Effect of errors on arm asymmetry of doubles
by
Dilip G Banhatti
[dilip.g.banhatti@gmail.com, banhatti@uni-muenster.de]
School of Physics
Madurai Kamaraj University
Madurai 625021
India
Abstract / Summary
Measured values of arm asymmetry parameter x = (θ> - θ<)/(θ> + θ<) of a double have
appreciable random errors due to errors in positions of radio peaks & of the optically
identified galaxy or quasar. These broaden the monotonic decreasing x-distribution g(x).
In addition, finite resolution & blending of complex structure leads to errors in
recognizing peaks leading to systematic overestimate of x. Thus both random &
systematic errors broaden g(x), & consequently broaden the distribution p(v) of derived
hotspot separation speed v, & shift its peak to larger v, since p(v) = -v.g’(v).
Keywords: active galaxies – double radio sources – bilateral symmetry – arm asymmetry
Fractional arm difference x
The arm asymmetry of a straight extended extragalactic edge-brightened (i.e., FR2
(Fanaroff & Riley 1974)) double radio source associated with an active galaxy is very
usefully specified by the fractional arm difference x = (θ> - θ<)/(θ> + θ<) (Banhatti 1979,
1980, 1984/5, Best et al 1995), defined to be the ratio of the difference of the two arms to
their sum, so that 0 ≤ x ≤ 1, ranging from fully symmetric double (x = 0) to fully
asymmetric one (x =1).
Errors affecting x = (θ> - θ<)/(θ> + θ<)
The lengths θ> & θ< of the two arms of a double are determined from the positions of the
hotspots & central radio cores or optical identifications, which have random errors of
measurement. Moreover, the location of a hotspot is systematically shifted towards inner
part of the linear source due to blending with lobe & coarse resolution. Thus, θ> & θ< are
systematic underestimates. Taking true values to be θ> + ∆ & θ< + ∆ (∆ > 0),
xtrue = (θ> - θ<)/(θ> + θ< + 2.∆) = x/(1 + 2.∆/θ) where θ> + θ< ≡ θ is the total angular size of
the straight double. Thus xtrue < xmeasured, so that the measured values of x are
overestimates. One may take ∆ to be independent of θ to first approximation. Then xtrue is
most different from xmeasured for smallest θ (= overall angular size). For the error on x to
be independent of x, x & θ should be uncorrelated. This indeed seems to be the case, as
tested by Teerikorpi’s (1986) Table 2.1 & Fig.2, using x vs θ scatter diagram.
The random errors on x are affected by many factors, & hence may be taken to be
Gaussian, equally likely to be positive or negative.
Banhatti (1979, 1980, 1984/5) uses the distribution g(x) of x to derive the distribution
p(v) of hotspot separation speeds v in an intrinsically symmetric model, where the
asymmetry is fully ascribed to double source orientation out of sky plane & (consequent)
light-travel time differences for radiation emanating from the two hotspots. He finds that
p(v) = - v.g’(v) where prime denotes differentiation. From observed samples of doubles,
the function g(x) is found to be decreasing from a maximum at x = 0 to zero at x = 1.
Considering the x-distribution g(x) as a histogram, the number of sources going out of a
bin due to random errors in x is proportional to the bin population. Since the bins
decrease in population toward large x, more x-values will shift toward larger x at every x,
so that g(x) widens due to random errors in x.
Thus both systematic & random errors in x cause widening of the x-distribution g(x).
This leads to a corresponding widening of the v-distribution p(v), & a shift of its peak to
a higher value. Thus the true p(v) is narrower & is peaked at smaller v than measured
(Banhatti 1984/5).
Bibliography
Banhatti, D G 1979 Bull Astr Soc India 7 116 Expnsn speeds in double radio sources
Banhatti, D G 1980 A&A 84 112-4 Expnsn speeds in extdd extragalactic dbl rad srcs fm
angular structure
Banhatti, D G 1984/5 PhD thesis TIFR / IIT Bombay Evolution of extragalactic rad srcs
Best, P N et al 1995 MNRAS 275 1171-84
Fanaroff, B L & Riley, J M 1974 MNRAS 167 31P- 35P … … …
Teerikorpi, P 1986 A&A 164 L11-L12 Rad asym vs size for qsrs with diffrt asym types
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