A TEMPORAL ANALYSIS INDICATES A MILDLY RELATIVISTIC COMPACT JET IN GRS 1915+105

In this subsection, we motivate a model of the fading compact jet associated with the decay curves in Figure 2. Note that the fading compact jet has never been imaged by VLBA. The full force compact jet has been imaged (see below), but the decay afterward is determined only by the light curves in Figure 2. Thus, we need some information to guide a speculative model of how a compact jet fades.

In the context of the decay curves in Figure 2, the flux density decays to low levels but not zero. Thus, we expect a low surface brightness jet to be the end state. There are some compact jet states that end with a major ejection. This is in distinction to the fade seen here. In this case, the compact jet either disappears or is too weak to be detected with a low sensitivity VLBA observation (Dwahan et al. 2004).

The fading of the compact jet or the disappearance of the compact jet is assumed to represent a decrease in the intrinsic power of the compact jet. The interior of the jet is hidden by SSA. The emission that is detected at 15 GHz is the $\tau \sim 1$ surface of the outgoing plasma. We assume that the power is drastically reduced on a timescale, T₀, which is much shorter than the time it takes the plasma to flow a scale length for the jet that represents a large fraction of the detected emission with VLBA, L. The abrupt transition to a state of highly reduced jet power begins at t= 0 and proceeds to completion in the time interval $0\lt t\lt {T}_{0}$. In the examples that follow, $L\gt 2$ mas. Namely, if the plasma velocity is V, then ${{VT}}_{0}\ll L$. This seems like a reasonable assumption, considering that the jet originates from a region near the black hole that is seven orders of magnitude smaller than the jet length, L (Punsly 2011). After the transition, a very weak jet is ejected from the central engine at $t\gt {T}_{0}$. However, there is still plasma from $-L/V\lt t\lt 0$ that occupies the conduit of the original HPS jet. The plasma that flows down this conduit is a strong transient feature that is approximately a step wave. In the language of magnetohydrodynamics, this is approximately a contact discontinuity (not a shock wave) that propagates at the speed, V, of the bulk flow of the jet plasma if the plasma is turbulent. The discontinuity might have to be supplemented by a more complicated rarefaction wave structure in principle, but this detail will not affect the emissivity and velocity of the preponderance of the remnant plasma from the HPS steady jet that is far upstream of the disturbance. The abrupt change that is assumed to occur is the fundamental assumption of the model of the jet fading that is manifest as a large transient discontinuity.

The plasma upstream of the propagating discontinuity has virtually the same plasma properties as the plasma ejected at $t\lt -L/V$ that was located at this same axial coordinate, x, along the jet during the steady HPS jet state. This includes the number density of particles and the magnetic field strength. The SSA opacity is determined by the local plasma state and is therefore virtually the same as it was at $t\lt -L/V$ during the steady jet phase of the HPS (Ginzburg & Syrovatskii 1969). The region for which this is true changes with time and is defined by the axial coordinate range, $x\gt {Vt},t\gt 0$. For this portion of the jet, the $\tau \sim 1$ surface and the intrinsic emissivity is virtually unchanged from the HPS steady jet. For $x\lt {Vt},t\gt 0$, the jet will be the weak new jet that was emitted from the central engine. Of course, an abrupt step function change is a mathematical simplification, but the transition region is much thinner than L. The change in emissivity is so drastic that for $t\lt L/(2V)$, the older plasma at $x\gt {Vt},t\gt 0$ is much more luminous than the newer plasma at $x\lt {Vt},t\gt 0$, and this dominates the integrated jet luminosity. Based on Figure 2 (and as verified in the analysis to follow) this describes the jet evolution and emissivity for many hours. At late times, a new weak jet may form and this is likely to have a more compact $\tau \sim 1$ surface because the number density and magnetic field are likely less. However, this is after the decay and is not germane to the discussion of the prompt decay of the 15 GHz flux density.

In summary, we are looking for a model of jet fading that is a consequence of a drastic reduction in jet power and, in general, leaves a weak jet behind. This simple model is a propagating discontinuity that separates the strong jet region from the weak jet region. Figure 3 describes our model. We emphasize that the simple dynamic depicted has never been directly observed. However, it is consistent with what we know about the compact jet from VLBA imagery and radio light curves. In Figure 3, t = 0 on the left indicates a steady HPS jet, which is the type of jet that we will describe later in Figures 4 and 5 and has been observed with VLBA (Fuchs et al. 2004; Ribo et al. 2004). The four steps in the grayscale shading crudely indicate the decay of luminosity along the jet length. The darker the gray, the brighter the jet. The power source for the jet is suddenly diminished in strength at t= 0, except for a weak (light gray) jet that persists during the event. For the sake of argument, the weak background residual jet has an integrated 15 GHz flux density ∼10 mJy, while the HPS jet was originally ∼100 mJy. The high emissivity plasma continues to propagate down the conduit formed by the steady state HPS jet after t = 0. The transition from light gray below to darker gray above locates the position of the discontinuity in the snapshots $\geqslant 2\;{\rm{hr}}$. As it propagates, the plasma upstream obeys the same effective emissivity (i.e., the resultant emissivity that includes the effects of Doppler beaming and SSA applied to the intrinsic emissivity) profile as the HPS steady state jet in this region, as determined empirically from the VLBA radio images. The original dark gray rectangle has moved downstream toward the top of the figure and occupies the position of the fainter second rectangle at t = 2 hr. The next time step proceeds similarly. No new flow of the high emissivity jet material leaves the central engine; the central engine for the jet is still nearly shutoff. The high emissivity plasma that once occupied the lowest rectangle at t= 0 continues advancing toward the top of the figure and occupies the third position at t= 4 hr. The jet luminosity decay continues and this is represented by the lighter shade of gray.

