Modeling of the Dynamical Evolution of Classical Radio Sources

EVOLUTION of CLASSICAL RADIO

SOURCES

PART I 1

1 Types of radio sour es 1

1.1 FRII typeradio sour es . . . 3

1.2 SizeofFRIItyperadiosour esfromGigahertz-Peaked-Spe trumSour es to GiantRadio Galaxies . . . 3

1.3 Cy les of the a tivity . . . 4

2 Theoreti al basis 5 2.1 Physi alpro esses inthe FRII typesour es . . . 5

2.2 Energy losses . . . 6

2.3 Sour e's sizes vs. other physi alfa tors . . . 8

3 Analyti al models of FRII type radio sour es 9 3.1 Sour e dynami s . . . 9

3.2 Sour e energeti s and luminosity evolution . . . 10

3.3 Determiningdynami alageand otherphysi alparametersforreal FRII type radio sour es . . . 12

3.4 Limitationsof the KDA model . . . 13

3.4.1 Asymmetry of radio sour es . . . 13

3.4.2 Unlimitedexternal density prole . . . 13

3.4.3 Limitationfor the jet a tivity . . . 14

PART II 15 4 Asymmetries of FR II type radio sour es predi ted with the KDA model 15 4.1 Lobe's length and total luminosity asymmetries . . . 15

4.2 Asymmetries resultingfrom dierent

β

exponent . . . 15

4.3 Asymmetries resultingfrom dierent axialratio

R

t

. . . 18

4.4 Asymmetries resultingfrom dierent

α

inj

parameter . . . 20

4.5 Asymmetries resultingfrom dierent

k

′

parameter . . . 22

4.6 Dis ussion and on lusions . . . 24

4.6.1 Modelpredi tions for the asymmetries . . . 24

4.6.2 Modelpredi tion in omparisonwith observationaldata. . . 25

5 The observational onstraint for the model of the radio-jets propaga-tion through the X-ray haloIGM interfa e (Kuligowska et al. 2009) 26 5.1 The base of the revised G-KW model . . . 27

5.3.1 Sample1 (GRGsample) . . . 31

5.3.2 Sample2 (Distant-sour e sample) . . . 31

5.3.3 Sample3 (3CRR sample) . . . 32

5.4 Ageing analysis of the samples'sour es . . . 36

5.5 Results of the modeling. . . 37

5.6 Observational onstrain of the model . . . 37

5.6.1 ComparisonoftheModel'sPredi tionwiththeObservationalData 38 5.6.2 Ageand physi al parameters of the samplesour es . . . 42

5.7 Dis ussion of the resultsand on lusion . . . 42

6 Modi ationof KDAmodelto itsversion appli able for radiosour es with non- ontinuous a tivity 51 6.1 The argument for anextension of the basi KDA model . . . 51

6.2 Extension of the original KDA model . . . 51

6.2.1 Generalbasis of the extended model . . . 51

6.2.2 Adiabati evolution of the o oon inthe ase of terminated nu- lear a tivity . . . 52

6.2.3 Spe tralageing inthe syn hrotrontheory. . . 53

6.2.4 The analyti alformulaforintegration of radiopower . . . 54

6.2.5 Predi tions of the extended model. . . 55

6.2.6 Dis ussion . . . 60

PART III 65 7 Appli ation of the extended model to a few sele ted radio sour es 65 7.1 Sample of examined radiosour es . . . 65

7.2 Fitting pro edure . . . 69

7.3 Appli ation of dierent dynami almodels to the sample sour es . . . . 70

7.4 Dis ussion and nal on lusions . . . 77

Even after a few de ades of resear h and investigations, the dynami al evolution of

powerfulradiogalaxiesisstillfarawayoffullunderstanding.The lassi aldynami alapproa h

toades riptionoftheintera tionbetweenthe"light"relativisti jets, arryinganoutstanding

amount ofenergy fromtheA tive Gala ti Nu leus(AGN)andintera ting withtheexternal

gaseous environment, annotprovide an uniquesolutionfor a numberof detailedparameters

des ribing the underlied physi al pro esses.The problem is that besides the observed linear

sizeofasour e(althoughproje tedonthesky)that anbeinterpretedasalengthofthejets,

theabove pro essesresult intime-dependent shape of theradiospe trumof agiven sour e.

All existinganalyti al models of dynami al evolution of lassi al radio sour es arebased

onsome un ertain assumptionsabout,for example: 1) initial power-law energy spe trum,2)

equipartition ofenergybetween therelativisti ele trons andmagneti eld,3) possible

rea - elerationoftheseele trons,4)parti ipationofprotonsintheenergylosses.However,re ently

thesemodels anbepartly onnedbyX-raymapping,unfortunatelyonlyfornearbyandthe

brightest sour es. In spite of the above, the existing models suer other severe de ien ies.

All of these models assume: 1) exponential prole of the ambient environment (simplied

King's model, King 1972), whi h is valid for a limited external halo and is not justied for

thelargest(thusold)radiogalaxies, 2) ontinuousinje tionofenergy viathejetsof onstant

powerthatisevidentlynot the asefortheobserved,so alleddouble-doubleradiostru tures

inwhi htwopairsoftwinlobesareobservedandinterpretedbyaterminationoftheprimary

jeta tivityand the appearan e ofa new, se ondary a tivity.

In this Thesis I addresssome of the above issues and modify thebasi analyti al model

of Kaiser, Dennet-Thorpe and Alexander, 1997 (hereafter referred to as KDA), as well as

ompareobservedspe traofafewexemplaryradiosour eswithpredi tionsofboththeKDA

modeland itsmodiedversions.

Theoutlineof theThesis isasfollows:Se tion 1summarize dierentapparent

morpholo-gies of radio sour es and dene the type of sour es for whi h analyti al models of their

dynami alevolution is onsidered here. Se tion 2 outlines physi al pro esses resulting inthe

observed morphology, brightness distribution, and spe tral shape of the sour es. Se tion 3

des ribesthe existing dynami almodels ofradio galaxy'sevolution and itslimitation due to

the real examples of observed radio galaxies. Se tion 4 presents the appli ation of original

KDAmodelin aseofstudyingthe observedasymmetriesofFRIItyperadiosour es.Se tion

5 is dedi ated to the revision of the existing model, assuming that radio sour es evolve in

two-media environment. Se tion 6 des ribes an attempt to extend the lassi al KDA model

ofthesour e'sevolutionwithfor assumed ontinousnu lear a tivitytoitsgeneralization

ap-pli able to the ase of sour es withnon- ontinous inje tion of relativisti ele trons. Finally,

Se tion7presentsappli ation oftheextendedmodelto afewrealradiogalaxieswithsteeply

1 Types of radio sour es

All existing radio sour es an be dened as elestial obje ts that emit rather strong

radio waves. However, they are basi allydivided into two general ases. The rst ase

in ludesallof "normal"galaxies inwhi h the dominantradiationinthe radiodomain

results from the star-formation pro esses, supernovae remnants (e.g. Cassiopeia A),

and the existen e of louds of ionized hydrogen (HII regions). The angular extent of

the sour e in this ase does not ex eed a solid angle of the opti al galaxy itself. The

se ond ase onsistsof the galaxieswith the dominantradio emissionoriginatingfrom

the nu lear a tivity - so- alled lassi al radio sour es. Their opti al spe tra are not

the simple superpositions of the spe tra of parti ular stars in the galaxy, but show

also hara teristi emissionlinesdue tothe high-energypro esses takingpla e intheir

enters. Su ha lassi alradiosour ehas linear,elongated,so- alled"double" stru ture

extendingusually far outside the parent galaxy.

Classi al radio sour es had been divided by Fanaro and Riley (1974) into two

morphologi altypes: FRI and FRII. This lassi ationin based onthe observed

mor-phology of the large-s ale radio stru tures. The FRI type radio sour es are brighter

towards its enter, whileFRIIsour es are thebrightest atthe edgesortheir two radio

lobes. The two FR types are also broadly separated by the luminosity; FRII sour es

aremoreluminousinthe radiowavelengthsthan FRItypes. Theoriginaldividingline

waspla edat

5 × 10

25

_{W Hz}

−

1

atobservingfrequen y of178 MHz.Thisdivision isdue

tothe e ien y of the energy transportinto the sour e's radio stru ture - the entral

engines of FRII sour es supply the lobes with the energeti parti les more e iently

than inthe FRI ase. The jets of the latter appear tobe subsoni lose tothe nu leus

and radiate asigni ant amount of their total energy on their way to the radio lobes.

