Stellar structure, magnetism and the variational principle

APS/123-QED Stellar structure, magnetism and the v ariational principle A. ˇ Cade ˇ z ∗ and A. Mohori ˇ c Dep artment of Physics, University of Ljubljana, Ljubljana, Slovenia M. Calv ani R etir e d fr om INAF, Astr onomic al Observatory of Padova, Padova, Italy (Dated: F ebruary 25, 2026) 1 Abstract Matter interacts with t wo fundamen tal long range forces: gra vity and electromagnetism. While elemen tary building blo c ks of matter alwa ys con tribute to the gravitational p oten tial, the electro- magnetic v ector p oten tial was expected to cancel out in large systems b ecause of the symmetry b et ween p ositiv e and negativ e charges. Y et, long range electromagnetic phenomena are present in astrophysical systems, so that the cancellation can not b e considered as p erfect. The aim of this pap er is to ﬁnd a possible mo del for stationary aggregation of matter to mak e a “star” whic h consisten tly includes angular momen tum and electromagnetic phenomena. W e recast the standard p olytropic stellar mo del as a v ariational problem and extend it to include the kinetic energy of (rigid) rotation and the electromagnetic energy of interaction b et ween positively and negatively c harged baryonic matter. W e argue that the electromagnetic contribution to the action should b e represen ted b y the minimal energy needed to induce the stellar magnetic dip ole moment. This energy consists of t w o parts: the pure electromagnetic energy , whic h can be expressed as a surface in tegral, and the free energy diﬀerence betw een the magnetized and unmagnetized state of matter. This second part of the action is form ulated by considering the magnetized quan tum state of a F ermi gas of electrons in a sea of cold ions. Diﬀerential forms calculus is a useful to ol to p erform the mathematical analysis ([35]). Our mathematical mo del includes electromagnetism to stellar mo deling in a wa y that is consisten t with the linearized v ersion of general relativit y . The complete mo del presents a complex op en b oundary problem. The problem can b e solved exactly under simplifying circumstances. The distribution of stellar ob jects in the phase diagram, based on the simpliﬁed mo del, exhibits a pattern which ma y prompt researc h to provide a deeper understanding of the natural balance b et ween matter, gra vitation, and electromagnetism. I. INTR ODUCTION The equations of stellar structure, whic h include the interaction of matter with the gravi- tational ﬁeld, the pro duction of energy by n uclear reactions, and the transp ort of energy via con v ection and radiation, hav e b een amazingly successful in explaining the observed prop- erties of stars and describing their history from birth to death. The long-range eﬀects of ∗ andrej.cadez@fmf.uni-lj.si 2 the electromagnetic force are traditionally neglected, since deviations from spherical symme- try caused by stellar rotation and magnetic ﬁelds app ear to hav e minimal eﬀects on stellar ev olution. Y et, stars and other celestial ob jects do spin and exhibit magnetic prop erties. The magnetism of pulsars seems to b e the most interesting prop ert y of neutron stars, while the magnetism of the Sun, Earth, and planets presen ts us with a n um b er of unanswered questions. This pap er presents a Hamiltonian action that leads to the basic equations of stellar structure through a v ariational principle. The action includes stellar rotation and magnetism in a straightforw ard w a y and leads to a phase diagram in whic h celestial ob jects are sorted according to the relative strength of their angular momentum and magnetic ﬁeld. The plan of the pap er is as follo ws: Section I I recapitulates the deriv ation of the Lane-Emden equation for a rotating p olytrope and expresses the energy released during the pro cess of forming a star from inﬁnitely rareﬁed gas in its ground state. This energy forms an action that, together with suitable constraints, leads bac k to the Lane-Emden equation via La- grange equations. Section I I I A in tro duces a brief formulation of Maxwell’s equations in a form conv enient for the analysis of diﬀerent conﬁgurations of curren ts driving the magnetic ﬁeld. The current distribution pro ducing a magnetic moment with the minimum amount of energy is introduced. Section I I I B treats the interaction of a degenerate gas of electrons with the electromagnetic ﬁeld. In section IV The electromagnetic energy is added to the action, which leads to generalized Lane-Emden equations that include the contribution of a stellar magnetic dip ole moment. A subsection IV A is dev oted to the fact that magnetic action is expressed by a surface integral and to the instability of this surface in the case of a strong magnetic ﬁeld. In section V, w e introduce rotation and dev elop a mo del of a rotating magnetic star and classify a num b er of stars according to this mo del. I I. R OT A TING ST ARS AND LANE-EMDEN EQUA TION THROUGH V ARIA- TIONAL PRINCIPLE P olytropic mo dels, described b y the La ne-Emden equation, are very simple stellar models. They are particularly useful in describing the mass-radius relation of compact stars th us pro viding a to ol to understand the state of degenerate matter in observed stellar ob jects. Lane-Emden equation follo ws from: lo cal static equilibrium, the la w of gravitation and the p olytropic equation of state. W e note that this equation can also b e derived from a 3 v ariational principle minimizing the total energy of the star which includes the en thalp y of stellar matter (the w ork done b y gra vitation to compress matter b elow the surface of the star) and the gra vitational energy . Since stars exhibit both rotation and magnetism, it seems natural that global eﬀects of angular momen tum and magnetism could also be described b y a generalized Lane-Emden equation that would follow from a v ariational principle minimizing the energy which includes also rotational and magnetic energy . The equations leading to the p olytropic equation for a rotating star are 1 : p olytropic equation of state p = K ρ 1+1 /n , lo cal equilibrium ρ d  v dt = − ρ ∇ Φ g − ∇ p , (1) gra vitational ﬁeld ∆Φ g = 4 π Gρ , together with the constraint that the velocity ﬁeld is that of rigid rotation:  v =  ω ×  r and d  v dt =  ω × (  ω ×  r ) = ∇  − 1 2 ω 2 R 2  . Here R is the distance from the axis of rotation. T aking in to accoun t also the equation of state and dividing the lo cal equilibrium equation by ρ , one obtains ∇  Φ g − 1 2 ω 2 R 2 + ( n + 1) K ρ 1 n  = 0 which is a constraining equation resulting from the requiremen t of rigid rotation. Since p and ρ are directly related, it is customary to in tro duce a new ﬁeld Θ such that ρ = ρ 0 Θ n and p = p 0 Θ n +1 , whic h brings the constraining equation in the simple form: Φ g = − ( n + 1) p 0 ρ 0 Θ + 1 2 ω 2 R 2 + β , (2) where β is an integration constan t. The Lane-Emden equation follows b y applying the gra vitational ﬁeld equation: n + 1 4 π Gρ 0 p 0 ρ 0 ∆Θ + Θ n − ω 2 2 π Gρ 0 = 0 . (3) The b oundary conditions on Θ follow from the general requirement that the divergence of stress energy tensor b elonging to all interacting ﬁelds and matter v anishes ev erywhere in space. The Lane-Emden problem for rotating polytrop es is th us an op en boundary problem. A t the surface of the star matter densit y and pressure v anish, but gra vitation extends b ey ond the surface, therefore the v anishing of stress-energy div ergence at the surface requires the 1 ∇ stands for gradient and ∆ for Laplacian op erator. 4 gra vitational ﬁeld and its deriv ativ e to b e contin uous at the b oundary - the surface of the star. In order to formulate the problem on the basis of v ariational principle, we write do wn the total energy of the star, i.e. all energies that hav e b een released or hav e done w ork during the pro cess of condensation of stellar matter from the inﬁnitely rariﬁed state to the current state of stable stellar equilibrium. Gra vitational energy can b e expressed in the following forms: W g = 1 2 Z V ρ Φ g dV = 1 8 π G Z V Φ g ∆Φ g dV = − 1 8 π G Z V ∇ Φ g · ∇ Φ g dV + 1 8 π G Z ∂ V Φ g ∇ Φ g · d  S , (4) kinetic energy of rotation as: W rot = Z V 1 2 ρ ω 2 R 2 dV = ω 2 8 π G Z V R 2 ∆Φ g dV = − ω 2 4 π G Z V  R · ∇ Φ g dV + ω 2 8 π G Z ∂ V R 2 ∇ Φ g · d  S , (5) and in ternal energy of stellar matter 2 : W H = Z V p dV − Z V T dS − → Z V p 0 Θ n +1 dV . (6) Here R V represen ts in tegral ov er the volume and R ∂ V in tegral o v er the outer surface of the star. Note that surface integrals result from v olume in tegrals o ver exterior region, whic h con tain in tegrands in the form of divergences going to zero at spatial inﬁnit y . The action for the corresp onding v ariational principle is the sum of all three energies taking in to account the rigid rotation constrain t (2) plus the mass ( M = 1 4 π G R ∂ V ∇ Φ g · d  S ) and angular momentum ( L = 2 ω W rot ) constrain ts with Lagrange m ultipliers λ M and λ L : A = − Z V p 0  ( n + 1) 2 p 0 8 π Gρ 2 0 ∇ Θ · ∇ Θ − ( n + 1)( λ L + ω ) ω 2 π Gρ 0  R · ∇ Θ − Θ n +1 + ( λ L + 3 ω 4 ) ω 3 2 π Gp 0  R ·  R  dV − 1 ω Z ∂ V ( n + 1) p 0 8 π G  p 0 ρ 0 (4( n + 1) λ L + (4 + 3 n ) ω ) Θ ∇ Φ g − 2  2( λ L + 3 4 ω )Φ g − (2 λ L + ω ) β + ω λ M  ∇ Φ g  · d  S . (7) 2 A p olytropic mo del assumes the entrop y p er particle to b e indep enden t of p osition, so that R V T dS = W S is a giv en constan t for a given mo del. This condition is a constraint of p olytropic mo del. 5 The ﬁeld equation (3) follows from the v olume part of the ab ov e action for λ L = − ω / 2. The surface part of action simpliﬁes, b ecause Θ ∂ V = 0, so the ﬁrst term in square brack ets v anishes and the surface action b ecomes Z ∂ V ( n + 1) p 0 8 π G [(Φ g + 2 λ M ) ∇ Φ g ] · d  S . The constraints are satisﬁed if − 2 λ M is the av erage gra vitational p oten tial on the surface. A. On solution of Lane-Emde equation The ﬁrst eﬀect of rotation of a star on its shape is ﬂattening - equatorial radius becomes larger than polar radius. The condition (2) requires the gra vitational p oten tial on the surface of the star to b e  1 2 ω 2 R 2 + β  , so that for slow rotation the dominan t term breaking the spherical symmetry is the quadrup ole. This suggests that elliptical shap e is a go od approximation for slowly rotating stars. T o illustrate the mec hanism, w e recapitulate the deriv ation of the classical Maclaurin model of gravitating incompressible ﬂuid, which is the p olytropic mo del with n = 0. Equation 3 tak es the very simple form 1 4 π Gρ p 0 ρ 0 ∆Θ + 1 − ω 2 2 π Gρ 0 = 0, which up on the introduction of constants Ω 2 = 2 π Gρ 0 , a 2 = p 0 2 ρ 0 Ω 2 , ˜ ω = ω / Ω and ˜ ∆ = a 2 ∆ b ecomes 1 − ˜ ω 2 + ˜ ∆Θ = 0 . (8) Since the surface of the star, where Θ = 0, is exp ected to b e elliptical, we seek the solution in elliptic co ordinates { ξ , θ , ϕ } deﬁned by:        x y z        =        a p ξ 2 + e 2 sin θ cos ϕ a p ξ 2 + e 2 sin θ sin ϕ a ξ cos θ        . (9) The Laplacian in these co ordinates is: ˜ ∆ = 1 ξ 2 + e 2 cos 2 θ  ∂ ∂ ξ  ( ξ 2 + e 2 ) ∂ ∂ ξ  1 sin θ  sin θ ∂ ∂ θ  + 1 ( ξ 2 + e 2 ) sin 2 θ ∂ 2 ∂ ϕ 2 . The complete set of cylindrically symmetric solutions of Laplace’s equation is: ψ ( int ) l = P l ( i ξ e ) P l (cos θ ) 0 < ξ < 1 , ψ ( ext ) l =  iQ l ( i x e ) + π 2 P l ( i ξ e )  P l (cos θ ) 1 < ξ < ∞ , (10) where P l ( x ) and Q l ( x ) are Legendre p olynomials and Legendre functions of order l for l = 0 , 1 , 2 . . . ∞ . The solution of eq.