Toward the Detection of Relativistic Image Doubling in Imaging Atmospheric Cherenkov Telescopes

Air showers may result when a high-energy particle enters the Earth's atmosphere. One such particle—hereafter assumed to be a gamma-ray for simplicity—interacts high up in the atmosphere and starts a cascade of secondary particles, many of which move faster than the local atmospheric speed of light. These superluminal charged secondaries trigger the isotropic emission of Cherenkov light (Morrison 1958) by the atmosphere immediately in their wake.

Practically, the gamma-ray must have an energy of at least a few hundred GeV to produce Cherenkov light visible on the ground (Porter & Weekes 1977). For a primary photon at one TeV energy, about 100 photons per m² are seen on the ground to a detection radius of about 125 m (Porter & Weekes 1977). The Cherenkov photons arrive within a very short time interval, a few nanoseconds (Holder et al. 2006). Images obtained with Imaging Atmospheric Cherenkov Telescopes (IACTs) show the projected track of the air shower, which points back to the celestial object where the incident gamma-ray originated. Past and current IACTs, including Whipple (Lewis 1990), HEGRA (Daum et al. 1997), VERITAS (Weekes et al. 2002), MAGIC (Baixeras et al. 2004), and H.E.S.S. (Aharonian et al. 2006), have been instrumental in discovering and studying many astronomical sources of high-energy radiations. A prominent future IACT is planned by CTA (Actis et al. 2011).

Recent advances in understanding the appearance of objects moving faster than light through a given medium have revealed an unusual optical phenomenon known as "relativistic image doubling" (RID; Cavaliere et al. 1971; Nemiroff 2015, 2018, 2019). Specifically, after the component of the speed of a superluminal object toward an observer (radial velocity) drops from superluminal to subluminal, an observer, here an IACT can perceive two images of that object simultaneously—with one image moving forward along the expected track, and a second image moving backward along the original track. RID effects have recently been hypothesized to help explain light curve features in gamma-ray bursts (Hakkila & Nemiroff 2019).

Although an example of RID has been recovered in the laboratory (Clerici et al. 2016), its occurrence has not been clearly isolated, as yet, elsewhere. Previously, papers about IACT imaging briefly indicated that such an effect could occur. For example, Hillas (1982) mentioned that "At about 100–150 m from the axis the radiation from several heights arrives simultaneously." The effect was also touched on in Hess et al. (1999) with the statement "In particular, the profiles in the 60 to 120 m distance range show a parabolic rather than linear shape, with photons from both the head and the tail end of the shower arriving late." This parabolic shape was evident in a few of the curves in their predictive Figure 9(a). Furthermore, Hess et al. (1999) indicated an actual detection by a HEGRA IACT in their Figure 13(b), although the plotted error bars appear to obscure its statistical significance.

More recently, Aliu et al. (2009) reported on Monte Carlo simulations in support of the MAGIC IACTs that show "In case of a small impact parameter (IP ≤ 60 m), the light emitted in the higher part of the shower (the shower head) will arrive delayed with respect to the light emitted in the lower part of the shower (the tail)" and "Events with an intermediate impact parameter show a flat time profile." Many of the simulated ovals in Figure 2 of Aliu et al. (2009) indicate that the central part of some IACT images from MAGIC are expected to arrive about a nanosecond before the outer parts of the image ovals.

The record of the Cherenkov light from an atmospheric shower by an individual telescope is typically a fleeting oval of illuminated pixels (Aharonian et al. 2006). Although these images are usually published without indication of time resolution beyond the likely duration of the event, which is a few nanoseconds, sub-event timing at a resolution of two-nanoseconds—or below—can be recorded (Holder et al. 2006; Maier & Holder 2017).

Although RID effects are inferred by these references, very few details are given. In particular, how and why two images of the shower are created together, how they diverge, how they are related to the Cherenkov cone, and the speeds and brightness of the individual images, have not been discussed.

Visually, a single observer or IACT will see Cherenkov emission at its brightest when the cluster of particles in the descending air shower is at the specific angle corresponding to the edge of the Cherenkov cone (Butslov et al. 1963). As will be made clear, RID effects create the Cherenkov cone, making these effects the rule for air shower detections rather than the exception. The double-image creation event may be obscured, however, near the creation point by air-shower width, such that some air showers will show diverging images more clearly than others.

