From Observations to the Central Engine - exul.ru Fireball Model 1.pdf · Lecture 3: The Fireball Model – C.Fryer (UA/LANL) What have we learned from Afterglows? ... Piran/index.html

From Observations to the Central Engine

Lecture 3: The Fireball Model – C.Fryer (UA/LANL)

What have we learned from Afterglows?

•  GRBs are in Star Forming Galaxies



•  GRBs occur in dusty (star-forming?) regions




•  GRBs trace the stellar distribution





•  Burst move through wind-swept media






•  Bursts include SN-like outbursts






•  Bursts include SN-like outbursts

•  Bursts have few foe Explosion energies and are beamed to a few percent

15 events with z and t_jet

Most SNe are not GRBs (but GRBs may all produce SNe)

VLA/ATCA survey of 34 Type Ib/c SNe to detect off-axis GRBs via radio emission

Berger PhD

•  Most nearby SNe Ib/c do not have

relativistic ejecta •  Two distinct populations •  Ek(GRB)<<1 foe (hydo

collapse) •  <10% are 1998bw-like

Binary Mergers

Collapsars

GR

Bs

as C

osm

olog

y P

robe

s

From Observations to the Central Engine

•  History of the Fireball Model : prediction vs. postdiction, external shock vs. internal shock Fireball model

•  Fireball Shock Basics: relativistic flows, synchrotron radiation, comparing to GRB spectra

•  Optical Bursts •  Jets

Fireball Model: Prediction vs. Postdiction

•  Prediction – (from Latin: prae- before + dicere to say): A foretelling on the basis of observation, experience or scientific reasoning.

•  Postdiction – (from Latin: post- after + dicere to say): To explain an observation after the fact.

•  If your model “predicts” all possible outcomes, it is not a prediction. This merely states that you can not constrain the answer with your current model.

Fireball Model

From IXth International Symposium on Particles, Strings and Cosmology

Tata Institute of

Fundamental Research, Mumbai, Inda

Tsvi Piran

http://theory.tifr.res.in/

pascos/Proceedings/Friday/ Piran/index.html

The Fireball model Assumes a relativistic Outburst and works With a variety of Mechanisms. We rely heavily on A review by Tsvi Piran.

The Fireball Model

compact source

Relativistic Particles Γ>100 or Poynting flux

~ 107 cm Shocks

g - rays

Goodman; Paczynski; Shemi & Piran, Narayan, Paczynski & Piran; Meszaros & Rees

Supernova Remnants (SNRs) - the Newtonian

Analogue •  ~ 10 solar masses are

ejected at ~10,000 km/sec during a supernova explosion.

•  The ejecta is slowed down by the interstellar medium (ISM) emitting x-ray and radio for ~10,000 years.

External Shocks   External shocks are shocks between

the relativistic ejecta and the ISM - just like in SNRs. Can be produced by a single explosion.

Recall, the first GRB model (Colgate 1968) Invoked SN Shocks to power GRBs!

External Shock Predictions For GRBs

•  The burst of gamma-rays should be accompanied by a burst of optical photons (within 1 second of the explosion).

•  Based on these predictions, fast slewing optical telescopes were developed (e.g. ROTSE, LOTUS).

GRB070228 – Optical Burst Seen! But observed much later than expected. This spelled the beginning

of the end for the external shock model! (Although some claimed this as a “prediction”!)

A NO GO THEOREM External shocks cannot produce a variable light

curve!!! (Sari & Piran 97)

Fire

ball

Theo

rists

then

foun

d a

lot

Wro

ng w

ith th

e ex

tern

al s

hock

mod

el!

Time to Revise the Theory

•  Gamma-rays and Afterglow arise from different sources

Birth of the Internal Shock Model

•  dT=R/cg2= d/c £ D/c=T

•  The observed light curve reflects the activity of the “inner engine”. ® Need TWO time scales.

•  To produce internal shocks the source must be active and highly variable over a “long” period.

D=cT

d=cdT

Internal Shocks Shocks between different shells of the ejected relativistic matter

T

dT

•  Internal shocks can convert only a fraction of the kinetic energy to radiation

(Sari and Piran 1997; Mochkovich et. al., 1997; Kobayashi, Piran & Sari 1997).

