3.9: Comparison of PT I flux-densities and VLA flux-densities for the
"radio cores of a subset of sources in our sample. Filled circles represent flat
spectrum cores, open circles are steep spectrum cores
(iii) in d ep en d an t of frequency,
(iv) can be used in th e largest possible fraction of th e d ata.
T hey also m ention a num ber of possible m eth o d s for e stim atin g source size. O ne m eth o d com m only used for class II radio sources is th e distan ce betw een th e h o tsp o ts at the o u ter edge of the lobes. T his m eth o d satisfies th e first th ree c rite ria listed above, b u t cannot be used for th e class I sources w hich com prise h alf of our sam ple.
A nother m ethod is to determ ine some ch aracteristic of th e visibility function such as th e sp atial frequency for which the visibility function of th e rad io source first falls to 0.5. A lternatively, one could calculate th e first m om ent of th e source d istrib u tio n function. These two m easures have th e advantage th a t they m ay be applied to p artially resolved d ata. Ekers and Miley also note th a t a bad m easure of source size is the distance betw een o u term o st contours since th is depends on receiver sensitivity and resolution.
We use a m eth o d which is essentially a h y b rid of th e m om ent m eth o d an d the visibility function m ethod. T he m eth o d involves a m om ent analysis of th e clean com ponents of th e radio m ap and is presented in detail in ap p en d ix B. We use this m e th o d for a num ber of reasons. M om ent analysis is best for class I sources since unlike class II sources they generally d o n ’t have a well defined edge. M om ent analysis can still be applied to class II sources an d it is desirable to use th e sam e m e th o d for all sources in th e sam ple. M om ent analysis is an objective m eth o d of estim atin g sizes which is consistent for th e en tire sam ple. M oreover, m om ent analysis is versatile. M om ent analysis can yield lengths, w id th s and position angles of radio sources. H igher m om ents of th e brightness d istrib u tio n give flux- density ratio s of radio lobes (a skewness p a ra m ete r) and a cen tral brig h tn ess p a ra m e te r to be discussed later.
Use of the CLEAN algorithm to deconvolve th e d irty beam from rad io m aps provides a n a tu ra l m eth o d to perform th e m om ent analysis because th e radio b rightness d istrib u tio n is m odelled as a set of po in t sources convolved w ith the clean beam . These point sources are referred to as th e “clean com ponents.” Schwarz (1978) has shown th a t th e c l e a n m eth o d is equivalent to a least squares fit to th e visibility d a ta. Because of this, a m om ent analysis is in d e p en d a n t of resolution (provided th e source is resolved). Schwarz (1979) advocates m om ent analysis of th e d istrib u tio n of clean com ponents to give th e to ta l flux (zeroth m om ent), position (first m om ent) and an g u lar size (second m om ent) of th e source.
In a seperate p ap er B u rn and Conway (1976) suggest a n u m b er of p a ra m ete rs derived from m om ents of a one dim ensional radio brightness d istrib u tio n . T hey go beyond the second m om ent to define a skewness p a ra m e te r (th ird m om ent)
an d a c en tral brightness p a ra m ete r (fourth m om ent). T hey suggest th a t the c en tral b rig h tn ess p a ra m ete r m ay be useful in distinguishing betw een class I and class II ra d io sources— p articu larly for poorly resolved sources.
In ap p en d ix B we generalise th e p aram eters of B u rn an d Conway (1976) to tw o dim ensions an d in co rp o rate Schw arz’ advocacy of clean com ponents to p ro d u ce a two dim ensional m om ent analysis of th e clean com ponents of our radio sources. E ach clean com ponent possesses a flux-density and p osition vector on th e sky. T h e zero point for the position vector is th e phase reference for th e m ap. T h e sum of flux-densities in th e clean com ponents yields th e to ta l (cleaned) flux- d ensity in th e source. In cases of overresolution this m ay differ from th e actu al flux-density of th e source. T he first m om ent of th e clean com ponents norm alised by th e to ta l flux-density is a vector representing the centroid of th e rad io source w ith resp ect to th e m ap phase centre. Before continuing to calculate higher m o m en ts th e coordinates are tra n sla te d so th a t th e origin (or phase reference) lies on th e centroid of th e radio source. If this step is no t carried o u t th e n , in general, th e resulting p a ram eters derived from higher m om ents will be biased.
T h e second m om ent yields (in the two dim ensional case) a 2 x 2 m atrix . T his m a trix is a two dim ensional analogue of th e m om ent of in e rtia ten so r used in m echanics to describe th e m ass d istrib u tio n of an extended body (e.g. G oldstein 1980 p l9 8 ). T he eigenvectors of this m atrix correspond to th e principle axes of th e brig h tn ess d istrib u tio n . T he corresponding eigenvalues are re la ted to th e len g th p a ra m ete r, R i , an d w id th p aram eter, R2, defined in th e appendix. T he
eigenvectors are used to derive the position-angle of th e radio source.
T h e eigenvectors are also essential to deriving higher order m om ents for the rad io source. In general th e n -th m om ent of th e d istrib u tio n is a sym m etric te n so r of ra n k n. To gain any physical insight from these m om ents th e tensors m u st be co n tracted onto a given vector. C o n tractin g th e m om ent ten so rs onto th e eigenvectors of th e brightness d istrib u tio n yields th e corresponding m om ents along th e principle axes of th e radio source. An exam ple of th is process for a sim ple brig h tn ess d istrib u tio n is given in appendix B.