White dwarfs (2014)Trabajo Inglés
|Universidad||Universidad de Barcelona (UB)|
|Grado||Física - 4º curso|
|Asignatura||Mecànica quàntica de N-cossos i sistemes ultrafreds|
|Año del apunte||2014|
|Fecha de subida||27/07/2014|
Quantum mechanical model for the structure of white dwarfs.
Vista previa del texto
Many-body Quantum Mechanics
University of Barcelona, Spring 2014
A simple model for the mechanical structure of a white dwarf is proposed, considering
classical equilibrium dynamics and non-interacting constituents. The equations are
manipulated to facilitate their computational resolution. Results are discussed and compared
to observational data.
1 Model 1.1 Mechanical structure The escape velocity for white dwarfs (WDs) is of the order of 0.02c  thus we can treat their dynamics classically. Furthermore we assume that they are cold, spherically symmetric, non-rotating objects in hydrostatic equilibrium.
Consider a small mass element ∆m = ρ∆V located at a distance r from the center of the star. As shown in figure 1, three forces act on it: a gravitational force, an inner pressure and an outer pressure. From Newton’s second law of motion P (r)∆S − F (r) − P (r + ∆r)∆S = ∆m d2 r .
dt2 (1) Using P (r + ∆r) = P (r) + dP ∆r dr (2) ∆V = ∆r∆S (3) and Newton’s law of gravitation F (r) = ∗ Gm(r)∆m , r2 (4) email@example.com 1 P (r + ∆r) F (r) dr P (r) r Figure 1: Sketch of the forces acting on a small mass element located at distance r.
equation (1) becomes − Gm(r)ρ(r) d2 r dP (r) ∆V − ∆V = ρ(r)∆V .
dr r2 dt2 Finally, in hydrostatic equilibrium (5) d2 r = 0, so dt2 dP (r) Gm(r)ρ(r) =− dr r2 (6) with P (r) and ρ(r) the pressure and the density at distance r and m(r) the mass enclosed in the sphere of radius r.
Equation (6) has introduced three radius-dependent magnitudes; we need two additional equations. One is obtained simply from the definition of density dm dm(r) dm = → = 4πr2 ρ(r) .
dV 4πr2 dr dr The third and last equation will be the equation of state, derived in the next section.
ρ= 1.2 (7) Material constituents We assume neutral matter, formed by atoms with mass number A (Z electrons, Z protons, A−Z neutrons). Furthermore the atoms are completely ionized, forming a homogenous plasma without any type of interaction. The nucleons provide the totality of the mass but don’t contribute to the energy. On the other hand, the electrons don’t contribute to the mass and provide all the energy and, consequently, pressure.
The electron gas can be modeled as a free non-interacting Fermi sea at T = 0, with singleparticle energy ε(k) = m2e c4 + 2 k 2 c2 (8) obeying the fundamental relation1 ρe = 1 Ne νk 3 = F2 Ω 6π (9) for an unpolarized system 2 with ρe the electron density, Ne the number of electrons, Ω the volume, ν the electron degeneracy and kF the Fermi momentum.
The interior temperature of a WD is Tint ∼ 107 K ; it’s rather counterintuitive to model the electron gas at T = 0. Nonetheless, the mean density is ρmean ∼ 108 kg m−3 ; taking into account that ρ = me ρe and using (8) and (9), this corresponds to a Fermi energy εF ∼ 10−13 J and Fermi temperature2 TF ∼ 1010 K. Thus only a tiny fraction (T /TF ∼ 10−3 ) of the electrons located near the Fermi surface can experiment thermal excitations, and can be disregarded when computing the pressure. However, the density is sufficiently high for the electron rest mass and kinetic energy to be comparable3 : no non-relativistic or ultra-relativistic limits may be considered.
1.3 Degeneracy pressure To calculate the electron gas’ pressure, we need its energy kF Ne E= ε(k) .
εi = ν i=1 (10) k Applying the continuous limit4 ∞ Ω 8π 3 Ω =ν 2 2π dk k 2 ε(k)Θ(kF − k) dω E=ν 0 (11) kF dk k 2 m2e c4 + 2 k 2 c2 0 with ω the solid angle. Introducing the dimensionless variable x = k/me c we compute the bulk energy E ν m4e c5 e= = 2 3 Ω 2π = a¯ e(xF ) xF √ dx x2 1 + x2 with xF = kF /me c, a = νm4e c5 /2π 2 xF e¯(xF ) = (12) 0 3 and √ dx x2 1 + x2 0 1 = (2x3F + xF ) 1 + x2F − sinh−1 (xF ) 8 2 3 εF = kB TF me c2 ∼ kF c ∼ 10−13 J kF 4 → k Ω (2π)3 d3 k Θ(kF − |k|) 3 (13) .