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One approach to describing the jet spatial variation is based on simple theoretical jet models. These were introduced into the literature in Blandford & Königl (1979), which give some general features of the effects of SSA and beaming in jets. However, the power law models of the parameters were found to be too simplistic to describe astrophysical jets, and more elaborate power law depictions of jets were created to deal with the constraints imposed by observation (Ghisellini et al. 1985; Ghisellini & Maraschi 1989; Ghisellini et al. 1996). These models were developed for extragalactic radio sources. Similar types of power law models for Galactic sources were applied to the jets in Cygnus X-1 and MAXI J1659-152 (Kaiser 2006; Paragi et al. 2013).

In the context of GRS 1915+105, models based on Ghisellini et al. (1985, 1996) and Ghisellini & Maraschi (1989) were applied to the compact jet (Punsly 2011). The models described the surface brightness profile of the compact jet considered here (in the next subsection and Figure 4) as well as the broadband spectral energy distribution. In order to satisfy all the constraints of the observations, the jet models were necessarily highly stratified in the transverse direction. The inner-most jet was the most collimated and the magnetic field was turbulent in equipartition with the plasma energy. The magnetic field became ordered outside this core and the jet expanded more rapidly with an increasing transverse distance from the core. The jet also flared non-uniformly in the axial direction. In the end, the parametric jet had 55 input parameters describing the various power laws.

Power law models incorporate the intrinsic luminosity, SSA, Doppler beaming, and the basic dimensions to create an effective emissivity. The complexity of the parametric model is not essential to understand in the present context. In order to implement the model of the compact jet fade in the last section, we want to understand the spatial variation of the effective emissivity of the steady state HPS jet. Thus, it is not advantageous to assume that actual jets are well described by numerous parameters that obey power laws. This has no benefit in the present context, and we strive to find something directly related to the observations. Instead, we choose to empirically determine the spatial variation of the source of effective emissivity responsible for the images of the compact jet in Fuchs et al. (2004) and Ribo et al. (2004) during HPSs. Namely, we want to determine the source function for the effective emissivity directly from the radio images, independent of the theoretical model of the jet.

Two deep 15 GHz VLBA (milliarcsecond—resolution) observations of a HPS compact jet have been published. The first was reported in Fuchs et al. (2003, 2004) and occurred on MJD 52731.6, which was 1.6 days before the HPS decay chronicled in the top frame of Figure 2. The second occurred on MJD 52748.4 (Ribo et al. 2004). We note that compact jets were observed in Dwahan et al. (2000), but these were not HPS jets. We comment on these observations at the end of this section. In order to estimate D, the total flux density as a function of displacement along the approaching jet is calculated from the radio image. The result of this integration for MJD 52731.6 is the black curve plotted in the left-hand frame of Figure 4. This is the "steady" state before the decay. The curve is the cumulative distribution of the detected 15 GHz flux density along the approaching jet. It starts as 0% at the radio core and equals 100% at the last contour in the radio image. The fact that this is not the actual end of the jet is not significant because the flux density is greatly reduced at these distances. The next step is to make a comparison with a radio luminosity source function. In order to do this we note three major facts:

• 1.  
  The jet is unresolved along its width so the source of effective emissivity for the VLBA image can be represented by a line source
• 2.  
  It should be noted that this image is blurred (made longer) by the finite beam width of the interferometer and interstellar scintillation. Based on the results of Dwahan et al. (2000), it is estimated that the effective "blurred beam width" (that includes the effects of interstellar scintillation) is 1.54 mas. The actual jet is effectively viewed through this "blurring aperture."
• 3.  
  One other caveat is that the brightness of the jet within the vicinity of the radio core depends on the jet velocity. The finite resolution of the interferometer will detect emission from both the counter jet and the approaching jet in this region. The observed flux density, ${F}_{\nu }\sim {\nu }^{-\alpha }$, in any beam is a Doppler enhancement of the intrinsic flux, ${F}_{\mathrm{intrinsic}}$ (Lind & Blandford 1985):

  $$\begin{eqnarray}&&{F}_{\nu }={\delta }^{2+\alpha }{F}_{\mathrm{intrinsic}},\end{eqnarray} \tag{ 2 }$$

  where

  $$\begin{eqnarray}&&\delta =\displaystyle \frac{\sqrt{1-{(v/c)}^{2}}}{1-(v/c)\mathrm{cos}\theta }.\end{eqnarray} \tag{ 3 }$$

