Radiation from Relativistic ChargesA. Marconi

Lecture 3Radiative Processes in AstrophysicsCourse on Relativistic Astrophysics AA 2016/2017

RADIATION FROM RELATIVISTIC

PARTICLES

y ,

it'

→

v

K'

* >

:K I,x£

Relativistic aberration

K'

is the reference frame Conomywith the path ,

kheeh has

velocity 8 in the K whenceframe of the observer

,

Lorentz 's transforms for any8

com be Wuhan as

t'

. rft . did )

He:*.at#IrlEt:¥¥ ,

where a and t dente the directions

Kluck are norollel and perpendicularto I

, resnectwely .

Differentiating We on obtain the

transforms for velocityIt -

g ( at 't III )dan - y ( dos

'

,+ of colt

'

)abet -

Iss 's

then we get

iii. - du£ .

Kobe ! + Fast'

)

←'

'

+ Ids )

if z Frias+ Fix

not - off -

day

Holt '+R÷deJ

u→± - t÷( s + Fein )

Let's now assume that ii'

and Pform on angle O

'

m thearmoury

frame ;the e angle between

µ→ and 8 m the observer frameis them obtained as follows

a-. - - -

ai%l%Ityo - ftp.t.INT

go.ir#egjtIITgrKi+cp4but We have :

/ ii 's 1- risen o

'

|u→ ! I z ii as e'

with KY - ii;

we finally obtain

tfg - txn÷( iiaso'

tv )this is the ulohivshe aberration

To understand itsmeaning

Consider a

among reference he'

where the pastel emits rsdagnaefly ( r ;e

. snhenad Wove fronts )I

:

*ate

so F

.→

I

* i xI

let 's consider the photons emitted m

K'

Luth o'

- the i.e. perpendicularly to

the direction of motion; theyobviously

have to 't - C.

In the u reference frame these photonswill move along a direction

forming on angle O with velocitysuch Ehret

tgo - Cxnu÷can +5- £

from the formula for the blotostz

aberration.

We can then find

me - tax#- F

if the particle is ultzarelahoshc

far and moxie that is

O ~ tar

herefne all the forward hemispherem K

' will be an cenhoted in a

come with aperture ~

ytaround

the direction of motion in K.

Relativistic bombe effect

Y^ 7

&o

D

1•e 2

IiZ

Consider the he zeferena frame ofthe observer

.

amoony hartle

emits monochromatic woohotwn

for 1 period whenmoving from

is to 2;

Bt is the period of the

e.mn.

wore emitted between I and 2.

Gang tocoming reference frame

rd we have

Dt a yBt

'

-

jT

'

-

:'

khere v'

is The frequency of the

hedioton.

The observer bees" the

pottle along a direction forminga & angle with its velocity .

Then,

the zodiothn embed in 2

reaches the observer with a

delay of ¥ with usnect to

that emitted in 1.

Therefore ,the difference in arrival

times between the impulse m L

and that m 2 is

Ata = @t+E) - ( Etd )A

time of arrival time of and

of 2 ( D distance ) of s ( started at to )

BE is the wheel of the sedation,

also the time interval between 1

and 2.

We have

Dta - Dt -d-C

d - lance a v At are

hence

Ata = Dt ( e - Iosco ) = Atf . faso)then the

"

observed"

frequency is

V -

feta = dt÷pas⇒ - yDt'÷p)wl T

W z

←Paso ).→

This is the formula for the

Zelokoishc Doppler effect check

tells us how the frequency ofthe emitted sedation varies fromthe

canonryto the observer

reference frame .

The termy

- 'is hurdy relativistic

and is connected to time

dilation while the term ( 1 - Paso )is

also present in the classical

Damper effect .For this reason

there is also a Demler effectfor motions t to the hne

of night ( aso - o ) : v - %

Qtvouonce with the classical

relation,

the relativistic one hasho mducotron of a reference system

for the Monaghan of the light :

it contains only the relative

velocity .

By manythe relativistic olcnotrm

formula to express and with

Cosco'

it is nomble to show

that

W - V'

j( I + Eased )

Radiation from a Relativism Paddle

To find The phonetics of the zadudhm

emitted by a relativistic pottle ,

We consider the k'

reference

armoring with the hotel .

'

Obviously ,due to its acceleration

,

the halide will not stay Anym the some Conomy myth

reference frame ;there will be

a time interval At m which

we have e non relativistic

pakele moony m k .

InIhs reference frame We com

useLarmer formula .

