Formation of Power Law Tail with Spectral Index -5 G. Gloeckler and L. A. Fisk Department of Atmospheric, Oceanic and Space Sciences University of Michigan, Ann Arbor, Michigan , USA SHINE 2008 Zermatt Resort and Spa, Midway Utah June 23, 2008

Power Law Tail with Spectral Index -5 during Quiet Times: Observations in the Helioshpere

Simple average of two ~ 1 year long time periods in the fast solar solar from the north and south polar coronal holes Three-component spectrum - Bulk Solar wind - Core particle (halo solar wind and pickup protons) - Suprathermal tail In the solar wind frame the distribution function of the suprathermal tail has the form f(v) = f o v –5 up to the speed limit of SWICS The tail to core pressure ratio is P t /P c = assuming a rollover at 3 MeV Fast, High-Latitude Solar Wind at ~3 AU

Ensemble average of many individual time periods during 1998 with low suprathermal tail fluxes Three-component spectrum - Bulk Solar wind - Core particle (halo solar wind and pickup protons) - Suprathermal tail In the solar wind frame the distribution function of the suprathermal tail has the form f(v) = f o v –5 up to the speed limit of SWICS The tail to core pressure ratio is P t /P c = 0.14 assuming a rollover at 3 MeV Quiet Slow Solar Wind at ~5 AU

Quiet Solar Wind at 1 AU: Protons Ensemble average of many individual time periods during 2007 with low solar wind speed Differential Intensity to ~1.5 MeV Three-component spectrum Spectrum rolls over at ~ 0.7 MeV In the solar wind frame the differential intensity of the suprathermal tail has the form dj/dE = j o E –1.5 exp[–(E/E o ) 0.63 ] E o = 0.72 MeV The tail to core pressure ratio is P t /P c = 0.01

Ensemble average of many individual time periods during 2007 with low solar wind speed Differential Intensity to ~1.5 MeV of 5 species with different mass/charge values (assumed to be that of solar wind ions measured my SWICS) Rollsovers observed in all spectra In the solar wind frame the differential intensity of all five suprathermal tails have the form dj/dE = j o E –1.5 exp[–(m/q)  (E/E o ) (1+  )/2 ] same  = 0.27 same E o = 0.72 MeV Quiet Solar Wind at 1 AU: H, He +, He ++, He, C, O, Fe

Power Law Tail with Spectral Index -5: Observations in Stationary Shocks and Corotating Interaction Regions

Upstream and downstream velocity distributions are measured above ~300 km/s Upstream Mach number is ~10.5 and corresponding R-H pressure and temperature jumps are ~135 and ~35 respectively The measured tail pressure jump of ~150 is a bit higher and the core temperature jump somewhat lower than R-H resulting from some core particles flowing into the tail In the solar wind frame the velocity distribution of the suprathermal tails have the form f(v) = f o v –5 exp[–(v/v o )  ] m p v o 2 ≈ 1 MeV See Gloeckler and Fisk, 6th IGPP/AIP, 2007 for details Magnetosheath of Jupiter’s Bow Shock 1212

Proton Spectra Upstream and Downstream of a CIR Reverse Shock Shock Parameters: M s = 5.2;  BN = 68±11° R-H density jump = ~3.6 R-H Pressure jump = ~34 Upstream spectrum (blue) DOY – Three-component spectrum -Anisotropic upstream beams dominate and eclipse the underlying quiet time tail Downstream (red) DOY – Three-component spectrum In the solar wind frame the velocity distribution of the suprathermal tails have the form f(v) = f o v –5 exp[–(v/v o )  ] m p v o 2 ≈ 1.7 MeV The measured tail pressure jump of ~100 is higher and the core pressure jump is lower than R-H because some core particles flowing into the tail 1212

Ensemble average of several individual time periods during 2007 with high (> 500 km/s) solar wind speed Model spectra (curves) of the form dj/dE = j o E –1.5 exp [ –(m/q) 0.43 (E/E o ) 0.71 ] provide good fits to all tails E c for the 2007 CIR is 0.28 MeV/n, lower than the quiet time 2007 value (0.72 MeV/n) Contributions of He + and He ++ to the tail He spectrum are about the same, thus (He/O) tail ≈ 2 (He/O) sw (C/O) tail ≈ 0.6 (Fe/O) tail ≈ 0.09 C/O approaches ~1 (observed in the 1970s) at high energies due to m/q dependence of roll over e-folding energy, E o Corotating interaction Regions at 1 AU

