New Mass Limit for White Dwarfs: Explains Super-Chandrasekhar Type Ia Supernovae

Upasana Das (left) and Banibrata Mukhopadhyay (right)







Authors: Upasana Das and Banibrata Mukhopadhyay

Affiliation: Dept of Physics, Indian Institute of Science, Bangalore, India

Background:

Extremely luminous explosions of white dwarfs, known as type Ia supernovae [1], have always been in the prime focus of natural science. Generally, they are believed to result from the violent thermonuclear explosion of a carbon-oxygen white dwarf, when its mass approaches the famous Chandrasekhar limit of 1.44M_⊙ [2], with M_⊙ being the mass of Sun. The observed luminosity is powered by the radioactive decay of nickel, produced in the thermonuclear explosion, to cobalt and then to iron. The characteristic nature of the variation of luminosity with time of these supernovae (see Figure 1) -- along with the consistent mass of the exploding white dwarf -- allows these supernovae to be used as a ‘standard’ for measuring far away distances (standard candle) and hence in understanding the expansion history of the universe.

Figure 1: Variation of luminosity as a function of time of a type Ia supernova [image courtesy: Wikipedia]

Observation and study of this very feature of distant supernovae led to the Nobel Prize in Physics in 2011 for the discovery of the accelerated expansion of the universe [3, 2Physics report]. Also, mainly because of the discovery of the limiting mass of white dwarfs, S. Chandrasekhar was awarded the Nobel Prize in Physics in 1983.

Chandrasekhar, by means of a remarkably simple calculation, was the first to obtain the maximum mass for a (non-magnetized, non-rotating) white dwarf [2]. So far, observations seemed to abide by this limit. However, the recent discovery of several peculiar type Ia supernovae -- namely, SN 2006gz, SN 2007if, SN 2009dc, SN 2003fg [4,5] -- provokes us to rethink the commonly accepted scenario. These supernovae are distinctly over-luminous compared to their standard counterparts, because of their higher than usual nickel mass. They also violate the ‘luminosity-stretch relation’ and exhibit a much lower velocity of the matter ejected during the explosion. However, these anomalies can be resolved, if super-Chandrasekhar white dwarfs, with masses in the range 2.1-2.8M_⊙, are assumed to be the mass of the exploding white dwarfs (progenitors of these peculiar supernovae). Nevertheless, these non-standard ‘super-Chandrasekhar supernovae’ can no longer be used as cosmic distance indicators. However, there is no estimate of an upper limit to the mass of these super-Chandrasekhar white dwarf candidates yet. Can they be arbitrarily large? Moreover, there has been no foundational level analysis performed so far, akin to that carried out by Chandrasekhar, in order to establish a super-Chandrasekhar mass white dwarf.

Our result at a glance:

We establish a new and generic mass limit for white dwarfs which is 2.58M_⊙ [6]. This is significantly different from that proposed by Chandrasekhar. Our discovery naturally explains the over-luminous, peculiar type Ia supernovae mentioned above. We arrive at this new mass limit by exploiting the effects of the magnetic field in compact objects. The motivation behind our approach lies in the discovery of several isolated magnetized white dwarfs through the Sloan Digital Sky Survey (SDSS) with surface fields 10⁵-10⁹ gauss [7,8]. Hence their expected central fields could be 2-3 orders of magnitude higher. Moreover, about 25% of accreting white dwarfs, namely cataclysmic variables (CVs), are found to have magnetic fields as high as 10⁷-10⁸ gauss [9].

Underlying theory:

We first recall the basic formation scenario of white dwarfs. In order to do so, we have to understand the properties of degenerate electrons. When different states of a particle correspond to the same energy in quantum mechanics, they are called degenerate states. Moreover, Pauli’s exclusion principle prohibits any two identical fermions (in the present context: electrons) to occupy the same quantum state. Now, when a normal star of mass less than or of the order of 5M_ʘ exhausts its nuclear fuel [10], it undergoes a collapse leading to a small volume consisting of a lot of electrons. Being in a small volume, many such electrons tend to occupy the same energy states, and hence they become degenerate, since the energy of a particle depends on its momentum which is determined by the total volume of the system. Hence, once all the energy levels up to the Fermi level, which is the maximum allowed energy of a fermion, are filled by the electrons, there is no available space for the remaining electrons in a small volume of the collapsing star. This expels the electrons to move out leading to an outward pressure. If the force due to the outward pressure is able to balance the inward gravitational force, then the collapse halts, forming the compact star white dwarf.

Figure 2: Landau quantization in presence of magnetic field B. [image courtesy: Warwick University, UK ]

For the current purpose, we have to also recall the properties of degenerate, relativistic electrons under the influence of a strong magnetic field, neglecting any form of interactions. The energy states of a free electron in a magnetic field are quantized into what is known as Landau orbitals [11]. Figure 2 shows that how the continuous energy levels split into discrete Landau levels with the increase of magnetic field in the direction perpendicular to the motion of the electron. Larger the magnetic field, smaller is the number of Landau levels occupied. Recent works [12-14] establish that Landau quantization due to a strong magnetic field modifies the equation of state (EoS), which relates the pressure (P) with density (ρ), of the electron degenerate gas. This should influence significantly the mass and radius of the underlying white dwarf (and hence the mass-radius relation). The main aim here is to obtain the maximum possible mass of such a white dwarf (which is magnetized), and therefore a (new) mass limit. Hence we look for the regime of high density of electron degenerate gas and the corresponding EoS, which further corresponds to the high Fermi energy (E_F) of the system. This is because the highest density corresponds to the lowest volume and hence, lowest radius, which further corresponds to the limiting mass [2]. Note that the maximum Fermi energy (E_Fmax) corresponds to the maximum central density of the star. Consequently, conservation of magnetic flux (technically speaking flux freezing theorem, which is generally applicable for a compact star) argues for the maximum possible field of the system, which implies that only the ground Landau level will be occupied by the electrons.

