Transcript 1/2 - Indico

From Nuclei to Stars:
A brief overview of Nuclear Astrophysics
D. N. Basu
Flow Chart:
 Lecture 1: Astrophysical Observations
• Hertzsprung-Russell Diagram & Mass-Luminosity Relations
•Main sequence stars, White Dwarfs, Red Giants
•Expansion of universe & Hubble’s Law
 Steady-state theory
• Calculation of mass generation
• Problems with Steady-state theory & Olber’s Paradox
 Lecture 2: Big-Bang theory
• Universal background radiation
• Standard Big-Bang model basic physics
• Hubble’s constant and its time dependence
• Radiation & matter dominated era
•Nuclear reaction rate; Nucleosynthesis
 Lecture 3: Compact Stars
•White Dwarfs, Neutron stars & Black Holes
• Lane-Emden Equation & Tolman-Oppenheimer-Volkoff Equation
• Polytropic Equation of State (EoS)
Lecture 4: White dwarfs
•Degenerate Electron gas & EoS for White Dwarfs
• Chandrashekhar’s Limit for White Dwarf mass & super-Chandrasekhar White Dwarfs
Lecture 5: Neutron Stars
• Modelling Neutron Stars
•β-equilibrated Neutron Star matter and Quark matter EoS
• Deconfinement phase transition: from nuclear matter to quark matter
• Calculations and Results: Masses & Radii of Neutron and Hybrid Stars
Hertzsprung-Russell Diagram & Mass-Luminosity Relations
Luminosity L is defined as the total energy emitted by a star: L = 4p R2 sT4
where s is the Stefan’s constant for black body radiation. In terms of sun’s properties this
relation leads to
L/L⊙ = (R/R⊙)2 (T/T⊙)4
An important application of H-R diagram is in the determination of
stellar distances. If the spectral class (temp) or color of a main
sequence star is known, its absolute luminosity is determined
from its position in the H-R diagram. This leads to distance of the
star by comparison with its observed brightness because the
brightness proportional to L/(distance)2 .
It is generally found that the most massive stars in the main
sequence are also most luminous which is not accidental but results
from the fundamental laws that govern the internal structures of the
stars. It is estimated that 90% of main sequence stars obey the
relation L proportional to M3.5 . The main sequence stars have
masses ~0.1 - 50 M⊙ and luminosity ~10-2 - 106 L⊙.
Although stars with masses ~10M⊙ initially possesses a much
larger reservoir of nuclear fuel than sun, it burns the fuel >1000
times faster, resulting in a much shorter life and a much quicker run
through the evolutionary stages. As a star grows older, it will run
through the H-R diagram on an evolutionary track fixed from the
outset by its initial mass and the mass loss along the way.
Expansion of universe & Hubble’s Law
Spectra of stars are generated by their surface chemical elements , each of which emits a characteristic
pattern of lines whose wavelengths are precisely known from the laboratory measurements. When a
galaxy (and hence its stars) are moving away from an observer, the wavelength of each spectral line
increases (shifts toward red) as a result of Doppler effect. The redshift Z is defined as
Z = Dl/l = [l(v) – l(0)] / l(0) = [l(v) / l(0)] – 1 = (n0 / nv) – 1
where n0 and nv are the emitted and observed frequencies, respectively. From relativistic Doppler effect
nv = n0(1-v/c)/(1-v2/c2)1/2 = n0[(1-v/c)/(1+v/c)]1/2
so that
Z = [(1+v/c)/(1-v/c)]1/2 – 1,
where v is the velocity with which a galaxy recedes and c is the velocity of light in vacuum. Thus an
observed Z shift can be used to calculate the velocity of the object relative to the observer.
Establishing distances of the galaxies, it was shown by Hubble that the velocities are recessional and the
velocity of a galaxy is directly proportional to its distance. Thus Hubble’s law is represented mathematically
v=Hr
Obviously, H has a dimension of time- inverse and hence 1/H is called the Hubble’s time. Since velocities
of all the galaxies are found to be recessional, it is concluded that our universe is ever expanding.
Steady-state theory
Steady-state theory is based on the following assumptions :
(1) Universe is unchanging, infinite and indefinitely old & (2) The density of matter remains constant
In order to explain cosmic expansion, the steady-state theory had to postulate continual creation of matter.
Calculation of mass generation: As the density r of the matter is assumed to be constant, the rate of
mass (M) change with time in a given volume of space (of radius r) is zero, implying
dM/dt = -4prr2 dr/dt + (4p/3) z r3 = 0
where z is rate of mass generation (from nothing). Using Hubble’s law dr/dt = Hr, above equation becomes
dM/dt = 0 = -4prr3 H + (4p/3) z r3,
implying
z = 3rH,
which for r = 2 x 10-31 gms/cc, H = 2 x 10-18 /sec, provides z = 1. 2 x 10-48 gms/cc/sec or 1 nucleon created
every 50 years per km3. This rate being extremely low may be treated as still undiscovered law of physics.
Problems with Steady-state theory & Olber’s Paradox: In steady-state model it is particularly
difficult to account for the microwave background radiation which has a spectral characteristics of thermal
radiation of 2.76 oK. It is satisfactorily explained as a relic of an epoch which was very hot and very dense.