It is the purpose of this work to conceptually isolate and study RID effects visible to IACTs, highlighting the temporal and angular scales that are most prominent. Section 2 will focus on the concepts behind RID, while 3 will present useful formulae. Subsequently, Section 4 will present the results of a simple example simulation, while Section 5 concludes with some discussion.

In the example scenario described, a cosmic-ray photon strikes the atmosphere at height z = h at t = 0. Initially, the angular height of the shower from the IACT vertex is $\theta =\arctan (h/L)$. The height of the shower at time t is therefore $z=h-{vt}$. The component of the shower's speed directly toward the IACT is ${v}_{r}=v\sin \theta$, where v_r is the shower's radial component toward the IACT.

A critical angle occurs when ${v}_{r}(z)={c}_{\mathrm{air}}$, where the IACT is at an angle $\sin {\theta }_{C}={c}_{\mathrm{air}}/v$ with respect to the path of the shower. This angle arises when the shower is at a critical height z_C, quantified below in Equation (5). Near this shower height, the speed of the shower toward the IACT closely matches the speed of the Cherenkov light toward the IACT. Therefore, a relatively large amount of this light arrives at the IACT "bunched up"—over a very short time. At height z_C, it can be said that the observer is on the "Cherenkov Cone." Note that since Cherenkov light is emitted uniformly and isotropically along the shower trajectory, the Cherenkov Cone is an observer-dependent phenomenon.

For most photons striking the Earth's atmosphere, an IACT will never be inside the shower's Cherenkov Cone for any value of z. Therefore, for these showers, the radial speed of the shower toward the IACT is never greater than c_air. These distant IACTs will see the shower go through the atmosphere, from sky to ground, monotonically. This occurs for all IACTs far away from the shower, specifically when $L\gt h\ \cot {\theta }_{C}$.

Conversely, for IACTs close to the shower base, where $L\lt h\ \cot {\theta }_{C}$, will be inside the Cherenkov Cone for at least some shower heights z. These IACTs can see an RID event. Specifically, there will be a region starting at the top of the atmosphere and extending down to z_C where v_r > c_air. In this region, the speed of the shower toward the IACT is faster than the speed of the Cherenkov radiation it triggers. When eventually seen by the IACT, this part of the shower will be seen time-backward, meaning that Cherenkov radiation emitted in shower locations increasingly earlier will reach the IACT increasingly later. This part of the shower will appear to go up from z_C. After the shower has descended down past z_C however, the radial speed of the shower will be slower than its Cherenkov light, so this part of the air shower will appear normally: time forward and headed down.

The first Cherenkov light detected by the IACT will be emitted when the shower is exactly at the height z = z_C, when the IACT is on the shower's Cherenkov Cone, such that v_r = c_air. After that, the IACT receives Cherenkov light from two locations: from both above and below z_C, at the same time. This is the basis for RID (Nemiroff & Zhong 2017; Nemiroff 2018).

We now derive equations and algorithms relevant to demonstrating how RID appears in air showers in the example scenario. The location of the actual shower at time t is not the location of the two images of the shower visible to the IACT at time t, because it takes a significant amount of time for light to go from the shower to the IACT. At height z, the actual radial velocity of the shower is

$$\begin{eqnarray}&&{v}_{r}(z)=\displaystyle \frac{{zv}}{\sqrt{{L}^{2}+{z}^{2}}},\end{eqnarray} \tag{ 1 }$$

with the shower's actual transverse velocity v_t(z) determined by ${v}^{2}={v}_{r}^{2}(z)+{v}_{t}^{2}(z)$. To calculate the perceived positions and perceived speeds of the shower images, light travel time must be incorporated. Starting at the top of the atmosphere, the total time t_total(z) for Cherenkov light to reach the observer is the addition of two parts: t_shower(z), the time it takes for the shower to reach height z, and t_light (z), the time it takes for Cherenkov light to go from height z to the IACT. Using straightforward geometry it is clear that

$$\begin{eqnarray}&&{t}_{\mathrm{shower}}=(h\mbox{--}z)/v,\end{eqnarray} \tag{ 2 }$$

while

$$\begin{eqnarray}&&{t}_{\mathrm{light}}=\sqrt{{L}^{2}+{z}^{2}}{/c}_{\mathrm{air}},\end{eqnarray} \tag{ 3 }$$

so that

$$\begin{eqnarray}&&{t}_{\mathrm{total}}={t}_{\mathrm{shower}}+{t}_{\mathrm{light}}.\end{eqnarray} \tag{ 4 }$$

