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Serge BOUQUET serge.bouquet@cea.fr CEA, DAM, DIF, F - 91297 Arpajon, France and

International Conference on High Energy Density Laboratory Astrophysics - HEDLA - Tallahassee (FL - USA), April 29 - May 04, 2012. SCALING ASTROPHYSICAL RADIATION HYDRODYNAMICS for the LABORATORY. Serge BOUQUET serge.bouquet@cea.fr CEA, DAM, DIF, F - 91297 Arpajon, France and

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Serge BOUQUET serge.bouquet@cea.fr CEA, DAM, DIF, F - 91297 Arpajon, France and

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  1. International Conference on High Energy Density Laboratory Astrophysics - HEDLA - Tallahassee (FL - USA), April 29 - May 04, 2012 SCALING ASTROPHYSICAL RADIATION HYDRODYNAMICS for the LABORATORY Serge BOUQUET serge.bouquet@cea.fr CEA, DAM, DIF, F - 91297 Arpajon, France and LUTH, Paris Observatory,, F- 92195 Meudon, France • Reviewpaper: S.B., E. Falize, C. Michaut, C. Gregory, B. Loupias, T. Vinci and M. Koenig, • High EnergyDensityPhysics (HEDP) 6(2010)368 • - E. Falize, C. Michaut and S.B., The Astrophysical Journal (ApJ) 730(2011)96 Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  2. CONTRIBUTORS CEA: Commissariat Energie Atomique – Energies Alternatives Emeric FALIZE and Bérénice LOUPIAS LUTH: Lab. Universe and Theory Claire MICHAUT LULI: Lab. Utilisation Lasers Intense Chris GREGORY, Michel KOENIG and Tommaso VINCI Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  3. ELEPHANTS in GREECE ! TILOS (THO): Dodecanese island (Turkey) ~ 10 miles Dwarf elephants (1.60 cm) Carbon-dated 7 000 – 4 000 BC Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  4. ELEPHANTS in GREECE ! Not homothetic ! No similarity or No scaling law Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  5. ELEPHANTS in GREECE ! • Mass – weight: P  • g: gravity  Force of a leg: F  (cross section) L increases: P increases faster than F ! C(g,L) increases with g and L  Another planet with g’ < g (same L): C’ = C(g’,L) < C(g,L) S’ < S The big elephant with g’ may become homothetic to the dwarf elephant with g Changing “external driving conditions” makes the systems homothetic Works for several astrophysical phenomena ! Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  6. RADIATION HYDRODYNAMICS - S.B., R. Teyssier, J.-P. Chièze, ApJS 127(2000)245 - X. Fleury et al., LPB 20(2002)263  Pressure condition: For example, satisfied for a shock wave provided: n: Particle density A: Atomic mass number m : Proton mass k: Boltzmann constant a: 1rst radiative constant proton  Flux condition: - P.A. Keiter et al., PRL 89(2002)165003 - R.P. Drake, PoP 14(2007)043301 c: Speed of light « Drake’s condition » For a given A (atomic mass number – dimensionless quantity), Therefore, it is easier to fulfill “flux condition” than “pressure condition” Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  7. OUTLINE 1) - SCALING LAW FORMALISMS 2) - YOUNG STELLAR OBJECT (YSO) JETS 3) - RADIATIVE SHOCKS 4) - CONCLUSION Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  8. GOAL OF SCALING LAWS H. Takabe, Prog. Th. Phys. Supp. 143 (2001) 202 - Resemblance of physics (same phenomena: shock in a gas/star), - Sameness of physics (EOS, opacity…), - Similarity of physical phenomena (preservation of the dimensionless numbers) Astrophysical Models HED plasma Models • - D. Ryutov et al., ApJ. 518 (1999) 821 • - D. Ryutov, R.P. Drake and B. Remington, ApJ. Supp. 127 (2000) 465 • D. Ryutov et al., Phys. Plasmas 8 (2001) 1804 • - D. Ryutov and B. Remington, Plasma Phys. Control. Fus. 44(2002)B40 • - D. Ryutov and B. Remington, Phys. Plasmas 10 (2003) 2629 • - D. Ryutov and B. Remington, Plasma Phys. Control. Fus. 48(2006)L23 • - D. Ryutov and B. Remington, Astrophys. Sp. Sci. 307(2007)291 • - D. Ryutov et al., ApJ. 698 (2009) 2144 • - Talk by Dmitri on Wednesdaymorning Astronomical observations Laser, Z-pinch experiments ??? CONNECTION BETWEEN LABORATORY EXPERIMENTS AND ASTRONOMICAL OBSERVATIONS Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  9. SIMILARITY and INVARIANCE D. Ryutov et al., ApJ 518 (1999) 821 … etc. … « Model » equation: Non-linear heat equation PDE (partial differential equation) T: temperature (D: diffusion coefficient = cst.) Solution: T = S(x,t) where S is a known function INVARIANCE under the transformation: AT (AT)n . Ax , AtandAT : Scaling parameters . = At (Ax)2 Independent variables: x and t x = Ax. , t = At. At .