On the Sensitivity of Apophis's 2029 Earth Approach to Small Asteroid Impacts

We begin our study at the earliest time at which Apophis could have had an undetected impact, set by the last observations of Apophis recorded before it moved too close to the Sun on the sky. These last observations represent the final times when the presence of dust from a hypervelocity impact onto Apophis could have been detected telescopically. At this writing, the Minor Planet Center shows that the last few observations of Apophis occurred from 2021 May through 2022 April. ⁵ The last observation from the major NEA surveys was 2021 May 12 for both PanSTARRS 1 Haleakala and Steward Observatory, Kitt Peak–Spacewatch, observations taken at a solar elongation angle of 63° (Minor Planet Center 2021a). Observatory B18–Terskol in Ukraine does report a additional position on 2021 May 20 at an elongation of 53° (Minor Planet Center 2021b), and there is an occultation observation listed in 2022 April (Minor Planet Center 2024).

The major NEA surveys consistently provide the most reliable observations of near-Earth objects. And given that the DART impact into Dimorphos caused a noticeable dust cloud and brightening within minutes to hours after the event (Kareta et al. 2023), we can expect that a sizable impact on Apophis would be almost immediately detected by the major surveys if observing conditions were good.

The observations taken of Apophis after those of the major surveys represent important contributions to the monitoring of Apophis, but it is unclear whether they would have detected dust near Apophis under difficult near-Sun viewing. Occultation observations, often collected by amateur astronomers under challenging conditions, represent valuable high-quality measurements of Apophis's on-sky position but are not optimized for dust detection. As a result, we will adopt the day after the final observation by the major NEA surveys, 2021 May 13, as the first date on which an unseen impact might have occurred. We will see that earlier impulses do as a rule produce larger deviations from Apophis's expected trajectory but that our results are not very sensitive to the precise timing of the impulse.

Based on the final recorded observations of Apophis, we will also adopt a solar elongation of 60° as the practical, though not absolute, limit at which Apophis observations can be made. We will use this value to determine when Apophis can be reobserved again when it reemerges from the daytime sky. We note that Apophis returned to solar elongations greater than 60° during 2021 October–2022 April at m_V ≈ 21, but its solar elongation did not exceed 80°. During this time, only the occultation observation mentioned earlier is recorded by the Minor Planet Center (see footnote 5).

We compute the path of Apophis within a model solar system, described below. To examine the effects of an asteroid impact on Apophis, we create a number of hypothetical versions of Apophis ("clones"). Each clone has the same initial conditions as Apophis at the start of the simulation, but each clone has a single velocity impulse applied to it. The impulse has a fixed magnitude Δv, which we assume is uniformly distributed on the sphere, though in reality, some directional asymmetry is expected (Le Feuvre & Wieczorek 2008; Robertson et al. 2021). In practice, the impulse direction is chosen by selecting three random numbers (R_x , R_y , R_z ) uniformly within the interval (−1, +1). If ${R}_{x}^{2}+{R}_{y}^{2}\,+{R}_{z}^{2}\gt 1$, the numbers are discarded and new ones drawn. If ${R}_{x}^{2}+{R}_{y}^{2}+{R}_{z}^{2}\lt =1$, then (R_x , R_y , R_z ) is normalized and defines the impulse direction of Δv in heliocentric Cartesian coordinates. This process ensures that the impulses are indeed uniformly distributed on the sphere. The impulse occurs at a time chosen at a uniformly random point within the time frame examined. This time frame extends from 2021 May 13 (JD 2459347.5) through to Apophis's next close approach on 2029 April 13 (JD 2462239.5).

Apophis and its clones are modeled within a solar system that includes the eight planets and the Moon. Planetary initial conditions are from the JPL DE405 ephemeris (Standish 1998). All particles are integrated with the RADAU (Everhart 1979) algorithm with a tolerance of 10⁻¹². Post-Newtonian effects are included, as is the Yarkovsky effect with parameters as determined by JPL and that are assumed to be unchanged by the small asteroid impact. Apophis and its clones are treated as massless test particles.

For testing purposes, we propagated our model forward 8 yr from our initial conditions to the close approach to Earth on 2029 April 13. Our model reports a close approach distance of 38,105 km, 93 km (<10⁻⁸ au) from JPL CNEOS's current value of 38,012 ± 4 km (see footnote 4), and occurring within 10 s of the time reported by JPL. Our model therefore matches the JPL values closely. JPL does not report the target b-plane position associated with their current orbital solution for Apophis, but our b-plane error is certainly similar to that in the close approach distance, about 100 km. We will see that this is sufficient to resolve changes of interest (which will be at their smallest about 200 km on the b-plane; see Section 3.1). Our uncertainties are somewhat larger than JPL's but sufficiently accurate for the sensitivity study presented in this paper.

