Mysterious Dust-emitting Object Orbiting TIC 400799224

TESS observed TIC 400799224 in Sector 10 (2019 March 26–April 22), 11 (2019 April 22–May 21), 37 (2021 April 2–28), and 38 (2021 April 28–May 26). Figure 2 shows the light curve from Sectors 10 and 11 of the TESS observation. The event of interest that drew our attention in the initial discovery of the object occurred at BJTD (BJD-2,457,000) ∼1575–1577 of Sector 10, with a close-up view of the event in the bottom panel of the same figure. The unusual shape of the dip demonstrates three distinct fadings over the course of the event, indicating a highly irregular occulting body, reaching a depth of ∼25%. We will show in Section 3.3, that the source light is composite from two stars, so the true fraction of light blocked from the host star is actually much larger.

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After learning of the periodicity of the dips (see Sections 2.2 and 2.3) we requested follow-up observations from LCO (discussed in Section 3.1) to coincide with the TESS sector 37 and 38 observations. Another dip event was detected at BJTD ∼2326 in Sector 37, which is shown with the TESS Sector 37 and 38 light curve in Figure 3.

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With the dip from Sector 10 demonstrating such a unique shape, we performed thorough pixel-level vetting of the signal and confirmed that both the Sector 10 and Sector 37 dip events are not due to TESS systematics and originated from the indicated source, so therefore must be astrophysical in nature. This vetting process is demonstrated in Figures 4 and 5. As seen from the figures, both events coincide with momentum dumps. However, the measured point-spread function (PSF) x- and y-photocenters do not show significant changes or discontinuities before, during, or after either event, indicating that their source is either the target star or the 062 separated star resolved by SOAR (see Section 3.3). The separation between the two stars is too small to resolve which of them is the source of the events based on the photocenter measurements. We note that the PSF x- and y-widths and orientation do change during the events. This is due to the redistribution of light inside the target's aperture as the brightness of the source of the dips decreases during the events.

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We also note from the TESS light curve that there is a clear modulation in the baseline flux from the system. Cleaning the light curve of the dip events, we produced the fold shown in Figure 6. This modulation is almost certainly related to the rotational period of one of the stars in the binary. A dip-like feature at BJTD ∼2335 is in phase with the modulation and we assess is most likely to be an anomalous feature on the rotation curve, though we do not rule out the possibility of this being out-of-phase collision debris (see Section 6).

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More than five years of ASAS-SN archival data (Shappee et al. 2014; Kochanek et al. 2017) are available for TIC 400799224. These data were of critical importance, as they allowed for the determination of a longer-term periodicity in the dips. The raw ASAS-SN photometric light curve is shown in the left panel of Figure 7. Numerous statistically significant dips are readily apparent (these are marked in red).

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In order to extract a possible period for these dips we utilized four different transform algorithms. These include a standard Box Least Squares ("BLS"; Kovács et al. 2002), Lomb–Scargle ("L-S"; Scargle 1982), a less common Plavchan transform (Plavchan et al. 2008), and a custom "Interval Match Transform" ("IMT"; see, e.g., Section 7 of Gary et al. 2017). The BLS transform is especially good in searching for periodic signals that involve narrow (i.e., low duty-cycle) features, such as planet transits and narrow eclipses. The L-S transform is designed to search for sinusoidal-like variations in unequally spaced data sets, and somewhat corrects for the window functions in the data set. The Plavchan periodogram is similar to a binless variation of the "phase dispersion minimization" algorithm (Stellingwerf 1978). Finally, the IMT is a brute force way of searching for common intervals in data sets where a relatively small number of "events" (e.g., dips) can be identified on an individual basis. Basically, it tests a large number of trial periods against all combinations of time differences in the event set.

The results of these four transforms applied to the ASAS-SN photometric data are shown in Figure 8. Each transform has its own type of artefacts, e.g., harmonics and subharmonics. What we see is that the most prominent peak in each transform (except for the L-S) is at a period close to 19.770 days. Nearly all of the remaining significant peaks are either harmonics or subharmonics of this period. In the case of the L-S transform, the first harmonic of the 19.77 day period is of comparable height to the base frequency.

