Benvenuti al Mera-TeV! 4-5- -6 Ottobre 2011 6 Ottobre 2011 Sala - PowerPoint PPT Presentation

Total losses Total losses 2e 2e 2 2e 2 2e P e = P’ = P’ e = 3c 3 a’ a’ 2 = 3c 3 (a’ (a’ 2 + a’ + a’ 2 ) 3c 3c = P’ e = 2e 2e 2 g 2 P e = P’ a 2 = g 2 a a’ a’ = a 3c 3c 3 ) = 2e 2e 4 a ’ || B 2 g 2 2 b 2 2 sin sin 2 q = = 0 P S ( q ) = || 3m 2 c 3 3m e v B B sin sin q a = a = g mc mc g 2 2 b 2 2 sin sin 2 q P S ( q ) = ) = 2 s T cU cU B <P <P S > = > = 4 If pitch angles are If pitch angles are g 2 2 b 2 4 s T cU cU B isotropic isotropic 3

S Log P S g 2 ~ E ~ E 2 Log P Why g 2 ?? Why ?? Log E Log E g 2 2 b 2 2 sin sin 2 q P S ( q ) = ) = 2 s T U B What happens when What happens when q  0 ? 0 ? Sure, but what happens to Sure, but what happens to the the received received power if you are power if you are in the beam of the particles? in the beam of the particles?

Synchrotron Spectrum Synchrotron Spectrum Characteristic frequency Characteristic frequency v 2 mc 2 b sin g mc sin q r L = = a eB eB e v B sin q e v B sin e e B n = 1/T = 1/T a = a = n B = g mc mc T = 2 p r L T = 2 L /v /v 2p g mc 2p This This is not is not the characteristic frequency the characteristic frequency

v<<c v<<c v ~ c v ~ c

D t A = ? = ?

1 = g 2 eB eB Compare with Compare with n B. n S = = B. = n B g 3 S = n S D t A 2 p mc mc b

The real stuff The real stuff x=

The real stuff The real stuff x=

Max Max synchro synchro frequency frequency Guilbert Guilbert Fabian Rees 1983 Fabian Rees 1983 p g m e p g m e e c 2 2 6 p 2 p q t syn syn = T = T  = c 2 g 2 eB eB s T B 2 2 1 g max max ~ B 1/2 1/2 2 ~ B g max = m e c 2 / a F = 70 MeV h n S,max S,max ~ B max = m = 70 MeV . shock shock (+ beaming) (+ beaming)

Emission from many particles Emission from many particles ) = K g -p The queen of relativistic N( N( g ) = K The queen of relativistic distributions distributions Log e (n ) Log N( g ) Log g Log n 1 e ( n ) d ) d n n = = N( N( g ) P ) P S d d g 4 p

Emission from many particles Emission from many particles ) = K g -p The queen of relativistic N( N( g ) = K The queen of relativistic distributions distributions Log e (n ) Log N( g ) d n d g Log g Log n Emission is peaked! Emission is peaked! 1 g n K g -p B 2 g 2 d e ( n ) ) ~ ~ d g g 2 eB eB = S 4 p n S d n 2 p mc mc

Emission from many particles Emission from many particles ) = K g -p The queen of relativistic N( N( g ) = K The queen of relativistic distributions distributions Log e (n ) Log N( g ) Log g Log n 1 K B (1+p)/2 (1+p)/2 n (1 (1-p)/2 p)/2 e ( n ) ) ~ ~ K B 4 p

Emission from many particles Emission from many particles ) = K g -p The queen of relativistic N( N( g ) = K The queen of relativistic distributions distributions Log e (n ) Log N( g ) Log g Log n p-1 1 K B a +1 +1 n - a a = e ( n ) ) ~ ~ K B 2 4 p

So, what? So, what? 1 K B a +1 +1 n - a e ( n ) ) ~ ~ K B 4 p 4 p Vol Vol e ( n ) ) K B a +1 +1 n - a 2 2 R F( F( n ) ~ ) ~ ~ q s R K B s 4 p d 2 Log F (n ) K B a +1 +1 K B If you know q s and R Two unknowns, one equation… we need another one Log n

Synchrotron self Synchrotron self- -absorption absorption • • If you can emit you can also absorb If you can emit you can also absorb • Synchrotron is no exception Synchrotron is no exception • • With Maxwellians it would be easy With Maxwellians it would be easy (Kirchhoff law) to get the absorption (Kirchhoff law) to get the absorption coefficient coefficient • • But with power laws? But with power laws? • • Help: electrons able to emit Help: electrons able to emit n are also the are also the ones that can absorb ones that can absorb n

A useful trick A useful trick g) Log N( g) Log N( g -p Many Many Maxwellians Maxwellians mc 2 with kT= g mc with kT= Log Log g g 2 eB eB = n kT n 2 /c I( n ) = 2 I( ) = 2 kT /c 2 2 p mc mc mc 2 n 2 /c = 2 g mc = 2 /c 2 B) 1/2 1/2 n 5/2 5/2 n/ B) g ~ ( n/ ~ There is no K ! There is no K ! B 1/2 1/2

From data to physical parameters From data to physical parameters n t belongs to thick and thin part. Then in principle one observation is enough insert B insert B get B get B and get K and get K

Inverse Compton Inverse Compton

Inverse Compton Inverse Compton  Scattering is one the basic interactions Scattering is one the basic interactions between matter and radiation. between matter and radiation.  At low photon frequencies it is a classical At low photon frequencies it is a classical process (i.e. process (i.e. e.m e.m. waves . waves )  At low frequencies the cross section is At low frequencies the cross section is called the Thomson cross section, and it is called the Thomson cross section, and it is a peanut. a peanut.  At high energies the electron recoils, and At high energies the electron recoils, and the cross section is the Klein the cross section is the Klein- -Nishina Nishina one. one.

