C. A Model for the Jet Kinematics
In this section I derive various constraints on the jet kinematics, and build a possible model for the jet. Some ingredients of the model have already been discussed in Section 2.A., where morphological considerations lead to a picture where the inner jet represents a high Mach number supersonic flow (knots D, E, F, and I), the jet's Mach number is reduced (slowed) at shocks in the transition region (knots A, B, and C), and finally the outer jet (knots G, H, lobes, etc.) is a region of unstable subsonic flow.
In section 2.C. we presented evidence for
apparent motions in the jet, and these
can be used to constrain both the speed and direction of motion for
these features.
Since it is possible that the visible "patterns" in the jet and the jet
"fluid" move with different speeds (e.g.,
Lind and Blandford
1985),
we consider the pattern and fluid speeds,
pattern
and
fluid,
separately. The usual relationship for superluminal motion
(Blandford and
Königl 1979b),
![]() |
can be used to derive constraints on the pattern speed
pattern
and angle between the direction of motion and the line of sight
from the observed speed
obs,
where all speeds
are in units of the velocity of light. For example, the observed speeds
of knot A, and
the fastest features in knots B and D, yield lower limits on
pattern
of 0.48c, 0.78c, and
0.95c, and lower limits on the Lorentz factor
pattern
of 1.1, 1.6, and 3.2, respectively.
The observed speeds here for knots A and B are not large enough to
constrain
, but
the speed for the fast region of knot D requires
D < 37° ±
5°, where the uncertainty
results from the formal uncertainty on the observed proper motion. Since
this feature
in knot D is seen to move directly away from the nucleus, it seems
reasonable to assume
its motion is along the jet axis, and hence the jet axis is positioned
within about 37° of the line of sight
(
jet axis ~ 37°).
An additional constraint can be garnered from the sharp feature or
"edge" seen in
knot A. Presumably this edge is a two-dimensional structure seen
apparently edge-on.
To see this sharp edge, the photons traveling toward the observer must
remain within
the plane of the structure as both they, and the two-dimensional
structure itself, move. This constraint requires
pattern
~ cos
(Biretta, Owen, and
Hardee 1983;
Eichler and Smith
1983);
otherwise, relativistic aberration will cause the plane containing the
two-dimensional structure to appear less "edge-on," or even "face-on." A
more generalized
geometry might allow the plane containing the edge to be at some angle
90° +
to the jet axis
(Reid et al. 1989;
BOC89)
and the resulting constraint is
![]() |
The observed angle between the edge and the jet axis measured on the
plane of the sky is ~ 72°, suggesting
|| could be at least
18°; we will adopt |
|
30° as a limiting
value. This constraint from the aberration of knot A is plotted in
Figure 21, along with
constraints from the observed proper motion of knot A's edge and the
fast feature in
knot D. The two constraints for knot A's edge result in
35° and
pattern
~ 0.4 for
knot A. Assuming the jet axis is straight between knots D and A, and
combining their
constraints, we estimate that the jet axis is about 40° from the
line-of-sight (
jet axis ~
40°).
So far we have considered only the pattern speeds of features in
the jet. The
more interesting parameter is the speed of the jet fluid
itself. Here the arguments are
unfortunately less direct. Non-relativistic numerical simulations of
jets (e.g.,
Norman, Winkler,
and Smarr 1984)
have found that the fluid speed tended to be at least twice
that of the visible patterns. From the derived pattern speeds
pattern
0.5 in much
of the jet, this would appear to imply at least
fluid
0.5, and perhaps
fluid ~ 1 if
relativistic effects can be ignored. Other indirect evidence can be
taken from statistical studies of superluminal quasars, where
Cohen (1990)
finds similar mean values for the Lorentz factors of the fluid and pattern.
We also note that "scissor effects," which are sometimes cited as a
possible cause
of pattern
>
fluid
(e.g.,
Hardee and Norman
1989;
Fraix-Burnet 1990),
must be relatively unimportant in the M87. A scissor effect might be
created, for example,
at the intersection of a pair of shock waves. While the individual waves
might move
slowly, their intersection point could, in principle, attain any
speed. However, we note
the moving features in knots D and A show motion directly down the
jet. And while
outward motion is seen in at least eight regions, no regions show
significant inward
motion. It would seem contrived if the observed motions, and especially
those in knots
D and A, were produced by the intersection of shock waves, rather than
some mechanism closely tied to the jet flow.
A constraint on the fluid speed may also be derived from the ratio of jet to counter-jet brightness and the usual formulae for relativistic beaming (Blandford and Königl 1979b). As we have seen (Sec. 2.D.) there is considerable evidence for the presence of a counter-jet of some sort. If we assume it has an intrinsic brightness equal to that of the visible radio jet, and use the limit on the brightness ratio R > 150 derived from 15 GHz VLA observations (BOC89) we have
![]() |
which may be solved for the limits
fluid
> 0.76
(
fluid
> 1.5) and
jet axis
< 40°. We
note that the stronger ratio R > 450 from optical measurements
(Stiavelli,
Möller, and Zeilinger 1992)
produces slightly weaker limits on the derived parameters, because of
the steeper optical spectral index. If a shock in knot A does reduce the
fluid speed, as
we have suggested, it maybe necessary to treat the "inner jet" and
"transition region"
separately, and this gives slightly weaker limits for each region than
given above. The
main uncertainty with these calculations, of course, is the assumption
that the unseen
counter-jet has an intrinsic brightness similar to the visible jet. It
is possible they have
intrinsic differences, or that there are apparent differences
caused by rapid evolution
and time delay between the jet and more distant counter-jet.