  Equation (2) provides an enhancement of the approaching jet flux and a reduction of the receding jet flux. The net effect of Equation (2) and the finite resolution of the restoring beam is to produce a velocity dependent offset of the peak of the flux density of the radio image from the true core position. The 1.54 mas effective beam width centered on the peak flux will detect major fractions of both the approaching and receding jets. The core shift is needed to allocate proper amounts of the peak flux density to each side of the jet. The fraction of the core flux assigned to the approaching jet by the implementing a core shift from the flux peak will be equal to the fraction of the total jet flux in the approaching side—if bilateral symmetry holds and the jet has a constant velocity. This offset is required to identify a starting point for the cumulative flux distribution of the approaching jet as observed with VLBA (the black curve in the left frame of Figure 4). Yet, the resultant decay length D that arises at the end of the analysis (combined with the decay time, T) must also produce the same velocity used in the derivation of the offset. So the problem is nonlinear. For example, in the analysis of this paper, the peak flux per beam is offset downstream of the true core position by ≈0.3 mas. In order to understand this core shift concept, and to appreciate the nonlinearity, we verify that it is of the correct magnitude explicitly after Equation (6).

We are actually interested in the empirical source function for the emissivity that was emitted by the source (not the image blurred by scatter-broadening and the finite resolution of the VLBA) in order to understand the length scale over which physical changes take place. We have created such an empirical source function and then mathematically blurred the image to replicate the image made by VLBA. The resultant cumulative distribution of flux density of the approaching jet is plotted as the red curve in the left-hand side of Figure 4. We can write the empirical source function, S(x), for the effective emissivity as

$$\begin{eqnarray}S(x) & = & ({F}_{0}/D){e}^{-x/D}\delta {(v)}^{2+\alpha }{\rm{\Theta }}(x)\\ & & +({F}_{0}/{D}_{\mathrm{rec}}){e}^{x/{D}_{\mathrm{rec}}}\delta {(-v)}^{2+\alpha }{\rm{\Theta }}(-x),\end{eqnarray} \tag{ 4 }$$

where x is the actual axial displacement of the jet plasma from the radio core and ${\rm{\Theta }}(x)$ is the Heaviside step function. The first term represents the approaching jet, D is the e-folding distance, and F₀ is the normalization. The second term is the source function of the receding jet. This is required because the wide effective beam width of the observation also samples the receding jet near the origin of the approaching jet. We introduced D_rec, the e-folding distance of the linear source function for the effective emissivity of the receding jet. In order to transform this quantity to the observed emissivity, ${ \mathcal E }(y)$, one must convolve with a Gaussian beam. If y is the observed displacement from the radio core,

$$\begin{eqnarray}&&{ \mathcal E }(y)={\int }_{-\infty }^{\infty }S(x)\displaystyle \frac{1}{\sqrt{2\pi }\sigma }{e}^{\tfrac{-{(x-y)}^{2}}{2{\sigma }^{2}}}\;{dx},\end{eqnarray} \tag{ 5 }$$

where the standard deviation, σ, is the effective beam width (1.54 mas) divided by 2.35 (the factor required to convert FWHM to standard deviation). The radio images were made from a circular beam. The cumulative distribution, ${ \mathcal F }(y)$, that is plotted in red in Figure 4 is

$$\begin{eqnarray}&&{ \mathcal F }(y)=\displaystyle \frac{{\int }_{0}^{y}{ \mathcal E }(z){dz}}{{\int }_{0}^{\infty }{ \mathcal E }(z){dz}}.\end{eqnarray} \tag{ 6 }$$

At this point it is easier for the reader to appreciate the core shift discussed in item (3) above. The empirical jet starting point is ambiguous because this inherently asymmetric source is sampled by a very wide effective beam (compared with the proper size of the core). The effective beam (including interstellar scintillation) is 1.54 mas (compare that with the e-folding distance derived by this analysis of 1.2 mas in Figure 4). Thus, the beam is large compared with the characteristic dimension of the jet. When the beam is centered on the peak flux density (${F}_{\nu }({\rm{peak}})$) location, it is capturing flux density from both the approaching (${F}_{\nu }(\mathrm{core}\ \mathrm{app})$) jet and the receding (${F}_{\nu }(\mathrm{core}\ \mathrm{rec})$) jet. We can write this as

$$\begin{eqnarray}&&{F}_{\nu }({\rm{peak}})={F}_{\nu }(\mathrm{core}\ \mathrm{app})+{F}_{\nu }(\mathrm{core}\ \mathrm{rec}).\end{eqnarray} \tag{ 7 }$$