Steady-state universe can not have had such conditions since all conditions, by definition, are unchanged.
Another problem is that the night sky can not remain dark. To prove this, let us assume that the absorption
of space can be neglected. The apparent luminosity of a star of absolute luminosity L at a distance r will be
L/4pr2. If the number density n of such stars is assumed to be constant, then the number of stars at a
distance between r and r+dr is 4pnr2dr, so that the total radiant energy density received at earth due to all
stars =∫(L/4pr2)4pnr2dr =Ln∫dr. The integral diverges for an infinite universe (limits 0 to infinity) as assumed
by the steady-state theory, leading to infinite energy density of starlight implying that the night sky should
never get dark. This is known as Olber’s Paradox which is avoided in the Big-Bang theory assuming finite
age and hence finite size of the universe.
Big-Bang theory
Standard Big-Bang model basic physics: If the recessional velocity of every galaxy remained unchanged
through all time, any galaxy now receding from us was once arbitrarily close and the time that has elapsed
since then is equal to the ratio of galaxy’s distance and its velocity. Since this ratio is 1/H which is same for
all the galaxies, all of them must have been crowded together at the same ancient time. In other words, at
some unique time in the past all the matter in the universe was compressed to an arbitrarily high density
everywhere. This time is approximately given by the Hubble’s time ~ 16 billion years – the age of the
universe. This simple calculation has led to the cosmological hypothesis that the world began with a great
primordial explosion called the “Big-Bang” involving all the matter and energy in the universe.
Universal background radiation: In 1965 radio astronomers Penzias and Wilson discovered another basic
and most important cosmological phenomenon. It is the low energy cosmological microwave radiation
background that apparently fills the universe and bathes the earth from all directions in space. This
microwave radiation has been shown to be highly isotropic. Intensity measurement of several wavelengths
show that this background radiation is consistent with that of a black body (Planck’s radiation law) at
temperature of 2.76 oK. Radiation of this type can not have been generated by known astronomical
objects. The current view is that this microwave radiation was there at the beginning of the universe. In the
first few seconds of the history of the universe the radiation had a temperature of ~1010 oK. The present day
temperature of 2.76 oK has come about because in the expansion of the universe the radiation has
constantly cooled from its initially extremely hot state.
Radiation & matter dominated era : Number density Ng & energy density eg of photons can be calculated as
Ng = ∫ nn dn = ∫(8pn2/c3) [1/{exp(hn/kT)-1}]dn = (p/13)(kT/ħc)3 = 20.25 T3
eg = ∫ hnnn dn = ∫(8phn3/c3) [1/{exp(hn/kT)-1}]dn = (p2/15)(kT)4/(ħc)3 = 4.72 x 10-3 T4 eV/cc
where T is in oK. The microwave background of 2.76 oK thus corresponds to photon gas filling the universe
with a number density of ~ 430 photons/cc. From the data of this background radiation, theorists are able to
calculate a new fundamental quantity: ratio of number of photons to baryons (protons & neutrons) in
universe which is ~109.
Although number density of photons is very high compared to baryons, photons being zero rest mass
particles have smaller energy density. The equivalent mass density of photons can be given by
rg = eg /c2 = 8.4 x 10-36 T4 gm/cc
which for present epoch (T=2.76 oK) turns out to be 4.9 x 10-34 gm/cc and is negligibly small compared to
the matter density r ~ 2 x 10-31 gm/cc (WMAP value 4.5 x 10-31 gm/cc). Therefore present epoch is called
“matter dominated”.
As universe expands like a container with expanding walls which is filled with particles, the Doppler effect
increases the wavelengths l of the particles in proportion to the size R. The Wein’s displacement law
yields lT constant and thus temperature T drops as 1/R. Since density r is proportional to1/R3 implies r is
proportional to T3. Therefore r = r0 (T/T0)3 where r0, T0 are baryonic matter density 2 x 10-31 gm/cc and
temperature 2.76 oK at the present epoch. For T = 1500 T0 = 4140 oK, one finds r ~ 6.75 x 10-22 gm/cc
and rg = 8.4 x 10-36 T4 gm/cc = = 2.47 x 10-21 gm/cc. Thus at T = 4140 oK, rg >> r and such an epoch is
called “radiation dominated”.
Hubble’s constant and its time dependence: We examine now the time dependence of the parameters of
the universe. Consider a sphere of radius R(t) enclosing a portion of the universe and suppose that at time
t a typical galaxy of mass m is at the surface of the sphere. Mass M contained in this sphere is given by
M = (4p/3) r(t)R3(t)
where r(t) is the mean density. It can be shown that the galaxy is gravitationally influenced only by the
matter within the sphere. The total energy E of the galaxy is then given by the sum of k.E. + p.E. as
E = -mMG/R(t)+(1/2)mv2(t) = mR2(t) [(1/2)H2(t) - (4p/3)Gr(t)]
: : using Hubble’s law v(t)=H(t)R(t)
