Andrew Cumming (McGill University). Zach Medin (McGill). Sanjay Reddy and
Shannon Holloway (LANL). Plumbing the depths of the neutron star ocean and ...
Plumbing the depths of the neutron star ocean and crust Andrew Cumming (McGill University) Zach Medin (McGill) Sanjay Reddy and Shannon Holloway (LANL)
rp process ash
ocean Schatz et al. 1999
carbon ignition chemical separation on freezing electron captures, pynonuclear reactions + ??
crust
Crust impurity parameter from observations of cooling transients
Brown & Cumming (2009)
for two sources, Qimp ~ few --- what sets this? Schatz et al. found rp-process ashes have Qimp~100
Problems understanding the properties of superbursts and other long duration thermonuclear flashes
stable burning
unstable burning
Cumming et al. (2006)
need extra heating
Schatz et al. (2003)
difficult to make enough carbon
Chemical separation on freezing
Horowitz, Berry and Brown (2007)
light elements get left behind in the liquid: what is the steady state? does it lead to a lower Qimp?
4
culation we set ∆fl = 0 and use the linear mixing mation: xi
i=1
"
flOCP (Γi )
$% # Zi . (11) + ln xi "Z#
or introduced by neglecting the ∆fl term in the ion for flTCP is discussed in Appendix B. ree energy of the solid phase of a two-component is given by
Free energy
Γ1 , x1 ) !
2 !
0.5
s
i=1
i
i
-0.5
-1.5 0
"Z#
+ ∆fs (Γ1 , x1 ) ,
0.4
Spindle type phase diagram
(13)
RZ ) =
C(RZ ) , √ √ √ √ 27(RZ −1) + 1+0.1(RZ −1) x( x − 0.3)( x − 0.7)( x − 1) (14)
0.6
0.8
1
x2
(12)
∆fs is the deviation from linear mixing in the solid. ∆fl , which is generally small even at large Γ1 dix B), ∆fs is comparable to the other terms in grows linearly with Γ1 ; we therefore expect ∆fs an important role in setting the phase transition ies. For charge ratios RZ = Z2 /Z1 in the range 1 : 5] the deviation is well fit by ∆fs (Γ1 , x1 ) ! Γ1 x1 x2 ∆g(x2 , Z2 /Z1 ) ,
0.2
1.6
FIG. 1: (Color online) An double-tangent conRZexample =34/26,of"fthe l=0 struction, for RZ = 34/8 and Γ1 = Γcrit /6 (cf. Figs. 4 and PCCDR 5). The stable compositions a2 and b2 (i.e., 1 − a1 and 1 − b1 ) are marked 1.4 by filled circles; here, one of the mixtures is stable in the liquid state and one is stable in the solid state. Note that the curves fl and fs plotted in this figure are given not by Eq. (11) and Eq. but by these equations !2 (12), respectively, OCP 1.2 minus the term (Γi ). The values of a2 and b2 i=1 xi fl obtained are the same whether flTCP and fsTCP or these modified expressions are used: adding terms constant or linear in the xi ’s to 1 both free energy curves has no effect on the results of the double-tangent construction.