We have two constraints. From Equation (2) and the assumption of bilateral symmetry

$$\begin{eqnarray}&&\displaystyle \frac{{F}_{\nu }(\mathrm{core}\ \mathrm{app})}{{F}_{\nu }(\mathrm{core}\ \mathrm{rec})}=\displaystyle \frac{\delta {(v)}^{2+\alpha }}{\delta {(-v)}^{2+\alpha }}.\end{eqnarray} \tag{ 8 }$$

We use the value of $\alpha =0.1\pm 0.2$ for the jet on MJD 52731.6 that was previously determined from VLBA images (Ribo et al. 2004). The other constraint derives from the constant velocity assumed for the jet, which leads to the approximately constant Doppler enhancement along the jet from Equations (2) and (3) if $\alpha \approx 0$, as observed. Thus, the total jet integrated flux density fractions in the approaching and receding jets should be approximately the same as the ratio of flux density from the core assigned to the approaching and receding jets

$$\begin{eqnarray}\displaystyle \frac{f}{1-f} & \equiv & \displaystyle \frac{{F}_{\nu }(\mathrm{total}\ \mathrm{jet}\ \mathrm{app})}{{F}_{\nu }(\mathrm{total}\ \mathrm{jet}\ \mathrm{rec})}\approx \displaystyle \frac{{F}_{\nu }(\mathrm{core}\ \mathrm{app})}{{F}_{\nu }(\mathrm{core}\ \mathrm{rec})}\\ & & =\displaystyle \frac{\delta {(v)}^{2+\alpha }}{\delta {(-v)}^{2+\alpha }}.\end{eqnarray} \tag{ 9 }$$

The next thing that needs to be computed is the effective core offset in the radio image that assigns a fraction, f, of the peak flux density to the approaching jet and a fraction, $1-f$, of the peak flux density to the receding jet. We need this offset to find out exactly where to start the integration of the cumulative approaching jet flux density that is computed in the left-hand side of Figure 4. From item (1), a line source for the effective emissivity is consistent with the data. Therefore, we can approximate the offset, Δ, of the start of the approaching jet from the center of the beam located at the point of peak flux density as

$$\begin{eqnarray}{\rm{\Delta }} & \approx & \displaystyle \frac{{F}_{\nu }(\mathrm{core}\ \mathrm{app})}{{F}_{\nu }({\rm{peak}})}(1.54\;{\rm{mas}})\\ & & -1.54\;{\rm{mas}}/2\approx (f-0.5)1.54\;{\rm{mas}}.\end{eqnarray} \tag{ 10 }$$

As an example, if one were to split the core evenly with both the receding jet and approaching jet, then $f\approx 0.5$. The jet would begin with zero offset (i.e., both the jet and counter jet start at the point of peak flux density). By Equation (9), this corresponds to $v\approx 0$. This linear approximation for the offset is only accurate if the beam is much wider than the size of the magnitude of the offset.

One can now appreciate how nonlinear an iterative solution is in general. One has five constraints, Equation (6) for both the jet and counter jet, the left-hand and right-hand equalities for the flux ratio in Equation (9) and Equation (10), and five unknowns (v, D, D_rec, f, and Δ). Thus, a solution should exist if the model is a reasonable representation of the physical situation. Trial values of v, D, D_rec, and Δ must be chosen. The value of Δ determines the start point of the integration of the jet and counter jet flux densities over their respective lengths. Using this starting point, one can compute the total flux density in both the approaching and receding jets, ${F}_{\nu }(\mathrm{core}\ \mathrm{app})$ and ${F}_{\nu }(\mathrm{total}\ \mathrm{jet}\ \mathrm{rec})$, respectively. From Equation (9) this determines f, which in turn determines Δ, by Equation (10) and hence the first aspect of the nonlinearity inherent to this problem. The other aspect of nonlinearity of this problem is that the chosen values of v (the Doppler boosting of both the jet and counter jet), D, and D_rec in conjunction with Equation (6), need to produce a cumulative flux density distribution that fits the observed cumulative flux density distribution determined from the VLBA observations. The observed cumulative flux density distribution determined from the VLBA observations is not independent of Δ (i.e., Δ changes the origin of integration). So, one is looking not just for an arbitrary solution for f and Δ, but one that can be fit with the simple exponential empirical model. Arbitrary values of Δ tend to produce observed cumulative flux density distributions determined from the VLBA observations with non-monotonic curvatures that are poorly fit by the exponential model. A third nonlinear aspect occurs when any of the parameters v, D, or D_rec are varied in the process of finding a solution; the other two parameters must be changed in order to fit the data due to the large beam size that captures both the approaching and receding jets near the core. Many iterations are required before an agreement of the constraints is reached.

