Flare&Loops& - Montana

Flare Loops

1d Gas dynamics & 0-‐d models Lecture 9

Feb. 15, 2017

u A

s=0 s=L

E = −u×B = 0

B ||u = u b̂b ^ Flow inside a staBc

tube: length coord s X-‐secBon A(s)

∂ρ∂t

= −∇⋅ (ρu) ∂ρ∂t

= −1A∂∂s(Aρu)

ρ∂∂t+u ⋅∇

$

%&

'

()u = −∇p+

1cJ×B+ ρg+∇⋅ µ∇u( )

ρ∂u∂t+u∂u

∂s"

#$

%

&'= −

∂p∂s+ ρg|| +

∂∂s

43 µ

∂u∂s

"

#$

%

&'component b̂

u A b ^

ρcv∂T∂t

+u ⋅∇T$

%&

'

()= −p ∇⋅u( )+µ ∇u 2

−∇⋅Fc − Lrad

32 p = ρcvT

Thermal energy density

ρcv∂T∂t

+u∂T∂s

"

#$

%

&'= −

pA∂∂s

Au( )+ 43 µ

∂u∂s

2

−1A∂∂s

AFc( )− Lrad

Energy equaBon

conducBve flux radiaBve loss

viscous heat p dV work

s=0 s=L

T(x)

x

nv

nv parBcle flux: same both direcBons

32 kbT+

32 kbT−

(3/2) nv kbT+

(3/2) nv kbT-‐

energy flux: greater from leU

Fc = 32 nvkb T+ −T−( )

Conduc*ve flux T− = T (x + 1

2 ℓmfp )

T+ = T (x − 12 ℓmfp )

ℓmfp

Fc ≈ − 32 nvkbℓmfp

∂T∂x

IF gradient is shallow or m.f.p. is small

!Fc = −κ∇T

= κ

Fourier’s law – classical heat flux

Fc

T(x)

x

T− = T (x + 12 ℓmfp )

T+ = T (x − 12 ℓmfp )

ℓmfp IF gradient is shallow or m.f.p. is small

= κ

κ = 32 kbvthnℓmfp = 3

2kbvthσ sc

σ sc ~q4

m2vth4

κ ~ kbm2vth

5

q4~ kb

7/2

m1/2q4T 5/2

e-‐s : smallest m most heat flux

κ = κ0 T5/2 [erg cm-‐2 s-‐1 K-‐1]

κ0 ≃ 10-‐6 erg cm-‐2 s-‐1 K-‐7/2

Conduc*ve flux Fc = 3

2 nvkb T+ −T−( )


∂T∂xRutherford sca\ering

T(x)

x

nv

nv parBcle flux: same both direcBons

32 kbT+

32 kbT−

(3/2) nv kbT+

(3/2) nv kbT-‐

energy flux: greater from leU

T− = T (x + 12 ℓmfp )

T+ = T (x − 12 ℓmfp )

ℓmfp

Fc < 32 nvth kbT+ = 3

2 nekb3/2

me1/2 Te

3/2

IF NOT

Free-‐streaming heat flux:

upper bound

=Ffs

Fc

Conduc*ve flux Fc = 3

2 nvkb T+ −T−( )

ℓmfp = 2rg


∂T∂x= κ

κ⊥ = 32 kbnveff ℓmfp = 6kbnrg

2νcol = 6kbnvth2

Ωc2 νcol

vth

vth rg =

vthΩc

= 6kbvthnvthνcol

!

"#

$

%&νcol

Ωc

!