(8) satisfying the b oundary condition Θ = 0 at the surface ξ = 1 is: Θ( ξ , θ ) = 1 − ˜ ω 2 2(3 + e 2 ) (1 − ξ 2 )(1 + e 2 cos 2 θ ) . (11) 6 The gra vitational p oten tial inside the star follows from eq.(2) and b ecomes: Φ ( int ) g = β − p 0 2 ρ 0  1 − ˜ ω 2 (3 + e 2 ) (1 − ξ 2 )(1 + e 2 cos 2 θ ) − 1 2 ˜ ω 2 ( ξ 2 + e 2 ) sin 2 θ  . (12) The constan t β and the relation betw een e and ω follow after the in ternal gravitational potential is smoothly joined to the external gravitational p oten tial Φ ( ext ) g = c 0 ψ ( ext ) 0 + c 2 ψ ( ext ) 2 at ξ = 1 (eq.10) written explicitly as: Φ ( ext ) g = c 0 arctan( e ξ ) + c 2  (3 ξ 2 + e 2 ) arctan( e ξ ) − 3 e ξ  P 2 (cos θ ) . (13) The b oundary conditions Φ ( ext ) g = Φ ( int ) g and d Φ ( ext ) g dξ = d Φ ( int ) g dξ at ξ = 1 lead to: ˜ ω 2 = 1 e 3  (3 + e 2 ) arctan( e ) − 3 e  → 4 15 e 2 − 8 35 e 4 + 4 21 e 6 + . . . , (14) β = p 0 2 ρ 0 e 3 (1 + e 2 )  e − (1 + e 2 ) arctan( e )  → p 0 2 ρ 0  − 2 3 − 8 15 e 2 + 8 105 e 4 − 8 315 e 6 + . . .  , c 0 = − p 0 ρ 0 1 + e 2 3 e , c 2 = − p 0 ρ 0 1 + e 2 6 e 3 . (15) This is equiv alent to MacLaurin solution if ε 2 M cl = e 2 1+ e 2 , ([22]) which is an exact solution (even if not dynamically stable for large e ). This solution can b e also obtained directly from v ariational principle for total energy , if one starts with the premise that a rotating self-gravitating incompressible ﬂuid takes the shape of a rotational ellipsoid, and the only unknown is the relation b et ween the angular momen tum and ellipticity of the bo dy . Gravitational energy and rotational energy can b e readily computed, while the in ternal energy of the incompressible ﬂuid is constan t, indep enden t of shap e. So, we ha v e: W g = − 3 G M 2 5 R ef (1 + e 2 ) 1 / 3 e arctan e − → e → 0 − 3 G M 2 5 R ef  1 − 1 45 e 4 + 64 2835 e 6 + . . .  , W rot = 1 5 M R 2 ef Ω 2 ˜ ω 2 (1 + e 2 ) 1 / 3 , (16) where the eﬀective radius is deﬁned so that 4 π 3 R 3 ef is the volume of the ﬂuid. The action A ( e , ˜ ω ) = W g + (1 + λ L 2 ˜ ω ) W rot has a minimum for a given angular momentum Γ = 2 5 M R 2 ef Ω ˜ ω (1 + e 2 ) 1 / 3 . The relation b et ween ˜ ω and e leading from this requirement is precisely (14). 7 The incompressible model is simple and leads to an exact solution of the problem, because the diﬀeren tial op erator in eq.(3) allo ws separation of v ariables in suc h a w ay that angular functions ( P l (cos θ )) are the same in the region inside the star and outside. This makes it p ossible to exactly join the in ternal and external gra vitational ﬁeld on the surface for all v alues of the parameter e . Other rotating p olytropes do not share this prop ert y . In general, their solutions may only b e calculated n umerically and one must tak e into accoun t the p ossibilit y that elliptic shap e do es not lead to the minimum energy solution. How ever, in the limit of slo w rotation, p olytropic models do con verge to ellipsoidal shape with small eccen tricity . The ﬁrst co eﬃcien t in relation (14) ( 4 15 ≈  ˜ ω e  2 ) is an increasing function of the p olytropic index. In the limit n = 5, when the mass is concentrated at the center of the star, it b ecomes 2 3 , while for the last marginally stable p olytrope n = 3 this co eﬃcien t is 0 . 65. In other words, hard p olytrop es are more deformed with resp ect to ˜ ω than soft p olytropes, but the factor  ˜ ω e  2 b et ween the hardest and softest is less than 3, ho w ever one m ust remember that the radius of a soft p olytrop es with a given mass is so muc h larger, therefore, as a consequence of decreasing their Ω, soft p olytropes b ecome unstable at smaller ω . I II. ON ELECTR OMA GNETIC ACTION Stellar magnetism may b e a complicated phenomenon, esp ecially if one w ould like to understand the v ariety of magnetic phenomena on our Sun. How ever, the common manifestation of stellar magnetic ﬁeld is through the magnetic dipole, which is usually expressed as “the magnetic ﬁeld at magnetic poles”. F rom this p erspective the magnetic dip ole ﬁeld app ears to b e the principal comp onen t of stellar magnetism entering in the balance b et ween gra vitation, pressure and rotation. The extension of p olytropic mo del to magnetism requires the addition of tw o energies to the v ariational action: a) magnetic dip ole energy b) the energy of elementary magnetic dip oles within matter interacting with magnetic ﬁeld. A. Magnetic dip ole energy Magnetic dip ole energy should, according to previous discussion, b e the energy needed to establish the giv en (measured) magnetic dipole momen t. W e express this energy through the w ork done b y currents inside the magnet v olume ( V m ) as: W E M = − Z t −∞ Z V m  j ·  E dV dt , which can b e transformed using Maxwell’s 8 equations 3 and v ector identities as follows: W E M = Z t −∞ Z V m −∇ ×  H + ∂  D ∂ t ! ·  E dV dt = Z t −∞ Z V m " −∇ ·   H ×  E  − ( ∇ ×  E ) ·  H + ∂  D ∂ t ·  E # dV dt = Z t −∞ Z ∂ V m   E ×  H  · d  S dt + Z t −∞ Z V m ∂  B ∂ t ·  H + ∂  D ∂ t ·  E ! dV dt . (17) The surface in tegral in the abov e expression represen ts the P oyn ting ﬂux propagating from the outer surface of the magnet and building the outer electromagnetic ﬁeld b y a w av e spreading in outer space, as the in terior ﬁeld is slowly rising to its ﬁnal v alue. Ev aluating this integral, it is con venien t to express EM ﬁeld with v ector p oten tial in rad iation gauge, deﬁned so that  E = − ∂  A ∂ t and  H = 1 µ 0 ∇ ×  A . The w av e, b y whic h magnetic ﬁeld propagates in outer space, satisﬁes the wa ve equation and the b oundary condition  A (  r s , t ) =  A S (  r s ) f ( t ) on the surface of the magnet. Here f ( t ) is a slowly rising function from 0 at t 0 to 1 at t f . At ﬁnal time t f the w av e has spread to the distance c ( t f − t 0 ) and is in any ﬁnite volume indistinguishable from the magnetic ﬁeld of a static magnet if t 0 → −∞ . The surface in tegral ev aluates to: W S = Z t −∞ Z ∂ V m   E ×  H  · d  S dt = − 1 µ 0 Z ∂ V m  A S × ( ∇ ×  A S ) · d  S Z t f t 0 f ′ ( t ) f ( t ) dt = − 1 2 µ 0 Z ∂ V m  A S × ( ∇ ×  A S ) · d  S . (18) The v olume integral in eq.17 app ears simple 4 , if the pro cess of magnetic ﬁeld generation can b e considered as adiabatic and the parameters µ and ε are constan ts. The result is simply: 1 2 µ 0 µ R V ( B 2 + µ ε c 2 E 2 ) dV . In this case magnetic energy can b e written as: W E M = − 1 2 µ 0 Z ∂ V  A × ( ∇ ×  A ) · d  S + µ 0 2 Z V  H · (  H +  M ) dV , (19) where  M is the contribution of internal polarization to magnetic ﬁeld (  B = µ 0 (  H +  M )). 3 The following form is used: ∇ ·  B = 0, ∇ ·  D = ρ , ∇ ×  E = − ∂  B ∂ t , ∇ ×  H =  j + ∂  D ∂ t , where  B = µµ 0  H and  D = εε 0  E . The ﬁelds  B and  E can also b e replaced b y v ector (  A ) and scalar (Φ) p otential:  B = ∇ ×  A ,  E = −∇ Φ − ∂  A ∂ t 4 Magnetic energy in exterior region is W e = 1 2 µ 0 Z V e  B ·  B dV = 1 2 µ 0 Z V e ( ∇ ×  A ) · ( ∇ ×  A ) dV . Use v ector iden tit y ( ∇ ×  A ) · ( ∇ ×  A ) = ∇ · (  A × ( ∇ ×  A )) −  A · ∆  A +  A · ∇ ( ∇ ·  A ), note that ∆  A = 1 c 2 ∂ 2  A ∂ t 2 in exterior region and ∇ ·  A = 0 in radiation gauge. W e consists of tw o terms, the ﬁrst is a volume in tegral of a divergence, which can b e transformed to the surface in tegral W S and the second term − 1 c 2 R V e  A · ∂ 2  A ∂ t 2 dV can b e made arbitrarily small in the limit t 0 → ∞ . 9 A t this p oin t one should remember that the phenomenological parameter µ describ es the amount of magnetization that matter contributes to magnetic ﬁeld, if matter is in a steady equilibrium with a given magnetic environmen t characterized by  H . How ev er, magnetization can not instantly follow magnetic ﬁeld, b ecause elementary charges, providing magnetizing currents, are sub ject to their own laws of dynamics. Th us, during the pro cess of alignment of magnetization with external magnetic ﬁeld µ is not a constant and the interaction of the t wo ﬁelds may exchange energy with the system of elementary charges. The energy transferred to the system of charges is part of the volume integral in eq.(19) but can only b e calculated if the p olarizabilit y including details of electromagnetic interaction b et ween magnetic ﬁeld and the system of c harges is fully taken in to account. The assumption µ = const neglects work needed to p olarize matter. The result (19) thus represents full magnetic energy , but the work needed to magnetize a magnet must include also the diﬀerence b et w een the free energy of p olarized and unp olarized matter. The work needed to adiabatically p olarize degenerate matter is calculated in the next section. A magnetic dip ole can b e generated in many diﬀeren t wa ys. Let us consider a spherical magnet made of paramagnetic material that is magnetized by a curren t ﬂowing on a spherical surface arranged so that it pro duces only a dip ole. Fig. 1 shows six conﬁgurations of magnetizing current in a spherical magnet pro ducing the same external magnetic ﬁeld but requiring diﬀerent energy input. It can b e shown quite generally that the energetically least demanding w ay to pro duce a given magnetic moment is to run the magnetizing curren t only on the surface of the magnet, distributed in such a wa y that it pro duces a homo- geneous magnetic ﬁeld inside. In this case the magnetic energy required to generate a magnetic momen t  M = R  M dV b ecomes W M = µ 0 M 2 4 V (1 + 2 µ ), where the ﬁrst term is the contribution of the surface in tegral and the second of the volume integral in (19). A p ermanen t magnet diﬀers from the current driven magnet by the fact that magnetization resulting from sp on taneous alignmen t of elementary magnetic dip oles along a common direction, is the only source of magnetism. Therefore, in this case µ → ∞ and the volume part of magnetic action v anishes. W e therefore consider the magnetic action to b e expressed only by the surface integral (18), an in tegral ov er the compact magnetic surface that encloses the v olume of complete magnetization. This circumv ents the classic problem of designing a classical dynamo mechanism ([23]) that pro duces the observed magnetic ﬁeld. W e prop ose instead that the ”dynamo” most likely works through the most eﬃcien t mec hanism possible to pro duce the observed ﬁeld. In taking the limit µ → ∞ w e do not assume the stellar material to b e a ferromagnet in the usual sense. Rather, this limit represen ts the eﬀectiv