P a r a m e te r S y m b o l N o r m a lis a tio n C o m m e n ts
Surface brightness I'u h/0
FW H M g' ctq
Spectral index Cooling factor
a
J ( a , X ) Jo
Age p aram eter M agnetic field
X 1/ 2
B B i c Inverse C om pton B field.
N orm alised density n' n0 D erived from norm alised D istance along lobe s' si = 1 kpc
qu an tities.
N orm alised by a rb itra ry scale. Velocity v' V \ ui = ß B 3/ c s \ v l /2(c\ s in # )- 1 / 2
(using equation 3.8). We estim ate th e ru n of norm alised electron density n' using equation (4.3) for th e surface brightness:
n / ( a + 2 ) / 3 £ / ( a + l ) / 2 _ 1 "
a' J ' (4.6)
where we have ad o p ted th e convention th a t prim ed q u a n tite s are norm alised by th eir initial values.
D ifferentiating eq u atio n (4.2) an d su b stitu tin g in equation (4.5) yields th e plasm a velocity at any p o in t in th e flow (lobe or jet):
vi
(1 + B 2) r d
B ' / i . X1/ 2 U s'In ( n n l
3B ' l2X l l 2) (4.7)
Here th e length p a ra m ete r (s' = s / si) is norm alised by some suitable length scale si and th e velocity (v' = v / v \ ) is norm alised by:
v\ = ß B \ ^ s \ v x^ ( c \ s'mß) 1/ 2. (4.8)
T he m agnetic field is norm alised by th e inverse C om pton equivalent m agnetic field so th a t B = B / B j c• T h e inverse C om pton equivalent m agnetic field is
Bj c = 3.2(1 + z 2) f.iG (de Young 1976). Since we use a variety of norm alisations,
we sum m arise all th e norm alised p aram eters in tab le 4.1. E q u atio n (4.7) is the basis for our m eth o d of determ ining lobe and jet velocities from m aps of spectral index and radio surface brightness. T he m eth o d an d results are discussed in the next section.
N ote th a t this m ethod for analysing sp ectral aging is su b stan tially different from th e m ethods applied by A lexander an d Leahy (1987) and M yers an d Span gler (1985) in th a t it takes in to account m agnetohydrodynam ical variations in th e flow which affect th e ra te a t which th e synchrotron sp ectru m evolves. We have already no ted th a t th e ra te of change of the sp ectral index w ith tim e de pends on the local value for th e electron density an d the m agnetic field. In their analysis of 3C R radio sources, A lexander and Leahy (1987) evaluate th e ra te of change of sp ectral index w ith distance along th e lobe, d a / d s w ith o u t regard for th e m agnetic field stre n g th in th e source and i t ’s effect on th e ra te of spectral index evolution. M yers and Spangler (1985) include th e m agnetic field in their analysis b u t fail to take in to account variations in th e m agnetic field stren g th along th e flow. B o th of these studies also neglect density variations along the flow and i t ’s effect on the ra te of sp ectral index evolution. M oreover, we exam ine th e self-consistency of o ur analysis by exam ining w hether th e continuity equation
(nvcr2 = c o n stan t) is satisfied w ith in th e flow.
4.2.2:
Velocity determinations
Applying th e theory outlined above we have a tte m p te d to o b tain velocity d eterm in atio n s for selected regions in 10 of th e sources in our sam ple.
T he startin g po in t in o u r analysis is a series of surface brightness profiles o b tain ed from th e 1.4 GHz m aps. We call these surface brightness profiles “slices.” In general, several slices are o b tain ed along each lobe following th e ru n of peak surface brightness along th e lobe. T he slices are o rien ted p erp en d icu lar to the lobe (or je t) an d are sep erated by approxim ately half a beam -w idth or less (see figure la ). These slices yield th e ru n of peak surface brightness an d full-w idth at half-m axim um (FW H M ) along th e lobe. D econvolution of th e FW H M and peak surface brightness is perform ed using th e p ro jected beam w id th along the slice and assum ing a gaussian sh ap e for b o th th e lobe cross-section an d th e beam (e.g. Killeen, Bicknell an d E kers 1986). Identical slices are also tak en across th e sp ectral index m aps. A m ean sp ectral index w eighted by th e 1.4 GHz surface brightness profile is calculated for each p o in t along th e lobes. T h e sp atial position of each po in t is tak en to be th e m id p o in t of th e slice. M agnetic fields are assum ed to be m inim um energy fields a n d are calculated from the peak surface brightness an d FW H M for each slice using eq uation (3.8).
From th e d a ta we have a list of num bers giving th e peak surface brightness, FW H M , m inim um energy m agnetic field an d the ru n of sp ectral index as a func tion of distance along th e lobe (jet). In an aging plasm a w ith no reacceleration th e sp ectru m steepens along th e flow. In a je t th e youngest plasm a is found n ear the core w ith sp ectral index steepening dow nstream to w ard th e lobes. In a
class II radio source w ith backflow from th e h o tsp o ts, th e youngest plasm a is gen erally found near th e h o tsp o ts w ith sp ectral index steepening dow n th e backflow tow ard th e galaxy. We take th e initial po in t on th e flow, w ith flattest spectral index to be th e fiducial po in t which gives us the initial (zero age) electron energy d istrib u tio n index a. Since we know th e sp ectral index, a at each point along the flow we m ay invert eq uation (5.4) relating sp ectral index to th e synchrotron cooling function J ( a , X ) to o b ta in a value for th e sy n ch ro tro n cooling function at each p o in t along th e flow. We calculate the age p a ra m ete r X 1//2 from J ( a , X ) using a tab le of J ( a , X ) as a function of th e zero age electron d istrib u tio n index,
a. This process is shown diagram atically in figure 4.1c.
