r-Process nucleosynthesis in neutron star mergers with

SkyNet

Jonas Lippuner

Luke Roberts, Rodrigo Fernandez, Francois Foucart,

Matt Duez, Christian Ott

NPCSM 2016, YTIP, Kyoto University, Kyoto, Japan

November 9, 2016

Outline

1. r-Process recap

2. SkyNet

3. Parametrized r-process study

4. r-Process in accretion disk outflow

5. r-Process in NSBH dynamical ejecta (time permitting)

2 Jonas Lippuner

r-Process recap

τn ≪ τβ− ∼ 10 ms – 10 s

65Cu

66Zn 67Zn 68Zn 70Zn

69Ga 71Ga

70Ge 72Ge 73Ge 74Ge 76Ge

75As

74Se 76Se 77Se 78Se 80Se 82Se

79Br 81Br

78Kr 80Kr 82Kr 83Kr 84Kr 86Kr

85Rb 87Rb

84Sr 86Sr 87Sr 88Sr

89Y

90Zr 91Zr 92Zr

neutron drip lineclosed neutron shell

3 Jonas Lippuner

r-Process recap

τn ≪ τβ− ∼ 10 ms – 10 s

65Cu

66Zn 67Zn 68Zn 70Zn

69Ga 71Ga

70Ge 72Ge 73Ge 74Ge 76Ge

75As

74Se 76Se 77Se 78Se 80Se 82Se

79Br 81Br

78Kr 80Kr 82Kr 83Kr 84Kr 86Kr

85Rb 87Rb

84Sr 86Sr 87Sr 88Sr

89Y

90Zr 91Zr 92Zr

neutron drip lineclosed neutron shell

3 Jonas Lippuner

r-Process recap

s-process: τβ− ≪ τn ∼ 102 − 105 yr

r-process: τn ≪ τβ− ∼ 10 ms – 10 s

neutron drip line

65Cu

66Zn 67Zn 68Zn 70Zn

69Ga 71Ga

70Ge 72Ge 73Ge 74Ge 76Ge

75As

74Se 76Se 77Se 78Se 80Se 82Se

79Br 81Br

78Kr 80Kr 82Kr 83Kr 84Kr 86Kr

85Rb 87Rb

84Sr 86Sr 87Sr 88Sr

89Y

90Zr 91Zr 92Zr

closed neutron shell4 Jonas Lippuner

r-Process recap

s-process: τβ− ≪ τn ∼ 102 − 105 yr

r-process: τn ≪ τβ− ∼ 10 ms – 10 s

neutron drip line

65Cu

66Zn 67Zn 68Zn 70Zn

69Ga 71Ga

70Ge 72Ge 73Ge 74Ge 76Ge

75As

74Se 76Se 77Se 78Se 80Se 82Se

79Br 81Br

78Kr 80Kr 82Kr 83Kr 84Kr 86Kr

85Rb 87Rb

84Sr 86Sr 87Sr 88Sr

89Y

90Zr 91Zr 92Zr

closed neutron shell4 Jonas Lippuner

r-Process recap

s-process: τβ− ≪ τn ∼ 102 − 105 yr

r-process: τn ≪ τβ− ∼ 10 ms – 10 s

neutron drip line

65Cu

66Zn 67Zn 68Zn 70Zn

69Ga 71Ga

70Ge 72Ge 73Ge 74Ge 76Ge

75As

74Se 76Se 77Se 78Se 80Se 82Se

79Br 81Br

78Kr 80Kr 82Kr 83Kr 84Kr 86Kr

85Rb 87Rb

84Sr 86Sr 87Sr 88Sr

89Y

90Zr 91Zr 92Zr

closed neutron shell4 Jonas Lippuner

r-Process recap

s-process: τβ− ≪ τn ∼ 102 − 105 yr

r-process: τn ≪ τβ− ∼ 10 ms – 10 s

neutron drip line

65Cu

66Zn 67Zn 68Zn 70Zn

69Ga 71Ga

70Ge 72Ge 73Ge 74Ge 76Ge

75As