"#

$

%&

2

κ⊥ = 6kbvthσ sc

νcol

Ωc

#

$%

&

'(

2

= 4κ ||νcol

Ωc

#

$%

&

'(

2

1/σsc

~10-‐14

ConducBve flux ⊥ to B

B

Heat flows ONLY || to B

Constraints on heat flux: !Q = −∇⋅

"Fc

dSdt

=!QTd3x

V∫ = −

∇⋅"FcT

d3xV∫ = −

"FcT⋅d#a −

∂V$∫

"Fc ⋅∇TT 2

V∫ d3x

Volumetric heaBng

Entropy change

2nd law of thermo: !Fc ⋅∇T ≤ 0

heat exchange across bndry

∇T ⋅Fc = − ∇T( ) ⋅

κ ⋅ ∇T( )Classical flux:

κ must have no nega*ve e-‐values

Z(+s) e-‐ vi

vf

hν=ΔE

•  e-‐ collides with element Z ionized +s •  e-‐ loses ΔE = me(vi2 -‐ vf2)/2 •  excites ion to state @ ΔE •  ion de-‐excites, emieng γ w/ hν=ΔE •  γ escapes to ∞ carrying away ΔE

opBcally thin radiaBon

γ ΔE collisional excitaBon

spontaneous emission

Energy loss by op*cally thin radia*on

•  excitaBon cross secBon: σZ,s(ve)

•  average γ energy εZ,s(ve)

•  rate of energy loss per e-‐


Ee = nZ ,sveσ Z ,s (ve )εZ ,s (ve )

nZ ,snee−meve

2 /2kbTe

(2πkbTe /me )3/2 veσ Z ,s (ve )εZ ,s (ve ) d

3ve∫

Assump*on I: e-‐s have Maxwellian dist’n w/ temp Te

RZ,s(Te) [erg cm3 s-‐1] volumetric energy loss rate [ erg cm-‐3 s-‐1] to Z(+s)

Z(+s) e-‐ vi

ΔE γ

LZ,s =

collision frequency

# density: nZ,s

sum over all transiBon!


LX,s =

Assump*on II: ionizaBon equilibrium: fracBon FZ,s(Te) of element Z ionized

to +s nZ,s = FZ,s(Te) nZ

X(+s) e-‐ ve

ΔE γ

RX,s(Te) [erg cm3 s-‐1] volumetric energy loss rate [ erg cm-‐3 s-‐1] to X(+s)

nX,snee−meve

2 /2kbTe

(2πkbTe /me )3/2 veσ X,s (ve )εX,s (ve ) d

3ve∫


FFe,s(T)

s=3

s=16




Z(+s) e-‐ ve

ΔE γ

Assump*on III: known abundances nZ = AZ nH

ne = nHAZ sFZ ,s ≈ nH 1+ 2AHe( )s∑

Z∑ ≈1.17nHH & He fully ionized:


nZ ,snee−meve

2 /2kbTe


3ve∫


LZ,s =




Z(+s) e-‐ ve

ΔE γ



Z∑ ≈1.17nHH & He fully ionized:


nZ ,snee−meve

2 /2kbTe


3ve∫


LZ,s =

volumetric energy loss rate to Z(+s)



Z(+s) e-‐ ve

ΔE γ


H & He fully ionized:

LZ ,s = nenZ ,sRZ ,s (Te ) =ne2

1.17AZFZ ,s (Te )RZ ,s (Te )

L = ne2 AZ

1.17FZ ,s (Te )RZ ,s (Te )

s∑

Z∑ = ne

2 Λ(Te )volumetric energy loss rate [ erg cm-‐3 s-‐1] RadiaBve loss funcBon Λ(Te)



Z∑ ≈1.17nH

Emission measure

e-‐

ΔEij γ λij = hc/ΔEij j

i

Corona & TR: mostly opBcally thin •  Collisional excitaBon •  Spontaneous emission

Iλ = ne2 (x)Gλ[Te(x)]∫ d3x ≈Gλ (Te ) ne

2 (x)∫ d3x

Emission measure: EM [ cm-‐3 ]

Regions of highest density emit most – ne ↑ × 10 I ~ EM ↑ × 100

a sidebar

The 1d flare loop

u A b ^

∂ρ∂t

= −1A∂∂s(Aρu)

ρ∂u∂t+u∂u

∂s"

#$

%

&'= −

∂p∂s+ ρg|| +

∂∂s

43 µ

∂u∂s

"

