthreedecadesbe-
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tween 1960 and 1990 (Roberts 1963; Papoyan et al. 1967;
0
. Blinnikov1972;Hubbard1974;Kozenko1975;Cunningham 2 ROTATING POLYTROPES WITH INDEX
1 1977; Zharkov & Trubitsyn 1978; Caimmi 1980; Williams UNITY
0
1988). A modern account of these works can be found in
7 Horedt (2004). Rigidlyrotatingaxisymmetricequilibriumconfigurationsof
1
Kongetal.(2015)assumethattheshapeofthebound- a self-gravitating perfect fluid are described by the Euler
:
v ary surface of such polytropes is an ellipsoid of revolution equations
i and construct analytic solutions of the corresponding equa-
X
tionforthedistributionofthedensity.Inthisnoteweshow ∂P ∂Φ
r −ρΩ2(cid:36)=−ρ , (1a)
a that they satisfy only approximately the coupled system ∂(cid:36) ∂(cid:36)
of Euler and Poisson equations describing equilibrium con- ∂P ∂Φ
=−ρ , (1b)
figurations of rigidly rotating polytropes with n = 1. We ∂z ∂z
alsoclarifythedifferencebetweensolvingthefree-boundary
and the Poisson equation for the gravitational potential Φ
Euler–Poisson problem and the corresponding equation for
the density that is solved in (Kong et al. 2015). ∆Φ=4πGρ. (2)
Solutions obtained by Kong et al. (2015) can still be
understood as approximations to the rigidly rotating poly- HereP denotesthepressure,ρisthemass-density,Ωisthe
tropeswithn=1,althoughthetrueshapeoftheboundary angularvelocityofthefluid,andGisthegravitationalcon-
surfaceofsuchpolytropesdeviatesfromtheellipsoidalone. stant.Weworkincylindricalcoordinates((cid:36),φ,z),assuming
Ingeneral,theshapeofanaxisymmetricrotatingpolytropic that the z axis coincides with the rotation axis of the fluid.
body(0<n<5)canbedescribedonlyapproximatelybya For barotropic equations of state P = P(ρ), one can
biaxialellipsoid.Theexactellipsoidaldescriptionispossible introduce the specific enthalpy h so that dh=dP/ρ. Then,
onlyforthehomogeneous,constantdensitypolytropen=0 in the region U where ρ>0, Eqs. (1) can be written as
(cf. Tassoul 1978, pp. 82–84).
∇(h+Φ +Φ)=0, (3)
c
or, equivalently,
1 This is actually the first of a series of four Chandrasekhar’s
papersondistortedpolytropes(Chandrasekhar1933a,b,c,d). h+Φc+Φ=C, (4)
(cid:13)c 0000RAS
2 J. Knopik, P. Mach, A. Odrzywo(cid:32)lek
where the centrifugal potential is In this section we give numerical examples of solutions
ofEqs.(2)and(6),obtainedwithoutassumingthatthesur-
1
Φc =−2Ω2(cid:36)2, (5) faceoftheconfigurationisellipsoidal.Ouralgorithmfollows
strictly the scheme proposed by Eriguchi & Mu¨ller (1985).
and C denotes an integration constant.
In short, it is based on Eq. (4), which is discretised on a
In this paper we deal exclusively with polytropic equa-
two-dimensionalgrid.Thevaluesofthegravitationalpoten-
tions of state of the form P =Kρ1+1/n, where K is a con-
tial Φ at the grid nodes are computed using the expansion
stant,andassumen=1.ThisyieldsP =Kρ2,andalinear
of the Green function for the Laplace operator in terms of
relation h=2Kρ.
Legendrepolynomials.InthiswayEqs.(2)and(6)arecon-
A rigidly rotating n=1 polytrope is thus described by
verted into a system of algebraic equations, which is then
the Poisson equation (2) and the equation
solved using standard numerical methods.
∇(2Kρ+Φ +Φ)=0, (6) The method of Eriguchi & Mu¨ller (1985) allows one to
c
obtain whole sequences of models, say with a fixed value
thatholdsintheregionU,andwhereΦcisgivenbyEq.(5). of the total mass and different values of the angular mo-
It is important to stress that the boundary of the region mentum. It is important to stress that due to the scaling
U, at which the density ρ tends to zero, is not known a symmetryofEqs.(2)and(6),onlyonesuchsequenceneeds
priori,buthastobeestablishedbytheprocedureofsolving to be computed (see, e.g., Aksenov & Blinnikov 1994).