Could the impulsive change to Apophis's velocity caused by a small asteroid impact push Apophis into a keyhole? The keyhole analysis of Farnocchia et al. (2013) reveals that the pattern of keyholes on the target plane is very complex. Our goal here is not to determine the circumstances that would put Apophis into a specific keyhole. Rather, we simply ask what size impulse would be needed to move Apophis substantially with respect to the most important keyholes and thus require a possible reevaluation of its future impact probability.

Farnocchia et al. (2013) reveal that Apophis would have to move approximately 200 km vertically on the target plane to reach the nearest important keyholes and about 1500 km vertically on the target plane to reach the complex of secondary keyholes surrounding the 2036 primary resonant return. We will adopt these numbers as representative of the target plane displacements of interest.

From our simulations, we find that a Δv of 3 × 10⁻⁴ m s⁻¹ is sufficient to move Apophis 200 km on the b-plane to the nearest keyholes, and 3 × 10⁻³ m s⁻¹ is sufficient to move it 1500 km on the b-plane to the more distant 2036 keyhole complex (see Figure 1). Asteroid impacts are found to efficiently move Apophis vertically on the target plane; however, any particular impulse is as likely to move Apophis away from a keyhole as toward it, depending on the direction in which it was applied. A plot of the median Δb/Δv over time in our simulations is presented in Figure 2. We conclude that a Δv ∼ 3 × 10⁻⁴ m s⁻¹ is the minimum impulse of concern; that is, it is the minimum necessary to deflect Apophis to one of the nearby keyholes. Impacts occurring later in time are less effective in moving Apophis on the target plane. Even if such an impulse should occur, the results would most likely be harmless. Nonetheless, monitoring Apophis telescopically for signs of such an event at the earliest possible date is recommended (see Section 3.3).

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Our simulations show that small asteroid impacts move Apophis efficiently in the vertical (ζ) direction on the target plane. Since the encounter geometry of Apophis with Earth is such that a sufficient vertical displacement on the target plane would bring about an impact with our planet, we examine this possibility here as well. An impulse Δv of 5 × 10⁻² m s⁻¹ is found to be the minimum value that could result in impact with Earth in 2029, though the impact would have to be applied at a very specific time and direction for this to occur (see Figure 3). At Δv ≳ 10⁻¹ m s⁻¹, a wider range of impulse directions and timings can produce an Earth impact. An animation of the scenario with Δv = 10 cm s⁻¹ is shown in Figure 4.

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For impulses in the 0.1–1 m s⁻¹ range (the largest impulses considered here), the resulting probability of Earth impact hovers near 5%, as only certain impulse directions can direct Apophis to our planet, and larger impulses eventually start to miss Earth by passing on the opposite side (see Figure 3).

Small asteroid impacts cannot easily push Apophis into a collision with the Moon in 2029. The nominal close approach distance of Apophis to the Moon is about 96,000 km (55 R_Moon). None of the cases examined here approach the Moon more closely than 40 R_Moon, and all remain outside the Moon's sphere of influence (37 R_Moon).

Having determined the magnitudes Δv of the impulses of interest, we now turn to an analysis of the probability of an asteroid of the necessary size striking Apophis.

What size asteroid would need to hit Apophis in order to generate the impulses described earlier? To an order of magnitude, the change in velocity Δv in Apophis due to an asteroid impact scales with the impactor's relative momentum,

$$\begin{eqnarray}\begin{array}{rcl}\beta {mv} & \sim & M{\rm{\Delta }}v,\\ {\rm{\Delta }}v & \sim & \displaystyle \frac{\beta {mv}}{M},\end{array}\end{eqnarray} \tag{ 1 }$$

where M is Apophis's mass, m ≪ M is the impactor mass, v is the speed of the impactor relative to the asteroid, and β is an enhancement factor resulting from momentum carried away by ejecta. For context, we consider the measured Δv of the DART impact into Dimorphos. Dimorphos has a 151 m diameter and a mass M = 4.3 × 10⁹ kg assuming a similar density to Didymos, while the DART impactor had a mass of 580 kg, hit at a relative speed of 6.1 km s⁻¹, and had a measured β between 2 and 5 (Daly et al. 2023). Adopting β = 3, Equation (1) yields Δv = 2.4 × 10⁻³ m s⁻¹, while the measured velocity change of the DART impact was 2.70 ± 0.10 × 10⁻³ m s⁻¹ (Cheng et al. 2023). Cheng et al. (2023) find a mean value of β = 3.6, which yields Δv = 2.88 × 10⁻³ m s⁻¹. These close matches reveal that Equation (1) is sufficient for our purposes.

