New mass limit of white dwarfs
aa r X i v : . [ a s t r o - ph . H E ] S e p New mass limit of white dwarfs
Upasana Das and Banibrata MukhopadhyayDepartment of Physics, Indian Institute of Science, Bangalore 560012, [email protected] , [email protected] 6, 2018
Essay received Honorable Mention in the Gravity Research Foundation2013 Awards for Essays on Gravitation
Abstract
Is the Chandrasekhar mass limit for white dwarfs (WDs) set in stone? Notanymore — recent observations of over-luminous, peculiar type Ia supernovae canbe explained if significantly super-Chandrasekhar WDs exist as their progenitors,thus barring them to be used as cosmic distance indicators. However, there is noestimate of a mass limit for these super-Chandrasekhar WD candidates yet. Canthey be arbitrarily large? In fact, the answer is no! We arrive at this revelation byexploiting the flux freezing theorem in observed, accreting, magnetized WDs, whichbrings in Landau quantization of the underlying electron degenerate gas. Thisessay presents the calculations which pave the way for the ultimate (significantlysuper-Chandrasekhar) mass limit of WDs, heralding a paradigm shift 80 years afterChandrasekhar’s discovery.
Keywords : white dwarfs; supernovae; stellar magnetic field; Landau levels;equation of state of gasesPACS Number(s): 97.20.Rp, 97.60.Bw, 97.10.Ld, 71.70.Di, 51.30.+i
Introduction
Chandrasekhar, in one of his celebrated papers [1], showed that the maximum possiblemass of non-rotating, non-magnetized white dwarfs (WD) is 1 . M ⊙ , when M ⊙ beingthe mass of Sun. He was awarded the Nobel Prize in Physics in 1983 mainly because ofthis discovery. This limiting mass (LM) is directly related to the luminosity of observedtype Ia supernovae which are used as standard candles for measuring far away distancesand hence in understanding the expansion history of the universe. The discovery of theaccelerated expansion of the universe led to the Nobel Prize in Physics in 2011 [2].1o far, observations seemed to abide by the Chandrasekhar limit. However, in orderto explain the recent discovery of several peculiar, anomalous, distinctly over-luminoustype Ia supernovae [3, 4] – namely, SN 2006gz, SN 2007if, SN 2009dc, SN 2003fg – the massof the exploding WDs (progenitors of supernovae) needs to be between 2 . − . M ⊙ , sig-nificantly super-Chandrasekhar. Nevertheless, these non-standard ‘super-Chandrasekharsupernovae’ can no longer be used as cosmic distance indicators. However, there is needof a foundational level analysis, akin to that carried out by Chandrasekhar, in order toestablish a super-Chandrasekhar mass WD. Moreover, there is no estimate of an uppermass limit for these super-Chandrasekhar WD candidates yet. Can they be arbitrarilylarge? These are some of the fundamental questions, we plan to resolve in the presentessay. Basic physical process rendering the new limit
We plan to exploit the effects of magnetic field to establish the new limit. Hence, weconsider the collapsing star to be magnetized and the resulting accreting WD to be highlymagnetized. This is in accordance with observations, which show that about 25% ofaccreting WDs, namely cataclysmic variables, are found to have surface magnetic fieldsas high as 10 − G [5]. Hence their expected central fields could be 2 − ∝ BR , when B is the magnetic fieldand R the WD’s radius, is conserved. Therefore, if the WD shrinks, its radius decreasesand hence magnetic field increases. This in turn increases the outward force balancing theincreased inward gravitational force, leading to a quasi-equilibrium situation. As accretionis a continuous process, the above process continues in a cycle and helps in increasing B above the critical value 4 . × G to bring in Landau quantization effects [6].Subsequently, the mass of the WD keeps increasing, even above the Chandrasekhar limit,until the gain of mass becomes so great that it attains a new limit. At this point the totaloutward pressure is unable to support the gravitational attraction any longer, leading toa supernova explosion. This we argue to observe as a peculiar, over-luminous type Iasupernova, in contrast to their normal counter parts.
