Lens Flare: Magnified X-Ray Binaries as Passive Beacons in SETI

The X-ray beacon can be a very simple optical system. Suppose a large, nonabsorbing lens with a very large focal length f is placed at distance ${x}_{\mathrm{em}}$ in front of a central illuminating source, in this case the NS in an LMXB (Figure 1, top). The thin lens equation gives the distance x in front of the lens at which a magnifying image is formed: $1/f-1/{x}_{\mathrm{em}}=1/x$. Now, if the lens is placed one focal length from the LMXB (${x}_{\mathrm{em}}=f$), the image will be formed at infinity, with infinite magnification. What this means is that all the light captured by the lens from one point on the LMXB's surface gets sent in one direction on the distant sky. The lens has become a collimator for the LMXB's light. From the observer's point of view, the entire lens appears to light up with the same surface brightness as whatever is directly behind its center (Figure 2, right). If a circular lens with a radius ${R}_{L}$ is viewed when it is directly in front of a spherical emitting region of radius ${R}_{\mathrm{em}}$, the luminosity will appear to increase by a factor of ${ \mathcal A }\equiv {({R}_{L}/{R}_{\mathrm{em}})}^{2}$, the ratio of the projected areas of the lens and the emitting region.

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Since we cannot resolve the system, an NS passing directly behind a large lens (${R}_{L}\gg {R}_{\mathrm{em}}$) appears as an eclipse, with an extremely bright flare in the middle. First, when the NS passes behind the edge of the lens, we lose the light from the NS (Figure 2, middle). Because the center of the lens is in front of empty space, we see the lack of light from that empty space instead. We have entered the shadow of the lens, which has angular radius ${\theta }_{\mathrm{shadow}}={R}_{L}/{x}_{\mathrm{em}}$. When the center of the lens occults the emitting region, the lens suddenly brightens to an apparent luminosity that is $\eta { \mathcal A }$ times brighter than the magnified source, where η is the efficiency of the lens. This corresponds to the image of the LMXB BL, which is beamed into a cone with angular radius ${\theta }_{\mathrm{image}}={R}_{\mathrm{em}}/{x}_{\mathrm{em}}$. Then, the center of the lens is in front of empty space again, even though the emitting region is still behind the lens periphery, and we are in eclipse again. Finally, the LMXB returns to the normal brightness once the lens is completely past the NS. Note that the BL only emits a fraction of the bolometric flux from the LMXB, which can be as high as ∼70% or as low as a few percent, but frequently is of order 50% (Sunyaev & Shakura 1986; Revnivtsev & Gilfanov 2006). The rest, from the larger accretion disk, will remain unaffected by the transit. The BL is expected to dominate the flux in harder X-rays of a few keV, though.

A lens trades the flare brightness for the fraction of the sky where the flare is visible, ensuring conservation of energy. If the illumination source has a flux of ${F}_{\mathrm{em}}$ and has a relative velocity ${v}_{\mathrm{rel}}$ with the lens, the lens shadow creates a fluence deficit of

$$\begin{eqnarray}&&| {\rm{\Delta }}{{ \mathcal F }}_{\mathrm{shadow}}| =\displaystyle \frac{{R}_{L}}{{v}_{\mathrm{rel}}}{F}_{\mathrm{em}}.\end{eqnarray} \tag{ 1 }$$

If we are fortunate, though, and the center of the lens has an impact parameter $\lt {R}_{\mathrm{em}}$, we observe a flare with fluence

$$\begin{eqnarray}&&{\rm{\Delta }}{{ \mathcal F }}_{\mathrm{flare}}=\displaystyle \frac{{R}_{\mathrm{em}}}{{v}_{\mathrm{rel}}}{F}_{\mathrm{em}}\times {\left(\displaystyle \frac{{R}_{L}}{{R}_{\mathrm{em}}}\right)}^{2}=\left(\displaystyle \frac{{R}_{L}}{{R}_{\mathrm{em}}}\right)| {\rm{\Delta }}{{ \mathcal F }}_{\mathrm{shadow}}| .\end{eqnarray} \tag{ 2 }$$

The flare has the much larger effect on fluence during its occultation. For most sight lines where the lens occults the NS, the lens center misses the NS, and only the short eclipse with no flares is observed. The probability of an eclipse without a flare is $1-{R}_{\mathrm{em}}/{R}_{L}$. This compensates for the brightness of the flare, so the average brightness of an LMXB transited by a dense, isotropically distributed swarm of orbiting efficient lenses is the same as an unlensed LMXB.

The X-ray index of refraction of atomic matter deviates by a small difference δ from 1. Although δ is small, it may be sufficient to build a simple lens because of the huge focal length, although the element would need to be a thin Fresnel lens to avoid X-ray absorption. Since δ is highly dependent on photon energy E, however, a single lens will suffer severe chromatic aberration. Typically, the dependence of δ is approximately ∝E⁻² (e.g., Braig & Predehl 2012). As a result, the focal length is $f{(E)\approx \bar{f}(E/\bar{E})}^{2}$, where $\bar{f}$ is some reference focal length for reference energy $\bar{E}$. I adopt the convention $\bar{f}={x}_{\mathrm{em}}$, so that the lens acts as a collimator for $\bar{E}$.

From the thin lens equation, the image is formed at a distance x from the lens, where x can be positive or negative. Either the light incident on the lens converges at x and then diverges beyond it, or it diverges from the virtual image at $-| x|$. Either way, the light from a point source ultimately forms a cone extending to infinity with opening angle

$$\begin{eqnarray}&&{\theta }_{\mathrm{image}}\approx \displaystyle \frac{{R}_{L}| {x}_{\mathrm{em}}-f| }{{{fx}}_{\mathrm{em}}},\end{eqnarray} \tag{ 3 }$$

assuming that ${x}_{\mathrm{em}}\gg {R}_{L}| {\bar{E}}^{2}-{E}^{2}| /{E}^{2}$, which is true except at very low (optical) energies. In addition, because the emitting source is not a point, this cone is broadened by an angle ${\theta }_{\mathrm{em}}={R}_{\mathrm{em}}/{x}_{\mathrm{em}}$, which contains all sight lines that pass from the emitting region through the center of the lens. Thus, if the emitting region has uniform brightness, the light intercepted by the lens diverges from the image as a cone with opening angle ${\theta }_{\infty }\equiv {\theta }_{\mathrm{image}}+{\theta }_{\mathrm{em}}$ (Figure 1, bottom).³ For an uncorrected lens, this angle varies as energy as

$$\begin{eqnarray}&&{\theta }_{\infty }\approx \displaystyle \frac{{R}_{\mathrm{em}}}{{x}_{\mathrm{em}}}\left(1+\displaystyle \frac{{R}_{L}}{{R}_{\mathrm{em}}}\displaystyle \frac{| {\bar{E}}^{2}-{E}^{2}| }{{E}^{2}}\right).\end{eqnarray} \tag{ 4 }$$

For photon energies very near $\bar{E}$, ${\theta }_{\infty }\approx {\theta }_{\mathrm{em}}$ as in the achromatic case. But away from that energy, the collected light diverges into a wider cone. If the lens is large (${R}_{L}\gg {R}_{\mathrm{em}}$), photons with $E\gg \bar{E}$ diverge into an angle $\sim {\theta }_{\mathrm{shadow}}$. The equality ${\theta }_{\infty }={\theta }_{\mathrm{shadow}}$ holds for $E=\bar{E}\sqrt{{R}_{L}/{R}_{\mathrm{em}}}$ and also $\bar{E}/\sqrt{2-{R}_{\mathrm{em}}/{R}_{L}}$, for which the lens essentially has no effect on a uniformly bright source. Furthermore, the light actually diverges into an angle much larger than ${\theta }_{\mathrm{shadow}}$ at low energies, with ${\theta }_{\infty }\gg {\theta }_{\mathrm{shadow}}$ when $E\ll \bar{E}/2$. In these energy ranges, the lens appears to glow weakly even if it is not covering any part of the emitting region.

Let d${L}_{\mathrm{em}}$/dE be the luminosity spectrum of the lensed region, and let Θ be the angle between the source, the lens center, and the sight line through the lens for an observer at infinity. The lens intercepts a specific luminosity of ${{dL}}_{\mathrm{em}}/{dE}\times (\pi {R}_{L}^{2})/(4\pi {x}_{\mathrm{em}}^{2})$, which is then emitted into a cone covering $\pi {\theta }_{\infty }^{2}$ sr. When ${\rm{\Theta }}\leqslant {\theta }_{\infty }$, the observed specific flux from the lens is ${{dL}}_{\mathrm{em}}/{dE}\times {R}_{L}^{2}/(4{x}_{\mathrm{em}}^{2})\times 1/(\pi {\theta }_{\infty }^{2}{d}^{2})$, or

$$\begin{eqnarray}&&\displaystyle \frac{{{dF}}_{\mathrm{lens}}}{{dE}}=\eta \displaystyle \frac{{{dL}}_{\mathrm{em}}/{dE}}{4\pi {d}^{2}}{\left(\displaystyle \frac{{R}_{L}/{R}_{\mathrm{em}}}{1+{R}_{L}/{R}_{\mathrm{em}}\times | {\bar{E}}^{2}-{E}^{2}| /E}\right)}^{2},\end{eqnarray} \tag{ 5 }$$

where d is the distance between the observer and the source. If ${\rm{\Theta }}\geqslant {\theta }_{\mathrm{shadow}}$, we also observe the flux from the unobscured emitting region itself, ${F}_{\mathrm{em}}=({{dL}}_{\mathrm{em}}/{dE})/(4\pi {d}^{2})$. Otherwise (${\rm{\Theta }}\lt {\theta }_{\mathrm{shadow}}$), the emitter is covered by the element, and only the lens contributes to the total observed flux. The excess flux ${\rm{\Delta }}{{dF}}_{\mathrm{obs}}/{dE}$ is found by summing the observed flux from the lens and the unobscured flux from the emitter and then subtracting the unmodified flux from the emitter ${{dL}}_{\mathrm{em}}/{dE}/(4\pi {d}^{2})$. An occultation event produces a change in the observed flux:

$$\begin{eqnarray}&&\begin{array}{l}{\rm{\Delta }}\displaystyle \frac{{{dF}}_{\mathrm{obs}}}{{dE}}=\displaystyle \frac{{{dL}}_{\mathrm{em}}/{dE}}{4\pi {d}^{2}}\\ \ \times \left\{\begin{array}{l}\eta {\left(\displaystyle \frac{{R}_{L}/{R}_{\mathrm{em}}}{1+{R}_{L}/{R}_{\mathrm{em}}\times | {\bar{E}}^{2}-{E}^{2}| /{E}^{2}}\right)}^{2}-1\\ & ({\rm{\Theta }}\leqslant {\theta }_{\mathrm{shadow}},{\theta }_{\infty })\\ \eta {\left(\displaystyle \frac{{R}_{L}/{R}_{\mathrm{em}}}{1+{R}_{L}/{R}_{\mathrm{em}}\times | {\bar{E}}^{2}-{E}^{2}| /{E}^{2}}\right)}^{2}-1 & \\ & ({\theta }_{\mathrm{shadow}}\leqslant {\rm{\Theta }}\leqslant {\theta }_{\infty })\\ & ({\theta }_{\infty }\lt {\rm{\Theta }}\leqslant {\theta }_{\mathrm{shadow}})\\ 0 & ({\theta }_{\infty },{\theta }_{\mathrm{shadow}}\lt {\rm{\Theta }}).\end{array}\right.\end{array}\,\end{eqnarray} \tag{ 6 }$$

At energies near $\bar{E}$, a chromatic lens flare proceeds much like an achromatic lens flare: a dip in the flux as the plate blocks the source, followed by a brilliant flash as the element lenses the source, and a return to the dip as the lens continues to move past the source, before ultimately returning to normal (see the animation in Figure 3). The intensity of the central flash is lessened and its duration lengthened away from perfect collimation. At energies far from $\bar{E}$, the specific flux actually remains in a dip if the lens is directly in front of the source, as the lens weakly glows with intensity <I_⋆ both during the occultation and at surrounding times.

