Possible formation of lowly luminous highly magnetized white dwarfs by accretion leading to SGRs/AXPs
B. Mukhopadhyay, M. Bhattacharya, A. R. Rao, S. Mukerjee, U. Das
aa r X i v : . [ a s t r o - ph . S R ] A ug Possible formation of lowly luminous highly magnetized white dwarfsby accretion leading to SGRs/AXPs
B. Mukhopadhyay ∗ , M. Bhattacharya , A. R. Rao , S. Mukerjee , U. Das
1. Indian Institute of Science, Bangalore 560012, India ∗ E-mail: [email protected]. University of Texas, Austin, USA3. Tata Institute of Fundamental Research, Mumbai, India4. University of Colorado, Boulder, USA
We sketch a possible evolutionary scenario by which a highly magnetized super-Chandrasekhar white dwarf could be formed by accretion on to a commonly observedmagnetized white dwarf. This is an exploratory study, when the physics in cataclysmicvariables (CVs) is very rich and complex. Based on this, we also explore the possibilitythat the white dwarf pulsar AR Sco acquired its high spin and magnetic field due torepeated episodes of accretion and spin-down. We show that strong magnetic field dra-matically decreases luminosity of highly magnetized white dwarf (B-WD), letting thembelow the current detection limit. The repetition of this cycle can eventually lead to aB-WD, recently postulated to be the reason for over-luminous type Ia supernovae. Aspinning B-WD could also be an ideal source for continuous gravitational radiation andsoft gamma-ray repeaters (SGRs) and anomalous X-ray pulsars (AXPs). SGRs/AXPsare generally believed to be highly magnetized, but observationally not confirmed yet,neutron stars. Invoking B-WDs does not require the magnetic field to be as high as forneutron star based model, however reproducing other observed properties intact.
Keywords : white dwarfs; strong magnetic fields; CVs; pulsars; SGRs/AXPs.
1. Introduction
Several independent observations repeatedly argued in recent past for the existenceof highly magnetized white dwarfs (B-WDs). Examples are overluminous type Iasupernovae , white dwarf pulsars etc. Also soft gamma-ray repeaters (SGRs)and anomalous X-ray pulsars (AXPs) could be explained as B-WDs , while theyare generally believed to be highly magnetized neutron stars without however anydirect detection of underlying required high surface field B s ∼ G. Interestingly,explaining SGR/AXP by a magnetized white dwarf requires a lower B s . G,which may however correspond to central field & G. Nevertheless, the originof such fields in a white dwarf remains a question, when the observed confirmedsurface field is . G.Here we explore a possible evolution of a conventionally observed magnetizedwhite dwarf to a B-WD by accretion, which may pass through a phase exhibitingcurrently observed AR Sco. This is an exploratory study, and the present ventureis based more on an idealized situation, when the physics in accreting white dwarfs,i.e. cataclysmic variables (CVs), is very rich and complex. We also show, based onsome assumption, that the thermal luminosity of such a B-WD could be very small,below their current detection limit. However, due to high field and rotation, theirspin-down luminosity could be quite high. Hence, they could exhibit SGRs/AXPs.
2. Accretion induced evolution
The detailed investigation of the accretion induced evolution faces several difficultiesincluding nova eruptions (hence nonsteady increase of mass) and the eruption andejection of accumulated shells. Nevertheless, the discovery of AR Sco, which is afast rotating magnetized white dwarf, argues for the possibility of episodic increaseof mass in a CV. Hence, we sketch a tentative evolutionary scenario with repeatedepisodes of accretion phase leading to the high magnetic field via flux freezing andspin-power phase decreasing field. Eventually this mechanism can plausibly leadto a B-WD. Note that there are already observational evidences for transitions be-tween spin-power and accretion-power phases in a binary millisecond pulsar . Theconservation laws controlling the accretion-power phase around the stellar surfaceof radius R and mass M , which could be inner edge of accretion disk, are given by l Ω( t ) R ( t ) = GM ( t ) R ( t ) , I ( t )Ω( t ) = constant , B s ( t ) R ( t ) = constant , (1)where l takes care of inequality due to dominance of gravitational force over thecentrifugal force in general, I is the moment of inertia of star and Ω the angularvelocity of the star which includes the additional contribution acquired due to ac-cretion as well. Solving the conservation laws given by equation (1) simultaneously,we obtain the time evolution of radius (or mass), magnetic field and angular velocityduring accretion. Accretion stops when − GMR = 1 ρ ddr (cid:18) B π (cid:19) | r = R ∼ − B s πRρ , (2)where ρ is the density of inner edge of disk.For a dipolar fixed field, ˙Ω ∝ Ω , where over-dot implies time derivative. Gen-eralizing for the present purpose it becomes ˙Ω = k Ω n with k being constant. There-fore, during the phase of spin-power pulsar (when accretion inhibits), the time evo-lution of angular velocity and surface magnetic field may be given byΩ = (cid:2) Ω − n − k (1 − n )( t − t ) (cid:3) − n , B s = r c Ik Ω n − m R sin α , (3)where Ω is the angular velocity when accretion just stops at the beginning of spin-power phase at time t = t , k is fixed to constrain B s at the beginning of firstspin-powered phase, which is determined from the field evolution in the precedingaccretion-power phase, α is the angle between magnetic and spin axes. Note that n = m = 3 corresponds to dipole field.Figure 1 shows a couple of representative possible evolutions of angular velocityand magnetic field with mass (time). It is seen that initial larger Ω with accretiondrops significantly during spin-power phase (when accretion stops and hence nochange of mass), followed by its increasing phase. Similar trend is seen in B s profileswith a sharp increasing trend (with value ∼ G) at the last cycle leading to theincrease of B c as well, forming a B-WD. At the end of evolution, it could be left out as a super-Chandrasekhar B-WD and/or a SGR/AXP candidate with a higherspin frequency. Of course, in reality they may depend on many other factors andthe current picture does not match exactly with what is expected in AR Sco itself. Fig. 1. Time evolution of (a) angular velocity in s − , (b) magnetic field in G, as functionsof mass in units of solar mass. The solid curves correspond to n = 3, m = 2 . ρ = 0 .
