Jet evolution, flux ratios and light-travel time effects

0.1 MB·English

by  James C. A. Miller-Jones

DraftofFebruary2,2008
PreprinttypesetusingLATEXstyleemulateapjv.11/12/01
JET EVOLUTION, FLUX RATIOS AND LIGHT-TRAVEL TIME EFFECTS
James C. A. Miller-Jones and Katherine M. Blundell
UniversityofOxford,Astrophysics,KebleRoad,Oxford,OX13RH,U.K.
and
Peter Duffy
DepartmentofMathematical Physics,UniversityCollegeDublin,Dublin4,Ireland
Draft of February 2, 2008
4
ABSTRACT
0
0 Studies of the knotty jets in both quasars and microquasars frequently make use of the ratio of the
2 intensities of corresponding knots on opposite sides of the nucleus in order to infer the product of the
intrinsic jet speed (β ) and the cosine of the angle the jet-axis makes with the line-of-sight (cosθ), via
n jet
a the formalismIa/Ir =((1+βjetcosθ)/(1−βjetcosθ))3+α, where α relatesthe intensity Iν as a function
J of frequency ν as I ∝ ν−α. In the cases where β cosθ is determined independently, it is found that
ν jet

7 the intensity ratio of a given pair of jet to counter-jet knots is over-predicted by the above formalism
comparedwiththeintensityratioactuallymeasuredfromradioimages. Asanexampleinthecaseofthe

1 microquasarCygnusX-3theoriginalformalismpredictsanintensityratioof∼185,whereastheobserved
v intensityratioatonesingleepochis∼3. Mirabel&Rodr´ıguez(1999)havepresentedarefinedapproach
2 to the original formalism which involves measuring the intensity ratio of knots when they are at equal
8
angular separations from the nucleus. This method is however only applicable where there is sufficient
0
time-sampling (with sufficient physical resolution) of the fading of the jet-knots so that interpolation of
1
their intensities at equal distances from the nucleus is possible. This method can therefore be difficult
0
to apply to microquasars and is impossible to apply to quasars. We demonstrate that inclusion of two
4
0 indisputable physical effects: (i) the light-traveltime between the knots and (ii) the simple evolution of
/ the knots themselves(e.g.viaadiabaticexpansion)reconcilesthis over-prediction(inthe caseofCygnus
h
X-3 quoted above, an intensity ratio of ∼3 is predicted) and renders the original formalism obsolete.
p
- Subject headings: methods: analytical — radiation mechanisms: non-thermal — relativity — ISM: jets
o
and outflows
r
t
s
1. introduction where t and t are the times at which light leaves
a app rec
: the approaching and receding knots respectively in order
v Relativisticjetsareobservedinbothquasarsandmicro-
to arriveatthe telescope at the same time. Unless the jet
i quasars,andareoftenseentoconsistofaseriesofdiscrete
X axisisperpendiculartothelineofsighthowever,thelight-
knots moving outwards from a central nucleus, believed
travel time between approaching and receding knots will
r to correspond to the compact object powering the out-
a mean that we see the receding jet as it was at an earlier
flow. Measurements of the proper motions of these knots
time, when it was more compact and hence intrinsically
are often used to constrain properties such as jet speeds
brighter (but also dimmed in the observer’s frame by its
and inclination angles, and source distance (e.g. Mirabel
recessionalmotion, taken into account by the originalfor-
&Rodr´ıguez1999;Hjellming&Rupen1995). Theratioof
malism), compared with the approaching jet seen at the
the intensities of approachingand receding knots (if there
same telescope time. To account for this effect, Mirabel
is sufficientspatialresolutionthatthese canbe accurately
& Rodr´ıguez (1999) proposed that the flux densities used
identified) have been used (e.g. Saripalli et al. 1997) to
to calculate the ratio should be measured at equal angu-
constrain their Lorentz factors, via
lar separations from the nucleus. This cannot always be
S µ k+α 1+βcosθ k+α implemented in practice however, since this will require
app app
= = , (1)
S (cid:18)µ (cid:19) (cid:18)1−βcosθ(cid:19) interpolation unless good temporal coverage of the jets is
rec rec
available, or unless the jet is a continuous flow, in which
where β = v/c is the jet speed, θ is the inclination angle
case the motion of individual knots cannot be tracked in
of the jet axis to the line of sight, α is the spectral index
any case. At early times it may also be difficult to sepa-
ofthe emission(definedbyS ∝ν−α,whereS isthe flux
ν ν rate the emissionfrommoving jet knots and a fading core
densityatfrequencyν),S andS arethefluxdensities
app rec if there is insufficient spatial resolution. Moreover, as a
ofacorrespondingpairofapproachingandrecedingknots,
result of opacity or the presence of a broken power law,
µ and µ are their proper motions, and k = 3 for a
app rec interpolation of the spectrum may not be straightforward
jet composed of discrete ejecta.
if in the observer’s frame we sample different parts of the
The luminosities L(t) of the knots change with time t,
spectrum at any given frequency.
astheknotsexpandandthe magneticfield,andhencethe In this Letter, we present a method of using the flux
synchrotronemissivity,decreases. Thusthetruefluxratio
ratios from a single image of a source to constrain the jet
is
speeds without resorting to interpolation via the Mirabel
k+α
S 1+βcosθ L (t )
app app app & Rodr´ıguez method.