 It should be followed by additional emission. “It ain't over till it's over” (Yogi Berra)

D=cT

d=cdT

Internal Shocks ⇒Afterglow

1) Compact Source, E>1051erg

2) Relativistic Kinetic Energy

3) Radiation due to Internal shocks = GRBs 4) Afterglow by external shocks The Central Compact Source is Hidden

Gamma-Ray Burst: 4 Stages

Plus burst of optical emission!

Inner Engine

Relativistic Wind

The Internal-External Fireball Model

There are no direct observations of the inner engine. The γ-rays light curve contains the best evidence on the inner engine’s activity.

External Shock

Afterglow

Internal Shocks

γ-rays

THE FIREBALL MODEL PREDICTED GRB

AFTERGLOW (late emission in lower wavelength that will

follow the GRB) Rhodes & Paczynksi, Katz,

Meszaros & Rees, Waxman, Vietri, Sari & Piran

Postdicted

Fireball Model History

•  External Shock model: relativistic ejecta from very energetic explosion shocks with the interstellar medium. Synchrotron radiation in shock produces gamma-rays and optical burst.

•  Optical burst appeared late – theorists move to internal shocks to explain gamma-rays.

Physics of GRB Observations under the Internal Shock Model

•  Basic Model (relativistic shock with synchrotron emission)

I)  Relativistic flows and Sedov analysis II)  Synchrotron Radiation and GRB

spectra

Afterglow Theory

Hydrodynamics: deceleration of the relativistic shell by collision with the surrounding medium (Blandford & McKee 1976) (Meszaros & Rees 1997, Waxman 1997, Sari 1997, Cohen, Piran & Sari 1998) Radiation: synchrotron + IC (?) (Sari, Piran & Narayan 98) Clean, well defined problem.

Few (?) parameters:

E, n, p, εe, εB

initial shell ISM

Sedov Explosion (non-relativistic dimensional analysis)

•  [r]=cm where r is the shock radius •  [t]=s where t is the time •  [E]=g cm2 s-2 where E is the explosion

energy •  [ρ]=g cm-3 where ρ is the density: ρ = ρ0r-ω

Primary Units:

Sedov Continued

[r]=[E/ρ]1/5[t]2/5=[E/ρ0]1/5rω/5t2/5

r (1-ω/5) ~ (E0/ρ0)1/5t2/5

r ~ (E0/ρ0)1/(5-ω)t2/(5-ω)

v = dr/dt = (E0/ρ0)1/(5-ω) 2/(5-ω) t(ω-3)/(5-ω)

= v0(t/t0)(ω-3)/(ω-5)

Sedov for Relativistic Flows •  Γβ = C [ρ1(r)rN+1]-a, where N=0,1,2 for plane,

cylindrical, and spherical shocks, respectively. For decelerating shocks [m(r)=-dlnρ1(r)/dlnr<N+1], a=1/2, for accelerating shocks [m(r)>N+1],a=1/5

•  Γβ=2/(3+N)[E/αρ1(r)c2]1/2r-(N+1)/2, for accelerating shocks (m(r)<N+1), where a=a(N,γ,Γ) is the similarity constant: α(2,4/3, ) =0.237 for an ultra-relativistic shock, while α(2,5/3,1)=0.603 for a non-relativistic shock.

•  Γβ=Γminβmin[ρ1(r1)/ρ1(r) (r1/r)N+1]1/5 for m(r)>N+1 (e.g. constant density)

8

Gnatyk 1985

What do Fireball Theorists use?

•  Γ=const r-g = const t-g/(1+2g)

•  r = const t1/(1+2g)

Where g is (3,3/2) for the radiative, adiabatic regime (assuming constant density). This is different than what we would expect for a constant density solution from Gnatyk but agrees with Gnatyk’s accelerating solution. Fireball Theorists are making the assumption that they are in the ultra-relativistic limit.

Comparison with Observations (Sari, Piran & Narayan 98; Wijers & Galama 98;

Granot, Piran & Sari 98; Panaitescu & Kumar 02)

Radio observations

Radio to X-ray

Synchrotron Radiation – Basics (see, for example, Rybicki & Lightman)

Synchroton radiation – relativistic particles accelerated in a magnetic field

d/dt(γmv)=q/c vxB d/dt(γmc2)=qv.E=0 γ=constant dvpar/dt =0, dvperp/dt=q/(γmc) vperpxB |vpar| = constant |vperp|=constant

Power from Accelerated Particles – Synchrotron Cont.

