Hydrodynamics in Astrophysics

Hydrodynamics
in
Astrophysics
M. Popov
LAPP/LAPTH - June 24 2011
Outline of the talk
1. Introduction: astrophysical phenomena
2. Fundamental equations in hydrodynamics
3. Supernova model
4. Supernova observations
Outline of the talk
1. Introduction: astrophysical phenomena
2. Fundamental equations in hydrodynamics
3. Supernova model
4. Supernova observation
Order of magnitudes
Stars are the buidling blocks
of our Universe
Comparison of star sizes
Antares
Rigel
Sirius A
Sun
Nova Eridani 2009
8.4
Artist illustration of nova flash
Supernova Remnant E0102-72
Jet from active galactic nuclei Centaurus A
Black Hole Candidate Cygnus X-1
Formation of protostellar cores in molecular clouds
A. Kritsuk, S.D. Ustyugov, P. Padoan, R.Wagner, M.L.Norman
Schematic view of our Sun
Outline of the talk
1. Introduction: astrophysical phenomena
2. Fundamental equation in hydrodynamics
3. Supernova model
4. Supernova observations
Leonardo da Vinci :
a precursor in hydrodynamics
Il a compris aussi que la vitesse de
l’eau est différente de la vitesse des
ondes qui se déplacent à la surface
libre :
« La vitesse de propagation des
ondulations (de surface) dépasse
toujours de beaucoup celle de
l’eau14... »
Léonard de Vinci a surtout été le premier, après Héron, à formuler le principe de conservation
de la masse, ou principe de continuité :
« Une rivière à chaque endroit de son cours et au même moment donne passage à une
même quantité d’eau, quelle que soit sa largeur, la profondeur, la pente, la rugosité, ou son
caractère plus ou moins tortueux » ; ce qui n’est exact qu’en écoulement permanent, bien
sûr. Ou encore : « Une rivière de profondeur constante aura un écoulement plus rapide
dans un passage étroit que dans un passage plus large, dans la mesure de ce que la plus
grande largeur excède la plus petite14. »
À Bâle, en Suisse, Daniel Bernoulli (17001782) et Leonhard Euler (1707-1783)
furent les auteurs des premières
traductions mathématiques des principes
de la mécanique des fluides. À partir des
principes de conservation de l’énergie
appliquée aux corps solides par Huygens
et Leibnitz, Bernoulli déduisit que dans un
fluide la somme de l’énergie potentielle
(représentée par la pression p et par
l’altitude z) et de l’énergie cinétique doit
rester constante
Equation of mass conservation
 w 

 z xy
w 
z


 u y  z
Volume mass:  xy z
 u 

x y z
u 
x



xy z
Rate of mass change:
z
y
x
 wxy
t
Net flow through the control volume faces:
 u 
 u

x y z   uy z 
xy z
u 
x
x


Euler equation
Euler , écrivit les équations différentielles qui
décrivent le mouvement d’un fluide, ainsi que
l’équation de continuité qui exprime la
conservation de la masse. Ce système est
toujours connu aujourd’hui comme les
équations d’Euler: principe de la dynamique
appliqué au mouvement d’un fluide de vitesse
V (de composantes notées u, v, w), sur lequel
s’exerce, par unité de masse, une force F de
composantes P, Q, R :





v


   v    v   p  F
 t

Navier-Stokes Equation




v 
1
 v   v   p  g   v
t

Equation of Energy conservation



 ( E )
 E v    p v    g v
t
Turbulence
•
Hydrodynamics equations are non linear
• Transition from laminar regime to turbulent
one (Osborne Reynolds) :
•
Andrei Nikolaevich Kolmogorov :
Turbulence
Instabilities
•
Kelvin-Helmholtz instability
Instabilities
Rayleigh–Taylor instability
Instabilities
•
Richtmyer-Meshkov instability
Outline of the talk
1. Introduction: astrophysical phenomena
2. Fundamental equations in hydrodynamics
3. Supernova model
4. Supernova observations
Piecewise Parabolic Method on a Local stencil
Linear advection equation:
q
q
a
0
t
x
q( x,0)  q0 ( x )
Solution is constant along the characteristic
x (t )  a
q( x, t )  q0 ( x  at )
Piecewise Parabolic Method on a Local stencil