Since the total energy E must remain constant at all times and since r(t) is proportional to1/R(t)3 the
potential energy goes to infinity at R(t) tending to 0, and thus two terms in the [ ] bracket must cancel in
this limit implying H2(t) = (8p/3)Gr(t). This relation defines the characteristic expansion time which is just
the inverse of Hubble’s constant: texp = [3/(8pr(t)G)]1/2. This result is also valid if the density includes
matter density as well as mass equivalent of radiation eg /c2 also.
In the matter dominated epoch the total mass within an expanding sphere remains constant and thus
matter density r(t) is proportional to1/R3(t). In contrast rg is proportional to T4 and since T is proportional to
1/R(t), it follows that rg is proportional to 1/R4(t). In general we can write r(t) is proportional to1/Rn(t) with
n= 3 & 4 for matter & radiation dominated eras respectively. As H (t) = [(8p/3)Gr(t)]1/2, it follows that
dR(t)/dt = H(t)R(t) is proportional to R(1-n/2)(t) whose solution provides
t2 - t1 = (2/n)[1/H(t2) - 1/H(t1)] = (2/n)[3/(8pG)]1/2 [r(t2)-1/2 - r(t1)-1/2]
Since at t1=0 the density is tends to infinity, the above result can be generalized to give the time t needed
to decrease to a density r(t) as
t = (2/n)[1/H(t)] = (2/n)[3/(8pr(t)G)]1/2
which is equal to texp /2 for n = 4 i.e. the radiation dominated era and equal to 2texp /3 for n = 3 i.e. the
present epoch of matter dominated era.
As the size of the universe is reduced (travelling backward in time), the temperature increases and the
radiation density becomes higher and higher. Above threshold temperature T ~ 1010 oK photons become
capable of producing electron-positron pair. Although particles and antiparticles with opposite charges are
created in this process the presence of matter in the universe clearly indicates that the baryon number of
universe is non-zero (NB /Ng ~ 10-9). The condition NB >0 means that the universe is not symmetric with
respect to matter and anti- matter. To explain this asymmetry, let us consider that equal numbers of Ko and
Ko-bar (anti particle of Ko) are produced by high energetic g’s. Their decays
Ko = e+ + p- + ne
Ko-bar = e- + p+ + anti-ne
are matter–antimatter correspondents. Yet the two sets of products are not equally produced as the Ko
decays about 0.7% more frequent than Ko-bar. Similarly generalizing for any X, X-bar bosons, when all X
bosons had disappeared, quarks and leptons would outnumber antiquarks and antileptons by some small
margin leading to a small excess of matter over antimatter, and all antiquarks (smaller in number than
quarks) would annihilate with quarks and antileptons with leptons winding up as high flux of photons.
The Source of Stellar Energy
Stars produce energy by nuclear fusion of
hydrogen into helium.
In the sun, this
happens
primarily
through the
proton-proton
(PP) chain
Stars: energy source: proton-proton chain
PPI (85% for Sun):
H1 + H1 -> D2 + e+ + nu(1) (1.442 MeV)
D2 + H1 -> He3 + gamma (5.493 MeV)
He3 + He3 -> He4 + 2H1 (12.859 MeV)
PPII (15% for Sun):
H1 + H1 -> D2 + e+ + nu(1) (1.442 MeV)
D2 + H1 -> He3 + gamma (5.493 MeV)
He3 + He4 -> Be7 + gamma (1.586 MeV)
Be7 + e- -> Li7 + nu(2) (0.861 MeV)
Li7 + H1 -> He4 + He4 (17.347 MeV)
PPIII (0.01% for Sun):
H1 + H1 -> D2 + e+ + nu(1) (1.442 MeV)
D2 + H1 -> He3 + gamma (5.493 MeV)
He3 + He4 -> Be7 + gamma (1.586 MeV)
Be7 + H1 -> B8 + gamma (0.135 MeV)
B8 -> Be8 + e+ + nu(3) (followed by spontaneous decay...)
Be8 -> 2He4 (18.074 MeV)
10
The CNO Cycle
In stars slightly more
massive than the sun
(>1.1 times), a more
powerful mechanism
for energy generation
than the PP chain
takes over:
The CNO Cycle.
The CNO Cycles
Mainly of interest for massive stars & nucleosynthesis
Basic mechanism for nucleosynthesis beyond iron: r- and s- process
Big-Bang Nucleosynthesis
The standard model of Big-Bang
Nucleosynthesis (BBN) builds upon
Friedman-Robertson-Walker cosmological
model. The standard model of Particle
Physics provide the constituent particles. It
sets any extra energy density and chemical
potentials of the three neutrinos to zero.
The thermal properties of each particle is
described by a temperature & a chemical
potential. The thermal coupling among
constituents equilibrates the temperature
while chemical coupling relates the
chemical potentials. Charge neutrality
provides another constraint. Taking the
neutron lifetime as 885.7(8) seconds, the
only adjustable/free parameter is the
baryon/photon ratio. Using this parameter
the system of coupled equations of the
rate of change of abundances of all the
species, is solved to get the abundances as
functions of time/temperature.
Reaction Rates: The fixation of baryon-tophoton ratio value from WMAP results leaves
only the reaction rates as adjustable inputs.
We replace 35 out of 88 reaction rates in the
Kawano-Wagoner model of standard BBN by
newer reaction rates which are meant to
supersede the earlier compilations.
The nuclear reaction inputs to BBN take the form of thermal rates which are the cross sections
averaged over a Maxwell-Boltzmann distribution.
1
E – centre of mass energy

