Cosmic Ray Connement and Transport Models for
Probing their Putative Sources
Mikhail Malkov
UCSD
Partially Supported by NASA and DoE
Ackn: R. Sagdeev, P. Diamond, I. Moskalenko, F. Aharonian
1 / 28
Outline
1
Two regimes of CR transport along magnetic eld:
-collectively self-impeded
-test particle transport in background-turbulence
2
CR self-connement and escape
Equation for CR Self-Connement
Self-Similar Solution
γ -ray
Spectra From CR-Illuminated Molecular Clouds
Environmental vs Transport-Induced Spectral Breaks
Summary and Conclusions
3
CR transport in background wave eld
Equation for Transport Driven by Pitch-Angle Scattering
CR Transport Equation and its Asymptotic Reduction
Chapman-Enskog expansion
conclusions (CR test particle transport)
2 / 28
Why need two dierent approaches to CR transport?
simple facts about CRs in SNR and in ISM
Dk ' DBohm (B/δB)2 →Dk varies between ∼ DB
5
SNR) and DISM ∼ 10 DBohm (quiet ISM).
(in CR source, e.g.
Away from the sources (SNR), CR energy is comparable to
magnetic energy, thermal plasma
&
star light and CMB all
∼ eV /cc
Therefore, in and around a CR source (SNR)
PCR B 2 /8π
Therefore, CRs cannot escape the source without driving strong
MHD waves, so
need to evolve CR from strong self-connement regime
D ∼ DBohm
in and around the source to their dissolution into the ambient ISM,
D ∼ DISM DBohm
→Interface
Problem
3 / 28
Problems addressed under self-connement/free diusion
regime
Self-connement
Free propagation
acceleration (Bell
CR background spectra
77)
CR chemical composition, spectral
propagation
anomalies
(Voelk, Wentzel,
p/He Pamela (Adriani 2011)
rev. ca 73-74)
illumination of
adjacent MC (e.g.,
Aharonian, Drury,
Voelk)
source morphology
Explanations: Drury; Ohira; Blasi;
Ptuskin; MM, Diamond, Sagdeev 2012
CR anisotropy, particularly sharp
∼ 10◦
Milagro (Abdo 2008)
Interpretations: Drury&Aharonian;
Desiati&Lazarian; MM, Diamond, Drury
source
and Sagdeev (ApJ 2010), Giacinti, Ahlers
calorimetry...
(coh. turb. realization, statistical eects),
(MM, PoP 2015)
4 / 28
CR escape from SNR/Problem setting/geometry
CR diuse along MF (⊥propagation may be important,
but too many issues, e.g. Kirk,
Duy and Gallant 96)
generate Alfvén waves that
suppress diusion
CR escape along MF from
two polar cusps of SNR
to obtain CR distribution both
processes are to be treated
self-consistently
result will determine MC
emissivity
5 / 28
Equations
CR propagation in self-excited waves
∂ κB ∂PCR
d
PCR (p) =
dt
∂z I ∂z
PCR (p) -partial pressure, I (p) -wave energy, resonance kp = eB0 /c ,
d/dt = [∂/∂t + (U + CA ) ∂/∂z] , κB ∼ crg (p)
Wave generation by
∇PCR
associated with the CR pitch-angle
anisotropy
∂PCR
d
I = −CA
− ΓI
dt
∂z
QL integral (Sagdeev et al '61)
κB ∂
I (z, t)
ln
PCR (z, t) = PCR0 z 0 −
CA ∂z
I0 (z 0 )
z 0 = z − (U + CA ) t
6 / 28
Equations
CR/Alfven wave coupling
∂W
∂ 1 ∂W
∂
−
= − P0 (z)
∂t
∂z W ∂z
∂z
W = CκABa(p)
I
(p)
|z/a| < 1
d/dt ≈ ∂/∂t , P0 -initial
-dimensionless wave energy,
Self-similar solution in variable
|z| > a,
CR distribution,
√
ζ = z/ t , W (z, t) = w (ζ)
for
outside initial CR cloud
d 1 dw
ζ dw
+
=0
dζ w dζ
2 dζ
solution depends on a background turbulence level
W 0 1,
and
integrated CR pressure in the cloud:
Z
Π=
0
1
P0 dz 1
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CR self-similar distribution
Comparing and Contrasting with
conventional TP predictions:
Π'3
CA a (p) P̄CR (p)
1
c rg (p) B02 /8π
considerable delay of CR
escape
narrower spatial
distribution of CR cloud
extended self-similar,
√
Self-connement
vs test-particle escape, tPCR vs
√
z/ t for dierent values of CR pressure Π (from
MM, P. Diamond, R. Sagdeev, F. Aharonian and I.