The first height where the shower is seen at the IACT, t_min, occurs when t_total is a minimum. The corresponding height z_C can be found from when ${{dt}}_{\mathrm{total}}/{dz}=0$ such that

$$\begin{eqnarray}&&{z}_{C}=\displaystyle \frac{{c}_{\mathrm{air}}L}{\sqrt{{v}^{2}-{c}_{\mathrm{air}}^{2}}}.\end{eqnarray} \tag{ 5 }$$

The value of t_min can be found by substituting z_C into Equation (4). At later times, two images of the shower are visible simultaneously that have heights both above and below z_C. These are found by solving Equation (4) for z, yielding

$$\begin{eqnarray}&&{z}_{\pm }=\displaystyle \frac{{c}_{\mathrm{air}}^{2}{t}_{\mathrm{total}}v-{c}_{\mathrm{air}}^{2}h\pm \sqrt{{c}_{\mathrm{air}}^{2}{h}^{2}{v}^{2}-2{c}_{\mathrm{air}}^{2}{{ht}}_{\mathrm{total}}{v}^{3}+{c}_{\mathrm{air}}^{2}{L}^{2}{v}^{2}+{c}_{\mathrm{air}}^{2}{t}_{\mathrm{total}}^{2}{v}^{4}-{L}^{2}{v}^{4}}}{{v}^{2}-{c}_{\mathrm{air}}^{2}},\end{eqnarray} \tag{ 6 }$$

where z₊ and z₋ are the height of the shower's images, with z₊ always being above z_C, and z₋ always being below.

The speed of each image can be computed by taking v_± = dz_±/dt_total, and the angular speeds ${\dot{\theta }}_{\pm }$ can be computed by dividing the transverse component of each image speed by its distance from the IACT. To estimate the apparent brightness of each image, we start from the assumption that the shower is intrinsically uniform and isotropic everywhere along its path. If the apparent brightness was proportional to the absolute brightness, it could be estimated by simply noting the uniformly bright path lengths visible to the observer over a uniform time interval. However, since relatively long path lengths are seen over a uniform time interval when v_r ∼ c_air, the apparent brightness b of each image is proportional to the angular speed of the image. Since image brightness also falls by the square of its distance from the IACT, then

$$\begin{eqnarray}&&{b}_{\pm }\propto \displaystyle \frac{{\dot{\theta }}_{\pm }}{{L}^{2}+{z}_{\pm }^{2}}.\end{eqnarray} \tag{ 7 }$$

The brightness formally diverges at the start, when ${v}_{r}={c}_{\mathrm{air}}$ which occurs at z_± = z_C, but this formal divergence is mitigated in practice by the shower being of finite angular size.

The instantaneous angular size of the shower is finite, of course, because secondary particles spread out as the shower descends. This creates showers images that, at any one time, have both width and length. Shower widths are frequently resolved by IACTs so that "Showers having impact parameters at some distance away are viewed partly from the side, and the width and length of the image reflect largely the width and length of the particle cascade" (Hillas 1985). Therefore, in cases where IACT image lengths exceed their widths, two RID-caused shower images should be temporally resolvable moving out from the center along the length axis.

It will be convenient to compute the relative image brightness of the shower rather than its absolute brightness. For IACT detectors that can already detect photon-triggered air showers at some heights, the relative brightness will indicate how detectable the separating images should be, in comparison, at other heights. Also, computing the absolute brightness of the shower is more strongly dependent on highly variable factors like the energy of the incident photon. It is problematic to normalize shower brightness to the RID event since not all IACTs will see the RID and since, in the approximation used in the example, RID events have formally infinite brightness. Rather, it was chosen to normalize shower brightness to the shower brightness at the ground: b_ground, as this location is unambiguous and ubiquitous.

Specific results from a simple example are now presented. Here the shower is assumed to begin at height h = 25 km and go straight down. The distance of the IACT from the projected floor of the shower is assumed to be L = 100 m. The index of refraction of the air is taken to be n_air = 1.00029, with the speed of light in air then being c_air = c/n_air. The speed of the shower is assumed to be constant at v = c. From Equation (5), the value of z_C, the height z where v_r(z) = c_air, is found to be z_C = 4.15 km. Note that this z_C is relatively low in the atmosphere compared to the 25 km height where the shower first developed.

A plot of t_total versus z_±, as computed from Equations (4) and (6), is shown in Figure 1. Inspection of this figure clearly shows that t_total is double-valued with respect to z. The first time t_total that the IACT sees the shower is when z = z_C, when RID first is seen.