In turn, the jets in FRII sour es remain relativisti on their entire length, and the

so- alledhot spotsvisibleontheendsofthe radiolobes an beinterpretedasthe

man-ifestation of supersoni sho ks formed when the jet abruptly terminates en ountering

the external intergala ti mediumof some density and pressure.

Figures(1) and (2)show ar hetypesof the abovetwo morphologies:radio galaxies

3C405 (Cygnus A) of FR II type and 3C31 of FRI type. The ontent of this thesis is

ourtesyof NRAO/AUI.

Figure 2: Example of FR I type radio sour e: 3C31 radio map at 5 GHz (VLA). Image

FR II type radio sour es are omposed of double, ellipsoidal,radio-emitting regions

(so- alled radio lobes) situated more or less symmetri ally on the both sides of their

host galaxieswhi hinmost ases are largeellipti algalaxies ontainingsupermassive

bla k holes (SMBH) of masses

10

6

_{− 10}

9

_M

⊙

intheir enter. Theselobe stru turesare

originated from twin jets outowing in opposite dire tions from the A tive Gala ti

Nu leus(hereafterreferred toasAGN).The radioemissionof these lobesresultsfrom

theele tronseje ted atnearlythe speedoflightthroughalongjetfromthe oreofthe

galaxy and deposited in radio lobes. The ele trons are trapped by the magneti eld

around the galaxy and produ e radio waves. In most of the powerful observed FR II

sour estheir jets end insigni antly brighter regions situatedatthe ends of the lobes

-hot spots learly seen in Figure(1).

The most famous example of FR II type radio sour es is Cygnus A. It has been

dis overedbyGroteReberin1939.CygnusAisoneof thestrongest radiosour esseen

from the Earth and, along with Cassiopeia A and Puppis A, had been the rst radio

sour e identied with opti al sour e (Baade and Minkowski, 1954) and, so, the rst

even observed lassi al radiogalaxy.

1.2 Size of FR II type radio sour es from

Gigahertz-Peaked-Spe trum Sour es to Giant Radio Galaxies

FRII type radio galaxies show a large variety of their linear sizes. The size of radio

galaxiesisdenedhereasthelineardistan ebetweenthebrightedgesofthebothradio

lobes. These sizes range from less then 100 p - Gigahertz-Peaked-Spe trum (GPS)

sour es up to about 4 Mp - Giant Radio Galaxies (GRG). Between these extreme

ases we observe a lot of medium size radio galaxies. Their typi al size is about 100

to400 kp . In general, this size is dependent of the age of the sour e - the older it is,

the more time the jets had to a t forming a large radio stru ture. On the other side,

the sour e's large size may resultfrom the lo allyo urred lowdensity ofthe ambient

mediumin whi h the sour e evolves( f. Se tion 3.1).

GPS sour eshaveamaximumof uxdensityintheirradio spe traatabout1GHz

andlinearsizes notex eeding 1kp .Theyare onsideredtobeyoung,re ently formed

sour es. Several studies show that they probably expand during their time evolution

and transform to Compa t Steep Spe trum (CSS) sour es with linear sizes of about

10 - 15 kp . Then, in a next stage of the evolution, they transform into the phase of

typi alFR I of FR II type radio sour e. In a spe i ondition they an evolve tothe

ultimatephase,i.e.GRGs.Theyaredenedasradiosour eswithlinearsizesofatleast

1megaparse (foraxedHubbleConstant

H

0

=0.71

km s

−

1

_Mpc

−

1

andthe osmologi al

parameters

Ω

m

= 0.27

and

Ω

Λ

= 0.73

(

Λ

CDM model). GRGs are interesting be ause of the fa t that their existen e brings out important questions about their evolution,

galaxies what suggests they an be the nal stage of sour e's evolution.

Until re ently the largest radio galaxywas 3C236that features a "double-double"

morphology onsisting of thegiantreli of

∼

4.4Mp size and aninnerstru ture of

∼

2 kp size whi h is a young, ompa t steep spe trum sour e. (Willis, Strom & Wilson

1974.) However, in 2005 the larger radio galaxy have been dis overed. J1420-0545 has

itslinearproje tedsizeof4.69Mp (Ma halskietal.2008).Despiteofthis,J1420-0545

isalso interesting due to itsrelatively high redshift (

z = 0.3067

).

An analysis of the rea hed sour es' size (and their age) in the fun tion of their

temporary radio luminosity was onsidered by Kaiser & Best (2007). A summary of

their onsiderationsis given in Se tion2.2.

1.3 Cy les of the a tivity

The a tivity of radio galaxies results from the pro esses of a retion of a matter

onthe bla k hole with magneti eld and angularmomentum. This a tivity evidently

annot be innite; it an be stopped aftera time dened as "jet ut-o" time. It was

estimatedtobenolongerthenabout

10

8

years(Komissarov&Gubanov1994). When

the a tivity ends even temporarily, the lobes are not longer powered with the inow

of relativisti parti les, soas the result, the luminosity of the radio stru ture tends to

ease. This ee t is observed mostly on higher radio frequen ies (

ν

>1000 MHz) and thisee tseemstobe ommoninthe aseofold,large-sizedradiosour es.Inastudyof

theradiostru turesof somegiantradiogalaxies,Subrahmanyan, Saripalli&Hunstead

(1996) drew attention to a variety of morphologi al features in the GRGs whi h were

indi ative of interrupted or episodi nu lear a tivity. It implies that GRGs may had

attained their large sizes as a result of restarting of their nu lear a tivity in multiple

phases.

Observations also indi ate the existen e of some spe i ase of radio sour es

-so alled Double - Double Radio Galaxies (DDRGs). Their extended radio stru tures

onsist of the inner, younger double stru tures with almost pure power-law spe tra

and the outer, larger, steepened spe trum stru ture, onsidered to be the older one.

Observations suggest alsothat su h double-double stru tures predominantly o ur in

the ase of the largest radio galaxies. However, only a small fra tion of GRGs has

double-double stru tures (S hoenmakers et al. 2000). Studies of the possible nu lear

re urren e in radio sour es are in their early days and the role of su h a re urrent

2.1 Physi al pro esses in the FRII type sour es

The radio galaxy's a tivity onne ted with jets formation results fromthe pro esses

of a retionof a matteronthe super massive bla k hole(hereafter referredto SMBH)

withmagneti eld andangularmomentum.AGNsare hara terizedby ahigh rate of

the a retion of gaseous matter situated lose to the entral SMBH. A reted matter

formsso- alleda retiondis swheretherotationalenergyoftheSMBHistransformed

into plasma'sinternal energy, and then, radiatedinthe wide range of the

ele tromag-neti spe trum. Magneti eld of the SMBH links the a retion dis with the plasma

pla ed outside the ergosphere. That implies that the high fra tion of the plasma an

be a eleratedin the dire tiondened by the spin of the AGN.

Jets (and double radiostru tures) are formed onlyin some fra tionof allobserved

AGNs. Wilson & Colbert (1995) and Blandford (1999) suggest that jets of so- alled

radio loud AGNs are only produ ed in the ase of entral SMBH with very strong

magneti eld and high spin. High values of the spin results probably from the

merg-ing event of two SMBH (Merrit & Ekers, 2002.) Physi al pro esses leading to the

observed stru ture of FR II type radio sour e are determined by so- alled "Standard

AGN model"(Blanford, Rees &S heuer 1974).The s hemati diagramofthe sour e's

omponentsis shown on Figure(3).

the host galaxy, then the medium of the luster ontaining that galaxy, and nally

it may rea h the Intergala ti Medium (hereafter IGM) whose density an be higher

than the density of the jet. Intera tion of the jet with those media produ es a bow

sho k.Thissho k formaninterfa ebetween thesho ked andtheunperturbed external

medium.A ompressionof relativisti jetmaterialhaspla e athot spots,the brightest

regionsof the ends of the jets where external mediumintera ts with the jet,resulting

in medium's parti les a elerations after their transition through the bow sho k. The

jet'spowerdepends onthetotal AGNrotationalenergy.The higherthis energyis,the

fastertherelativisti parti lesofthejetsmove.IftheAGNisrelativelyweak,theradio

lobesare powered with poorly ollimatedjets in whi h turbulent for es o ur and the

velo ity of the relativisti parti lesslows down.

