Detection of Superluminal Motion in the X-Ray Jet of M87

Having found evidence of movement in the difference map, several approaches were tried for determining the proper motions of the knots. First, akin to the method used to align the images in Section 2, the cross-correlation of knot images from the two epochs was generated to determine relative offsets. A unique region was defined for each knot where all other jet features and background point sources were masked. Each region was varied by size, position, and orientation multiple times to ensure the results were not biased by the initial region selection. A two-dimensional cross-correlation function was calculated between epochs for each knot, where all possible alignments of the two images were sampled in the cross-correlation function. The maximum peak of the function was fitted with a two-dimensional Lorentzian profile to determine the relative image offsets.

Using the cross-correlation method for each knot, a clear offset was measured for Knot D with a radial shift Δ_∥ = 0.046 ± 0011 and transverse shift Δ_⊥ = 0.002 ± 0006. This shift equals a speed of Δ_∥ = 9.2 ± 2.3 mas yr⁻¹ and Δ_⊥ = 0.4 ± 1.2 mas yr⁻¹, or β_∥ = 2.4 ± 0.6 and β_⊥ = 0.1 ± 0.3. Unfortunately, the cross-correlation method failed to identify proper motions for the remaining knots in the jet due to low count statistics or issues separating the knot's emission from adjacent sources, the latter of which is discussed further in Section 4.2.

Having attempted measurement of the proper motions within the jet, upper limits were calculated for the remaining knots using the standard deviation in centroid positions. This technique was chosen because it provides accurate uncertainties for knots with low count statistics. Additionally, the centroid uncertainties were consistent with the cross-correlation uncertainties of knots with high count rates, such as Knot A. To measure the centroid uncertainty, a region was again defined surrounding each knot based on the criteria described in Section 2, and the CIAO task dmstat was used to locate the centroid position. The centroid position, $\bar{x}$, of a region is defined as the count-weighted mean,

$$\begin{eqnarray}&&\bar{x}=\displaystyle \sum _{\mathrm{pixel}\ i}\displaystyle \frac{{n}_{i}{x}_{i}}{N},\end{eqnarray} \tag{ 3 }$$

where x_i is the position of pixel i, n_i is the number of events in this pixel, and N is the total event counts for the region. The standard deviation in the centroid position is estimated as

$$\begin{eqnarray}&&{\rm{\Delta }}x=\sqrt{\displaystyle \frac{1}{N(N-1)}\left(\displaystyle \sum _{\mathrm{pixel}\ i}{n}_{i}{x}_{i}^{2}-N{\bar{x}}^{2}\right)}.\end{eqnarray} \tag{ 4 }$$

For each knot, the statistical uncertainties of both epochs were combined in quadrature to estimate the total proper motion uncertainty. See Table 3 for the proper motion upper limits for the remaining knots in the X-ray jet.

Using the methods discussed in Section 4.1, direct measurement of proper motion was not obtained for HST-1. Due to HST-1's close proximity to the AGN and its decrease in brightness of 73% from 2012 to 2017, we were unable to establish good determinations of its centroid locations for the two epochs. A different measurement technique was required that would be insensitive to significant brightness changes at small separation angles. We therefore opted to generate a model image of the AGN/HST-1 complex from which we may measure offsets between the two epochs.

To begin, a 4'' × 4'' region surrounding the AGN/HST-1 complex was extracted from the image for each epoch, avoiding background point sources and other jet features. Both observations were binned at a subpixel size of 0066 pix⁻¹. Synthetic images of the region were generated by modeling the AGN and HST-1 each as a two-dimensional Gaussian where the x-position, y-position, amplitude, and FWHM parameters were allowed to freely vary. A constant background component was also included in the model. The model was fit to each epoch using the Sherpa fitting package (Freeman et al. 2001; Fruscione et al. 2006), with the Nelder–Mead method using Cash statistics. The observations are compared to the best-fit synthetic images in Figure 2, while the final fitted parameters for the model are available in the Appendix. For both epochs, the model image agrees well with the observed structure.

