5.1. Component Proper Motions in the Mas-Scale Jet of J1826+3431
To identify the jet components throughout the full time span of the VLBI observations, we considered their positions, FWHM sizes, and flux densities. The components that are identified across the different epochs are denoted by the same identifiers in
Table A2,
Table A3,
Table A4. The core component is numbered with 0 at each epoch and frequency; thus, it is denoted as L0, C0, D0, and J0 at
,
,
, and
GHz, respectively.
For further analysis, we assumed the core to be stationary and we derived the positions of the fitted jet components relative to it (
Table A1,
Table A2,
Table A3,
Table A4). The jet components are assumed to move away from the core at constant speeds along a straight trajectory. Thus, we fitted their core separations with linear functions. The derived angular proper motions were converted to apparent speeds in units of the speed of light,
c, using the relation [
1]
where
is the angular proper motion of the component in rad s
,
is the luminosity distance in m,
c is the speed of light in m s
, and
z is the redshift of the source.
At
GHz, at most five components were needed to describe the jet structure. The inner ≤10 mas region of the jet could not be resolved into two distinct components in
, unlike at later epochs (components C1 and C2). Therefore, the single fitted inner jet component, Ca, cannot be unambiguously identified with either of the components detected later. Thus, the motion of C1 and C2 can only be tracked at the relatively closely-spaced observing epochs of the 2010s. Because of the short time baseline, these two components show no significant change in their core separation during the available observing epochs. On the other hand, component C3 could be unambiguously identified through all epochs. It shows an angular proper motion of
mas yr
, which corresponds to an apparent speed of
. At large core separations, component C4 could only be detected at two epochs. Therefore, it is unsuitable for further proper motion calculations. The core separations for each component and the fitted linear proper motion curve for C3 are shown in
Figure 3.
At
GHz, three jet components, D2, D3, and D4, can be detected at all four epochs, while D1 can only be detected at the last three epochs. This may indicate that it was ejected some time between the epochs
and
. No discernible proper motions could be detected for components D1 and D2, while D3 and D4 show apparent superluminal motions (
Table 3). The core separations and the fitted linear curves for D3 and D4 are shown in
Figure 3. As we performed proper motion and jet parameter calculations using D3 and D4, we present contour maps of J1826+3431 made at this frequency at all four epochs in
Figure 4, indicating the fitted model components.
Figure 4.
Naturally-weighted VLBI images of J1826+3431 at epochs (
a) 1996.370; (
b) 2015.064; (
c) 2017.227; and (
d) 2018.437 taken at
GHz restored with the Gaussian model components fitted to the visibility data. The location and size (FWHM) of the model components are indicated with circles. In each image, the elliptical Gaussian restoring beam (FWHM) is shown at the lower left corner. Further details of the images are given in
Table 4.
Figure 4.
Naturally-weighted VLBI images of J1826+3431 at epochs (
a) 1996.370; (
b) 2015.064; (
c) 2017.227; and (
d) 2018.437 taken at
GHz restored with the Gaussian model components fitted to the visibility data. The location and size (FWHM) of the model components are indicated with circles. In each image, the elliptical Gaussian restoring beam (FWHM) is shown at the lower left corner. Further details of the images are given in
Table 4.
At GHz, only three years elapsed between the first and last observations. This time span is too short for detecting proper motions in any of the fitted components. Component J1 could be first detected at epoch . This may indicate that J1 was ejected some time between the first and second epochs. Later, as it moved farther away from the core, this component became detectable even at a lower frequency, GHz, at the epoch as D1.
To summarize, apparent superluminal motions could be detected in three components at relatively larger core separations (C3, D3 and D4) in the jet of J1826+3431. Interestingly, the highest apparent speed is shown by a component (D4) located at the largest core separation (∼10 mas) implying acceleration in the jet. Acceleration is often detected in the jets of blazars (e.g., [
25]), and although it usually takes place at smaller core separations, there are examples for faster components further down the jets (e.g., PMN J0405−1308 [
25,
26], PKS 2201+171 [
27]).
5.2. Relativistic Beaming in J1826+3431
At the highest observing frequency,
GHz, which provides the best angular resolution, we derived the brightness temperature of the core component using the following equation [
28]:
where
S is the flux density of the fitted component in Jy,
is the FWHM diameter of the circular Gaussian in mas, and
is the observing frequency in GHz.