05 gmcm − , l = 1 . n = 3, m = 2, ρ = 0 . − , l = 2 . k = 10 − CGS,˙ M = 10 − M ⊙ Yr − , α = 10 o and R = 10 km at t = 0. This is reproduced from a previous work .
3. Luminosity
With the increase of mass, the radius of white dwarfs, hence B-WDs, becomes verysmall . Indeed, the increase of magnetic field is due to decreasing radius viaflux-freezing. Now due to smaller radius, UV-luminosity of B-WDs turns out to bevery small if the surface temperature is same as their nonmagnetic counterpart .However more interestingly, from the conservation of energy, it is expected thatthe presence of strong magnetic field enforces decreasing thermal energy and hencelowering luminosity in stable equilibrium.Combining the magnetostatic and photon diffusion equations in the presence ofmagnetic field but ignoring tension, we obtain ddT ( P + P B ) = 4 ac πGML T κ , (4)which we solve to obtain the envelop properties. Here P is the matter pressure, P B the magnetic pressure, κ the opacity, T the temperature, a the radiation constant, M the mass of white dwarf within the core radius r , which is practically the wholemass of white dwarf because the envelop is very thin, and L is the luminosity.For the strong field considered here, the radiative opacity variation with B can bemodelled similarly to neutron stars as κ = κ B ≈ . × ρT − . B − cm g − . We use a field profile proposed earlier for neutron stars to enumerate the fieldmagnitude at a given density (radius), irrespective of other complicated effects,given by B (cid:18) ρρ (cid:19) = B s + B (cid:20) − exp (cid:18) − η (cid:18) ρρ (cid:19) γ (cid:19)(cid:21) , (5)where B (similar to central field) is a parameter with the dimension of B , otherparameters are set as η = 0 . γ = 0 . ρ = 10 g cm − for all the calculations.Further equating the electron pressure for the non-relativistic electrons on bothsides of the core-envelop interface gives ρ ∗ ( B ∗ ) ≈ . × − T / ∗ B s (6)at interface. Now we solve equation (4) along with the photon diffusion equation dTdr = − ac κ ( ρ + ρ B ) T L πr , (7)with boundary conditions ρ ( T s ) = 10 − g cm − , r ( T s ) = R = 5000 km and M = M ⊙ , where T s is the surface temperature, and obtain ρ − T and r − T profiles.Further T ∗ and ρ ∗ can be obtained by solving for the ρ − T profile along withequation (6), as shown in Fig. 2, and knowing T ∗ , we can obtain r ∗ from the r − T profile. We see that interface moves inwards ( r ∗ decreases) with increasing B and L . But ρ ∗ increases with increasing L and/or B , as ρ ∗ ∝ T / ∗ B . (cid:1) (cid:2)(cid:3)(cid:4) × (cid:5)(cid:1) (cid:1) (cid:4)(cid:3) × (cid:5)(cid:1) (cid:1) (cid:6)(cid:3)(cid:4) × (cid:5)(cid:1) (cid:1) (cid:5)(cid:3) × (cid:5)(cid:1) (cid:2) (cid:5)(cid:3)(cid:2)(cid:4) × (cid:5)(cid:1) (cid:2) (cid:1)(cid:2)(cid:1)(cid:1)(cid:1)(cid:7)(cid:1)(cid:1)(cid:1)(cid:8)(cid:1)(cid:1)(cid:1)(cid:9)(cid:1)(cid:1)(cid:1)(cid:5)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1) / (cid:10) ρ / (cid:1) (cid:2) (cid:3) - (cid:1) Fig. 2. Variation of density with temperature for B ≡ ( B s , B ) = (10 G , G) and L =10 − L ⊙ (dashed line), 10 − L ⊙ (dotted line) and 10 − L ⊙ (dot-dashed line). The solid line repre-sents equation (6). This is reproduced from a previous work . With above benchmarking, we now explore, based energy conservation, if theluminosity of a magnetized white dwarf or B-WD changes. For B = 0 and L =10 − L ⊙ , we have r ∗ = 0 . R , ρ ∗ = 170 . − and T ∗ = 2 . × K. Usingthe same boundary condition as described above, we now solve equations (4) and(7) with B = 0, but vary L in order to fix r ∗ = 0 . R . We find interestinglythat L decreases for B = 0, as shown in Table 1. Physically this corresponds toincreasing B , and thence magnetic energy, is compensated by decreasing thermalenergy (decreasing T ∗ ) and thence L , when total energy is conserved. Similarly, increasing B may be compensated by decreasing gravitational energy (decreasing r ∗ ). In either of the cases, L decreases.Table 1: Variation of luminosity with magnetic field for fixed r ∗ = 0 . RB/