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From the modeled results, centroid coordinates for the AGN and HST-1 were obtained to high accuracy. As any motion of the AGN is assumed to be unresolved (Kovalev et al. 2007), the AGN was treated as a stationary point in the system. We may therefore measure the distance from the AGN to HST-1 for both epochs and compare the results to determine the motion of the knot. In comparing the relative distances, HST-1 was found to move Δ_∥ = 0.121 ± 0008 and Δ_⊥ = 0.054 ± 0003 with respect to the jet axis over the 5 yr time span. This shift corresponds to a proper motion speed of Δ_∥ = 24.1 ± 1.6 mas yr⁻¹ and Δ_⊥ = 10.9 ± 0.6 mas yr⁻¹, or β_∥ = 6.3 ± 0.4 and β_⊥ = 2.9 ± 0.2.

The proper motion results from Section 4 provide two equally probable interpretations of the system. The first interpretation is that the measurements are due to motion of the X-ray knots, while the second assumes that brightening and/or fading of substructure within the knots gives the appearance of motion at Chandra's resolution. Validation that the observed shifts are indeed motion requires the Chandra observations to be compared with contemporaneous higher-resolution observations to ensure that: (1) the X-ray data traces the emission regions to the desired spatial accuracy, and (2) the measured proper motion is consistent between the high- and low-resolution data sets.

To test whether the knot motions reflect material motion in the jet, the Chandra observations were compared to contemporaneous, archival HST images of M87. Previous analyses of M87 with HST have shown evidence of proper motions for all knots within the jet (Biretta et al. 1995, 1999; Meyer et al. 2013), and the spatial resolution of HST is a factor of ∼3 better than HRC-I, making it an excellent reference for comparison. Archival WFC3/UVIS HST images at F275W were used (PropIDs #12989 and #14618), where the pivot wavelength for the filter is 270.4 nm. These HST observations, obtained on 2012 December 25 and 2017 March 3, were selected to be the closest match to the Chandra epochs (2012 April 14 and 2017 March 2). Figure 3 compares the Chandra observations with the HST data included as overlays. Visual comparison of the UV and X-ray knots shows their positions to be consistent at the two epochs. Centroid positions of the knots were compared between the X-ray and UV data sets, with both in agreement to the positions reported in Table 3 to within the centroid position error determined from the HRC-I observations. Figure 3 also illustrates the centroid position of the X-ray knots for which proper motion was clearly detected, HST-1 and Knot D, overlaid as dashed lines. Using the lines as reference, a shift in the centroid position is observed for each knot, and these position shifts are visually consistent with the proper motions from HST.

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Beyond visual comparisons, we may compare the measured X-ray proper motions to previous results from HST observations. M87 proper motion measurement from Biretta et al. (1999) and Meyer et al. (2013) were selected for comparison, as shown in Figure 4. HST and Chandra proper motion measurements both parallel and perpendicular to the jet are broadly consistent for the knots examined. The only notable discrepancy observed is for the transverse motion of HST-1, which is discussed further in Section 5.2. Altogether, these results indicate that the X-ray and optical/UV knots track the same physical locations in the jet of M87, to within the spatial accuracy of the X-ray measurements.

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The jet velocity of M87 has been extensively studied on both parsec and kiloparsec scales using radio and optical observations. These measurements include thorough proper motion analyses of the knots within the jet. It is therefore interesting to compare the proper motion results from X-rays with independent measurements from other wavelength bands. See Figure 4 for a comparison of reported proper motions in M87, including the X-ray results reported here.

Among the knots investigated, HST-1 is by far the fastest-moving in all bands observed. X-ray and optical observations both show HST-1 to move at a speed of ∼6c along the jet axis (Biretta et al. 1999), while VLBI observations place an upper limit of 5c (Cheung et al. 2007). Given the reasonable agreement in speeds of HST-1 before and after its 2002–2008 flaring period (Harris et al. 2003, 2009), the flaring event appears to have had little impact on the average proper motion of the knot along the jet axis.

