The

Up to the Chandrasekhar limit, we say the white dwarf exhibits "electron degeneracy". As we saw in a previous blog:

http://brane-space.blogspot.com/2010/06/some-more-quantum-mechanics.html

the assorted solutions to the Schrodinger wave equation, yield different wave functions each of which displays a certain set of "quantum numbers". These are:

n, the

l, the

and m_l,

In terms of the wave function solutions ("eigenfunctions") each set (n, l, m_l) determines a particular energy level in an atom, for example the hydrogen atom. NO two electrons (which are fermions) can have the same set of quantum numbers. This is the basis for what we call the

In the case of the white dwarf this applies and one has

Thus, the condition of "electron degeneracy" applies and this means that electron occupancy can be such that many more electrons can be packed into the same space or volume. This is what gives rise to the enormous attendant density, for example that we find in white dwarfs .

To get to the degeneracy state in the first place, we must have a violation of one of the foremost conditions of normal stellar structure:

In this condition, the pressure of hot radiation from the deep interior of the star balances the weight of its overlying layers (think of a star like an onion with layers and layers each of different density- the greater density toward the center). We show this condition as the differential equation (recall our venture into these last month):

dP/ dr = - G M(r)rho/ r^2 = - (rho)g

where M(r) is the mass of a particular layer, G is the Newtonian gravitational constant (G = 6.62 x 10^-11N-m/kg^2) and rho is the density.

The path to the white dwarf follows because of the interior of a star being cooled, due to less efficient nucelar fusion. Because of this cooling, the outward-directed supporting gas -radiation pressure decreases and the overlying layers collapses up to the degeneracy level, since the downward (or inner-directed) weight overcomes the radiation pressure.

Interestingly, when we work out the formula for the white dwarf mass (M(wd)) it turns out to be a function of the ratio (Z/A) where Z = protons, and A = number of nucleons (neutrons + protons). We find when this ratio (Z/A) = 0.5 for M(wd), the white dwarf is at the Chandrasekhar limit. We also have found NO white dwarf mass can exceed this.

Thus, no stable white dwarf can exist beyond the limit because even a theoretically "infinite" central pressure would not keep the star from further collapse.

Now, however, a new finding appears to flout this limit evidently pushing the Chandrasekhar limit beyond 1.44 solar masses. This was reported in the

Could this be possible? The data appear to bear it out (see graph). In the graph of "Ni mass" vs. luminosity (in ergs/s and bear in mind 1 J = 10^7 ergs), we see four data points labelled SN2006gz, SN2003fg, SN2007if and SN 2009dc (from the

Of course, because the data points are few much more work needs to be done to provide confirmation. There are also peculiarities that need to be explained, for example SN2003fg appears to have contained 1.3 M(s) of Ni alone. It also displays carbon absorption lines that are weak or absent in typical Type 1a supernovae. Also its shell of material was cast off at a relatively slow velocity: ~ 8,000 km/s rather than the more usual 10,000- 12,000 km/s.

How could a white dwarf attain such mass, if the data are to be believed? Three main hypotheses are up for consideration:

1) The white dwarf may have been rotating unusually rapidly so that the centripetal force produced helped to balance the gravity - thereby preventing the halting of collapse at the usual 1.44 limit .

2) The supernovae could have resulted from TWO white dwarfs merging and exploding.

3) It may have been an ordinary white dwarf exploding asymmetrically with its birght side facing Earth, thereby yielding anomalous L readings.

We'll have to dig further to see what it all means, but in the meantime, the data provide an intriguing puzzle and possible test of what we think we really know about white dwarf physics.