= , (2)
S (cid:18)1−βcosθ(cid:19) L (t )
rec rec rec
1
2 Miller-Jones et al.
2. flux ratios R−1, so the plasmon emissivity has a simple dependence
on frequency and size given by
2.1. Simple Scalings
J(ν)∝ν(1−p)/2R(1−3p)/2. (12)
Asynchrotron-emittingplasmonwheretheparticlesun-
dergo adiabatic expansion will have a power law decay in The ratio of flux densities as seen by the observer is then
intensity, L(t)∝t−ζ, in which case equation2 becomes (1−3p)/2 k+(p−1)/2
S R(t ) 1+βcosθ
app app
Sapp = 1+βcosθ k+α tapp −ζ, (3) Srec =(cid:18)R(trec)(cid:19) (cid:18)1−βcosθ(cid:19) .
S (cid:18)1−βcosθ(cid:19) (cid:18)t (cid:19) (13)
rec rec
Althoughwecouldtaketheexpansionoftheplasmontobe
and this scaling will apply to any process which gives a
ofthe formR∝tη, itis particularlyinstructive to lookat
power law decay in intensity. We consider symmetric ap-
the case of linear expansion, η = 1, for which equation13
proaching and receding jets, in which case after ejection
becomes
at t = 0, the epochs at which photons leave correspond- k−p
S 1+βcosθ
ing points of the front and back plasmons, tapp and trec app = . (14)
respectively, are related by Srec (cid:18)1−βcosθ(cid:19)
t 1+βcosθ This is the flux ratio observed at a given instant by the
app
= . (4)
t 1−βcosθ telescope as opposed to the interpolated flux at equal an-
rec
gularseparations. As asimple genericcase,emissionfrom
So in this simple case equation3 becomes
a jet composed of discrete ejecta (k =3) from a spectrum
S 1+βcosθ k+α−ζ of electron index p = 2 will give an exponent of unity
app
= . (5) for equation14. By way of contrast, obtaining an inter-
S (cid:18)1−βcosθ(cid:19)
rec polated estimate at equal angular separations will, in this
While this ratio is applicable to any process that gives case, give an exponent of k+α = 3.5. The flux ratios to
a power law decay, in the adiabatically expanding syn- bemeasuredinthetwocaseswould,however,differ,being
chrotroncasetheparametersαandζ arenotindependent measuredinasingleimageintheformercaseandatequal
so that the determination of the flux ratios by equation 5 angular separations in the latter. A comparison of both
is actually not introducing an extra parameter. methods in any given source would of course be a useful
means of inferring a possible asymmetry in the approach-
2.2. Optically thin synchrotron emission and adiabatic ing and receding jets (Atoyan & Aharonian 1997).
expansion
2.3. Synchrotron Self-Absorption and Spectral Breaks
Thetotalsynchrotronemissivityfromasingle,optically
thin jet knot scales as (e.g. Longair 1994) Whentheparticlespectrumcontainsabreakoraturnover
we must adapt the above discussion. For example, in
J(ν)∝B3/2N(γ)γ2ν−1/2, (6) Cygnus X-3 (Miller-Jones et al. (2004)) we observe two
discreteknots,oneoneachsideofthecentralnucleus,with
where B is the magnetic field strength, γ is the Lorentz
evidenceforaturnoverduetosynchrotronself-absorption.