Relativistic Particles P = 2q2/3c3(a’perp

2+a’par2)

(convert to observer’s frame and set apar=0)

= 2q2/3c3 γ4 (aperp2)

Psynchrotron = 2q2/3c3 γ4 q2B2/(γ2m2c2)vperp

2

B

vperp

vpar

Power from Accelerated Particles – Synchrotron Cont.

Relativistic Particles Psynchrotron = 2q2/3c3 γ4

q2B2/(γ2m2c2)vperp2

= 2/3 r02cβperp

2γ2B2

where r0=e2/mc2

Psynch=4/3σTcβ2γ2B2/(8π) where σT=8πr0

2/3 is the Thompson Cross-Section

B

vperp

vpar

Isotropic velocities <βperp2>=2β2/3

Synchrotron Spectra

P(ω)=(3/2π)1/2q3Bsinα/(mc2) F(ω/ωc)

Where ωc=3γ2qBsinα/

(2mc)

γ-1

Radiation is beamed with an opening angle = γ-1

Spectral shape depends only on the frequency of gyration.

Spectrum from Power-Law Electron Distribution

Assume N(E)dE=CE-pdE, E1<E<E2

Ptot(ω) is the proportional to the integral of F(ω/ωc)γ-pdγ

x=ω/ωc, γ=c1x1/2, dγ=c2x-1/2dx Ptot(ω) = Cω-(p-1)/2

Log ν

Log

l ν

ω-(p-1)/2

Spectrum: Power Law + Self Absorption

Self Absorption occurs when a photon interacts with a charge in magnetic field. This process is related to stimulated emission through Einstein coefficients.

A21=2hν3/c2 B21 Assumes thermodynamic equilibrium.

Kirchhoff’s law related emission to absorbtion in a thermal emiiter: jν=ανBν

•  A21=transition probability per unit time for spontaneous emission

•  B12J=transition probability per unit time for absorption where J is the integral of the flux times the line profile.

•  B21J=transition probablity per unit time for stimulated emission.

Spectrum: Power Law + Self Absorption

αν=hν/(4π)ΣΣ[n(E1)B12-n(E2)B21]φ21(ν)

P(ν,E2)=hνΣA21φ21(ν) = (2hν3/c2)hνΣB21φ21(ν)

αν=c2/(8πhν3)Σ[n(E2-hν)-n(E2)]P(ν,E2)

With some work, we get: Source Function =P(ν)/4παν=Cν5/2

Log ν

Log

l ν

ν-(p-1)/2 ν5/2

Spectrum: Power Law + Self Abs. + Bound Free Absorption

At even lower energies, bound-free absorption can change the slope yet again.

Thermal Emission in an optically thick region.

ανbf=Dν-3

Log ν

Log

l ν

ν-(p-1)/2 ν5/2

ν2

GRBs – 2 additional Constraints with n(γe) =γγe

2qeB/(2πmec): Below some frequency, we are below the minimum electron Lorentz factor: γm=εe(p-2)/(p-1) mp/meγ Above a critical frequency, the electron loses a significant fraction of its energy to radiation: γc=6πmec/(σTγB2t)

Sari, Piran, & Narayan 1998

GRBs – 2 phases γm>γc: All the electrons cool down to γc: fast cooling. γc>γm: Only electrons with γe>γc can cool: slow cooling.


•  R(t)=

•  γ(t)=

Time Evolution

(17Et)1/4/ (4πmpnc)1/4

(4ct/L)1/7L

adiabatic radiative

(17E)1/8/ (1024πmp nc5t3)1/8

(4ct/L)-3/7 adiabatic radiative


Time Evolution νc=2.7x1012εB

-3/2E52-1/2n1

-1td-1/2Hz

νm=5.7x1014εB1/2εe

2E521/2td-3/2Hz

Fν,max=1.1x105εB1/2E52n1

1/2

D28-2µJy

νc=1.3x1013εB-3/2E52

-4/7γ24/7

n1-13/14td-2/7Hz

νm=1.2x1014εB1/2εe

2E524/7γ2

-4/7

n1-1/14td-12/7Hz

Fν,max=4.5x103εB1/2E52

8/7n15/14

γ2-8/7D28

-2td-3/7µJy


The Early Afterglow and the Optical Flash

  The late afterglow observations confirmed relativistic motion.

  But what is the value of g during the GRB phase?