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A plot of the relative brightness of each image as a function of z_±, computed from Equation (7), is shown in Figure 2. This figure shows a sharp peak in the brightness of each image when v_r = c_air at z = z_C. The lower image is also relatively bright when near the IACT.

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The apparent brightness of each image is shown as a function of time t_total in Figure 3. This figure shows the light curves that would be seen by the IACT if the IACT had negligible size. The figure shows that at first the IACT sees nothing, but then suddenly, the shower is seen at height z_C = 4.15 km, with a sharp spike in brightness. This corresponds to the image-pair creation episode of the RID, when the IACT is on the shower's Cherenkov Cone. As the images move away from z_C—one image moves up toward the top of the atmosphere, while simultaneously the second image moves toward the ground. The top image disappears when it reaches the top of the atmosphere, while the bottom image disappears when it hits the ground. The time axes of Figures 1 and 3 are truncated when the upper image exits the top of the atmosphere.

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Codes that simulate the example in this section are listed in the Appendix.

Although the concept has been alluded to previously, few details about the appearance of RID in air showers have been published and therefore may be generally unknown. Some of these novel, interesting, and potentially useful details are summarized here. First, the relative brightness of the two images that appear in the descending air shower have never been noted previously. For the simple example case explored here, relative image brightness can be computed from Equation (7) and is evident in Figures 2 and 3. In general, the two images in RID will appear to be created with equal apparent brightness but the perceived ratio of their brightnesses will quickly diverge. This attribute of air shower RID can be important for identifying and verifying images resulting from RID. Furthermore, individually, both RID images are brightest just when they appear to be created, but the brightness of both images will appear to fade rapidly. In the simple example case studied, this fading is shown in Figures 2 and 3.

The sudden appearance of the RID images in air showers is novel, without classical precedent, and has not been published previously. Counterintuitively, there will be absolutely no flux from an air shower received by an IACT before it detects the shower's RID image creation event. It is not that the shower appears undetectably faint by the IACT at earlier times, but rather that no light at all from the descending shower will have reached the IACT before the arrival of the RID event. Then, suddenly, a bright RID event becomes visible. This is evident, for example, in Figure 1.

The kinematic context of why air shower RID images are so bright has also never been discussed before in the detail described here. The underlying physical reason for the high brightness of the RID images is that the radial speed of the air shower toward the IACT drops from superluminal to subluminal. The closer the radial speed of the shower to the speed of light toward the IACT, the larger the segment of the air shower seen by the IACT in a given amount of time. This corresponds to the passing of the Cherenkov cone over the IACT, but the underlying kinematics should not be obscured by citing this association abstrusively. In fact, the Cherenkov cone is a simple observer effect created precisely by this velocity coincidence (Nemiroff 2018).

It has also not been previously noted that the RID images appear with formally infinite angular speed. After creation, the speed of each diverging image appears to drop rapidly. This behavior is evident from the discussion surrounding Equation (7). This is important for detection and tracking of RID images by IACTs.

Another novel but previously unnoted attribute of air shower RID images is that one image of the RID pair is seen moving time-backward—back up into the upper atmosphere. This image moves up from the z_C because it is being seen at the IACT at progressively earlier times. In the simple example explored here, this behavior is clear from Equation (6) and Figure 1. However, the time-backward nature of upward moving images is generic to all shower RIDs. This may be important, for example, because it shows that even though early parts of the air shower are over, they may not be gone from view. Information about this early part of the shower may still be recovered in the next few nanoseconds because the images above z_C always arrive late.

Another newly noted feature of tracking individual images of an air shower RID is that the downward moving image (only) may pass relatively near the IACT as it approaches the ground—and thus may appear to rebrighten. Therefore, if this image becomes too dim to track, it may subsequently become visible to this IACT for a second time. This second visibility may occur even though this image is not related to the Cherenkov cone of the descending shower. This behavior results solely from the decreasing r in the 1/r² brightness falloff, and is evident in the example in Equation (6) and the leftmost part of Figure 2.

RID effects may be thought immeasurable because both images fade so rapidly. However, even tracking the images for a short angular distance would be interesting. It is unclear from a previous reference whether an RID effect has actually already been observed. Inspection of Figure 13(b) of the HEGRA paper Hess et al. (1999) does indicate such an effect, but the error bars are large and its statistical significance was not estimated—it could well be marginal. Calling new attention to this effect and giving previously unpublished details may make it a higher priority for existing and future IACTs to be equipped with sensitive and high speed imagers that could better detect RID and track the two diverging images.