Eqs. (2) and (6) itself. In mathematical literature, such a InFigs.1–3weplottheshapesofthreeoftheobtained
formulation is known as a free-boundary problem. It is a configurations. Because Kong et al. (2015) discuss a model
remarkable property of the set of Eqs. (2) and (6) that it of α Eri with an ellipsoidal surface, we believe that a com-
allowsforthedeterminationoftheboundaryofU.Inother parisonofthenumericalsolutionandtheiranalyticapprox-
words,Eqs.(2)and(6)donotadmitregularsolutionswith imation for the observed parameters of α Eri would be in
a non-negative density ρ for a wrong guess of the shape U. order. Such a comparison is shown in the first of the pre-
The approximation presented by Kong et al. (2015) sented figures.
(andadvertisedasanexactsolution)isbasedonthefollow-
OurnumericalsolutionfromFig.1wascomputedwith
ing, well known, observation. By computing the divergence
the observed mass M =4.9M , the equatorial radius R =
(cid:12) e
ofEq.(6)andcombiningtheresultwithEq.(2),oneobtains 12.0R , and the angular velocity Ω = 2.9725×10−5s−1,
(cid:12)
an equation for the density
as quoted by Kong et al. (2015, Table 1). The values of
2πG Ω2 solar parameters used in the code are: M(cid:12) = 1.98855×
∆ρ+ ρ= . (7) 1033 g, R = 6.95700 × 1010 cm. The assumed value of
K K (cid:12)
the gravitational constant reads in cgs units G=6.67384×
The above equation also holds in the region U, and it is
10−8cm3s−2g−1.
assumedthatρtendstozeroattheboundaryofU.Equation
Ourcomputedratioofpolarandequatorialradiiforthe
(7) is equivalent to Eq. (8) of Kong et al. (2015), which is
αErimodelisRp/Re=0.624,andthepolytropicconstant
writteninoblatespheroidalcoordinates,andinfactitisEq.
K =1.54×1016cm5g−1s−2. Kong et al. (2015) report the
(7)andnotthesystemofEqs.(2)and(6)thatissolvedby
value K = 1.7500×1016cm5g−1s−2 and the value of the
Kongetal.(2015).IncontrasttothesystemofEqs.(2)and
eccentricity E = 0.745356, which, up to the used precision,
(6), it admits solutions with the boundary of U being an √
is equivalent to R /R = 1−E2 = 2/3. In our computa-
ellipsoid of revolution. p e
tion the value of K is obtained together with the structure
It is clear from the above derivation that solutions of
of the configuration as one of the unknowns (Eriguchi &
Eq. (7) and the distribution of the density ρ obtained by
Mu¨ller 1985). The obtained value differs by 12% from that
solving Eqs. (2) and (6) may differ by a function f satisfy-
ofKongetal.(2015)becauseofthedifferencesinthedensity
ing ∆f = 0. It is also easy to see that Eq. (7) yields the
stratification. The obtained shape of the boundary surface
propersolution,i.e.asolutionequaltothatofEqs.(2)and
is plotted in Fig. 1. For comparison, we also plot an ellipse
(6), provided that the true shape of the boundary of U is
withthesemi-axesthatwereobtainedbyKongetal.(2015)
assumed. The fact that this boundary cannot be an ellip-
and an ellipse fitted to the obtained shape, i.e., an ellipse
soid of revolution follows from the classic theorem that can
with semi-axes equal to the values of R and R of our nu-
be found in the textbook by Tassoul (1978, pp. 82–84). It p e
mericalmodel.Thereisacleardiscrepancybetweenallthese
statesthattherearenorigidlyrotating,barotropicsolutions
shapes.
of Eqs. (1) and (2) with an ellipsoidal boundary, except for
The solution presented in Fig. 1 here is relatively close
the configurations of constant density.2
to the limiting configuration characterized by the maximal
allowed (critical) rotation. Finding the ratio of R /R cor-
p e
responding to the critical configuration is a subtle numer-
3 NUMERICAL SOLUTIONS ical issue. James (1964, Table 4) and Hurley & Roberts
(1964,Table1)givethevalues0.558and0.493,respectively.
BecauseguessingoftherightshapeoftheregionU inEq.(6)
Cooketal.(1992,Table7),whoconsidergeneral-relativistic
isdifficult(ifatallpossible),onehastoresorttonumerical
configurations report a Newtonian-limit polytropic solution
methodsoracombinationofnumericalandanalyticschemes
with n = 1 and R /R = 0.552. The shape of an (almost)
(see, e.g., Odrzywo(cid:32)lek 2003). p e
critically rotating n = 1 polytrope found in our numeri-
cal computations is shown in Fig. 2. In this example we
2 Thetheoremisactuallymoregeneral:itexcludesallbarotropic keep the same values of the total mass M = 4.90M(cid:12) and
configurationswithanellipsoidalstratification. thepolytropicconstantK asbefore.Theremainingparam-
(cid:13)c 0000RAS,MNRAS000,000–000
The shape of a rapidly rotating n=1 polytrope 3
8 Ω=2.973⨯10-5 s-1
6
]
⊙
R
[
z 4
2
0
0 2 4 6 8 10 12
ϖ [R ]
⊙
Figure 1. Meridional cross-section of a rigidly rotating n=1 polytrope obtained numerically for the model of α Eri, as explained in
thetext.Thethicksolidlineshowsthecomputedsurface,whilethethinoneisapartofanellipsewithsemi-axesthatareidenticalto
√
thepolarandequatorialradiioftheKongetal.(2015),i.e.,Rp/Re = 1−E2 =2/3.Thethindashedlinedepictsanellipsewiththe
semi-axesequaltothevaluesofRp andRe thatwereobtainedforournumericalmodelofαEri.Thethickdot-dashedlinedepictsthe
shapeofanon-rotatingconfigurationwiththesamemassandthesamevalueofthepolytropicconstantK.