74Se 76Se 77Se 78Se 80Se 82Se

79Br 81Br

78Kr 80Kr 82Kr 83Kr 84Kr 86Kr

85Rb 87Rb

84Sr 86Sr 87Sr 88Sr

89Y

90Zr 91Zr 92Zr

closed neutron shell4 Jonas Lippuner

r-Process recap

s-process: τβ− ≪ τn ∼ 102 − 105 yr

r-process: τn ≪ τβ− ∼ 10 ms – 10 s

neutron drip line

65Cu

66Zn 67Zn 68Zn 70Zn

69Ga 71Ga

70Ge 72Ge 73Ge 74Ge 76Ge

75As

74Se 76Se 77Se 78Se 80Se 82Se

79Br 81Br

78Kr 80Kr 82Kr 83Kr 84Kr 86Kr

85Rb 87Rb

84Sr 86Sr 87Sr 88Sr

89Y

90Zr 91Zr 92Zr

closed neutron shell4 Jonas Lippuner

r-Process recap

s-process: τβ− ≪ τn ∼ 102 − 105 yr

r-process: τn ≪ τβ− ∼ 10 ms – 10 s

neutron drip line

65Cu

66Zn 67Zn 68Zn 70Zn

69Ga 71Ga

70Ge 72Ge 73Ge 74Ge 76Ge

75As

74Se 76Se 77Se 78Se 80Se 82Se

79Br 81Br

78Kr 80Kr 82Kr 83Kr 84Kr 86Kr

85Rb 87Rb

84Sr 86Sr 87Sr 88Sr

89Y

90Zr 91Zr 92Zr

closed neutron shell4 Jonas Lippuner

Solar system abundances

r sN = 50

r sN = 82

r sN = 126

logrelative

abundan

ce(Si=

106)

Mass number A

even A

odd A

iron-peak

−2

0

2

4

6

8

10

0 25 50 75 100 125 150 175 200 225

Data credit: Katharina Lodders, ApJ 591, 1220 (2003)

5 Jonas Lippuner

SkyNet

▸ General-purpose nuclear reaction network

▸ ∼8000 isotopes, ∼140,000 nuclear reactions

▸ Evolves temperature and entropy based on nuclear reactions

▸ Input: ρ(t), initial composition, initial entropy or temperature

▸ Open source (soon)

JL, Roberts 2016, in prep.

6 Jonas Lippuner

SkyNet

Define abundance

Yi = ni

nB. (1)

Consider reaction

p + 7Li→ 2 4He (2)

with rate λ = λ(T , ρ). ThenY4He = 2λYpY7Li +⋯,Yp = −λYpY7Li +⋯,

Y7Li = −λYpY7Li +⋯ (3)

7 Jonas Lippuner

SkyNet reaction types

Strong

▸ Ordinary: n + 196Au → 197Au (REACLIB, Cyburt+10)

▸ Neutron induced fission: n + 235U → 118Pd + 118Pd (Panov+10,

Mamdouh+01, Wahl02)

▸ Spontaneous fission: 301Md → 121Ag + 180Xe (Frankel+47)

Weak

▸ Beta decays: 86Br → 86Kr + e− + νe (REACLIB, Fuller+82)

▸ Electron capture: 26Al + e− → 26Mg + νe (REACLIB, Fuller+82)

▸ Neutrino interactions and e−/e+ capture on free nucleons:n + νe → p + e− (Arcones+02)

▸ λνe ∝ ∫ ∞wecdE E2(E −Q)2(1 − fe)fνe

8 Jonas Lippuner

SkyNet additional features

Science

▸ Expanded Helmholtz equation of state

▸ Calculate nuclear statistical equilibrium (NSE)