e situation in which the in trinsic magnetization of degenerate matter 10 is large compared to the auxiliary ﬁeld H , so that B ≃ µ 0 M and the volume term µ 0 H ( H + M ) b ecomes negligible. The internal magnetic energy is then contained in the free energy of the magnetized electron gas, while the work as sociated with the external dip ole ﬁeld arises solely from the surface integral. This limit serves as a conv enient mathematical device for isolating the minimal action required to generate the observ able magnetic dip ole. B. Magnetic in teraction energy The ab o v e discussion shows that the magnetic contribution to the action can b e reduced to a surface term once the in ternal magnetization of matter is treated as part of its equilibrium free energy . In this macroscopic description the volume term in volving H ( H + M ) v anishes in the eﬀective limit H ≪ M , whic h characterizes a medium whose magnetization is intrinsic rather than maintained b y driven currents. T o justify this assumption on microscopic grounds, we now turn to the quantum mechanical prop erties of degenerate matter. F ollowing Chandrasekhar’s approach to the electron gas , we show that a dense electron–ion plasma naturally dev elops a magnetized ground state through the combined eﬀects of Landau quan tization, charge p olarization, and the mass asymmetry b et ween electrons and ions. This provides the in ternal mechanism resp onsible for the sp on taneous magnetization that underlies the surface representation of magnetic action derived in the previous subsection. Matter interacts with magnetic ﬁeld through elementary magnetic moments of its constituent electrons ( µ B ), protons ( µ N ), neutrons ( µ N ). Let us consider the equation of state of cold white dwarf matter - degenerate matter - under given pressure and immersed in a giv en magnetic ﬁeld. In the absence of magnetic ﬁeld, one starts by describing electrons in a cubic b o x with a wa ve function, whic h is an antisymmetric linear com bination of single particle w av e functions satisfying the Sc hr¨ odinger equation − ℏ 2 2 m e ∆Ψ = E Ψ. The single particle wa v efunctions in a b o x with sides L are ψ pq rs ∝ sin( pπ L x ) sin( q π L y ) sin( rπ L z ) | s ⟩ with p , q , r integers and the spin quantum num b er s takes on the v alues ± 1 2 . Energy eigen v alues of these states are: E pq ns = ℏ 2 m e  π L  2  p 2 + q 2 + n 2  . The ground state of electron gas in this b o x is the state in which all the energy states up to the F ermi energy ( E F ) are ﬁlled, i.e. if E pq ns ≤ E F for an y com bination of p , q , r and s , then this state is o ccupied, otherwise it is not. The main results of this analysis are the relations b etw een the F ermi energy and electron num b er density and the relation b etw een degenerate gas kinetic energy density and F ermi energy and degenerate gas pressure 11 ([15]): n e = 1 3 π 2 ℏ 3 (2 m e E F ) 3 / 2 , (20) w e = 3 5 n e E F = 3 5 (3 π 2 ) 2 / 3 ℏ 2 2 m e n 5 / 3 e , (21) p g = (3 π 2 ) 2 / 3 ℏ 2 5 m e n 5 / 3 e . (22) T o extend this treatmen t to the presence of magnetic ﬁeld, one must include the magnetic interaction in the Sc hr¨ odinger Hamiltonian, as w ell as the W eiss ﬁeld, which takes account of exchange in teraction ([18]). Th us, the single particle Hamiltonian must be of the form H = − 1 2 m e  − i ℏ ∇ − e  A  2 + e m e B · σ + H W , where H W represen ts the exchange interaction via W eiss ﬁeld. In the absence of exchange interaction electron gas is diamagnetic, since electrons gyrate in magnetic ﬁeld about a ﬁxed axis in such w ay as to mak e a curren t lo op opp osing external magnetic ﬁeld. T o contribute to magnetic ﬁeld, an electron must close a loop in the opp osite direction. This is made p ossible in the presence of a radial electric ﬁeld, as in Fig.2, which pulls the electron tow ard the radial center. The classical Hamiltonian describing such a scenario can b e written as : ˜ H class = 1 2 m e   p − e  A  2 + m e ˜ ε ˜ ω 2 c ( x 2 + y 2 ) , (23) where  A = {− m e ˜ ω c 2 e y , m e ˜ ω c 2 e x, 0 } , ω c = e B m e and √ ˜ ε ˜ ω c is angular velocity of gyration center circulating in the direction opp osite to cyclotron gyration. The last term of the ab o ve Hamiltonian is understo o d to represen t the (av erage) exc hange in teraction b et ween the electrons and ions and is represen ted b y the electric p olarization:  E r = 2 ˜ ε m e e ˜ ω 2 c  R , (24) where |  R | = p x 2 + y 2 .W e sho w at the end of this section that suc h electric ﬁeld ma y reasonably be exp ected as a result of p olarization b et ween the co ol gas of heavy p ositively c harged ions and the degenerate gas of ligh t electrons who ose degenerate pressure must be counterbalanced b y electrostatic interaction betw een the t wo gases. In order to quan tize the ab o ve Hamiltonian, it is conv enient to ﬁrst express the classical Hamiltonian with another set of dynamic v ariables α = p x − i p y − i 2 m e ω c ( x − i y ) √ 2 m ˜ h ω c , α ⋆ = p x + i p y + i 2 m e ω c ( x + i y ) p 2 m e ˜ h ω c , β = p x + i p y − i 2 m e ω c ( x + i y ) √ 2 m e ˜ h ω c , β ⋆ = p x − i p y + i 2 m e ω c ( x − i y ) p 2 m ˜ h ω c , 12 whic h b elong to algebra of Poisson brac kets analogous to the algebra of creation and annihilation op erators in quan tum mechanics 5 . With these op erators the Hamiltonian: H class = ℏ ω c α α ⋆ + 2 ε ℏ ω c ( α α ⋆ + β β ⋆ ) + p 2 z 2 m e (25) is the same as (23) if ˜ h = ℏ 1+4 ε , ˜ ε = ε (1 + 2 ε ) and ˜ ω c = ω c 1+4 ε . The z comp onent of angular momentum l z = m ( x ˙ y − y ˙ x ), expressed with α , β , b ecomes: l z = ℏ ( − 2(1 + 2 ε ) α α ⋆ + 4 ε β β ⋆ + ( α β + α ⋆ β ⋆ )) . (26) The quantum Hamiltonian and angular momentum op erator are no w readily obtained by replacing dynamic v ariables α , α ⋆ , β , β ⋆ with annihilation and creation op erators a , a ⋆ , b , b ⋆ and adding the spin interaction ( σ z ) with magnetic ﬁeld 6 . H = ℏ ω c  a ⋆ a + 1 2 + 2 ε ( b ⋆ b + a ⋆ a + 1) + σ z  − ℏ 2 2 m e ∂ 2 ∂ z 2 . (27) W e consider the degenerate electron gas to b e constrained in a cylinder of hight L and radius R 0 . The eigenfunctions of (27) separate in cylindrical co ordinates into pro ducts | n s k z σ z ⟩ ∝ sin( k z z ) | σ z ⟩| n, s ⟩ . The function sin( k z z ) ensures that the wa ve function v anishes at the b ottom ( z = 0) and at the top ( z = L ) of cylindrical enclosure if k z = π L q for q an integer. The radial part of single electron wa v e function | n, s ⟩ can b e generated by repeated action of op erators a + and b + on the v acuum state | 0 ⟩ : | n s ⟩ = e i k z z ( n ! s !) − 1 / 2 a + n b + s | 0 ⟩ . (28) The quan tum num b er n can b e thought of as quantizing the kinetic energy of the electron gyrating ab out a ﬁxed axis with angular frequency ω c , while the quan tum n umber s can b e though t of as quantizing the radial position of the axis of circular motion, which in this case mov es (with angular frequency √ ε ω c ) on a circular orbit with respect to the center of rotation of degenerate gas. The radial p osition R r of the axis of gyration corresp onding to eigenfunction | n s k z σ z ⟩ can b e deﬁned by: R 2 r = ⟨ n s k z σ | ( x 2 + y 2 ) | n s k z σ ⟩ = ℏ 2 m e ω c ⟨ n s k z σ | 4( a ⋆ − b )( a − b ⋆ ) | n s k z σ ⟩ = ℏ 2 m e ω c 4(1 + n + s ) (29) 5 In the classical treatment ˜ h is as arbitrary constan t 6 In replacing, w e symmetrize the pro ducts α α ⋆ → 1 2 ( a a ⋆ + a ⋆ a ) → a ⋆ a + 1 2 , taking in to accoun t the commutation relations for a and b op erators. 13 and the eﬀective width of the wa ve is the gyration radius whic h is (for small ε ): δ R r ∼ q ℏ 2 m e ω c n . If the radius R 0 of the cylindrical enclosure is suﬃciently large with resp ect to δ R r , then the presence of an electron inside the enclosure can b e quite accurately conﬁrmed if R r ≤ R 0 and denied if R r > R 0 . This condition, together with the requirement R 0 ≫ δ R r , replaces the commonly used b oundary condition Ψ = 0 at b oundary surface. The v alidity of this approach is supported by the iden tity: ∞ X s =0 ⟨ n, s, k z , σ z | δ 2 ( R − R 0 ) δ ( z − z 0 ) | n, s, k z , σ z ⟩ = m e ω c 2 π ℏ L , (30) whic h guarantees the completeness of Hilb ert space, but also assures that the incoherent density matrix ∞ X s =0 | n, s, k z , σ z ⟩⟨ n, s, k z , σ z | pro duces a homogeneous density of electrons in the slab 0 < z < L [6]. T o calculate the num b er of particles, the energy and angular momentum of electrons in the ground state inside the cylindrical b o x, we m ust ﬁll all the levels below the F ermi energy according to zero temp erature F ermi-Dirac distribution, noting that the single state energy lev el must b e gauged with resp ect to ground lev el of chemical p oten tial. The eigenfunctions and eigenv alues of the single state Hamiltonian are: H| n s k z σ z ⟩ = (31)  ℏ ω c  n + 1 2 + 2 ε ( n + s + 1) + σ z  + ℏ 2 2 m e L 2 q 2 π 2  | n s k z σ z ⟩ . It is conv enient to introduce the quantized dimensionless radial co ordinate R 2 = m e ω c 4 ℏ R 2 = ( n + s + 1) and the scale factor ζ 2 =  π L  2 ℏ 2 m e ω c , whic h casts the eigenv alues of the Hamiltonian in the form: E n, R ,q ,σ z = ℏ ω c  n + 1 2 + ε R 2 + ζ 2 q 2 + σ z  . (32) In order to determine the ground lev el of c hemical potential, the quan tum Hamiltonian must be split in to the kinetic and p oten tial part. W e do this in analogy with the equiv alent pro cedure in classical mechanics b y ﬁrst deﬁning the quantum equiv alent of v elo cit y comp onents as: ˆ v x = − i ℏ ( x H − H x ) = r ℏ ω c 2 m e ((1 + 2 ε )( a + a ⋆ ) + 2 ε ( b + b ⋆ )) , ˆ v y = − i ℏ ( y H − H y ) = i r ℏ ω c 2 m e ((1 + 2 ε )( a − a ⋆ ) + 2 ε ( b − b ⋆ )) , ˆ v z = − i ℏ ∂ ∂ z . (33) 14 With these the quantum Hamiltonian (27) can b e written as: H = m 1 + 4 ε  1 2 ( ˆ v 2 x + ˆ v 2 y + (1 + 4 ε )ˆ v 2 z ) + ε (1 + 2 ε ) ω 2 c ( ˆ x 2 + ˆ y 2 )  + ℏ ω c σ z =  ℏ ω c  a ⋆ a + 1 2 + 4 ε 2 1 + 4 ε ( a ⋆ a + b ⋆ b + 1) +2 ε 1 + 2 ε 1 + 4 ε ( a b + a ⋆ b ⋆ )  + 1 2 m ˆ v 2 z  +  ε 1 + 2 ε 1 + 4 ε m ω 2 c ( ˆ x 2 + ˆ y 2 )  + ℏ ω c ( σ z − 1 2 ) . (34) The expression in the ﬁrst square brac kets represents the kinetic energy , and the one in the second the p oten tial energy due to electric p olarization. In the same manner we deriv e the op erator corresp onding to the z comp onent of angular momentum: ˆ l z = m ( x ˆ v y − y ˆ v x ) = 2 ℏ  (1 + 2 ε ) a ⋆ a − 2 ε b ⋆ b − 1 2 ( a b + a ⋆ b ⋆ ) + 1 2  . (35) The single state energy with resp ect to the ground level of chemical p oten tial is the kinetic part of energy eigen v alue ⟨ n s k z σ z |H kin | n s k z σ z ⟩ = ℏ ω c  n + 1 2 + ζ 2 q 2 + σ z  . The n umber of electrons in the cylinder can now be expressed with the sum: N = m e ω c 2 ℏ R 2 0 X R 2 =0 X n,q ,σ z        1 , if E F ℏ ω c ≥ n + 1 2 + ζ 2 q 2 + σ z 0 , otherwise = m e ω c R 2 0 3 ℏ ζ  E F ℏ ω c  3 / 2 +  E F ℏ ω c − 1  3 / 2 − 2 ζ 3 ! = 1 3 π 2 ℏ 3 (2 m e E F ) 3 / 2  1 − 3 4 ℏ ω c E F + . . .   