Chapter 16 White Dwarfs

Chapter 16 White Dwarfs The end product of stellar evolution depends on the mass of the initial configuration. Observational data and theoretical calculations indicate that stars with mass M < ∼ 4M! after ejecting part of their mass in the form of a planetary nebula give birth to a white dwarf, with typical mass, radius and density M ∼ 1M! , R ∼ 5000 km, and ρ ∼ 106 gr/cm3 . White dwarfs are composed largely of helium, carbon and oxygen, because the progenitors masses are such that the temperature never becomes high enough to burn much beyond carbon, and even if burning may, in principle, proceed all the way to iron the time needed to complete the process would be longer than the Universe age. As we shall later show, white dwarfs of mass exceeding the critical value MCH ∼ 1.4M! cannot exist. Neutron stars or black holes are thought to be the leftover of the gravitational collapse, following a supernova explosion, of stars whose mass is greater than 4M! , but the mechanism that may produce one or the other is still unclear. (For a review, see for instance A. Heger, C. L. Fryer, S. E. Woosley, N. Langer, D.H. Hartmann “How massive single stars end their life”, The astrophysical Journal 59, 288-300, 2003. Also available at http://www.journals.uchicago.edu/ApJ/journal/issues/ApJ/v591n1/57419/57419.html). Numerical simulations indicate that if the mass of the progenitor star is smaller than ∼ [20, 30]M! , a neutron star should form, whereas bigger masses would produce a black hole. As for white dwarfs, a critical mass exists also for neutron stars. The absolute upper limit is in the range ∼ 2 − 3M! ; the value of the critical mass depends on the equation of state which is chosen to describe matter at supranuclear densities, as those prevailing in the core of a neutron stars. Neutron stars have been observed in binary systems or as isolated objects. Typical parameters are M ∼ 1 − 3 M! , R ∼ 10 km, and ρ ∼ 1012 gr/cm3. Black holes of astrophysical origin can have very different masses, ranging from a few solar masses of the “stellar black holes”, born in the gravitational collapse of big stars or in the coalescence or accretion driven processes in binary systems, to supermassive black holes, with masses M ∼ 106 − 108 M! , which sit at the center of several galaxies. In this chapter we shall focus on the study of white dwarfs, whose structure can be described using the equations of newtonian gravity; in the next chapter we shall derive the equations of stellar structure in general relativity, needed to describe neutron stars.

216

CHAPTER 16. WHITE DWARFS

16.1

217

The discovery of white dwarfs

The first white dwarf, Sirius B, was observed in 1915 by Adams. He found that the spectrum of the stellar object orbiting around Sirius, named Sirius B was that of a white star, not very different from the spectrum of Sirius. The mass of the newly discovered star was found by applying third Kepler’s law GMSB ω 2r = , r2 and it was estimated to be in the range 0.75 − 0.95M! . Knowing the distance of the system from Earth, from the observed flux of radiation it was possible to estimate the effective 4 temperature, that in this case was ∼ 8000 K. Since for a black-body emission L ∼ R2 Tef f, from spectral measurements it was then possible to estimate the radius of the star, which was, surprisingly, RSB = 18.800 km, much smaller than that of the Sun! The actual values of the mass and radius are MSB = 1.034 ± 0.026 M! and RSB = 0.084 ± 0.00025 R! (i.e. RSB ∼ 5850 km). At that time this result was really a surprise because a star having a mass comparable to that of the Sun but a radius nearly forthy times smaller had never been observed. In addition, although the gravitational redshift predicted by Einstein’s theory of Relativity had already been measured in the famous Eddington expedition in 1919, the redshift of spectral lines of Sirius B measured by Adams in 1925 provided a much better verification of the theory, and in fact in his book The internal constitution of stars Sir Arthur Eddington wrote “Professor Adams has killed two birds with one stone: he has carried out a new test of Einstein’s general theory of relativity, and he has confirmed our suspicion that matter 2000 times denser than platinum is not only possible, but it is actually present in our universe”. The discovery of such an extremely dense star raised a main question: how can this “white dwarf”, as it was named, support its matter against collapse? Indeed, if the matter composing the star were a perfect gas its temperature would be too low to prevent the collapse, i.e. the corresponding pressure gradient would not be sufficient to balance the gravitational attraction. About this problem Eddington wrote “It seems likely that the ordinary failure of the gas laws due to finite sizes of molecules will occur at these high densities, and I do not suppose that the white dwarfs behave like perfect gas”. What is then that keeps white dwarfs in equilibrium? The answer to this question came a few years later, when Dirac formulated the Fermi-Dirac statistics (August 1926), R.H. Fowler identified the pressure holding up a white dwarf from collapsing with the electron degeneracy pressure (December 1926). This was the crucial step toward the formulation of a consistent theory of these stars that led S. Chandrasekhar to predict the existence of a critical mass above which no stable white dwarf could exist. In order to formulate the theory, let us briefly recall some basic equations of degenerete gases.