2.2 Energy losses

The FR II type sour es have predominantly power-law spe tra. Non-thermal

on-tinuum radio emission of the lobes is due to both syn hrotron pro ess and

inverse-Comptons atteringofambientphotons ofthe osmi mi rowaveba kground(CMBR).

Thesyn hrotronradiationarisesfromultrarelativisti hargedparti les(with Lorentz

fa tor of

γ ∼ 10

4

) thatintera twith the magneti eld.This emissionin highly

polar-izedand anbeobserved withintheentireele tromagneti spe trum,soit anbeeasily

re ognized.A ording to lassi alele trodynami sthe life-timeofrelativisti ele trons

emitting mediumand short radio waves has to be relatively short. Be ause the radio

lobes are stable, long-living stru tures (asthe observations indi ate), they have to be

ontinuously powered with the amountsof new parti les. The sour eof theseparti les

an beonly the entral partof the galaxy -AGN.Due toso- alledsyn hrotron losses,

themostenergeti parti lesloosetheirenergy inthe fastestrate.Itimpliesthat,inthe

absen e of the onstant inje tion of the new parti les, syn hrotron losses in the radio

lobes result a violent steepening of the observed radio spe tra of the sour es. Most

energeti ele trons preferentially emit at high frequen ies, so the high-frequen y part

of the spe trum rapidly steepens and nally uts o.

The Inverse Compton ee t originates from the intera tion of the jet's relativisti

parti les with CMBR photons. A ording to Kardashev (1962), this pro ess need to

be taken into a ount in the analysis of the sour e's total energeti s, as well as the

adiabati expansions ofthe lobes(resulting inboth energylosses and gains),Coulomb

losses (o urring when relativisti parti les ollide with thermal ele trons of a given

density), and energy a quisition by systemati or sto hasti a eleration. It had been

proved that Coulomb losses play a major role in total energy losses only at very low

energies(

γ < 10

).Bothradiativeandadiabati lossesare dominantathigherparti le's energies, espe iallyfor

γ > 10

3

(Murgia 1996).

L

D

L − D

reprodu ed inFigure (4).

Figure4:The FRII type sour e's luminosityevolution of alobe.

The sour e's luminosityis linked to astage of the itslife and the dominant energy

losses at a this stage. For the ase of dominant adiabati loses, the radiation losses

ould be negle ted. In this regime the radio luminosity depends on sour es size as

L ∝ D

(8−7β)/12

. It implies that at this stage the expe ted luminosity an de rease or in rease depending on whether

β >

8

7

or

β <

8

7

, respe tively. The syn hrotron losses are the most important at the early time of sour e's life be ause the energy density

of the magneti eld in whi h parti les radiates de rease while the sour e grows and

grows older. The radio luminosity in this regime is the onstant fun tion of its size

L ∝ D

0

_{∼ const}

. The external density prole is now approximated by a power law

and the relation between the length D and the sour e age t depends on the value of

thepower-lawexponentβ.Inthis regimesyn hrotronlosses dominateovertheInverse

Compton ee t.

As the magneti eld de reases due to the ageing of the sour e, Inverse Compton

lossesremain onstantand nallydominateoversyn hrotronlosses.Atthisphasevery

strong totalenergy losses are observed

L ∝ D

(−4−β)/(5−β)

Thelobe'slengthisnotonlythesimplefun tionofitsage.Thetotalsizedepends also

on amount of energy released from the AGN and the properties of the IGM inwhi h

thesour eevolves. ForexampleGRGsareonlyafra tionof 6

%

ofthetotalpopulation of luminous radio galaxies (Laing, Riley & Longair, 1983). It may imply that their

enormous sizes result from the "density voids" in the IGM. Additionally, only about

10

%

ofknownGRGslieatredshiftlargerthan0.5.Themaximumlinearsizeofobserved radio galaxies de reases as

D ∝ (1 + z)

−

3

(Gopal-Krishna & Wiita 1987). This fa t

is in a goodagreement with the theory assuming that in the adiabati allyexpanding

Universe, lled up with hot, uniform IGM, its density in reases as

̺

IGM

∝ (1 + z)

3

,

(Kapahi1989) anditskineti temperaturegrows up proportionallyto

T

k

∝ (1 + z)

2

.It

impliesthat the pressure of IGM hanges as

p ∝ (1 + z)

5

(Cotter1998). The pressure

of the external medium likely de ides how far the radio stru ture an expand. That

explain why most of the GRGs are situated lose to us. However, nding GRGs at

mu hhigherredshiftmay provethatparts oftheIGM, withsigni antlylowerdensity

then an average one, are alsoin early osmologi al epo hs. Forthis reason GRGs are

usefultoolfor studyingtheevolutionofradiosour esitself,aswellasthe osmologi al

All analyti al models take into a ount known physi al pro esses o uring in FR II

type radio sour es. They hara terize these sour es in terms of their dynami s and

energeti s onne tedtothe luminosityevolution.Allof themoriginatefromthe

"stan-dard model" fordouble radio sour es (e.g.Blandford &Rees 1974)in whi h lobesare

omposed of a sho ked jet and IGM material.The axiallength of a lobe results from

thebalan ebetween the jet'sthrustand theram pressureof theexternalmedium,and

itswidth is afun tion of the jet's parti lesinternal pressure.

3.1 Sour e dynami s

This balan eisa base forthe sour e's dynami sdes ribed by the moresophisti ated

analyti al model of Kaiser & Alexander (1997; hereafter referred to KA). The jets

thrust,

Π

jet

,and itsbalan ewith theram pressure ofthe gaseousenvironmentisgiven by

Π

jet

∼

 Q

jet

v

jet



≈ ρ

a

v

h

2

A

h

,

(1)

where

Q

jet

and

v

jet

arethejet'spowerandspeed;thelateroneisassumedtobe loseto thespeedoflight(

v

jet

≈ c

).

ρ

a

istheambientdensity,

v

h

isvelo ityofthejet'shead( f. Figure(3)), and

A

h

isthe ross-se tion areaof the bowsho k there the time-averaged

Π

jet

is dis harged over.

Transforming Eq.(1) we have

v

h

=

d

dt

r

j

≈



Q

jet

ρ

a

cA

h



1/2

.

(2)

Taking into a ount that

A

h

is an in reasing fun tion of time ( f. Murgia 1996), and assumingthatthedensitydistributionoftheunperturbed ambientgassurroundingthe

radio sour eshas a power-law radialdensity distribution(the simpliedKing's (1972)

prole)s aling with the distan e r from the enter of the host galaxy as:

ρ

a

(r) = ρ

0

 r

a

0



−

β

,

(3)

where

ρ

0

is entral density of the radio ore,

a

0

is its radius, and

β

is exponent of the power-lawdensityprole inthe King'smodel. Integration of Eg.(2)givesthe total

r

j

(t) = c

1

 Q

jet

ρ

0

a

β

0



1/5−β

t

3/5−β

.

(4)

where

r

j

is identied with one half of the sour e's linear size,

r

j

= D/2

. A value of the oe ient

c

1

is a omplex fun tion of equations of state of the unperturbed externalmediumand sho ked jetand surroundingenvironmentmaterial( f. Kaiser&

Alexander 1997). The sho ked jet material inates a spa e volume (hereafter referred

to as o oon) within, so- alled, onta t dis ontinuity. The observed lobes of an FR II

type sour e are identied with the radiating parts of this o oon ( ross-hat hed area

inFigure (3)).

The lobeexpand due to the hotspotplasma pressure,

p

h

, and the o oonpressure itself-

p

c

.Thepressure ratioisproportionaltothe axialratio-

R

t

, denedasthe ratio of lobe's length toits base diameter:

P

hc

≡

 p

h

p

c



≃ 4 R

t

.

