formation of power law tail with spectral index -5

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 48109-2143, 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

10-6

10-4

10-2

100

102

104

106

FW H+ N+S correctedBulk SWTailPIHalo SW

4 107 6 107 108 3 108

Phase Space Density (s

3/km

6)

Ulysses SWICS

H+

f = fov

-5

Coronal Hole <R> ≈ 3 AU

BulkSolarWind

HaloSolarWind

PickupProtons

Suprathermal Tail

Proton Speed (cm/s)

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) = fov –5

up to the speed limit of SWICS The tail to core pressure ratio is

Pt /Pc = 0.044

assuming a rollover at 3 MeV

Fast, High-Latitude Solar Wind at ~3 AU

10-4

10-2

100

102

104

106

108

4 107 6 107 108 3 108 5 108

FW H+ bkgd_correctedFWPITailBulk SWHalo SWM21H1d|w0.90-1.10|M/Q1.00-1.00|


3/km

6)

Proton Speed (cm/s)

Quiet SlowSolar Wind

<R> ≈ 5.2 AU

Ulysses SWICS

f = fov

-5

Suprathermal Tail

BulkSolarWind

HaloSolarWind

PickupProtons

H+

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) = fov –5

up to the speed limit of SWICS The tail to core pressure ratio is

Pt /Pc = 0.14

assuming a rollover at 3 MeV

Quiet Slow Solar Wind at ~5 AU

10-3

10-1

101

103

105

107

109

1011

0.001 0.01 0.1 1

*b* H+ SWICS corrected 2007 4:46:50 PM 3/9/08

dj/dE

*dj/dE

SWbulk

Halo

SUM

PIH+

Tail

Differential Intensity 1/(cm

2 sr s MeV/n)

Energy/nucleon (MeV/n)

ACE SWICS and ULEIS

2007.0 – 2008.0Quiet times

Vsw

< 320 km/s

H+

He++

He+

OCFe

He = He+ + He

++

Y = M0*XM1

0.051466M0

-2.4943M1

0.99995R

Suprathermal Tail

dj/dE = joE

–1.5

in the spacecraft frame

SWICS ULEIS

κ = 3

H+Bulk

Solar Wind

HaloSolarWind

PickupProtons

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 = joE –1.5exp[–(E/Eo)0.63]

Eo = 0.72 MeV

The tail to core pressure ratio is

Pt/Pc = 0.01

Ensemble average of many individual time periods during 2007 with low solar wind speed

Differential Intensity to ~1.5 MeV of5 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 = joE –1.5exp[–(m/q)(E/Eo)(1+ )/2]

same = 0.27same Eo = 0.72 MeV

Quiet Solar Wind at 1 AU:H, He+, He++, He, C, O, Fe

10-7

10-5

10-3

10-1

101

103

105

107

109

1011

0.001 0.01 0.1 1

*a* H+tails W>1.45 SWCSnew 5:48:28 PM 3/5/08

dj/dEHOFeCdj/dEdj/dEdj/dE 1/(cm2 sr s MeV/n)dj/dE 1/(cm2 sr s MeV/n)dj/dE 1/(cm2 sr s MeV/n)dj/dE 1/(cm2 sr s MeV/n)dj/dE 1/(cm2 sr s MeV/n)HeHfitHefitOfitFefitCfitdj/dE300dj/dE


2 sr s MeV/n)

ACE SWICS and ULEIS

H+

He++

He+

OCFe

He = He+ + He

++

dj/dE = joE

–1.5exp[–(E/E

o)0.8

](in solar wind frame)

Y = M0*XM1

0.051466M0

-2.4943M1

0.99995R

H+

He++

He+

H

He

OC Fedj/dE = j

oE

–1.5exp[-(m/q)

0.27(E/E

c)

0.63]in the solar wind frame


H

H

He+

He++

HeO

Fe

C

2007.0 – 2008.0Quiet times

Vsw

< 320 km/s

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Power Law Tail with Spectral Index -5:

Observations in Stationary Shocks and Corotating Interaction Regions

10-7

10-5

10-3

10-1

101

103

105

107

100 1000 104

Jup in Sheath SWICS 2:51:35 PM 6/20/07

FWH+*FH+P 30.6*H+ FW 97 [0-2]a.4*H+ FW 97 [0-2]b*H+ FW 97 [0-2]cFWPI .65*FWPIa up quiet 97FWcore upFWPI down V115 vth370 k6FWtail down*FWtail up.55*FWtail R_HFW core down0.09*FWPI down*FWSW main downFWSW main upcore up sum of swHalo and PIFWtail up below 2500 km/s.65*FWPI c up quiet 97FW M1/1H1dProton Speed (km/s)

SWICS and HISCALE on Ulysses

Jupiter's Bow Shock

Downstream

Upstream

f(v) = f v –5

o

tail

core

bulk solar wind

pickupprotons

sd su

tutd

cucd

tr

roll over

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) = fov –5exp[–(v/vo)]

mpvo2 ≈ 1 MeV

See Gloeckler and Fisk, 6th IGPP/AIP, 2007 for details

Magnetosheath ofJupiter’s Bow Shock

12

10-9

10-7

10-5

10-3

10-1

101

103

105

107

1 10

upM21H1d|w0.91-1.09|M/Q1.00-1.00|

FWH+ satcor down

upFWH+ satcor

M21H1d|w0.91-1.09|M/Q1.00-1.00|

*FH+P' 120/P2'

*FH+W/P3'

*FWH+/P' LEMS 120

FWH+/W

HI down*FW_H+

HI up *FW_H+

fit up sw

sw downfit

*FW up PIcorefw

uptailFW(total)

fw_PI tail

down PIcoreFW(total)

FWdowntail

*fw_PI H+core down Phase Space Density [s

3/km

6]

W Ion Speed/Solar Wind Speed

Ulysses SWICS and HISCALE

1992 285Reverse Shock

H+Solar

Wind

Downstream

Upstream

f = fow

-5

Quiet-time tailf = f

ow

-5Upstream beams

PickupIons

4 FW BW H+ 92.285 Rsh

Proton Spectra Upstream and Downstream of a CIR Reverse Shock

Shock Parameters:Ms = 5.2; BN = 68±11°

R-H density jump = ~3.6 R-H Pressure jump = ~34

Upstream spectrum (blue) - 1992 DOY 285.42 – 290.42

- Three-component spectrum- Anisotropic upstream beams

dominate and eclipse theunderlying quiet time tail

Downstream (red) - 1992 DOY 283.83 – 285.38

- Three-component spectrum



mpvo2 ≈ 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

12

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 = joE –1.5exp[–(m/q)0.43(E/Eo)0.71]

provide good fits to all tails

Ec 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 overe-folding energy, Eo

10-5

10-3

10-1

101

103

105

107

109

0.001 0.01 0.1 1

*d* j_dE ULEIS 2007 all H-Fe 8:24:23 PM 3/2/08

dj/dE 1/(cm2 sr s MeV/n)





H+

He

C

O

Fe

He+

He++

Eo

dj/dE

dj/dE

dj/dE

dj/dE H+tailfit

dj/dE He+tailfit

dj/dE He++tailfit


2 sr s MeV/n)


ACE SWICS and ULEIS

2007.0 – 2008.0CIR

VSW

> 500 km/s

Y = M0*XM1

0.28219M0

-0.59859M1

0.99612R

H+

He++

He+

OCFe

He = He+ + He++

dj/dE = joE

–1.5exp[-( m/q)

0.43( E/E

o)

0.71 ]

in the solar wind frame

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

10-11

10-9

10-7

10-5

10-3

10-1

101

103

105

107

107 108 109 1010

H 2004.0-07.7 4:18:34 PM 5/18/08

FWH+FW TS to 170 days after TS LECPFW mass loadedFWPI higher bprodFWSW Up 85.6 AUFWSW HSFWPI HS FWTail HSFWTail Upf bg=(min+mean)/2