The fluence from the event consists of two contributions, a deficit as the shadow of the lens passes over the observer and blocks the source, and a gain when the observer is in the cone of light that is created by the lens. The lens blocks the source for a time ${t}_{\mathrm{shadow}}=\sqrt{{R}_{L}^{2}-{b}^{2}}/{v}_{\mathrm{rel}}$ and transmits light for ${t}_{\mathrm{flare}}=\sqrt{{\theta }_{\infty }^{2}{x}_{\mathrm{em}}^{2}-{b}^{2}}/{v}_{\mathrm{rel}}$, where $b\equiv {x}_{\mathrm{em}}\min {\rm{\Theta }}$ is the projected impact parameter of the source. Consider the case when the source passes directly behind the lens (b = 0) with constant relative speed ${v}_{\mathrm{rel}}$. The lens then blocks the source for a time ${t}_{\mathrm{shadow}}=2{R}_{L}/{v}_{\mathrm{rel}}$, neglecting the small size of the source, causing a fluence deficit of $d{{ \mathcal F }}_{\mathrm{shadow}}/{dE}={dF}/{dE}\times {t}_{\mathrm{shadow}}$, or

$$\begin{eqnarray}&&\displaystyle \frac{d{{ \mathcal F }}_{\mathrm{shadow}}}{{dE}}=\displaystyle \frac{{{dL}}_{\mathrm{em}}/{dE}}{4\pi {d}^{2}}\displaystyle \frac{2{R}_{L}}{{v}_{\mathrm{rel}}}.\end{eqnarray} \tag{ 7 }$$

We observe transmitted light through the lens for a time ${t}_{\mathrm{flare}}=2{\theta }_{\infty }{x}_{\mathrm{em}}/{v}_{\mathrm{rel}}$, and the fluence of this transmitted light is

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{d{{ \mathcal F }}_{\mathrm{flare}}}{{dE}} & = & \eta \displaystyle \frac{{{dL}}_{\mathrm{em}}/{dE}}{4\pi {d}^{2}}{\left(\displaystyle \frac{{R}_{L}}{{R}_{\mathrm{em}}}\right)}^{2}\displaystyle \frac{2{R}_{\mathrm{em}}}{{v}_{\mathrm{rel}}}\\ & & \times \,{\left(1+\displaystyle \frac{{R}_{L}}{{R}_{\mathrm{em}}}\displaystyle \frac{| {\bar{E}}^{2}-{E}^{2}| }{{E}^{2}}\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 8 }$$

So, if the lens center passes exactly over the source, the net fluence excess caused by this lens flare event is

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{d{{ \mathcal F }}_{\mathrm{excess}}}{{dE}} & = & \displaystyle \frac{{{dL}}_{\mathrm{em}}/{dE}}{4\pi {d}^{2}}\displaystyle \frac{2{R}_{L}}{{v}_{\mathrm{rel}}}\\ & & \times \,\left(\displaystyle \frac{\eta {R}_{L}/{R}_{\mathrm{em}}}{1+({R}_{L}/{R}_{\mathrm{em}})| {\bar{E}}^{2}-{E}^{2}| /{E}^{2}}-1\right).\end{array}\end{eqnarray} \tag{ 9 }$$

Using a single lens with chromatic aberration instead of an achromatic system may be easier and produces a unique spectral evolution that might be recognized by observers that either are nearby or have powerful instruments. The lens flare event is much dimmer, however, limiting its use as a beacon. Figure 3 shows an example of how the spectrum of an emission region changes during a lensing event. When the lens is perfectly aligned with the source, the resultant spectrum is strongly peaked (thin black line). Although it reaches the achromatic specific luminosity at $E=\bar{E}$, the bolometric luminosity of the lens is far less than in the achromatic case (solid gray line), and at low energies, it is less than the unobscured source (dashed gray line). For the chromatic lens shown in Figure 3, the peak amplification of the bolometric luminosity is 70 instead of 10⁴. The shadow of the lens manifests at ${\rm{\Theta }}\gtrsim {\theta }_{\mathrm{em}}$ as the suppression of the flux in an energy range containing this peak. The weak emission of the zone plate at low energies, which is visible even when the element does not cover the emission region, is minor.

The Fresnel zone plate is a very simple optical system capable of forming X-ray images at very long focal lengths. It consists simply of alternating, concentric opaque and transparent rings, where the ring width decreases farther from the center. Variants like Phase Fresnel Lenses delay the phases of incoming electromagnetic waves instead of absorbing them and attain a high efficiency (Skinner 2002, 2010). The whole disk then has a focal length that depends on the wavelength of the incident light.

Diffractive optics also suffers from chromatic aberration. The focal length is $f(E)=\bar{f}E/\bar{E}$, where $\bar{f}$ is some reference focal length for reference energy $\bar{E}$ (e.g., Skinner 2002). Thus, photons of a particular energy are emitted into a cone with angular radius

$$\begin{eqnarray}&&{\theta }_{\infty }\approx \displaystyle \frac{{R}_{\mathrm{em}}}{{x}_{\mathrm{em}}}\left(1+\displaystyle \frac{{R}_{L}}{{R}_{\mathrm{em}}}\displaystyle \frac{| \bar{E}-E| }{E}\right),\end{eqnarray} \tag{ 10 }$$

and the flux and fluence can be calculated similarly as a lens with chromatic aberration. The observed flux from the element itself is

$$\begin{eqnarray}&&\displaystyle \frac{{{dF}}_{\mathrm{lens}}}{{dE}}=\eta \displaystyle \frac{{{dL}}_{\mathrm{em}}/{dE}}{4\pi {d}^{2}}{\left(\displaystyle \frac{{R}_{L}/{R}_{\mathrm{em}}}{1+{R}_{L}/{R}_{\mathrm{em}}\times | \bar{E}-E| /E}\right)}^{2},\end{eqnarray} \tag{ 11 }$$

resulting in a flux change of

$$\begin{eqnarray}&&\begin{array}{l}{\rm{\Delta }}\displaystyle \frac{{{dF}}_{\mathrm{obs}}}{{dE}}=\displaystyle \frac{{{dL}}_{\mathrm{em}}/{dE}}{4\pi {d}^{2}}\\ \times \left\{\begin{array}{ll}\eta {\left(\displaystyle \frac{{R}_{L}/{R}_{\mathrm{em}}}{1+{R}_{L}/{R}_{\mathrm{em}}\times | \bar{E}-E| /E}\right)}^{2}-1 & \\ & ({\rm{\Theta }}\leqslant {\theta }_{\mathrm{shadow}},{\theta }_{\infty })\\ \eta {\left(\displaystyle \frac{{R}_{L}/{R}_{\mathrm{em}}}{1+{R}_{L}/{R}_{\mathrm{em}}\times | \bar{E}-E| /E}\right)}^{2}-1 & \\ & ({\theta }_{\mathrm{shadow}}\leqslant {\rm{\Theta }}\leqslant {\theta }_{\infty })\\ & ({\theta }_{\infty }\lt {\rm{\Theta }}\leqslant {\theta }_{\mathrm{shadow}})\\ 0 & ({\theta }_{\infty },{\theta }_{\mathrm{shadow}}\lt {\rm{\Theta }}).\end{array}\right.\end{array}\end{eqnarray} \tag{ 12 }$$

The net fluence of the event is

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{d{{ \mathcal F }}_{\mathrm{excess}}}{{dE}} & = & \displaystyle \frac{{{dL}}_{\mathrm{em}}/{dE}}{4\pi {d}^{2}}\displaystyle \frac{2{R}_{L}}{{v}_{\mathrm{rel}}}\\ & & \times \ \left(\displaystyle \frac{\eta {R}_{L}/{R}_{\mathrm{em}}}{1+({R}_{L}/{R}_{\mathrm{em}})| \bar{E}-E| /E}-1\right),\end{array}\end{eqnarray} \tag{ 13 }$$

if the source is small and it passes directly behind the lens (b = 0).

The time-dependent spectrum observed when a diffractive element passes directly in front of the NS is illustrated in Figure 3. Qualitatively, it behaves similarly to a single refractive lens with chromatic aberration, but its spectral peak is wider and the event is brighter since f has a weaker dependence on E. It attains a maximum bolometric amplification of 130.

An isolated NS may be considered as stationary relative to the orbits of lenses. If ETIs want to set up an isotropic beacon to draw the attention of others in all directions, they need many lenses in orbit of the LMXB to ensure that at least one passes in front of the NS as viewed from each direction. Furthermore, a recipient with relatively poor X-ray detection abilities, like ourselves, is unlikely to detect individual eclipses at large distances because eclipses are so short and the photon fluence deficit is so small.⁴ Thus, they would have to ensure that at least one lens transits with impact parameter $\lt {R}_{\mathrm{em}}$ of the NS as viewed from each direction to ensure that flares of peak luminosity are observed. This requires a swarm of at least ${N}_{L}\gtrsim \pi {x}_{\mathrm{em}}/{R}_{\mathrm{em}}$ lenses. A lens flare then occurs once per orbital period.

The ETIs may want to ensure a higher lens flare rate if the lenses have very long orbital periods ${P}_{L}$, with correspondingly large ${a}_{L}$ (assumed equal to ${x}_{\mathrm{em}}$). To ensure a flare rate of ${{\rm{\Gamma }}}_{\mathrm{flare}}$ ($\gt 1/{P}_{L}$) as defined by passages within ${R}_{\mathrm{em}}$, the necessary number of lenses is $\pi {a}_{L}/{R}_{\mathrm{em}}\times {P}_{L}{{\rm{\Gamma }}}_{\mathrm{flare}}$, which is

$$\begin{eqnarray}&&{N}_{L}\approx \displaystyle \frac{2{\pi }^{2}{{\rm{\Gamma }}}_{\mathrm{flare}}}{{R}_{\mathrm{em}}}\sqrt{\displaystyle \frac{{a}_{L}^{5}}{(1-\beta ){{GM}}_{\mathrm{NS}}}},\end{eqnarray} \tag{ 14 }$$

where ${M}_{\mathrm{NS}}$ is the NS mass, G is Newton's gravitational constant, and $1-\beta$ is a factor that parameterizes the effect of radiation pressure (Section 2.6). For ${a}_{L}\sim 30\ \mathrm{au}$ and ${{\rm{\Gamma }}}_{\mathrm{flare}}\sim 1/(30\ \mathrm{yr})$, $\sim {10}^{10}$ lenses are necessary.

The flare duration ${t}_{\mathrm{flare}}$ is determined by the lens orbital speed and is ${R}_{\mathrm{em}}/{v}_{L}\approx {R}_{\mathrm{em}}\sqrt{{a}_{L}/[(1-\beta ){{GM}}_{\mathrm{NS}}]}$, or

$$\begin{eqnarray}&&{t}_{\mathrm{flare}}\approx 1.6\ {\rm{s}}\,\left(\displaystyle \frac{{R}_{\mathrm{em}}}{10\ \mathrm{km}}\right){\left(\displaystyle \frac{{a}_{L}}{30\mathrm{au}}\right)}^{1/2}\times {\left(\displaystyle \frac{(1-\beta ){M}_{\mathrm{NS}}}{1.4{M}_{\odot }}\right)}^{-1/2}.\end{eqnarray} \tag{ 15 }$$

LMXBs are binary systems, and thus the NS itself moves. In a typical LMXB system, the donor is a low-mass dwarf or possibly subgiant with mass ${M}_{\mathrm{donor}}\sim 0.5\ {M}_{\odot }$, with an orbital period ${P}_{\mathrm{em}}$ of several hours to 1 day (e.g., Pfahl et al. 2003). From most inclinations, the NS traces out an elliptical path on the sky as viewed by the observer, with a projected semimajor axis ${a}_{\mathrm{NS}}$ of $\lesssim 1\ {R}_{\odot }$. The NS motion therefore is more important than the lens's orbital motion.