(5)

Inparti ular,the oe ient

c

1

isdependenton

P

hc

also( f.Eqs.(32) and(38) inKA). It is worth noti e that in the KA model the rate at whi h parti les are transported

fromthe AGN to the hot spotsis onstant during the lifetimeof the sour e. Iftwo of

itsbasi freeparameters:

Q

jet

and

ρ

a

(r) = f (ρ

0

, β)

,arespe ied,this simpliedmodel predi ts evolution of sour e's geometri alparameters only.

3.2 Sour e energeti s and luminosity evolution

Inordertoprovideanother,independentrelationbetweentheparameters

Q

jet

and

ρ

0

, Kaiser,Dennett-Thorpe & Alexander(1997) ombinedthe pure dynami alKAmodel

with the analyti al model (hereafter KDA model) for the total energeti s of a FR II

type sour e, i.e.forexpe tedradio emissionof its lobes(or o oon) under inuen e of

theenergylosspro esses:adiabati lossesduetotheirexpansion,syn hrotronemission

inthemagneti eld,andinverseComptons atteringoftheCMBRphotons.Theexa t

formulafor the radio power is not an analyti ally solvable integral inthis model, but

it an be solved numeri ally for assumed values of the modelfree parameters, tra ing

allof the sour e's energy losses.

The initialenergy distribution of inje ted parti lesis given by

n(γ

i

) = n

0

γ

−

p

. The

additionalmodel free parameteres that have to be xed are:

a

0

,

β

,

θ

(the in lination angleofthejetaxistotheobserver'sdire tionofsight),

γ

min

and

γ

max

(Lorentzfa tors determining the energy range of the relativisti parti les),

p

(initial power-law expo-nent),

Γ

j

,

Γ

a

,

Γ

B

and

Γ

c

(adiabati indi esintheequationof stateforthejetmaterial,

the ratio ofthe energy density of the magneti eld andrelativisti parti les, given by

r = u

B

/u

e

= (1 + p)/4(1 + k

′

)

, assuming energy equipartition ondition.

k

′

is a ratio

of thermalto relativisti parti les, and

p = 2α

inj

+ 1

.

Anensemble of

n(γ)

relativisti ele trons withLorentzfa tor

γ

,pla edinavolume V,inthepresen e ofmagneti eld

B

,emitssyn hrotronpowerperunitfrequen y and unit solid angle a ording tothe relation:

P

ν

=

σ

T

c

6π

B

2

2µ

0

γ

3

ν

n(γ)V,

(6)

where

σ

T

isthe Thomson ross-se tionand

µ

0

isthe permeability of freespa e. These relativisti ele trons are supposed tobeinje ted into the lobe fromthe hotspot by its

head (extended regionof turbulent a eleration around the hotspot.) Radio spe trum

evolves with time due to energy losses. The pro ess of slowing down the parti lesthe

Lorentz fa tor

γ

an be des ribed by the formula:

dγ

dt

= −

a

1

3

γ

t

−

4

3

σ

T

m

e

c

γ

2

(u

b

+ u

c

),

(7)

wheretherstterm(fromtheright)referstotheadiabati lossesintheexpandinglobes

and the se ond - to the ombined syn hrotron and inverse Compton ee t radiation.

Inthis formula,

m

e

is the ele tron mass,

u

b

- energy density of the magneti eld, and

u

c

- energy density of CMBR.

IntegratingEq.(7)overtimeandperformingsomeothertransformationsone nally

obtainsthe formulaforthe radiopowerof innitesimalvolumeelementsofthe o oon

atgiven frequen y. Bysumming the total ontributionof allof those elements we an

numeri ally al ulatethis integralovertheinje tiontime

t

i

.However,

t

i

annotbeless than a minimum inje tion time,

t

min

, at whi h the o oon material(still radiatingat frequen y

ν

) has

γ

i

≤ γ

max

. Thus:

P

ν

(t) =

t

Z

t

min

dt

i

σ

T

cr

6πν(r + 1)

Q

jet

n

0

(4R

2

t

)

(1−Γ

c

)/Γ

c

×

γ

3−p

_t

a

1

/3(p−2)

i

[t

−

a

1

/3

− a

2

(t, t

i

)γ)]

2−p

·

 t

t

i



−

a

1

(1/3+Γ

B

)

.

(8)

Asithavebeenmentioned,thereare examplesofmoreadvan eddynami almodels

basedontheoriginalKAmodel.Theyareallrelatedtothe samebasi onsiderationof

sour e'slengthvs.agedependen eandtheme hanismofradiationandradiativelosses.

energydistributiondenedabove.Inthissimplieds enario,theradiatingparti lesare

inje ted from the hotspot into the lobe. Some authors argue that this approximation

maybein orre t.Forexample,intheBRW(Blundell,Rawlings&Willot1999)model,

this inje tion index is not expe ted to be onstant and varies between the dierent

energyregimesduetosomebreak frequen ies.Onthe ontrary,theMK(Manolakou&

Kirk2002)assumesthatthisindexisalso onstant,buttheparti le'sbehaviourismore

ompli atedthanina aseof pureKDA modeland theyare additionallyrea elerated

in the lobe's head. The both models are widely des ribed and ompared by Barai &

Wiita(2006).

3.3 Determining dynami al age and other physi al parameters

for real FRII type radio sour es

The originalKDA modelallows predi tion of the jets length (i.e. the linear size of a

sour e) and its radio power at a given frequen y. This is possible if the values of the

allfreeparametersof themodelare spe ied.However, aspe ialtoolhas been desired

tosolve the "reverse problem", i.e.to estimate model's physi al parameters for a real

radio sour e with all the observational parameters derived from radio maps: the size

and axial ratio of its lobes and their radio spe trum, i.e. the luminosity at a number

of observingfrequen ies.

In order to solve su h "reverse problem" Ma halski, Chy»y, Stawarz & Kozieª

(2007a; hereafter referred to as MCSK) elaborated the algorithm "DYNAGE" whi h

allows toderivevaluesof fourof the sour e's unknown parameters (i.e.age, jet power,

entral density and the initial inje ted energy) from the t to the known observables

(sour e's size, volume, the radio power at given frequen y and the shape of the

spe -trum).This numeri al approa hdemands multifrequen y radio observations in luding

atleast three dierent ux densitiesembra ing possibly wide range of the radio

spe -trum. The "DYNAGE" algorithm demands also xing values of the remaining free

modelparameters, i.e. values of

a

0

,

β

,

θ

,

γ

min

,

γ

max

,

Γ

j

,

Γ

a

,

Γ

B

,

Γ

c

, and

k

′

indi ating

ratioof energydensityof the thermalparti lestothat ofthe relativisti parti les.

Set-ting all of these parameters, one an estimate values of

t

,

Q

jet

,

ρ

0

, and

α

inj

for every individualFR II type radio sour e.

Espe ially,the dynami alage ofagivensour e anbe estimated.The "DYNAGE"

algorithm provides the re ipe for tting "the best" model radio spe trum to the

ob-served data. This pro edure is shown in details in MCSK. So- alled "age solution" is

found knowing that numeri ally tted values of the

Q

jet

in rease with de reasing age and, on the ontrary,

ρ

0

tend to in rease with age onthe

Q

jet



ρ

0

diagram ( f. their gure (1)). Therefore the rossing point for all of the frequen y urves on these

dia-grams(or, thenumeri allyted age orrespondingtotheminimumdispersionof these

dependssigni antlyon

α

inj

value.Thus,the"DYNAGE"algorithmpredi tsthe"best age solution"resultingfromthet tothe best "ee tive"value of the initialenergy of

inje ted parti lesprovidingaminimum of the jet's kineti energy,

Q

jet

· t

3.4 Limitations of the KDA model

The KDA model suers three serious limitations in its des riptions of the physi al

pro essesand onditionsgoverningthedynami alevolutionofradiosour es.Theseare:

1. Negligen e of observed asymmetries in the lobe's length, axial ratio, and radio

spe trum,

2. Unlimitedde rease of the externaldensity prole,

3. Constant relativisti parti les energy distribution and the jet power during the

lifetimeof asour e.