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From these searches we assess that there is a single unique period in the ASAS-SN data of 19.77 days. In the right-hand panel of Figure 7 we show the folded ASAS-SN light curve at this period. A clear dip of up to ∼24% in flux is evident at phase 1.0 on the plot. However, several other features of this fold are evident that make it quite different from typical profiles of binary eclipses or planet transits. (i) The width of the main dip in flux is ≃ 15% in phase of the 19.77 days period. (ii) The statistics are not optimum, but the profile of the dip does not look as if there is a monotonic decrease in flux during the ingress, followed by a monotonic rise in flux during the egress. (iii) There are clearly quite a number of low flux points that are not in phase with the main dip. (iv) There are points with a seemingly unperturbed flux in the phase region where the dip occurs. Therefore, we can conclude immediately that this is not an eclipse or transit of a hard body on a simple Keplerian orbit.

Analyzing this further, we isolated the ASAS-SN fluxes within the region of each expected transit (phase 0.89–1.05, based on the right panel of Figure 7) and measured their depth. The results are shown in Figure 9 as a cumulative distribution of flux during the eclipse region. It can be seen from this distribution that, though the object is clearly periodic at 19.77 days, a measurable reduction in flux will occur in only approximately one out of every three to five transits.

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We can conclude from this that the orbiting object is either too small to cause any noticeable decrease in flux when transiting, or it is itself not transiting the star from our perspective. The former leads to the conclusion that the object is most likely an asteroid rather than a disintegrating planet, while the latter allows for a planet or other orbiting body. In any case, an occulting cloud of substantial opacity is sporadically present and amorphous, causing a varying depth, duration, and shape of the dips in the light curve.

We obtained four years of Evryscope (Law et al. 2014; Ratzloff et al. 2019) archival data for TIC 400799224, ranging from 2017 January to 2021 January. Forced aperture photometry was performed on each of 85,394 Sloan G-band images using the Evryscope Fast Transient Engine on-demand photometry pipeline as described in Corbett et al. (2020), using a 268 aperture radius. The full light curve is shown in Figure 10.

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Though these data have a substantial signal due to the lunar sidereal and synodic periods, the strongest peak in the BLS occurs at 19.757 days and confirms the period found in the ASAS-SN data. The binned and averaged fold of the Evryscope data at this period is shown in the right panel of Figure 11. The epoch time of phase zero on this plot is the same as that of the fold for the ASAS-SN data. In examination of the folds of the Evryscope and ASAS-SN data, we determined that the areas within the dips differ by approximately 1.6; this could be accounted for by the differences in filters used and seasons sampled. It is quite possible, even, that Evryscope and ASAS-SN observed an only partially overlapping set of transits. When considering the variable presence, depth, and duration of the transits, it is strong confirmation of the behavior that the periodicity was even found in both data sets.

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Following the determination of the periodicity of the dips from ASAS-SN and Evryscope data, we requested photometric follow-up from the LCO Global Telescope (LCOGT) 1 m network (Brown et al. 2013). LCOGT attempted to observe the target in Sloan $g^{\prime}$ and Pan-STARRS z-short bands every 3 ± 1 hr during three separate windows of ±1.25 days from a t₀ corresponding to our calculations of predicted events at 18:17 UT on 2021 April 22, 12:44 UT on 2021 May 12, and 07:12 UT on 2021 June 1. The 4096 × 4096 LCOGT SINISTRO cameras have an image scale of 0389 per pixel, resulting in a $26^{\prime} \times 26^{\prime}$ field of view. The images were calibrated by the standard LCOGT BANZAI pipeline (McCully et al. 2018), and photometric data were extracted with AstroImageJ (Collins et al. 2017) using apertures with radius 27.

The 2021 April 22 (BJTD ∼2326) dip event, overlapping with TESS Sector 37, is shown in Figure 3, with a close-up view in the bottom panel. While the expected dip in flux at this time is only ∼3% deep in the TESS data, the LCO photometry seems to follow the same behavior, at least to within its statistical precision. The 2021 May 12 (BJTD ∼2346) observation window coincided with the TESS Sector 38 data downlink gap, though there does not appear to have been a detectable dip during that time. The 2021 June 1 (BJTD ∼2366) observation window, which occurred after the completion of TESS Sector 38, shows a ∼4% dip, which we assess to be likely another small event, similar to the 2021 April 22 (BJTD ∼2326) dip event.