π R 2 0 L  + π 2 3 √ 2 R 2 0 L 2 , (36) whic h agrees with (20) for ω c → 0. In a similar manner we calculate the energy (using notation R 0 = p m e ω c 2 ℏ R 0 ): W e = m e ω c 2 ℏ R 2 0 X R 2 =0 X n,q ,σ z        E n, R ,q ,σ z , if E F ℏ ω c ≥ n + 1 2 + ζ 2 q 2 + σ z 0 , otherwise =  π R 2 0 L  1 15 √ 2 π 2  m e ω c ℏ  3 / 2 ℏ ω c "  6 E F ℏ ω c + 5 ε R 2 0   E F ℏ ω c  3 / 2 +  6 E F ℏ ω c + 5 ε R 2 0 + 4   E F ℏ ω c − 1  3 / 2 # =  π R 2 0 L  3 5 E F 1 3 π 2 ℏ 3 (2 m e E F ) 3 / 2 " 1 + 5 12 (2 ε R 2 0 − 1) ℏ ω c E F − 5 16 (2 ε R 2 0 − 1)  ℏ ω c E F  2 + . . . # , (37) 15 whic h b ecomes (21) if ω c → 0 and ε → 0. The calculation of angular momentum of electron gas requires a closer scrutiny , since the eigenfunc- tions of energy | n, s, q , σ z ⟩ are not eigenfunctions of the quantized version of angular momentum op erator (35). The signiﬁcance of this fact can b e understo o d by the analysis of the classical Hamiltonian sys- tem ˜ H class , which leads to the solution: α = a n e iω 1 t , α ⋆ = a ⋆ n e − iω 1 t , β = b s e iω 2 t and β ⋆ = b ⋆ s e − iω 2 t , where ω 1 = 1 2  1 + √ 1 + 8 ε  ω c and ω 2 = 1 2  1 − √ 1 + 8 ε  ω c , so that l z = 2 ℏ ( −| α n | 2 + ˜ ε | β s | 2 + ( 1 2 − ˜ ε )( α n β s e i ω c t + α ⋆ n β ⋆ s e − i ω c t )). Th us, the angular momentu m of an electron oscillates ab out the av erage v alue l z = 2 ℏ ( −| α n | 2 + ˜ ε | β s | 2 ) with the frequency ω c . This means that a single electron exchanges an- gular momentum with electromagnetic ﬁeld 7 . As a consequence, electrons in a magnetized degenerate gas exchange angular momentum among themselves via this in teraction. A rigorous treatmen t of mag- netized degenerate gas may require quantizing electromagnetic ﬁeld as well. Ho w ever, since the inter- action of electrons with magnetic ﬁeld in a magnet do es not pro duce real photons, it seems justiﬁed to assume that angular momentum is exchanged b et w een electrons with virtual photons only , representing a static electric and magnetic ﬁeld. Th us, we can assign to each electron the av erage angular momentum ⟨ l z ⟩ = ℏ ⟨ n, s, q , σ z | ( − 2 a a ⋆ + 4 ˜ εb b ⋆ + σ z ) | n, s, q , σ z ⟩ = ℏ  − 2(1 + 4 ε )( n + 1 2 ) + 2 ε R 2 + σ z  . This allows us to write down the angular momentum density as 8 : λ z = 1 π R 2 0 L m e ω c 2 ℏ R 2 0 X R 2 =0 X n,q ,σ z        ⟨ l z ⟩ , if E F ℏ ω c ≥ n + 1 2 + ζ 2 q 2 + σ z 0 , otherwise =  m e ω c ℏ  3 / 2 " √ 2 3 π 2 ε R 2  E F ℏ ω c − 1  3 / 2 +  E F ℏ ω c  3 / 2 ! − 1 15 √ 2 π 2  E F ℏ ω c − 1  3 / 2  8 E F ℏ ω c − 3  − 1 15 √ 2 π 2  E F ℏ ω c  3 / 2  8 E F ℏ ω c + 15  + 4 √ 2 15 π 2 ε  E F ℏ ω c − 1  3 / 2  4 E F ℏ ω c + 1  +  E F ℏ ω c  3 / 2  4 E F ℏ ω c + 5  !# . 7 This mec hanism is used in magnetrons to generate electromagnetic wa ves 8 One should k eep in mind that the limit ω c → ∞ do es not exist, since in this case the densit y matrix (30) b ecomes singular, therefore, the b oundary condition δ R ≪ R 0 can not b e met. Ho w ev er, for all ω c  = 0 the angular momentum density has a deﬁnite v alue for magnets that are muc h, muc h larger than the cyclotron radius 2 ℏ m e ω c . 16 T aking into accoun t that an electron in a state with angular momentum l z = ℏ l b eha ves as a magnetic dip ole µ z = µ B l (where µ B is Bohr magneton), one ﬁnds that M z = µ B λ z / ℏ is the magnetization generated b y the degenerate gas. If µ 0 M z = B , then the only source of magnetic ﬁeld is the degenerate gas itself. This condition is met if ε = 1 2  E F ℏ ω c − 1  3 / 2  8 E F ℏ ω c − 3  +  E F ℏ ω c  3 / 2  8 E F ℏ ω c + 15  + 30 π 2 √ 2 τ 0 ω c  E F ℏ ω c − 1  3 / 2  5 R 2 − 16 E F ℏ ω c − 4  +  E F ℏ ω c  3 / 2  5 R 2 − 16 E F ℏ ω c − 20  , (38) where τ 0 = 4 m e ℏ π 2 r 2 cl , r cl is classical electron radius. The energy of degenerate electron gas, whic h supp orts its own magnetic ﬁeld can thus b e expressed with (37) by replacing ε with (38) and taking the limit R 2 ≫ E F ℏ ω . After some algebra we obtain ( w e = W e π R 2 0 L ): w e = E F 1 24 π 2 ℏ 3 (2 m e E F ) 3 / 2 "  1 − ℏ ω c E F  3 / 2  4 + ℏ ω c E F  +  4 + 3 ℏ ω c E F  + 6 π 2 √ 2 τ 0 ω c  ℏ ω c E F  5 / 2 + E F m e c 2 O  8 c 5 R 0 ω c  2 # → E F  1 3 π 2 ℏ 3 (2 m e E F ) 3 / 2   1 − 1 4 ℏ ω c E F + 3 π 2 √ ω c τ 0  ℏ ω c 2 E F  5 / 2 + · · · + E F m e c 2 O  8 c 5 R 0 ω c  2 # . (39) In a similar w ay we express the comp onen ts of electron gas stress tensor by quantizing the classi- cal expression T i j = n e X k =1 m e ( v k ) i ( v k ) j , whic h is a sum ov er particles in a unit volume. The comp onen ts T ij are expressed in the same form as the energy (37) if one replaces E n, R ,q ,σ z with exp ectation v alues ⟨ n, s, k z , σ z | m e ˆ v i ˆ v j | n, s, k z , σ z ⟩ , where electron velocity comp onen ts are expressed with eqs.(33), so that: m e  ˆ v 2 x + ˆ v 2 y  = ℏ ω c  1 + 4 ε 2 1 + 4 ε   a ⋆ a + 1 2  + 4 ε 2 1 + 4 ε  b ⋆ b + 1 2  + 2 ε 1 + 2 ε 1 + 4 ε ( a b + a ⋆ b ⋆ ) , m e ˆ v 2 z = − 1 2 ℏ 2 ∂ 2 ∂ z 2 , m e ˆ v x ˆ v y = i 2 ℏ ω c  (1 + 2 ε ) 2 ( a a − a ⋆ a ⋆ ) +4 ε (1 + 2 ε ) ( b ⋆ a − b a ⋆ ) + 4 ε 2 ( b ⋆ b ⋆ − b b )  . The exp ectation v alues of these op erators lead to the follo wing stress tensor comp onen ts: T x,x = T y ,y = 2 15 π 2  2 m e ℏ 2  3 / 2 E 5 / 2 f 1 + 5 64  ℏ ω c E F  3 + . . . ! , (40) T z ,z = 2 15 π 2  2 m e ℏ 2  3 / 2 E 5 / 2 f 1 − 5 4 ℏ ω c E F + 15 16  ℏ ω c E F  2 + . . . ! . (41) 17 Note that equations (40) and (41) reduce to (22) for ω c → 0. The main lesson taugh t by inclusion of angular momentum in the interaction of electrons with magnetic ﬁeld is the recognition that energy eigenstates are not eigenstates of angular momentum, since in the presence of magnetic ﬁeld electrons exchange angular momentum with magnetic ﬁeld as well as with the electric ﬁeld of ions. Electrons, having more energy p er particle than heavy ions in a degenerate gas, o ccup y a slightly larger volume than ions, creating a surface c harge lay er, as sho wn in Fig.4. This in teraction generates the static electric p olarization (24). T o estimate the thic kness ( δ R ) of the surface c harge lay er, note that the charge imbalance δρ = e ( n e − n i ) = ε 0 ∇ ·  E r . Expressing ˜ ε ≈ ε in (24) with (38) for very large R 2 and assuming E F ≫ ℏ ω c , w e obtain δ ρ = 16 5 ε 0 1 R 2 m e ω c E F e ℏ = 16 5 ε 0 E F e R 2 ele . Overall c harge neutrality requires δ ρ e n e = 2 δ R R , so that the thic kness of the surface charge lay er is δ R ∼ (48 λ c ) 2 R ele q m e c 2 E F , where λ c = h m e c . This represents a very thin la yer as can b e seen by expressing the radius of the sample in terms of the num b er of cyclotron radii ( N c ) as R ele ∼ N c q E F m e ω 2 c and F ermi energy as E F = n max ℏ ω c , then δ R ∼ 33 . 9 N c n max λ c , i.e. it is only a fraction of Compton w av elength, since N c ≫ 1 and n max > 1. Finally , we must express the total energy and pressure of the magnetized degenerate electron gas as a function of electron density and magnetic ﬁeld density . W e ﬁrst express F ermi energy by inv erting eq.(36) in a series expansion: E F = (3 π 2 ) 2 / 3 ℏ 2 2 m e n 2 / 3 e   1 − 1 (9 π ) 2 / 3 ω c λ c c n 2 / 3 e ! 2 (42) − 1 (9 π ) 4 / 3 ω c λ c c n 2 / 3 e ! 4 − 7 3(9 π ) 6 / 3 ω c λ c c n 2 / 3 e ! 6 + . . .   + 1 2 ℏ ω c . The energy density (39) then b ecomes: w e = (3 π 2 ) 2 / 3 ℏ 2 2 m e n 5 / 3 e + n e ℏ ω c + m e ω 2 c 4 π r cl 1 +  n e r 3 cl 9 π  1 / 3 ! − m 3 e ω 4 c 96 π 4 ℏ 2 n e + . . . (43) = (3 π 2 ) 2 / 3 ℏ 2 2 m e n 5 / 3 e + B 2 µ 0 " 1 + 2 π α 2 f m e c 2 µ B B ( n e r 3 cl ) +  1 9 π n e r 3 cl  1 / 3 − α 2 f 6 π 2 B 2 /µ 0 n e m e c 2 # + . . . 18 and the comp onen ts of pressure tensor (40, 41 ) turn into: T x,x = T y ,y = (3 π 2 ) 2 / 3 ℏ 2 5 m e n 5 / 3 e + 1 2 ℏ ω c n e + 1 4 m e ω 2 c  n e 9 π 4  1 / 3 − m 3 e ω 4 c 72 π 4 ℏ 2 n e + . . . (44) T z ,z = (3 π 2 ) 2 / 3 ℏ 2 5 m e n 5 / 3 e + 1 4 m e ω 2 c  n e 9 π 4  1 / 3 + + m 2 e ω 3 c 12(3 π 8 n e ) 1 / 3 ℏ − m 3 e ω 4 c 72 π 4 ℏ 2 n e + . . . (45) Comparing eq.(20) with eq.(42) and eq.(22) with eqs.(44,45), one notes that in the absence of magnetic ﬁeld they are the same. W e note that the anisotrop y of the stress comp onen ts in Eqs. (44) and (45) is precisely the form exp ected from magnetostriction, with magnetic tension along the ﬁeld direction and enhanced transverse pressure. How ever, equation (21) assigns the energy 3 5 E F to the av erage electron, while in eq.(43) the a v erage energy p er electron is E F . This diﬀerence is clearly due to the fact that eq.(21) only includes the kinetic energy of electrons, while eq.(43) includes also the p oten tial energy of electrons with resp ect to electric p olarization, which must b e presen t in the degenerate gas of charged electrons immersed in the cold gas of opp ositely c harged ions. This electric p olarization is needed to allow electrons to o ccupy states with b oth p ositive and negative angular momentum. Energy densit y (43) as a function of electron n umber densit y for electron gas in equilibrium with magnetic ﬁeld of diﬀerent strenghts is sho wn in Fig.5. According to this ﬁgure one can iden tify three distinct parts of the curve for each magnetic ﬁeld: 1) at lo w electron num b er densit y the last term in eq.(43) prev ails, so that the electron gas can not supp ort the required magnetic ﬁeld; 2) at higher densities the term m e ω c 4 π r cl = B 2 µ 0 prev ails, making the electron gas magnetic ﬁeld dominated and can not be controlled by c hanging its pressure; 3) only at high enough density the ﬁrst term in eq.(43) prev ails, making the degeneracy pressure the dominant force, yet in this regime the magnetic contribution ( n e ℏ ω c ) starts increasing linearly with densit y (dotted lines in Fig.5) and is only ov ertaken b y degeneracy pressure b ecause of its 5 / 3 exp onen t. W e note that the state of degenerate electron gas in the cores of white dw arfs is in this third regime where gra vitational balance essentially dictates the lo cal v alue of F ermi energy . In this case the energy density can b e expressed with eq.