Note that the method above is not guaranteed to produce a good fit if the empirical model is inherently inaccurate. However, we have found some rather tight fits to the data. Our best fit is shown on the left-hand side of Figure 4. The parameters are D = 1.2 mas for the approaching jet. This model also required $v=0.35c$, which corresponds to an apparent velocity of the approaching jet of ${V}_{\mathrm{app}}=0.36c$. A similar fit to the receding jet yielded ${D}_{\mathrm{rec}}=0.6$ mas, which is consistent with the foreshortening of the receding jet that is readily visible in the radio image. The velocity $v=0.35c$ corresponds to an apparent velocity of the receding jet of ${V}_{\mathrm{app}}=-0.25c$. The fraction of the flux in the approaching jet was found self-consistently to be $f\approx 69\%$. This agrees with the results of Ribo et al. (2004), but the agreement need not occur in general (see Figure 5 and the related discussion). The offset of the peak flux relative to the start of the jet is ${\rm{\Delta }}\approx 0.3\;{\rm{mas}}$.

As an aside, we note that Equation (10) is not the same calculation as the core offset due to SSA (Blandford & Königl 1979). We are not comparing core positions at different frequencies as in a discussion of SSA opacity. We are estimating the fraction of the peak flux in the very wide beam (compared with the size of the jet) that is attributable to the approaching jet at one frequency, 15 GHz. Certainly, there is a relationship between the two ideas, and the offset estimate in Equations (7)–(10) can be considered a crude independent method of estimating the magnitude of the shift in position of the origin of the jet relative to the peak of the flux density, as is the intent of SSA core shift analysis. This method requires

• 1.  
  a knowledge of the angle of the jet to the line of sight,
• 2.  
  the velocity of the jet and counter jet,
• 3.  
  a detected jet and counter jet,
• 4.  
  and a constant spectral index along the jet.
• 5.  
  the effective beam width is comparable to the spatial scale of variation (e.g., D)
• 6.  
  the effective beam width is much larger than the computed offset
• 7.  
  The jet and counter jet are unresolved in the transverse direction

The method assumes that

• 1.  
  the velocity is constant along the detected jet and counter jet,
• 2.  
  intrinsic bilateral symmetry.

In general, one does not have sufficient astrometric accuracy and numerous high frequencies of sampling in order to accurately find the core position by extrapolating the core shift to infinite frequencies (in the SSA based method). Equations (7)–(10) can provide a crude estimate of the core shift if the seven requirements listed above are achieved.

The computation of V_app is based on D, which is the e-folding distance of the linear source of effective emissivity. We now invoke the model of the flare decay posited in Section 2.1 and illustrated in Figure 3. This model is equivalent to identifying the time dependent line source for the effective emissivity of the approaching jet, ${S}_{{\rm{app}}}(x,t)$, after the central engine power has been drastically reduced as

$$\begin{eqnarray}{S}_{{\rm{app}}}(x,t) & = & {S}_{0}(0.7/D){e}^{-x/D}{\rm{\Theta }}(x-{V}_{\mathrm{app}}t)\\ & & +\epsilon (x){\rm{\Theta }}(x),\end{eqnarray} \tag{ 11 }$$

where ${\rm{\Theta }}(x-{V}_{\mathrm{app}}t)$ is the Heaviside step function, $\epsilon (x)$ is the low level background residual jet (i.e., the jet does not turn all the way off, in general), and S₀ is the overall normalization. The coefficient of 0.7 results from the fact that 70% of the 15 GHz flux in the jet 1.6 days earlier (on MJD 52731.6, the left-hand frame of Figure 4) is in the approaching jet and 30% in the receding jet (Ribo et al. 2004). We assume this same fraction. The light curves in Figure 2 represent the total integrated luminosity. Therefore, we also need the contribution from the receding jet. For the receding jet, one has

$$\begin{eqnarray}{S}_{{\rm{rec}}}(x,t) & = & {S}_{0}(0.3/{D}_{\mathrm{rec}}){e}^{x/{D}_{\mathrm{rec}}}{\rm{\Theta }}(-x+{V}_{\mathrm{rec}}t)\\ & & +{\epsilon }_{{\rm{rec}}}(x){\rm{\Theta }}(-x).\end{eqnarray} \tag{ 12 }$$

Assuming bilateral symmetry in Equation (1) and the line of sight of 60^◦ (i.e., $-120^\circ$ for the receding jet) determined in Reid et al. (2014), the receding apparent velocity is V_rec = −0.25c. At any time t, after the jet central engine was reduced in power, the total flux density, ${F}_{\nu }$, emitted from the compact jets is

$$\begin{eqnarray}{F}_{\nu }(t) & = & {\displaystyle \int }_{-\infty }^{\infty }{S}_{{\rm{app}}}(x,t)+{S}_{{\rm{rec}}}(x,t)\;{dx}\approx 0.7{S}_{0}{e}^{-{V}_{\mathrm{app}}t/D}\\ & & +0.3{S}_{0}{e}^{{V}_{\mathrm{rec}}t/{D}_{\mathrm{rec}}}.\end{eqnarray} \tag{ 13 }$$