Finally, we have a brief look at the kinematics of knot A under the
assumption that
it is a strong shock. If we presume that the fluid speed of the inner
jet is not too different
from the derived pattern speed of the fast region in knot D, then we
have fluid
3
for the inner jet. Knot A must therefore be treated as a relativistic
shock, for which
material has a downstream velocity of c/3 relative to the shock
(Blandford and Rees
1974).
When added to the observed motion of knot A, and assuming
jet axis ~ 40°,
we obtain an apparent speed of 0.9c for the material in the
downstream region. This
is at least roughly consistent with speeds in 4 out of 5 regions of knot
B, which range
from O.6c ± 0.2c to 1.3c ° 0.2c.
Therefore, to summarize our proposed model of the kinematics
(Fig. 22): The inner
jet (nucleus to knot A) is a high Mach number flow with a Lorentz factor
3 and
oriented about 40° from the line-of-sight. There is a shock at knot
A, about 1 kpc from the nucleus, where the flow speed is reduced to
fluid ~
1.4 relative to the nucleus.
A second shock in knot C further reduces the jet speed such that beaming
becomes
relatively unimportant, and at larger distances the jet is a subsonic,
buoyant plume.
It seems virtually certain that a counter-jet is present, though of
course, details of its
structure are unknown. This model is built upon, and explains, the
following evidence:
(1) overall jet morphology, consisting of straight inner jet (< 1.0
kpc), gradual bending
in transition region, and sharp bends in outer jet (> 1.6 kpc); (2)
presence of transverse
features in knots A and C; (3) magnetic field normal to jet axis in
knots A and C; (4)
superluminal speeds up to 3c for knot D; (5) motion and
appearance of sharp edge in
knot A; (6) lower observed speeds in most of knot B (0.6c to
1.3c); and (7) symmetric
source structure (i.e. two-sided structure) on scales
2.0 kpc. It is
interesting to reflect
that many elements of this model are suggested in early papers (e.g.,
Rees 1978).
![]() |
Figure 22. Suggested model for M87 jet and
counter-jet. Initially the jet is relativistic with
|
A model of this type may have several additional benefits. As mentioned
in Section 3.B., time dilation effects
associated with relativistic flow along the inner jet might
allow energetic electrons to travel farther from the nucleus, thereby
reducing the particle
lifetime problem posed by the optical emission. The knot spectra are
similar until knot
A, at which point they steepen systematically with increasing distance -
which could
be attributed to a sudden reduction in the jet velocity knot A. In terms
of other FR-I
radio sources, a relativistic "inner jet" might account for the
one-sided jet "bases" seen in many of these sources
(Bridle 1986).
These one-sided bases are typically straight,
well collimated, and 2 to 20 kpc in length (e.g.,
Bridle 1984;
Eilek et al. 1984;
Leahy, Jagers, and
Pooley 1986;
O'Dea and Owen 1986),
and thus resemble the inner jet of
M87. Only on larger scales does the structure become
two-sided and
poorly collimated, and suggests non-relativistic flow (e.g.,
Scheuer 1987).
Furthermore, a relativistic inner
jet would be consistent with
5 on the parsec scale,
as required by unified models
which propose that FR-I radio sources are the parent population of BL
Lac objects
(Urry, Padovani,
Stickel 1991,
and references therein). We note that
fluid
5 is not
ruled out by our observations, and that such regions would appear very
dim due to
beaming and
jet axis
~ 40°.
One concern about the model proposed here is the slow velocity seen in
the nucleus
by VLBI techniques. Component N2 appears to move outward at only ~ 0.3c
(Reid et al. 1989).
This could be attributed to
pattern
<<
fluid,
but this makes M87 rather different from quasar nuclei where
pattern
~
fluid
seems common, although cases of "slow" features are not unknown
(Shaffer and
Marscher 1987).
Also, the large
angle we propose between the jet axis and line of sight would make any
features with
pattern
fluid
5 appear very
dim. Future VLBI monitoring may clarify this
situation, if as components are seen in the M87 nucleus. Another concern
regards
the disposition of the inner jet's bulk kinetic energy after knot A. If
the jet is suddenly
decelerated at knot A, where is this energy going? It does not seem to
go into radiation,
since the luminosity of knot A is not that much greater than the other
knot's; and the
jet remains collimated, so it probably does not go into internal
pressure. However, it is
unclear how much kinetic energy is being carried by the inner jet; it is
possible that the
jet is very light (i.e. low mass density) and therefore carries
relatively little bulk kinetic
energy. A related concern is the high luminosity of the lobes, which is
presumably
supplied by the jet. However, this luminosity is dominated by low
frequency radio
emission, and these electrons have extremely long lives. It may be
adequate to merely
supply energetic electrons, rather than bulk kinetic energy, to the
lobes. If we ignore
adiabatic expansion, and assume luminous plasma flowing through knot B
at
~ 0.7
merely inflates the lobes, then the lobe luminosity could be supplied in
105 years (which
is less than electron lifetimes at low radio frequencies).