16.1.1

218

Degenerate gas in quantum mechanics

A perfect gas is said to be ‘degenerate’ if its behaviour differs from the classical behaviour due to the quantum properties of the system of particles. Since degenerate gases are important in the study of the internal structure of compact stars, we shall outline some basic elements of the theory. Consider a gas composed by particles all belonging to the same species. In general, the system will be completely described if we assign the number of particles per unit phase-space volume, i.e. the number density in the phase space dN d3 xd3 p

=

g f (x, p), h3

(16.1)

where h3 is the volume of a cell in the phase-space, g = 2s + 1 is the number of states of a particle with a given value of the 3-momentum p, s is the spin, and f (x, p) is the probability density function, i.e. the probability of finding a particle at a position between x and x + dx and with a 3-momentum between p and p + dp. 1 If the rest mass of a 1 particle is m, its total energy is E = [p2 c2 + m2 c4 ] 2 and the total energy density of the gas is ! ! dN g E= E 3 3 d3 p = 3 E f (x, p) d3 p . (16.4) d xd p h The distribution function for an ideal gas of fermions or boson in equilibrium is f=

1 e

Ec −µ kT

±1

,

(16.5)

where the + sign holds for fermions (Fermi-Dirac statistics) and the - for bosons (BoseEinstein statistics). Some useful relations: The 4-momentum of a relativistic particle is 1

pα = (mcγ, p), where p = mγv is the 3-momentum. Moreover, remember that the total energy of the particle is E = p0 c. Since pα pα = −m2 c2 , it follows that E2 − 2 + p2 = −m2 c2 , c 2 where p is the norm of the 3-momentum, and consequently the total energy of the particle can be written as #1/2 " . (16.2) E = p 2 c2 + m 2 c4 From this equation it follows that, since E = mc2 γ "

p 2 c2 + m 2 c4 γ= mc2

#1/2

and since the norm of the particle velocity is v = p/(mγ), v=

pc2 1/2

[p2 c2 + m2 c4 ]

.

(16.3)


219 1

In eq. (16.5) Ec is the particle kinetic energy Ec = [p2 c2 + m2 c4 ] 2 − mc2 and µ is the chemical potential, which is the partial derivative of any thermodynamical potential of the system (the enthalpy, the internal energy, etc.) with respect to the number of moles, keeping fixed the number of moles of the other species of particles if present, and the state parameters in terms of which the potential is expressed. For example µi =

$

∂H ∂ni

%

= S, P, nk =const

$

∂U ∂ni

%

,

(16.6)

S, V, nk =const

where H is the enthalpy and U the internal energy. From eq. (16.5) we see that, since f must be positive, the chemical potential of fermions can take any real value, either positive or negative, whereas that of bosons is bounded to be µ < Ec . If the temperature is high, or the energy is low (E 65 , Θ(ξ) vanishes for some ξ = ξ1 . When Θ = 0 both the density and the pressure vanish, therefore ξ1 is the boundary of the star, which can be determined numerically. The procedure to find the stellar structure can be summarized as follows. • Choose a value of γ (for instance γ = 53 or γ = 43 ), find the corresponding polytropic 1 index n = γ−1 , and integrate numerically eq. (16.43) with the initial conditions (16.45) up to the value ξ = ξ1 where Θ = 0. For instance, for γ = 53 and γ = 43 we would find γ=

5 3

n=

ξ1 = 3.65375

ξ12 Θ$ (ξ1 ) = −2.71406

(16.46)

γ=

4 3

n = 3 ξ1 = 6.89685

ξ12 Θ$ (ξ1 ) = −2.01824

(16.47)

3 2

It should be noted that Θ is a monotonically decreasing function of ξ, that is why its first derivative at the boundary is negative.