The particle m he'

ants energydw 'm time mtewol It '

;

for I'

- o ( k' centred on the north )

we have

It a rdt'

hit the some relation is also

valid for dlo' check the temporal

port of the energy - momentum

four .vector

.

The four vector of the emitted

Zeduetwn is

dpn'

-

fee'

,dF'

)d F - o because the emission process

m K'

is the classical one and the

antenna molten is centrally symezt.ua with respect to the halide

.

Hence

dwt - jdw'

The nodded power in Ie is

P -old

-

fdw'

At #- dart - P

'

the radiated power is Lorentz

inoouant !

We can then use Lonmin 's formulaO

'

- }g÷ lamp

the four - acceleration an . dae

is such that

Anny

- o

but mthe pottle reference fame

un'

a ( c,

o )and

, if

an'

- ( a'

.,£

'

)the property oiiua -0 implies

that ol.

a °.

We can them write

an'

qui e - di + a'

. d'

e

- @'

. I'

- 1 E'

12

/ @'

12 is them Lorentz invariant

because it is equal to the

module of the four . acceleration.

Finally We have

P - 2¥ dan

Written m a Lorentz moouontfrm

Let 's now use the classical

vector acceleration instead of The

four -accelerations

.

Form the review of special

relativity by Prof .

Del tome

ue know that

Al"

- j3 a.,

a

'

+ a j2

A+

P - O'

- In lamp . ¥,Cai tai )

finally

P - 2¥ j ( at + pa :)1- and a refer to the pottle

velocity .This is the relativistic

Larmer 's formula .

If We consider the complete

expression of The Zedokbe field

Eras - ¥ x×[ city xp ]find 5 - € / EP m→ and integrate

on the spherical surface Cotti

Zedrus R centered on the

pottle ( always m the on R >>R<i.e. well outside of the Coulomb

upon ) we obtain the zelotude

expression for the hound 's famebe . By many

The zdotoshe

Monona of P We have

avoided e lot of Gleulotwns !

The angular distribution of emitted

and absorbed power

In the he'

reference ,

do'

is emitted

m the sold angle old'

- xnddo 'd& '

along a direction forming on

onyle o' with velocity

Odr '

10' >P > se

'

Let's use the notation

me and µ'

a as O'

d R - dud to dr'

- diidt '

The energy - impulse four uectrr ofzadhotun is

pn . ( E , F)and its temporal hat transforms

as

Earle + I . pytrot

:¥ - y ( dew't pp

'ass

'

)p

'is

the momentum of the

emitted radiation which We

Comdr consider mull on overagebecause We one mthested on

the Zadidtum emitted m a givendirection

once p'

is the hadiothmmoney

turnpi < at

hen

: to - r ( stpn '

)

duhWe found that

�1� & - tinyfiasco' +5

which,

for u'

- c,

becomes

tyg -sent

fcrso 't

%)

Gso -

1-

ff- Gay

1 + Ease'

bbuch, putting µ - cod

,µ

'

- csd

becomes

µ- n¥

fn'

differentiating

die -

dm'←- fuyzonce dt - At

'

( rotation around

velocity 8) we con write

dr -

ItatipKluck

,

combined with

dkr -

y ( s + pri )dw'

mnlies

ddkf.jp (s+fij3dW'←'

the power emitted m k'

ohmydirection I

'

is simply

d¥,

- dW'←dt '

but mk We have two possiblechokes of the time mtewoe

to find thepower

1) dt - jolt'

This is the the

interval M cchrch emission

takes need in K;

this mohdes

the emitted power m he, Pe

2) dta - f ( s - fm )dt'

This is the time

interval in check the zedbthm

is zeceived from a stationeryobserver mik

. ( s - pay is the

delay factor due to the source

Motrin ; the definition of dta is

just the lelotvbhz Doppler effectThis definition provides the

received power m K, Pre

It should be nded that

) ofptdr a Jeff dr a P

with Pgiven by the ielokushe

Lama formula .

We have

Fez . 84rem' Paff . ftp.adf

date . system 's 'adI÷ . standoffindeed

, from m-

n¥

We obtainltfu

'

µ'

a µ#n

and stfu'

e

÷fm )

Lhrch one should we use ?

Pr is the one measured by the

observer,

hence it seems the fest

choice.

In favour of Pr,

there

is olfo the feet Fhefthe inverse

transform can be obtained

by inverting the variables m.

wth the others,

and changing

signto

p.