Moskalenko ApJ 2013)
P ∝ 1/z
region
8 / 28
Results/CR pressure distribution
CR partial pressure (found in closed but implicit form) is well
approximated by:
√
h
i−3/5
2
tP = 2 ζ 5/3 + (DNL )5/6
e −W0 ζ /4 ,
particle diusivity is strongly suppressed by
√
ζ = z/ t
self-connement
eect:
DNL ∼ DISM e −Π ,
integrated CR pressure parameter is typically large:
Π'3
CA a (p) P̄CR (p)
1
c rg (p) B02 /8π
9 / 28
Breaks in Spectra of Escaping CRs
normalized partial pressure
P (p)
approximation
n
o−3/5
P ≈ 2 z 5/3 + [DNL (p) t]5/6
for
at
DNL (p) < z 2 /t
DNL (pbr ) = z 2 /t ,
p = pbr ,
δ
from
if
exp (−Π) ∝
P
DISM (p)
is at at
if δ < 0, P
p > pbr
momentum independent (DSA TP
DNL ∝ p δ ,
and CR pressure
p −σ and
p < pbr
for
raises with
DISM ∝
δ>0
p
f ∼ p −4 )
break index
= δ/2
Π (p)
p λ at
p ∼ pbr ,
so
δ = λ − σ,
−δ/2 at
and steepens to p
−δ/2 at
as p
p < pbr
then
p = pbr .
and it levels o at
10 / 28
Morphological Signatures of CR Self-Connement
central source (magenta radio image)
emission is masked
bi-polar morphology of escaping CR is
clearly seen
not everywhere correlated with the
dense gas (green contours) distribution:
strong
γ -ux
is expected from
overlapping regions of CR and gas
density
Fermi-LAT
W44,
γ -image
of SNR
Uchiyama et al 2012
strong indication of eld aligned
propagation
CR diusivity is suppressed by up to a
e.g. Uchiyama et al 2012)
factor of ten (
11 / 28
Spectral Signatures of the DSA and subsequent escape
from W 44
presumably from dense MC
illuminated by CR
best t is given by a TP source
spectrum
E −q , q = 2,
no
cut-o required within
CR subjected to propagation
in evanescent Alfven waves
inside MC
wave evanescence and damping
are due to ion-neutral
collisions in MC
γ
-emission from MC near
SNR W44
→break
in the CR spectrum of
index unity
E −q → E −q−1
12 / 28
Escape Summary and Conclusions
escape of CR from accelerator is treated
self-consistently with
self-generated Alfven waves
resulting CR distribution is obtained in a closed form
strong
self-connement of CR is demonstrated for
PCR B 2 /8π
results are consistent with recent observations of W44 by
Fermi-LAT
escape spectra are roughly DSA-like power laws with breaks (not
peaked at
Emax )
environment (target for
pp
reactions) is equally important to
interpret observed emission from adjacent MC
13 / 28
Test Particle Transport: Basic Equation
CR transport driven by pitch-angle scattering, gyro-phase averaged
z
-along
B;
∂f
∂f
∂
∂f
+ vµ
=
1 − µ2 D (µ)
∂t
∂z
∂µ
∂µ
µ
-cosine of CR pitch angle
need evolution equation for
1
f0 (t, z) ≡ hf i (t, z) ≡
2
Z1
f (µ, t, z) dµ.
−1
Answer deems well known (e.g., Parker 65, Jokipii 66): average and
expand in
1/D :
∂f0
v ∂
=−
∂t
2 ∂z
∂f
1−µ
∂µ
2
,
v ∂f0
∂f
'−
∂µ
2D ∂z
14 / 28
Master Equation
equation for f0
∂f0
∂ ∂f0
v2
=
κ
κ=
∂t
∂z ∂z
4
Critical step:
1 − µ2
D
∂f /∂t is neglected compared to v ∂f /∂z
for Dt & 1 anisotropic part f˜ = f − f0 must
Justication:
−λ1 Dt
decay ∝ e
higher orders
→
suggest retaining
∂f /∂t →∂ 2 f0 /∂t 2
largely
and higher
derivative terms in master equation, e.g. Earl 1973, Litvinenko &
Schlickeiser 2013
arrive at telegrapher's equation:
where
∂f0
∂ ∂f0
∂ 2 f0
−
κ
+τ 2 =0
∂t
∂z ∂z
∂t
2
τ ∼ 1/D , κ ∼ v /D
15 / 28
Pros and Cons of telegraph equation
∂f0
∂t
−
∂ ∂f0
∂z κ ∂z
2
+ τ ∂∂tf20 = 0
Pros
ameliorates the major defect of diusive approximation, innite
propagation velocity (UHECR: Aloisio $ Berezinsky, 2013)
allows for a wave-like transport of CR clouds
keeps the transport description simple
Cons
parabolic equation becomes hyperbolic
no longer an evolution equation
cannot be solved with no recourse to lower-level equation (needed
to compute
∂f0 /∂t
at
t = 0)
n
n
innite sequence of ∂ f0 /∂t -terms will result from iterations
smells of not properly handled (eliminated) secular terms
16 / 28
Looking ahead: Is the telegraph term real? TE:
∂f0
∂t
−
∂ ∂f0
∂z κ ∂z
2
+ τ ∂∂tf20 = 0
In a sense... yes, BUT!