2


v – relative velocity of reactants




8
σv  
  σ(E)E exp  E k T dE
m- reduced mass of reactants
3
B 



 
 πμ k T  
S – astrophysical S-factor
 B  

Z Z e2
z – Sommerfeld parameter  12v
s is the cross section, which for
S(E)exp(  2πz)
low energies is factorized as σ E  
E
Results:
Conclusion:
There is a little effect of new reaction rates on the standard BBN abundance yields.
 The chances of solving either of lithium problems by conventional means are unlikely.
Future Plans:
• Study of deep sub-barrier fusion reactions & astrophysical S-factor calculations
• Calculation of reaction rates of astrophysical importance
• Investigation of effects of some more new reaction rates on BBN
• Study of White Dwarf EoS under strong magnetic field & super-Chandrasekhar limit
Compact Stars: Introduction
Black Hole
When the sign of E is negative no equilibrium exists, the total energy / fermion can be decreased without
bound by decreasing R and gravitational collapse sets in resulting in Black Hole from which nothing can
escape not even light. Since gravitational Doppler effect is given by
n= n0 [1 – 2GM / Rc2]1/2
this happens when
2GM / Rc2 > 1.
By setting 2GM/Rc2 = 1, one obtains the criteria for Black-Hole formation and then
R = 2GM / c2
Is called the Schwarzschild radius for a particular mass M.
White Dwarfs
A white dwarf, also called a degenerate dwarf, is a stellar remnant composed mostly of electrondegenerate matter. They are very dense; a white dwarf's mass is comparable to that of the sun, and its
volume is comparable to that of the Earth. Its faint luminosity comes from the emission of stored thermal
energy.
White dwarfs are thought to be the final evolutionary state of all stars whose mass is not high enough to
become a neutron star—over 97% of the stars in the Milky Way. After the hydrogen–fusing lifetime of a
main-sequence star of low or medium mass ends, it will expand to a red giant which fuses helium to
carbon and oxygen in its core by the triple-alpha process. If a red giant has insufficient mass to generate
the core temperatures required to fuse carbon, around 1 billion K, an inert mass of carbon and oxygen
will build up at its center. After shedding its outer layers to form a planetary nebula, it will leave behind
this core, which forms the remnant white dwarf. Usually, therefore, white dwarfs are composed of carbon
and oxygen. If the mass of the progenitor is between 8 and 10.5 solar masses, the core temperature is
sufficient to fuse carbon but not neon, in which case an oxygen-neon–magnesium white dwarf may be
formed. Also, some helium white dwarfs appear to have been formed by mass loss in binary systems.
The material in a white dwarf no longer undergoes fusion reactions, so the star has no source of energy,
nor is it supported by the heat generated by fusion against gravitational collapse. It is supported only by
electron degeneracy pressure, causing it to be extremely dense. The physics of degeneracy yields a
maximum mass for a non-rotating white dwarf, the Chandrasekhar limit—approximately 1.4 solar
masses—beyond which it cannot be supported by electron degeneracy pressure. A carbon-oxygen white
dwarf that approaches this mass limit, typically by mass transfer from a companion star, may explode as
a Type Ia supernova via a process known as carbon detonation.
Chandrashekhar Mass Limit for White Dwarfs
• The “critical mass” at which the white dwarf detonates is nothing
but the Chandrasekhar mass .
• The white dwarf detonates due to runaway thermonuclear fusion
reactions.
• Since the Chandrasekhar limit is constant, the amount of energy
output and hence the “luminosity” or absolute brightness of every
Type Ia supernova is the same.
• This fact makes Type Ia supernovae as probes or “standard
candles” for the measurement of the acceleration of the universe.
Illustration of a Type Ia supernova:
Chandrasekhar limit for white dwarfs : Let there be N fermions in a star of radius R, so that the
number density of fermions is n ~ N / R3 . The volume per fermion is 1/n so that the uncertainty relation
(between momentum & length) provides momentum of a fermion as ħ n1/3 . Thus Fermi energy of a gas
particle in the relativistic regime is
EF ~ ħn1/3 c ~ ħcN1/3 / R
The gravitational energy per fermion is EG ~ -GMmB /R
where M = NmB , mB is the baryonic mass
and remembering that even if pressure comes from electrons, most of the mass is in baryons. The
equilibrium is achieved at a minimum of the total energy / fermion E given by
E = EF + EG = ħcN1/3 /R - GMmB /R.
When the sign of E is negative, the total energy / fermion can be decreased without bound by decreasing
R and gravitational collapse sets in. When the sign of E is positive, it can be decreased by increasing R.
This decreases EF and electrons tend to become non-relativistic with EF ~ pF2 ~1/R2 . Eventually EG
dominates over EF with increasing R, and then E becomes negative, increasing to zero as R tends to
infinity. There will be therefore a stable equilibrium at a finite value of R. The maximum baryon number for
equilibrium is therefore determined by setting E =0:
Nmax ~ [ħc/GmB2]2/3 which yields maximum mass Mmax ~ mBNmax ~ 1.4 solar mass.
A white dwarf is very hot when it is formed, but since it has no source of energy, it will gradually radiate
away its energy and cool. This means that its radiation, which initially has a high color temperature, will
lessen and redden with time. Over a very long time, a white dwarf will cool to temperatures at which it will
no longer emit significant heat or light and it will become a cold black dwarf. However, the length of time it
takes for a white dwarf to reach this state is calculated to be longer than current age of universe.
Highly Magnetized Super-Chandrashekhar White Dwarfs
•
Form the endpoint of stellar evolution of low to intermediate mass stars upto ~10MSun.
•
Supported by relativistic electron degeneracy pressure against gravitational collapse.
•
Radius of the order of the radius of the earth and mass of the order of stellar mass.
•
Maximum mass consisting of “free” electron gas is given by Chandrasekhar limit ~1.4 MSun
•
Observations have revealed that 10% of all identified white dwarfs have high magnetic fields ~ 1 MG.
•
Of these, some of them have surface magnetic fields as high as 109 G (PG 1031+234) , comparable to
the magnetic fields of neutron stars.
•
Moreover, observations have shown that ultramagnetized white dwarfs are ultramassive, close to the
Chandrasekhar limit. For example, EUVE J0317-853 has a mass of 1.37MSun and a surface field
strength of 800 MG.
•
These white dwarfs are expected to have a magnetic field strength ~ 1012 – 1015 G at their centres.
Landau Quantization:
Electron Gas in a strong Magnetic field
The energy states of a free nonrelativistic electron in a uniform
magnetic field are quantized into what is known as Landau
levels, which define the motion of the electron in a plane
perpendicular to the magnetic field. The dispersion relation is :
where
Eν,p z
p z2
 νωc 
2me
1
n  j  s
where
2
j = 0,1,2,… is the principal quantum number
of the Landau level and σ = ±1/2 is the spin of the electron.
•The electron gas at zero temperature can become relativistic in two ways:(i)
If the Fermi energy equals the rest-mass energy of the electron at
sufficiently high density, or
(ii)
If the cyclotron frequency equals the rest-mass energy of the electron.
•
In the second case the strength of the magnetic field can be found from the
equation
2 3
m
eB
2
2
ec
c  me c  
 me c  B  Bc 
 4.414 1013 G
me c
e
Both conditions are easily satisfied in the interior of strongly magnetized
white dwarfs.
Relativistic Fermi gas in strong magnetic field
•The Landau level dispersion relation of the Dirac equation is given by
En , pz
 2 2
B 
2 4

  p z c  me c 1  2n
Bc 


1/ 2
•The density of states in unit momentum interval of a degenerate
magnetized electron gas at absolute zero is 2eBgn / h 2 c where gν = 1
for ν = 0 and gν = 2 for ν ≥1.
2eB
2
• Hence total density of states should be 2 g v  dp z instead of 3  d 3 p .
h
n h c
B
pF (n )
EF (n )
B

;
x
(
n
)

;
e
(
n
)

Let
D
F
F
2
Bc
me c
me c
•The number density is then given by
nm
2eB
ne   2 g v
n 0 h c
pF (n )
nm
2eB
g v pF (n )
2
n 0 h c
 dpz  
0
•The energy density is given by
nm
2eB
e e   2 gv
n 0 h c
pF (n )
nm

2BD
xF (n ) 
2

E
dp

m
c
g
(
1

2
n
B
)

D
0 n , pz z (2p )2 le3 e n0 v
 (1  2nB )1/ 2 
D


•The pressure is given by
nm

d  ee 
2 BD
xF (n ) 
2
   e e  ne EF 


Pe  n
m
c
g
(
1

2
n
B
)


e
v
D
3
1/ 2 
2

dne  ne 
(2p ) le
n 0
 (1  2nBD ) 
2
e
where
z
1
2
1
2
1
2
1
2
 ( z )   1  y 2 dy  z 1  z 2  ln( z  1  z 2 ) and  ( z )  z 1  z 2  ln( z  1  z 2 )
0
•The Fermi energy of the Landau level ν is given by
 2
B 
2
2 4