The exponential line source from Equation (4) for the effective emissivity with an e-folding distance for the approaching (receding) jet of D = 1.2 mas (${D}_{\mathrm{rec}}=0.6$ mas) and a bulk velocity V_app = 0.36c applied to Equations (11)–(13) can be used to quantify the simple fading jet model of Section 2.1. Equation (13) yields the fit to the decay data presented in Figure 2 in the right-hand side of Figure 4. We plot the curve two ways to investigate the affect of shifting the origin of time. The solid red curve is a plot of the decay if it started when the observation started. The overall normalization is ${S}_{0}=94$ mJy. The heavy dashed black curve had a starting flux density of 135 mJy when the decay started, similar to the level the day before (see Figure 2). This means the observation had to have started 1.2–1.4 hr before the observation in order to achieve a good fit. The decay is not that sensitive to the start time for an exponential decay, as can be seen from Equation (13). The larger term from the approaching jet also comes from the longer jet with the larger decay time, so this larger exponential function dominates more as the time elapses and the decay gets closer to a pure exponential. There is a slight excess at late times to the fit in the right-hand side of Figure 4, which might represent emission from the weak new jet. In summary, this empirical source function for the effective emissivity reproduces the compact jet in the HPS on MJD 52731.6 and (in the context of the simple jet fading model) the subsequent decay on MJD 52733 just 1.6 days after the jet was imaged.

Even though a major advantage of studying HPS is to provide a stable state to analyze, it should be noted that in general 1.6 days is a long time interval for dynamic change in GRS 1915+105. Major changes can occur in the X-ray and radio on timescales of seconds or minutes (Klein-Wolt et al. 2002). Although, we have no evidence that this occurs in HPS, it might not be valid to assume that the D value determined in Figure 4 typified the jet just prior to the decay. Without simultaneous data, this can only be estimated by studying multiple VLBA images of HPS. This is addressed below.

The most sensitive radio imaging of an HPS jet was on MJD 52748.4 (Ribo et al. 2004). Figure 5 shows the fit of a linear decaying source of the effective emissivity to the cumulative flux distribution along both the approaching and receding jets. The fit is defined by $D={D}_{\mathrm{rec}}$ = 1.5 mas and v = 0.15c. This jet is much more symmetric than the jet on MJD 52731.6 and therefore requires a much smaller bulk velocity. The decay length is slightly larger than that on MJD 52731.6 in Figure 4. However, there is one interesting feature that is unique to this radio image: a weak discrete ejection appears from 2 to 4 mas beyond the end of the compact jet. Comparing with the 8.3 GHz radio image in Ribo et al. (2004) indicates that it is optically thick (flat spectrum). The feature is 10 times the rms noise in the image and seems to be real, but without a deeper image showing a similar feature we cannot rule out that this is an artifact of inadequate u–v coverage. Thus, we must consider this to be tentative detection of a discrete component. A third unpublished HPS compact jet observation with comparable u–v coverage and sensitivity to the two images analyzed in this paper was also performed³ and occurred on MJD 52722; it is also consistent with $D\;\approx$ 1.2–1.5 mas.

The range of 1.2 mas $\lt \;D\;\lt$ 1.5 mas is consistent with the two other 15 GHz VLBA published observations of compact jets with comparable u–v coverage, flux density, and sensitivity on MJD 50914 and MJD 50935 (Dwahan et al. 2000). However, these jets were not produced during a HPS. It is worth noting that the much weaker (45 mJy) compact jet observed on MJD 50744 reported in Dwahan et al. (2000) was fit with an e-folding distance of $\lt 1$ mas. This small value might be an artifact of the insufficient sensitivity to detect the faint jet extremities.

The 1.6 days between the VLBA observation on MJD 52731.6 and the decay on MJD 52733 is quite long for GRS 1915+105, so one cannot assume that this was the jet length during the decay as noted above. From Figure 2, the e-folding time is bound by the existing observations by 3.8 hr $\;\lt \;T\;\lt$ 8.6 hr. From Figures 4 and 5 and the discussion that followed, 1.2 mas $\;\lt \;D\;\lt$ 1.5 mas. Then from Equation (1), this creates a range of apparent jet velocities, $0.16c\lt {V}_{\mathrm{app}}\lt 0.46c$, and a range of intrinsic jet velocities of $0.17c\lt v\lt 0.43c$. The high end of the range is corroborated by other data. It is driven by the rapid decay on MJD 52733 of only 3.6 hr. It is very likely that this rapid decay is the result of this being a high velocity jet state. It was noted in Ribo et al. (2004) that this was the most asymmetric compact jet ever detected. Thus, their analysis of jet speed from Doppler asymmetry produced the largest v of any compact jet by a considerable amount. Correcting their result for the new line of sight (see the Discussion) from the parallax measurements of Reid et al. (2014), yields $v=0.37\pm 0.04c$, which is in agreement with the maximum estimated v attained in this study. Another argument for this being the high end of the velocity range is that Table 1 indicates that this HPS was much more luminous than any other observed HPS.