(39), which shows that under said conditions the energy density has a minim um with resp ect to magnetic ﬁeld ( ω c ) at: B min = m e 3 π 2 √ 2 e s τ 0  E F ℏ  3 = 1 4 µ 0 µ B n e , (46) 19 whic h is by 1 µ 0 B 2 b elo w the v alue at B → 0. W e also note that any realistic stellar mo del has an atmosp ere, where electron density is to o lo w to contribute its own magnetization. In summary , the microscopic analysis of the magnetized degenerate electron gas conﬁrms the macroscopic assumptions introduced in subsection II I A. The in teraction b et ween Landau quan tization, the small charge im balance induced by the mass asymmetry of electrons and ions, and the resulting p olarization ﬁeld leads to a magnetized ground state in which B ≃ µ 0 M and the auxiliary ﬁeld H is negligible. Thus the internal magnetic energy is naturally incorporated in to the free energy of matter, while the w ork required to establish the observ able magnetic dip ole resides in the surface term identiﬁed earlier. The tw o approaches therefore describ e the same physical phenomenon at diﬀerent lev els: subsection I II A provides the minimal-action macroscopic representation, and subsection I I I B supplies the corresp onding microscopic mechanism that pro duces intrinsic magnetization in degenerate stellar matter. IV. “LANE-EMDE EQUA TION” F OR A MAGNETIC ST AR In subsection (I II A) we argued that the contribution of magnetic ﬁeld to the action can b e expressed only by the surface integral, while in subsection (I II B) we show ed that the degenerate electron gas can generate magnetization in equilibrium with electromagnetic ﬁeld. The main eﬀect of such equilibrium is the shift of internal energy of electron gas, but it do es not essentially c hange the equation of state (eq.22 vs. 44,45 ) except for the pressure anisotropy . Lane-Emde equation, which follo ws from the lo cal equilibrium equations (1), is not exp ected to c hange if magnetic ﬁeld is added to action. Ho wev er, at the magnetic surface lo cal equilibrium changes, since the electromagnetic stress is discontinous at the magnetic surface and pro duces additional pressure, which m ust b e comp ensated b y gravit y . The matching conditions at the magnetic surface follow directly from the requiremen t that the total stress–energy tensor of matter, gra vity , and electromagnetism has v anishing divergence in the linearized theory [21]. Consider a pillbox near the surface of a star. F or an y box completely inside or completely outside the star the sum of forces acting on a pillb ox v anish. This statement can b e expressed by surface integrals of their corresp onding stress energy tensors gra vitational, matter 9 and electromagnetic as: R S  T g + T p + T m  · d  S = 0. In other words, the sum of div ergencies of these tensors v anishes T g µν . ν + T p µν . ν + T m µν . ν = 0 . (47) 9 here designated with p, b ecause its space comp onen ts represen t pressure 20 A t the magnetic surface the sum of stress energy tensors can b e expressed as T all µν =  T g µν + T p µν + T m µν  ext Θ( ξ − 1) +  T g µν + T p µν + T m µν  int Θ(1 − ξ ), where Θ represents the Hea viside function as the function of the radial co ordinate ξ , such that ξ = 1 is at the surface of the star. Since the divergence of T all v anishes except at the surface, the divergence at the surface b oils down to T all µν . ν =  T m µξ + T p µξ  ext −  T p µξ + T m µξ  int  δ ( ξ − 1) , (48) where we hav e taken into account the fact that the gravitational stress-energy tensor is contin uous across the b oundary . In order to express these stress tensors, it is conv enient to use the cov ariant formalism of linearized gra vity . The matter tensor can b e considerd as the sum of the isotropic pressure tensor, expressed with resp ect to arbitrary static co ordinates ( x µ ) as T p µν = ρc 2 + p c 2 u µ u ν + pg µν and a weak anisotropic comp o- nen t describ ed b y the the anisotropic pressure induced by electromagnetic ﬁeld as in eqs.(44,45). Here g µν is the metric tensor and pressure p ( x µ ) is a scalar. The cov ariant comp onents of gravitational stress- energy tensor, which replaces the Einstein tensor in linearized gravit y , can b e expressed with the gra vi- tational p oten tial Φ g as: ( T g ) µν = 1 4 π G  ∂ Φ g ∂ x µ ∂ Φ g ∂ x ν − g µν ∂ Φ g ∂ x λ g λσ ∂ Φ g ∂ x σ  , and the contra v ariant comp onen ts of electromagnetic stress ene rgy tensor can b e expressed with electromagnetic vector p oten tial 1-form ( A ) as: ( T m ) µν = ⋆ [( d x µ ∧ d A ) ∧ ⋆ ( d x ν ∧ d A )] − 1 2 g µν ( d A ∧ ⋆ d A ). The dynamic equilibrium condition can b e expressed in Minko wski co ordinates ( t, x , y , z ) in vector form as: ( T g ) µν . ν → ( 1 4 π G ∆Φ g ) ∇ Φ g = ρ ∇ Φ g ; ( T m ) µν . ν → − ρ e ∇ Φ e −  j e ×  B , where ρ e and  j e are charge and current density and Φ e is electric p oten tial; ( T p ) µν . ν → ∇ p . T o obtain analytic insight in to interaction of magnetic ﬁeld with gravit y , we again return to the simplest p olytropic mo del, the one for n = 0. The basic equations (2) and (3) with n = 0 and ω = 0 still apply together with the Poisson equation for gra vitational ﬁeld, ho wev er, boundary conditions change. The incompressible ﬂuid mo del mak es b oundary conditions particularly simple, b ecause the magnetic surface ma y (almost) coincide with the surface of the star to make the atmosphere inﬁnitely thin. W e show that in this case an oblate elliptical surface satisﬁes the stress energy condition. 21 In elliptic co ordinates 9, the in ternal and external gravitational p oten tials (12,13) can b e written as: Φ g = 1 3 a 2 Ω 2  ξ 2 − 1 − 2(1 + e 2 ) arctan e e +  e − (1 + e 2 ) arctan e e + e (3 + 2 e 2 ) − 3(1 + e 2 ) arctan e e 3 ξ 2  P 2 (cos θ )  ξ ≤ 1 → 1 3 a 2 Ω 2  − 3 + ξ 2 + e 2  − 4 3 + 2 15 (3 ξ 2 − 5) P 2 (cos θ )  + O ( e 4 ) Φ g = − 1 3 a 2 Ω 2  2(1 + e 2 ) arctan e ξ e + (1 + e 2 ) (3 ξ 2 + e 2 ) arctan e ξ − 3 e ξ e 3 P 2 (cos θ ) ! ξ ≥ 1 (49) → 2 3 a 2 Ω 2  − 1 ξ − e 2 3 ξ 3  − 1 + 3 ξ 2 + 2 5 P 2 (cos θ )  + O ( e 4 ) The electromagnetic ﬁeld of an ellipsoidal dip ole can b est b e expressed in terms of the v ector p oten tial one form. The main ingredient of the external ﬁeld must b e the static magnetic dip ole ﬁeld satisfying the ﬁeld equation ⋆ d ⋆ d A = 0. How ever, the discussion of chapter 3 suggests that a (small) electric quadrup ole comp onen t ( A 0 d t ) , generated by the surface charge, must also b e present. The same discussion suggests that an internal electric p olarization is required to allow ele ctrons to o ccupy nonzero angular momen tum orbits. Thus, the internal solution must com bine a pure magnetic ﬁeld, an electric comp onen t representing the chage imbalance, and a sourceless electric quadrup ole generating the surface charge whic h makes the system electrically neutral. The ﬁeld, which satisﬁes the ab o ve prop ositions, can b e expressed as follo ws: A = − B 0 2 a 2 ( ξ 2 + e 2 ) sin 2 θ d ϕ + S a 2 ( ξ 2 − 1)  1 + 2 e 2 3 + e 2 P 2 (cos θ )  c d t a + Qa 2  ξ 2 + e 2 3  P 2 (cos θ ) c d t a ξ ≤ 1 A = − B 0 2 a 2 (1 + e 2 ) ( ξ 2 + e 2 ) arctan e ξ − e ξ (1 + e 2 ) arctan e − e sin 2 θ d ϕ + Qa 2 (1 + e 2 3 ) (3 ξ 2 + e 2 ) arctan e ξ − 3 e ξ (3 + e 2 ) arctan e − 3 e P 2 (cos θ ) c d t a ξ ≥ 1 (50) In the next step we express stress energy tensors ( T g ) µν , ( T m ) µν and ( T p ) µν . The calculation of the ﬁrst t wo is straightforw ard but quite messy and needs not b e repro duced here in full detail, except for the fact 22 that the divergence of the electromagnetic stress energy tensor c an b e written as: ( T m ) ν µ . ν = 18 S 2 µ 0 (3 + 2 e 2 )           0 − ξ (2 + e 2 + e 2 cos 2 θ ) e 2 (1 − ξ 2 ) sin 2 θ 0           − 3 Q S µ 0           0 − ξ (1 + 3 cos 2 θ ) (3 ξ 2 + e 2 ) sin 2 θ 0           (51) whic h is a gradient of a scalar expressed in the formalism of diﬀerential forms as: d x µ ( T m ) ν µ . ν = 1 µ 0 d  9 S 2 3 + e 2  (2 + e 2 ) ξ 2 + e 2 ( ξ 2 − 1) cos(2 θ )  + 3 Q S 2  ξ 2 + (3 ξ 2 + e 2 ) cos(2 θ )   . (52) The in terior equilibrium condition can, therefore, b e expressed only by the exterior deriv ative of a 0-form: d x µ ( T g ) ν µ . ν + d x µ ( T p ) ν µ . ν + d x µ ( T m ) ν µ . ν = d ( ρ Φ g ) + d p + 1 µ 0 d  9 S 2 3 + e 2  (2 + e 2 ) ξ 2 + e 2 ( ξ 2 − 1) cos(2 θ )  + 3 Q S 2  ξ 2 + (3 ξ 2 + e 2 ) cos(2 θ )   , (53) whic h m ust v anish. This equation deﬁnes pressure up to a constan t p 1 . One notes, how ever, that the in terior condition can still b e satisﬁed if a divergence-free tensor is added to pressure. By adding β a 2 Ω 4 π G ( T z z ), where ( T z z ) µν d x µ d x ν = d z 2 , one can mo del magnetically induced pressure anisotropy (cf. Eqs 40 and 41), where β is a dimensionless constant determining its strength. Boundary conditions (48) that remain to b e solv ed are: ∆ T µξ =  T m µξ  ext −  T p µξ + T m µξ  int = 0 for µ → ξ and µ → θ , while the other tw o components are iden tically zero. W e in tro duce dimensionless v ariables p 1 = ρa 2 Ω 2 π 1 , B 0 = a Ω 2 r µ 0 π G B , (54) Q = a Ω 2 r µ 0 π G Q , S = a Ω 2 r µ 0 π G S , and obtain the following expression for the ( ξ , θ ) comp onen t: ∆ T ξθ = ( c 0 + c 2 cos(2 θ )) sin(2 θ ) , (55) where: c 0 = − 6 e 3 (1 + e 2 ) (1 + e 2 ) arctan e − e B 2 − 6 e 5 + 2 e 7 (3 + e 2 ) arctan e − 3 e Q 2 − 18(1 + e 2 )(2 + e 2 ) Q S + 6(1 + e 2 ) β , c 2 = − 6 e 5 (3 + e 2 ) (3 + e 2 ) arctan e − 3 e Q 2 − 18( e 2 + e 4 ) QS . (56) 23 The ( ξ , ξ ) comp onen t of the b oundary condition is expressed in the form: ∆ T ξξ = 1 2 d 0 + 5 2 d 2 P 2 (cos θ ) + 9 2 d 4 P 4 (cos θ ) , (57) with the co eﬃcien ts: d 0 = − 8 e 3 (1 + e 2 )(2 e + e 3 − 2(1 + e 2 ) arctan e ) 3((1 + e 2 ) arctan e − e ) 2 B 2 + 64 e 5 (9 e + 9 e 3 + e 5 − 3(3 + 4 e 2 + e 4 ) arctan e ) 45((3 + e 2 ) arctan e − 3 e ) 2 Q 2 + 8 e 2 (19 + 29 e 2 + 13 e 4 + 3 e 6 ) 5(3 + e 2 ) QS + 24(1 + e 2 ) 2 (60 + 85 e 2 + 34 e 4 + 5 e 6 ) 5(3 + e 2 ) 2 S 2 + 4 3 (1 + e 2 )(3 + e 2 ) π 1 + 8 3 (1 + e 2 ) 2 β + 8(1 + e 2 ) 2 (2(3 + e 2 ) arctan e − e ) 15 e (58) d 2 = 8 e 3 (1 + e 2 )(2 e + e 3 − 2(1 + e 2 ) arctan e ) 15((1 + e 2 ) arctan e − e ) 2 B 2 + 128 e 5 (9 e + 9 e 3 + e 5 − 3(3 + 4 e 2 + e 4 ) arctan e ) 315((3 + e 2 ) arctan e − 3 e ) 2 Q 2 + 16(1 + e 2 )(21 + 11 e 2 + 14 e 4 + 6 e 6 ) 35(3 + e 2 ) QS + 48 e 2 (1 + e 2 ) 2 (14 + 23 e 2 + 7 e 4 ) 35(3 + e 2 ) 2 S 2 + 8 e 2 15 (1 + e 2 ) π 1 + 16 15 (1 + e 2 ) 2 β + 4(1 + e 2 ) 2 ((21 + 18 e 2 + 13 e 4 ) arctan e − 21 e − 11 e 3 ) 105 e 3 (59) d 4 = 128 e 5 (9 e + 9 e 3 + e 5 − 3(3 + 4 e 2 + e 4 ) arctan e ) 315((3 + e 2 ) arctan e − 3 e ) Q 2 + 64 e 2 (3 + 3 e 2 + e 4 + e 6 ) 105(3 + e 2 ) QS − 128 e 4 (1 + e 2 ) 2 35(3 + e 2 ) 2 S 2 + 16(1 + e 2 ) 2 ((3 + e 2 ) arctan e − 3 e ) 315 e . (60) The ﬁve co eﬃcien ts ( c 0 , c 2 , d 0 , d 2 and d 4 ) must v anish, which gives ﬁve nonlinear equations for the ﬁve constan ts B , Q , S , π 1 and β . These complicated equations hav e surprisingly simple solutions. F or e = 0 the solution b ecomes: π 1 = − 2 3 (1 + 12 S 2 ) , B = 0 , Q = 0 , β = 0. F or S = 0 this is the clasical spherical gravitational solution. F or S  = 0 the solution also con tains an internal radial electric ﬁeld ( A = S Ω 2 p µ 0 π G a 3 ( ξ 2 − 1) c d t a ) which is screened by negative surface c harge, as discussed in the previous c hapter. Noting that A 0 is the negativ e electrostatic p oten tial, one can express the electric charge im balance 24 densit y as − ε 0 ∆( A 0 ) = 12 π √ ε 0 Gρ S . F urthermore, by expressing mass density as ρ = A m m p n A and electron n umber density as Z n A , the ratio of charge imbalance density to electron num b er density can b e written as 12 A m Z ( √ π G ε 0 m p e ) S ≈ 5 . 