EF (n )   pF (n )c  me c 1  2n
Bc 


1/ 2
•The maximum number of Landau levels can be found from the
condition
pF2 (n )  0
n 
e F2  1
2 BD
•So we see that vm  e F2 max
and
vm 
1
B
or n m 
e F2 max  1
2 BD
if B is kept constant.
if e F max is kept constant.
Equation of state , mass-radius relation & stability
Figure 1: Graph of EOS of ultramagnetized
super-Chandrasekhar WD having εFmax =
20.
Pe = PD (2.668*1027 erg/cc)
ρe = ρD (2*109 gm/cc)
Ref: U. Das and B. Mukhopadhyay, Strongly
magnetized cold degenerate electron gas:
Mass-radius relation of the magnetized
white dwarf, Phys. Rev. D 86 (2012)
042001 [arXiv:1204.1262].
Figure 2: Mass-Radius plot of ultramagnetized
super-Chandrasekhar WD having εFmax =
20.
M = MD (MSun)
R= RD (108 cm)
Ref: U. Das and B. Mukhopadhyay, Strongly
magnetized cold degenerate electron gas:
Mass-radius relation of the magnetized
white dwarf, Phys. Rev. D 86 (2012)
042001 [arXiv:1204.1262].
Future works
Inclusion of electron-electron & electron-nucleus Coulomb interactions
in the magnetic EOS of electron gas using the Thomas-Fermi and
Feynman-Metropolis-Teller statistical treatments and the inclusion
of the modified EOS in TOV equation to find the mass-radius
relation, stability of magnetized super-Chandrasekhar white dwarfs.
•It is basically a statistical theory of highly compressed atoms below
neutron drip density.
•Incorporates the effects of nuclear-electron and electron-electron
Coulomb interactions in an atom.
•Assumes the electrons in an atom as a Fermi gas of non-uniform
density and radially dependent Fermi energy. Hence the theory is
more suitable for heavy atoms and compressed atoms.
•Not all electrons give the degeneracy pressure. Pressure is given only
by the electrons at the boundary of the atoms.
Neutron Stars
A neutron star (NS) is the remnant of a large star after core collapse, where
• there is no thermonuclear fusion
• gravitational pull is stalled by the pressure of degenerate neutrons
• the matter is in beta-equilibrium
Neutron stars are hard to see but can be detected as they rotate.
Rotating stars are important for
 Their bulk properties constrain the high-density Equation of State (EoS).
(Mass and radius are among the physical properties which can be
determined from observations also.)
 Changes in their rotational periods provide information about the physical
processes going inside or of Cosmological relevance.
 The rotational instabilities are predicted to produce gravitational waves
which are expected to be detected and enrich the observation based
knowledge.
The beta equilibrium proton fraction calculated with NSE obtained from
present work is plotted as a function of r/r0.
0.05
Fractionproton
0.04
0.03
0.02
0.01
0.00
0
1
2
3
r/r0
4
5
At temperatures sufficiently lower than typical Fermi temp (TF ~1012 K),
n, p, e must have momenta close to Fermi momenta (PF ).
The condition for momentum conservation for direct URCA is:
PFp +PFe ≥ PFn neglecting the neutrino and antineutrino’s momenta.
Using charge neutrality condition np=ne and PF =(1.5p2r)1/3 , baryon
density= r=rn+rp one can find at threshold rn= 8rp  x=rp/r=xth=1/9.
From the condition of beta equilibrium in degenerate matter we have chemical
potential (m) of electron me= mn- mp=-є/x where є (r, x) =energy per
baryon.ħc(1.5p2rx)1/3= 4Sv(r)(1-2x) .
The density rth at which proton fraction x=xth=1/9 can be found from
Svα rq.in the above relation.
An alternative rapid cooling path: Direct Hyperon URCA
The hyperons begin to populate the central region of the star at the density
n≈2n0, (RBHF calculation) where n0=0.16 fm-3= normal nuclear matter density.
[Ref. H. Huber et al., nucl-th/9711025].
Presence of hyperons might lead to direct hyperon URCA even if the proton
fraction is too small (<11%). [M. Prakash et al. Astrophys. J 390, 1992, L77]
Besides the direct nucleon URCA process the most important Direct Hyperon
URCA processes (with their inverse at same rate) are :
S-→L+ e-+ νe, L→p + e- + νe,
S-→n + e- + νe, X-→L+ e- + νe.
Modelling Neutron Star
 The observations of pulsar glitches show that for a Neutron Star the departures
from perfect fluid are of the order 10-5 .
 As the star cools, several mechanisms like viscosity, turbulent motions,
magnetic effects, etc. act to enforce uniform rotation on a small time scale .
 Within a year of its formation, the temperature (<< 1 MeV) of an NS becomes
much smaller than the Fermi energies (> > 60 MeV) of the interior.
After ~1 year of its formation, the bulk properties of an
isolated rotating relativistic star can be modelled accurately by a
uniformly rotating, zero-temperature perfect fluid.
The matter can then be characterized by the energy-momentum tensor
Tμν  ξ  P u μ u ν  g μν P
x = energy-density
P = pressure
uμ,ν = four-velocity
gμν = metric tensor
Numerical scheme of Stergioulas et al.
If rapidly rotating compact stars were non-axisymmetric, they would emit gravitational
waves in a very short time scale and settle down to axisymmetric configurations.
Assumptions:




Matter can be described as a perfect fluid.
Matter distribution and the spacetime are axisymmetric.
The matter and the spacetime are in a stationary state.
The matter has only circular motions (no meridional motions)
represented by angular velocity.
 Angular velocity is constant, as seen by a distant observer.
 Under these assumptions the metric can be expressed as