,
( 6)
L
R
L
q( x )  qi   qi  qi  qi 1   


0  1

( 6)
L
R
L
qi 1 / 2 (t   )  qi   qi  qi  qi 1   
n
x  xi 1 / 2

, x  xi 1 / 2  a
x
for
a0
similarly for a<0 characteristic comes from the zone i+1
Our code vs Athena: Rayleigh-Taylor instability test
Our
Athena
hydrostatic equilibrium
  1 for y  0
  2 for y  0
Unpub-
lished
material
v y  0.011  cos2 x / Lx  
1  cos2 y / Ly  4
grid 300 x 900
Our code vs Athena: Bow shock
 1  x  1,  1.5  y  1.5
p  0.0015
  100 for r  0.0625
  0.01 for r  0.0625

where r   x  0.75  y
2

2 1/ 2
A uniform supersonic x-velocity in
all regions except for the overdense
region and to the right of it.
“The tails are not exactly symmetric about
the x-axis even though the bow shock and
flattened sphere maintain good symmetry.
The reason for this behavior is currently
unknown.”
Jim Stone, Princeton Univ.
Our code vs Athena: Bow shock
Unpublished material
Astrophysical bow shock
Great Nebula in Orion (1500 light-years from Earth):
bow shock around the very young star LL Ori.
Presupernova initial configuration
S. Woosley, A. Heger, T. Weaver "The evolution and explosion of
massive stars", Rev. Mod. Phys., 74, 1015, 2002.
Evolution of central temperature and density of 15 and 25 M stars.
Presupernova initial configuration
Presupernova initial configuration
25 M star parameters at helium burning stage:
 c  762 g / cm
8
Tc  1.96  10 K
3
R  1030 R
Main source of energy is nuclear reactions:
4
12
3 He  C
   7.281 MeV
C  He  O    7.150 MeV
12
4
16
O16  He4  Ne20    4.750 MeV
Energy generation is very sensitive to temperature:
E
7 2 3
28
 1.66  10  YHe 4 T8
ergs /( g s )
“Normal” helium burning time is 839 000 years
Asymmetric explosion model
K. Maeda, T. Nakamura, K. Nomoto, P. Mazzali, F. Patat, I. Hachisu
"Explosive Nucleosynthesis in Aspherical Hypernova Explosions and
Late-Time Spectra of SN 1998bw", ApJ, 565, 405, 2002.
Energy 1 1052 ergs was deposit below 0.17 R (contains 2 M )

p
~

,   4/3
Radiation pressure dominated:
Cylindrical coordinate system
Grid: 800 x 800 (1 quadrant)
Energy distribution: 50% thermal, 50% kinetic
Assymetric distribution of kinetic energy
v z   z, vr   r,  :   8 : 1
Cylindrical coordinate system problems
B. Fryxell, D. Arnett, E. Muller "Instabilities and clumping in SN 1987A. I
- Early evolution in two dimensions", ApJ, 367, 619, 1991.
S. Couch, D. Pooley, J. Wheeler, M. Milosavljevic "Aspherical Supernova
Shock Breakout and the Observations of Supernova 2008D", Accepted
to ApJ, Nov. 2010
• Numerical artifacts in the high-resolution simulations.
• Artificially accelerated growth of instabilities near the axis.
• Symmetry violation between the north and the south hemispheres.
Explosion dynamics (20 seconds of simulation)
3