A key assumption of this study is the scenario for jet fading described in Section 2.1 and illustrated schematically in Figure 3. In this scenario, the jet fading is a transient decay of a strong jet that was abruptly suppressed at its source. There is no direct observational evidence to support this idea and the scenario is not unique. Alternatively, one might think of the jet fading as a slow turnoff of the jet source in which one of the steady state power law solutions of Punsly (2011) are slowly varied and no strong transient or contact discontinuity is created. The lone restriction here is that the plasma from the full power jet episode must propagate sufficiently far down the jet so that its contribution to the total flux density is negligible after an e-folding time T (i.e., ${V}_{\mathrm{app}}\gg D/T\approx 0.36c$). If this were not true, there would have to be a conspiratory connection between turning down the power source at the base of the jet and the rate that previously ejected high emissivity plasma is losing its surface brightness in order to produce an exponential decay in time—not only once but on both MJD 52733 and MJD 53122.

In order to assess the plausibility of these ideas, we first note the nontrivial successes of the jet fading scenario presented here:

• 1.  
  The model provides a physical connection between the spatial exponential decay profile of the source of effective emissivity of the steady state HPS jet and the temporal exponential decay of the fading of the jet emission 1.6 days later
• 2.  
  The constraint imposed by the above physical connection implies a jet velocity that agrees very closely with the jet velocity found in Ribo et al. (2004) by considering jet bilateral asymmetry due to Doppler abberation of an intrinsically bilaterally symmetric system.

By contrast, consider these issues in the context of a slowly varying jet scenario.

• 1.  
  The spatial exponential decay profile of the source of effective emissivity of the steady state HPS jet and the temporal exponential decay of the fading of the jet emission 1.6 days later are coincidental. The temporal decay profile is a manifestation of how the local physics at the base of the jet alters the parameters injected into the jet model. It is a coincidence that this follows an exponential profile on both MJD 52733 and MJD 53122.
• 2.  
  The constraint, ${V}_{\mathrm{app}}\gg D/T\approx 0.36c$, noted above (that is required to produce the temporal exponential decay) disagrees with the jet velocity found in Ribo et al. (2004) by considering jet bilateral asymmetry due to Doppler abberation of an intrinsically bilaterally symmetric system

One other curious feature of the slowly varying jet scenario is evident if the details of the power law models of Punsly (2011) are considered. The base of the jet that produces the majority of the 15 GHz flux density is located $\lt 3\times {10}^{7}$ cm from the central black hole (see Table 1 and Figure 2 of Punsly 2011). Considering the timescales for decay of 3.8 and 8.6 hr from Figure 2, turning down the power of the jet by moving the plasma in the disk at the base of the jet corresponds to a rather suspect slow radial velocity of ∼10³ cm s⁻¹.

In summary, the transient model implies one value of V_app that is consistent with the cumulative flux distribution, the rate of compact jet decay, and the bilateral asymmetry. The slowly varying jet model is not consistent with bilateral asymmetry, and the variations in time and space are unrelated observable parameters. We choose to consider the interpretation that has descriptive power and is more consistent with other observations, although we cannot rule out the slowly varying jet model.

In order to characterize the source X-ray behavior during the HPS, we looked at data that were simultaneous with the HPS that is best followed at radio frequency. We then reduced and analyzed RXTE/PCA data taken during observations that began on MJD 53100.230, 53117.114, 53122.107, and 53122.179.

The data were reduced in a very standard manner (as in e.g., Rodriguez et al. 2008) with the HEASOFT V6.16 software suite. Energy spectra were obtained from the top layer of the Proportional Counter Unit (PCU) #2, background spectra were produced from the "bright source" model, and responses produced with pcarsp for the appropriate PCU, layer, and anode. We added 0.6% systematic uncertainties to all spectral channels and analyzed the resultant spectra within XSPEC v12.9.0.

High resolution ($7.8125\times {10}^{-3}$ s) light curves covering the full PCA spectral range were obtained from different data modes in different observations. On MJD 53100.230, we used the GOOD XENON data, while for the other three observations we combined BINNED and EVENT that, respectively, cover the low (0–35 or ∼2–15 keV) and high (36–255, $\gt 15$ keV) spectral channels. PDS from each individual observation were produced with POWSPEC on intervals of 64 s; all intervals were further averaged.

In order to see the spectral evolution, we show a plot of the spectral energy distributions (SEDs) in the top frame of Figure 6. The first epoch is MJD 53117.114 in red, which occurs during a HPS (see Figure 1). It is a much harder spectrum than the two spectra obtained on MJD 53122.107 (green) and MJD 53122.179 (blue) during the decay of the same HPS (see Figure 2). The plot shows that the spectral softening appears to continue between MJD 53122.107 and MJD 53122.179.