▸ Calculate inverse rates from detailed balance to be consistent with NSE

▸ NSE evolution mode

▸ Implementing screening with chemical potential corrections

Code

▸ Adaptive time stepping

▸ Python bindings

▸ Extendible reaction class

▸ Make movie with chart of nuclides

9 Jonas Lippuner

Parametrized r-process study

10 Jonas Lippuner

Parametrized r-process

Lippuner & Roberts, 2015, ApJ, 815, 82, arXiv:1508.03133

Parameters

0.01 ≤ Ye ≤ 0.50 initial electron fraction

1kB baryon−1 ≤ s ≤ 100kB baryon−1 initial specific entropy

0.1ms ≤ τ ≤ 500ms expansion time scale

Density profile

ρ(t, τ) =⎧⎪⎪⎪⎨⎪⎪⎪⎩

ρ0e−t/τ

t ≤ 3τρ0 (3τ

te)3 t ≥ 3τ t

=3τ

Density

ρ/ρ 0

Time t/τ10−9

10−6

10−3

1

10−3 10−2 10−1 1 101 102 103

Initial conditions

▸ Choose initial temperature T0 = 6GK▸ Find ρ0 by solving for NSE at T0 and Ye that produces specified s

11 Jonas Lippuner

Movies

http://lippuner.ca/skynet/SkyNet_Ye_0.010_s_010.000_tau_007.100.mp4

http://lippuner.ca/skynet/SkyNet_Ye_0.250_s_010.000_tau_007.100.mp4

12 Jonas Lippuner

Final abundances vs. electron fraction

τ = 7.1mss = 10kB baryon−1

Relativefinal

abundan

ce

Mass number A

Ye = 0.01Ye = 0.19

Ye = 0.25Ye = 0.50

LanthanidesActinidesSolar r-process

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

0 50 100 150 200 250

13 Jonas Lippuner

Final abundances vs. entropy

τ = 7.1msYe = 0.19

Relativefinal

abundan

ce

Mass number A

skB = 1skB = 3.2

skB = 10skB = 100

LanthanidesActinidesSolar r-process

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

0 50 100 150 200 250

14 Jonas Lippuner

Impact of electron fraction

s = 10kB baryon−1

τ = 1ms

logX

Number

offissioncycles

Electron fraction Ye

XLa

XAc

XLa+Ac

Number of fission cycles

−5

−4

−3

−2

−1

0

0.0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

7

8

9

10

15 Jonas Lippuner

Example light curves

s = 10kB baryon−1

τ = 7.1ms

M = 0.01M⊙

Luminosity,heatingrate

[erg

s−1]

Time [day]

Ye = 0.01Ye = 0.13Ye = 0.25

LuminosityHeating rate

1039

1040

1041

1042

0 5 10 15

16 Jonas Lippuner

r-Process in accretion disk outflow

17 Jonas Lippuner

Ejecta mass

τττ [ms] MejMejMej [10−3

M⊙10−3 M⊙10−3M⊙] Mej,Ye≤0.25Mej,Ye≤0.25Mej,Ye≤0.25 [10−3 M⊙10−3 M⊙10−3 M⊙]

0 1.8 1.36

10 1.9 1.07

30 3.3 0.83

100 7.8 0.52

300 18.0 0.67

∞ 29.6 0.69

JL, Fernandez, Roberts, et al. 2016, in prep.

18 Jonas Lippuner

YYYeee distribution vs. HMNS lifetimeEjectamass[10−3M⊙]

Electron fraction Ye

τ = 0 ms

τ = 10 ms

τ = 30 ms

τ = 100 ms

τ = 300 ms

τ = ∞

0.0

0.5

1.0

1.5

2.0

2.5

0.1 0.2 0.3 0.4 0.5

19 Jonas Lippuner

Final abundances vs. HMNS lifetimeFinal

mass×a

bundan

ce(M

ejY

i)

Mass number A

τ = 0 msτ = 10 ms

τ = 30 msτ = 100 msτ = 300 msτ = ∞ msSolar r-process

10−6

10−5

10−4

10−3

10−2

10−1

0 50 100 150 200 250

20 Jonas Lippuner

τ = 300τ = 300τ = 300 ms ejecta properties

0.1 0.2 0.3 0.4 0.50

20

40

60

80

100

Ye,5GK

s5GK

[kBbaryon−1]

