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