Examination of the measured transverse proper motions for HST-1 shows a speed of +2.9 ± 0.2c for X-rays, while the prior VLBI observations show an average transverse speed of −0.5c. Transverse motion measurements were not reported for the HST data. The transverse motion detected from X-rays may be due to variations in brightness of substructure not resolvable with Chandra, though it is surprising that the VLBI, which is able to resolve significantly more internal structure of the knot, indicates motion in the opposite direction. In contrast, the centroid positions of HST-1 agree well between Chandra and HST at both epochs (Figure 3), and there is no evidence of brightness changes in the optical/UV or of substructure within it that would be unresolved by Chandra. This comparison supports the conclusion that optical/UV and X-rays track the same physical feature in the jet. Follow-up observations with high-resolution interferometry, such as the Atacama Large Millimeter/submillimeter Array (ALMA) or Square Kilometer Array, would help determine whether the measured transverse shift is proper motion or is instead related to substructure changes at spatial scales smaller than the resolution of HST. Additional Chandra observations will be useful to further verify the transverse motion.

Moving down the jet, Knot D is demonstrably slower than HST-1 as its proper motion parallel to the jet is only 2.4 ± 0.6c in X-rays. This result is in excellent agreement with the average motion from VLA and HST observations (Biretta et al. 1995, 1999; Meyer et al. 2013). Transverse motions also agree well, showing no evidence for motion away from the jet axis. Knot D has shown remarkable consistency in velocity over the 23 yr probed through proper motion observations.

Upper limits on proper motion for the remaining knots were also extracted from the X-ray data and may be compared to previous estimates. The X-ray result for Knot E places limits 50%–75% lower than those found from HST data in Biretta et al. (1999) and Meyer et al. (2013). This is unsurprising given that the HST analyses probed fast-moving edges of the knot, whereas the X-ray data provide the average motion due to the lower spatial resolution of Chandra. Limits established for Knots F, A, and B are each consistent with the average proper motions reported from HST and VLA observations. The less restrictive limits for Knots I and C also agree with independent measurements, as expected given the poor X-ray count statistics, and subsequent large centroid uncertainty, for each feature. Follow-up Chandra observations will provide larger time differences between epochs that will reduce the X-ray limits and, ultimately, resolve proper motions for the remaining knots. Based on average speeds from other wavelengths, Chandra observations as early as 2020 would provide a sufficient baseline to detect proper motion in Knots E, F, A, and B.

The decreases in brightness observed for HST-1 and Knot A (Section 3) might be explained by adiabatic expansion of the radiating region over the examined time interval. As described in Snios et al. (2019), synchrotron flux F_E at a fixed energy will scale as F_E ∝ V^−2p/3 under isotropic expansion, where V is the volume of the emitting region and the electron energy distribution has the form dN/dγ ∝ γ^−p for electron Lorentz factor γ.

If the X-ray fading is due to adiabatic losses, we should expect to see a similar reduction in brightness in other bands. The 2012 and 2017 HST observations from Section 5.1 were therefore studied for brightness changes comparable to those observed in X-rays. Using the archival HST observations, HST-1 and Knot A were each found to have the same UV intensity in both epochs to within their respective uncertainties. The lack of evidence for any change in the optical/UV brightness of the knots strongly disfavors adiabatic loss as the primary cause of the fading observed in both HST-1 and Knot A.