4 × 10 − 18 A m Z S . F or nonzero ellipticity tw o types of solutions exist: ellipticity may b e induced sollely b y internal electric p olarization: π 1 = − (1 + e 2 )(12 + 17 e 2 + 10 e 4 + e 6 ) arctan e − e (12 + 13 e 3 + 3 e 4 ) 4 e 5 → − 14 15 − 27 35 e 2 + 2 105 e 4 + . . . (61) S = (3 + e 3 ) p (3 + e 2 ) arctan e − 3 e 6 e 2 √ 2 e → 1 √ 30 − 1 21 r 2 15 e 2 + 1 49 √ 65 e 4 + . . . (62) B = 0 , Q = 0 , β = 0 , (63) or b y a combination of electric p olarization and magnetic ﬁeld: π 1 = (6 − e 2 − 7 e 4 ) arctan e − e (6 + 13 e 2 + 9 e 4 ) 18 e 3 (64) → − 8 9 − 217 270 e 2 + 67 945 e 4 + . . . β = e (9 + 7 e 2 ) − (9 + 10 e 2 + e 4 ) arctan e 18 e 3 → 4 135 e 2 − 4 189 e 4 + . . . B = 1 3 (1 + e 2 ) arctan e − e e 2 → 2 9 e − 2 45 e 3 + . . . (65) S = 1 6 (66) Q = (1 + e 2 )((3 + e 2 ) arctan e ) − 3 e 2 e 3 (3 + e 2 ) → − 2 45 e 2 + 8 945 e 4 + . . . (67) The relation b et w een ellipticity and magnetic ﬁeld can b e obtained also by applying the v ariational principle. W e rep eat the argument used in deriving eq.16 and replace rotational energy by magnetic energy as deriv ed in section 3.1. The gravitational energy , magnetic energy and the action can b e written as: W g = − 3 G M 2 5 R ef (1 + e 2 ) 1 / 3 arctan e e W mag = B 2 0 2 µ 0  4 π 3 R 3 ef  ( e − arctan e ) (1 + e 2 ) arctan e − e A = W g + W mag + λ 1 µ 0 B 0  4 π 3 R 3 ef  , (68) 25 where R ef = (1 + e 2 ) 1 / 6 a . Solving ∂ A ∂ B 0 = ∂ A ∂ e = 0 and expressing B 0 with dimensionless B , one obtains 10 : B = r 2 15 (1 + e 2 ) arctan e − e e 2 (69) Comparing eqs. 69 and 65 one ﬁnds the same functional dep endence of B on e , only the factor q 2 15 ab o ve is ab out 10% higher than 1 3 from eq. 65. Noting that in the p olytropic mo del the scale of B is inv ersely prop ortional to cen tral pressure: B = q π G µ 0 1 a Ω 2 = 1 √ µ 0 π Gp c B 0 , where p c is the cen tral pressure as deﬁned b y p olytropic mo del (cf deriv ation of eq.8) the discrepancy may b e plausible, since in the v ariational problem w e did not take into account that the equation of state (53) is mo diﬁed by additional pressure generated by electric p olarization in the presence of electromagnetic ﬁeld. A. Magnetic surface instability In the previous section we ha ve sho wn that a star can reach equilibrium with magnetic ﬁeld if it assumes elliptic shap e. Y et, the ab o v e analysis could not prov e that the op en b oundary gravito-magnetic problem has a unique solution. Exp erimen ts with magnetic ﬂuid in the presence of magnetic ﬁeld suggest that the surface of magnetic ﬂuid develops a spiny structure whic h disp erses a simple dip olar ﬁeld into a high order m ulip ole ﬁeld whic h shields the in ternal ﬁeld from spreading aw ay from the source. A detailed analysis of this phenomenon presents a rather diﬃcult problem. F or the present analysis we assume, as in the n = 0 mo del, that the magnetic surface coincides with the physical surface of the star. T o get some idea how the magnetic surface can b e deformed by forming wrinkles, we ask what other surface shapes can also minimize the energy of the star with a given magnetic moment. In principle an y inﬁnitesimally deformed sphere can be describ ed in spherical coordinates by expanding the radius in terms of spherical harmonics as r S ( θ , ϕ ) = R + P l,m a l,m Y m l ( θ , ϕ ). The deformation generally increases gra vitational energy , while magnetic energy for a giv en magnetic momen t decreases, if the magnetic ﬁeld of a dipole is dispersed in to higher m ultip oles. Higher m ultip oles conﬁne the magnetic ﬁeld more tigh tly to the vicinity of the surface, reducing the volume of space ﬁlled with magnetic ﬂux and therefore low ering the external magnetic energy for a ﬁxed magnetic moment. If magnetic pressure can do enough work during the deformation to comp ensate for the increase of gravitational energy , then such a deformation leads to a lo wer energy state. 10 Note that during the v ariational pro cess the v olume and density of the star m ust remain constan t, therefore, during this pro cess, w e in tro duce a new dimensionless magnetic ﬁeld with B 0 = R ef Ω 2 p µ 0 π G ˜ B . 26 Because of complexity of the general problem, we limit the analysis to cylindrically symmetric de- formations of the form r S ( θ ) = a (1 + δ λ P λ (cos θ )) for even p ositiv e integers λ . W e solv e gravitational and electromagnetic equations in a system of co ordinates ξ , ζ , ϕ deﬁned so that ζ = cos θ and x = r S ( ζ ) ξ p 1 − ζ 2 cos ϕ, y = r S ( ζ ) ξ p 1 − ζ 2 sin ϕ, z = r S ( ζ ) ξ ζ , such that ξ = 1 represents the surface. F unctions ψ ( − ) l = ξ l r S ( ζ ) l P l ( ζ ) and ψ (+) l = 1 ξ l r S ( ζ ) l +1 P l ( ζ ) solve the Poisson equation for ξ < 1 and ξ > 1 resp ectiv ely , while 1-forms α ( − ) l = (1 − ζ 2 ) ξ l +1 r l +1 S ( ζ ) dP l ( ζ ) dζ d ϕ and α (+) l = 1 − ζ 2 ξ l r l S dP l ( ζ ) dζ d ϕ solve the electro- magnetic equation d ⋆ d α = 0. T ogether with the particular solution of gravitational Poisson equation and these functions one can construct solutions of gravitational and magnetic ﬁeld equiv alent to eqs.(49) and (50). The only diﬃculty is that eigenfunctions ψ ( − ) l , ψ (+) l and 1-forms α ( − ) l , α (+) l do not ha ve the same angular dep endence at the surface, since r S is a function of ζ . Therefore, matching in terior to exterior solution requires expanding eigenfunctions at the surface in a common basis of Legendre p olynomials. W e use the ﬁrst order expansion in terms of parameter δ . Numerical pro cedure is straigh tforward, but quite tedious and must b e done n umerically . Numerical results lead to expansion co eﬃcien ts k ( g ) λ and k ( m ) λ through which energy is expressed as a function of parameter δ in the form W g = − 3 5 G M 2 R ef (1 − k ( g ) λ δ 2 ) (70) W m = 3 4 G M 2 R ef B 2  1 + k ( m ) λ δ 2  . (71) The case λ = 2 is particular. Here the sphere is deformed into a prolate ellipsoid for small p ositiv e δ and an oblate one for negativ e δ . This solution naturally leads to results of previous section noting that magnetic momen t v aries with δ and that δ = (2( − 1 + √ 1 + e 2 )) / (2 + √ 1 + e 2 ) ∼ e 2 3 − 5 e 4 36 + . . . . F or all λ > 2 b oth energies hav e an extrem um at δ = 0 - a minim um for gravitation and a maximum for magnetism. If  3 5 k ( g ) λ + 3 4 k ( m ) λ B 2  δ 2 > 0 then gravit y prev ails, and the energy has a minimum whic h mak es the system stable against perturbation, otherwise the c hosen deformation allows a low er energy state making the system unstable with resp ect to this p erturbation. 27 The numerically calculated co eﬃcien ts k ( m ) λ and k ( g ) λ are shown in Fig.7 together with extrap olations for large 11 λ . Since k ( m ) λ approac hes a constant v alue ( ∼ 3 . 27) for large λ and k ( g ) λ decreases in absolute v alue, the magnetic surface is unstable with resp ect to deformations with λ greater than the solution of equation B = 2 √ λ − 1 q k ( m ) λ (2 λ +1) ∼ 1 √ 3 . 27 λ . Stellar magnetic ﬁelds are typically quite small, which means that magnetic surface can only b e unstable with resp ect to deformations of high order λ crit ≈ 1 | k ( m ) λ →∞ | B 2 , (72) so that deformations corrugate the surface on a small length scale. Predicting the shap e of such corrugations is a diﬃcult task, which is b ey ond the scop e of this article. Ho wev er, a similar Rosensweig instability [26] is observed in exp erimen ts with ferroﬂuids whose surface in a jar b ecomes vertically corrugated if a suﬃcien tly strong magnet is approached from b elow. Let us calculate the typical size of corrugations that might b e exp ected on a magnetic ob ject the size of Earth. Expressing B with p olar ﬁeld B 0 according to eq.(54), w e obtain λ crit ≈ µ 0 a 2 Ω 4 π G k ( m ) λ B 2 0 ∼ 4 × 10 5 ( B 0 / T esla) 2 , whic h is to sa y that the magnetic ﬁeld of ∼ 100 gauss w ould generate corrugations of ab out 2 π R E arth /λ crit ∼ 1cm . This is roughly comparable to the size of ripples observed in exp erimen ts with magnetic ﬂuid. Corrugation of magnetic surface may app ear unimp ortan t regarding the magneto-mechanical equilibrium of a star, which is dominated by the magnetic dip ole moment deﬁned b y the integral of magnetization o ver the volume of the star. How ever, stellar magnetic ﬁelds are usually deduced from the dip ole moment deduced from the external magnetic ﬁeld. These t wo dip ole moments are the same only for spherical magnets. Corrugations disperse magnetic energy be t ween high m ultip oles, which mak es the exterior dipole comp onen t w eaker than the one deﬁned by in terior magnetization. The phenomenon is illustrated in Fig.8, which represents magnetic ﬁeld of a solid ideal ferromagnet whose surface is describ ed by r S = 1 + 0 . 1 P 10 (cos θ ). Corrugations generate additional mini-p oles which redistribute the internal ﬁeld to the 10-p ole and other higher multipoles. 11 The expression k ( g ) λ = 5 2(2 λ +1)  1 − 3 2 λ +1  ﬁts the ﬁrst ﬁve numerically obtained v alues to nine signiﬁcant ﬁgures and approaches asymptotically the 1 /λ dep endence whic h follo ws from ﬁrst order p erturbation analysis. In the case of magnetic co eﬃcien t k ( m ) λ w e don’t hav e a ﬁrm argument for asymptotic b eha viour, except for the fact that calculated parameters can b e quite accurately ﬁt with a T a ylor series of in v erse p o wers of λ . 28 The details of a free surface of a star are muc h more complicated, b ecause we do not kno w the shap e and the length of corrugations. The fact that the magnetic surface must be in most cases b elo w the actual surface of the star complicates the matter even further. Y et it seems quite safe to say that the internal magnetic ﬁeld of stars is stronger, if not m uch stronger than the ﬁeld deduced from measurements, which are usually based on the assumption of magnetic dip ole. Finally , we must add a cav eat. The ab o ve analysis is v alid only if the magnetic surface is at the physical surface of the star. More realistic mo dels m ust take in to account the fact that, a given magnetization can b e sustained inside the star, only up to a certain level to which the state of matter is adequately represented b y equations (43,44, 45) in the domain of pressure domination (cf. Fig.