In , we were unable to resolve the core region into two components as at the later epochs. As a test, we tried to fit the innermost region with an elliptical Gaussian component and the resulting fit confirmed that the core was elongated in the direction of component J1 detected at the later epochs. Therefore, J0 and J1 were blended in , thus the FWHM size and the flux density are overestimates of the core parameters at this epoch. Since the brightness temperature is inversely proportional to the square of the FWHM size, its derived value can be regarded as a lower limit, ≥ K. In , the obtained brightness temperature is K. At epoch , we could only derive an upper limit for the size of the core component, thus only a lower limit for the brightness temperature can be given, ≥12 × 10 K. The lower limits agree with the estimate made at the middle epoch within its errors.
All brightness temperature values exceed the equipartition limit,
K [
29], indicating relativistic beaming in the source. The relativistic beaming can be quantified by the Doppler boosting factor,
, which can be estimated from the measured
value and the equipartition brightness temperature limit using the following relation [
1]:
The obtained brightness temperature value corresponds to a Doppler factor of .
5.3. Jet Parameters
The jet parameters, the bulk Lorentz factor (
), and the inclination angle with respect to the line of sight (
) can be calculated from the Doppler factor and the apparent jet speed as [
1]
and
The component close to the innermost core region with the best-defined positions and thus proper motion is D3; therefore, we used the apparent superluminal speed obtained for this jet feature, , to derive the jet parameters. Using the Doppler factor obtained from the -GHz measurement at epoch , we get a Lorentz factor of , implying relativistic bulk jet velocity in units of the speed of light, . The corresponding jet inclination angle is .
5.4. Radio Spectrum of J1826+3431
We collected the radio flux density measurements of J1826+3431 from the literature (
Table 5). We fitted a power–law curve to these data points to obtain the spectral index (
; defined as
, where
S is the flux density and
is the observing frequency). Since the flux density measurements are not simultaneous, source variability may affect the spectral index calculation, so the value can be considered as tentative. The resulting fit is shown with a black line in
Figure 5, the obtained spectral index is
, indicating a flat spectrum.
For comparison, we also show the sum of the VLBI Gaussian model component flux densities with blue squares and upward arrows in
Figure 5. For each observing frequency, we have chosen the observation that is the closest to the
-GHz EVN measurement. VLBI observations are insensitive to the large (arcsec) scale radio structure; therefore, in the presence of such emission, they underestimate the total flux density of a source. This can be seen in
Figure 5 where all VLBI points are indeed below the fitted spectrum.
The simultaneous
- and
-GHz observations allow us to map the spectral index distribution of the mas-scale radio emission in J1826+3431 using VIMAP [
36]. We present a spectral index image in
Figure 6. We used our data from the epoch 2015.064. After using elliptical masks to exclude the cores from the images, we matched the
- and
-GHz maps of J1826+3431 by using two-dimensional correlation. Considering that the frequencies are relatively close to each other and that the measurements were performed at the same time, it was not necessary to shift the brightness peak. The spectral index image (
Figure 6) suggests a flat-spectrum radio emission (
) in the core region and along the jet closer to the core. It is in good agreement with the spectral index calculated using the archival total flux density data (
Figure 5) since the dominant emission feature of the source is the core. The spectrum gradually steepens further away from the core along the jet, as expected from optically thin synchrotron radio emission.
5.6. -Ray Properties of 3EG J1824+3441
3EG J1824+3441 was detected with a weak
-ray flux of
photon s
cm
by EGRET [
6]. However, an alternate version of the EGRET catalogue [
37] (see also [
7]) which used reprocessed data assuming a different model for the Galactic interstellar emission does not contain as many as 107 sources previously detected in [
6], including also 3EG J1824+3441. In that sense, it is not surprising that no
-ray source is listed in the most recent fourth
Fermi/LAT catalogue [
3] within a
radius of the position of the putative EGRET
-ray source.
Alternatively, even if the source exists, long-term variability of its
-ray flux can also explain the
Fermi non-detection. Indeed, a comparison of high-confidence (>4
) EGRET detections with the third
Fermi/LAT catalogue revealed 10 extragalactic
-ray sources missing from the latter [
38]. Subsequent analysis of seven years of
Fermi data showed that five sources were in a low-luminosity state during the
Fermi observations compared to their luminosity during the EGRET observations [
38]. The generally decreasing long-term trend we see in the radio flux density of the possible counterpart, the blazar J1826+3431, with respect to the 1990s when CGRO was operational (see e.g., our VLA results and
Figure 7) is also consistent with a
-ray flux decrease.