Thomson scattering Thomson scattering n 1 n 0 q = scattering angle = scattering angle q • hv << m e c 2 hv 0 << m • tennis ball against a wall tennis ball against a wall • The wall doesn’t move The wall doesn’t move • The ball bounces back with the same speed The ball bounces back with the same speed (if it is elastic) (if it is elastic) n 1 = = n 0

Thomson cross section Thomson cross section 2 r 0 d s T (1+cos 2 q ) = 2 (1+cos a peanut a peanut d W 8 p e 2 2 r 0 s T = r 0 = 3 m e c 2

Why a peanut? Why a peanut?

Why a peanut? Why a peanut?

Why a peanut? Why a peanut? B E

Why a peanut? Why a peanut? dP dP e 2 a 2 = Remember: Remember: p c 3 sin sin 2 Q d W W 4 p

2 1 2 d s T r 0 (1+cos 2 q ) = (1+cos d W 2

Direct Compton Direct Compton x 0 h n x = x = m e c 2 q x 1 x 0 x 1 = = 1+x 0 (1 1+x (1-cos cos q ) Klein Klein-Nishina cross section Nishina cross section

Klein Klein-Nishina cross section Nishina cross section

Klein Klein-Nishina cross section Nishina cross section ~ E -1 ~ E

Inverse Compton: typical frequencies Inverse Compton: typical frequencies Thomson regime Thomson regime x x x’ x’ x’ x’ 1 =x’ =x’ x 1 Lab frame K Rest frame K’

Min and max frequencies Min and max frequencies 180 o  = 180  = 0 o =0 o o =180 o o  1 =0  1 =180 =4 g 2 x =x/4 g 2 x 1 =4 x 1 =x/4

Total loss rate Total loss rate Everything in the lab frame Everything in the lab frame n( n( e ) = density of seed photons of energy ) = density of seed photons of energy e =h =h n v rel rel = “relative velocity” between photon and electron = “relative velocity” between photon and electron v rel rel = c = c-vcos vcos  =  = c(1 c(1- b cos cos  ) s T

Total loss rate Total loss rate s T There are many There are many e 1 , because there are many , because there are many  1 .. .. We must average the term 1 We must average the term 1- b cos cos  1 , getting , getting

Total loss rate Total loss rate There are many There are many e 1 , because there are many , because there are many  1 .. .. We must average the term 1 We must average the term 1- b cos cos  1 , getting , getting { U rad rad

Total loss rate Total loss rate { U rad rad If seed are isotropic, average over If seed are isotropic, average over , , and take out and take out the power of the incoming radiation, to get the the power of the incoming radiation, to get the net electron losses: net electron losses: <P <P c > = > = 4 Compare with Compare with g 2 2 b 2 4 s T cU rad rad synchrotron losses: synchrotron losses: 3 <P <P S > = > = 4 g 2 2 b 2 2 4 s T cU B 3

Inverse Compton spectrum Inverse Compton spectrum The typical frequency is: The typical frequency is: n = g 2 n 0  Going to the rest frame of the e Going to the rest frame of the e- we see we see gn gn 0  There the scattered radiation is isotropized There the scattered radiation is isotropized  Going back to lab we add another Going back to lab we add another g -factor. factor.

The real stuff The real stuff down down upscattering upscattering

The real stuff The real stuff down down upscattering upscattering 75% 75%

Emission from many particles Emission from many particles ) = K g -p The queen of relativistic N( N( g ) = K The queen of relativistic distributions distributions Log e (n ) Log N( g ) Log g Log n 1 e ( n ) d ) d n n = = N( N( g ) P ) P C d d g 4 p

Emission from many particles Emission from many particles ) = K g -p The queen of relativistic N( N( g ) = K The queen of relativistic distributions distributions Log e (n ) Log N( g ) d n d g Log g Log n Emission is peaked! Emission is peaked! 1 g n K g -p U rad rad g 2 d e ( n ) ) ~ ~ d g 4 = 4 p g 2 2 n 0 n d n 3

Emission from many particles Emission from many particles ) = K g -p The queen of relativistic N( N( g ) = K The queen of relativistic distributions distributions Log e (n ) Log N( g ) Log g Log n 1 rad n (2 (2-p)/2 p)/2 n -1/2 1/2 e ( n ) ) ~ ~ KU KU rad 4 p

Emission from many particles Emission from many particles ) = K g -p The queen of relativistic N( N( g ) = K The queen of relativistic distributions distributions Log e (n ) Log N( g ) Log g Log n p-1 1 rad n - a a = e ( n ) ) ~ ~ KU KU rad 2 4 p

Synchrotron Self Compton: SSC Synchrotron Self Compton: SSC Due to synchro, then Due to synchro, then t c B a +1 +1 n - a proportional to: proportional to: c B a +1 - a t 2 +1 n c e c ( n ) ~ ) ~ Electrons work twice Electrons work twice

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