In the binary case, a large number of lens orbits pass over the NS orbit from the perspective of the observer. Typically, the NS will be at the wrong orbital phase to be occulted when a lens crosses its orbit. The probability of a lens flare during any one lens passage is set by the ratio of the time window for the flare and the NS orbital period:

$$\begin{eqnarray}&&{{ \mathcal P }}_{\mathrm{flare}}\approx \displaystyle \frac{2{R}_{\mathrm{em}}}{{v}_{L}{P}_{\mathrm{bin}}}\approx \displaystyle \frac{{R}_{\mathrm{em}}}{\pi }\sqrt{\displaystyle \frac{{a}_{L}}{(1-\beta ){a}_{\mathrm{bin}}^{3}}},\end{eqnarray} \tag{ 16 }$$

which is of order ${10}^{-5}\mbox{--}{10}^{-4}$, where ${a}_{\mathrm{bin}}={a}_{\mathrm{NS}}{M}_{\mathrm{bin}}/{M}_{\mathrm{donor}}$. The probability of a shadow event is ${R}_{L}$/${R}_{\mathrm{em}}$ times bigger. On the other hand, only one lens needs to cross the entire $\sim {a}_{\mathrm{NS}}$ width of the orbit to ensure that a lens flare eventually occurs, unlike the stationary source case, where one lens per $\sim {R}_{\mathrm{em}}$ is needed to ensure a lens flare.

Many fewer lenses are needed to achieve an observed peak luminosity flare rate ${{\rm{\Gamma }}}_{\mathrm{flare}}$ when the source is in a binary:

$$\begin{eqnarray}&&{N}_{L}\approx \displaystyle \frac{\pi {a}_{L}}{{a}_{\mathrm{NS}}}\displaystyle \frac{{P}_{L}{{\rm{\Gamma }}}_{\mathrm{flare}}}{{{ \mathcal P }}_{\mathrm{flare}}}\approx \displaystyle \frac{2{\pi }^{3}{a}_{L}^{2}{{\rm{\Gamma }}}_{\mathrm{flare}}}{{a}_{\mathrm{NS}}{R}_{\mathrm{em}}}\sqrt{\displaystyle \frac{{a}_{\mathrm{bin}}^{3}}{{{GM}}_{\mathrm{bin}}}}.\end{eqnarray} \tag{ 17 }$$

For ${a}_{L}\sim 30\ \mathrm{au}$, ${a}_{\mathrm{NS}}\sim {R}_{\odot }$, and ${{\rm{\Gamma }}}_{\mathrm{flare}}\sim 1/(30\ \mathrm{yr})$, about 2 × 10⁹ lenses are needed. Flare events are shorter because the NS rapidly crosses the center of the lens in a time ${R}_{\mathrm{em}}({M}_{\mathrm{bin}}/{M}_{\mathrm{donor}})\sqrt{{a}_{\mathrm{bin}}/({{GM}}_{\mathrm{bin}})}$, or

$$\begin{eqnarray}&&{t}_{\mathrm{flare}}\approx 0.13\ {\rm{s}}\,\left(\displaystyle \frac{{R}_{\mathrm{em}}}{10\ \mathrm{km}}\right){\left(\displaystyle \frac{{a}_{\mathrm{NS}}}{{R}_{\odot }}\right)}^{1/2}\left(\displaystyle \frac{{M}_{\mathrm{bin}}}{1.9\ {M}_{\odot }}\right)\times {\left(\displaystyle \frac{{M}_{\mathrm{donor}}}{0.5{M}_{\odot }}\right)}^{-3/2}.\end{eqnarray} \tag{ 18 }$$

A further consideration is that the NS orbit requires a minimum field-of-view ${a}_{\mathrm{NS}}/{a}_{L}$, set by the distance between the NS and the barycenter, assuming that the lens has synchronous rotation. Diffraction-based lenses like zone plates have narrow fields of view, especially severe for huge zone plates in hard X-rays. Field curvature and astigmatism imply aberration-free fields of view of just $\sqrt{f\lambda }/{R}_{L}$, and coma limits the aberration-free field to $\lambda {f}^{2}/(2{R}_{L}^{3})$ (Young 1972). A zone plate must therefore have a minimum orbital distance of

$$\begin{eqnarray}&&\begin{array}{l}{a}_{L}\gt {\left(\displaystyle \frac{{a}_{\mathrm{NS}}^{2}{R}_{L}^{2}E}{{hc}}\right)}^{1/3}\\ \,\gt 105\ \mathrm{au}{\left(\displaystyle \frac{{a}_{\mathrm{NS}}}{{R}_{\odot }}\right)}^{2/3}\times {\left(\displaystyle \frac{{R}_{L}}{1000\mathrm{km}}\right)}^{2/3}{\left(\displaystyle \frac{E}{10\mathrm{keV}}\right)}^{1/3}.\end{array}\end{eqnarray} \tag{ 19 }$$

Because of the extremely low tolerance for misaim, it may be easier to replace a singular plate with many smaller elements. These would require wedge prisms to align light before entering the smaller zone plates, which themselves would introduce severe chromatic aberration.

The extreme luminosity of the LMXB necessitates that the lenses be placed very far from the binary to avoid overheating and sublimating. This is especially true for zone plates, which absorb a large fraction of the luminosity incident on them. If a lens is maintained at temperature ${T}_{L}$ around an LMXB of luminosity ${L}_{\mathrm{em}}$, and it absorbs a fraction ${\zeta }_{a}$ of incident flux, its distance from the LMXB must be ${x}_{\mathrm{em}}=\sqrt{({\zeta }_{a}{L}_{\mathrm{em}})/(4\pi {\sigma }_{\mathrm{SB}}{T}_{L}^{4})}$, or

$$\begin{eqnarray}&&{x}_{\mathrm{em}}=6.3\ \mathrm{au}{\left(\displaystyle \frac{{\zeta }_{a}{L}_{\mathrm{em}}}{{10}^{38}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{1/2}{\left(\displaystyle \frac{{T}_{L}}{2000{\rm{K}}}\right)}^{-2}.\end{eqnarray} \tag{ 20 }$$

In principle, ${\zeta }_{a}$ might be very small, as might be achieved with phase Fresnel lenses or phase zone plates (Skinner 2010).

Practical considerations may require the lenses to be placed much farther still. Because LMXB luminosities are variable on multiple timescales (e.g., van der Klis 1989), the lens will have an erratic temperature, heating and cooling as the LMXB fluctuates. The structures therefore will expand and contract. In order to reduce the enormous stresses, they will then need expansion joints or some other solution. The thermal fluctuation in the lens size might also change the focal length of the lens. Thermal stresses also pose a threat to the precise engineering needed for an optical element, causing distortions in the lens shape or misalignment of sublenses.

Radiation pressure from a bright LMXB is also a challenge. The ratio of the radiation pressure force and gravitational force on the lens is

$$\begin{eqnarray}&&\beta =\frac{{\zeta }_{a}{L}_{\mathrm{em}}}{4\pi {{GM}}_{\mathrm{bin}}{{\rm{\Sigma }}}_{L}c}=0.10\,\left(\frac{{\zeta }_{a}{L}_{\mathrm{em}}}{{10}^{38}\ \mathrm{erg}\,{{\rm{s}}}^{-1}}\right)\times {\left(\frac{{M}_{\mathrm{bin}}}{1.9{M}_{\odot }}\right)}^{-1}{\left(\frac{{{\rm{\Sigma }}}_{L}}{10{\rm{g}}{\mathrm{cm}}^{-2}}\right)}^{-1},\end{eqnarray} \tag{ 21 }$$

where ${{\rm{\Sigma }}}_{L}$ is the projected surface density of the entire lens structure. The X-ray optical depth of a material is generally set by its composition and surface density. For optically thin materials, ${\zeta }_{a}\propto {{\rm{\Sigma }}}_{L}$, and the energy-loss grammage of 10 keV photons is $\sim 3\ {\rm{g}}\,{\mathrm{cm}}^{-2}$ for lithium, $0.4\ {\rm{g}}\,{\mathrm{cm}}^{-2}$ for carbon, $0.03\ {\rm{g}}\,{\mathrm{cm}}^{-2}$ for silicon, and $0.005\mbox{--}0.01\ {\rm{g}}\,{\mathrm{cm}}^{-2}$ for iron and heavier metals (Hubbell & Seltzer 1996; Tanabashi et al. 2018). Note that a significant fraction of the LMXB luminosity is the softer emission from the accretion disk, which should couple even more efficiently to the structure. Large thin plates made of these materials would be blown away by the radiation pressure. The problem is particularly acute for ULXs.

If the X-ray source is steady, β ≤ 1 might be sufficient: the lenses would then be levitated by radiation pressure and could move very slowly in their orbits as "quasites" (Kipping 2019). The fluctuating radiation pressure from bright LMXBs would cause erratic changes to the orbits of the lenses, however. Therefore, practical lens structures around the brightest LMXBs need to be weighted down. The "ballast" could consist of long, thin pegs with minimal cross section scattered in the lens. Additional mass is likely to be contributed by supporting structures for the lens, as well as counterweights needed to ensure rotational stability (see Appendix C). The problem may also be somewhat ameliorated if thermal emission from the lens can be confined to a small cone in the direction opposite the LMXB, to push the lens back, although the practicality depends on the lens temperature.

The lenses can be thought of as thin structures, with a thickness of only a few centimeters, despite being hundreds of kilometers wide. To this is added the mass of support structures, counterweights, and ballast. The total mass of a single lens is $\pi {R}_{L}^{2}{{\rm{\Sigma }}}_{L}=\pi { \mathcal A }{R}_{\mathrm{em}}^{2}{{\rm{\Sigma }}}_{L}$, or

$$\begin{eqnarray}&&{M}_{L}\approx 3.1\times {10}^{17}\ {\rm{g}}\,{\left(\displaystyle \frac{{R}_{L}}{1000\mathrm{km}}\right)}^{2}{\left(\displaystyle \frac{{R}_{\mathrm{em}}}{10\mathrm{km}}\right)}^{2}\times \left(\displaystyle \frac{{{\rm{\Sigma }}}_{L}}{10\ {\rm{g}}\,{\mathrm{cm}}^{-2}}\right).\end{eqnarray} \tag{ 22 }$$

Individually, this is quite small compared to other proposed megastructures—hundreds of gigatons per lens, about the mass of an asteroid with a radius of 3 km. It is especially impressive when one considers that this artifact is capable of repeatedly producing a signal potentially observable at intergalactic distances.