eters of this solution are: R = 7.42R , R = 13.3R , model and the data for Jupiter. [Note that it is explicitly
p (cid:12) e (cid:12)
R /R =0.557, Ω=3.004×10−5s−1. stated in Cao & Stevenson (2015) that the outer boundary
p e
As expected, the ellipsoidal approximation is much in the polytropic model differs from an exact ellipsoid of
better for slowly rotating polytropes. A comparison be- revolution.] The values of n close to unity can be probably
tween the actual shape and the matching ellipsoid for Ω = alsoassumedforbrowndwarfsandsomeneutronstars.For
0.2146×10−5s−1 isshowninFig.3.Hereagainwekeepthe the star in the radiative equilibrium (like α Eri) the value
same values of M and K. The remaining parameters read: n≈3seemstobemoreappropriate.Thisfactwasactually
R =8.19R , R = 9.47R , R /R = 0.865. Note that all admitted by Kong et al. (2015).
p (cid:12) e (cid:12) p e
solutions presented in Figs. 1–3 form a sequence character-
It is possible to search for exact solutions representing
ized by the same mass and the same equation of state, but
rotating polytropes of index n = 1 assuming an ellipsoidal
different angular velocities.
boundary surface and differential rotation. Such solutions
The numerical grid used in our computations is also
were constructed by Cunningham (1977) (He uses the term
depicted in each of Figs. 1-3. We use 31 grid points in the
‘spheroids’ referring to ellipsoids of revolution).
angulardirectionand23intheradialone.Thehighestorder
of Legendre polynomials used in the numerical code is 40. Itisalsoworthpointingoutthatbecauseeverysolution
ofEqs.(2)and(6)satisfiesEq.(7),thelattercanbeactually
used to infer important characteristics of solutions of Eqs.
4 FINAL REMARKS (2) and (6). This is also true in nonlinear cases (e.g., for
polytropes with n (cid:54)= 1); results of this type were obtained
Polytropic solutions with index unity serve as suitable ap-
by Mach & Malec (2011).
proximations of the structure of Jovian planets (Hubbard
1974).InarecentworkCao&Stevenson(2015)providean Stability of the rotating n = 1 polytropes was investi-
interesting comparison between parameters of a polytropic gated by Bini et al. (2008).
(cid:13)c 0000RAS,MNRAS000,000–000
4 J. Knopik, P. Mach, A. Odrzywo(cid:32)lek
8 Ω=3.004⨯10-5 s-1
6
]
⊙
R
[
z 4
2
0
0 2 4 6 8 10 12
ϖ [R ]
⊙
Figure 2. SameasFig.1,butforthemodelofanalmostcriticallyrotatingpolytrope.HereΩ=3.004×10−5s−1.
ACKNOWLEDGEMENTS James R. A., 1964, ApJ, 140, 552
Kong D., Zhang K., Schubert G., 2015, MNRAS, 448, 456
We are grateful to Georg Horedt for his comments on the
Kopal Z., 1937, ZAp, 14, 135
first version of this article. A.O. was supported by the
Kopal Z., 1939, MNRAS, 99, 266
Kosciuszko Foundation.
Kozenko A. V., 1975, AZh, 52, 887
Mach P., Malec E., 2011, MNRAS, 411, 1313
Odrzywo(cid:32)lek A., 2003, MNRAS, 345, 497
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(cid:13)c 0000RAS,MNRAS000,000–000
The shape of a rapidly rotating n=1 polytrope 5
8 Ω=2.146⨯10-5 s-1
6
]
⊙
R
[
z 4
2
0
0 2 4 6 8 10 12
ϖ [R ]
⊙
Figure 3. Same as Fig. 1, but for the model of a slowly rotating polytrope with Ω = 2.146×10−5s−1. The difference between the
obtainedshapeoftheboundarysurfaceandtheellipsoidalfitisnotvisibleinthiscase.
(cid:13)c 0000RAS,MNRAS000,000–000