#$

%

&'

ρ(s,t), u(s,t) & T(s,t)

Fixed A(s) & g||(s)

p = kbmρT cv =

32kbm

Source of flare energy

S=L S=0

ρcv∂T∂t

+u∂T∂s

"

#$

%

&'= −

pA∂∂s

Au( )+ 43 µ

∂u∂s

2

+1A∂∂s

Aκ ∂T∂s

)

*+,

-.− ne

2Λ(T )+ h

Integral quanBBes

Str

S=L

M = ρ(s, t)A(s)dsstr

L/2

∫ L/2

S=0

dMdt

= ρ(str )u(str )A(str )

Etot = 12 ρu

2 + 32 p+ ρΨ"# $%Ads

str

L/2

∫

dEtot

dt≈ − ne

2Λ(T )Adsstr

L/2

∫ + 12 u

3Atr+ 52 puA tr

−κ∂T∂s

Atr

+ hAdsstr

L/2

∫enthalpy flux

radiaBve loss conducBve flux

flare heat

Their evoluBon: 0d models

M = ρ(s, t)A(s)dsstr

L/2

∫ ≡ L2 Ampne

dnedt

= 2L ne,trutr

Etot = 12 ρu

2 + 32 p!" #$Ads ≡ 3L

4 A pstr

L/2

∫ enthalpy flux

radiaBve loss conducBve flux

Ffl

ddt

32 p( ) ≈ − 2

L ne2Λ(T )ds

str

L/2

∫ + 5L ptr utr − 2

Lκ∂T∂s tr

+ 2L hdsstr

L/2

∫

≈ −ne2Λ(T )

≈8κ07L2

T 7/2

ddt

32 p( ) ≈ −ne2 Λ(T )+ 5

L ptr utr −8κ07L2

T 7/2 + 2L Ffl

What will be the value of Ffl?

vA vA

B

Ffl = ςB2

8πvA =

ς2 ρ

B2

4π!

"#

$

%&

3/2

B = 100 G ρ = 10-‐15 g cm-‐3

⇒ Ffl = ς 3×1011ergs−1 cm−2

Compare to upward fluxes: -‐ Steady AR:

F ~ 107 erg cm-‐2 s-‐1 -‐  Luminosity (white light)

L⊙/4πR⊙2 = 6 × 1010

τ rad ≡32 p

ne2Λ(T )

=3ne kbTne2Λ(T )

~ 3kb1.2×10−19

T 3/2

ne

ddt

32 p( ) ≈ −ne2 Λ(T )+ 5


T 7/2 + 2L Ffl

τ cond ≡32 p

8κ07L2T 7/2

=21kb8×10−6

neL2

T 5/2

1τ cool

=1τ rad

+1

τ cond

Cooling rate:

p = 2nekbT

Fully-‐ionized H plasma

τ rad ~3kb

1.2×10−19T 3/2

ne

τ cond =21kb8×10−6

neL2

T 5/2

ddt

32 p( ) ≈ −ne2 Λ(T )+ 5


T 7/2 + 2L Ffl

Tpk =7LFfl4κ0

!

"#

$

%&

2/7

Tpk=50 MK

F fl = 101

1 erg cm

-‐2 s-‐

1

heaB

ng

EvaporaBon*: dnedt

= 2L ne,trutr

ddt

32 p( ) ≈ −ne2 Λ(T )+ 5


T 7/2 + 2L Ffl

mass flux increases ne

enthalpy flux

cond

ucBve flu

x

E cor = (3

/2) p

L

Etr

•  Heat flows into TR – conducBve flux: Fc •  TR expands into corona – enthalpy flux: Fe •  No change in Etr & no losses (i.e. rad)  No change in Ecor   p ~ ne × T = const.

á la AnBochos & Sturrock 1978; Cargill et al. 1995 Fe

Fc

Ecor

* Historical term based on analogy. Not genuine evaporaBon

ddt

32 p( ) ≈ −ne2 Λ(T )+ 5


T 7/2 + 2L Ffl

≃ 0 p ~ ne × T ≃ const.