3.4.1 Asymmetry of radio sour es

A ording to the lassi al approa hbased on standard model of FR II type sour es,

every radio sour e forwhi h modelingis performed isstri tly symmetri al. Therefore,

the KDA model assumes that the two lobes have the same length, axial ratio, and

total radio power at a given frequen y. In the ase of real radio sour es more or less

evident asymmetries inboth: lobe's length and theirradio luminosity are observed. In

the standard modelingthese dieren es are negle ted and the values of

D

and

P

ν

,the opposite for lobes, are averaged. This simpli ation implies that estimated values of

t

,

Q

jet

and

ρ

0

, as well as

β

exponent, are always equal for both lobes of the sour e. In the Se tion 4, I analyse dependen es of the ratios of the lobe size,

D

1

/D

2

, and mono hromati power,

P

1

/P

2

,resulting fromthe KDA model, ondierentvalues of a fewof itsbasi free parametersassumed for opposite lobes of a du ialsour e.

3.4.2 Unlimited external density prole

The original KDA model assumesthat radio galaxyevolves in one-medium

environ-ment: ahalo with de reasing density des ribed by the simpliedKing's (1972)prole.

Gopal-Krishna& Wiita(1987) proposed a more sophisti ated modelof the jet

propa-gation inwhi h host galaxy issurrounded by two-media environment onsisting of an

X-ray halo aroundthe parent galaxy with gas density de reasing with radial distan e

from the galaxy and a hot IGM with onstant density. In Se tion 5, I revised their

modelintrodu ing ontemporaryvaluesforthe density and temperatureofthe

onsid-ered media, as well as ompare the revised model dynami alpredi tions with the age

andlinearsize ofseveral FR IItypesour es(formingthreedierentsamples)resulting

KDA model has been also onstru ted on the assumption that the jets a tivity

of the radio sour e is ontinuous, the power-law distribution of inje ted relativisti

parti les is onstant, and the lobes are powered with onstant amount of the new

radiatingparti les. This approa h is a good approximation only in the ase of young

radio sour es. It alsoimplies that originalKDA modelis appli able for young sour es

with regular spe tra only. However, the observations indi ate that there are many

examplesof radio sour es (in luding giant radio galaxies)in whi h the high-frequen y

parts of their spe tra are steeper than expe ted for the ase of the KDA

ontinuum-inje tion model,

α

inj

+ 0.5

. This strongly suggests that their a tivity stopped some time ago. In su h the ase, the use of "DYNAGE" algorithm results a di ulty in

determiningasatisfa tory agesolutionfor su hasour e by the best tofmodel'sfree

parameters to the observables. The attempt of extending this algorithm to a version

appli ableforFRIItyperadiosour eswithhigh-frequen yspe trasteeperthan

α

inj

+

4 Asymmetries of FR II type radio sour es predi ted

with the KDA model

4.1 Lobe's length and total luminosity asymmetries

As numerous observations indi ate, the asymmetries of FR II type radio sour e's

lobes (their morphology, size, brightness distribution, total luminosity, spe trum and

polarization of radio emission an be meaningful. In this dissertation I onsider the

lobes' length,

D

,and mono hromati radio powers,

P

ν

, only.

In most ases these observed asymmetries annotbe explainedby asimple

proje -tion ee t only- the situationwhen one of the lobes, that isphysi ally situated loser

to the observer (when the jet's axis is signi antly in lined to the dire tion of view)

seems to be longer and fainter than the opposite one, a ording to the dieren es in

the lighttravel time (Longair& Riley, 1979).

ThereforeinthisSe tion,Ianalyseasymmetriesbetweenthelobe'slengthandradio

luminosity predi ted by the KDA model. They appear in the model when the values

of some of it's free parameters are dierent for the opposite lobes of a given sour e.

Hereafter these model-predi ted asymmetries are des ribed by the ratios

D

1

/D

2

and

P

ν1

/P

ν2

(the latter at the observingfrequen y of 178 MHz.

Eqs.(4)and(8)inSe tions 3.1and3.2indi atethemodel'sparametersfromwhi h

the lobe's length,

D/2

, and its radio luminosity,

P

ν

depends. Be ause the parameters

Q

jet

,

ρ

0

,

a

0

and

t

areexpe tedtoberather onstantforthegivensour e-relativevalues of

D/2

willbe onlydependent onthe oe ient

c

1

and

β

parameter, respe tively. As it had been mentioned before, oe ient

c

1

is a fun tion of the following model's parameters:

Γ

a

,

Γ

c

,

β

and

P

hc

≃ 4 R

2

t

( f. Eq. 5). However, one an assume that only values of

β

and

R

t

an be dierent in the opposite lobes. Similarly, relative values of

P

ν

depend onfollowingparameters:

Γ

B

,

Γ

c

,

R

t

,

β

,

α

inj

and

r = f (k

′

)

( f.Se tion 3.2), wherethe lastfour values onlyare supposed tobe dierent inthe opposite lobes. It is

worth notingthat

R

t

value parametrizesthe assumed ylindri al geometryof the lobe ( o oon) and is used to al ulateits volume,

V

.

The remaining values of the KDA model's free parameters have been set up as

follows:

a

0

=

10 kp ,

γ

min

= 1

,

γ

max

= 10

7

,

θ = 90

◦

,

Γ

j

=

Γ

B

=

Γ

c

=

Γ

a

= 5/3

.

4.2 Asymmetries resulting from dierent

β

exponent

One an expe t that

β

exponentinthe power-lawdensity distributionofthe external mediumsurrounding the radio sour e may have dierent values in dire tions of

prop-dieren es will ause dierent values of both: lobe's size and its radio luminosity at

a given frequen y without referring to any internal asymmetries due to physi al

pro- esses inthe sour e itself.In orderto analysethe above asymmetries, I used adu ial

sour e with xed values of

Q

jet

= 10

38

W,

ρ

0

= 10

−

22

kg/m

3

,

α

inj

= 0.51

(implying

p = 2.02

),

R

t

= 3.0

and

z = 0.5

.

The al ulationisperformedfortwosele ted sour e'sages:

t =

10and100 Myr(for

D

1

/D

2

asymmetry) and for three s enarios inthe ase of

P

ν,1

/P

ν,2

asymmetry:

t = 10

Myr and

k

′

= 0

,

t = 100

Myr and

k

′

= 0

, and

t = 100

Myr and

k

′

= 10

. Following Kaiseretal.(1997)I usedthe frequen y of 178MHz. Itisassumed that the valueof

β

may varyfrom1.0to 1.9, withthe highestdieren e

β

1

− β

2

= 0.45

between the lobes of this sour e.

Figure (5) presents predi ted ratios of

D

1

/D

2

(plot a) and

P

ν,1

/P

ν,2

(in log s ale, plot b), respe tively, versus the dieren e

β

1

− β

2

. The urves on both plots indi ate median values of the above ratios in a set of the model solutions for

D

1

and

D

2

, as wellas for

P

ν,1

and

P

ν,2

that are dependent on varying values of

β

providingthe same dieren e of

β

1

− β

2

. The verti al bars show the standard deviation fromthe median value.

The diagramsshow that the lengthand luminosity asymmetries in rease with age

of the sour e. Besides, the asymmetry inluminosity in reases with in reasing fra tion

ofnon-relativisti (thermal) parti lesintheradiolobes.Open ir lesonboth diagrams

marktheobserved ratios of

D

1

/D

2

and

P

ν,1

/P

ν,2

versusdieren e of

β

1

− β

2

published forthesampleof30giant-sizedradiogalaxiesby sampleof30giant-sizedradiogalaxies

byMa halskietal.(2009)andMa halski(2011).Intheabovepapers,

P

ν,i

(i = 1, 2)

are given at a number of observing frequen y of 151 MHzas the losest one to the model

frequen y of 178 MHz. For the sample sour es (or lobes) withoutdata at 151 MHz, I

use the radio power al ulated with ux densities interpolated between neighbouring

data points, e.g. between 74MHz and 325 MHz. The values of

β

are taken from their "self onsistent" solution for the opposite lobes, i.e. from the DYNAGE t of their

radio spe tra with a model assuming ommon values of its free parameters

Q

jet

and

ρ

0

.