factorofanindividualelectronassumedtoberadiatingat
Inthis casethe spectrumwillhavetheformJ ∝ν5/2 be-
a single frequency ν
low the turnover frequency ν when the knot radius is R
0 0
γ2eB while above this frequency the spectrum is optically thin,
ν = , (7) J ∝ ν−(p−1)/2. As the knot expands to a radius R its
(cid:18)2πm (cid:19) ν
e emission will then take the self-absorbed form
andN(γ)is the totalnumberofelectronswithenergiesin
the range (γ, γ+dγ) in the plasmon, given by 5/2
ν
N(γ,t )=Aγ−p, (8) J(ν,R)=Jmax(R) , ν ≤νc(R), (15)
0 (cid:18)ν (R)(cid:19)
c
wherepistheelectronindex,t0issomearbitraryreference while in the optically thin regime the intensity scales like
time,andAisthenormalisationconstant. Astheplasmon
−(p−1)/2
expands, nonrelativistically, from a radius R0(t0) to R(t) J(ν,R)=J (R) ν , ν ≥ν (R). (16)
max c
the electron energy scales as (cid:18)ν (R)(cid:19)
c
R The critical frequency beyond which the emissivity be-
0
γ = γ , (9)
R 0 comes optically thin is determined by
if synchrotron losses are negligible. The spectrum then R −(3p+4)/(p+4)
evolves according to νc(R)=ν0 , (17)
(cid:18)R (cid:19)
0
R R
N(γ,t)= N γ ,t . (10) and the emission at that frequency, which is the peak of
0 0
(cid:18)R0(cid:19) (cid:18)R0 (cid:19) the knot spectrum, becomes
Putting all of the above together we find that the syn- −5p/(p+4)
R
chrotron emissivity of an expanding plasmon is given by J (R)=J . (18)
max 0
(cid:18)R (cid:19)
J(ν)∝ν(1−p)/2B(1+p)/2R1−p. (11) 0
Turning now to the flux ratios observed at a given in-
As the plasmon expands the magnetic field strength will stantandfrequencyitisclearthatatsufficientlyearlyand
decrease and, in the case of a tangled field, we have B ∝ late times we will have two extremes. In the former case,
Flux ratios and light-travel time effects 3
when the observed emission from each knot is optically the approaching and receding knots would then have to
thick we will see a J ∝ R5/2ν5/2 spectrum from each be measured when the knots were both in the same ex-
ν
and the flux ratio exponent is k = 3 for discrete ejecta. pansion regime. Unless the transition radius were known,
However, it may be difficult to observe actual knots in this would require actually measuring (as opposed to in-
this regime without mixing in possible nuclear emission. terpolating) the flux densities at equal angular separation
At observed frequency ν the emission will become opti- from the core. We note that if there is significant deceler-
callythinfromtheapproachingknotwhenitsradiusisR ation of the expanding plasmons due to interaction with
1
which is determined by surrounding material, as mentioned by Hjellming & Han
R −(3p+4)/(p+4) (1995),then(R/R0)∝(t/t0)η,whereη <1,andequation
ν =(1+βcosθ)ν 1 , (19) 14 then requires modification. We also draw attention to
0
(cid:18)R (cid:19)
0 Fender (2003), which presents caveats to be considered
while the emission from the receding knot will remain op- when using proper motions to place limits on the bulk
tically thick, atthis frequency,until the approachingknot Lorentz factors of jets; any Lorentz factors thus derived
has a radius of R2 which can be easily shown to be are strictly only lower limits.
2p/(3p+4)
1+βcosθ
R = R . (20) 3. comparison with observations
2 (cid:18)1−βcosθ(cid:19) 1
Therefore, when the approaching knot has a radius R The VLBA observations of Cygnus X-3 presented by
satisfying R ≤ R ≤ R the observed emission at fre- Miller-Jones et al. (2004) show a jet which at 5GHz and
1 2
15GHzappearstobe composedoftwoseparatingdiscrete
quency ν will be a mix of Doppler boosted, optically thin
emission from the forward knot and optically thick emis- knots, which were interpreted as approaching and reced-
sion from the receding component. The flux ratio in this ing plasmons. A precession modelling analysis yielded a
value βcosθ = 0.62±0.11, and the spectral index of the
regime is now dependent on time, i.e. knot radius, and is
given by emissionwasfoundtobeα=0.60±0.05. Assuminglinear
expansion of the jet knots, we would thus predict a flux
Sapp 1+βcosθ k R −(3p+4)/2 density ratio of 3.2±1.0. For the last two epochs (2001
= . (21)
S (cid:18)1−βcosθ(cid:19) (cid:18)R (cid:19) September 20 and 21), the measured flux ratios are given
rec 1
At R=R allof the emissionbecomes optically thin at in Table1. While not matching the theoretical prediction
2
this frequency and the flux ratios predicted by equations perfectly, they are now of the correct order, in contrast
14 and 21 are equal. Therefore, the flux ratio exponent with the predictions of the original formalism, which is
drops from a value of k to k −p as the front and then wrongbytwoordersofmagnitude. Therearevariouspos-
the recedingknotbecomeoptically thin. During this time sible explanations for the slight discrepancy. Most impor-
the spectrumatfrequencyν alsoevolvesfromν5/2 toν−α tantly, the measurement of the flux densities themselves