−3

−2

−1

logvfinal[c]

0

1

2

Mej,−3⊙

0.0

0.5

1.0

1.5 Mej,−

3⊙

t5GK [s]

−3 −2 −1

log vfinal [c]

0.51.01.52.0 M

ej,−

3⊙

0 2 4 6

Mej,−3⊙

0.03 0.1 1 30.3

21 Jonas Lippuner

r-Process in NSBH dynamical ejecta

22 Jonas Lippuner

Neutron star–black hole merger

1. Full GR simulation of NS–BH

Francois Foucart (LBL), Foucart+14

2. Ejecta in SPH code,

Matt Duez (WSU)

3. Nucleosynthesis with SkyNet and

varying neutrino luminosity

JL and Luke Roberts (Caltech)

Roberts, JL, Duez, et al. 2016, MNRAS in

press, arXiv:1601.07942

Figure credit: F. Foucart

23 Jonas Lippuner

BHNS: Final abundances vs. neutrino luminosityRelativefinal

abundan

ce

Mass number A

Lνe ,52 = 0.2Lνe ,52 = 1

Lνe ,52 = 25Solar r-process

10−8

10−7

10−6

10−5

10−4

10−3

10−2

0 50 100 150 200 250

24 Jonas Lippuner

BHNS: Electron fraction distributionMass[M⊙]

Electron fraction Ye

Lνe ,52 = 0.2Lνe ,52 = 1Lνe ,52 = 25

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.00 0.05 0.10 0.15 0.20 0.25 0.30

25 Jonas Lippuner

BHNS: New first peak production mechanism

▸ Original seeds: A ∼ 80 → full r-process

▸ With neutrinos:

▸ νe + n→ p + e−▸ 2p + 2n→ 4He▸ 3 4He + n→ 12C + n

▸ Additional low-mass seed nuclei → enhanced 1st peak

▸ No combination of complete and incomplete r-process

26 Jonas Lippuner

Summary

▸ SkyNet is a flexible reaction network that will be open source

▸ Ye ∼ 0.25 is the critical value for lanthanide production

▸ Heating rate is fairly uniform

▸ Disk outflow after neutron star merger produces 3rd peak regardless of τ ,

but 3rd peak under-produced for τ ≳ 10 ms

▸ Black hole-neutron star merger produces very strong 3rd peak

▸ Neutrino irradiation can enhance 1st peak via low-mass seed nuclei

27 Jonas Lippuner

Extra slides

28 Jonas Lippuner

YYYeee slices

0.1

1

10

100

500Ye = 0.01

τ[m

s]

Ye = 0.25 Ye = 0.50

1 10 1000.1

1

10

100

500

s [kB baryon−1]

τ[m

s]

1 10 100s [kB baryon−1]

1 10 100s [kB baryon−1]

10−5

10−4

10−3

10−2

10−1

0.3

finalX

La+Ac

1037

1038

1039

1040

1041

1042

Mǫat1day

[erg

s−1]

29 Jonas Lippuner

Nuclear reaction network

Consider reaction

[j] + [k]→ [m] (4)

cross section = σ = # of reactions per target [j] per second

flux of projectiles [k]

= R/(Vnj)nkv

= r

njnkv, (5)

and so

r = R

V= σvnjnk =# of reactions per second per volume, (6)

where

R =# of reactions per second,

V = volume,

nj,k = number density of species [j], [k],

v = relative speed between [j] and [k].

30 Jonas Lippuner

Nuclear reaction network

In general

rj,k = ∫ σ(∥v j − v k∥)∥v j − v k∥d3njd

3nk , (7)

using Boltzmann distribution

rj,k = njnk⟨σv⟩j,k = njnk ( 8

µπ)1/2 (kBT)−3/2 ∫

∞

0Eσ(E)e−E/(kBT)

dE , (8)

where

µ = reduced mass = mjmk

mj +mk

,

T = temperature,

kB = Boltzmann constant.