762
g
/
cm
Density
0
Temperature T0  1.96  108 K
Magnetic field influence
B0
Density
Temperature
Bz  0
Magnetic field distribution
2
B
Emag 
2
SN 1987 A
Initial composition
S. Woosley, A. Heger, T. Weaver " The evolution and explosion of
massive stars “, Rev. Mod. Phys., 74, 1015, 2002
X (C12 )  1.05  102 , X ( Ne20 )  1.1  102 , X (' iron ' )  1.3  103
A simplified network of nuclear reactions
Nuclides connected by all possible reactions with p and
30 nuclides in total.
 - particles,
Tracer particles method
S. Nagataki, M. Hashimoto,K. Sato, S. Yamada "Explosive
Nucleosynthesis in Axisymmetrically Deformed Type II Supernovae",
ApJ, 486, 1026, 1997.
• A Lagrangian component in an Eulerian grid code.
• Tracers are massless - does not couple to the flow via gravity or
inertia.
• Advected by the flow, recording the history of conditions along their
path.
• Isotopic yield is calculated as a post-processing step over the
recorded density and temperature.
• Each tracer represents the same amount of mass.
•128 tracers per axis give the accuracy better than 2% for nuclides
with mass fraction >10 -5
Density reconstructed from tracers data
128 2 = 16384 tracers.
Each tracer represents 0.001 M
Unpublished material
Explosive nucleosynthesis: detailed yields
Effect of initial composition
The distribution of main nuclides
Unpublished material
Outline of the talk
1. Introduction: astrophysical phenomena
2. Fundamental equations in hydrodynamics
3. Supernova model
4. Supernova observations
Why nickel is important?
56
Ni
6.1 d
56
 Co
77.7 d
  Fe56
• Explanation of supernova light curves, nickel decay defines the peak
of the light emission when the expanding shell becomes optically thin.
• p and  heat and ionize the ejecta, the energy is reemitted in
optical and infrared wavelengths.
• Detailed description of the chemical and physical structure of the
ejecta required for spectral synthesis calculations.
Nickel production in observations
S. Smartt, J. Eldridge, R. Crockett, J. Maund "The death of massive stars
- I. Observational constraints on the progenitors of type II-P supernovae",
Mon. Not. R. Astron. Soc., 395, 1409, 2009.
No progenitors SNe II-P above 17 M (Statistically significant at 2.4 
confidence.) The red supergiant problem!
SN 2008D
A. Soderberg, E. Berger, K.L. Page et al. "An extremely luminous X-ray
outburst at the birth of a supernova", Nature, 453, 469, 2008.
SN 2008D (Type Ib) in NGC 2770 galaxy - 88 million light years from the
Earth. 5-minute X-ray outburst was detected!
Polarimetric monitoring SN2008D vs SN 2007uy
J. Gorosabel, A. Postigo, A. Castro-Tirado et al. "Simultaneous polarization
monitoring of supernovae SN2008D/XT 080109 and SN2007uy: isolating geometry
from dust", A&A, 522, A14, 2010.
Polarization in supernovae originates by Thompson photon scattering
through an aspherical photospheric expansion.
Optical polarimetric conclusions
First time optical polarimetric monitoring done simultaneously
for 2 supernovae.
• The probabilities that Q and U are simultaneously constant are
0.563 for SN 2007uy and 4.3  104 for SN 2008D.
• SN 2007uy polarization is consistent with constant eccentricity.
• A dominant symmetry axis in SN 2008D exist.
• The symmetry axis could be explained by an axisymmetric
aspherical expansion with variable eccentricity.
Spectrum of SN 2008D compared with other SNe
M. Tanaka, M. Yamanaka, K.
Maeda et al. "Nebular Phase
Observations of the Type Ib
Supernova 2008D/X-ray
Transient 080109: Side-viewed
Bipolar Explosion", Astrop. J.,
700, 1680, 2009.
Double-peaked profile in [O I]
line ( 6300 Å, 6364 Å ) can be
explained by a torus-like
distribution of excited O,I
viewed from the line of sight
is > 50 from the polar
direction!
Schematic bipolar explosion model for SN~2008D
M. Tanaka, M. Yamanaka, K. Maeda et al. "Nebular Phase Observations of the Type
Ib Supernova 2008D/X-ray Transient 080109: Side-viewed Bipolar Explosion",
Astrop. J., 700, 1680, 2009.
Another asymmetrical model
M.V. Popov, S.D. Ustyugov, V.M. Chechetkin Development of the Geometric Structure
of the Thermonuclear-Deflagration Front in Type Ia Supernovae, Astron. Rep., 48,
921, 2004
•
THANK
YOU