coecient
τ
at
∂ 2 f0 /∂t 2 -term
obtained by simple iteration in
1/D ,
is numerically incorrect
neglected
∼
∼ ∂ 4 f0 /∂z 4 -term
contributes to the same order
∂ 3 f0 /∂z 3 contribute to even lower order unless
D (µ)
is
symmetric and the term zeroes out
being converted into
τ -term
∂ 2 f0 /∂t 2
term the term
∼ ∂ 4 f0 /∂z 4
is subdominant compared to the other two for
alters
Dt & 1
τ
and
may largely be ignored altogether in the long time transport
17 / 28
Basic Equation with magnetic focusing
gyro-phase averaged equation with magnetic focusing
∂f
∂f
∂f
σ
∂
∂f
+ vµ
+v
=
νD (µ) 1 − µ2
1 − µ2
∂t
∂z
2
∂µ
∂µ
∂µ
σ = −B −1 ∂B/∂z
- magnetic mirror inverse scale;
scattering rate, while
ν
-pitch angle
D (µ) ∼ 1
small parameter
ε=
λ
CR m.f .p.
v
= =
1
lν
l
problem scale
l - scale of the problem;
τ ν → τ ; z/l → z ; σl → σ ∼ 1
∂f
∂
∂f
σ
2 ∂f
2 ∂f
1−µ
−
D (µ) 1 − µ
= −ε µ
+
∂t
∂µ
∂µ
∂z
2
∂µ
18 / 28
Formal Expansion in
ε1
f = f0 + εf1 + ε2 f2 + · · · ≡ f0 + f˜
where
hf i = f0 ,
with
1
h·i =
2
Z1
(·) dµ
−1
Master equation
D E
∂
∂f0
µf˜
= −ε
+σ
∂t
∂z
X
∞
ε2 ∂
n−1
2 ∂fn
=
+σ
ε
1−µ
2 ∂z
∂µ
n=1
xed evolutionary structure
similarly to Lorenz's gas (Gurevich 61, Kruskal &Bernstein) f0
depends on slow time
t2 = ε2 t
rather than on
t.
19 / 28
Regular Expansion
slow time in f0 evolution suggests to attribute time derivative term
to higher orders, so term
∝ ∂ 2 f0 /∂t 2
appears, converting the
convection - diusion equation into a telegraph equation.
However,
f˜ does
depend on
t
(as on the fast time)
Ordering should be dierent (f
= f0 + εf1 + · · · ≡ f0 + f˜):
∂fn
∂fn
∂
−
D (µ) 1 − µ2
∂t
∂µ
∂µ
= −µ
∂fn−1
∂fn−1 σ
−
1 − µ2
≡ Φn−1 (t, µ, z)
∂z
2
∂µ
20 / 28
Chapman-Enskog analysis
Solubility condition for f2 :
hΦ1 i =
1
2
(∂/∂z + σ)
1 − µ2 /D ∂f0 /∂z = 0
Too strong restriction...
However, f0 depends on slow time
t2
suggesting multi-time
expansion:
CE:
∂/∂t = (∂/∂t)0 + ε (∂/∂t)1 + . . .
more customary is a hierarchy of independent variables
t → t0 , t1 , . . .
∂
∂
∂
∂
=
+ε
+ ε2
...