Heliosheath

Solar Wind

n = 0.003 cm –3

V = 150 km/sVth = 50 km/s

0.1– 100 AUcompressed by TS

94– 100 AU

pickup ion pressuretotal pressure = 0.75

Solar Wind

V = 100 km/s

n = 0.0038 cm-3

Vth = 35 km/s

P = 3.9*10-14

dyne/cm2


3/km

6)

Proton Speed (cm/s)

Solar Wind

PickupProtons

f = fov

-5

Suprathermal Tail

<R> ≈ 95 AU

H+

Voyager LECP

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 windupstream: extrapolations from

Voyager 2 measurementsdownstream: Voyager 2 measurements in heliosheath

Pickup hydrogenupstream: model calculationsdownstream: STEREO measurements of ENAs



mpvo2 = 4 MeV

Pt /Pc = 0.15

12

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 accelerationmechanism, 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:

∂f

∂t=

1

v4

∂

∂v

δu2

9κv

∂

∂vv5 f( )

⎛

⎝⎜

⎞

⎠⎟

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 .κ δu2

δu2 κ

In the supersonic solar wind:Adiabatic deceleration due to the mean flow competes

with our acceleration process.

j = joE−1.5 exp −

EEo

⎡

⎣⎢

⎤

⎦⎥ Eo =

δu2

vorgo

QA

rousw

mpvo2

2

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; rgo is the particle gyro radius at a reference speed vo; mp 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.

λ

κ =κ oAα E α +1( )/2

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:

j ∝ E−1.5 exp −5⋅4 2 +1( )

AE 1+( )/2

λro

1−5⋅4 2 +1( )A

Eo +1( )

roλ

E +1( )/2 +1⎡

⎣⎢

⎤

⎦⎥

−1⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥exp −A E

Eo

⎛

⎝⎜⎞

⎠⎟

1+( )/2⎡

⎣⎢⎢

⎤

⎦⎥⎥

The only two free parameters, Eo 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, λ.

10-8

10-6

10-4

10-2

100

102

0.1 1 10 100 1000

ACR & modACR fits 9:35:27 PM 6/13/08

H ModACR

He ModACR

N ModACR

O ModACR

Ne ModACR

Ar ModACR

H tail

He tail

N tal

O tal

Ne tal

Ar talParticles/(cm

2 s sr MeV/nuc)

Energy/nucleon (MeV/nuc)

H

He

Oxygen

N

Ne

Ar

dj/dE = joE

–1.5exp[-(m/q)

•( /E Eo)(1+)/2]

105.7 AU

Modulated ACR

Local TailSpectrum

1 10 100 1000

j H V1 2008_053-104 9:35:27 PM 6/13/08

H Particles/cm2 s sr MeV

He Particles/cm2 s sr MeV

O Particles/cm2 s sr MeV

j H/(cm2 sr s MeV/nuc)

j He/(cm2 sr s MeV/nuc)

j O/(cm2 sr s MeV/nuc)

j N/(cm2 sr s MeV/nuc)

j Ne/(cm2 sr s MeV/nuc)

j Ar/(cm2 sr s MeV/nuc)

H sum

He sum

O sum

N sum

Ne sum

Ar sum

Energy/nucleon (MeV/nuc)

H

He

O

N

Ne

Ar

dj/dE = joE

–1.5exp[-(m/q)

•( /E Eo)(1+)/2]

2008 053-104CRS circles 2008 053-104LECP squares

1Voyager106.7 AU

j ∝ E−1.5 exp −5⋅4 2 +1( )

AE 1+( )/2

λro

1−5⋅4 2 +1( )A

Eo +1( )

roλ

E +1( )/2 +1⎡

⎣⎢

⎤

⎦⎥

−1⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥exp −A E

Eo

⎛

⎝⎜⎞

⎠⎟

1+( )/2⎡

⎣⎢⎢

⎤

⎦⎥⎥

formation of power law tail with spectral index -5

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