In order for lens flares of peak brightness to be visible from any direction, millions of lenses are needed, and billions are needed to ensure that they occur frequently (Section 2.5). The total mass of all the lenses is ${M}_{\mathrm{swarm}}={N}_{L}{M}_{L}$, or

$$\begin{eqnarray}\begin{array}{rcl}{M}_{\mathrm{swarm}} & \approx & \displaystyle \frac{2{\pi }^{4}{a}_{L}^{2}{R}_{L}^{2}{{\rm{\Sigma }}}_{L}{{\rm{\Gamma }}}_{\mathrm{flare}}}{{a}_{\mathrm{NS}}{R}_{\mathrm{em}}}\sqrt{\displaystyle \frac{{a}_{\mathrm{bin}}^{3}}{{{GM}}_{\mathrm{bin}}}}\\ & \approx & \,5.1\times {10}^{26}\ {\rm{g}}{\left(\displaystyle \frac{{a}_{L}}{30\mathrm{au}}\right)}^{2}{\left(\displaystyle \frac{{R}_{L}}{1000\mathrm{km}}\right)}^{2}\\ & & \times \,\left(\displaystyle \frac{{{\rm{\Sigma }}}_{L}}{10\ {\rm{g}}\,{\mathrm{cm}}^{-2}}\right)\left(\displaystyle \frac{{{\rm{\Gamma }}}_{\mathrm{flare}}}{{\left(30\mathrm{yr}\right)}^{-1}}\right)\left(\displaystyle \frac{{a}_{\mathrm{NS}}}{{R}_{\odot }}\right){\left(\displaystyle \frac{{R}_{\mathrm{em}}}{10\mathrm{km}}\right)}^{-1}\\ & & \times \,\left(\displaystyle \frac{{M}_{\mathrm{bin}}}{1.9\ {M}_{\odot }}\right){\left(\displaystyle \frac{{M}_{\mathrm{donor}}}{0.5{M}_{\odot }}\right)}^{-3/2}.\end{array}\end{eqnarray} \tag{ 23 }$$

This is about the mass of Mars. Simple occulting systems have much less onerous requirements, since even an off-center eclipse blocks the central source effectively, reducing the necessary mass by ${R}_{\mathrm{em}}$/${R}_{\mathrm{occult}}$.

The material requirements are likely to be among the greatest challenges of building this system. Any solid materials originally present in the system face a double onslaught: first, from the supernova that accompanied the birth of the NS, and second, from the LMXB luminosity itself. Even if solid planets survived a supernova or formed from the remnant, an Eddington luminosity LMXB would sublimate all rocky material within about 10 au (Miller & Hamilton 2001). Asteroid belts without planets around pulsars, including millisecond pulsars, have been proposed to explain timing anomalies, nulling, and radio transients (as in Cordes & Shannon 2008; Campana et al. 2011; Mottez et al. 2013; Shannon et al. 2013; Brook et al. 2014; Huang & Geng 2014; Geng & Huang 2015). These could be mined if they are present during the LMXB phase.

The planetary system around the millisecond pulsar PSR B1257+12 is possible evidence that X-ray binaries can host solid materials (Wolszczan & Frail 1992), as the pulsar may have passed through an X-ray binary phase during its spin-up (Alpar et al. 1982). Many theories as to how these mysterious planets formed have been advanced (Phinney & Hansen 1993; Podsiadlowski 1993; Martin et al. 2016). While it remains possible that the planets formed out of the supernova remnant through a fallback disk (e.g., Lin et al. 1991; Currie & Hansen 2007; Hansen et al. 2009), the planets may instead have formed from the disruption of a companion star, which would only happen after (or at the end of) any LMXB phase (e.g., Rasio et al. 1992; Stevens et al. 1992; Banit et al. 1993; Martin et al. 2016; Margalit & Metzger 2017). Tavani & Brookshaw (1992) suggested that PSR B1257+12's planets formed during an LMXB phase, with the accretion disk acting as a shield against the intense X-ray emission. Miller & Hamilton (2001) instead hypothesized that PSR B1257+12 was born a millisecond pulsar, and so avoided any binary phase. If LMXB planets do exist, they might be found by observing their transits (Imara & Di Stefano 2018).

The prospective engineers would be left with few other choices. They might import material from distances $\gtrsim 10\ \mathrm{au}$—from surviving distant planets or Oort clouds, post-supernova Kuiper belts formed from fallback disks, or even from other star systems. The solar Oort Cloud's mass has been estimated at one to tens of Earth masses, for example, most within an inner cloud a few thousand astronomical units out (Weissman 1996; Boe et al. 2019). If a fallback disk with rocky material forms, they may also build the lenses after the supernova but before the LMXB achieves full luminosity, though it could be millions of years before it became useful. Planets around high-mass stars remain unconstrained, and fallback disks and pulsar planets seem to be rare (Wang et al. 2014; Kerr et al. 2015). On the other hand, transporting $\sim {10}^{27}\ {\rm{g}}$ across interstellar distances requires vast amounts of energy and/or time. A more radical option would be to mine the accretion disk or donor star itself. They might more easily mine planetesimals in a circumbinary disk during the LMXB phase, if they exist (Tavani & Brookshaw 1992).

Another possibility is to use a much less luminous X-ray-emitting system as the light source, reducing temperature and radiation pressure constraints (Section 2.6). Fainter LMXBs number in the hundreds in the Milky Way (Grimm et al. 2002). High-mass X-ray binaries with NSs in the Milky Way also have luminosities less than $3\times {10}^{37}\ \mathrm{erg}\,{{\rm{s}}}^{-1}$ (with fainter ones being more abundant), but they are rarer than LMXBs (Grimm et al. 2002). Nonaccreting NSs emit thermal X-rays, as they maintain temperatures of $\sim {10}^{6}\ {\rm{K}}$ for about 1 Myr (e.g., Pavlov et al. 2002; Yakovlev & Pethick 2004). Older NSs of ages $\gt 1\ \mathrm{Myr}$ may have rotationally powered X-rays, apparently from small hot spots, though the luminosities are low, around ${10}^{28}\mbox{--}{10}^{34}\ \mathrm{erg}\,{{\rm{s}}}^{-1}$ (as in Pavlov et al. 2009; Posselt et al. 2012). They are far less practical as beacons visible across the Galaxy or between galaxies.

Although the lenses do not require power to continue beaming the LMXB luminosity, the swarm presents additional challenges that could require maintenance.

First, it is necessary to ensure that the lenses do not crash into one another. Suppose the lenses are distributed randomly in a shell with thickness ${{\rm{\Delta }}}_{a}{a}_{L}\ll {a}_{L}$ and have random velocities ${v}_{\mathrm{rand}}$. With a collisional cross-sectional area ${\zeta }_{g}\pi {R}_{L}^{2}$, the rate of collisions is ${{\rm{\Gamma }}}_{\mathrm{col}}\approx {N}_{L}{\zeta }_{g}{R}_{L}^{2}/(4{{\rm{\Delta }}}_{a}{a}_{L}^{3})\times \sqrt{(1-\beta ){{GM}}_{\mathrm{bin}}/{a}_{L}}\times ({v}_{\mathrm{rand}}/{v}_{L})$. From Equation (20) relating lens orbital distance to temperature from Equation (17),

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{\mathrm{col}} & \approx & \displaystyle \frac{{\pi }^{3}{\zeta }_{g}{R}_{L}^{2}{{\rm{\Gamma }}}_{\mathrm{flare}}}{2{{\rm{\Delta }}}_{a}{a}_{\mathrm{NS}}{R}_{\mathrm{em}}}\sqrt{(1-\beta )\displaystyle \frac{{a}_{\mathrm{bin}}^{3}}{{a}_{L}^{3}}}\left(\displaystyle \frac{{v}_{\mathrm{rand}}}{{v}_{L}}\right)\\ & \approx & {\left(94\mathrm{kyr}\right)}^{-1}{\zeta }_{g}\sqrt{1-\beta }{\left(\displaystyle \frac{{R}_{L}}{1000\mathrm{km}}\right)}^{2}\left(\displaystyle \frac{{{\rm{\Gamma }}}_{\mathrm{flare}}}{{\left(30\mathrm{yr}\right)}^{-1}}\right)\\ & & \times {\left(\displaystyle \frac{{{\rm{\Delta }}}_{a}}{0.1}\right)}^{-1}{\left(\displaystyle \frac{{a}_{\mathrm{NS}}}{{R}_{\odot }}\right)}^{1/2}{\left(\displaystyle \frac{{R}_{\mathrm{em}}}{10\mathrm{km}}\right)}^{-1}{\left(\displaystyle \frac{{a}_{L}}{30\mathrm{au}}\right)}^{-3/2}\\ & & \times {\left(\displaystyle \frac{{M}_{\mathrm{bin}}}{1.9{M}_{\odot }}\right)}^{3/2}{\left(\displaystyle \frac{{M}_{\mathrm{donor}}}{0.5{M}_{\odot }}\right)}^{-3/2}{\left(\displaystyle \frac{{v}_{\mathrm{rand}}}{{v}_{L}}\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 24 }$$

Actually, because each collision produces many pieces of potentially hazardous debris, the timescale for a collisional cascade could be one or two orders of magnitude shorter (Kessler & Cour-Palais 1978; see Appendix B).

Destruction by a collisional cascade may be inhibited by not placing the lenses on random orbits. Lenses might be placed at different distances from the LMXB, each with an appropriate focal length, to spread out the space between them. Furthermore, subsets of lenses with different orbital planes can have highly ordered velocities, with minimal dispersions, so encounter speeds are much slower. The small dispersions reduce both the rate and severity of collisions. The true lifetime of a lens swarm may be determined by how quickly the lens orbits are perturbed, or are subject to its own gravitational instabilities. I estimate the relevant timescales due to the quadrupole field of the binary, Jeans instability, and lens–lens encounters in Appendix A, finding that they take many millennia to play out. Fluctuating radiation pressure is likely to randomize the orbits at least somewhat, although radiation pressure also blows out debris from collisions. This traffic control problem is a frequent one that arises in the consideration of ETI megastructures (Carrigan 2009; Lacki 2016; Sallmen et al. 2019). Alternatively, the collisional life span may be extended by placing lenses even farther from the NS at the cost of more lenses, or by using smaller lenses. Simple occulters also have a reduced collision problem because fewer are needed.

Guidance may also be necessary to ensure that the lenses collimate the LMXB optimally. At the very least, a counterweight in the form of a central pole or enclosing tube is needed to ensure the rotational stability of the lens (see Appendix C). If the orbit of the lens attains an eccentricity of e, then even a synchronously rotating lens will not be pointed directly at the system barycenter, but will deviate by up to 2e radians. Thus, the lenses need to either maintain very low eccentricity or have relatively wide fields of view. The limited depth of field is another reason for large, achromatic lenses to maintain circular orbits. This issue is not a problem for elements that already suffer from chromatic aberration, with different photon energies already having different focal lengths. Collimators with many small lenses have much greater tolerance for radial drift.

Finally, the fabrication of the lenses would require precise shaping on scales far beyond our capabilities. Even small deviations in the lens or zone element shape would degrade performance, or even suppress the focusing ability entirely. For example, a single zone plate with radius 1000 km and focal length 30 au at 10 keV would have about two billion zones, the outermost with width 0.03 cm. If the assembly actually consists of many smaller zone plates, the requirements are less severe: a zone plate subplate with radius 0.1 km only needs about 20 zones, the outermost with width 300 cm. However, the output beams from these subplates need to be aligned within an angle $\lesssim {\theta }_{\mathrm{em}}\approx 0.5\ \mathrm{mas}{({R}_{\mathrm{em}}/10\mathrm{km})(f/30\mathrm{au})}^{-1}$, equivalent to a deviation of less than $\sim 2\ \mu {\rm{m}}\,{\mathrm{km}}^{-1}{({R}_{\mathrm{em}}/10\mathrm{km})(f/30\mathrm{au})}^{-1}$. This may require the subelements to be shifted slightly over time using some internal system. The extreme thermal stresses likely induced by variability of bright LMXBs make this an especially difficult challenge. Of course, the structure should still act as an occulter even if the optical system fails, but the resultant signal would be harder to detect and more ambiguous.

How effective are LMXB lenses as beacons? Could our instruments detect them across the Galaxy, or even at intergalactic distances?