∂t
∂t0
∂t1
∂t2
∂fn
∂fn
∂
−
D (µ) 1 − µ2
= Ln−1 [f ] (t0 , . . . , tn ; µ, z)
∂t0 ∂µ
∂µ
n
∂fn−1 X ∂fn−k
∂fn−1 σ
≡ −µ
−
1 − µ2
−
∂z
2
∂µ
∂tk
k=1
21 / 28
Chapman-Enskog analysis
Solution can be written as
fn = f n (t2 , . . . ; µ) + f˜n (t0 , . . . ; µ)
where f˜n and
fn
satisfy, respectively, the following equations:
h i
∂ f˜n
∂ f˜n
∂
−
D (µ) 1 − µ2
= Ln−1 f˜ (t0 , . . . , tn ; µ, z)
∂t0 ∂µ
∂µ
and
−
∂ f¯n
∂
D (µ) 1 − µ2
= Ln−1 f¯ (t2 , . . . , tn ; µ, z)
∂µ
∂µ
The solution for f˜n takes the form
22 / 28
Chapman-Enskog analysis
f˜n =
∞
X
(n)
Ck (t) e −λk t0 ψk (µ)
k=1
can be easily found for any
−
n
using eigenfunctions of diusion operator
∂ψk
∂
D (µ) 1 − µ2
= λk ψk ,
∂µ
∂µ
For D = 1, for example, ψk are
λk = k (k + 1), k = 0, 1, . . . .
constants
(n)
Ck
the Legendre polynomials with
are determined by initial conditions for f˜n
all f˜n exponentially decay in time for
Starting from
t&1
n=0
∂ f¯0
=0
∂t0
23 / 28
Chapman-Enskog analysis
solubility condition for f¯1
∂ f¯0
= 0.
∂t1
1 ∂f0
f¯1 = − W
2 ∂z
where
∂W
1
= , hW i = 0.
∂µ
D
solubility condition for f2 yields nontrivial result for master eq., the
leading term of the
∂f0
1
=
∂t2
4
∂f0 /∂t
expansion in
ε1
*
+
1 − µ2
∂
∂f0
+σ m
, m=
.
∂z
∂z
D
24 / 28
Chapman-Enskog analysis
solubility condition for f3 and f4 , ...
∂f0
∂t3
∂f0
∂t4
(
∂
σ
+
∂z
2
2
∂f0
1 ∂
∂
σ = −
+σ
+
µW 2
4 ∂z
∂z
2
∂z
1 ∂
=
+σ ×
8 ∂z
2 0
W U −m +
∂
+σ
∂z
∂
∂z
*
[m (1 − µ) + U]2
2 D ( 1 − µ2 )
+)
∂f0
∂z
where
Zµ
U≡
−1
1 − µ2
dµ,
D
U 0 = ∂U/∂µ
25 / 28
Chapman-Enskog analysis
process may be continued
1 − µ2 ∂fn /∂µ
ad innitum since terms containing
can be expressed through fn−1 ,
interested in evolving f0 on time scales
contributions of
f˜n
and retain only
master equation up to
f¯n 's:
t2 & 1
or
fn−2 , ...
t & ε− 2
neglect
ε4 :
∂f0
ε2 = ∂z0 m − ε∂z00 µW 2 +
∂t
4
*
"
+#)
2 0
ε2
[m (1 − µ) + U]2
∂f0
00 2
0
W U − m + ∂z ∂z
∂z
2
2D (1 − µ2 )
∂z
here
∂z0 = ∂z + σ
and
∂z00 = ∂z + σ/2.
26 / 28
Is Telegraph Equation recoverable?
The structure of ME (magnetic focusing, asymmetry dropped)
∂t f = ε2 ∂z2 1 − τ ε2 ∂z2 f
or, to the leading order in
ε,
formally
ε2 ∂z2 f ≈ ∂t f , ε4 ∂z4 f ≈ ∂t2 f
∂f0
∂ 2 f0
∂ 2 f0
− ε2 2 + τ 2 = 0
∂t
∂z
∂t
τ -term
belongs to the fast time part of the CE reduction scheme
f , ε → 0:
(1 + τ ∂t )∂t f = O ε2
associated with the anisotropic part of
t&τ
f = const for ε = 0
First solution: transient phenomenon, decays at
Second solution: long time evolution
27 / 28
Conclusions (Test Particle CR transport)
CR transport, constrained by scattering on magnetic irregularities
revisited
Chapman-Enskog approach revealed convective terms arising from
the magnetic focusing eect, only
no telegrapher (second order time derivative) term emerges in
any order of the proper asymptotic expansion
the telegraph
∂ 2 /∂t 2
-term may formally be
back-converted from
the fourth order of expansion, (usually not entertained in the
literature) as well as higher time derivative terms but they do not
play any signicant role in the CR transport within the method's
validity range
28 / 28