Optical spectra of TIC 400799224 were recorded using the CHIRON spectrograph at the 1.5 m telescope at Cerro Tololo Inter-American Observatory (Tokovinin et al. 2013) operated by the SMARTS consortium. The observations were made in the fiber mode (spectral resolution 25K) on 2021 February 11, 12, 19, and 22 with 15 minutes exposure time, accompanied by the ThAr comparison spectra. Moreover, three consecutive 10 minute exposures (also in the fiber mode) were taken on 2021 February 15. The 15 minute spectra had an average signal of about 600 e⁻ pixel⁻¹, or a signal to noise ratio (S/N) of ∼25. The cumulative exposure time was 1.5 hr. The spectra were reduced using the standard CHIRON pipeline. They were cross-correlated with the binary solar-spectrum mask. All cross-correlation functions (CCFs) had a single dip with a constant radial velocity (RV), having a mean value of 9.54 km s⁻¹ and an rms scatter of 0.24 km s⁻¹. The best-fit RVs for each observation are shown in Table 3. We assess the RV to be constant within error, implying that the star is not a close binary. If, as we believe, the 19.77 day period is associated with an orbiting body, its mass is too small to cause measurable reflex motion of the host star. The shape of the CCF profile also does not change, as shown in Figure 12. In Section 3.4 we extract other information from the spectra.

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TIC 400799224 was observed by the speckle camera at the 4.1 m SOAR telescope on 2021 February 26 and resolved as a 062 pair with components of comparable flux. The speckle instrument and the data processing are described in Tokovinin (2018). It was determined that the photometry and the spectra presented above refer to the sum of two stars (the CHIRON fiber diameter is 27). The resulting speckle measurements are shown in Table 4. Overall, four data cubes in the I-band (with and without binning) and two data cubes in the V-band were recorded. Figure 13 illustrates the resolution.

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The two stars resolved by speckle are most likely mutually bound because the probability of having two unrelated stars of similar brightness so close in the sky is very small. We note that a magnitude difference of ΔG ∼1 mag and angular separation of 062 are right at the edge of Gaia's contrast sensitivity (see e.g., Brandeker & Cataldi 2019). This system is not present in Gaia DR2, and its absence could be related to its double nature, although DR2 normally provides positions and photometry for such double stars. The large excess astrometric noise in Gaia EDR3, where the system is identified, is also likely caused by the superposition of two stars. To evaluate the probability that the two sources detected in SOAR data are unrelated field stars that just happen to be so close to each other on the sky by coincidence, we use the Gaia EDR3 catalog as follows.

A query of the catalog shows that there are 13 sources within ΔG ∼1 mag of TIC 400799224 inside a 15 × 15 TESS pixel array centered on the target. None of these sources is within an arcmin of the target (corresponding to projected separation of ∼28,000 au), or have comparable parallax or proper motion, which indicates that they are unlikely to be gravitationally bound to TIC 400799224. Assuming that this is the representative density for such sources in the field of view, it follows that there are 13/225 = 0.058 such sources per TESS pixel, or ∼0.00013 sources per square arcsec. Thus the probability that there is a random field star with ΔG ∼1 mag from TIC 400799224 and at an angular separation of 062 is only ∼0.0036.

The average parameters of the CCF are: amplitude a = 0.107, rms width σ = 13.85 km s⁻¹, estimated projected rotation speed $V\sin i=24.5$ km s⁻¹, equivalent width (product of a σ) equal to 1.49 km s⁻¹. According to the calibration of CHIRON CCFs on synthetic spectra, a star of solar metallicity and T ≃ 6400 K would yield such a CCF.

Despite the low S/N, the spectra show the lithium doublet at 6707.7 Å with an equivalent width of 154 ± 14 mÅ, and the dispersion is 13.8 km s⁻¹ (same as the CCF width by coincidence: the CCF is broadened by the mask, while the Li line width is due to blending of its two components). The presence of lithium will be important for the analysis in Section 4, as it leads us to strongly suspect that the system is young. No emission in the Hα line is seen.

