grb afterglow spectra daniel perley astro 250 19 september* 2005 * international talk like a pirate...

GRB Afterglow Spectra

Daniel Perley

Astro 250

19 September* 2005

* International Talk Like a Pirate Day

Background

Daniel Perley 19 September 2005GRB Afterglow Spectra

The GRB Standard Model

ISM

Shocked Gas

Earth

SH

OC

KS

HO

CK

Background GRB Standard Model Relativistic Shock

Energy Deposition Proton Energy Electron Energy Electron Distribution Magnetic Energy

Uncooled Synchrotron Emission Mechanisms Relativistic Cyclotron Synchrotron Beaming 1e- Spectrum Multi-e- Spectrum

Cooled Synchrotron Cooling Time 1e- Spectrum Cooling Regimes Fast Cooling Slow Cooling Cooling Comparison Self-Absorption Complete Comparison

Spectral Observation Observation vs. Theory Observation Parameters Intervening ISM Effects

Background


Relativistic Shock

SH

OC

KS

HO

CK

ISM

Γ

number density

no

energy density

Eo = no mp c2

energy per particle

Eo/no = mp c2

From Brian’s lecture…

n′ = 4 no

E′ = 4 2no mp

c2

E′/n′ = mp c2

= Γ √2

>Compression< by 4

Energy Increase by factor

Deceleration by factor √ 2







Energy Deposition

Where does the energy go?

energy per particle

Eo/no = mp c2E′/n′ = mp c2 Energy Increase

by factor

• Protons

• Electrons

• Magnetic field

• Other particles?

Ep = εp E′

Ee = εe E′

B = εB E′

Energy DepositionBackground GRB Standard Model Relativistic Shock





Energy Deposition


Proton/Electron Energy

SH

OC

KS

HO

CK

ISMShocked Gas

Γ

e

Extreme (relativistic) ‘temperature’ of shocked gas described by p, e

Bulk motion of shocked gas relative to observer

Particle energy deposited in random motions.







Proton Energy

Not particularly interesting on its own.

Protons necessarily drag electrons with them at the same bulk velocity.

Share energy with electrons: electron factors necessarily much higher.







Electron Energy

Faster-moving electrons will radiate more efficiently by all important processes.

e







Electron Energy Distribution

Q: How is electron energy distributed?

A:

…

…

…

?

Hypothesis: Power-law? (Seen in SNe, NR shocks)







Log

N

Log

N α -p

N α [Complicated]

Model as power-law:

Energy Deposition Electron Energy DistributionBackground GRB Standard Model Relativistic Shock







Simplify: cut-off power law at minimum energy

Log

N

Log

N α -

p

N α [Complicated]

m

Minimum energy








Mimimum energy determined by total energy density:

n = ∫ Ne de Ee = me c2 ∫ e Ne

de

= C m1-p = me c2 C m

2-p

Infinite if p<2

N

e

e-p

m

n

eN

e

m

e1-p

E

11-p

C = (1-p) mp-1 n

12-p

= me c2 m n1-p2-p

m p-2p-1

Ee

n me c2

p-2p-1

mp

me εe ≈ 610 εe







Magnetic Energy

Strong post-shock magnetic field expected from equipartition.

Generation mechanism unknown/complicated – various plasma effects

B2

8π= εB E′

= εB 4 2no mp c2

= 32π εB 2no mp

c2

B2

B = 32π εB no mp c

≈ (0.4 gauss) εB1/2 ( )1/2

no

cm-3

B







Emission Mechanisms

How does it cool?

Bremsstrahlung

P α e3/2 n2

Inverse Compton

P = σTcβ2e2Uph

SynchrotronP = σTcβ2e

2UB

Uncooled Synchrotron






43

43


Relativistic Cyclotron

Relativistic modification to cyclotron frequency:

ωcyc = e B

m c

Most emission is not at this frequency.








Synchrotron Beaming

Emission is highly pulsed – we see emission for only 1/2 of total emission time.