ds 2  e (γ ρ)dt 2  e 2α dr 2  r 2 dθ 2  e (γ ρ) r 2sin 2θdφ  ωdt 
Metric potentials γ, ρ, α and ω depend on r and θ only.
 Now the different components of Einstein’s field equations are solved
with a specified EoS of form ξ = ξ(P).
N. Stergioulas et al. The Astrophysical Journal 444 (1995) 306
T. Nozawa et al. Astronomy & Astrophysics Supplement Series 132 (1998) 431
H. Komatsu et al. Monthly Notices of Royal Astronomical Society 237 (1989) 255
G. B. Cook et al. The Astrophysical Journal 422 (1994) 227
2
Equations of State
A compact star can be broadly divided into two main regions‐ crust and core.
Crust: accounts for ~5% of mass and ~10% of radius
Equations of state applied are:
i. Feynman ‐ Metropolis ‐ Teller (FMT) : based on Thomas‐Fermi model;
applicable at densities below r ~104 g/cm3 where a fraction of electrons are
bound to nuclei.
ii. Baym ‐ Pethick ‐ Sutherland (BPS) : applicable above r ~104 g/cm3 and
up to neutron drip, r ~4.3x1011g/cm3 = 2.6x10-4 baryons/fm3.
iii. Baym ‐ Bethe ‐ Pethick (BBP) : based on compressible liquid drop model
to calculate energy of nuclear matter, as a function of density and
proton concentration. Applicable at densities above 4.3x1011 g/cm3.
Core: accounts for major part of the star
:contains nucleons (mainly neutrons) from sub-nuclear density (r0/3 - r0/2)
to supra-nuclear density (~10r0) and sometimes quarks and strangeness.
Equation of State for core: b  equilibrated nuclear matter for neutron star core
FMT: R. P. Feynman et al. Physical Review 75 (1949) 1561
BPS: G. Baym et al.
The Astrophysical Journal 170 (1971) 299
BBP: G. Baym et al.
Nuclear Physics A 175 (1971) 225
Equation of State for core: For hadronic part we use our β-equilibrated NS matter EoS
Energy per nucleon e for interacting Fermi gas of neutrons and protons turns out to be
E/A= e = [3ħ2k2F /10m]F(X) + C (1-b r2/3) r Jv /2
with F(X) = [(1 + X)5/3 + (1 − X)5/3]/2 where X is the isospin asymmetry, where Jv= Jv00 + X2 Jv01
= ∫∫∫ [t00M3Y + X2 t01M3Y ]d3s and considering energy variation of zero range potential to vary with
the kinetic energy part of ekin of e.

c 3π 2ρx β

13
 4Esym 1  2x β 
where xβ = β-equilibrium proton fraction and
the symmetry energy Esym= e(x=1) – e(x=0)
[Basu et al. Nuclear Physics A 811 (2008) 140]
For quark matter, we use EoS of Kurkela et al.
[A. Kurkela et al. Physical Review D 81 (2010) 105021]
Constraints on EoS from HIC
Danielewicz et al. Science 298 (2002) 1592
Calculation and Results
Bounds:- Static models, Keplerian rotation and instabilities
Keplerian frequency = mass shed limit
Instabilities: hinder the star to rotate with Keplerian frequency
I. Pure Neutron Star
up to ρ ≈ 0.05 fm-3 or equivalently
energy-density up to ξ ≈ 7.7x1013 g/cm3 , then
beta EoS: up to ρ ≈ 1.4 fm-3 or
up to ξ ≈ 3.5x1015 g/cm3
crust EoS:
The common tangent is drawn for the energy density versus density plots where pressure is
the negative intercept of the tangent to energy density versus density plot. However, as
obvious from the figure, the phase co-existence region is negligibly small which is
represented by part of the common tangent between the points of contact on the two
plots implying constant pressure throughout the phase transition.
Energy density (energy/vol.) ξ can be defined as
ξ = r(e  mc2)
Hence:
dξ/dr = (e  mc2)  r de/dr
Or,
rdξ/dr = ξ  r2de/dr
Since pressure P is given by
implies that:
P = r2de/dr
ξ = (dξ/dr r  P
Thus the negative intercept of the tangent (having slope dξ/dr) drawn at a point to
the energy density versus density plot represents pressure at that point.
II. Neutron Star with quark core
The energy-density of quark matter is
lower than nucleonic EoS at densities
higher than 0.4 fm-3.
So we take
crust EoS: up to ρ ≈ 0.05 fm-3
beta EoS: up to ρ ≈ 0.4 fm-3 or
ξ ≈ 7x1014 g/cm3
quark EoS: up to ρ ≈ 1.37 fm-3 or
ξ ≈ 3x1015 g/cm3
Summary
 For pure Neutron stars, maximum mass and radius for
•
•
•
•
static case – 1.92 solar mass and 9.7 km
rotational period 2.0 ms – 1.94 solar mass and 9.8 km
rotational period 1.5 ms – 1.95 solar mass and 9.9 km
Keplerian limit – 2.27 solar mass and 13.1 km
 The energy-density of quark matter is lower than that of β-equilibrated NS matter
at densities above ≈0.4 fm-3 showing a hadron to quark transition inside.
 For Neutron stars with quark core, maximum mass and radius for
•
•
•
•
static case – 1.68 solar mass and 10.4 km
rotational period 2.0 ms – 1.71 solar mass and 10.6 km
rotational period 1.5 ms – 1.72 solar mass and 10.7 km
Keplerian limit – 2.02 solar mass and 14.3 km.
Conclusion
The nucleon-nucleon effective interaction used in the present work, which is found to
provide a unified description of elastic and inelastic scattering, various radioactivities and
nuclear matter properties, also provides an excellent description of the β-equilibrated NS
matter which is stiff enough at high densities to reconcile with the recent observations of the
massive compact stars 2 M⊙ while the corresponding symmetry energy is supersoft as
preferred by the FOPI/GSI experimental data.
Although compact stars with quark cores rotating with Kepler’s frequency have masses up
to 2 M⊙, but if the maximum frequency is limited by the r-mode instability, the maximum
mass 1.7 M⊙ turns out to be lower than the observed mass of 1.97±0.04 M⊙, by far the
highest yet measured with such certainty, implying exclusion of quark cores for such massive
pulsars.
In order not to conflict with the mass measurement, either there must be some
mechanism to prevent nuclear matter to deconfine into quark matter or the quark EoS
(which was obtained from rigorous ab-initio calculation and not phenomenological bag
model) should be made stiffer by several possible realistic improvements.
It is worthwhile to mention in the present context that a pulsar (PSR J17482446ad)
rotating faster than the limit set by the r-mode instability has been already observed.
T
H
A
N
K
Y
O
U
Isoscalar and isovector components of the effective interaction
The central part of the effective interaction between two nucleons 1 and 2
can be written as:
v12(s) = v00(s) + v01(s) t1.t2 + v10(s) s1.s2 + v11(s) s1.s2 t1.t2
where t1, t2 are the isospins, s1, s2 are the spins and s is the distance
between nucleons 1, 2.
 For spin symmetric nucleons v10 and v11 do not contribute
Z-component of Isospins (Iz) of proton and neutron are +1 and -1.
tn.tp= tp.tn = -1 and tn.tn= tp.tp = +1.
 For SNM only the first term, the isoscalar term, contributes.
For IANM the first two terms the isoscalar and the isovector (Lane)
terms contribute.
 The n-n and p-p interactions are vnn= vpp= v00+v01
The n-p and p-n interactions are vnp= vpn= v00-v01
Effective NN interaction potential

