1. A Crust with Nuggets Sanjay Reddy Los Alamos National Laboratory Jaikumar, Reddy & Steiner, PRL 96, 041101 (2006) SQM, UCLA, March (2006)

2. Strange Quark Matter Hypothesis Bodmer, Witten, Terezawa nucleon hyperon Hyper nuclei Fe nuclei Strange Nugget E/A n s /n B

3. Strange Quark Stars 1)Composed entirely of Strange Quark Matter 2) Can be ultra-compact (R~5-8 km) 3) Expected to have a very exotic surface: (i) Large density gradient ~10 26 g/cm 4 ! (ii) Enormous electric field (as electrons spill out) (iii) Electric field can suspend a tiny nuclear crust

4. Strange Star Surface + + + + + + + + + + + e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- e- 10 fm1000 fm Alcock, Farhi, Olinto, Ap. J. 310, 261, (1986) R~10 km M~1.4 M solar

5. Is dense matter homogeneous ? Matter at sub-nuclear density is heterogeneous - charged nuclei are embedded in negatively charge background of electrons. 1.In general strong interactions and Fermi degeneracy energy favor an electrically charged state. 2.Debye screening will ensure that charged regions have finite spatial extent (set by the D ) 3.Surface to Volume ratio of the charge neutral unit-cell is finite - consequently a large surface tension can disfavor the heterogeneous state.

6. Frustration in Dense matter: Two Conserved Charges: Baryon number and electric charge Baryon number fixed by  and  e ensures total charge is zero Pressure (=-free energy) changes with  e for fixed  Locally neutral state with  e = n Q /  Q has lower pressure than the charged state with  e =0. Energy cost per quark:

7. SQM + Electron Gas Mixed Phase Ravenhall, Pethick & Wilson, Phys. Rev. Lett. 50, 2066 (1983) (nuclei -> nuclear matter) Glendenning, Phys. Rev. D46, 1274 (1992), (nuclear -> quark) Alford, Rajagopal, Reddy & Wilczek, Phys. Rev. D64, 074017 (2001) (nuclear ->CFL) Gibbs Equilibrium:

8. If Surface Tension is Small then..

9. …..Strange Stars have Strange Crusts Quark Matter Electron Gas Quark “Nuggets”~ 10 fm Electron Voids ~ 100 fm Radial Extent  R~50-100 m

10. Density gradient is reduced Electron chemical potential drops to zero Poor thermal conductor (electron- nugget scattering) Solid ! Density Profile at the Surface

11. Energetics of the Mixed Phase Gain in Gibbs energy per quark: Surface and Coulomb energy cost per quark: Mixed Phase wins when: In the bag model where: n Q = m s 2  2  Q = 2    2

12. Screening length D =1/(4  e 2  Q ) 1/2 ~ 5-10 fm Large droplets (R >> D ) are neutral inside no energy gain Surface to volume ratio of small droplets is large surface effects are enhanced Surface Energy per quark in a droplet with radius R= D : Debye Screening Mixed phase favored if : Alford, Rajagopal, Reddy, Steiner, in Perp.

13. Is the mixed phase favored ? In the Bag Model: Bag Model Calculations of Surface Tension: (Berger & Jaffe Phys.Rev. C35, 213 (1987)) For m s =150 MeV  MeV/fm 2 For m s =200 MeV  MeV/fm 2

14. Beyond the Bag Model: Physics that can change the electric charge density (n Q ) and the charge susceptibility (  Q ) include: (1)Pairing Correlations: Pairing is strongest in the flavor anti-symmetric channel. Pairing energy between up and down quarks  ~ 100 MeV. This will act like the symmetry energy in nuclear physics. (2)Other strong interaction correlations: Largely unknown. Clearly warrants further work.

15. Role of Pairing or Color Superconductivity uds e - P F Local charge neutrality and strange quark mass split the Fermi levels If pairing energy  m s 2 /2  then the Color-Flavor-Locked (CFL) state is favored. Here n u = n d = n s is enforced by strong interactions. The consequence is n Q =0 ! Rajagopal & Wilczek, PRL 86, 3492 (2001) Steiner, Reddy & Prakash, PRD 66, 094007 (2002)

16. Up-Down Pairing (2SC) When  m s 2 /4  pairing does not involve the strange quarks. However, when we abandon the condition of local neutrality Up-down pairing is inevitable ! This will result in an additional energy gain for the heterogeneous nugget phase (compared to the homogeneous state): Pairing will also tend to increase the Debye Screening length. These effects combine to enhance the stability of the nugget phase.

17. Conclusions If the heterogeneous “Nugget & Void” phase is favored: Strange stars do not have large electric fields at the surface and are not “bare” - normal radiation and no suspended normal crusts (cannot hide them) ! Strange stars can potentially mimic surface behavior of normal stars - they can glitch, burst, have an insulating surface layer etc. But, can they mimic this behavior just right ? Stability of the heterogeneous phase: Likely if m s > 200 MeV and  < few MeV (in the bag model). Need to study the role of Debye screening and curvature energy. Address physics that can change n Q and  Q. The phase competition seems to depend sensitively on the (poorly known) details - cannot exclude this possibility.