Using known RVs and barycentric corrections, the spectra were order-merged with continuum normalization, shifted to zero velocity, and averaged with weights proportional to fluxes. The first 10 minute spectrum was discarded, as it gives an abnormal CCF, probably due to an un-removed cosmic ray spike. The average spectrum was compared to synthetic spectra with temperatures of 5500, 6000, 6500 K, solar metallicity, and gravity log g = 4.4. The synthetic spectra were rotationally broadened by $V\sin i=25$ km s⁻¹. The best match between the observed and synthetic spectra, shown in Figure 14, is found for T = 6500 K. The sodium-D doublet in TIC 400799224 is deeper than in the synthetic spectra, possibly because of additional interstellar absorption.

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The CCF of TIC 400799224 looks like a superposition of narrow and broad profiles, while the CCFs of the synthetic spectra are closer to a Gaussian. Given that the spectra actually belong to two stars, this is natural. We tentatively identify the two components seen in the spectra with the two stars resolved with SOAR. Approximating individual CCFs by the sums of two Gaussians does not provide evidence for RV variability of either the broad or narrow component, and the separation of the two heavily blended components is not reliable. Assuming that the RVs of both stars were constant during our observations, the CCF of the average spectrum was approximated by the double Gaussian. This works better than fitting the individual CCFs, as shown in Figure 15, with parameters of the individual Gaussians shown in Table 5.

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Table 5. Parameters of Two Gaussians Approximating the CCF

RV	a	σ	a σ	$V\sin i$
(km s−1)		(km s−1)	(km s−1)	(km s−1)
2.30	0.047	18.27	0.86	34.3
−0.59	0.062	9.97	0.62	16.1

Note. a—amplitude, σ—rms width, $V\sin i$—projected rotation speed estimated from σ. The areas of the Gaussians are proportional to a σ.

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There is a 3 km s⁻¹ difference between the RVs of the two components. An RV difference of a few km s⁻¹ is expected for a binary with a separation of a few hundred au, supporting the conclusion that the two stars form a bound system. The ratio of the areas of the two CCF components corresponds to Δm = 0.36 mag. Both the CCF area and the rotation of main-sequence stars depend on their spectral type and change rapidly around F5V owing to transition from radiative to convective envelopes. The brighter star in the TIC 400799224 pair is likely hotter and a faster rotator than the fainter star. Therefore, the CCF areas differ less than the actual magnitude difference measured at SOAR, ≃ 0.8 mag in V.

If we assume that the occultations in TIC 400799224 are due to dust, then we can estimate the minimum amount of dust required to block 37%–75% of the light from the host star. Consider a uniform layer of dust with area

$$\begin{eqnarray}&&{A}_{\mathrm{dust}}\simeq 4\pi {dhf}\tau \end{eqnarray} \tag{ 1 }$$

where d is the orbital radius of the dust-emitting body, h is the height of the dust perpendicular to the orbital plane, f is the fraction of the orbit over which the occultation is observed, and τ is the optical depth of the dust in the visible band. In the optically thin limit, the total cross section of all the dust particles, σ_tot, for a given total mass in dust, M_dust is

$$\begin{eqnarray}&&{\sigma }_{\mathrm{tot}}\simeq {N}_{\mathrm{grain}}{\sigma }_{\mathrm{grain}}\simeq \displaystyle \frac{3{M}_{\mathrm{dust}}{\sigma }_{\mathrm{grain}}}{4\pi {\rho }_{d}{s}^{3}}\end{eqnarray} \tag{ 2 }$$

where N_grain is the total number of dust grains, each of cross section σ_grain, blocking light from the host star, s is the mean effective size of a dust grain, and ρ_d is the mean bulk density of the dust particles. We can equate Equations (1) and (2) to find a general expression for the minimum required mass in dust:

$$\begin{eqnarray}&&{M}_{\mathrm{dust}}\gtrsim \displaystyle \frac{16\pi \,d\,s\,h\,f\,\tau \,{\rho }_{d}}{3({\sigma }_{\mathrm{grain}}/{\sigma }_{\mathrm{geom}})}\end{eqnarray} \tag{ 3 }$$

where the grain cross section is now normalized to its geometric cross section. Since the most efficient grain scattering cross sections per unit mass usually occur near fractional micron size particles, where σ_grain is still close to its geometric cross section in the visible band we find