Computing the new mass limit
In the presence of strong magnetic field, the equation of state (EoS) of degenerate electrongas for WDs can be recast, at least in the piecewise zones of density ( ρ ), in the polytropicform: P = K m ρ Γ , when P is the pressure, K m and Γ = 1 + 1 /n are piecewise constantsin different regimes of ρ [6]. At the highest density regime (which also corresponds tothe highest magnetic field regime), Γ = 2. Now we recall the condition for magnetostaticequilibrium and estimate of mass, assuming the WD to be spherical, as1 ρ ddr (cid:18) P + B π (cid:19) = F gr + ~B · ∇ ~B πρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r , dMdr = 4 πr ρ, (1)when r is the radial distance from the center of WD and F gr the gravitational force. We2ote here that the choice of a Newtonian framework is justified in our case as the densitycorresponding to the degenerate pressure is much smaller than the matter density to con-tribute significantly to the effective mass of the WD (see Figure 1 of [6]). Moreover, inorder to correctly include the effect of strong magnetic field in a general relativistic hy-dromagnetic balance equation, the Tolman-Oppenheimer-Volkov (TOV) equation, whichis true only for the non-magnetic case, itself has to be modified first. Assuming B variesvery slowly around the center of the WD which is the regime of interest, we obtain [7]1 r ddr (cid:18) r ρ dPdr (cid:19) = − πGρ, (2)where G is Newton’s gravitation constant. This further can be recast, with the use ofEoS, into 1 ξ ddξ (cid:18) ξ dθdξ (cid:19) = − θ n with ρ = ρ c θ n , (3)where θ is a dimensionless variable and ξ = r/a, a = " ( n + 1) K m ρ − nn c πG / . (4)Equation (3) can be solved with the boundary conditions θ ( ξ = 0) = 1 , (cid:18) dθdξ (cid:19) ξ =0 = 0 . (5)Note that for n < θ becomes zero for a finite value of ξ , say ξ , which basicallycorresponds to the surface of the WD such that its radius R = aξ . (6)Also the mass of the WD can be obtained as M = 4 πa ρ c ξ Z ξ θ n dξ. (7)Now, the scalings of mass and radius of the WD with its central density ( ρ c ) are easilyobtained as M ∝ K / m ρ (3 − n ) / nc , R ∝ K / m ρ (1 − n ) / nc . (8)Clearly n = 3 (Γ = 4 /
3) corresponds to M independent of ρ c (provided K m is independentof ρ c ) and hence LM. Therefore, we have to find out the condition for which n = 3 and thecorresponding proportionality constant for the scaling of M . Note, however, that n = 1for the extremely magnetized, highly dense, degenerate electron gas EoS, which is thepresent regime of interest. Below we explore the generic mass limit of WDs consideringtwo scenarios. 3 Modeling WDs with varying magnetic field inside
Magnetized WDs are likely to have a varying B profile, with an approximately constantfield in the central region (CR), falling off from the central to surface region. As WDsevolve by accreting mass, their central and surface B s, along with the density, increase.This enables the WDs to hold more mass and hence they deviate from Chandrasekhar’smass-radius relation. Note that a large B corresponds to a large outward magnetic pres-sure (along with a magnetic tension). Hence, an equilibrium solution depends on thenature of variation of B within the WD such that it might no longer remain spherical.Moreover, how fast the field inside the WD decays to a smaller surface value affects itsmass and radius. In order to give rise to a stable super-Chandrasekhar WD, the fieldneeds to remain constant up to a certain region from the center, so that enough massis accumulated due to the Landau quantized EoS. However, at the limiting (very large)density, when the mass becomes independent of (central) density, the basic trend of themass-radius relation has to be same as that of Chandrasekhar, except for the larger mass,in order to achieve n = 3 in EoS. At this situation, WDs will become, theoretically, verysmall such that R = 0 (and hence the spherical assumption of its shape does not alterthe result), as that obtained by Chandrasekhar in the absence of B . Hence, WDs closeto the LM practically should have a constant B throughout. However, at high density, K m ∝ B − ∝ ρ − / c , unlike the non-magnetized EoS when K m is independent of B (and ρ c ). Hence, when ρ → ρ c , the highly magnetized EoS reduces to P = Kρ / when K = c ~ π / / ( m H µ e ) / , (9)where c is the speed of light, ~ the reduced Planck’s constant, m H the mass of proton, µ e the mean molecular weight per electron. Now combining equations (4), (7) for K m = K and n = 3, we obtain the LM M l = 10 . µ e c ~ Gm / H ! / . (10)For carbon-oxygen WDs, µ e = 2 and hence the LM becomes 4 . M ⊙ . • Modeling the central part of WDs
Here we consider only the CR of WDs and estimate the mass of this region. Of course thesize of the CR changes as WDs evolve [8], which in turn determines how underestimatedour result is with respect to the total mass (and total radius) of WDs. In CR [9] K m = Kρ − / c , (11)and hence from equation (8) M ∝ ρ − n ) / nc , R ∝ ρ (3 − n ) / nc , (12)revealing M independent of ρ c for n = 1, when the radius becomes independent of themass in the mass-radius relation [8, 9]. Now combining equations (4), (7) with n = 1, we4btain the value of LM M l = 5 . µ e c ~ Gm / H ! / . (13)For µ e = 2 the LM becomes 2 . M ⊙ . Note that M l is arrived at by considering aconstant B and hence is naturally smaller than M l which additionally counts the massaccumulated outside the CR, for a varying B . Conclusions
We summarize the findings of this work as follows: • More than 80 years after the proposal of Chandrasekhar mass limit, this new limitperhaps heralds the onset of a paradigm shift. • The masses of WDs are measured from their luminosities assuming Chandrasekhar’smass-radius relation, as of now. These results may have to be re-examined based onthe new mass-radius relation, at least for some peculiar objects (e.g. over-luminoustype Ia supernovae). • Some peculiar known objects, like magnetars (highly magnetized compact objects,supposedly neutron stars, as of now), should be examined based on the above con-siderations, which could actually be super-Chandrasekhar WDs. • This new mass limit should lead to establishing the underlying peculiar supernovaeas a new standard candle for cosmic distance measurement. • In order to correctly interpret the expansion history of the universe (and then darkenergy), one might need to carefully sample the observed data from the supernovaeexplosions, especially if the peculiar type Ia supernovae are eventually found to beenormous in number. However, it is probably too early to comment whether ourdiscovery has any direct implication on the current dark energy scenario, which isbased on the observation of ordinary type Ia supernovae. • Importantly, one now needs to carry out complete self-consistent calculations forthe structure of WDs by generalizing the TOV equation to account for the strongmagnetic field and pressure anisotropy, which has not been performed yet, in orderto confirm the LM of WDs derived here.
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