Also, equation (9) becomes ρe = νm3e c3 x3F .
6π 2 3 (14) To compute the pressure we use the relation ∂(eΩ) ∂Ω Ne ∂e ∂e ∂ρe = −e − Ω = −e − Ω ∂Ω ∂ρe ∂Ω Ne ∂e ∂e − 2 = −e + ρe .
= −e − Ω ∂ρe Ω ∂ρe P =− ∂E ∂Ω =− (15) We’ve effectively found the equation of state: e can be expressed in terms of ρe using (14) and ρe is related to ρ. However, it’s more convenient to re-write equations (6-7) and (15) in terms of xF , which is done in the next section.
2 2.1 Computation Dimensionless equations The density is locally dependent on the radius, ρ = ρ(r), thus also xF = xF (r). For simplicity, hereon forward we drop the subindex F : xF → x.
For starters we express (15) in terms of x P (x) = −a¯ e + ρe ∂(a¯ e) ∂¯ e ∂x = a ρe − e¯ ∂ρe ∂x ∂ρe (16) = aP¯ (x) and after some algebra √ 1 P¯ (x) = (2x3 − 3x) 1 + x2 + 3 sinh−1 (x) .
24 (17) Now we transform the differential equation for P (r) into a differential equation for x(r) dP dP dx = dr dx dr (18) using the equation of state (16-17) dP P¯ (x) a x4 =a = √ .
dx dx 3 1 + x2 (19) 4 Substituting in (6) yields dx Gmρ 3G mρ =− 2 =− dr r (dP/dx) a r2 √ 1 + x2 x4 (20) with m = m(r) and ρ = ρ(r).
Next we express ρ in terms of x, starting off with the electron density ρe = Ne /Ω = ZNA /Ω, where NA is the number of atoms forming the star. The total mass of the star is M = Nn mn + Np mp = NA (A − Z)mn + NA Zmp = NA Z((A/Z − 1)mn + mp ) = NA Z m ˜N (21) where we’ve defined an effective nucleon mass m ˜ N = (y −1 − 1)mn + mp , with y = Z/A the usual symmetry factor. Since ρ = M/Ω → ρe = ρ/m ˜ N ; the local density will have the same form.
3 Equating to (14) yields ρ(r) = bx with b = ν m ˜ N m3e c3 /6π 2 3 . Thus, the differential equations to be solved are dm = 4bπr2 x3 (22) dr √ 3Gb m 1 + x2 dx =− (23) dr a r2 x Finally we introduce two dimensionless variables: m ¯ = m/m0 and r¯ = r/r0 . Setting m0 = m ˜N 3 π 3 2ν αG r0 = me c with αG = Gm ˜ 2N / c, we obtain two very easy equations dm ¯ = r¯2 x3 d¯ r 2.2 m ¯ dx =− d¯ r r¯ √ 3 π 2ν αG 1 + x3 .
x (24) (25) Numerical resolution To solve the two differential equations given in (25) we apply the fourth-order Runge-Kutta method. The appropriate boundary conditions are m(¯ ¯ r = 0) = 0 and x(¯ r = 0) = xC ; different values of xC will give rise to stars with different (total) mass and radius.
The radius of the star R is defined as the distance where the pressure vanishes: P (r = R) = 0; this value also defines the star’s mass: M = m(r = R).
We’re not computing the pressure, thus we must find a condition for x. Since x is related to the density5 , which is defined positive, we have the condition x > 0. Furthermore, x decreases monotonically with r¯; implying that the algorithm must stop when x ≤ 06 .
Care must be taken at the first iteration as dx/d¯ r → ∞ for r¯ → 0. However, if we consider 3 m(¯ ¯ r → 0) ∼ r¯ , the divergence disappears and dx/d¯ r = 0 at r¯ = 0. This approximation is supported by the fact that the pressure and the density (and consequently x) present a maximum at the center of the star.
5 6 √ x∼ 3ρ This is backed by the fact that (17) has only one root (x = 0) and increases monotonically with x .
5 3 Results The values of the various physical constants and parameters7 involved in our model have been retrieved from  and .
3.1 Plots and discussion Figure 2 shows the computed results of our model, for different values of the symmetry factor.
We’ve computed masses as low as 0.08 M , which is the minimum mass for a protostar to enter the main sequence  and, thus, the theoretical lower bound for a WD’s mass.