Tpk=50 MK

F fl = 101

1 erg cm

-‐2 s-‐

1

heaB

ng

ddt

32 p( ) ≈ −ne2 Λ(T )+ 5


T 7/2 + 2L Ffl

draining < 0

Tpk=50 MK

F fl = 101

1 erg cm

-‐2 s-‐

1

heaB

ng

EvaporaBon: dnedt

= 2L utrne,tr ≈ 2

L utrne

ddt

32 p( ) ≈ −ne2 Λ(T )+ 5


T 7/2 + 2L Ffl

AnBochos & Sturrock 1978; Cargill et al. 1995

≃ 0 p ~ ne × T ≃ const.

5L ptr utr = 5

L putr =8κ07L2

T 7/2 =32 pτ c,0

TT0

!

"#

$

%&

7/2

=32 pτ c,0

nene,0

!

"##

$

%&&

−7/2

2L utr = 3

51τ c,0

nene,0

!

"##

$

%&&

−7/2

ddt

nene,0

!

"##

$

%&&=

35

τ c,0

nene,0

!

"##

$

%&&

−5/2

ne(t) = ne,0 2110t − t0τ c,0

+1"

#$$

%

&''

2/7

T (t) = T0 2110t − t0τ c,0

+1"

#$$

%

&''

−2/7

RadiaBve cooling ddt

32 p( ) ≈ −ne2 Λ(T )+ 5


T 7/2 + 2L Ffl

Cargill et al. 1995

ne(t) = ne,1 1− 23t − t1τ r,1

"

#$$

%

&''

1/2

T ∝ne2 p = 2ne kbT = p1

nene,1

!

"##

$

%&&

3

3 nene,1

!

"##

$

%&&

2ddt

nene,1

!

"##

$

%&&= −

1τ r,1

nene,1

!

"##

$

%&& T (t) = T1 1− 2

3t − t1τ r,1

"

#$$

%

&''

I. draining (empirical)

heaB

ng

radiaBve. evap. heaBng

t0 t1

t0

t1

τ rad = τ r,1nene,1

!

"##

$

%&&

−1TT1

!

"#

$

%&

3/2

= τ r,1nene,1

!

"##

$

%&&

2

RadiaBve cooling ddt

32 p( ) ≈ −ne2 Λ(T )+ 5


T 7/2 + 2L Ffl

ne(t) = ne,1 1+ 13t − t1τ r,1

"

#$$

%

&''

4

T ∝ne1/2 p = 2ne kbT = p1

TT1

!

"#

$

%&

3

3 TT!

"#

$

%&

2ddt

TT!

"#

$

%&= −

2τ r,1

TT!

"#

$

%&

7/2

T (t) = T1 1+ 13t − t1τ r,1

"

#$$

%

&''

−2

II. mechanical equilibrium

heaB

ng

radiaBve. evap. heaBng

t0 t1

t0

t1

τ rad = τ r,1nene,1

!

"##

$

%&&

−1TT1

!

"#

$

%&

3/2

= τ r,1TT1

!

"#

$

%&

−1/2

remain equal

EvaporaBon again: ddt

32 p( ) ≈ −ne2 Λ(T )+ 5


T 7/2 + 2L Ffl

enthalpy flux

cond

ucBve flu

x

Etr

•  Heat flows into TR – conducBve flux: Fc •  TR expands into corona – enthalpy flux: Fe •  TR radiates Rtr = c1Rco •  No change in Etr

è Fe = Fc – Rtr = Fc – c1Rco

EBTEL – Klimchuk et al. 2008

Fe

Fc

Rco

ddt

32 p( ) ≈ −(1+ c1)ne2 Λ(T )+ 2

L Ffl

TR ra

diaB

on

2L utr =

25p

Fe =25p

8κ07L2

T 7/2 − c1ne2 Λ(T )