!crit/!1
i
1
0
-1
double tangent # " $% 2 ! construction Z i CP x f OCP (Γ ) + ln x (Γ , x ) = 1
fl fs fmin
0
0.2
0.4
0.6 x2
0.8
agrams agree agreement is half of each d our group an main differen regions at hig ing RZ ) trans in the diagram spindle-type t RZ ! 1.2 ! 2 value of RZ t or DeWitt et
1
Appendix C
m i=1 f " Z i the MCP is The free"energy of the solid phase of OCP (Γi ) + ln xi xi fl (23) #Z$ and i=1 MCP fs (Γ1 , x1 , . . . , xm−1 ) or ! m to the ' %& plasmas and # of one and two$component whereIdea: #Z$use = fitsi=1 xi Zfree-energies .m i" Zi extend to higher dimensions fl (!a) + OCP The free energy of " the solid of f (Γthe ) +MCP ln xis x phase
i
i
s
i
i=1
#Z$
fsMCP (Γ1 , x1 , . . . , xm−1 ) +#∆fs (Γ1 , x1 , . . $ . , xm−1%& ). m " Zi OCP f (Γ ) + ln x " x i i According to Ogatai ets al. [15], the deviation #Z$ i=1
or (24)
dfl ( fl (!a) + dxi
dfl of the solid (!a) − !a · ∇ f (!a) + from linear mixing ∆fs for a three-component plasma lis dxi and ∆fs (Γ1 ,by x1 , . . . , xm−1 ) . (24) given to good + accuracy According to Ogata et al. [15], the deviation of the solid ∆fs (Γ1 , x1 , . . . , xm−1 ) from linear mixing ∆fs for a three-component plasma is $ % m−1 m " " by given to good accuracy xj Zj
Γi xi xj ∆g
"
∆fs (Γ1 , x1 , . . . , xi=1 ) m−1j=i+1 m−1 "
,
xi + xj Zi
and
, (25)
fl (
Using Eqs solve becom
fl (!a) − !a · ∇
$ % m " xjdeviation Zj from linear mixing, use Ogata et al.’s fitting formula for the Γi xi xj ∆g " , , (25) Using Eqs. (23) an x + x Z applied between all pairs of nuclei in the mixture i j i i=1 j=i+1 solve becomes
δf
OCP
$
Zi (Γi ) + ln ai #Z$ a
%
Medin & Cumming (2010) $ % Zi Zi Zi − = ln bi − − ∆fs (Γ1 , x #Z$ a #Z$ b #Z$ b
olid-to-liquid ratio is greater than unity". 1.6
Current work HBB TCP approx.
xs/xl
1.2
0.8
0.4
0
0
10
20
30
40
50
Z
FIG. 3. !Color online" Ratio of the solid abundance to the liquid
f chemical hich accre, and conositionallyructure of dels: first, osed of Se a mixture se heavierect of the the heat-
1.6
1.4
!crit/!Fe
(2009) inrust of the 29 and KS quiescence. al heating In this pahat extent arbon and that could e observa-
What is the steady-state for the ocean?
L
1.2
S
1 0
0.2
0.4
0.6 xSe
11 9
L
0.8
1
Mixing length theory for convection 2 vconv ≈ gl(∇ − ∇ad )
F ≈ ρvcP T (∇ − ∇ad ) ≈ ρv 3 Often the convection is efficient
v � cs
∇ ≈ ∇ad
Similar idea here, except now we have a composition flux rather than heat flux - the result is
∇ad χT ∇µ ≈ χµ
For conditions in the ocean
kB T χT ∼ EF
1 0.3
XFe
0.25
Fe/Se
0.2
0.15 Initial liquid Final liquid
0.1
6
8
10
10
10
12
10
10
2
y (g/cm ) 0.35 0.3 0.25 0.2 XO
O/Se
0.15 0.1 0.05 0
Initial liquid Final liquid 4
10
6
10
8
10
10
10
12
10
14
10
2
y (g/cm )
FIG. 1: Mass fraction of the light element X1 for A1 , A2 = 56, 79 [i.e., Z1 , Z2 = 26, 34, X1 = XFe ] (top panel) and A1 , A2 = 16, 79
duct the latent heat away is ∇ > ∇ad , so that the latent heat release drives thermal convection (Stevenson 1981). When chemical separation drives convection in the ocean, we must also consider the convective heat flux in addition the to theconvection conductive heattransports flux. In mixing heat length theory the heat flux is
inwards
ξ (41) Fconv = ρvconv cP T (∇ − ∇ad ) , 2 where cP is thein heat capacity. Using Eq. (23) forby theoutwards => steady state balanced convective velocity together with the fact that ρvaccr = m, ˙ we find conduction Fconv = −Fconv rˆ (42)
hows that the inwards convective flux in-
T8
t by the ions; using the internal energy exwith& Slattery (2003) we find cP ∼ (1– DeWitt n−1 ] giving cP T /mp ∼ (0.1–0.6)T cP8T[60/"A#] m ˙ %keV Xi − Xi,06 F = Fcrust + Fconv where the range of values the Fconv = is across χ . (43) Xi F = Fcrust X ocean. Although cP T /mp is χmuch smaller T i i=1 convective flux has an additional factor of /(kHere extraunit factorvector. of 10– Note that 4 Fconv > 0 B T ). This rˆ is gives the an radial −2 −1 he final convective flux is ∼ 10 transports –10 MeV heat inwards. This is so that the convection which can be comparable to Qb . For the O-convection, ∇ < ∇ so because, unlike thermally-driven