Most WDs’ composition is mono-elemental, commonly carbon or oxygen, although some helium WDs have been discovered . The different plots would correspond, respectively8 , to 4 He/12 C/16 O, 3 He, 13 C, 17 O and 18 O. Clearly symmetric matter (y = 1/2) adjusts best to the Chandrasekhar limit mass (MCL = 1.44 M ). Odd A values can be discarded as additional nuclear isospin degeneracy should be considered. This is probably also the case for 18 O, which, on the other hand, has a very low natural abundance (∼ 0.2% ).
In all cases, however, the total radius decreases with the total mass. This is explained by the equation of state being “soft”, a direct consequence of the Pauli exclusion principle. Compressing a WD would increase the electron density and consequently the electrons’ energy. This would augment the pressure and, in hydrostatic equilibrium, also the star’s mass. Nonetheless, since electrons are massive particles, their energy is upper-bound by special relativity. There comes a moment, thus, when the degeneracy pressure is unable to balance the gravitational pull, explaining the existence of the Chandrasekhar limit mass. An additional interpretation is that at very high densities, electrons and nuclei are forced very closely together and reverse beta decay sets in, causing the star to be unstable.
Figure 3 shows the interior mass profile of three symmetric WDs and figure 4 the interior density profile for the same three stars.
The mass increases very quickly in massive stars and quite slowly in lighter stars. However, all three cases present a low exponential initial trend (roughly up to 0.2R) followed by a linear central increase and a slow final convergence (slower for lighter stars).
The density drops very quickly in the outer regions of all three stars, but the central decrease has a higher slope for more massive stars.
3.2 Comparison to observations For completeness, in figure 5 we compare our results to some observational data, taken from two different sources: HOB  and PSHT . This comparison is not intended, by any means, to be exhaustive or statistically relevant, but serves to illustrate a rough agreement between our model and observations, at least in the range 0.45 − 1.00 M .
7 8 ν=2 in order of appearance in the figure’s key 6 Mass profiles 1/2 2/3 6/13 8/17 8/18 1.0 1.5 2.0 M (M ) m(r) (M ) 30 25 20 15 10 5 0 0.08 0.5 MCL R (×103 km) Mass-radius relation 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 2.5 0 1 2 3 4 5 6 7 8 9 10 r (×103 km) Figure 2: Total radius vs. total mass for WDs, for various symmetry factors.
Figure 3: Mass profile m(r) for three symmetric WDs (y = 1/2).
Density profiles Mass-radius relation 25 0.5 M 1.0 M 1.4 M R (×103 km) ρ(r) (kg m−3 ) 14 10 1012 1010 108 106 104 102 100 0.5 M 1M 1.4 M HOB PSHT model 20 15 10 5 0 0 1 2 3 4 5 6 7 8 9 10 r (×103 km) 0.25 0.5 Figure 4: Density profile ρ(r) for three symmetric WDs (y = 1/2).
0.75 1.0 1.25 M (M ) 1.5 Figure 5: Comparison between our model and observational data.
References  http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit3/extreme.html, retrieved 2014-04-29.
 Kutner, M. L. (2003). Astronomy: A physical perspective. Cambridge University Press.
 http://www.wolframalpha.com/input/?i=%282x%5E3-3x%29*sqrt%281%2Bx%5E2%29% 2B3*asinh%28x%29, retrieved 2014-05-01.
 http://physics.nist.gov/constants, retrieved 2014-04-30.
 http://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html, retrieved 2014-05-02.
7  http://www.thphys.may.ie/staff/mshadmehri/AstroNotes/MINIMUM.pdf, retrieved 2014-05-02.
 http://research.amnh.org/~bro/WD/faq.html, retrieved 2014-05-02.
 http://hyperphysics.phy-astr.gsu.edu/hbase/astro/whdwar.html#c5, retrieved 2014-05-02.
 http://environmentalchemistry.com/yogi/periodic/O-pg2.html, retrieved 2014-05-02.
 Holberg, J. B.; Oswalt, T. D.; Barstow, M. A. (2012). “Observational Constraints on the Degenerate Mass-Radius Relation”. The Astronomical Journal 143:68. DOI: 10.1088/00046256/143/3/68.
 Provencal, J. L.; Shipman, H. L.; Høg, E.; Thejll, P. (1998). “Testing the White Dwarf Mass-Radius Relation with Hipparcos”. The Astrophysical Journal 494:759. DOI: 10.1086/305238.