During a detectable lens flare, a number of photons arrive nearly simultaneously from the LMXB. The near coincidence of their arrival times would stand out compared to the relatively steady background flux. In addition, there's a much smaller photon deficit during the shadow phase of the occultation. Both the number of photons and their coincidence are diluted if the optical element has chromatic aberration. It's extremely unlikely, however, that a lens flare would occur while a narrow-field detector happened to observe a given LMXB. For this reason, I shall consider observability with several wide-field detectors that observe ≳1 sr at a time in addition to the more sensitive narrow-field instruments. These instruments, generally designed to locate gamma-ray bursts, have effective areas that can be ∼10–100 times smaller than sensitive narrow-field instruments.

The signal-to-noise ratio is calculated for a lens flare event with a source passing directly behind the lens (b = 0). I also make the simplifying assumption that the source is small compared to the lens, with all parts of the source being occluded or magnified simultaneously. For a detector with effective area ${A}_{\mathrm{eff}}(E)$, the photon excess detected during a time window ${t}_{\mathrm{window}}$ is

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}{N}_{\mathrm{BL}} & = & \displaystyle \int \left[\displaystyle \frac{{{dF}}_{\mathrm{flare}}}{{dE}}\min \left({t}_{\mathrm{window}},\displaystyle \frac{2{\theta }_{\infty }(E){x}_{\mathrm{em}}}{{v}_{\mathrm{rel}}}\right)\right.\\ & & -\,\left.\displaystyle \frac{{{dF}}_{\mathrm{shadow}}}{{dE}}\min \left({t}_{\mathrm{window}},\displaystyle \frac{2{R}_{L}}{{v}_{\mathrm{rel}}}\right)\right]\displaystyle \frac{{A}_{\mathrm{eff}}(E)}{E}{dE},\end{array}\end{eqnarray} \tag{ 25 }$$

where the excess fluence spectrum follows from the luminosity spectrum and the lens characteristics. I consider two window lengths, one for the "peak" of the flare and one for the occultation length. After taking into account the minimum time resolution ${t}_{\mathrm{res}}$ of the instrument, these are ${t}_{\mathrm{window}}^{\mathrm{flare}}=\min ({t}_{\mathrm{flare}},{t}_{\mathrm{res}})$ and ${t}_{\mathrm{window}}^{\mathrm{shadow}}=\min ({t}_{\mathrm{shadow}},{t}_{\mathrm{res}})$. There is only a miniscule difference in the case of an achromatic lens, but the peak accounts for only a fraction of the excess fluence for zone plates.

The BL of a highly accreting LMXB is Compton-opaque and has a Wien spectrum (Popham & Sunyaev 2001; Gilfanov et al. 2003). From Rybicki & Lightman (1979),

$$\begin{eqnarray}&&\displaystyle \frac{{{dL}}_{\mathrm{BL}}}{{dE}}=\displaystyle \frac{{L}_{\mathrm{BL}}{E}^{3}}{6{\left({k}_{B}{T}_{\mathrm{em}}\right)}^{4}}\exp \left(-\displaystyle \frac{E}{{k}_{B}{T}_{\mathrm{BL}}}\right).\end{eqnarray} \tag{ 26 }$$

The temperature of the BL has a common mean temperature of ${k}_{B}{T}_{\mathrm{BL}}=2.4\ \mathrm{keV}$ (Gilfanov et al. 2003; Revnivtsev et al. 2013). The unlensed flux from the accretion disk serves as an unavoidable background. I use the multicolor model to estimate its observed spectrum:

$$\begin{eqnarray}&&\displaystyle \frac{{{dL}}_{\mathrm{disk}}^{\mathrm{app}}}{{dE}}\propto {\int }_{0}^{{T}_{\mathrm{disk}}^{\mathrm{in}}}\displaystyle \frac{{E}^{3}}{\exp [E/({k}_{B}T)]-1}{\left(\displaystyle \frac{T}{{T}_{\mathrm{disk}}^{\mathrm{in}}}\right)}^{-11/3}\displaystyle \frac{{dT}}{{T}_{\mathrm{disk}}^{\mathrm{in}}},\end{eqnarray} \tag{ 27 }$$

where ${{kT}}_{\mathrm{disk}}^{\mathrm{in}}=1.4\ \mathrm{keV}$ is typical of LMXBs and the spectrum is normalized to an apparent luminosity ${L}_{\mathrm{disk}}^{\mathrm{app}}$ (Mitsuda et al. 1984).

Wide-field X-ray instruments without focusing optics have high background rates because background events from many parts of the sky are confused with those from the source. X-ray instruments with focusing optics generally have low background rates because only those falling within the point-spread function of the source contribute to the noise. The background rate $\dot{{N}_{\mathrm{back}}^{\mathrm{eff}}}$ for each considered instrument is listed in Table 1; it dominates over the normal flux from the BL by one to three orders of magnitude.

Table 1.  Signal-to-noise Ratio during Lens Flare

Facility	Instrument	${t}_{\mathrm{res}}$	$\dot{\overline{{N}_{\mathrm{BL}}}}({{\rm{s}}}^{-1})$	$\dot{{N}_{\mathrm{disk}}}({{\rm{s}}}^{-1})$	$\dot{{N}_{\mathrm{back}}^{\mathrm{eff}}}({{\rm{s}}}^{-1})$	Shadow	Achromatic	Chromatic Lens		Diffractive Element
						${t}_{\mathrm{window}}^{\mathrm{shadow}}$	${t}_{\mathrm{window}}^{\mathrm{flare}}$	${t}_{\mathrm{window}}^{\mathrm{flare}}$	${t}_{\mathrm{window}}^{\mathrm{shadow}}$	${t}_{\mathrm{window}}^{\mathrm{flare}}$	${t}_{\mathrm{window}}^{\mathrm{shadow}}$
Wide-field Instruments
HETE-2	FREGATEa	$6.4\ \mu {\rm{s}}$	66	26	2200	6.9	410	32	22	49	39
	WXMa	$256\ \mu {\rm{s}}$	50	49	530	10	350	30	26	43	46
RHESSIa		$100\ \mathrm{ms}$	11	1.8	1800	1.3	160	8.0	3.8	14	7.0
Swift	BAT	$140\ \mu {\rm{s}}$	40	0.4	40,550	1.0	300	0.05	0.2	0.2	0.7
Intermediate-field Instruments
MAXI	GSC	$100\ \mu {\rm{s}}$	5.9	6.6	120	2.6	120	9.5	6.6	14	12
Narrow-field Instruments
Athena	WFIb	$450\ \mu {\rm{s}}$	2840	36,435	0.001	72	2700	140	55	220	160
Chandra	ACIS-I	$3.2\ {\rm{s}}$	133	717	1E-6	23	580	52	16	81	47
	HRC-I	$16\ \mu {\rm{s}}$	25	375	5E-6	6.2	250	13	4.6	20.0	13
eROSITA		$50\ \mathrm{ms}$	125	2863	0.00485	11	560	16	1.1	27	9.3
NICER		$100\ \mathrm{ns}$	328	3736	0.17	26	910	45	15	69	49
NuSTAR		$2\ \mu {\rm{s}}$	444	277	0.001	83	1100	99	150	140	230
RXTE	PCAa	$1\ \mu {\rm{s}}$	3244	2401	96	210	2800	260	390	370	600
XMM-Newton	EPIC-pnCCD	$73.3\ \mathrm{ms}$	477	3170	0.0002	39	1100	87	73	130	140

Notes. Assumed parameters: ${L}_{\mathrm{BL}}={10}^{38}\ \mathrm{erg}\,{{\rm{s}}}^{-1}$, ${k}_{B}{T}_{\mathrm{BL}}=2.4\ \mathrm{keV}$, ${R}_{\mathrm{em}}=10\ \mathrm{km}$, ${L}_{\mathrm{disk}}^{\mathrm{app}}={10}^{38}\ \mathrm{erg}\,{{\rm{s}}}^{-1}$, ${k}_{B}{T}_{\mathrm{disk}}^{\mathrm{in}}=1.4\ \mathrm{keV}$, ${v}_{\mathrm{rel}}=80\ \mathrm{km}\,{{\rm{s}}}^{-1}$, $\eta =1$, ${R}_{L}\,=1000\ \mathrm{km}$, $\bar{E}=3{k}_{B}{T}_{\mathrm{BL}}$, $d=10\ \mathrm{kpc}$, b = 0. Used window listed under element type.

^aDefunct instrument. ^bPlanned instrument.

References. Athena: Barcons et al. (2015) (${A}_{\mathrm{eff}}$), Rau et al. (2013), Figure 10 (${N}_{\mathrm{back}}^{\mathrm{eff}}$, ${t}_{\mathrm{res}}$); Chandra ACIS-I: Schwartz (2014) (${A}_{\mathrm{eff}}$), Garmire et al. (2003) (${N}_{\mathrm{back}}^{\mathrm{eff}}$, ${t}_{\mathrm{res}}$); Chandra HRC-I: Schwartz (2014) (${A}_{\mathrm{eff}}$), Weisskopf et al. (2003) (${N}_{\mathrm{back}}^{\mathrm{eff}}$, ${t}_{\mathrm{res}}$); eROSITA: Merloni et al. (2012) (${N}_{\mathrm{back}}^{\mathrm{eff}}$ summed particle and photon background over square arcminute); HETE-2 FREGATE: Atteia et al. (2003), ${N}_{\mathrm{back}}^{\mathrm{eff}}$ sum of detectors A and B; HETE-2 WXM: Shirasaki et al. (2003); MAXI-GSC: Matsuoka et al. (2009) (${A}_{\mathrm{eff}}$ from detection efficiency, scaled to $10\ {\mathrm{cm}}^{2}$ from Sugizaki et al. 2011; background rate from idealized $12\times 10\ {{\rm{s}}}^{-1}$); NICER: Arzoumanian et al. (2014) (${A}_{\mathrm{eff}}$), Gendreau et al. (2012) (${N}_{\mathrm{back}}^{\mathrm{eff}}$ from $0.05\ {{\rm{s}}}^{-1}$ plus $12\ \mathrm{keV}\times 0.01\ {{\rm{s}}}^{-1}\ {\mathrm{keV}}^{-1}$ from particles, ${t}_{\mathrm{res}}$); NuSTAR: Harrison et al. (2013); RHESSI: Lin et al. (2002) (${A}_{\mathrm{eff}}$, ${t}_{\mathrm{res}}$), Smith et al. (2002), Figure 9 (${N}_{\mathrm{back}}^{\mathrm{eff}}$, integrated to 50 keV); RXTE-PCA: Jahoda et al. (1996) (${N}_{\mathrm{back}}^{\mathrm{eff}}$ summed over all bands in Table 5), Bradt et al. (1993) (${t}_{\mathrm{res}}$); Swift-BAT: Swift-BAT Team (2005) (${A}_{\mathrm{eff}}$, ${t}_{\mathrm{res}}$), Baumgartner et al. (2013) (${N}_{\mathrm{back}}^{\mathrm{eff}}$, estimated from Equation (8)); XMM-Newton: XMM-Newton SOC (2018) (${A}_{\mathrm{eff}}$), Strüder et al. (2001) (${N}_{\mathrm{back}}^{\mathrm{eff}}$, ${t}_{\mathrm{res}}$).

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I then estimate the expected signal-to-noise ratio of the event from ${\rm{\Delta }}{N}_{\mathrm{BL}}$ detected and the effective background count ${N}_{\mathrm{back}}^{\mathrm{eff}}=\dot{{N}_{\mathrm{back}}^{\mathrm{eff}}}{t}_{\mathrm{window}}$, giving

$$\begin{eqnarray}&&\left({\rm{S}}/{\rm{N}}\right)=\displaystyle \frac{| {\rm{\Delta }}{N}_{\mathrm{BL}}| }{\sqrt{\overline{{N}_{\mathrm{BL}}}+\max ({\rm{\Delta }}{N}_{\mathrm{BL}},0)+{N}_{\mathrm{disk}}+{N}_{\mathrm{back}}^{\mathrm{eff}}}},\end{eqnarray} \tag{ 28 }$$

where $\overline{{N}_{\mathrm{BL}}}$ is the expected number of photons detected from the BL when no event occurs and ${N}_{\mathrm{disk}}$ is the expected number from the accretion disk during the window.