ωcyc = e B

m c

E

t

1/ωcyc

1/2ωcyc

- One factor of from beaming angle

- Additional factor of from "Doppler" boost 1/

1/







1e- Synchrotron Spectrum

E

t

1/ωcyc

1/2ωcyc

= E

t

δ(t-n/ωcyc)

=








E

t

= E

t

δ(ω-nωcyc)^

^

Fourier transformed:

ωcyc

2ωcyc

2ωcyc

2ωcyc

2ωcyc









log

P

1/3 e-/ωcyc 1/2

pk log

More precisely…

Shocked frame:

α e2

α e2

α const

Total Power: P = σTcβ2e2UB

Peak Freq.: pk ≈ e2ωcyc / 2π

Peak Power: Ppk ≈ P / pk

43









log

P

log

1/3 e-/ωcyc 1/2

pk pk

Shocked frame:

α e2

α e2

α const




43









log

P

log

1/3 e-/ωcyc 1/2

pk pk




Observer frame:

α e22

α e2

α

43








Uncooled Multi-e- Spectrum

log

P

log

1/3

pk

exp

Material contains many electrons at different velocities (e) – true spectrum is a combination of individual spectra, according to electron energy distribution.

log

N

log e

-p

m

Electron distributionElectron spectrum

Uncooled Sychrotron








Can just do a weighted sum (convolution) – but need to convert x-axis from e to pk.

log

N

log e

-p

m

Electron distribution

From before,

pk α e2

log

N

log pkm

e- distribution:

Nα e-p

Solve:

N = N

α e-p e

-1

α -(p+1)/2

dd

dd α e

-(p-1)/2

Sign error??









Electron distribution

log

N

log pkm

-(p-1)/2

log

P

log

1/3

pk

exp

Electron spectrum

Total Spectrum








log

N

log pkm

-(p-1)/2

log

P

log

1/3

pk

exp

log

P

1/3


-(p-1)/2

m









"Broken" Power law:

• Below m, emission dominated by low- e-

• Above m, emission from electrons with peak() = m

log

P

1/3 -(p-1)/2

m log pk







Cooled Synchrotron Characteristic Cooling Time


This analysis is too simplistic for GRBs.

Calculate characteristic cooling time:

log

P

1/3 -(p-1)/2

m

tcool = E / P = mec / σTcβ22UB

≈ 4 × 10-3 s ( )-2 -1

43

Bgauss

Potentially much shorter than time since GRB (shock passage)

log pk






Cooled Synchrotron


Cooling e- Spectrum

If an electron's energy changes significantly over the time since the energy injection, use an "averaged" spectrum for that electron.

e = Initial electron energy (at injection)

c ≡ Final electron energy (after cooling) ≈ Energy of the highest-e- that hasn't cooled

Determined by observational timescale: tobs = cmec / σTcβ2c

2UB

c =

43

6π mecσTB2tobs






Cooled Synchrotron


Cooling e- Spectrum

Electron radiates as it cools, with a simple synchrotron spectrum corresponding to the instantaneous energy i .

e = Initial electron energy

c ≡ Final electron energy

log

P

log

1/3

pk

exp

Instantaneous spectrum

i







Cooling e- Spectrum

Peak power radiated at each i is the same:

log

P

log

1/3

pk(i)

exp

Instantaneous spectrum

e = Initial electron energy

c ≡ Final electron energy

i P(i) = const

log

P

ec

Electron evolution

log i

Cooled Synchrotron







Cooling e- Spectrum

From before,

pk α e2

Power distribution:

P= const

Solve:

P = P

= -1

= -1/2

dd

dd α e

log

P

ec

Another convolution - need to transform e to pk.

log

P

c

Electron evolution

const

-1/2

e log i

Cooled Synchrotron







Cooling e- Spectrum

log

P

ec

log

P

log

1/3

pk

exp

Instantaneous spectrum Electron evolution

-1/2

Cooled Synchrotron






Cooling e- Spectrum

log

P

ec

log

P

log

1/3

pk

exp

Instantaneous spectrum Electron evolution

log

P

1/3 -1/2


-1/2

Cooled Synchrotron






Cooling e- Spectrum


log

P

1/3-1/2

c e

Broken power law:

• > e : Exponential cut-off (model as no emission)