One might argue that RID is a trivial effect. Who cares that two images of the same shower are sometimes visible simultaneously? In our view, however, RID is a novel optical effect caused not by lenses but solely by relativistic kinematics—and this is interesting basic physics even without a demonstrated usefulness. However, discovering and tracking air shower RID effects may even prove useful. Such observations could add information that better allows shower orientation to be recovered, or give an independent method of confirming air shower geometry. In the simple case discussed above, the ground location of the air shower L might be better located using Equation (7).

Studying the basic science of other multiple-imaging optical effects is common practice in gravitational lensing (Nemiroff 1987; Schneider et al. 1992) and temperature inversion effects in the Earth's atmosphere (Young 1999), for example.

Along this line, it is interesting to further compare, conceptually, RID to gravitational lensing. Multiple-imaging in gravitational lensing is created by similarities in the magnitudes of the time delay caused by the increased slowing of time near a gravitational mass, and the time delay caused by the increased path length further from the line of sight. These similarities create different locations in the plane of the lens that have the same total time delay, including critical points on the time-delay surface where images form.

Similarly, RID in air showers is created by the similarities in magnitude of the time delay for the shower to reach given heights, and the time delay caused by the different path lengths from those heights to the IACT. This similarity creates different locations along the shower's path that have the same total time delay, including critical locations where images form.

Why have RID effects remained so obscure? One reason is that they have no classical analog—they depend crucially on the finiteness of the speed of light. Classical thinking may allow scientists to accurately visualize how the air shower itself descends, but not always how it appears as it descends.

Another reason for the sparsity of previous RID analyses is that RID events happen locally on timescales too short for humans to notice: nanoseconds, as shown by Figure 1. However, as computer technology and miniaturization are increasing the frame rate that can be captured, imagers appearing are becoming capable of recording sub-nanosecond events (Clerici et al. 2016).

There are also attributes of realistic IACTs that obscure RID effects. One is that the IACT dish itself is not just a point—it may be so large that light travel time across its combined mirrors is significant when compared to t_light (B. Humensky 2019, private communication). Then, any light curve that the IACT measures will convolve the size and shape of the IACT, not just geometry inherent to the air shower. However, even in this case, if the IACT's angular point-spread function is smaller than the angular trajectory of the shower on the IACT's imager, then the dish-crossing time will not compromise the general character of the shower's temporal development on the IACT's imager. In general, no matter the size and geometry of the IACT, the central angular point of the shower at z = z_C will begin to light up first, after which two images of the shower will appear to diverge.

Another obscuring practical outcome is that RID effects will look different for individual IACTs, even for the same air shower. The ground locations of individual IACTs are paramount—only those close enough to the base of the air shower such that z_C < h will have a Cherenkov cone pass over them and be able to see an RID effect. For the example given, from Equation (1) and h = 25 km, any IACT with L > 603 m will not see the shower start with a RID. Therefore, simply adding together the images of multiple IACTs will likely not enhance the detection of RID effects, and may even convolute it beyond recognition. However, a careful reconstruction accounting for timing separate RID events as seen by different IACTs should capable of enhancing RID detection.

There are many interesting RID-related effects that were deemed too complex to be incorporated in the primarily conceptual work. One such interesting case occurs when the bulk speed of the particles in the air shower decreases significantly along its path to the surface. This may happen even if the shower remains superluminal and continually triggers Cherenkov radiation. If v decreases, then v_r will decrease proportionately, forcing z_C to occur higher in the atmosphere. Additionally, air density and ionization could cause significant variations in c_air. A full treatment of these effects would likely require more detailed and complex computer modeling.

In sum, resolving RID events in time and space would recover a novel type of optics caused not by lenses but by relativistic kinematics alone. Furthermore, tracking simultaneous images as they emerge and diverge in angle, angular speed, and relative brightness could resolve or independently confirm information about the air shower, including shower trajectory, speed, and brightness along its path as deconvolved through the equations in Section 3. These may, in turn, result in an increased accuracy in the direction of the originally incident photon or cosmic ray that started the air shower.

We thank Oindabi Mukherjee, Rishi Babu, and the anonymous referee for valuable comments.

The example in this work was computed by each coauthor separately as a check on the accuracy of the results. R.J.N.'s FORTRAN code is listed first, and N.K.'s Python code is listed afterward.

A.1. FORTRAN Code

A.2. Python Code