We also require th e norm alised p lasm a density n' . We estim ate this a t each po in t along th e flow from the o th e r p aram eters using eq u atio n (4.7) which relates th e norm alised plasm a density an d the norm alised m agnetic field, B' to obervable param eters. E rrors for each of these p aram eters are derived from th e surface brightness w eighted sta n d a rd deviation of th e sp ectral index a t each p oint.
So, given th e age p a ra m ete r, X 1/ 2 a t a n u m b er of p o in ts along the flow we proceed to estim ate th e velocity of th e flow which depends on th e derivative of the p ara m ete r In (n ,1/ 3# 1/ 2X 1/ 2) w ith respect to th e norm alised distance along the flow s' . T here are generally only a few resolution elem ents along any one radio lobe (or je t) yielding a t b est, several d a ta po in ts for each lobe. In view of the sparseness of th e d a ta , we use a least squares stra ig h t line fit to ln (n ,1/ 3ß 1/ 2X 1/ 2) as a function of s' to o b tain a m ean value for ^ 7ln (n ,1/ 3# 1/ 2X 1/ 2) along the flow. This avoids evaluating th e derivative directly from th e d a ta which would introduce u n acceptable noise in to our velocity d eterm in atio n . We ignore the initial slice (w here we s' = 0 an d X = 0) in our least squares fit because of the singularity in ln (n ,1/ 3ß 1/ 2X 1/ 2). For d a ta which is b e tte r sam pled, th e initial p a rt of th e flow could be b e tte r handled using th e lim iting form of equation (4.7) for th e flow velocity:
T he least squares fit to ln( n 11/ 3 B 1/ 2X 1/ 2) is w eighted by th e p a ra m ete r error at each po in t an d th e error for th e fitted slope is derived from th e sta n d a rd deviation of th e po in ts aro u n d th e fit. T h e slope is used to derive a velocity a t each point along th e lobe and these velocities are used to o b ta in a m ean velocity for the lobe.
A schem atic diagram of this process for estim atin g flow velocities from spectral index gradients is presen ted in figure 4.1c.
We note here th a t th e velocities we derive for th e class II backflows are th e velocities of th e flow w ith respect to the h o tsp o t. One can identify two com ponents to th e flow velocity in a radio lobe. T he h o tsp o t of th e radio source advances th ro u g h th e interg alactic m edium w ith a velocity determ ined by ram pressure balance (e.g. B landford an d Rees 1974). In add itio n light je ts w ith high M ach num b er exhibit backflow in th e lobe w ith respect to th e intergalactic m edium (N orm an et al. 1982, W illiam s 1985). Since th e flow velocity determ ined from th e sp ectral index variatio n is m easured w ith respect to th e h o tsp o t, the observed flow is th e sum of th e velocity of advance of th e h o tsp o t w ith respect to the in terg alactic m edium , vh plus th e velocity of th e backflow w ith respect to th e intergalactic m edium , v f .
^ ob served — T
In th e next section we generally refer to th e “backflow” as being th e observed flow velocity in a class II radio lobe. In section 5.3 when we discuss th e energy budget of radio sources, backflow refers solely to th e velocity of th e backflow w ith respect to th e interg alactic m edium .
In m ost cases p resen ted below, th e sc a tte r of th e velocities ab o u t th e m ean is very m uch sm aller th a n th e error bars suggesting th a t we have been overly pessim istic in th e original erro r d eterm in atio n from th e sc a tte r of sp ectral index along each slice. For this reason we take th e final error of th e m ean velocity to be th e sta n d a rd deviation of th e s c a tte r a b o u t the m ean, w eighted by th e inverse of th e errors on each po in t. Figures 4.1 th ro u g h to 4.30 present diagram s of the tre n d of relevant p aram eters for ou r velocity derivations. E ach panel, labelled a) th ro u g h to j) illu strates th e following.
P a n e l a ) th e ru n of th e peak surface brightness as a function of distance along th e lobe (jet). T h e u n its are erg s -1 c m -2 H z ~ l s r~ l . In all cases th e distance p lo tte d along th e x-axis is m easured along th e lobe in th e direction of th e inferred aging— away from the h o tsp o t in class II sources an d away from th e core in class I sources such as PK S 0344-345.
P a n e l b ) T h e ru n of th e full w id th a t half m axim um of th e lobe (jet) as a function of distan ce along th e lobe (jet).
P a n e l c) T h e ru n of sp ectral index a (S„ oc v ~ a ) p lo tte d as a function of distance along th e lobe (jet).
N o r t h e r n l o b e
S o u t h e r n l o b e
F ig u re 4 .1 a : A rtis t’s im pression of a class II radio source illu stratin g th e slices taken for th e aging analysis. T h e m id p o in ts of th e slices follow th e peak surface brightness of th e lobe. The slices are ta k en p erpendicular to th e peak surface brightness of th e lobe. T he to ta l in te n sity m aps yield surface brightness profiles across th e lobe (or je t). T he corresponding sp ectral index m aps yield th e ru n of spectral index along th e slice.
Minimum energy. Age parameter. Synchrotron cooling funct. Magnetic field. Normalised number density. velocity Radio lobe
width. Radio spectral
index. Radio intensity.
Linear fit to evaluate a v e r a g e :
Zero age
electron energy distbn power
index.
Variation along the flow of the parameter:
as a function of s