$$\begin{eqnarray}&&\begin{array}{l}{M}_{\mathrm{dust}}\gtrsim 16\,d\,s\,h\,{\rho }_{d}\,f\tau \\ \quad \simeq 2-4\,\times \,{10}^{19}\left(\displaystyle \frac{d}{33\,{R}_{\odot }}\right)\left(\displaystyle \frac{f\tau h/{R}_{* }}{0.03}\right)\,g\end{array}\end{eqnarray} \tag{ 4 }$$

and we have taken s ≃ 0.2 μm, ρ_d ≃ 3 g cc^–1, h τ ≃ 0.30–0.65R_*, and f ≃ 0.15. This is a substantial amount of dust, e.g., equal to that of an asteroid of radius 10 km. Given that the dust activity in this source seems to change dramatically on a timescale of about 100 days, if we use that as a proxy for its replenishment lifetime, then the rate at which dust is produced must be of the order of $\dot{M}\simeq 3\times {10}^{12}$ g s⁻¹ (5 M_⊕ Gyr⁻¹). If one dismantled the asteroid Ceres (R = 500 km) at this rate it would last for ∼8000 yr.

This raises the question of how such quantities of dust can be produced in the TIC 400799224 system.

In the case of the disintegrating planets, dust production has been ascribed to a thermal (Parker) wind generated by surface irradiation and evaporation (Parker 1960; Rappaport et al. 2012; Perez-Becker & Chiang 2013; van Lieshout & Rappaport 2018). This interpretation is not viable for TIC 400799224. At a temperature T = 1525 K (Table 1), the vapor pressure of silicates is too low for any size object to yield the desired mass loss rate of ∼10¹² g s⁻¹. At fixed temperature, the highest rates of mass loss are obtained in the free-streaming limit, when the thermal speed of molecules v_th exceeds the surface escape velocity v_esc from the planet and gravity can be ignored. In this limit $\dot{M}\sim {P}_{\mathrm{vap}}{R}^{2}/{v}_{\mathrm{th}}$ where P_vap(T) is the equilibrium surface vapor pressure (which is exponentially sensitive to T via the Clausius–Clapeyron relation) and R is the body radius. For T = 1525 K, we have v_th ∼ 1 km s⁻¹ (assuming a mean molecular weight of μ = 30 appropriate for a silicate gas), P_vap ∼ 10⁻⁴ dynes cm⁻² (Perez-Becker & Chiang 2013, their Figure 1 for olivine), and R ∼ 600 km as given by the condition v_esc ∼ v_th. Then $\dot{M}\sim 4\times {10}^{6}$ g s⁻¹. Considering larger R only strengthens surface gravity and decreases $\dot{M}$ (see, e.g., Figure 2 of Perez-Becker & Chiang 2013). Thus thermal mass loss from a single planet fails to explain the inferred $\dot{M}$ by 5–6 orders of magnitude.

Regarding the detectability of the occulting dust cloud in the NIR, we first note that the expected equilibrium temperature of dust in a 19.77 day orbit of ≃ 1525 K does not match the bump seen in the SED (Figure 16) near 10 μm. In Figure 21 we show schematically a comparison among the λ F(λ) curves for blackbody spectra from (i) a host star of T_eff = 6000 K, (ii) a cool dust component with T_eq = 450 K at 2 au, (iii) an occulting dust cloud with T_eq = 1500 K and the size required to block 37% of the light from the host star, and (iv) a debris disk that extends from 35 R_⊙ to 500 R_⊙ with a 5° tilt. We conclude from this that such an occulting dust cloud at 1500 K would not be readily visible in the SED, especially if it is only present about one quarter of the time and is thereby diluted in the average of the WISE measurements. On the other hand, there is a significant excess in the SED that corresponds to ∼450 K dust, which likely would be located at ∼2 au from the host star. The debris disk (blue curve) in Figure 21 will be important in Section 6.5.