The density dependent M3Y interaction potential is used for isoscalar and
isovector part:
v00 (s,r,ekin)= t00M3Y(s,ekin) g(r),

v01 (s,r,ekin) = t01M3Y(s,ekin) g(r )
Isoscalar t00M3Y and isovector t01M3Y components of M3Y interaction
supplemented by zero range potential representing the single nucleon exchange
term are given as
M 3Y
t00
 7999
exp( 4 s )
exp( 2.5s )
 2134
 276(1  e kin ) ( s )
4s
2.5s
M 3Y
t01
 4886
exp( 4 s )
exp( 2.5s )
 1176
 228(1  e kin ) ( s )
4s
2.5s
where the energy dependence parameter  = 0.005 MeV-1. Strengths of the Yukawas are
extracted by fitting its matrix elements in an oscillator basis to those elements of G-matrix
obtained with Reid-Elliott soft core NN interaction. The density dependence of the form
g(r)= C[ 1-br2/3] accounts Pauli blocking effects and higher order exchange effects.
Symmetric and isospin asymmetric nuclear matter calculations
For a single neutron interacting with rest of nuclear matter with isospin
asymmetry X=(ρn-ρp)/(ρn+ρp), the interaction energy per unit volume at s is:
ρnvnn(s) + ρpvnp (s) = ρn [v00(s) + v01(s)] + ρp [v00(s) - v01(s)]
= [v00(s) + v01(s)X]ρ
Similarly, for the case of proton the interaction energy per unit volume
ρnvpn(s) + ρpvpp (s) = [v00(s) - v01(s)X]ρ
 Asymmetric nuclear EOS can be applied to study the pure neutron matter
with isospin asymmetry X=1.
 The bulk properties of neutron matter such as the nuclear incompressibility (Ko), the
energy density (x), the pressure (P) and the velocity of sound in nuclear medium can be
used to study the cold compact stellar object like neutron star.
Kinetic & Potential energy of a nucleon in
symmetric nuclear matter
• K.E. /A = [ 0∫kF (ħ2k2/2m)(4d3p/h3) ] / [ 0∫kF4d3p/h3 ]
•
= (3/5) ħ2kF2/2m
• P.E. / A = (1/2) ∫ 0∫kF { ∫ 0∫kFv00(s) (4d3p2/h3) d3s} (4d3p1/h3) d3r
/ [ ∫ 0∫kF(4d3p1/h3 )d3r]
= (1/2) ∫ 0∫kFv00(s) (4d3p2/h3) d3s
= (1/2) ∫ v00(s)d3s 0∫kF(4d3p/h3)
= (1/2) r ∫ v00(s)d3s
= (1/2) r gr) Jv00
since r = 0∫kF(4d3p/h3)
where Jv00 = ∫ t00M3Y(s,ekin)d3s
Calculations for Symmetric Nuclear Matter
e  [3ħ2k2F/10m] +C ( 1-brn ) r Jv00 /2
e/r = ħ2k2F/5mr +C [ 1- (n+1) brn] Jv00 /2
-J00C ( 1-brn )[ħ2k2F/10m]
where =0.005/MeV and J00 =-276MeV
Using: 1. e/r  0 at r=r0 2. e  e0 at r=r0
where saturation energy per nucleon = e0
and the saturation density  r0
b  r0-n[(1-p)+{q-(3q/p)}] / [(3n+1)-(n+1)p+{q-(3q/p)}]
where p = 10me0/ħ2k2F0 and q = 2e0J00/ J0v00
C  -(2ħ2k2F0/5mr0J0v00)/[1-(n+1) br0n-{qħ2k2F0(1-br0n)/10me0}]
where J0v00 = Jv00 at ekin =e0kin
Isospin Asymmetric Nuclear Matter (IANM)
The isospin asymmetry parameter X=(rn-rp)/ (rn+rp) with density r= rn+rp.
For X=0
p
n
p
n
n
p
n
p
Symmetric Nuclear Matter SNM, X=0,
contains same no. of n and p (rn=rp).
Towards asymmetry
Change X from 0
For X positive fraction
Isospin Asymmetric Nuclear Matter, X≠0,
contains different no. of n and p (rn≠rp).
Range of X: -1 ≤ X ≤ 1
p
n
n
p
n
n
p
n
Equation of State for IANM
Assuming interacting Fermi gas of neutrons and protons, the kinetic energy per
nucleon ekin turns out to be
ekin = [3ħ2k2F /10m]F(X)
with
F(X) = [(1 + X)5/3 + (1 − X)5/3]/2
The EoS for IANM:
E/A= e = [3ħ2k2F /10m]F(X) + C (1-b rn) r Jv /2
where Jv = Jv00 + X2 Jv01 = ∫∫∫ [t00M3Y + t01M3Y X2]d3s
considering energy variation of zero range potential to vary
with ekin.
E/Ae of NM with different X as functions of r/r0 for present calc.
x=0.0 (SNM)
250
x=1.0 (PNM)
200
ePNM>0 always unbound
by nuclear interaction
E/A (MeV)
150
100
50
0
emin=-15.26±0.52 MeV<0 for SNM
-50
E/A for SNM is negative up to 2r0 (Bound)
-100
0
1
2
3
4
r/r0
5
6
7
8
The pressure P of SNM as a function of r/ro is consistent with
experimental flow data for SNM
RMF NL3
3 curves of this
work for
e=-15.26 ±0.52
MeV
P (MeV fm-3)
100
Expt flow data
(Ref: P.Danielewicz et al.,
Science298, (2002) 1592
10
Akmal et al.
1
1
2
3
4
r/r0
5
6
7
The pressure P of PNM as a function of r/ro is consistent with experimental flow data
for PNM with weak (soft NM) and strong (stiff NM) r-dependence.
3 curves of this
work for
e=-15.26 ±0.52
MeV
P (MeV fm-3)
100
- - - Akmal
____ This Work
__ __ Soft NM
_____ Stiff NM
10
1
1
2
3
4
r/r0
5
6
7
Kτ versus K0 (Kinf ) for the present calculation using DDM3Y effective interaction and
comparison with the other predictions. The dotted rectangular region encompasses the
values of K0 = 250–270 MeV [1] and Kτ = −370±120 MeV [5].
 E sym 
L  3ρ 0 