A omparison between the modelpredi tion and the above data for real sour es is

0,0 0,2 0,4 0,6 0,60 0,80 1,00 1,20 1,40 1,60 t = 10 Myr t = 100 Myr D D 1 - 2 0,0 0,2 0,4 0,6 -1,00 -0,80 -0,60 -0,40 -0,20 0,00 0,20 0,40 0,60 t = 10 Myr, k' = 0 t = 100 Myr, k' = 0 t = 100 Myr, k' = 10 l o g P l o g P 1 - 2 b

Figure5:

D

1

/D

2

(plot a)and

logP

ν1

− logP

ν2

(plotb)vs.varyingdieren e

β

1

− β

2

forthe xedvalueof:

α

inj

,

r

,

R

t

andtwodierentvaluesofdynami alagesand

k

′

.The urvesindi ate

mediansofthe distributionsof allindividualmodelsolutions for

D

1

/D

2

and

P

ν,1

/P

ν,2

ratios, al ulated asthefun tionof in reasing dieren esbetween the

β

parameters in theopposite lobes. The error bars are the standard deviation in these distributions. Cir les represent

4.3 Asymmetries resulting from dierent axial ratio

R

t

Observations shows that the opposite lobesmay have signi antly dierent values of

R

t

. Following Kaiser (2000), it is asummed that the axial and transversal expansion of the radio lobe, governed by the presssure ratio

P

hc

, is related to its axial ratio

R

t

( f. Eq. 5). Therefore, observed dieren es in

R

t

values in the opposite lobes suggest dierent ratios of the pressure inside the external environment along and a ross the

jets (dierent values of

P

hc

at the ends of the opposite jets of the same sour e), thus dierent physi al onditions in the IGM surrounding the jet's material in luding its

inhomogenity.

Dierent values of

R

t

inuen es both the lobe's length and luminosity, besides dierent

β

exponent an ause an observed asymmetry of these lobe's parameters. I assume that

R

t

varies from 1.5 to 6 with the maximum ratio of 2. Figure (6) shows expe ted ratios of

D

1

/D

2

(plot (a) and

P

ν,1

/P

ν,2

(in log s ale, plot (b), respe tively, versus theratioofdieren eof

R

t,1

/R

t,2

al ulatedfortwodierentagesofthedu ial sour es -

t = 10

and

t = 100

Myr. Additionaly, plot b shows the model predi tion for

t = 100

Myr and

k

′

= 10

.Open ir les markthesame observationaldata asinSe tion 4.2.

0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 0,60 0,80 1,00 1,20 1,40 1,60 t = 10 Myr t = 100 Myr D D R T1 / R T2 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 -1,00 -0,80 -0,60 -0,40 -0,20 0,00 0,20 0,40 0,60 t = 10 Myr, k' = 0 t = 100 Myr, k' = 0 t = 100 Myr, k' = 10 l o g P l o g P R T1 / R T2 b

Figure6:

D

1

/D

2

(plota)and

logP

ν,1

−logP

ν,2

(plotb)vs.varyingratioof

R

t

parameterfora xedvaluesof:

α

inj

,

r

,

β

andtwodierentvaluesofdynami alagesand

k

′

.The urvesindi ate

medians of the distributions of all individual modelsolutions (as in Se tion 4.2.) al ulated

asthefun tion ofin reasing dieren esbetween the

β

parameters intheoppositelobes. The error bars are the standard deviation in these distributions. Cir les represent observational

4.4 Asymmetries resulting from dierent

α

inj

parameter

In ontrast to the parameters

β

and

R

t

,

α

inj

ae ts onlythe value of the mono hro-mati radio power at a given frequen y. It does not have any inuen e for the size

of the lobe. Diagrams in this se tion show how a dieren e in the "ee tive" density

distribution of relativisti parti les a ross the bow-sho k at the head of the opposite

jets an result in a diverse syn hrotron emission of the relevant lobes and expe ted

symmtetryof

P

ν,1

/P

ν,2

.The al ulationsare performedwithidenti alvaluesofthefree parameters of the modelas inprevious twose tions, but hanging values of

α

inj

only. It is assumed that the values of

α

inj

may hange from 0.5 to 0.8, with the maximum dieren e

α

inj,1

− α

inj,2

= 0.15

for the opposite lobesof the du ial sour e.

Resulting diagram is presented in Figure (7). As previously, the urves show the

median values of

P

ν,1

/P

ν,2

in a set of the model solutions of this ratio for a given dieren e of

α

inj,1

− α

inj,2

. Again, the verti al bars show the standard deviation from these median values. Open ir les markthe observational data as in Se tions 4.2 and

-0,05 0,00 0,05 0,10 0,15 0,20 -1,00 -0,80 -0,60 -0,40 -0,20 0,00 0,20 0,40 t = 10 Myr, k' = 0 t = 100 Myr, k' = 0 t = 100 Myr, k' = 10 l o g P l o g P inj1 - inj2

Figure7:

logP

ν,1

− logP

ν,2

vs.varyingdieren eof

α

inj,1

− α

inj,2

parameterforaxedvalues of::

β

,

r

,

R

t

andfordierentvaluesofdynami alagesand

k

′

.The urvesindi atemediansof

thedistributionsofallindividualmodelsolutions(asinSe tion4.2.) al ulatedasthefun tion

ofin reasingdieren esbetweenthe

α

inj

parametersintheoppositelobes.Theerrorbarsare the standard deviation in these distributions. Cir les represent observational data available

4.5 Asymmetries resulting from dierent

k

′

parameter

Dierentvaluesof

k

′

(theratioofenergydensityofthethermalparti lestothatofthe

relativisti parti lesinthelobe'smaterial)also auseanasymmetryinradioluminosity

of the opposite lobes. It ae ts only the value of radio power (Se tion 3.2) and has no

inuen e intolinear size of the lobe. Anestimation of the value of

k

′

inreal sour esis

verydi ult.Forexample,Bro ksoppetal.(2011),modellingthe doubledoubleradio

galaxyB1450

+

333, onsidered

k

′

values from0 to100.

The al ulations are performed with identi al values of the free parameters of the

model as in previous two se tions, but hanging values of

k

′

only. It is assumed that

they may hange from0 ( orrespondingto the ase of nothermal parti lesin the jet)

to30,with themaximumratio

k

′

1

/k

′

2

= 15

for theoppositelobesof thedu ialsour e. Resulting diagram is presented in Figure (8). The urves show the median values of

P

ν,1

/P

ν,2

ina set of the modelsolutionsof this ratio for a given ratiosof

k

′

1

/k

′

2

.

0 2 4 6 8 10 12 14 16 18 -1,00 -0,80 -0,60 -0,40 -0,20 l o g P l o g P t = 10 Myr t = 100 Myr k 1 ' - k 2 '

Figure8:

logP

ν,1

− logP

ν,2

vs.varyingratioof

k

′

parameterforaxedvaluesof::

β

,

r

,

R

t

and for two dierent values of dynami al ages. The urves indi ate medians of the distributions

of all individual model solutions (asin Se tion 4.2.) al ulated asthe fun tion of in reasing

dieren esbetween the

k

′

parameters in theopposite lobes. Theerror bars arethestandard

Observed dieren es in linearsizes and mono hromati radiopowers of the opposite

lobes of FR II type radio sour es are often too large to be explained by the sele tion

ee ts or proje tion of the sour e on the plane of the sky only. The al ulations

per-formed in Se tions 4.2, 4.3, 4.4 and 4.5 showed that su h asymmetries an be easily

predi ted with the KDA model assuming dierent physi al onditions in these lobes

and/ortheir environment.

These al ulations pre ise the inuen e of dierent values of

β

,

R

t

and

t

model parametersonthe lobe'slengthasymmetry

D

1

/D

2

(asexpe tedfromEg.4),aswellas onrmasuppositionthatasigni antinuen eontheluminosityasymmetry,

P

ν1

/P

ν2

, have themodel parameters

β

,

R

t

,

α

inj

,

k

′

and

t

.Althoughdierentage ofthe opposite lobesisrather not admissible,this is worth emphasizingthat the inuen e ofabsolute

age value of the du ialsour e onthe asymmetries onsideredhere ismeaningful.

4.6.1 Model predi tions for the asymmetries

Asymmetryoflobe's lengthdepends onthe valuesofmodelparameters

β

and

R

t

.This inuen eis stronger inthe ase of

R

t

parameterdeterminingthe pressure ratiosinthe lobe,

P

hc

( f.Figure(6a)).In turn,Figure(5a)presents very strongdependen yofthe sour e's age onthe resulting

D

1

/D

2

asymmetry. These asymmetries in rease with the age of a given sour e.