(5)). Magnetic surface must b e at the lev el where electron densit y falls below the critical density at whic h the degenerate gas changes from pressure to magnetic ﬁeld dominated. Beyond this level pressure is magnetic ﬁeld dominated, but density is not. Therefore, magnetic pressure do es not push against gravit y . As a result the magnetic surface can deform without muc h change in gravitational energy . Therefore, one should exp ect that the magnetic instability sets in at muc h, muc h larger wa v elengths than predicted by eq.72. V. R OT A TING MA GNETIC ST AR The extension of ”Lane-Emde equation” for a magnetic star to a rotating magnetic star is straightforw ard. Rotation adds ”nondiagonal” terms to the matter tensor and cen trifugal pressure ( 1 2 ρ ω 2 R 2 ) to pressure. In Mink owski co ordinates the stress tensor of moving matter is written as: T p µν = ρc 2 + p c 2 u µ u ν + pη µν , where u µ are comp onen ts of 4-velocity and η µν represen ts the Minko wsky tensor. This form is the standard stress tensor of a rigidly rotating ﬂuid in the linerized (Newtonian) limit, the velocity comp onen ts of matter rotating ab out the z axis are: v x = − ω y , v y = ω x , so that the comp onents of rotating matter stress-energy tensor are:  T p  µν − →           − ρ c 2 + p 0 0 0 0 p 0 0 0 0 p 0 0 0 0 p           +           1 2 ρ ω 2 ( x 2 + y 2 ) − ρc ω y ρc ω x 0 − ρc ω y ρ ω 2 y 2 − ρ ω 2 x y 0 ρc ω x − ρ ω 2 x y ρ ω 2 x 2 0 0 0 0 0           The divergence of the second tensor in the ab o ve expression T rot only has space-like comp onents and can b e written as  T rot  µν . ν − → ρ ω 2  R + ρ c ( d ω dt ×  R ), where  R stands for { x, y , 0 } . Here the cen trifugal term con tributes only to the diagonal spatial comp onen ts, reﬂecting the additional pressure generated b y rigid 29 rotation. If dω dt = 0, the problem simpliﬁes, since the space-time comp onen ts of T rot con tribute no force to rotating mater. Suc h a simpliﬁcation o ccurs only if the electromagnetic stress tensor has no space-time comp onen ts to b e comp ensated b y matter, i.e. if the magnetic ﬁeld is aligned with rotation axis. W e restrict our analysis to the case where the magnetic ﬁeld is aligned with the rotation axis, since any misalignment pro duces time-dep enden t stresses that can not b e treated witin the stationary v ariational framework used here. Our discussion of rotating magnetic stars is limited to this simple example. Expressing spatial comp onen ts of T rot as a bilinear 1-form and translating it in the language of elliptic co ordinates: d x i  T rot  ij d x j = ρ ω 2 ( x d y − y d x ) 2 = ρ ω 2 ( ξ 2 + e 2 ) 2 sin 4 θ d ϕ 2 , one ﬁnds that the only nonzero comp onen t is T ϕϕ rot = ρ ω 2 . The equilibrium of the star is again expressed b y eq.(47),where gravitational and magnetic ﬁeld tensors follo w from (49) and (50), while T p has the additional ω 2 term. The equation 53 now turns into d x µ ( T g ) ν µ . ν + d x µ ( T p ) ν µ . ν + d x µ ( T m ) ν µ . ν = (73) d ( ρ Φ g ) + d  p + ρ ω 2 2 ( ξ 2 + e 2 ) sin 2 θ  + 1 µ 0 d  9 S 2 3 + e 2  (2 + e 2 ) ξ 2 + e 2 ( ξ 2 − 1) cos(2 θ )  + 3 Q S 2  ξ 2 + (3 ξ 2 + e 2 ) cos(2 θ )   The b oundary condition (55) is unchanged by rotation, so that the corresp onding conditions (56) remain unc hanged, while the conditions (57) are changed to: d 0 → d 0 + 4 3 (1 + e 2 ) 2 (1 + 1 5 e 2 )  ω Ω  2 , d 2 → d 2 − 4 15 (1 + e 2 ) 2 (1 − 1 7 e 2 )  ω Ω  2 and d 4 → d 4 − 16 315 e 2 (1 + e 2 ) 2  ω Ω  2 . The solution of the new equations, equiv alent to 64 - 67, b ecomes: π 1 = (6 − e 2 − 7 e 4 ) arctan e − e (6 + 13 e 2 + 9 e 4 ) 18 e 3 − (24 + 35 e 2 + 11 e 4 ) arctan e − e (24 + 31 e 2 + 9 e 4 ) 18 ((3 + e 2 ) arctan e − 3 e ) ˜ ω 2 (74) β = e (9 + 7 e 2 ) − (9 + 10 e 2 + e 4 ) arctan e 18 e 3 + (9 + 10 e 2 + e 4 ) arctan e − e (9 + 7 e 2 ) 18 ((3 + e 2 ) arctan e − 3 e ) (75) B 2 =  1 3 (1 + e 2 ) arctan e − e e 2  2 −  (1 + e 2 ) arctan e − e  2 9 e ((3 + e 2 ) arctan e − 3 e ) ˜ ω 2 (76) 30 S 2 =  1 6 2 − e 3 36 ((3 + e 2 ) arctan e − 3 e ) ˜ ω 2  (77) Q 2 =  1 + e 2 3 + e 2 (3 + e 2 ) arctan e − 3 e 2 e 3  2 −  1 + e 2 3 + e 2  2 (3 + e 2 ) arctan e − 3 e 4 e 3 ˜ ω 2 (78) F or the purely magnetic case ( ˜ ω = ω Ω = 0) this solution go es to the solution 64 - 67, while for pure rotation ( B = 0) it leads to the McLaurin solution 14 and sets β = 0, S = 0 and Q = 0. 68 as W = W g + W mag + W rot , and W 0 is the gravitational energy of spherical star. Figure 9 shows a phase diagram, based on the exact n = 0 p olytropic solution, for stars with av ailable measuremen ts of rotation and magnetic ﬁeld. The axes are the magnetic parameter B , the dimensionless spin parameter, ˜ ω and energy excess W . W e deﬁne B as the ratio of the magnetic ﬂux through the equatorial cross-section to the stellar mass, ˜ ω as the ratio of the observed stellar angular velocity , ω , to the critical breakup v alue, Ω K = √ 2 π Gρ , and W = W g + W mag + W rot (cf. eq.17). A broad range of stellar ob jects is found to cluster in the same region of this diagram. This coincidence supp orts the interpretation that the electromagnetic ﬁeld preserves its long-range character through magnetism. The Sun, Earth Jupiter and Saturn are represented by blue p oints on dotted lines whose p ositions corresp ond to the measured angular velocity ˜ ω . P arallel to those lines are lines (red for Sun, green for Earth, magen ta for Jupiter and brown for Saturn) corresp onding to their measured ellipticity parameter e . Note that calculated ellipticity parameters for Earth and Jupiter are quite close to measured v alues represen ted by coloured lines and can easily b e understo od within the p olytropic mo del with 0 < n < 1 . 5. Ho wev er, the ellipticit y of the Sun is less than even the softest p olytropic mo del with n = 3 would predict and may even b e correlated with solar cycle activit y [24]. The rest of the diagram is mostly p opulated b y pulsars, w ell known for their rotation and magnetic ﬁeld. The p ositions of pulsars in the diagram were calculated from their p ositions in the P-Pdot diagram assuming their mass and radius to b e 1.3 M ⊙ and 13.5km respectively . T o indicate the p ossible uncertaint y in calculating ˜ ω and B , the Crab pulsar has four small satellites, whic h were calculated by assuming pulsar masses to b e 2.09, 2.03, 1.7, .67 M ⊙ , with resp ectiv e radii (10, 12, 13, 14 kilometers), which follow from a recen t neutron star mass-radius model [4]. The pulsar p opulation is represen ted b y mem b ers of the b oundary in the P-Pdot diagram shown at right of Fig. 9; pulsars ha ve the same color in b oth ﬁgures. Magnetars presen ted in ([8]) are also shown in blac k, together with their recently disco vered un usual pulsar PSR J0901- 4046 assuming that their mass is the largest p ossible neutron star mass 2.13 M ⊙ according to [4]. The same 31 magnetars are shown as smaller grey balls if their mass is assumed to b e 1.3 M ⊙ . According to [30] the magnetic ﬁeld of PSR J0901-4046 may b e more than 100 times stronger, which would bring it to the level of ”Gleam” (GLEAM-X J162759.5-523504.3 [8]). White dwarfs are known to b e magnetic and do rotate. Ho wev er, less data give their measured mass together with rotation p eriod and magnetic ﬁeld. Tw o suc h examples are white dw arfs J221141.80 [25] and J1745 [32] with known rotation rates (P=614 s , M=0.92 M ⊙ , B > 7 × 10 6 gauss) and (P=70.36, M=1.268 M ⊙ , B = 15 × 10 6 gauss) 12 are shown in Fig.9 as small blue spheres num b ered 1 and 2. Magnetic ﬁeld can also b e detected on some white dwarfs through Zeeman splitting of hydrogen and helium lines ([3],[10],[13],[14]), ho wev er data on their rotation are generally not av ailable. Nicola Pietro Gentile F usillo from Universit y of T rieste has b een kind enough to provide a priv ate collection of a v ailable data containing information on mass, temp erature, parallax, sp ectral data and magnetic ﬁeld of 139 white dwarfs. In Fig.10 we display these data as the distribution of magnetic white dwarfs with resp ect to mass and temp erature 13 . It was quite surprising to ﬁnd out that white dwarfs are distributed on the same interv al of magnetic moment B as all other celestial ob jects shown in Fig.9. A recent pap er [17] quotes rotational p erio ds of a num b er of magnetic white dwarfs, four of which (WD 1 : J0043-1000, WD 2 : J1214-1724, WD 3 : J1659-4401 and WD 4 : J2257-0755) are also members of Nicola’s collection. They are p oin ted at by arrows in Fig.10 and are also in tro duced in Fig.9 as blue balls. A recent article [16] quotes rotation p eriods masses and magnetic ﬁeld of ten more white dw arfs (WD1: 0011-134, WD 0009+501, WD 2359-434, WD 0011-721, WD J075328.47- 511436.98, WD J171652.09-590636.29, WD 0912+536, WD 0041-102, WD 2138-332, LSPM J0107+2650). They are represented by blue balls and are n umbered from 3 to 12 in Fig.9. A recently studied W olf Ray et star HD 45166 with v ery high magnetic ﬁeld ([28]) is also shown in Fig.9 as a blue ball. The high B v alue of this star with resp ect to white dw arfs is due to the fact that HD is a “normal size” star ( R = 2 . 6 R ⊙ , M = 3 . 4 M ⊙ ), while white dwarfs are more than 100 times smaller in radius, therefore, their B v alue (which measures magnetic ﬂux p er mass of the star) may b e less even if their magnetic ﬁeld is higher. 12 Since white dw arf radii are not given in the quoted sources, we use the simple mass- radius relation R wd ∼ ℏ 2 2 m e G M 1 / 3 ( Z m p ) 5 / 3 with Z = 1 / 2 (n um b er of electrons p er nucleon) 13 The magnetic parameter B was calculated as ab o ve, cf. fo otnote 12 32 VI. DISCUSSION The phase diagram in Fig.9 may b e considered as a demonstration of interpla y betw een magnetism and rotation in breaking the symmetry of gravit y . W e ﬁnd that v ery diﬀerent celestial ob jects, planets, normal stars, white dw arfs and pulsars o ccup y more or less the same, relativ ely limited region in this diagram, where angular v elo cit y ( ˜ ω ) is scaled by characteristic rotational breakup velocity (Ω) and magnetic ﬁeld parameter B representing magnetic ﬂux p er mass of the star. On a linear scale this region would app ear as an extremely narrow band, only ab out tw o orders of magnitude wide but extending o v er nearly eight orders of magnitude in length, which makes the observ ed clustering even more striking. Remem b ering that rotational and magnetic dip ole and total energy of a star can b e expressed as W rot ∼ 1 5 M a 2 Ω 2 ˜ ω 2 , W mag ∼ 1 2 M a 2 Ω 2 B 2 , and W total ∼ − 1 3 M a 2 Ω 2 , it can b e understo od that the phase diagram represen ts a correlation b et ween magnetic, rotational and total energy of stars. In this resp ect we note that for every white dwarf w e can ﬁnd a nearb y pulsar in the diagram, for example 2 and Crab, 3 and V ela etc. It is also interesting that white dw arfs (9) can b e found near the pulsar-magnetar division and white dwarfs 7, 6 and 11 are very close to pulsar’s death line. Fig.9 also suggests that slo wer (older?) pulsars hav e stronger magnetic ﬁeld then fast ones. This app earance is usually explained b y the fact that pulsars reac hing the death line (mark ed in magen ta) b ecome unobserv able, while fast pulsars with strongest magnetic ﬁeld (marked in brown) slow down so fast that few are observed b oth with high magnetic ﬁeld and fast rotation. How ever, if a similar correlation b etw een high magnetic ﬁeld and slow rotation p ersists in the case of white dwarf p opulation, the ab o v e argument can not be applied to justify the shortage of slo w lo w magnetic ﬁeld white dwarfs. Firstly , the magnetic ﬁeld of white dw arfs is not deduced from their slo w-do wn rate and secondly , the characteristic decay time of white dw arf slow down due to magnetic dip ole radiation is incomparably long with resp ect to pulsars 14 . W e also note that our simple mo del requires a rederiv ation ot the slow-do wn law for pulsars, since the energy lost by dip