and the forward knot expands by a factor R /R , where was often difficult. It is also possible that the plasmon
2 1
both radii are frequency dependent. The time taken for expansion was not exactly linear with time, which would
this expansion is determined by the expansion velocity V altertheexponentinequation14andchangethepredicted
of the knot. flux density ratio. Moreover,the measured spectral index
In reality however, it is unlikely that the turnover in α was for the integrated spectrum; the values of α and p
the spectrum would occur at one single frequency. There could in principle differ for the individual jet knots. The
would be a finite turnover region in which the spectrum qualityofthedatamakesitdifficulttointerpolatebackto
evolvedfromaν5/2 power-lawtoν−(p−1)/2. Dependingon thefluxdensitiesatequalangularseparationsinthiscase,
the width of the turnover region and the value of βcosθ, but our best attempts gave flux ratios between 1.58 and
the receding knot could be in the turnover region of the 10.63. Insuchcases,ourdirectmeasurementmethodgives
spectrum by the time the approaching knot had become a much more accurate determination of the expected flux
opticallythin. Inthiscase,theaboveresultswouldnotbe ratio,for moremeaningfulcomparisonwiththe jetspeeds
strictly applicable. andinclinationanglefoundbydifferentmethods. Wenote
Nonetheless, these general points apply to any source that if the spectral index of the jet material is known,
of opacity which changes the spectral shape or indeed to our method requires only a single image to determine the
any broken power law that might be attributable to the valueofβcosθ,whereastheinterpolationmethodrequires
acceleration mechanism. It presents the possibility that at least two images taken at different times. This frees it
the evolution of the spectrum from flares in microquasars fromthe uncertainty inherentincomparingVLBI images,
may well be influenced by the light-traveltime differences particularly if the imaging is difficult, as was the case in
between approaching and receding knots, as outlined in these observations (Miller-Jones et al. 2004). For a sin-
Miller-Jones et al. (2004). gle image, the ratio of two flux densities is set, whereas
when comparing different images, in order to be able to
2.4. Caveats interpolate accurately, one has to be confident that one
has recoveredthe same fractionof the true flux density in
Careshouldbe takenifthe expansionmodeofthe plas-
both images in order to be able to take an accurate flux
monschangespriortotheobservationfromwhichtheflux
density ratio.
ratio is derived. In such a case, for example a transi-
This theory couldalsobe appliedto the observationsof
tion fromslowedto free expansion(Hjellming & Johnston
GRS1915+105 detailed by Mirabel & Rodr´ıguez (1994).
1988), the time decay of the flux density would change
They observeddiscrete radio ejecta moving outward from
(steepen with time in this case). In order to use flux ra-
the nucleus over a period of ∼ 1 month. Again, we take
tios to constrain the value of βcosθ, the flux densities of
4 Miller-Jones et al.
the flux density ratio of their observed knots once they
had clearly separated from one another and from the nu-
cleus, and we only compare corresponding pairs of ejecta.
From their derived value of βcosθ = 0.323±0.016 and
their quoted spectral index of α=0.84±0.03, we predict
a flux density ratio of 1.24±0.05. For the later epochs
(1994 April 16, 23 and 30), the measured flux density ra-
tios are 2.33, 2.63 and 1.80 respectively. Again, this is
slightly greater than we predict, but is of the right order.
Underpredicting the flux density ratio implies the expo-
nentshouldbelargerinequation14,whichrequiresη >1,
i.e. the expansion scales slightly more rapidly with time
than R∝t.
4. conclusions
We have considered the evolution of synchrotron bub-
bles (plasmons) in oppositely-directed microquasar jets.
We have found that our new formalism can explain the
observedfluxdensityratiosinmicroquasarjetsinsystems
in whichthe synchrotronbubble modelisapplicable,such
as Cygnus X-3. In contrast, the original formalism con-
siderably overpredicts the observed flux density ratio in
observations of this system. In the case of free (linear)
expansion, (R/R )∝(t/t ), we found that the flux ratios
0 0
of the approaching and receding plasmons are given by
S /S =((1+βcosθ)/(1−βcosθ))k−p.
app rec
J.C.A.M.-J.thankstheUKParticlePhysicsandAstron-
omy Research Council for a Studentship. K.M.B. thanks
the Royal Society for a University Research Fellowship.
K.M.B.andP.D.acknowledgeajointBritishCouncil/Enterprise
Ireland exchange grant.
Flux ratios and light-travel time effects 5
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Table 1
Cygnus X-3 VLBA flux density ratios measured from observations of 2001 September outburst of Cygnus X-3
Date(UT) ObservingFrequency(GHz) Fluxdensityratio(South/North)
September20 5 1.43±0.05
September21 5 2.39±0.10
September20 15 1.14±0.19
September21 15 3.09±0.14