Asymmetries of lobe's radio brightness may result from dierent values number

of the model parameters. In Se tions 4.2, 4.3, 4.4 and 4.5 I analyzed the dependen e

of su h asymmetry on the values of parameters

β

,

α

inj

,

R

t

and

k

′

. A variation of

the

β

parameter gives relatively low

P

ν,1

/P

ν,2

asymmetries slowly in reasing with the in reasing age of the sour e,while for parameters

α

inj

,

R

t

and

k

′

the predi ted degree

of the asymmetry isnearly the same and highlydependent onthe sour e's age.

As on erns the

P

ν,1

/P

ν,2

asymmetry, one an ompare its dependen e on the pa-rameters

β

,

α

inj

and

R

t

with that resulting from dierent values of

k

′

. It is almost

equally strong as the impa t of the hange of the sour e's age on shape of the model

urves. At the same time it isdi ultto learly determine whi h one of these

param-eter inuen es the greatest asymmetry. In the ase of the parameters

β

and

α

inj

it is the age, but in turn for

R

t

it isa hange of the parameter

k

′

.

It is learly seen that whilein the ase ofasymmetry inthe lengthof the lobesthe

number of fa tors that may ause them is relativelysmall, whereas the asymmetry of

their radio brightness may be aused by dierent values of several model parameters

and thusit isnot easy todetermine whi hone of them has a de isiveinuen e inthis

Asymmetries of lobe's length and brightness predi ted by the model are onfronted

with the observed asymmetries in the smalland heterogeneous (only available in the

literature)sampleof30"giant"radiogalaxies(Ma halskietal.2009,Ma halski2011).

In both of these publi ations the observed asymmetries

D

1

/D

2

and

P

ν,1

/P

ν,2

are re-produ ed by varying the values of only two model parameters, namely

β

and

α

inj

. In parti ular,dierentvaluesof

β

inthe oppositelobes(where usually dierent values of

R

t

are known from observations) are used to explain the observed asymmetry in the lengthof the lobes. In the next step, the observed asymmetry in the brightness of the

oppositelobesof radio galaxiesfromthose samplesisreprodu edby a variationofthe

α

inj

parameter in these lobes. In this way the observed ratio of

P

ν,1

/P

ν,2

is des ribed by a diversity of not the only one, but three parameters of the model,

R

t

,

β

and

α

inj

. Thus these observations onrm the on lusions from the previous paragraph that a

givenasymmetryoflobe'slengthandbrightness annotbedes ribedbydierentvalues

of a single modelparameter, however it an be provided by a ombination of several

dierent values of its free parameters only. This is in a ordan e with the physi s of

theFR IItyperadiosour e,be ausetheradio brightnessatagiven frequen y depends

onenergydistributionofrelativisti parti les, strengthandorientationofthe magneti

eld, and the volume of the sour e (or its lobes), thus on mu h larger number of the

model'sfree parametersthan the lengthof the lobes. Itis alsoevident fromthe

exam-pleof the observationaldataof thesamplesour es. The diagramsillustrating

P

ν,1

/P

ν,2

asymmetry show mu h more noti eable dispersion of the sample sour es around the

model urvesthan the diagrams onfronting the observed

D

1

/D

2

asymmetry withthe modelpredi tions.

Onlythesmallfra tionofobservedasymmetries orrespondstothepredi tionofthe

modelpresented onFigures(5), (6)and(7). Itis worthnoting thattheseasymmetries

in real radio sour es mainly orrespond to the model urves al ulated for sour e's

age of100 Myr and for dierent values of

k

′

. Itis understandablebe ausethe samples

omprisesour eswithverylargelinearsizesandagesofabout

≃

100Myr,notin luding, however, smalland relativelyyoung sour es.

Inthe aseofthediagramspresentingthe

P

ν,1

/P

ν,2

asymmetryone annoti estrong deviationsofthenotablepartoftheobservedsour esfromthemodelpredi tions.These

sour es are not even lose to the urves al ulated for the sour e age of 100 Myr (for

both values of

k

′

), though their tted dynami al ages al ulated for the sour es have

typi alvalues of this range and are not ex eeding the value of 250 Myr.

Theabove on lusionsarenot omprehensiveinthesensethatthesampleisnot

rep-resentative for the entire FR II typesour es population. As itwas alreadymentioned,

thesele tionee t ausesthatit onsistoflargeradiogalaxiesonly.Therefore,afuture

resear h shouldfo us onthe omparisonof modelpredi tions for the asymmetry with

radio-jets propagation through the X-ray haloIGM

interfa e (Kuligowska et al. 2009)

Kuligowska, Jamrozy, KozieªWierzbowska & Ma halski,

2009, A A, 59, 431

Extended large-sized radio sour es are not easy to re ognize be ause of their

relatively low radio brightness and a di ulty to dete t eventual bridge onne ting

brighter parts (lobes) of a ommonradio stru ture. Several observational eorts show

that most of known GRGs lie at low redshifts of z

<

0.25. For a long time this aused a presumption that su h extragala ti double radio sour es, espe iallythose of

FRII-type, did not exist at redshifts higher than about one be ause of the expe ted strong

evolutionofauniformIGM,

ρ

IGM

∝ (1+z)

3

, onningthelobesofsour es(e.g.Kapahi

1989). The situation hanged over 10 years ago when Cotter, Rawlings & Saunders

(1996)and Cotter(1998) presented anunbiasedsample ofgiantradio sour essele ted

from the 7C survey (M Gil hrist et al. 1990). Their sample omprised 12 large-size

sour es with 0.3

<

z

<

0.9. The list of known GRGs with z

>

0.5 and D

>

1 Mp is very short.The undertaken sear hforsu h GRGsonthe southern sky hemispherewiththe

11m SALT teles opeduring the Performan e Veri ation (PV) phase has resulted in

the dete tion of 21GRGswith the proje ted linear size greaterthan 1Mp .However,

one anfoundthattheirredshiftsdonotex eedthevalueof0.4andtheenergydensity

in only two of them is less than

10

−

₁₄

Jm

−

₃

. One of them, J1420-0545, is the largest

known GRGin the Universe ( f.Ma halski etal. 2008).

Thedynami alevolutionofaFRIIradio sour estronglydependson hara teristi s

of the ambient medium. Gopal-Krishna & Wiita (1987) proposed the two-medium

model onsistingofanX-rayhaloaroundtheparentgalaxywithgasdensityde reasing

with radial distan e from the galaxy and a mu h hotter intergala ti medium (IGM)

with onstantdensity.Thesetwomediawere on eivedtobepressure-mat hedattheir

interfa e.Their modelallowed topredi tlimiting(maxima)values forthesour e's age

andlinearsizedependingontheenvironment onditions,thejetpower, andthe osmi

epo h hara terized by the sour e's redshift. However, our re ent dete tions of very

large-sized radio sour es with z

>

1 and ex eeding the limits predi ted by their model (hereafterreferredtoasG-KWmodel),suggeststhatsomeofitsfreeparametersshould

be modied.

In this Se tion an observational onstraint for the G-KW model is analyzed. For

this purpose, aneort todetermine the highestsizes anddynami alages of FRII-type

radio sour es at redshifts 1

<

z

<

2 is undertaken. The original G-KW model is briey des ribed and modied adopting modern ( ontemporary) values for thermodynami

head and the age are al ulated.

Theobservationaldatausedto onstrainthetwo-mediummodelarepresented. The

smallsampleof themost distantgiant-sizedradio sour esisrevisedand supplemented

withtwootherlimitedsamplesofFRII-typesour es omprising:(i)sour eslargerthan

400 kp within the redshift range 1

<

z

<

2, most of them found in this paper, and (ii) sele ted3CRRsour esinmajoritysmallerthan400kp atz

>

0.5forminga omparison sampleof "normal"-sizedradiosour es. Physi alparametersofthe samplesour es:the

dynami al age, the jet power, the entral radio- ore density and the IGM density,

and others, are derived using the "DYNAGE" algorithm (Ma halski et al. 2007a).