Power Law Tail with Spectral Index -5: Observations in the Heliosheath

Proton Spectra Upstream and Downstream of the Termination Shock The three-component spectra upstream (blue) and downstream (red) consist of: - bulk solar wind - core (pickup H and some halo solar wind) -suprathermal tail Solar wind upstream: extrapolations from Voyager 2 measurements downstream: Voyager 2 measurements in heliosheath Pickup hydrogen upstream: model calculations downstream: STEREO measurements of ENAs In the solar wind frame the velocity distribution of the suprathermal tails have the form f(v) = f o v –5 exp[–(v/v o )  ] m p v o 2 = 4 MeV P t /P c =

Power Law Tail with Spectral Index -5: Brief Summary of Theoretical Concepts

The fact that the common spectral shape can occur in the quiet solar wind, far from shocks, suggests that the acceleration mechanism is some form of stochastic acceleration. It cannot, however, be a traditional stochastic acceleration mechanism, which in general has a governing equation that is a diffusion in velocity space. Many different solutions to the diffusion equation are possible, including power law solutions. But the solutions are dependent on the choice of the diffusion coefficient, which is unlikely to be the same in all the different conditions where the common spectral shape occurs.

The underlying assumption of the theory is that the acceleration occurs in thermally isolated compressional turbulence, which we demonstrate is equivalent to spatially homogeneous compressional turbulence -- conditions that may be common in the solar wind. With this assumption it is necessary to treat the statistics of the problem differently from what is normally done in deriving the diffusion equation that governs standard stochastic acceleration. In a normal diffusion derivation the behavior of particles at one location is unrelated to the behavior elsewhere in the volume. To maintain thermal isolation the behavior of particles in different parts of the volume has to be related to each other. This fundamentally different approach alters the statistics and guarantees that the accelerated spectrum is always a power law with spectral index of -5.

The governing equation for this acceleration process The equation that governs the time evolution of the distribution function, f, in the frame of the solar wind, can be shown to be: is the mean square turbulent flow speed; is the spatial diffusion coefficient, v is particle speed. Note that the equilibrium spectrum is a power law with spectral index of -5, independent of the choice of and.

In the supersonic solar wind: Adiabatic deceleration due to the mean flow competes with our acceleration process. If we add the competing adiabatic deceleration, and make the assumption that the diffusion coefficient is a standard cross-field diffusion coefficient, particle speed times particle gyro-radius, we find that the accelerated spectrum, expressed as differential intensity, is: Here, E is particle kinetic energy per nucleon; r go is the particle gyro radius at a reference speed v o ; m p is the mass of a proton; A is mass number; Q is charge number. Note: the cutoff has a specific mass-to-charge dependence and magnitude, which can be compared with observations. It is also independent of radial distance. where

The model for the acceleration of ACRs in the Heliosheath The same governing acceleration equation as in the supersonic solar wind. -No limit to the rollover e-folding energy due to adiabatic deceleration. -The limit is due to the ability of the particles to escape by diffusion. The spatial diffusion coefficient is taken to be the following form, particle speed times a power law in particle rigidity [recall ACRs are singly charged]. It is independent of radial distance, and can be normalized so that it yields the observed spatial gradient of 5%/AU for 16 MeV/nucleon Helium. The solar wind speed is taken to decline with a characteristic length scale. This is a crude approximation that makes the math tractable. The mean square random speed of the turbulence is taken to be independent of radial distance, consistent with a constant turbulent pressure.

The resulting ACR spectra The rollover e-folding energies of the suprathermal tails grow towards higher energies as you go further into the heliosheath. The growth is limited by the ability of the particles to escape by diffusion. Particles diffuse inward to form the ACRs, and are subject to standard convection-diffusion modulation. The resulting spectra for the ACRs are: The only two free parameters, E o and  can be chosen to fit the high energy rollovers in the ACRs. This leaves only one adjustable parameter, the characteristic fall-off distance of the solar wind speed,.