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The spectral evolution can be formally characterized by parametric spectral fits. To look for signs of spectral evolution we tried models that required an accretion disk, but there is no evidence of a disk in any of the data. Our model had the following parameters: The absorbing column of neutral hydrogen, N_H, was allowed to vary for MJD 53122.107 and MJD 53122.179; in the models described in Table 2, we found values of $5.69\pm 0.27\times {10}^{22}\;{\mathrm{cm}}^{-2}$ and $5.88\;\pm 0.26\times {10}^{22}\;{\mathrm{cm}}^{-2}$, respectively. Because the values were similar, we fixed ${N}_{{\rm{H}}}=5.7\times {10}^{22}\;{\mathrm{cm}}^{-2}$ for MJD 53117.114, which constrained the power law better in the more complicated fit for this epoch in Table 2. We note that if N_H was left free in the fit on MJD 53117.114, it optimized at ${N}_{{\rm{H}}}=6.3\times {10}^{22}\;{\mathrm{cm}}^{-2}$. Columns (2)–(4) of Table 2 describe the properties of the Gaussian line used in the fit. Columns (5) and (6) are the cutoff energy and e-folding energy required by the fit, respectively. The particle number density power law spectral index, Γ, is tabulated in Column (7). The epoch MJD 53117.114, during the HPS, has a high energy excess and requires a second power law that is poorly constrained by the data, as indicated by the large relative uncertainty in column (8). The hard second power law comprises only 1.5% of the flux from 1.5–150 keV (extrapolated down to 1.5 keV). The total observed flux from 1.5 to 150 keV (extrapolated down to 1.5 keV) is listed in column (9). The intrinsic flux (corrected for N_H) from 1.5 to 150 keV is tabulated in column (10). The reduced ${\chi }^{2}$ and degrees of freedom of the model are in the last column. The most striking evolution of the X-ray spectrum from the fully powered up jet state to the jet decay is provided by the spectral index of the underlying power law in column (7). The power law steepens from ${\rm{\Gamma }}=2.11$ to ${\rm{\Gamma }}=2.55$. Perhaps the best way to characterize the steepening of the X-ray spectrum during the jet decay is by looking at the intrinsic flux ratio, $R={F}_{1.5-3}/{F}_{12-50}$, where ${F}_{1.5-3}$ is the intrinsic flux in the interval 1.5–3 keV and ${F}_{12-50}$ is the flux in the interval 12–50 keV. The fluxes were computed from PCA observations with the assumption (in contrast to Table 2) that ${N}_{{\rm{H}}}=5.7\times {10}^{22}\;{\mathrm{cm}}^{-2}$ and models otherwise similar to those described in Table 2. The fluxes below 2.5 keV were extrapolated from the model and would not accurately depict any low energy contribution from a thermal disk component. These data reductions and extrapolations were originally performed in Punsly & Rodriguez (2013a) as part of a method that calibrated counts from the RXTE All Sky Monitor with PCA determined fluxes. On MJD 53117.114, R = 0.42 compared to R = 1.14 on MJD 53122.179. This softness ratio, R, is plotted during the course of the HPS and during its decay in the bottom frame of Figure 6. There is a dramatic change during the decay. Considering the top frame of Figure 6, this affect is driven primarily by a change in the soft X-ray flux. The physical interpretation would seem to be that there are less low energy Comptonizing electrons in the jet state and more low energy Comptoninzing electrons in the decay state.

Table 2.  X-Ray Spectral Fits

1	2	3	4	5	6	7	8	9	10	11
Epoch	Line	Line	Line	E	E	Power	Power	Flux	Flux	Reduced
Date	Center	Sigma	Norm	Cutoff	Fold	Law	Law	10−8	Intrinsic	${\chi }^{2}$
(MJD)	(keV)	(keV)		(keV)	(keV)	1	2	$(\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}$	${10}^{-8}\;\mathrm{erg}\;{{\rm{s}}}^{-1}\;{\mathrm{cm}}^{-2}$	(dof)
53117.114	5.86 ± 0.10	1.61 ± 0.08	0.13 ± 0.01	13.9 ± 0.2	25.4 ± 1.0	2.11 ± 0.06	0.54 ± 0.69	2.12 ± 0.03	3.15 ± 0.05	1.46(100)
53122.107	5.69 ± 0.27	1.24 ± 0.10	0.08 ± 0.01	14.3 ± 0.4	34.9 ± 2.9	2.51 ± 0.02	...	2.14 ± 0.03	3.80 ± 0.06	1.00(96)
53122.179	6.29 ± 0.13	1.10 ± 0.10	0.08 ± 0.01	14.7 ± 0.4	35.2 ± 1.8	2.55 ± 0.02	...	2.14 ± 0.03	3.80 ± 0.06	1.30(87)

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Figure 7 contains plots of the power spectral density (PSD) on MJD 53117.114 in red with the PSD of the combined data from MJD 53122.107 and MJD 53122.179 in black. Energy in the temporal variations migrates from a lower frequency during the HPS to a significantly higher frequency during the jet decay. As a reflection of this, the fundamental quasi-periodic oscillation (QPO) frequency changes by a factor of 3.5: from 0.69 Hz to 2.43 Hz. We also plot another PSD during the HPS on MJD 53100.230. During the entire HPS, the QPO fundamental frequency is less than 1 Hz and this frequency dramatically changes during the decay, as noted above.

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