ole radiation is released by all av ailable energy reservoirs. The fundamental law to b e tak en in to account is the law of conserv ation of angular momentum stating dL dt = − P rad ω , where L is the angular momentum of the star and P rad = 1 12 π q µ 0 ε 0  ω c  4 M 2 is the dip ole radiation p ow er. Expressing magnetic momen t ( M ) and angular velocity ω with dimensionless quantites B and ˜ ω , the dip ole radiation p o wer b ecomes: P rad = 1 3 c 3 a 2 G M 2 Ω 4 B 2 ˜ ω 4 . Expressing the angular momentum as L = 2 W rot ω , one obtains 14 in terms of dimensionless ( ˜ ω ) and ( B ), the radiation braking decay time can b e expressed as: τ d = ω ˙ ω = 4 5 a c  a c 2 GM  2 1 B 2 ˜ ω 2 , so it is roughly ∼ (100) 3 times longer for white dw arfs. 33 dL dt = 2 5 M a 2 Ω 2 (1 + e 2 )  (1 + e 2 ) d ˜ ω dt + 4 ˜ ω e de dt  . The term e de dt can b e expressed as e de ≈ 15 4 ˜ ω d ˜ ω + 81 4 B d B b y taking the total deriv ative of equation (76) if terms of higher order in B and ˜ ω are neglected. Ob viously d ˜ ω dt ≫ ˜ ω e de dt , so to ﬁrst order the angular momentum conserv ation law leads to d ˜ ω dt = − 5 6(1+ e 2 ) 2 G M Ω 2 c 3 B 2 ˜ ω 3 , whic h is the classical result, except for the fact that e is a function of time. T aking this into accoun t in calculating the second deriv ative of rotational frequency , we obtain: d 2 ˜ ω dt 2 = 2 d ˜ ω dt  d B B dt + 3 2 d ˜ ω ˜ ω dt  . The braking index ( n b = ˜ ω ¨ ˜ ω ˜ ω 2 ) so b ecomes 15 : n b ≈ 3 −  12 5 c 3 G M Ω 2 1 ˜ ω 2 B 3 d B dt + 15 ˜ ω 2  . The discussion so far is based on the simplest n = 0 p olytropic mo del, assuming that the magnetic ﬁeld deduced from stellar sp ectra and p olarization prop erties applies to the observ ed dip ole comp onen t. Since this mo del deals with incompressible ﬂuid, the magnetic surface can only b e at the surface of the star, which results in very strong coupling b et ween gravit y and magnetism. As a result of such strong coupling the magnetic surface instability sets in only for very high order spherical harmonics expansion. This assumption may seem acceptable regarding magnetic ﬁelds of old neutron stars, white dwarfs and planets, but less so regarding the solar magnetic ﬁeld, whic h appears in the ab o ve phase diagram m uc h too w eak to ha ve any eﬀect on oblateness. Y et the solar oblateness [5] is to o small even with resp ect to the softest p olytropic mo del, while [12] rep orted a slight v ariation that could b e c orrelated with solar activity . F urthermore, the solar magnetic ﬁeld is c haracterized b y spik es of in tense magnetic ﬁeld in sunsp ots [20], whic h may b e reminiscent of magnetic surface instability forming ”magnetic hair” on magnetic surface, deep b elo w the photosphere. Conv ection, diﬀerential rotation and meridional ﬂow ma y tangle magnetic hair app earing at sunsp ots and drive solar magnetic activity . In this resp ect the magnetic W olf Ray et star HD 45166 with magnetic ﬁeld of 43kG is interesting. WR stars are known as stars stripp ed of their en velope. Noting that the magnetic ﬁeld 43kG is less than 10 times stronger then the magnetic ﬁeld in sunspots [20], one ma y ask if this is an indication, that HD 45166 w as stripp ed do wn to its magnetic surface. The wide v ariety of pulsar’s pulse shap es at high energies [1] also suggests that the dip ole ﬁeld alone can not b e res ponsible for sharp edges of gamma ra y pulses. In particular, the detailed study of V ela’s gamma– ra y pulse at energies from 50 MeV to 30 GeV( [27],[2]) reveals the increasing complexit y of pulse shape with increasing energy , which is consistent with the idea that the high– energy pulse is generated closer to the pulsar surface where the higher multipole order of the magnetic ﬁeld dominates ov er the dip ole. The curious pulse structure of in termittend 10 . 4 s pulsar PSR J1710 − 3452 ([31]), with a magnetic ﬁeld possibly as m uch as 10 4 times as strong as the Crab, adds to the mistery of the m ultip ole structure of h ighly magnetized stars. 15 Ha ving ab o ve neglected terms of higher order, w e understand 1 + e 2 ∼ 1 34 VI I. CONCLUSION This work was stimulated by a simple question: what is the diﬀerence, appart from scale, b et ween magnetism of a p ermanen t magnet and the magnetism of a star? According to everyda y experience, matter needs to b e magnetized b y driv en curren ts to b ecome magnetic. Y et, once magnetized, a magnet as w ell as a star pers ist in b eeing magnetic indeﬁnitely . This suggests that the state of b eeing magnetic is a characteristic of equilibrium of a material b ody with its electro-magnetic en vironment. In a similar wa y , spin may b e charaterizing the wa y in which a b o dy is imbedded into its surroundings. Therefore, we attempt to include rotation and electromagnetic interaction in the basic discussion of stellar structure. The starting p oint of this endeav our is the p olytropic mo del which predicts the mass-radius relation for a celestial b ody based on the law of gravitation and a rather simple equation of state for stellar material. W e formulate the solution to this problem through a v ariational principle and extend it to include also spin and magnetic moment as tw o additional observ able parameters. The emergence of magnetic momen t is demonstrated by deriving the equation of state for degenerate matter consisting of cold heavy ions and the F ermi gas of light electrons. W e sho w that such a quan tum system has a magnetized ground state. A similar conclusion, yet b y diﬀerent means, w as obtained by [34] in their study of sp on taneous magnetiza tion of collisionless plasma 16 . An exact solution of the extended gravito-magnetic p olytropic problem was found for the simplest poly- tropic model with n = 0. Mo dels with a soft equation of state are m uch more demanding, since the magnetic con tribution to action is form ulated as a surface integral ov er a generally unstable magnetic surface, which lies b elo w the physical surface of the star. The instability of magnetic surface is brieﬂy discussed. In Fig.9 we distribute stars with av ailable data on their rotation and magnetic ﬁeld in a phase diagram based on the exact n = 0 p olytropic solution. The axes of this diagram are magnetic parameter ( B ), the spin parameter ( ˜ ω ) and energy excess W . B represents the ratio of magnetic ﬂux through the equatorial section and mass of the star, ( ˜ ω ) is the ratio of the measured rotational velocity of the star and the breakup v elo cit y Ω = √ 2 π Gρ and W is the free energy of the rotating magnetic star with resp ect to the free energy of non-rotating and no-magnetic star. It is quite remark able that so many diﬀerent celestial ob jects o ccup y 16 The requirement that electrons are mo ving almost ump erturb ed in the electromagnetic en vironmen t is the common feature needed in b oth deriv ations to obtain the magnetic ground state. 35 the same region in this diagram. 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Left: b ottom: paramagnetic shell (0 . 85 r o < r < r o ) with µ =5 mag- netized b y a coil on the inner surface, Right: top: paramagnetic sphere with radius 0.75 r o and µ = 10 3 inside spherical coil with radius r o , Right:middle: the same paramagnetic sphere as ab o ve but driven by current on the surface. Right: b ottom: paramagnetic shell as to the left magnetized b y a coil on the outer surface. Driving curren t is represented by blue (for incoming) and red (for outgoing) dots with diameter proportional to curren t. Black streamlines follo w magnetic ﬁeld lines, while the grayish logarithmically scaled contour lines and the blue to brown color shading plots indicate the distribution of B 2 . 38 FIG. 2. Classical orbit corresp onding to dynamical solution of H class for ε > 0 (left) and ε < 0 (righ t). Orbit starting p oin t is denoted by a red dot. Arro ws in magen ta depict electric ﬁeld (24). 20 40 60 80 100 - 0.10 - 0.05 0.05 0.10 0.15 FIG. 3. W av e functions | n, s ⟩ for (n=10, s=100), (n=40, s=2000) and (n=80, s=4000) . Red p oin ts are at the center ˜ R c = p 2( n + s − 1) and green p oin ts mark the width at ˜ R c ± 3 2 √ n . R ions R ele FIG. 4. The gas of ions (red) o ccupies a smaller cylinder than the gas of electrons (blue). F rom n e R 2 ele − n i R 2 ion = 0 it follows: ( n e − n i ) R 2 ele ≈ 2 n i R ele δ R , where δ R = R ele − R ions . 39 .1 gauss 10 gauss 10 3 gauss 10 5 gauss Iron white dwarfs 1 × 10 7 1 × 10 11 1 × 10 15 1 × 10 19 1 × 10 23 1 × 10 27 10 - 12 0.01 10 8 10 18 n e [ cm - 3 ] w e [ Pa ] FIG. 5. Energy density of magnetized degenerate gas of electrons for diﬀeren t v alues of magnetic ﬁeld as a function of electron densit y . Dotted lines represen t the part due to magnetization. The gra y b o xes indicate conduction electron density in iron and typical av erage electron density in white dwarfs. - 0.02 - 0.01 0.00 0.01 0.02 0.994 0.996 0.998 1.000 1.002 1.004 δ U λ ( δ ) FIG. 6. Numerical results of calculating magnetic energy W m,λ ( δ ) to determine co eﬃcien ts k ( m ) λ ( δ ) for λ = 4 , 8 , 10 , 12 , 14 , 30. The steep linear slop e corresp onds to U ( m ) 2 ( δ ) ∝  1 − 9 5 δ  . 40 10 20 30 40 50 60 - 3.2 - 3.0 - 2.8 - 2.6 λ k λ ( m ) 10 20 30 40 50 60 0.02 0.05 0.10 0.20 0.50 λ k λ ( g ) FIG. 7. left: numerical v alues of k ( m ) λ (blue p oin ts) and ﬁt with inv erse p o wers of λ ; righ t: k ( g ) λ n umerical v alues (magen ta), asymptotic approximation (green), k ( g ) λ = 5 2(2 λ +1)  1 − 3 2 λ +1  (grey) FIG. 8. Magnetic ﬁeld of a spherical magnet deformed with r S = 1 + 0 . 1 P 10 (cos θ ). Magnetic energy density contours and red and blue dots representing the surface current are co ded as in Fig.1 41 FIG. 9. Magnetism and rotation of the Sun, planets, pulsars, magnetars and white dw arfs. Colored pulsars from the b order of the P-Pdot diagram on the righ t are represented with the same color in the left phase diagram (orange for high magnetic ﬁeld pulsars, magen ta for death line pulsars, green for young pulsars, and yello w for millisecond pulsars). The smaller satellites of the Crab pulsar represen t its p osition on the diagram assuming its diﬀerent masses from .67 to 2.09 M ⊙ . Magnetars (shown in grey) are the only ob jects in this diagram with their elipticity controled b y magnetic ﬁeld. White dwarfs are represented as n um b ered (WD 1 to WD 4 and 1 to 12) blue balls 42 WD 1 WD 2 WD 3 WD 4 ℬ 4  10 - 4 1  10 - 4 2.5  10 - 5 6.25  10 - 6 5000 10 000 15 000 20 000 25 000 30 000 35 000 0.4 0.6 0.8 1.0 1.2 Temperature [ K ] Mass [ M ⊙ ] FIG. 10. Mass, temp erature and magnetic parameter B of magnetic white dw arfs from Gentile’s collection, with arro ws p oin ting to the four with kno wn rotation rate from [17]. Magnetic parameter is co ded b y size and color as shown in inset; stars with B < 3 × 10 − 4 are represented b y pale greenish dots. 43

Stellar structure, magnetism and the variational principle

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