#

$%

&

'(

dnedt

= 2L utrne =

15kbT

8κ07L2

T 7/2 − c1ne2 Λ(T )

#

$%

&

'(

2 eqns. for unknowns p(t) & ne(t) — T = p/(2kbne)

EBTEL

heaB

ng

Cargill et al. 1995

τ rad ~3kb

1.2×10−19T 3/2

ne

τ cond =21kb8×10−6

neL2

T 5/2

τcond = τrad

τ radτ cond

=8×10−6

7×1.2×10−19T 2

ne L=3×106

(2kb )2p2

ne3L

= 4×1037 p2

ne3L

ne = 3×1012 p2/3

L1/3

ne,max = 2.6×1012 E 2/3

V 2/3L1/3

Warren & AnBochos 2004

p = 2E3V

ne,max

Tem,pk All flare energy, E, kept through evap phase

Tem,pk = 930ELV

!

"#

$

%&1/3

Tpk =7LFfl4κ0

!

"#

$

%&

2/7

= 5×107K

L = 5 × 109 cm = 50 Mm Ffl = 1011 erg/s/cm2

EA= 2 Ffl dt∫ = 4×1011 ergcm-2

Tem,pk = 930EA!

"#

$

%&1/3

= 7×106K

ne,max = 2.6×1012 (E / A)2/3

L= 3×1010cm-3

ne,max = 2.6×1012 (E / A)2/3

L= 8×1010cm-3

Tpk =7LFfl4κ0

!

"#

$

%&

2/7

= 5×107K

Ffl = 1011 erg/s/cm2

EA= 2 Ffl dt∫ = 2×1012 ergcm-2

Tem,pk = 930EA!

"#

$

%&1/3

=1.2×107K

L = 5 × 109 cm = 50 Mm

ne,max = 2.6×1012 E 2/3

V 2/3L1/3

Warren & AnBochos 2004

Tem,pk = 930ELV

!

"#

$

%&1/3

IF flare were a single loop

max(EM ) =V ne,max2 = 7×1024 E 4/3

V 1/3L2/3= 7×1049cm3 E30

4/3

V271/3L9

2/3

Tem,pk = 930ELV

!

"#

$

%&1/3

= 9×106K E30L9V27

!

"#

$

%&

1/3

F1−8 ≈10−63 Wm2 EM T 5/4 = 4×10−5 W

m2 ⋅E307/4

L91/4V27

3/4GOES peak:

E30 = E/1030 ergs

M4 flare

~ E7/4

BUT a real flare is built from many diff. loops – each evolving independently

1000 loops: •  Ffl = 1011 erg/s/cm2

•  L = 5 × 109 cm •  Ei = 2 × 1028 erg

max(EM ) = 7×1049cm3 E304/3

V271/3L9

2/3 = 0.03×1049cm3

×6

max(EM ) = 7×1049cm3 E304/3

V271/3L9

2/3 = 30×1049cm3

Each loop:

Flare as 1 loop:

Example model:

heated in groups of n

n(t) has decaying envelope

n=26 Hori et al. 1997, Warren et al. 2002, Reeves & Warren 2002, Warren 2006, Qiu et al. 2012, 2013, …

E = 2 × 1031 erg

GOES

F1−8 ≈ 4×10−5 Wm2 ⋅

E307/4

L91/4V27

3/4 = 3×10−4 Wm2

Flare as 1 loop:

flare as 1 loop

1000 loop composite

T =1EM

EMii∑ Ti

T derived from GOES

flare as 1 loop

max(EM) = 2 × 1050 cm3

Summary •  EvoluBon following energy release – governed by 1d gas dynamics

•  Single loop experiences 3 phases: –  HeaBng ( T ↑ @ ~ const. ne ) –  EvaporaBon ( T ↓ , ne ↑ , p ~ const. or ↓ ) –  RadiaBve cooling (T ↓ , ne ↓ )

•  Real flare may be composite of many loops

Next *me:

EvoluBon in one dimension

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