**refers to the upper limit on mass for a stellar collapsed core, beyond which the matter becomes "***Chandrasekhar limit**degenerate*". The limit is typically taken to be 1.44 solar masses (M(s)). Generally it applies to the end stage of normal stellar masses associated with binary star systems. It is the threshold beyond which the white dwarf can no longer support itself with any outward pressure, and it becomes a "collapsar" - generally meaning a neutron star.Up to the Chandrasekhar limit, we say the white dwarf exhibits "electron degeneracy". As we saw in a previous blog:

http://brane-space.blogspot.com/2010/06/some-more-quantum-mechanics.html

the assorted solutions to the Schrodinger wave equation, yield different wave functions each of which displays a certain set of "quantum numbers". These are:

n, the

*principal quantum number*l, the

*angular momentum quantum number*,and m_l,

*the magnetic quantum number*.In terms of the wave function solutions ("eigenfunctions") each set (n, l, m_l) determines a particular energy level in an atom, for example the hydrogen atom. NO two electrons (which are fermions) can have the same set of quantum numbers. This is the basis for what we call the

*"Pauli Exclusion Principle*". In astrophysics, however, this isn't an ironclad rule as it is in normal lab chemistry, but can be violated in extreme conditions of gravitational collapse.In the case of the white dwarf this applies and one has

*degenerate matter*, which accounts for why it's often stated that a "teaspoonful of a white dwarf would weigh 15 tons". This could only happen if many more electrons are squeezed into the same energy states such that they share the same sets of quantum numbers.Thus, the condition of "electron degeneracy" applies and this means that electron occupancy can be such that many more electrons can be packed into the same space or volume. This is what gives rise to the enormous attendant density, for example that we find in white dwarfs .

To get to the degeneracy state in the first place, we must have a violation of one of the foremost conditions of normal stellar structure:

*hydrostatic equilibrium or pressure-gravity balance*.In this condition, the pressure of hot radiation from the deep interior of the star balances the weight of its overlying layers (think of a star like an onion with layers and layers each of different density- the greater density toward the center). We show this condition as the differential equation (recall our venture into these last month):

dP/ dr = - G M(r)rho/ r^2 = - (rho)g

where M(r) is the mass of a particular layer, G is the Newtonian gravitational constant (G = 6.62 x 10^-11N-m/kg^2) and rho is the density.

The path to the white dwarf follows because of the interior of a star being cooled, due to less efficient nucelar fusion. Because of this cooling, the outward-directed supporting gas -radiation pressure decreases and the overlying layers collapses up to the degeneracy level, since the downward (or inner-directed) weight overcomes the radiation pressure.

Interestingly, when we work out the formula for the white dwarf mass (M(wd)) it turns out to be a function of the ratio (Z/A) where Z = protons, and A = number of nucleons (neutrons + protons). We find when this ratio (Z/A) = 0.5 for M(wd), the white dwarf is at the Chandrasekhar limit. We also have found NO white dwarf mass can exceed this.

Thus, no stable white dwarf can exist beyond the limit because even a theoretically "infinite" central pressure would not keep the star from further collapse.

Now, however, a new finding appears to flout this limit evidently pushing the Chandrasekhar limit beyond 1.44 solar masses. This was reported in the

**Physics Today**issue of May, 2010 (p. 11) in the article*'Overluminous Supernovae Push the Chandrasekhar Limit'*.Could this be possible? The data appear to bear it out (see graph). In the graph of "Ni mass" vs. luminosity (in ergs/s and bear in mind 1 J = 10^7 ergs), we see four data points labelled SN2006gz, SN2003fg, SN2007if and SN 2009dc (from the

*Supernova Legacy Survey*) that disclose an anomalously large luminosity relative to the bulk of Type 1a supernovae.Of course, because the data points are few much more work needs to be done to provide confirmation. There are also peculiarities that need to be explained, for example SN2003fg appears to have contained 1.3 M(s) of Ni alone. It also displays carbon absorption lines that are weak or absent in typical Type 1a supernovae. Also its shell of material was cast off at a relatively slow velocity: ~ 8,000 km/s rather than the more usual 10,000- 12,000 km/s.

How could a white dwarf attain such mass, if the data are to be believed? Three main hypotheses are up for consideration:

1) The white dwarf may have been rotating unusually rapidly so that the centripetal force produced helped to balance the gravity - thereby preventing the halting of collapse at the usual 1.44 limit .

2) The supernovae could have resulted from TWO white dwarfs merging and exploding.

3) It may have been an ordinary white dwarf exploding asymmetrically with its birght side facing Earth, thereby yielding anomalous L readings.

We'll have to dig further to see what it all means, but in the meantime, the data provide an intriguing puzzle and possible test of what we think we really know about white dwarf physics.

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