  100 < γ0 = E0/M0 < 105

  “dirty” “clean”

  This could be tested   by early afterglow   observations (Sari &   Piran: Rome, Oct 1998;   Astro-ph/11/1/1999):

A very strong optical flash coinciding with the GRB optical

x-rays

γ-rays

Inner Engine

Relativistic Wind

The Internal-External Fireball Model

External Shock

Afterglow

Internal Shocks

γ-rays

OPTICAL FLASH

Optical Flash Revisited

•  Do internal shocks also predict prompt bursts?

Predictions of the Optical

Flash

Depending upon the Exact details of the Explosion (width of Shock, structure of Internal shocks in Addition to all of the Many Fireball free Parameters)

Sari & Piran 1999

The Parameter space allowed for Optical Flashes

The amount of energy In the optical flash Varies over many Orders of magnitude. Prediction – depending Upon the model, you will/will not see the flash.

Sari & Piran 1999

GRB 990123 - The Prompt Optical Flash

ROTSE’s detection of a 9th magnitude prompt optical flash.

Moving away from Spherical Symmetry

•  GRB engines predict jets, not spherical explosions

I)  What are the consequences on the fireball model

II)  Do these consequences affect the observations

GRB Energies

  Etot - The total energy   Eγ - Observer γ-ray energy

γγεEEtot

1=

The Resolution of the Energy Crisis

  Etot - The total energy   Eγ iso - Observed (iostropic) γ-ray energy

isotot EE γγε1−=

isotot EEE γγγγ

θεε

2

211 −− ==

Beaming:  Eγ - Actual γ-ray energy

The two most powerful BeppoSAX bursts are jets (Sari, Piran & Halpern; 1999).

JETS and BEAMING

Jets with an opening angle θ expand forwards until γ= θ-1 and then expand sideways rapidly lowering quickly the observed flux (Piran, 1995; Rhoads, 1997; Wijers et al, 1997; Panaitescu & Meszaros 1998).

γ = θ -1

Particles spreads sideways quickly

Radiation is “beamed” into a large cone

Particles remain within initial cone

Radiation is “beamed” into a narrow cone

Jets – Predicted by the Collapsar Model

νm=C t-2, jet

t-3/2, spherical

νc=C const, jet

t-1/2, spherical

Fν,max=C t-1, jet

const, spherical

νa=C t-1/5, jet

const, spherical

Sari, Piran, & Halpern 1999

Fν<νa=C const, jet

t1/2, spherical

Jets – Predicted by the Collapsar Model

Fνa<ν<νm=C t-1/3, jet

t1/2, spherical

Fνm<ν<νc=C t-p, jet

t-3(p-1)/4, spherical

Fνa<ν<νm=C t-p, jet

t-3p/4+1/2, spherical

Sari, Piran, & Halpern 1999

A Sharp (Not Achromatic) Break!

* synch emission does not include the effects of cooling.

Fireball Model - Summary •  Thusfar, the fireball model has made very few

true predictions •  Current Favorite form of the Fireball model uses

internal and external shocks.

Inner Engine

Relativistic Wind

External Shock

Afterglow

Internal Shocks

γ-rays

Fireball Model - Summary •  Basic Fireball model

simple – Relativistic shocks with synchrotron + inverse Compton emission

Fireball Model - Summary •  Basic Fireball model

simple – Relativistic shocks with synchrotron + inverse Compton emission

•  Internal Shocks produce optical burst and gamma-rays, External Shocks produce afterglow

•  Jets alter the spectra in an observable way.

References •  Blast Wave Overview: P. Meszaros, Annual

Review of Astronomy and Astrophysics, 2002, 40, 137

•  Relativistic Blast Waves for GRBs: Tan, Matzner & McKee 2001, ApJ, 551, 946

•  GRB Spectra and Light Curves: Sari, Piran, & Narayan 1999, ApJ, 497, L17

•  GRB Spectra and Light Curves: E. Waxman 197, ApJ, 485, L5

•  GRB Jet Spectra and Light Curves: Sari, Piran & Halpern 1999, ApJ, 519, L17

Tomorrow – Past and Current Models of the Central Engine

•  Energy Sources in Astrophysics •  Arguments for compact objects •  Black hole models (Collapsar and compact

binary mergers) versus magnetar (highly magnetized neutron star) models.

Documents

From Observations to the Central Engine - exul.ru Fireball Model 1.pdf · Lecture 3: The Fireball Model – C.Fryer (UA/LANL) What have we learned from Afterglows? ... Piran/index.html