Generally the EoS can be recast in the polytropic form of P=Kρ^Γ, when K is a constant and Γ (=1+1/n) is the polytropic index. At the highest density regime (which also corresponds to the highest magnetic field regime), Γ=2. Now combining the above EoS with the condition of magnetostatic equilibrium (when net outward force is balanced by the inward force), we obtain the mass and radius of the white dwarf to scale with its central density (ρ_c) as M ∝ K^(3/2) ρ_c^(3-n)/2n and R ∝ K^(1/2) ρ_c^(1-n)/2n respectively [6]. For Γ = 2, which corresponds to the case of limiting mass, K ∝ ρ_c^(-2/3) and hence M becomes independent of ρ_c and R becomes zero. Substituting the proportionality constants, for Γ = 2 we obtain exactly [6]:

where h is the Planck’s constant, c the speed of light, G Newton’s gravitation constant, μ_e the mean molecular weight per electron and m_H the mass of hydrogen atom. For μ_e=2, which is the case for a carbon-oxygen white dwarf, M≈2.58M_⊙. To compare with Chandrasekhar’s result [2], we recall the limiting mass obtained by him as

which for μ_e =2 is 1.44M_⊙.

Figure 3: Mass-radius relation of a white dwarf. Solid line – Chandrasekhar’s relation; dashed line – our relation.

For a better reference, we include a comparison between the mass-radius relation of the white dwarf obtained by Chandrasekhar and that obtained by us in Figure 3.

Justification of high magnetic field and its effect to hold more mass:

The presence of magnetic field in a white dwarf creates an additional outward pressure apart from that due to degenerate electrons, which is however modified in presence of a strong field in it. On the other hand, the inward (gravitational) force is proportional to the mass of the white dwarf. Hence, when the star is magnetized, a larger outward force can balance a larger inward force, allowing it to have more mass.

However, the effect of Landau quantization becomes significant only at a high field B ≥ B_c = 4.414×10¹³ gauss. How can we justify such a high field in a white dwarf? Let us consider the commonly observed phenomenon of a magnetized white dwarf attracting mass from its companion star (called accretion). Now the surface field of an accreting white dwarf, as observed, could be 10⁹ gauss (≪ B_c) [7]. Its central field, however, can be several orders of magnitude higher ∼ 10¹² gauss, which is also less than B_c. Naturally, such a magnetized CV, still follows the mass-radius relation obtained by Chandrasekhar. However, in contrast with Chandrasekhar’s work (which did not include a magnetic field in the calculations), we obtain that, a nonzero initial field in the white dwarf, however ineffective for rendering Landau quantization effects, proves to be crucial in supporting the additional mass accumulated due to accretion.

As an above-mentioned magnetized white dwarf first gains mass due to accretion, its total mass increases which in turn increases the gravitational power and hence the white dwarf contracts in size due to the increased gravitational pull. However, the total magnetic flux in a white dwarf is understood to be conserved, which is magnetic field times the square of its radius. Therefore, if the white dwarf shrinks, its radius decreases and hence magnetic field increases. This in turn increases the outward force balancing the increased inward gravitational force (due to increase of its mass), which leads to a quasi-equilibrium situation. As the accretion is a continuous process, above process of shrinking the white dwarf, increasing the magnetic field and holding more mass, goes in a cycle. This continues until the gain of mass becomes so great that total outward pressure is unable to support the gravitational attraction. This finally leads to a supernova explosion, which we observe as a peculiar, over-luminous type Ia supernova, in contrast to their normal counter parts.

Punch lines:

More than 80 years after the proposal of Chandrasekhar mass limit, this new limit perhaps heralds the onset of a paradigm shift. This discovery has several consequences as briefly described below.

The masses of white dwarfs are measured from their luminosities assuming Chandrasekhar's mass-radius relation, as of now. These results may have to be re-examined based on the new mass-radius relation, at least for some peculiar objects (e.g. over-luminous type Ia supernovae). Further, some peculiar known objects, like magnetars (highly magnetized compact objects, supposedly neutron stars, as of now) should be examined based on the above considerations, which could actually be super-Chandrasekhar white dwarfs.

This new mass limit may also lead to establishing the underlying peculiar supernovae as a new standard candle for cosmic distance measurement. Hence, in order to correctly interpret the expansion history of the universe (and then dark energy), one might need to carefully sample the observed data from the supernovae explosions, especially if the peculiar type Ia supernovae are eventually found to be enormous in number. However, it is probably too early to comment whether our discovery has any direct implication on the current dark energy scenario, which is based on the observation of ordinary type Ia supernovae.

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