Beyond adiabatic expansion, synchrotron cooling is another potential mechanism that may explain the observed decreases in brightness of the knots. The synchrotron cooling rate for an X-ray knot may be estimated with the Kardashev–Pacholczyk (KP) model (Kardashev 1962; Pacholczyk 1970). As discussed in Snios et al. (2019), the cooling rate is maximized when the particles move perpendicular to the magnetic field and particle scattering is assumed to be negligible. Under these assumptions, we can estimate the minimum magnetic field strength required to account for the observed fading of HST-1 and Knot A. The initial electron distribution for each knot was taken to have the form dN/dγ = Kγ^−p, with p = 3.6 for HST-1 and p = 4.2 for Knot A based on independent measurements (Kataoka & Stawarz 2005; Perlman & Wilson 2005). A maximum Lorentz factor of 10⁹ was used for the model, making the initial X-ray spectrum a power law to well above the energies accessible to Chandra. The electron distribution was evolved, allowing for the effects of synchrotron cooling, over the 5 yr time span of the X-ray observations to determine the final X-ray spectrum, which was folded with the HRC-I response to determine the change in count rate over this interval. The magnetic field strength was adjusted to obtain the observed reduction in count rate. Relativistic effects (time dilation and light travel delay), which are discussed further below, were ignored initially for simplicity. Under these assumptions, HST-1 required a minimum magnetic field strength of ∼ 800 μG, while Knot A required a minimum of ∼ 250 μG. These values are consistent with estimates of the equipartition magnetic field (Harris et al. 2003, 2006, 2009; Kataoka & Stawarz 2005; Stawarz et al. 2005). Under synchrotron cooling, the optical/UV emission of the knots should remain constant over the 5 yr span of these observations, consistent with our findings for the archival HST observations (Section 5.3).

As noted in Section 5.3, the constant optical/UV knot emission argues against adiabatic loss models. Our determination of the minimum magnetic field strength from the synchrotron cooling model relies on the same electron population being responsible for the X-ray emission at both observing epochs, requiring the knot material to move at relativistic speeds between the observations. This is noteworthy because it implies that the speeds of the jet knots reflect bulk relativistic motion of jet plasma, not just of a disturbance, such as a wave or shock front in the jet. To avoid this conclusion, the jet plasma emitting at the initial knot position would have to cool even faster than assumed, requiring a substantially greater magnetic field. Since the required field strength would then be larger than the equipartition value, this seems unlikely. The simplest conclusion is that the motion of the jet knots directly reflects the bulk speed of the jet plasma.

For a knot moving at speed βc, the proper duration of a process, t_p, is boosted by time dilation to γt_p, where $\gamma =1/\sqrt{1-{\beta }^{2}}$, in the rest frame. Light travel delay reduces the duration measured by a stationary observer by an additional factor of $1-\beta \cos \theta$, where θ is the angle between the velocity of the knot and the observer's line of sight (θ = 0 for a knot approaching directly). Thus, the stationary observer measures the duration

$$\begin{eqnarray}&&{t}_{o}=\gamma (1-\beta \cos \theta ){t}_{p}={t}_{p}/\delta \,,\end{eqnarray} \tag{ 5 }$$

where δ is the jet Doppler factor. The synchrotron cooling time of electrons in the knot that radiate photons of energy _p scales as ${t}_{p}\propto {({\epsilon }_{p}{B}^{3})}^{-1/2}$ and the energy of those photons in the observer's frame would be _o = _pδ. Combining these scalings, the relativistic correction to our estimate of the minimum magnetic field strength in the knot's rest frame would be a factor of δ^−1/3. While the Doppler factor is poorly constrained, the apparent knot speed, ${\beta }_{p}=\beta \sin \theta /(1-\beta \cos \theta )$ is known, placing a joint constraint on β and θ. For fixed β, the expression for β_p is maximized as a function of θ when $\cos \theta =\beta$, therefore the minimum possible value for the knot momentum is βγ = β_p. Although the Doppler factor could still have any positive value, if higher knot momenta are less likely, reasonable values of the Doppler factor will be comparable to the value corresponding to the minimum momentum, i.e., ${\delta }_{m}=\sqrt{1+{\beta }_{p}^{2}}$.

Applying the Doppler factor δ_m to the observed 5 yr time interval increases the cooling time for HST-1 to 35.1 yr and Knot A to 6.2 yr. Using the revised timespans, HST-1 required a minimum magnetic field strength of ∼ 420 μG, and Knot A required a minimum of ∼ 230 μG. These revised lower limits are again consistent with previous equipartition magnetic field estimates, and the synchrotron cooling agrees with the constant optical/UV intensity observed via HST. Given the consistency between observations and the models in both cases tested, synchrotron cooling is therefore the most probable mechanism to explain the observed brightness decreases for HST-1 and Knot A.