I present calculated $({\rm{S}}/{\rm{N}})$ for several X-ray instruments in Table 1. Of course, the actual number detected depends on a variety of parameters, especially ${R}_{L}$, the distance, and the properties of the LMXB BL. I assumed that the lens system is in a bright LMXB system with BL luminosity ${L}_{\mathrm{BL}}={10}^{38}\ \mathrm{erg}\,{{\rm{s}}}^{-1}$ and an equal accretion disk apparent luminosity. The Galaxy hosts two NS LMXBs with total luminosity greater than $2\times {10}^{38}\ \mathrm{erg}\,{{\rm{s}}}^{-1}$ (Grimm et al. 2002):⁵ Cir X-1 (Linares et al. 2010) and Sco X-1 (e.g., Mata Sánchez et al. 2015).⁶ When the luminosity is this high, the BL is expected to cover much of the NS (Inogamov & Sunyaev 1999; Popham & Sunyaev 2001), so I adopt ${R}_{\mathrm{em}}=10\ \mathrm{km}$. I also use ${R}_{L}=1000\ \mathrm{km}$, which is generally large enough to ensure that a flare from a chromatic lens is detectable at the fiducial distance of $10\ \mathrm{kpc}$, while remaining significantly smaller than the planet-size structures of Arnold (2005).

Efficient (η = 1) achromatic lenses this big around a bright LMXB produce flares that are easily detectable throughout the Galaxy. After all, the apparent luminosity of the fiducial achromatic lens is comparable to a soft gamma repeater (SGR) flare, although it lasts only a fraction of a second. I find signal-to-noise ratios in excess of 120 for all considered instruments. They could also be detected by all considered instruments within the Large Magellanic Cloud, at a distance of 49.45 kpc, with (S/N) ≥ 24 (Pietrzyński et al. 2019). In addition, these flares could be detected by all considered narrow-field instruments with (S/N) ≥ 5 even in M31, 785 kpc away (McConnachie et al. 2005). ATHENA-WFI and RXTE-PCA could make an (S/N) ∼ 7 detection from M81 (3.63 Mpc; Karachentsev et al. 2002) and other galaxies like NGC 5128, NGC 253, and M82.

Of course, maybe ETIs would use smaller lenses, which use less construction material and collide less frequently. The signal-to-noise ratio's dependence on ${R}_{L}$ is shown in Figure 4 for RXTE-PCA (blue) and HETE-2 FREGATE (black/gray), when all other parameters are fixed. The minimum radius for $({\rm{S}}/{\rm{N}})\geqslant 5$ with HETE-2 FREGATE is ∼40 km. ETIs might also choose fainter LMXBs as the illumination source, because they are much more common, less mass is needed as ballast against radiation pressure, and the swarm can be more compact while avoiding overheating if collisions can be prevented. Fainter LMXBs have smaller BL emitting areas. Under the assumption that the effective temperature of the BL is constant, ${R}_{\mathrm{em}}\propto \sqrt{{L}_{\mathrm{em}}}$. Then, the peak apparent luminosity of an achromatic lens is invariant, but flares are shorter and it is more unlikely for any given lens to transit the NS on any given pass. As seen in Figure 4, $({\rm{S}}/{\rm{N}})\propto {L}_{\mathrm{em}}^{1/4}$, so achromatic lenses around faint LMXBs are still detectable throughout the Galaxy.

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Zone plates and lenses with chromatic aberration with $\bar{E}=3{k}_{B}{T}_{\mathrm{em}}$ and ${R}_{L}=1000\ \mathrm{km}$ produce far fewer excess photons and thus have much lower signal-to-noise ratio. Swift-BAT is particularly insensitive to these events, failing to detect a lens flare even from the bright, nearby Sco X-1 (2.8 kpc; Bradshaw et al. 1999), because it can only detect the high-energy tail of the LMXB spectrum. If $\bar{E}$ is significantly higher (perhaps 10 keV), its prospects should improve. RHESSI, on the other hand, would have achieved $({\rm{S}}/{\rm{N}})\gt 28$ for Sco X-1 with the same assumptions. I nonetheless find (S/N) ≥ 5 for all instruments except Swift-BAT and sometimes RHESSI, even for lenses with strong chromatic aberration. (S/N) > 10 is achieved for LMC lens flares for a few sensitive narrow-field instruments like Chandra ACIS and XMM-Newton EPIC-pnCCD; HETE-2 WXM was marginally sensitive with (S/N) ∼ 5. Smaller lenses may very well be missed, as (S/N) ≥ 5 is achieved only for HETE-2 FREGATE when ${R}_{L}\gtrsim 70\ \mathrm{km}$. In addition, they would be difficult to detect around the more numerous fainter X-ray binaries. A luminosity $\gtrsim {10}^{37}\ \mathrm{erg}\,{{\rm{s}}}^{-1}$ was necessary for HETE-2 FREGATE to achieve $({\rm{S}}/{\rm{N}})\sim 5$ for a chromatic lens flare. Although these events are much dimmer, they may be detected far more often because some photons are reemitted into a larger cone. I tested the effects of shifting b and found that HETE-2 FREGATE could detect the fiducial chromatic lens for $b\lesssim 120\ \mathrm{km}$ for the ${t}_{\mathrm{window}}^{\mathrm{shadow}}$ window, raising the frequency of detectable events by ≳10. Further gains in (S/N) could be achieved by taking advantage of spectral information, as chromatic elements produce an unusual spectral evolution.

In addition to the rare lens flares, there should be occultation events when the lens passes in front of the BL but magnifies empty space.⁷ These are much more numerous, roughly by a factor $\sim {R}_{L}/{R}_{\mathrm{em}}$, but also are much harder to detect, since the photon deficit is bounded by the luminosity of the LMXB and event duration. From Table 1, these events are easily detectable by narrow-field instruments that are observing an occulted LMXB, but the $({\rm{S}}/{\rm{N}})$ of a shadow event is low (≲10) for a wide-field instrument. Small occulters are detectable by an instrument like RXTE-PCA (long-dashed lines in Figure 4). In addition, 1000 km radius occulters were detectable around Sco X-1 with $({\rm{S}}/{\rm{N}})\sim 70$ for HETE-2, ∼20 for MAXI-GSC, ∼16 for RHESSI, and ∼13 for Swift-BAT. For the fiducial parameters of Section 2.5, the occultation rate is one every ≳10 Ms, and it is conceivable that one might be detected during a very long observation of Sco X-1 or some other nearby bright LMXB.

LMXBs are known to flare. During the common type I X-ray burst phenomenon, the X-ray luminosity can rise by a factor of 100 in a fraction of a second, with characteristic temperatures of $2\mbox{--}3\ \mathrm{keV}$. Unlike lens flares, though, type I bursts take tens of seconds to decay, and they are caused by thermonuclear burning on the NS surface (Strohmayer & Bildsten 2003; Galloway et al. 2008). Longer types of bursts, like superbursts, also exist and are due to thermonuclear burning (Strohmayer & Bildsten 2003; in't Zand et al. 2019). A lens flare would be characterized by a symmetric time profile. Curiously, though, some bursts have long (∼100 s) "eclipse"-like dips, sometimes accompanied by faster (∼1 s) variability (in't Zand et al. 2019). It is unlikely that occultations would happen only during X-ray bursts, demonstrating that searches for true lens eclipses may have to deal with false positives.

A few X-ray sources display type II X-ray bursts, classically explained as accretion instabilities. Type II bursts can have super-Eddington luminosities (${10}^{38}\mbox{--}{10}^{39}\ \mathrm{erg}\,{{\rm{s}}}^{-1}$) and can last between a few seconds and a few minutes (Lewin et al. 1993). The longer bursts display a "flat top" light curve, with a roughly constant flux followed by a steep decline and a faint tail. Although vaguely similar to lens flare light curves, long type II bursts have a faint tail not expected with lens flares, which break the symmetry of the light curve. In addition, they last much longer than predicted lens flares; short type II bursts decay over a prolonged time. Furthermore, type II bursts recur frequently, with just a few seconds to a few hours between them (Lewin et al. 1993).

SGRs, presumed to be magnetars, can emit very powerful X-ray/γ-ray flares. These start with an extremely short peak, rising within a millisecond and decaying over a fraction of a second. They lack the symmetric time curve of lens flare, are much too hot ($\gtrsim 100\ \mathrm{keV}$), and can be followed by tail emission that decays over minutes (Cline et al. 1980; Hurley et al. 2005; Palmer et al. 2005). SGRs also emit smaller bursts that do resemble lens flares more: they are cooler ($\sim 9\ \mathrm{keV}$), short ($\sim 0.1\ {\rm{s}}$), randomly recurring, and some have "flat top" light curves with an unresolved rise and fall (Norris et al. 1991; Thompson & Duncan 1995). The main arguments against these bursts being lens flares are that (1) they are somewhat hotter than expected; (2) the sources are not accreting NSs but magnetars; (3) the sources emit non-time-symmetric bursts, indicating that they naturally flare; and (4) no long shadow events are observed.

Other types of X-ray transients have been discovered, but they last too long to be lens flares. Among these are X-ray flashes, likely off-axis gamma-ray bursts, with durations in the tens of seconds (e.g., Lamb et al. 2005). Novel extragalactic X-ray transients, including XRT 000519 (Jonker et al. 2013), CDF-S XT1 (Bauer et al. 2017), and CDF-S XT2 (Xue et al. 2019), decay slower than they rise and also last for about a minute. The so-called fast X-ray transients are heterogeneous but last for hours to a day (e.g., Sguera et al. 2005). Some extragalactic ULXs flare, brightening ∼100 times, but they also last too long and have asymmetric light curves (Sivakoff et al. 2005; Irwin et al. 2016).

Although most X-ray instruments have had narrow fields of view, the RHESSI (2002–2018) could have detected flares from most of the sky (Lin et al. 2002). Thus, a Galactic lens flare from a large lens around a bright LMXB in this time frame would probably have been observed. The lack of such flares suggests that (1) isotropic X-ray lens beacons do not exist around the brighter LMXBs in the Galaxy; (2) the lenses are significantly smaller than 1000 km; (3) the number of lenses in actual swarms is very small, possibly compounded by long orbital periods; or (4) they all have severe chromatic aberration.