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Once dust has been released, regardless of the mechanism, there is the question of what radiation pressure subsequently does to it. Each dust grain that is exposed to the radiation flux of the host star is subjected to a radiation pressure force, F_rad equal to L σ_grain/(4π cd²). The ratio of F_rad/F_grav is defined as β which is independent of d. σ_grain in this case is the effective radiation pressure cross section for a photon to impart outward momentum to the dust grain. Formally, the full expression for F_rad is written as an integral over wavelength-dependent Mie scattering and absorption cross sections and the spectral luminosity of the host star (see Kimura et al. 2002 for the details). For the other six systems discussed and listed in Table 1, the luminosity is lower than for the present star with L ≃ 6L_⊙. In particular, our estimates for the luminosities of the two stars in the TIC 400799224 system, from Table 6 are 6.32 and 3.25 L_⊙ (pre-MS solution) and 5.35 and 2.64 L_⊙ (post-MS solution). Thus, compared to the other systems in Table 1, the highest luminosity of which is 0.17 L_⊙, we expect substantial values of β for the dust grains in TIC 400799224.

In Figure 22 we show how β in this system might vary for a generic dust grain as a function of its size. The reasons for the uncertainty are that we do not know (i) which of the two stars in TIC 400799224 hosts the dips and (ii) the chemical composition of the dust (in particular the indices of refraction of the dust grains are unknown). We also show in the right-hand panel of Figure 22 the cumulative distribution of values of β for an assumed particle size distribution of dp/ds ∝ s⁻³. For values of β > 1/2 the dust particles will be unbound from the system. For lower values of β the dust particles will go into somewhat eccentric orbits with δ P_d /P_plan ≃ 2β where δ P_d is the difference between the period of the dust orbit and the emitting planetesimal orbit (for β ≪ 1). In the rest frame of the dust-emitting planetesimal, the dust orbits form rosette patterns with petals marking each time they go all the way around to the azimuth from which they were initially launched (see, e.g., Figure 7 of Rappaport et al. 2014). The number of petals in the rosette will be roughly equal to 1/2β.

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In Figure 23 we show two illustrative dust-tail images that might exist in the TIC 400799224. In these simulations, we assume that all the particles are ejected from the vicinity of a lunar-size planetesimal at the escape speed and directed into a 30° cone centered on the direction of the host star. The size of each particle is chosen at random from a particle size distribution proportional to s⁻³. The particles are assumed, quite arbitrarily, to sublimate on the timescale of the 20 day orbit. The dust orbits are shown in the reference frame of the orbiting planetesimal. The left panel shows the paths of 5000 dust particles for the case where the particle sizes are in the range of 10 μm > s > 0.01 μm. Due to the steep power-law size dependence most of the particles are small and β is near a constant value of ∼0.15. One can see the formation of something like a rosette pattern, but one where the particles sublimate as time goes on. In the right panel we truncate the particles below a size of 1 μm. Because there are now few small particles there is actually a larger range of values for β. Each different value of β < 1/2 leads to a different orbit, while grains with β > 1/2 leave the system. In spite of the complex nature of the pattern, there is still a clear concentration of dust over a small fraction of the orbit near the planetesimal which could cause dips in the stellar flux if the dust is actively being produced.

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Another possibility for stochastically producing dust in copious quantities is developed by Jackson et al. (2014) in the context of giant impacts in optically thin, collisional debris disks orbiting young stars. The scenario, which is mentioned by Vanderbosch et al. (2021) as being possibly relevant for dust production in ZTF J0328-1219, involves catastrophic collisions among large bodies in a debris disk. A long-term (at least years) phase coherence in the dips requires a principal body that is undergoing collisions with minor bodies, i.e., ones that (i) do not destroy it, and (ii) do not even change its basic orbital period. The collisions must be fairly regular (at least 20–30 over the last 6 years) and occur at the same orbital phase of the principal body. Consider, for example, that there is a 100 km asteroid in a 20 day orbit around TIC 400799224. Further suppose there are numerous other substantial, but smaller (e.g., ≲1/10th the radius), asteroids in near and crossing orbits. Perhaps this condition was set up in the first place by a massive collision between two larger bodies. Once there has been such a collision, all the debris returns on the next orbit to nearly the same region in space. This high concentration of bodies naturally leads to subsequent collisions at the same orbital phase. Each subsequent collision produces a debris cloud, presumably containing considerable dust and small particles, which expands and contracts vertically, while spreading azimuthally, as time goes on. This may be sufficient to make one or two dusty transits before the cloud spreads and dissipates. A new collision is then required to make a new dusty transit.