(AT)n -1 Dependent variable: T(x,t) . = (Ax)2 T = AT. .(AT)n -1 At The equation is invariant under the transformation = 1 = (Ax)2 Solution: but AT = [(Ax)2 / (At)]1/(n -1) The solution isinvariant Ax and At are arbitrary ! The solution is the same at both scales Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  10. LIE GROUP POINT of VIEW (LIE SYMMETRIES) Sophus LIE (1888, 1890, 1893) Ovsjannikov (1962), Ibragimov (1985), Bluman (1974), Olver (1986) Sophus LIE (1842 - 1899) , Norwegian mathematician The equation is invariant under x = Ax. , t = At. , T = AT. the transformation: AT = [(Ax)2 / (At)]1/(n -1) holds. provided the relation Two free parameters (!!!) to make the 2 systems homothetic . 1/(n -1) T(x,t) . t 1/(n -1) I INVARIANT of the transformation (Lie group) = = = ( 2)1/(n -1) (x2)1/(n -1) It is not a dimensionless number !!! SYSTEMATIC AND RIGOUROUS CONNECTION BETWEEN LABORATORY PLASMA QUANTITIES and the CORRESPONDING ASTRO. QUANTITIES Access to « hidden » astrophysical quantities from experimental measurements Several INVARIANTS for several PDE’s or for a PDE with several terms Remark: Dimensionless numbers, combine x and t to obtain Self-Similar Solutions (SSS’s), EDP’s become ODE’s, Analytic solutions … Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  11. EXTENSION of STRICT INVARIANCE x, t and T have been rescaled; D has remained the same STRICT INVARIANCE plus D = AD.  x = Ax. , t = At. , T = AT. .(AT)n -1 At At .(AT)n -1 . AD. . AD = 1 = (Ax)2 (Ax)2 . 1/(n -1) T(x,t) . (D.t) 1/(n -1) AT = [(Ax)2 / (AD.At)]1/(n -1) I = = ( 2)1/(n -1) (x2)1/(n -1) 3 free parameters WEAK INVARIANCE  GLOBAL INVARIANCE , D = x = Ax. , t = At. , T = AT. (STRICT) Falize et al., ApSS 322(2009)107 Specify the microscopic dependence of D, e.g. D() = Cdiffusion . m Add = A andCdiffusion= ACtherefore Example for a fluid: Ideal gas EOS p = CEOS. .T but STRICT > WEAK > GLOBAL Remark: strict = global + invariance of Cdiffusion ( ) Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  12. 1) - SCALING LAW FORMALISMS 2) - YOUNG STELLAR OBJECT (YSO) JETS 3) - RADIATIVE SHOCKS 4) - CONCLUSION Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  13. YOUNG STELLAR OBJECTS (YSO) JETS Unit : 1 000 a.u.(1 a.u. = 1.5 1013 cm and 1 pc = 2 105 a.u.) Herbig - Haro Objects Length from 1 000 au to 0.1 pc Differences in the structure, as well Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  14. Piston 500 m Laser Washer 100 m diameter Foam 50 mg/cc 100 mg/cc 250 m FOAM-FILLED-CONE JETS at LULI LASER FACILITY (LULI2000) Shadowgraphy 10 ns after the shock break-out Ombroscopy diagnostic L = 800 m at t =10 ns B. Loupias et al., Phys. Rev. Lett. 99 (2007) 265001 C. Michaut et al., IAU Symposium 275 (2011) 402 B. Loupias et al., Plasma Phys. Controlled Fus. 51 (2009) 124027 M. Koenig et al., Plasma Fus. Research 4 (2009) 044 C.D. Gregory et al., PoP 17 (2010) 052708 Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  15. INVARIANCE in RADIATION HYDRODYNAMICS ? Optically thin radiation hydrodynamics • N=0: plane, N=1: cylindrical, N=2: spherical geometry and • = constant = cooling function power law form exponents  and  CEOS= constant I.G.: = = 1 exponents  and  and  = ( - )/( - 1), Invariance ? Scaling parameters: Aq(q: any physical quantity) • vastro = Av. vlab Tastro = AT. Tlab CEOS,astro = ACEOS.CEOS,lab tastro = At. tlab astro = A. lab 0,astro = A0. 0,lab pastro = Ap. plab xastro = Ax. xlab Yes ! astro = A. lab Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  16. WEAK INVARIANCE Three free parameters Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  17. SCALING LAWS Experimental data: 3 free parameters Values obtained from WEAK INVARIANCE Protostellar jet  HH111 low low low Remarks: - Not necessary to specify the type of cooling - Ideal gas EOS to get T from pressure and density Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  18. 