227

• Assign a value to κ, i.e. the number of nucleons per free electrons, then find K from eq. (16.36) or (16.37). Choose a central density ρ0 . Knowing K and ρ0 the radius of the star can be found using the definition of ξ given in eqs. (16.42) R = αξ1

→

R = ξ1

4

(n + 1)K 4πG

51 2

1−n

· ρ02n .

(16.48)

• The mass of the star can now be determined as follows M =

! R 0

4πr 2 ρ(r)dr = 4πα3 ρ0 '

! ξ1

! ξ1

(

0

ξ 2 Θn dξ

d dΘ = −4πα ρ0 ξ2 dξ dξ dξ 0 = −4πα3 ρ0 ξ12 Θ$ (ξ1 ) 3

where use has been made of eq. (16.43). Finally, the value of M as function of K and ρ0 can be found by using the expression of α given in (16.42) M = 4π ξ12 |Θ$ (ξ1 )| Let us define

4

5

(n + 1)K , A= 4πG

4

(n + 1)K 4πG

53 2

3−n

· ρ02n .

B = 4π ξ12 |Θ$ (ξ1 )|,

so that

(16.49)

(16.50)

1−n

R = ξ1 · A1/2 · ρ02n , and

(16.51)

3−n

M = B · A3/2 · ρ02n . Combinig eqs. (16.51) and (16.52), a relation between M and derived ) * n−3 3−n n M = B · A n−1 · ξ11−n · R 1−n .

(16.52) R can easily be (16.53)

From the procedure outlined above we understand that, having fixed the number of nucleons per free electrons, κ, and the polytropic index n, once we have found ξ1 and Θ$ (ξ1 ) by numerical integration of the Lane-Emden equation we obtain a family of solutions parametrized with different values of the central density ρ0 , the radii and masses of which are given by (16.48) and (16.49). Conversely, if we change the number of nucleons per free electrons, the new configuration can easily be obtained by rescaling the various quantities in the following way κ" ρ, κ 6 72 κ M(r), κ"

ρ$ = M(r)$ =

P $ = P, κ r $ = $ r. κ

(16.54)


16.1.5

228

A note on the numerical integration of eq. (16.43)

Although the initial conditions (16.45) are correct, it would be impossible to integrate eq. (16.43) numerically starting from ξ = 0 with these conditions. Indeed, since ξ = 0 is a singular point, running the code we would get immediately an overflow. However this problem can be overcome if we start the numerical integration at some small, but finite, value of ξ = ξstart and use as initial values for the function Θ(ξ) a suitable Taylor expansion. Let us do it step by step. Since we know from (16.45) that Θ(0) = 1 and Θ$ (0) = 0, we can write the approximate solution near ξ = 0 as a power series Θ(ξ) ∼ 1 + Θ2 ξ 2 + Θ3 ξ 3 + Θ4 ξ 4 + O(ξ 5),

(16.55)

(we can keep as many terms we want, but let us stop here). Θ1 , Θ2 , Θ3 .. are the constants we need to find using eq. (16.43), therefore we also need to Taylor-expand the function Θn on the right hand side, i.e. Θn ∼ 1 + nΘ2 ξ 2 + O(ξ 3); (16.56) by substituting in eq. (16.43) the expansions (16.55) and (16.56) we find 6Θ2 + 12Θ3ξ + 20Θ4 ξ 2 + ... = −[1 + nΘ2 ξ 2 ] + ...

(16.57)

and this equation is satisfied only if the coefficients of the same power of ξ vanish, i.e. 1 = −6Θ2

Θ3 = 0

20 Θ4 = −nΘ2

→

Θ2 = − →

Θ4 =

1 6

n ; 120

the expansion has only even powers of ξ (this is true also at higher order). Thus the approximate solution for Θ and Θ$ near the origin is 1 n 4 ξ + O(ξ 6) Θ(ξ) ∼ 1 − ξ 2 + 6 120 n 1 Θ$ (ξ) ∼ − ξ + ξ 3 + O(ξ 5). 3 30

(16.58)

We now have all we need to numerically integrate the Lane-Emden equation, because we can start at, say, ξstart = 10−4 using as initial values the functions (16.59) computed at ξstart .