ad Fig. 5, the convective flux is ≈ 0.2 MeV per that a fluid element adiabatically2 outwards is e base of the convection zone. displaced For Fe-Se, the its surroundings at its new location. ast cooler betweenthan the heavy and light elements smaller Forthe twoconvective species, the flux compoWeflux. show as Fconv mp /m ˙ with6 m ˙ = 7 8 9 10 10 10 10 FX3=×m(X ˙104 − so sthat −1 we can write 0 ),−2 gX cm in Fig. 5. In the ocean, the heat 3 ! (g/cm ) P T χX /(XχT ) = FX cP (∂ ln T /∂ ln X|P,ρ ). ve heat flux corresponds to the rate of ernal energy given the flux of composition h [compare equation (2) of Montgomery et Fig. 6.— (Color online) The thermal profile
9
104 2×
g cm−2 s−1 108 K at ρ =
10
10
for a m ˙ = 3× ocean composed of and with T = 8 × 105 g/cm3 , both when the convective flux is 16 O, 79 Se
Crust reactions for a single species Haensel & Zdunik
Brown & Cumming (2009)
Haensel & Zdunik (2008)
PRL 101, 231101 (2008)
PHYSICAL REVIEW
80
Distribution of N and Z
70 60 50
HZ08 1σ and 2σ limits Mean
40 30
Abs[log10 (Y )]
20 10 25
3.0
3.5 4.0
4.5
26 27 28 29 30 31 32 33 34 Electron Chemical Potential EF (MeV)
35
Gupta et al. (2008) FIG. 1 (color). Abundance distributions YðNÞ and YðZÞ shown for absðlog10 ðYÞÞ < 5 at !EF ¼ 0:1 intervals in the EF range 25– 35 MeV for an MCP calculation starting with pure 106 Pd.
from betw 106 Ge lease scena from of the much space that neutr 80 Cr after supe branc
A simple model for evolution of a mixture in the crust with Sanjay Reddy
Can we understand the behavior seen by Gupta et al, in a simple model? What is Q and how does it depend on initial composition?
Follow N nuclei to increasing pressure. At each pressure, allow reactions to occur if energetically allowed: neutron captures and emission electron capture/beta decay Use FRDM mass model of Moller et al. 1995
neutron separation energy
3
Fig. 2.— Neutron separation energy in MeV for each mass chain A as a function of µe . Moving from left to right for a given mass chain, we allow the nucleus to lower its energy by electron capture as µe increases, stopping each sequence when neutron drip occurs (Sn < 0) (or when we reach µe = 29 MeV).
Simple 2 estimate of tunnelling threshold
Fig. 1.— Neutron separation energy below which tunnelling reactions will occur as a function of electron chemical potential and density. The curves from left to right are for Z = 20, 30 and 40. An accretion rate of m ˙ = 104 g cm−2 s−1 is assumed.
function to th fitting the odd effects. This m surface energy Our approa ically N = 1 and Z chosen start at a pres crust (the cor ≈ 3 × 108 g c pressure in sm the density an of the followin any nucleus: e emission, 1 or ture of neutro (tunnelling). W mixture, and getically favor chemical pote
5
Initial composition 56Fe
Q in the range 1-10 Fig. 4.— The same starting nuclei as in Figure 3, but now including 100 nuclei in the calculation. This allows a range of reactions and different nuclei to appear in the simulation.
6
Initial composition 106Pd
Fig. 5.— The same starting nuclei as in Figure 3, but now including 100 nuclei in the calculation. This allows a range of reactions and different nuclei to appear in the simulation.
7
Fig. 6.— An initial mixture of 4 species, A = 106, 80, 56, 38 and Z = 46, 36, 26, 20 with equal numbers of each nucleus.
The impurity level in the inner crust seems to be robustly ~ 3-10 independent of starting composition Still need to: improve mass model (e.g. need surface term) pycnonuclear reactions non ground-state captures with Shannon Holloway, Sanjay Reddy (LANL)
Conclusions Observations of crust cooling and long thermonuclear flashes allow us to probe deep into the ocean and crust Cross talk between nuclear and astrophysics In the ocean, chemical separation leads to heating (up to ~0.2 MeV/ nucleon) and enriches the ocean in light elements ---> implications for superbursts? In the crust, multiple species open up new reaction channels not considered previously. Evolution is towards a state with impurity parameter ~few --> in good agreement with crust cooling observations