Henle, from now on We will

use p a Preas we noted already ,

the

relativistic aberration will

Concentrate emwmrm olvny the

direction of motion

µ- are re e - ok for & - for I

p - ( i - ⇒ "2x 1 -

say

* wxE÷yand this function is strongly

peaked for Oto.

In the u'

reference the noneis non -

vlotwshe and its

emission, given by Looms 's

formula ,is

fled,

- gulag xi O'

0¥61

@'

m the angle between the

aeeelerdhm and the emission

dheothm, os We have seen m

the warriors lectures,

WE also found that

d'

z Bin + Be 't

ain a Panhit a j2 at

hence

d÷rsExRfv÷÷8YYEEgITia

ofi÷seIe¥mo .

How we need To relate @'

, anglebetween a

'

and A due drum ofemission

,

with o'

( no ) , anglebetween The mouth velocity 8

and m→

^

rig'

ofQtr ii-

se' L

&

In general ,

thisis a complex taste

therefore we will consider onlyspecial cases

1) a→ 1/8 -7 @'

- O'

µ a µ'

+# ,

is are - Goin- Paso '

from which go'

- are- Pass

shed - senior'

- , -

arietta( I - Paso )2

-

z

1 +P2w2o -

#e- Chee -

P2#ne÷aso )2

-@. P )2( s - are )

¥2z Him( I -fy2

Zepleany then expression m DI

and toknymto account that•^

A+

-° We finally obtain

dpn e2- z

#AT send

dr ( l - paso)6

the redrawn diagram mthe k

'

and

K reference frames is therefore

K'

k

We should note that The dipole

diagram m K'

is dsthted in

K firm the zeldtvde obemotum

that bends it toward the direction

of motion

3) It T

E'

- in'

( take A'

along Ee'

,The )

m→- send ast

'I

'

+ send sen to 'T 't aso' Q

As On'

-re .

I'

* I

We obtain

as @'

a send as &'

hence ( &'

- & )sm2@

'

- I- xn2&g2÷( I fm)2

⇐ .e÷ataer¥÷dfr a fixed to ,

we have the

following diagrams

K"

*

In the ultwrelohnshz Gmt

( r > > , ) ,1 . pm is small

and the zllotvshe boastingm the direction of motion is

extreme.

We can show that

1- fm = 1+862=2

by xltmy aso = ( s . ok ) f- i - HeWe them have

of±r= islet ford(1+8202)

6

often = h±q;÷ grhzrttasztttht( It p2oy6

these expressions depend upona

only through to ;the reek of

lmrssum is them for 8 ~ Yyand this is the result of the

Uhokvstc aberration.

To conclude,

let's now find how

the meafcmttnnty of the bdiohm

varies from K'

to K.

at first ,

let's find how a those

space volume element varies.

Consider a

group of horttdesmthe

among reference frame he '

;

volume elements where nobles are loaded

d3x→'

- dsddg' Is

'

( space )

d3p'

- dpsidpjdpj ( momenta )

d38'

and d3p'

are infinitesimalhence hates have a well definedposition and momentum

.

Hence,

at first order,

frtheuenergies

dw'

a - dpo

'

- o

this is because

wihemochtCYFP

but

we . meat c2( IFII +2 Fidp )181.2 is

theoverage momentum

and Dftonbe neglected .

Let 's bobassume that the

observer has velocity p with aspect

to k'

,

and let's also assume

that The motion is along ahs x.

Consider d 38,

volume as seen bythe observer ;

mdhm is ohmgsehence dy - dy

'

,

alt - Is ' and

Sse - g-' dse '

givingd3e→

- g-' d3sI '

Gnnder d3F ,

its Transform →

that of a four veotk hence

dpjdpj -

dpydpzdbe- 8 ( d Poe

'

+ Pdp .

'

)

but since the halide energy is the

some m the rest frame df .

to

→ dpse - jd Pla and finally

d3p→ - yd3p'

The phase space volume element

IT - d3x→d3f- f

'

d3I'gd3f'

-dv

'

is a Lorentz moouant

DE - u, ( r ) drdvd ✓ a If drdvdv

⇒ ¥ dXd#=hvf#p2dp#

p a he ⇒ dp a he du

IE dyehvf k¥ he#

finally f a f÷If3f is Lorentz mount

,

hence

If isLorentz invariant

¥3 . Ifa . FI s - rata[ Dappled

from which we obtain that

Iu - Iij3(i - Paso )3