ρ

 ρ ρ 0
K τ  K sym  6L  Q 0 K 0 L
  2 E sym 
K sym  9ρ 
2 
 ρ  ρ ρ
0
  3ε SNM 
Q 0  27ρ 

3
 ρ  ρ ρ0
2
0
3
0
 
Isobaric incompressibility
Model
Expt.
K0(X=0)
K 0 (X)  K 0  K τ X 2  O X 4
Esym(ρ₀)
L
Ksym
-
Kt
125
 500 100
250-270 [1]
30-34 [2]
45-75 [3]
274.7±7.4
30.71±0.26
45.11±0.02
-183.7±3.6
-408.97±3.01
NL3
271.56
37.29
118.20
100.90
-697.36
DDME1
244.04
33.10
54.51
-103.00
-559.49
DDME2
250.28
33.10
50.44
-89.05
-541.58
FSUGold
230.00
32.59
60.50
-51.30
-276.77
This work
[1]. M. M. Sharma NPA 816 (2009) 65
[2]. From different experimental studies
[3]. M. Warda et al. PRC 80 (2009) 024316
[4]. M. Centelles et al. PRL 102 (2009) 122502
[5]. Lie-Wen Chen et al. PRC 80 (2009) 014322
[4]
The nuclear symmetry energy Esym (r) represents a penalty levied on the system as
it departs from the symmetric limit of equal number of protons and neutrons and
can be defined as the energy required per nucleon to change the symmetric
nuclear matter to pure neutron matter and hence E sym ρ   ερ, X  1  ερ, X  0
From the most physical definition of the nuclear symmetry energy as defined
above, the resent calculation gives a value of Esym (r0) = 30.71± 0.26 MeV that is
consistent with the empirical value extracted by fitting the droplet model to the
measured atomic mass excesses using maximum likelihood estimator method.
If we use the alternative definition of Esym (r) = (1/2)[2e(r,X)/X2]X=0 the
value of nuclear symmetry energy remains almost same 30.03±0.26 MeV.
Elastic and inelastic scattering of protons
•
ds/dW  [ No. of events/time/solid angle ] / [ No. of incident particles/time/area ]
•
Yb = exp(ik.r) b + fb(q [exp(ikrb)/rb]
asymptotically
•
No. of events/time/solid angle = {vb | fb(q [exp(ikrb)/rb] |2 rb2 dW }/dW
•
No. of incident particles/time/area = v | exp(ik.r) | 2
• Elastic scattering :
• Inelastic scattering :
ds/dW  |f(q|2
ds/dW vb/v|fb(q|2
PWA for elastic & DWBA for inelastic scattering
Elastic : f(q obtained from partial wave
analysis solving the radial equation :
d2ul/dr2 + [2m/ħ2(E-V0) -l(l+1)/r2]ul = 0
•
V0(R)=VRfR+iWVfV+4iaSWS(d/dr)fS
+2(ħ/mpc)2VSO[1/r(d/dr)fSO]L.S+VC
For VRfR+iWVfV  Folded DDM3Y has
been used, rest phenomenological
f(q  (2ik)-1 S(2l+1)[ exp(2il)-1] Pl(cosq)
2
where k2=2mE/ ħ
l= rl ∫
[k2 -2m
2
r2]1/2
V(r)/ ħ - l(l+1) /
dr
-rl ∫ [k2 - l(l+1) / r2]1/2 dr
where rl is the classical turning point.
Inelastic :
•
•
H=h+K+V
H – Hi = V
•
Tb  -2pħ2/mb) fb
Hi = h + K
h fb  eb fb
K exp(ikb.rb) = (E – eb  exp(ikb.rb)
Y = Sg fg
Tb  < exp(ikb.rb) fb | V |  >
exact
•
Gell-mann--Goldberger transformation:
•
Tb  < b fb | V – V0 |  >
•
DWBA
•
Tb  < b fb | V – V0 |  f >
exact
Inelastic scattering and nuclear deformation
• For deformed target nucleus :
•
R = R0 [1 + bv Y 02  q, f  ]
•
V(r,R) = V0( r,R0 ) – bv R0 Y 02  q, f  dV0/dr
•
•
R is the collective co-ordinate of target nucleus whereas r is the relative co-ordinate
between proton and target.
First term is the usual spherical folded potential and the second term is responsible
for the coupling between elastic and inelastic channels.
• Tb  < b fb|V–V0| f >  < b fb| –bvR0Y 02 q, f  dV0/dr | f >
From E2 transition: br(1+0.16 br+0.20br2+. . ) = 4p[B(E2)]1/2/(3Z R20r)
•
•
•
•
V0( r,R0 ) is obtained by folding DDM3Y and hence dV0/dr as well
Deformation b is then obtained by fitting inelastic scattering data
Renormalization constant for inelastic sc.(dV0/dr part) related to bv2 R02
Potential deformation and density deformation are related : br R0r = bV R0V
Elastic & Inelastic Scattering of Protons using folded potentials of DDM3Y
whose density dependence is determined from nuclear matter calculation
Nuclear Deformation Parameters
Extracted from Inelastic Scattering:
O-18
O-20
O-22
Ne-18
b
b
b
b
0.33
0.46
0.26
0.40
Nuclear Deformation Parameters
Extracted from BE(2) values:
O-18
O-20
O-22
Ne-18
b
b
b
b
0.355 8 *
0.261 9 * / 0.50 4 **
0.208 41 *
0.694 34 *
* S. Raman et al., At. Data Nucl. Data Tables 78
(2001) 1.
* * J.K. Jewell, et al., Phys. Lett. B 454 (1999)
181.
Proton Radioactivity Half Lives using DDM3Y whose
density dependence is determined from nuclear matter calculation
Alpha Radioactivity Half Lives using DDM3Y whose
density dependence is determined from nuclear matter calculation