16.2

The Chandrasekhar limit

In section 16.1.3 we have shown that if the density is much smaller than the critical density, electrons behave as a polytropic gas with γ = 53 . In this regime eqs. (16.50) give A = 2.9562 · 10−19 κ−5/3 ,

B = 34.1059


229

and using eq.(16.25) and (16.52) we can write the mass of the star in this form −5/2

M = 2.73 κ

'

ρ0 ρc

(1/2

M! ,

(16.59)

where M! = 1.989 · 1033 g is the mass of the Sun. This equation shows that the mass of the star increases with the central density. As the central density increases above the critical density, the electrons start to behave as a relativistic gas with a polytropic equation of state with γ = 43 . Equation (16.49) shows that in this limit the mass becomes independent of the central density ρ0 and takes the value M = MCH = 5.74 κ−2 M! .

(16.60)

This is a critical mass above which no stable configuration for a white dwarf can exist, and it is called the Chandrasekhar limit, as it was derived by Subrahmanyan Chandrasekhar in 1931 3 . It should be noted that the information on the internal composition is contained entirely in the parameter κ. For instance, if we set κ = 2 we find MCH = 1.435 M! .

(16.61)

The fact that a critical mass should exist can also be understood from the following qualitative considerations. A given configuration of matter will be in equilibrium if the gradient of pressure is balanced by the gravitational attraction. In the non relativistic case 5

a)

P ∼ρ

5 3

→

M3 P ∼ 5 R

→

M3 P ∼ 4 R

5

→

dP M3 ∼ 6. dr R

→

M3 dP ∼ 5. dr R

(16.62)

In the ultra-relativistic case 4

b)

P ∼ρ

4 3

4

(16.63)

The gravitational force per unit volume behaves like Gm(r)ρ M2 ∼ . r2 R5

(16.64)

If the star is in equilibrium dP Gm(r)ρ =− ; dr r2 in the non-relativistic case the gradient of pressure (16.62) and the gravitational force (16.64) depend on the radius to a different power thus, for a given value of the mass, the star can The concept of a limiting mass for white dwarfs was first introduced by Chandrasekhar in a paper published in 1931: “The Maximum Mass of Ideal White Dwarfs”, in The Astrophysical Journal, 74 n.1, 81. The problem was subsequently investigated in a series of papers, and a complete account can be found in the book Chandrasekhar wrote on the subject in 1939: “An introduction to the study of stellar structure”, University of Chicago Press, Chicago Illinois 3


230

‘adjust’ the radius until the two forces are equal. Conversely, in the ultra-relativistic case the gradient of pressure (16.63) and the gravitational force (16.64) have the same dependence on the radius, and therefore the equilibrium is possible only for one value of the mass, i.e. for the critical mass. If M > MCH the gravitational attraction exceeds the gradient of pressure and stable configurations are no longer possible. Although the existence of a critical mass for white dwarfs seems an obvious consequence of the theory today, it was not accepted when Chandrasekhar found it. The prejudice at that time was that white dwarfs do represent the final state of a star, and that they could have any mass (neutron stars were discovered much later in 1965). The famous astronomer Sir Arthur Eddington was the strongest opponent to the new theory, and called it “a stellar buffonery”. Nobody at that time gave to Chandrasekhar any public support, although a few, as for example Rosenfeld, told him in private that they thought his result was correct 4 . It should be stressed that this limit is a static limit, i.e. it refers only to the equilibrium configuration. It says that stars with a mass exceeding the critical mass cannot exist. However, even if a star is in equilibrium it may become unstable against small perturbations. In this case we would call it a dynamical instability. A second point which should be noted is that in the derivation of the critical mass general relativity plays no role. The basic ingredients are special relativity and the Fermi-Dirac statistics.

An interesting account of the controversy between Eddington and Chandrasekhar on white dwarfs maximum mass can be found in the book ”Chandra: a biography of S. Chandrasekhar”, University of Chicago Press 1991 4