Note that ⟨σv⟩j,k = ⟨σv⟩j,k(T).

31 Jonas Lippuner

Nuclear reaction network

Define abundance

Yi = ni

nB= # of species [i]

# of baryons, (9)

where nB is baryon number density, then for [j] + [k]→ [m]

Ym = rj,kV

# of baryons= rj,k

nB= YjnBYknB⟨σv⟩j,k

nB= YjYkλj,k , (10)

where

λj,k = nB⟨σv⟩j,k = NAρ⟨σv⟩j,k(T) = λj,k(T , ρ), (11)

where NA is Avogadro’s number, and ρ is the mass density.

And, of course

Yj = Yk = −Ym. (12)

32 Jonas Lippuner

SkyNet

In general

Yi = ∑α

Nαi λα(T , ρ) ∏

m∈Rα

Y∣Nα

m ∣m , (13)

where

Yi = ni/nB = abundance of species [i],

α = index running over all reactions,

Nαi =# of species [i] destroyed/created in α,

λα = reaction rate,

Rα = set of reactants of α.Example:

Y4He = ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶decay

4He→ 2d

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶producing reaction

p + 7Li→ 2 4He

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶destroying reaction

n + p + 2 4He → 7Li + 3He

+ ⋯ (14)

33 Jonas Lippuner

SkyNet

In general

Yi = ∑α

Nαi λα(T , ρ) ∏

m∈Rα

Y∣Nα

m ∣m , (13)

where

Yi = ni/nB = abundance of species [i],

α = index running over all reactions,

Nαi =# of species [i] destroyed/created in α,

λα = reaction rate,

Rα = set of reactants of α.Example:

Y4He = − λ4He´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶decay

4He→ 2d

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶producing reaction

p + 7Li→ 2 4He

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶destroying reaction

n + p + 2 4He → 7Li + 3He

+ ⋯ (14)

33 Jonas Lippuner

SkyNet

In general

Yi = ∑α

Nαi λα(T , ρ) ∏

m∈Rα

Y∣Nα

m ∣m , (13)

where

Yi = ni/nB = abundance of species [i],

α = index running over all reactions,

Nαi =# of species [i] destroyed/created in α,

λα = reaction rate,

Rα = set of reactants of α.Example:

Y4He = − λ4HeY4He´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶decay

4He→ 2d

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶producing reaction

p + 7Li→ 2 4He

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶destroying reaction

n + p + 2 4He → 7Li + 3He

+ ⋯ (14)

33 Jonas Lippuner

SkyNet

In general

Yi = ∑α

Nαi λα(T , ρ) ∏

m∈Rα

Y∣Nα

m ∣m , (13)

where

Yi = ni/nB = abundance of species [i],

α = index running over all reactions,

Nαi =# of species [i] destroyed/created in α,

λα = reaction rate,

Rα = set of reactants of α.Example:

Y4He = − λ4HeY4He´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶decay

4He→ 2d

+ 2λp,7Li´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶producing reaction

p + 7Li→ 2 4He

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶destroying reaction

n + p + 2 4He → 7Li + 3He

+ ⋯ (14)

33 Jonas Lippuner

SkyNet

In general

Yi = ∑α

Nαi λα(T , ρ) ∏

m∈Rα

Y∣Nα

m ∣m , (13)

where

Yi = ni/nB = abundance of species [i],

α = index running over all reactions,

Nαi =# of species [i] destroyed/created in α,

λα = reaction rate,

Rα = set of reactants of α.Example:

Y4He = − λ4HeY4He´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶decay

4He→ 2d

+ 2λp,7LiYpY7Li´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶producing reaction

p + 7Li→ 2 4He

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶destroying reaction

n + p + 2 4He → 7Li + 3He

+ ⋯ (14)