1) - SCALING LAW FORMALISMS 2) - YOUNG STELLAR OBJECT (YSO) JETS 3) - RADIATIVE SHOCKS 4) - CONCLUSION Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  19. RADIATIVE SHOCKS in SNe or SNR ’s Cassiopeia A (1685) Crab Nebula (1054) SN1987a (Feb. 23, 1987) RS FS Muller et al. Astron. Astrophys. (1991) Motion of the Forward Radiative Shock T decreases due to the cooling (radiative flux ahead the sock) Also in supernovae: downstr. /upst. = 7 ( = 4/3) precursor therefore  increases (first the compression is 4 and becomes much larger) H. Bethe, Astrophys. J. (1997) The velocity is normalized to the shock velocity Radiative precursor:the energy goes through the discontinuity and heats the medium ahead of the shock. B.T. Draine & C.F. McKee, Annual Rev. Astron. Astrophys. 31 (1993) 373 Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  20. GOI (Gated Optical Imager) Exposure time: 100ps Series of Snapshots for initial pressure PXe=0.1 bar 1.8 mm 1.8 mm +5 ns +6 ns +7 ns +8 ns +10 ns +9 ns Shock front Precursor front Time: 5 ns RADIATIVE SHOCKS at LULI LASER FACILITY (LULI2000) Paul DRAKE experiments (OMEGA): Xe is denser (atmospheric pressure) - Keiter et al. PRL (2002) - no precursor, BUT - collapsed dense layer behind the shock front: radiative losses (density)power are important and compression rate ~ 50-100 S. Bouquet et al., PRL 92 (2004) 225001 C. Michaut et al., Astrophys. Space Science 322 (2009) 77 A.R. Reighard et al., Phys. Plasmas 13 (2006) 082901 T. Vinci et al., Phys. Plasmas 13 (2006) 010702 M. Koenig et al., Phys. Plasmas 13 (2006) 056504 Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  21. LASER TARGETS FILLED WITH XENON LULI2000 targets (Paris observatory and LULI) Claire MICHAUT group Patrice BARROSO Front du choc radiatif Simu DUED Précurseur t = 11 ns Tommaso Vinci (LULI) + Stefano Atzeni (ROMA) Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  22. OPTICALLY THICK RADIATION HYDRODYNAMICS E. Falize et al., Journal Physics: Conf. Series 112 (2008) 042016 E. Falize et al., Astrophys. Sp. Sc., 322 (2009) 107 • , •  =  = 1, Ideal Gas • m = -2, n = 13/2 Kramers opacity Free-free absorption (Stellar structure, stellar evolution) , Invariance ? • Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  23. SCALING INVARIANCE … YES !!! STRICT INVARIANCE 2 free parameters No Only Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  24. SCALING LAWS – RADIATIVE SHOCK Experimental data: STRICT INVARIANCE (Ideal gas + Kramers) x 10 too high BAD ! Small-scale laboratory models are not achievable with the kJ-class lasers  NIF, LMJ Experimental precursor 10 times longer (1 mm instead 100 microns): Much better Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  25. 1) - SCALING LAW FORMALISM 2) - YOUNG STELLAR OBJECT (YSO) JETS 3) - RADIATIVE SHOCKS 4) - CONCLUSION Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  26. CONCLUSION 1) – Rigorous derivation of scaling laws has been made and the connection between experiments and astrophysical objects is 1 to 1 2) – Laboratory astrophysics is a relevant approach in spite of some difficulties: Rad. Shocks, for instance. BUT: Erad and Prad neglected in the rescaling (not yet achieved in laboratory experiments MOREOVER: Strict invariance has been used but deviation to SNe due to the use of Xenon should be taken into account: CEOS and Copacityare not invariant ! Global invariance should be used … 3) – Use scaling laws in the reverse way to determine the right laboratory plasmas Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  27.  Boltzmann number: Michaut et al., ApSS 322(2009)77 Radiation hydrodynamics: Bo << 1 Same role as Peclet number, Pe, for non radiating fluids: : kinematic viscosity D: thermal diff. coeff. Prandtl number: : heat capacity per unit mass Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  28. Falize et al., ApSS 322(2009)107  Mihalas number: Michaut et al., ApSS 322(2009)77 Radiation hydrodynamics: R << 1 Radiation flux dominated flow: Bo << 1 but R > 1 Zeldovich and Raizer (1966) p. 526 Radiation pressure dominated flow: Bo << 1 and R << 1 Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

  29.  Strouhal number:  Euler number:  Diffusion approximation:  Small photon mfp (thick):  Small photon mfp (thin): Serge Bouquet, HEDLA’12, April 29 - May 04, 2012, Tallahassee, Florida, USA

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