We illustrate how this scenario might work in Figure 24. In this example there is an asteroid orbiting the host star every 19.77 days when it undergoes a collision that launches 10,000 particles in random directions with a thermal distribution of speeds. The velocity dispersion, σ, in units of the orbital speed, is 0.03. In Figure 24 we show eight snapshots of the full simulation in increments of orbital phase equal to 90°. At each phase we show three panels which represent the views of observers situated at large distances along the −y, +z, and +x axes. They show how the debris cloud expands both in the azimuthal and the vertical direction (i.e., perpendicular to the orbital plane). What is significant is that all the particles pass periodically (but not simultaneously), through nearly the same {x, y, z} point from which they were launched, and therefore the motion in the Z direction is periodic. The high concentration of debris particles within this "funnel" region could facilitate further disruptive collisions. In that case, all further debris clouds would be initiated at the same orbit phase as the original "master" asteroid.

A variant of the debris disk scenario involves shepherding of debris by the gravitational force exerted by a master asteroid or planet. We consider this less likely than the above stochastic collision scenario. Consider for example how Neptune gravitationally sculpts the dusty debris in the Edgeworth–Kuiper Belt (Liou & Zook 1999), or how the Earth traps particles into orbital co-rotation (Dermott et al. 1994). A planet can corral solid particles into its mean-motion resonances (e.g., 1:1, 6:7, 5:6, 4:5, 3:4, 2:3, 1:2, with the orbit of the perturbing planet either inside or outside the orbits of the particles), creating azimuthally localized dust concentrations that occult the Sun when viewed edge-on from without, with wave pattern speeds equal to the planet's orbital frequency. While the occultations would occur with frequencies equal to integer multiples of the planet's orbital frequency, they would also be highly extended in orbital phase (see e.g., Figures 4 and 5 of Liou & Zook 1999) and produce only gradual, sub-percent modulations in extinction (e.g., Figure 5 of Dermott et al. 1994, and note that the maximum density of the dust clump trailing the Earth is only ∼10% greater than the background density). A fine-tuned viewing geometry could reduce the duration of the transit to better match the event durations observed for TIC 400799224, but at the expense of further muting the transit amplitude.

Another way to produce shorter-duration events is to situate the planet in an optically thick disk of particles whose viscosity damps away gravitationally driven perturbations before they travel too far afield. This picture applies to "propeller" moonlets in Saturn's rings where particle density disturbances are localized to the vicinity of the satellite's Hill sphere (e.g., Tiscareno et al. 2006). However, an optically thick particle disk is disallowed for TIC 400799224—see Figure 21 (blue curve) which shows that such a disk would produce excess emission above 5 μm wavelengths, which is not observed. A final objection to all of the gravitational shepherding scenarios considered above is that they would predict light curves that would be consistent from transit to transit.

When launched with a speed v with respect to an orbiting body of orbital speed V, the tilt of the new orbit of the small particle with respect to the original orbital plane is

$$\begin{eqnarray*}&&\tan \mu =\displaystyle \frac{(v/V)\cos \theta }{1+(v/V)\sin \theta \cos \phi }\approx (v/V)\cos \theta \end{eqnarray*}$$

where θ is the launch angle from the normal to the orbit and ϕ is the azimuthal launch angle (where ϕ = 0 corresponds to the orbital direction). Therefore, the maximum vertical height attained by the ejected particle is $h\simeq d\,v/V\cos \theta$ one quarter of an orbit after its release. Expressing h in units of the host-star radius, we have: $h/{R}_{* }\simeq (d/{R}_{* })(v/V)\cos \theta$. ²⁸ But, from Table 1 we see that d/R_* for TIC 400799224 is ≃15. This implies that in order for h/R_* to be as large as 20% (for dips that deep), dust or particle ejection speeds of $v/V\cos \theta \sim 0.013$ are required. Finally, for orbital speeds of V ∼ 85 km s⁻¹ corresponding a 20 day orbit about TIC 400799224, launch speeds of ∼1 km s⁻¹ are implied. Collision ejecta speeds could be of this order.