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|
Phase Space Density (s
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
Differential Intensity 1/(cm
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
Energy/nucleon (MeV/n)
H
H
He+
He++
HeO
Fe
C
2007.0 – 2008.0Quiet times
Vsw
< 320 km/s
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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
In the solar wind frame the velocity distribution of the suprathermal tails have the form
f(v) = fov –5exp[–(v/vo)]
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)
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)
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
Differential Intensity 1/(cm
2 sr s MeV/n)
Energy/nucleon (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
Phase Space Density (s
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
In the solar wind frame the velocity distribution of the suprathermal tails have the form
f(v) = fov –5exp[–(v/vo)]
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⎡
⎣⎢⎢
⎤
⎦⎥⎥