• c < < e : Instantaneous emission when electron passed through appropriate

• < c : Post-cooling emission

log

Cooled Synchrotron






Cooling e- Spectrum


log

P

1/3-1/2

m e

Higher initial energy simply extends the curve to higher frequencies.

log

Cooled Synchrotron







Cooling Regimes

Two possibilities for multi-electron spectra:

log

N

log e

-p

m

log

N

log e

-p

m

c < m

c

c

c > m

ALL electrons will cool on given timescale :Fast cooling

SOME electrons will cool on given timescale :Slow cooling

Cooled Synchrotron







Fast Cooling

log

N

log e

-p

m

c < m

c

ALL electrons will cool on given timescale :Fast cooling

Cooled Synchrotron






Fast Cooling

log

N

log e

-(p-1)/2

mc

log

P

1/3-1/2

Cooled synchrotron spectrum Electron distribution

Sum for multi-e- using the new spectrum:

-p/2

c e

Fractionof N >

log

P

1/3 -1/2

log


c m

-1/2

-(p-2)/2 ??

Cooled Synchrotron







Fast Cooling

log

P

1/3 -1/2

c m

-p/2

Broken power law:

• > m : Emission from electrons with e > , during passage through appropriate

• c < < m : Emission from all electrons, during passage through appropriate

• < c : Emission from all electrons at all times

log

Cooled Synchrotron







Slow Cooling

log

N

log e

-p

m c

c > m

SOME electrons will cool on given timescale :Slow cooling

Cooled Synchrotron







Slow Cooling

Fast-cooling electrons have fast-cooling spectrum, but with effective m → c (no -1/2 segment)

log

N

log e

-p

m c

-p/2

1/3

c

log

P

log

Cooled Synchrotron







Slow Cooling

Non-cooling electrons have an uncooled-population spectrum, but cut off at c.

log

N

log e

-p

m c1/3

-(p-1)/2

m c log

log

P

Cooled Synchrotron







Slow Cooling

By their powers combined…

log

N

log e

-p

m c1/3

m c

-(p-1)/2

-p/2

1/3

1/3-(p-1)/2

-p/2 log

P

log

Cooled Synchrotron







Slow Cooling

log

P

1/3 -(p-1)/2

m c

-p/2

Broken power law:

• > c : Emission from cooling electrons with e > during passage through appropriate

• m < < c : Emission from slow electrons with initial (constant) energy

• < m : Emission from slow electrons with min. m

log

Cooled Synchrotron







Cooling Comparison

log

P

1/3 -(p-1)/2

m c

-p/2

log

P

1/3 -1/2

c m

-p/2

Fast cooling

Slow cooling

log

log

Cooled Synchrotron






Cooled Synchrotron


Synchrotron Self-Absorption

Photon can be re-absorbed to excite an electron in a magnetic field (inverse of synchrotron emission.)

Synchrotron emission/absorption will be in equilibrium below a certain frequency a: below this point the shocked gas is optically thick and will radiate as a blackbody (P α 2)

1/3

log

P

log

2

a






Synchrotron Summary


Complete Comparison

log

P

1/3 -(p-1)/2

m c

-p/2

log

P

1/3 -1/2

c m

-p/2

Fast cooling

Slow cooling

2

2

log

log

a

a






Observing


Theory vs. Observations

GRB970508 – Galama et al. 1998 tburst = 12.1 days

-0.6

0.44>1.1

-1.12






Subject


Observable Parameters

An instantaneous spectrum gives several key pieces of information:

a

c

m

p Fpk

z

εe εB no

E'






Subject


Intervening ISM Effects

Cosmological redshift will not affect power-law - all radiation scaled down by (1+z)

Will see deviation from power-law in some frequency ranges in some cases:

Galactic extinction (can be calculated/removed)

Host extinction (similar to Galactic, but at higher frequencies, and cannot be estimated independently of GRB)

Hydrogen absorption features (associated with high-z)






grb afterglow spectra daniel perley astro 250 19 september* 2005 * international talk like a pirate...

Documents

p e e e

log e e

e c t c

energy increase

t cool e

t critical e

synchrotron cooling

particle e o n o