Further constraints on these beacons come from the lack of observed occultations by offset lenses. Some LMXBs do display partial and total eclipses; the recurrence times are hours, and they probably result from accretion disk structures (Frank et al. 1987). Sco X-1, one of the Galaxy's brightest LMXBs, has been the subject of several searches for millisecond flux drops, which might result from occultations by Kuiper Belt objects. After some initial controversy, the occultation rate is now constrained to less than one per megasecond (Chang et al. 2016). Occultations by misaligned lenses would be much more spectacular than Kuiper Belt object occultations, lasting several seconds for ${R}_{L}=1000\ \mathrm{km}$. They should be detectable in Sco X-1 even by all-sky monitors. Reported light curves from these instruments, however, generally bin fluxes into daily measurements (e.g., McNamara et al. 1998; Krimm et al. 2013). Some, like Fermi-GBM, are nonimaging and cannot distinguish individual nearly steady sources (Case et al. 2011). Although flares should be caught by event triggers, it is not clear that flux decrements from occultations are adequately constrained yet. RXTE observed Sco X-1 for $1\ \mathrm{Ms}$ (Chang et al. 2016), although the time between eclipses can easily be an order of magnitude longer. INTEGRAL's total exposure on Sco X-1 is about 4 Ms (Revnivtsev et al. 2014). Because it only measured the flux at 17 keV and above, it could constrain occulters or achromatic lenses, but probably not misaligned elements with chromatic aberration. Ariel 5 observed Sco X-1 for 5 yr, during which 10 short dips in the flux were observed, but these have been attributed to instrumental effects (Priedhorsky & Holt 1987). The Ariel 5 light curve is binned by 6000 s orbits (Priedhorsky & Holt 1987), compared to the $\sim 10\ {\rm{s}}$ duration of occultations for ${R}_{L}=1000\ \mathrm{km}$, so any actual shadow events would be diluted unless the lenses were much bigger than Earth. The Monitor of All-sky X-ray Image (MAXI) observes Sco X-1 for about a minute during each orbit, and after 10 yr of operation, it has accumulated several megaseconds of exposure in the relevant energy band. The effective area is small ($\sim 10\ {\mathrm{cm}}^{2}$ for the Gas Slit Camera; Sugizaki et al. 2011), but sufficient to detect occultations of Sco X-1, and the time resolution is high. Wide-Field MAXI, which covers 20% of the sky with a greater effective area (Kawai et al. 2014), should provide far more powerful constraints on Sco X-1 occultations.

The large variety of X-ray transients (Section 3.2), including those from LMXBs, poses a recognition problem for lens flares. This is of course a generic problem faced by most proposed technosignatures. Lens flares have a few characteristics that are not generally expected from X-ray bursts. Perhaps most importantly, the flares should be accompanied by a larger number of occultation events. For an achromatic lens, both the rate of occultation events and their duration are proportional to the square root of the peak luminosity enhancement; chromatic elements yield a predictable distribution of spectrum, duration, and peak luminosity for events. Flares also should occur in the middle of an occultation event. The shape of the light curve and possibly spectral variation are also atypical for X-ray transients, although perhaps these could be explained naturally. ETIs may engineer secondary technosignatues that are less ambiguous: perhaps instead of having a single $\bar{E}$, they use subelements with $\bar{E}$ in ratios of prime numbers, or they use square elements whose shape could be inferred from the light curve. Nonetheless, a candidate event might be explained as some new natural phenomenon, especially since a detection may be too marginal to infer detailed characteristics or to detect occultations. Follow-up work with a more powerful instrument would be necessary to gain confidence in the artificial nature of the event.

Ambiguous events still allow us to set upper limits on the rate of lens flares, just as we can set upper limits on the prevalence of other technosignatures. The lack of events with the expected characteristics already allows us to constrain these events, demonstrating that they are rare or faint in the Milky Way.

X-ray emitters, particularly NSs, are among the most compact long-lived sources of luminosity that exist. Therefore, they produce the largest flux variations when they are modulated by a structure with a fixed size. Large X-ray lenses can magnify NSs to briefly boost their apparent luminosities by orders of magnitude, a lens flare transient. In this paper, I considered LMXBs, which give the biggest bang for the buck: an ${R}_{L}=1000\ \mathrm{km}$ achromatic lens can produce flares with luminosities of a billion Suns.

As a "beacon" built by ETIs, lens flares have the advantage of being largely passive. Like simple occulting structures, they require no electronic systems or power to "broadcast." The sheer brilliance possible with a collimating lens is the greatest allowed by geometrical optics. The disadvantage of a lens beacon, however, is that the center of an achromatic lens must pass directly in front of the emitting source to produce a flare. The effective cross section of the lens for a flare is ${R}_{\mathrm{em}}$/${R}_{L}$ times the geometrical area. Thus, many lenses—about 10⁹ with fiducial parameters—are required to ensure that lens flares are frequent enough. In turn, the implied mass of the lens system is comparable to a planet; it is not at all clear whether enough solid material exists in LMXB systems. The large number of lenses may prove a guidance problem, if their velocities start to randomize and they begin to crash into one another. Although the lenses do not need to generate vast amounts of power or use complicated beamforming electronics, they could require maintenance to repair damage or deformations that degrade lens performance, particularly due to thermal stress.

In principle, a flare from an achromatic lens with ${R}_{L}=1000\ \mathrm{km}$ could be detected at intergalactic distances if our most sensitive X-ray instruments were observing the LMXB at the time. The same detectors could detect a flare from a zone plate of similar size within the Galaxy if it was looking in the right place. However, lens flares are likely to be extremely rare, occurring perhaps once per decade, and lasting only for about ∼0.1 s. Past all-sky X-ray monitors could detect lens flares within the Milky Way even for ${R}_{L}\sim 100\ \mathrm{km}$. These have included RHESSI and FREGATE on HETE-2, which have operated over most of this century so far. They would have had trouble detecting the much more common shadow events when a misaligned lens simply blocks the NS instead of magnifying it, but dedicated observations with narrow-field instruments can easily detect these. Lens flares do not match the properties of observed X-ray bursts of LMXBs.

Lens flares may be spectacular and attention grabbing, but would ETIs go to the effort to build a lens system in a hostile, distant environment even if they could overcome the many inherent technical problems? Our current detector sensitivity limits may be but a passing phase to such a long-lived interstellar society, as emphasized by Sagan (1973). The Drake equation suggests that ETIs will only be prevalent if their lifetime is very long, and thus their populations may be dominated by societies much older than our own. On the one hand, the technical capabilities of a typical millennia-old spacefaring society could mitigate the limitations of lens flares. Over thousands of years, they could accumulate decades of exposure time for all nearby galaxies with X-ray telescopes like our own, or build all-sky X-ray instruments as sensitive as RXTE, for example. Typical Galactic ETIs might find it easy to detect flares in M31 or further, and conversely, Galactic lens swarm systems may have been built primarily as intergalactic beacons.

On the other hand, the same technological advancement may eliminate the need for such brilliant events. It is probably more economical and practical on their end to use much fainter X-ray sources like young NSs, or to use occulters, small lenses, or zone plates instead of achromatic lenses (see Chennamangalam et al. 2015; Imara & Di Stefano 2018). If the typical recipient has capabilities many orders of magnitude beyond our own, LMXBs could be overpowered. For that matter, there are passive optical beacon systems around normal stars, like the transiting structures described in Arnold (2005). Although they produce only tiny fluctuations in stellar flux, with our own technology, it is relatively easy to do precision photometry for large fields of stars in the optical. If most ETIs judge it easier to build optical telescopes able to detect these variations over large distances than all-sky X-ray monitors somewhat better than ours, they could prefer stellar occulters. Vast improvements would be necessary to detect stellar occulters over intergalactic distances; we are incapable of detecting a G dwarf in M31, for example.

It depends on the trade-offs and intent of the hypothetical ETIs, of course, which we cannot really determine a priori—and likewise, their determination of their audience's capabilities. The advantage of lenses around LMXBs, however, is that it maximizes the thermal luminosity that can be modulated with kilometer to megameter size structures. This makes it easier to draw attention over intergalactic distances. Even if a typical interstellar society has X-ray telescopes a million times more sensitive than ours, for example, using LMXBs could be advantageous because they could be detected over gigaparsec distances. The use of bright but rare LMXBs could also indicate how much attention the builders want directed, like a monument or advertisement; perhaps they would be preferred locations of "galactic clubs" to indicate their reach. Lens swarms may also be placed around LMXBs for other purposes, namely, beamed power transmission, with the lenses parked in a "beacon" constellation in between uses.

Whatever the case, passive beacons like lens flares are a promising route for SETI. Although they require a large investment to build, they require less maintenance and might last for geological epochs if deformation and collisions can be either repaired or avoided. They can also be detected commensally, through the use of wide-field variability observations. The challenge on our end may simply be patience.

The gravitational field of the binary deviates from that of a single spherical object, which complicates the dynamics of the swarm. The lens orbit is not significantly affected by the time variability of the potential introduced by the binary's mutual orbit, because the binary orbital period is so much shorter than that of the lens. After averaging over the inner binary's phase for the secular evolution, the potential still has some higher-order terms deviating from a monopole. The first is a quadrupole term, with magnitude ${{GM}}_{\star }/{a}_{L}\times {({a}_{\mathrm{NS}}/{a}_{L})}^{2}$. If the lens or the binary ever develops a highly eccentric orbit, then the octupole term also becomes significant. This generally should not happen, though, since the binary is very close together and circularized by tidal forces, while the lens would be placed in a circular orbit for attitude maintenance.

A lens and the inner binary form a hierarchical triple system, with the outer component being nearly massless. The dynamics of these systems has been intensively studied, particularly the Kozai–Lidov mechanism when one member of the inner binary is much smaller than the other two. Naoz et al. (2013) derived the quadrupole- and octupole-level evolution of the orbits. In the relevant quadrupole approximation, the eccentricity e_O and inclination i_Q of the outer orbit remain constant. Instead, the orbital plane and orbital pericenter both precess. The ascending node h_O precesses at a rate

$$\begin{eqnarray}&&| \dot{{h}_{O}}| =\displaystyle \frac{-6{C}_{2}\sin (2{i}_{\mathrm{tot}})}{{L}_{2}\sqrt{1-{e}_{O}^{2}}}\lt \displaystyle \frac{12{C}_{2}}{{L}_{2}\sqrt{1-{e}_{O}^{2}}}\end{eqnarray} \tag{ A2 }$$

if the inner eccentricity is zero. Applying the definitions of C₂ and L₂ in Naoz et al. (2013) to the lens–binary triple, I find

$$\begin{eqnarray}&&| \dot{{h}_{O}}| \lt \displaystyle \frac{3}{4}\displaystyle \frac{q}{{\left(1+q\right)}^{2}}\displaystyle \frac{\sqrt{{{GM}}_{\mathrm{bin}}}{a}_{\mathrm{bin}}^{2}}{{a}_{L}^{7/2}},\end{eqnarray} \tag{ A3 }$$

where q is the mass ratio ${M}_{\mathrm{NS}}$/${M}_{\mathrm{donor}}$ of the inner binary. The drift timescale is then

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{drift}}(s) & \gtrsim & \displaystyle \frac{s}{{a}_{L}| \dot{{h}_{0}}| }\approx 60\ \mathrm{kyr}\left(\displaystyle \frac{q/{\left(1+q\right)}^{2}}{3/16}\right)\left(\displaystyle \frac{s}{{R}_{\odot }}\right){\left(\displaystyle \frac{{a}_{L}}{30\mathrm{au}}\right)}^{5/2}\\ & & \times \ {\left(\displaystyle \frac{{a}_{\mathrm{NS}}}{{R}_{\odot }}\right)}^{-2}{\left(\displaystyle \frac{{M}_{\mathrm{bin}}}{1.9{M}_{\odot }}\right)}^{-5/2}{\left(\displaystyle \frac{{M}_{\mathrm{donor}}}{0.5{M}_{\odot }}\right)}^{2}.\end{array}\end{eqnarray} \tag{ A4 }$$

If the orbit starts out eccentric, its pericenter drifts at a rate

$$\begin{eqnarray}&&| \dot{{g}_{O}}| =3{C}_{2}\left[\displaystyle \frac{4\cos {i}_{\mathrm{tot}}}{{G}_{1}}+\displaystyle \frac{10{\cos }^{2}{i}_{\mathrm{tot}}-2}{{L}_{2}\sqrt{1-{e}_{O}^{2}}}\right],\end{eqnarray} \tag{ A5 }$$

if the inner eccentricity is zero. From the definition of G₁ in Naoz et al. (2013), the 1/G₂ term is by far more important, so $| \dot{{g}_{O}}| \lt 24{C}_{2}/{G}_{2}$. The drift time is therefore

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{drift}}(s) & \gtrsim & \displaystyle \frac{s}{{a}_{L}| \dot{{g}_{0}}| }\approx 30\ \mathrm{kyr}\left(\displaystyle \frac{q/{\left(1+q\right)}^{2}}{3/16}\right)\left(\displaystyle \frac{s}{{R}_{\odot }}\right){\left(\displaystyle \frac{{a}_{L}}{30\mathrm{au}}\right)}^{5/2}\\ & & \times \ {\left(\displaystyle \frac{{a}_{\mathrm{NS}}}{{R}_{\odot }}\right)}^{-2}{\left(\displaystyle \frac{{M}_{\mathrm{bin}}}{1.9{M}_{\odot }}\right)}^{-5/2}{\left(\displaystyle \frac{{M}_{\mathrm{donor}}}{0.5{M}_{\odot }}\right)}^{2}.\end{array}\end{eqnarray} \tag{ A6 }$$

Equivalently, the estimated velocity drift is $\sim 0.1\ \mathrm{cm}\,{{\rm{s}}}^{-1}$. This is a conservative estimate, however, since the orbits are likely to be nearly circular. Furthermore, the octupole equations in Naoz et al. (2013) imply that the timescale for e_O's evolution is trillions of years.