Theappli ation ofthis algorithmtothesamplesour es andthe resultingvalues ofthe

sour e'sparametersaredes ribedinSe tion5.2.A omparisonofthemodelpredi tions

with the observational data is presented and dis ussed inSe tion 5.3.

5.1 The base of the revised G-KW model

Inthe G-KW model,the jetpropagates intoatwo- omponentmedium omprised of:

thegaseoushalowithapower-lawdensityprole

ρ

h

(d) = ρ

0

[1 + (d/a

0

)

2

_]

−

δ

bound

totheparentopti algalaxy,where

ρ

0

and

a

0

arethedensityandtheradiusofthe entral radio ore, respe tively, and

δ

=5/6. This distribution is assumed to be invariant with redshift. It is also assumed that this halo has nearly uniform ele tron temperature

(kT )

h

[keV℄ (medium1),and

 the surrounding hotter IGM of uniform density,

ρ

IGM

, with the temperature

(kT )

IGM

(1 + z)

2

[keV℄ (medium2).

SimilarilytoGopal-Krishna&Wiita(1987)itisne essery toassume hara teristi

values for the density and temperature of the onsidered media. The values adopted

hereafterfor the two omponentsare based on the followingdata:

(1)The radio oreradius,

a

0

=3kp is basedonthe tted X-ray surfa e-brightness proleofninenearby,low-luminosityradiogalaxiesre entlyobserved byCroston etal.

(2008). This value of the radius is derived from the observed angular radius of about

10ar se .

(2) The halos' gas temperature have been determined in a number of papers. A

uniform temperature

(kT )

h

=0.7 keV was measured for a few nearby, X-ray luminous ellipti al galaxies with the Chandra Observatory by Allenet al. (2006). Using

XXM-Newtonand Chandra observations,the values from1to5keV with amedianofabout

2.1 keV was found by Belsole etal.(2007) for the X-ray lusters surrounding 20

lumi-nous 3CRR radio sour es. For the low-luminosity radio galaxies analysed by Croston

et al. (2008), a medianof the tted temperatures is about 1.4 keV. Taking the above

data intoa ount one an estimate:

(kT )

h

=1.4 keV. (3)The halos'gas(proton) density of (1 2)

×10

4

m

−

3

istted to X-ray ountsby

pressure againstthepressure distributioninthehalo. Anon-relativisti gasinthermal

equilibriumthat hasanele tron density

n

e

[m

−

₃

℄and temperature

(kT )

e

[keV℄ willhave anele tron pressure

p

e

=

n

e

(kT )

e

[Pa℄. Expressingele tron density by the mass density,

ρ

=

n µ m

H

, this balan ewill have pla e atthe halo'sradius

R

h

al ulated from

ρ

0

µ

h

m

H

1 + (R

h

/a

0

)

2



−

δ

(kT )

h

=

ρ

IGM

µ

IGM

m

H

(kT )

IGM

,

(9)

where

µ

and

m

H

are the mean mole ular weight and the mass of hydrogen atom, respe tively.

µ

h

is assumed to

µ

h

=0.5 and

µ

IGM

=1.4. Besides, for the halo (medium 1)

n

p

=

1.5 × 10

4

m

−

3

is adopted (i.e. a mean proton density of the values given by

Belsole et al. (2007), whi h orresponds to

ρ

0

=

10

−

22.6

kgm

−

3

, and the temperature

(kT )

h

=1.4 keV). Forthe IGM density 50% of the osmi matter density is taken, i.e.

ρ

IGM

=

0.5Ω

m

h

2

_ρ

clos

=

0.5 × 0.27 × 0.71

2

_{× (3 H}

2

0

)/(4π G)

, whi h gives

ρ

IGM

=

10

−

26.9

kgm

−

₃

. For the IGM temperature the values of

(kT )

IGM

=25 keV is adopted. Substi-tuting the above values into Eq.(9) one an nd

R

h

=642 kp . This radius of X-ray halo is ompatible with the radii determined by Cassano et al. (2007) for 15 Abell

luster radio haloes with the mean of

∼ 560±

170 kp . This is worth to noti e that thit radiusof 642kp ismu hlargerthan 171 kp used by Gopal-Krishna&Wiita.In

anexpanding and uniform IGM this radius should evolveas

R

h

(z) = 642(1 + z)

−

_5/(2δ)

kp , i.e.

642(1 + z)

−

₃

kp for

δ

=5/6.

Figure (9) (a), (b), ( ) present the basi hara teristi s of the two-media model:

the mass density

r(d)

, the ele tron temperature

kT (d)

, and the resulting ele tron gas pressure

p(d)

, as fun tionsof the distan e from the host galaxy ( ompa t radio ore), respe tively. Note that the balan e

p

h

(R

h

) = p

IGM

at

d = R

h

orresponds to a rapid transitionbetween

r(R

h

)

and

r

IGM

,aswellasbetween

kT (R

h

)

and

kT

IGM

,and auses an unphysi al ee t shown in Se tion 5.6.1. The dashed urves in Figure (9) (a) and

(b) indi ate desired smooth transitions between the relevant parameters whi h would

5.2 Predi tions of the model

For

d ≤ R

h

it is assumed that the jet propagate (through the medium 1) with a onstant opening angle,

θ

. Under this ondition, the ram pressure balan e results in thefollowingdependen es forthe jetlength(theradiolobesize,

D

)ontime(thelobe's age,

t

) and the jet's head expansionvelo ity,

v

h

, on

D

or

t

:

D(t) = [(2 − δ) A t]

2−δ

1

_,

(10)

v

h

(D) = A D

(δ−1)

,

(11)

v

h

(t) =

h

(2 − δ) A

δ−1

1

t

i

δ−1

_2−δ

,

(12) where

A ≡



4 c

1

Q

jet

πθ

2

_{c ρ}

0

a

2δ

0



1

₂

.

(13)

Here

c

1

isa onstantwithavaluebetween1.5and3.8dependingonthesour e's(lobe's) geometrydes ribedbyitsaxialratio

R

t

(Kaiser&Alexander1997),while

c ≈ v

jet

isthe speed oflight.Thejet'sopening angleisalsodes ribed by

R

t

,

θ

2

_{= c}

2

/(4 R

t

)

,where

c

2

isa onstantwithavaluebetween3.6to4.1dependingonspe i heatsforthematerial

inthe jetand the lobe( o oon), (Eq.(17) inKaiser&Alexander 1997).At

d = R

h

(z)

the jet enters the hotter IGM (medium 2) at least an order of magnitude less dense

butpressure-mat hed,asshown inFigure(9).Inordertoanalyzethe jet'spropagation

overthis regime,Gopal-Krishna& Wiitahave onsidered two likely extremes enarios

for the lobe's expansion:

S enario Awherethe jet openingangle,

θ

,is onserved. Due toarapidde rease of the ambient density at the interfa e,

ρ

IGM

≪ ρ

h

(R

h

)

, a su ient ram-pressure will be provided only if the jet's head velo ity,

v

hs

, in reases abruptly at

d

=

R

h

and then graduallyapproa hes the

v

h

∝ d

−

₁

lawexpe tedfora onstantdensitymedium.Inthis

s enario:

D(t) =

(

2

K(z) + a

δ

₀

A



ρ

0

ρ

IGM

(1 + z)

3



1/2

t

!)

1/2

and

(14)

v

h

(t) = a

δ

0

A



ρ

0

ρ

IGM

(1 + z)

3



1/2

/D(t),

(15) where

K(z) =

1

2

R

2

h

(z) − R

(2−δ)

h

(z)

a

δ

0

2 − δ



ρ

0

ρ

IGM

(1 + z)

3



1/2

(16)

is a redshift-dependent onstant providing that the time orresponding to

D

=

R

h

inEqs.(10) and (14) is the same.

S enario B wherethe jet'sheadvelo ity a rossthe interfa e remains ontinuous

and thereforemat hed tothe value given by Eq.(11) for

D

=

R

h

.This an be a hieved onlywith anabrupt aringof the jet's opening angle.Under this ondition the model

predi ts:

D(t) =



2



A R

δ

_h

(z) t − R

2

_h

(z)

δ

2(2 − δ)



1/2

and,

(17)

v

h

(t) = A R

δ

h

(z)/D(t).

(18)