G = (B s / G , B / G) L/L ⊙ T ∗ / K ρ ∗ / g cm − T s / K(0 ,
0) 1 . × − . × . × . × (10 , × ) 2 . × − . × . × . × (5 × , × ) 3 . × − . × . × . × (10 , ) 1 . × − . × . × . × (2 × , × ) 4 . × − . × . × . × (5 × , × ) 2 . × − . × . × . × (5 × , ) 2 . × − . × . × . ×
4. SGRs/AXPs as B-WDs
Paczynski and Usov independently proposed that SGRs and AXPs are mod-erately magnetized white dwarfs but following Chandrasekhar’s mass-radius rela-tion . Many features of SGRs/AXPs are explained by their model at relativelylower magnetic fields, while the more popular magnetar model requires field & G, which is not observationally well established yet. Nevertheless, such a whitedwarf based model suffers from a deep upper limit on the optical counterparts ofsome AXPs/SGRs, e.g. SGR 0418+5729, due to their larger moment of inertia.Now B-WDs established here could be quite smaller in size and hence havesmaller moment of inertia. Therefore, their optical counterparts, with very lowUV-luminosities, are quite in accordance with observation. Hence, the idea of B-WD brings a new scope of explain SGRs/AXPs at smaller magnetic fields, whichare observationally inferable, compared to highly magnetized magnetar model. Fordetails see the work by Mukhopadhyay & Rao .
5. Continuous Gravitational Radiation
Due to smaller size compared to their regular counterpart, B-WDs rotate relativelyfaster. Now if the rotation and magnetic axes are misaligned, they serve as goodcandidates for continuous gravitational radiation due to their quadrupole moment,characterized by the amplitude h + ( t ) = h α ) cos Φ( t ) , h × ( t ) = h cos α sin Φ( t ) , h = 4 π GI zz ǫc P s D , (8)where α is the inclination of the star’s rotation axis with respect to the observer,Φ( t ) is the signal phase function, ǫ amounts the ellipticity of the star, I zz is themoment of inertial about z-axis, and D is the distance between the star and detector. A B-WD of mass ∼ M ⊙ , polar radius ∼
700 km, spin period P s ∼ , ǫ ∼ × − and D ∼
100 pc would produce h ∼ − , which is within thesensitivity of the Einstein@Home search for early Laser Interferometer GravitationalWave Observatory (LIGO) S5 data . However, DECIGO/BBO would give a firmconfirmation of their gravitational wave because they are more sensitive in theirfrequency range. In fact, if the polar radius is ∼ P s ∼
10 s and otherparameters intact, DECIGO/BBO can detect it with h ∼ − . Nevertheless,(highly) magnetized rotating white dwarfs approaching B-WDs are expected to becommon and such white dwarfs of radius ∼ P s ∼
20 sec and D ∼
10 pccould produce h & − which is detectable by LISA.
6. Summary
The idea of B-WD has been proposed early this decade, mainly to explain observedpeculiar type Ia supernovae inferring super-Chandrasekhar progenitor mass. Latelyit has been found with various other applications, e.g. SGRs/AXPs, white dwarfpulsars like AR Sco, continuous gravitational wave etc. Here we have attemptedto sketch a plausible evolution scenario to explain the formation of such a highlymagnetized, smaller size white dwarf. In our simplistic picture, ignoring manycomplicated CV features, we are able to show that a commonly observed magnetizedwhite dwarf could be evolved to a B-WD via accretion. Hence, the existence ofhighly magnetized, rotating, smaller white dwarfs is quite plausible.
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