Thus, it takes many millennia before the quadropole potential terms may start negatively impacting the lens swarm. Since all the lenses of a given ${a}_{L}$ and ${i}_{\mathrm{tot}}$ precess at the same rate, the collision hazard should not be significantly enhanced, and only the aim is affected. Tidal locking or large fields of view would prolong the useful period of the lens. Of course, an LMXB differs from the typical detached binary considered in hierarchical triple dynamics, because the system may have mass loss and the donor star is tidally distorted, but those complications should be minor and are beyond the scope of this paper.

The lens swarm might be thought of as a "gas" of lenses, at least on smaller scales entirely within the shell. The swarm could thus experience a gravitational Jeans-like instability. The timescale is $1/\sqrt{{{Gn}}_{L}{M}_{L}}$, or

$$\begin{eqnarray}\begin{array}{rcl}{t}_{J} & \approx & {\left(\displaystyle \frac{{M}_{\mathrm{bin}}}{{{Ga}}_{\mathrm{bin}}}\right)}^{1/4}{\left(\displaystyle \frac{{\pi }^{3}{R}_{L}^{2}{{\rm{\Sigma }}}_{L}{{\rm{\Gamma }}}_{\mathrm{flare}}}{2{{\rm{\Delta }}}_{a}{a}_{\mathrm{NS}}{a}_{L}{R}_{\mathrm{em}}}\right)}^{-1/2}\\ & \approx & 58\ \mathrm{kyr}{\left(\displaystyle \frac{{M}_{\mathrm{donor}}}{0.5{M}_{\odot }}\right)}^{3/4}\\ & & \times \ {\left(\displaystyle \frac{{M}_{\mathrm{bin}}}{1.9{M}_{\odot }}\right)}^{-1/2}{\left(\displaystyle \frac{{a}_{\mathrm{NS}}}{{R}_{\odot }}\right)}^{-1/4}\\ & & \times \ {\left(\displaystyle \frac{{R}_{L}}{1000\mathrm{km}}\right)}^{-1}{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{L}}{10{\rm{g}}{\mathrm{cm}}^{-2}}\right)}^{-1/2}{\left(\displaystyle \frac{{{\rm{\Gamma }}}_{\mathrm{flare}}}{{\left(30\mathrm{yr}\right)}^{-1}}\right)}^{-1/2}\\ & & \times \ {\left(\displaystyle \frac{{{\rm{\Delta }}}_{a}}{0.1}\right)}^{1/2}{\left(\displaystyle \frac{{a}_{L}}{30\mathrm{au}}\right)}^{1/2}{\left(\displaystyle \frac{{R}_{\mathrm{em}}}{10\mathrm{km}}\right)}^{1/2}.\end{array}\end{eqnarray} \tag{ A7 }$$

In addition, the instability is easily thwarted with a small internal velocity dispersion. The necessary dispersion is of order ${a}_{L}/{t}_{J}$, which is

$$\begin{eqnarray}\begin{array}{rcl}{v}_{J} & \approx & 2\ {\rm{m}}\,{{\rm{s}}}^{-1}{\left(\displaystyle \frac{{a}_{L}}{30\mathrm{au}}\right)}^{1/2}{\left(\displaystyle \frac{{M}_{\mathrm{donor}}}{0.5{M}_{\odot }}\right)}^{-3/4}{\left(\displaystyle \frac{{M}_{\mathrm{bin}}}{1.9{M}_{\odot }}\right)}^{1/2}{\left(\displaystyle \frac{{a}_{\mathrm{NS}}}{{R}_{\odot }}\right)}^{1/4}\\ & & \times \left(\displaystyle \frac{{R}_{L}}{1000\ \mathrm{km}}\right){\left(\displaystyle \frac{{{\rm{\Sigma }}}_{L}}{10{\rm{g}}{\mathrm{cm}}^{-2}}\right)}^{1/2}{\left(\displaystyle \frac{{{\rm{\Gamma }}}_{\mathrm{flare}}}{{\left(30\mathrm{yr}\right)}^{-1}}\right)}^{1/2}{\left(\displaystyle \frac{{{\rm{\Delta }}}_{a}}{0.1}\right)}^{-1/2}\\ & & \times \ {\left(\displaystyle \frac{{R}_{\mathrm{em}}}{10\mathrm{km}}\right)}^{-1/2}.\end{array}\end{eqnarray} \tag{ A8 }$$

A final effect I consider here is individual lens–lens interactions. When lenses pass each other without colliding, they still exert a gravitational force on each other. Over time, the cumulative effect of these encounters will both induce drift in the lenses and increase their velocity dispersion.

I treat the effect of a lens–lens encounter using an impulse approximation, in which the encounter's overall effect is a small impulse in velocity toward the other lens at the point of closest approach. An incoming lens projectile with velocity v and impact parameter b induces a velocity impulse of

$$\begin{eqnarray}&&{\rm{\Delta }}v\approx \displaystyle \frac{2{{GM}}_{L}}{{{bv}}_{\mathrm{rand}}}.\end{eqnarray} \tag{ A9 }$$

Over a time t, the lens experiences many such encounters at a variety of distances. Because these are random encounters, the total velocity change from these encounters is roughly a Gaussian with a variance equal to the sum of the squares of the velocity impulses:

$$\begin{eqnarray}&&{\sigma }_{v}^{2}(t)\approx {\int }_{{b}_{\min }}^{{b}_{\max }}t\displaystyle \frac{d{\rm{\Gamma }}}{{db}}{\left(\displaystyle \frac{2{{GM}}_{L}}{{{bv}}_{\mathrm{rand}}}\right)}^{2}{db},\end{eqnarray} \tag{ A10 }$$

where $d{\rm{\Gamma }}/{db}=2\pi {n}_{L}{{bv}}_{\mathrm{rand}}$ is the rate of encounters with impact parameter b. The minimum impact parameter is given by the mean closest b over an interval t and is roughly $1/\sqrt{\pi {n}_{L}{{tv}}_{\mathrm{rand}}}$. The maximum impact parameter is set by the requirement that an encounter is shorter than the mean time between encounters and is thus $1/{(4\pi {n}_{L})}^{1/3}$, of order the mean distance between lenses. The time to heat the lens population by σ_v is

$$\begin{eqnarray}\begin{array}{rcl}{t}_{v} & \approx & \displaystyle \frac{{\sigma }_{v}^{2}{v}_{\mathrm{rand}}}{8\pi {G}^{2}{n}_{L}{M}_{L}^{2}\mathrm{ln}{\rm{\Lambda }}}\approx {\sigma }_{v}^{2}\displaystyle \frac{{v}_{\mathrm{rand}}}{{v}_{L}}\sqrt{\displaystyle \frac{(1-\beta ){a}_{L}}{{a}_{\mathrm{bin}}^{3}}}\\ & & \times \,\displaystyle \frac{{{\rm{\Delta }}}_{a}{M}_{\mathrm{bin}}{a}_{\mathrm{NS}}{R}_{\mathrm{em}}}{4{\pi }^{5}G{{\rm{\Sigma }}}_{L}^{2}{R}_{L}^{4}{{\rm{\Gamma }}}_{\mathrm{flare}}\mathrm{ln}{\rm{\Lambda }}},\end{array}\end{eqnarray} \tag{ A11 }$$

with ${\rm{\Lambda }}\equiv {b}_{\max }/{b}_{\min }$. This time is nearly a quadrillion years even for ${\sigma }_{v}=1\ {\rm{m}}\,{{\rm{s}}}^{-1}$.

But even the small velocity jolts might add up over geological epochs. Each encounter produces a mean displacement of ${\rm{\Delta }}{vt}/2$ over a time t, since the typical encounter occurs halfway through the interval. The total displacement is also roughly a Gaussian with a variance:

$$\begin{eqnarray}&&{\sigma }_{s}^{2}(t)\approx {\int }_{{b}_{\min }}^{{b}_{\max }}\displaystyle \frac{{t}^{3}}{4}\displaystyle \frac{d{\rm{\Gamma }}}{{db}}{\left(\displaystyle \frac{2{{GM}}_{L}}{{bv}}\right)}^{2}{db}.\end{eqnarray} \tag{ A12 }$$

Finally, the drift time is

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{drift}} & \approx & {\left[{\sigma }_{s}^{2}\displaystyle \frac{{v}_{\mathrm{rand}}}{{v}_{L}}\sqrt{\displaystyle \frac{(1-\beta ){a}_{L}}{{a}_{\mathrm{bin}}^{3}}}\displaystyle \frac{{{\rm{\Delta }}}_{a}{M}_{\mathrm{bin}}{a}_{\mathrm{NS}}{R}_{\mathrm{em}}}{{\pi }^{5}G{{\rm{\Sigma }}}_{L}^{2}{R}_{L}^{4}{{\rm{\Gamma }}}_{\mathrm{flare}}\mathrm{ln}{\rm{\Lambda }}}\right]}^{1/3}\\ & \approx & 0.66\ \mathrm{Myr}{\left(1-\beta \right)}^{1/6}{\left(\displaystyle \frac{{\sigma }_{s}}{{R}_{\odot }}\right)}^{2/3}{\left(\displaystyle \frac{{v}_{\mathrm{rand}}}{{v}_{L}}\right)}^{1/3}{\left(\displaystyle \frac{{a}_{L}}{30\mathrm{au}}\right)}^{1/6}\\ & & \times \,{\left(\displaystyle \frac{{a}_{\mathrm{NS}}}{{R}_{\odot }}\right)}^{-1/6}{\left(\displaystyle \frac{{M}_{\mathrm{bin}}}{1.9{M}_{\odot }}\right)}^{-1/6}{\left(\displaystyle \frac{{M}_{\mathrm{donor}}}{0.5{M}_{\odot }}\right)}^{1/2}{\left(\displaystyle \frac{{{\rm{\Delta }}}_{a}}{0.1}\right)}^{1/3}\\ & & \times \ {\left(\displaystyle \frac{{R}_{\mathrm{em}}}{10\mathrm{km}}\right)}^{1/3}{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{L}}{10{\rm{g}}{\mathrm{cm}}^{-2}}\right)}^{-2/3}{\left(\displaystyle \frac{{R}_{L}}{1000\mathrm{km}}\right)}^{-4/3}\\ & & \times \,{\left(\displaystyle \frac{{{\rm{\Gamma }}}_{\mathrm{flare}}}{{\left(30\mathrm{yr}\right)}^{-1}}\right)}^{-1/3}{\left(\displaystyle \frac{\mathrm{ln}{\rm{\Lambda }}}{10}\right)}^{-1/3}.\end{array}\end{eqnarray} \tag{ A13 }$$

The lens swarm should therefore be safe from this effect for hundreds of millennia.