33 Jonas Lippuner

SkyNet

In general

Yi = ∑α

Nαi λα(T , ρ) ∏

m∈Rα

Y∣Nα

m ∣m , (13)

where

Yi = ni/nB = abundance of species [i],

α = index running over all reactions,

Nαi =# of species [i] destroyed/created in α,

λα = reaction rate,

Rα = set of reactants of α.Example:

Y4He = − λ4HeY4He´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶decay

4He→ 2d

+ 2λp,7LiYpY7Li´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶producing reaction

p + 7Li→ 2 4He

− 2λn,p,2 4He´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶destroying reaction

n + p + 2 4He → 7Li + 3He

+ ⋯ (14)

33 Jonas Lippuner

SkyNet

In general

Yi = ∑α

Nαi λα(T , ρ) ∏

m∈Rα

Y∣Nα

m ∣m , (13)

where

Yi = ni/nB = abundance of species [i],

α = index running over all reactions,

Nαi =# of species [i] destroyed/created in α,

λα = reaction rate,

Rα = set of reactants of α.Example:

Y4He = − λ4HeY4He´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶decay

4He→ 2d

+ 2λp,7LiYpY7Li´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶producing reaction

p + 7Li→ 2 4He

− 2λn,p,2 4HeYnYpY24He´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

destroying reaction

n + p + 2 4He → 7Li + 3He

+ ⋯ (14)

33 Jonas Lippuner

Time stepping method

inputs: ρ(t), initial T or s, initial Y

calculate ∆t from previous ∆Y

calculate reaction rates from T , ρ(t +∆t)

initial guess x0 → Y

calculate Y (xn), J(xn) for guess xn,

xn+1 → xn − J−1 ( xn−Y∆t− Y )

converged?n > 10?

Y → xn+1, calculate ∆s from ∆Y ,

s → s +∆s, T → EOS(s, ρ(t +∆t),Y )

t → t +∆t

∆t →∆t/2

yes

no

no

yes

34 Jonas Lippuner

τττ slices

1

10

100τ = 0.10ms

s[k

Bbaryon−1]

τ = 7.1ms τ = 500ms

0.01 0.1 0.2 0.3 0.4 0.51

10

100

Ye

s[k

Bbaryon−1]

0.01 0.1 0.2 0.3 0.4 0.5Ye

0.01 0.1 0.2 0.3 0.4 0.5Ye

10−5

10−4

10−3

10−2

10−1

0.3

finalX

La+Ac

1037

1038

1039

1040

1041

1042

Mǫat1day

[erg

s−1]

35 Jonas Lippuner

Light curves vs. electron fraction

s = 10kB baryon−1

τ = 1ms

6days

1day1600K

6000K

logX

La+Ac,tp/3−5

,T

eff/30

00−4

.5

logPeakluminosity

Electron fraction Ye

Lanthanide and actinide mass fraction XLa+Ac

Peak time tp [day]Peak effective temperature Teff [K]

Peak Luminosity [erg s−1]

−5

−4

−3

−2

−1

0

0.0 0.1 0.2 0.3 0.4 0.51038

1039

1040

1041

1042

36 Jonas Lippuner

Light curves vs. electron fraction

−5

−4

−3

−2

−1

0

logX

i,logtp,logTeff

0.0 0.1 0.2 0.3 0.4−5

−4

−3

−2

−1

0

Ye

logX

i,logtp,logTeff

0.5 0.0 0.1 0.2 0.3 0.4

Ye

0.5 0.0 0.1 0.2 0.3 0.4 0.5

Ye

skB= 10, τ = 0.1 ms

skB= 30, τ = 0.1 ms

skB= 10, τ = 1 ms

skB= 30, τ = 1 ms

37

38

39

40

41

42

logM

ǫ,logLp

skB= 10, τ = 10 ms

37

38

39

40

41

42

logM

ǫ,logLp

skB= 30, τ = 10 ms

final XLa+Ac

Xn at 10min

Mǫ at peak [erg s−1]

Lp [erg s−1]

log tp − 4 [day]

logTeff − 6 [K]

37 Jonas Lippuner