The diameter d needed for an impacting asteroid to create a particular Δv in Apophis can be found from Equation (1), assuming a roughly spherical shape and density ρ, as

$$\begin{eqnarray}&&\begin{array}{l}d\sim 9.2{\left(\tfrac{{\rm{\Delta }}v}{1\,{\rm{m}}\,{{\rm{s}}}^{-1}}\right)}^{\tfrac{1}{3}}{\left(\tfrac{\beta }{3}\right)}^{-\tfrac{1}{3}}{\left(\displaystyle \frac{\rho }{3500\,\mathrm{kg}\,{{\rm{m}}}^{3}}\right)}^{-1/3}\\ \,\times \,{\left(\tfrac{v}{17\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right)}^{-\tfrac{1}{3}}\left(\displaystyle \frac{D}{340\,{\rm{m}}}\right){\rm{m}},\end{array}\end{eqnarray} \tag{ 2 }$$

where D = 340 m is Apophis's diameter (Brozović et al. 2018), and its density is taken to be 3500 kg m³ (Binzel et al. 2009). The average impact velocity of bolides arriving at Earth is v ≈ 20 km s⁻¹ (Grün et al. 1985; Brown et al. 2002; Greenstreet et al. 2012; Drolshagen et al. 2020), which includes a 11 km s⁻¹ component due to Earth's gravitational attraction that will not apply at Apophis, so we adopt $v\approx \sqrt{{20}^{2}-{11}^{2}}\approx 17$ km s⁻¹ for the typical relative velocity.

Equation (2) reveals that an impact able to move Apophis significantly relative to the 2029 keyholes (Δv ≳ 3 ×10⁻⁴ m s⁻¹) would necessitate an impactor with d ≳ 0.6 m. In order to create the possibility of an impact with Earth itself in 2029, a Δv ≳ 5 × 10⁻² m s⁻¹ is needed, translating to d ≳ 3.4 m.

What are the probabilities of asteroids of sufficient size striking Apophis? Adopting the measured flux of meter-sized objects at Earth from Brown et al. (2002), we see that approximately 140 0.6 m diameter or larger objects and only a single 3.4 m or larger object strike the Earth each year, on average. However, Apophis's physical cross section is only (0.17 km/6378 km)² ≈ 7 × 10⁻¹⁰ that of Earth, so the rate of impacts is reduced by this factor. This puts the odds of Apophis being struck by an asteroid large enough to deflect it significantly relative to the 2029 keyholes at less than one in 1 million. Given that not all impulses of a given size move Apophis toward a keyhole, the odds of a small asteroid impact deflecting Apophis to a dangerous post-2029 encounter over the 8 yr time frame examined here are minuscule. The odds of an unseen small asteroid deflecting Apophis enough to direct it into a collision with Earth in 2029 (d ≳ 3.4 m, Δv > 5 × 10⁻² m s⁻¹) are approximately 10⁻⁸. Given that only 5% of such impulses are in the correct direction to generate an Earth impact (see Section 3.1), the overall probability of a small impact directing Apophis into a collision with the Earth is less than one in 2 billion.

Our results depend on the densities ρ and velocities v of the small impactors at a given size. If either of these is higher than we assumed here, then the resulting Δv at a given diameter is also increased, and smaller impactors could create the necessary impulses. This could increase the probability of the deflections considered here occurring, since smaller impactors are more abundant. However, Equation (2) depends on v and ρ only to the negative one-third power, and so the probabilities discussed here are not very sensitive to the details of the impactor population.

We showed that the probability of a small asteroid deflecting Apophis substantially is exceedingly small. For any other asteroid, the eventuality might be dismissed as negligible. However, given the disproportionate risks associated with Apophis, we turn to an examination of how and when we could determine whether such a deflection has occurred.

Apophis is as of this writing in the daytime sky and cannot be monitored telescopically. If Apophis suffers an impact before it returns to visibility in 2027, the most easily detectable aftereffects (i.e., optical brightening due the impact-produced dust cloud) could have dissipated. If that is the case, the effect of a small asteroid impact would perhaps only be revealed to telescopes on Earth by a deviation of Apophis from its expected position in the sky. Changes in Apophis's expected magnitude or even its distance could also be useful; we will discuss those later in Section 3.3.3.

The detailed determination of an asteroid's trajectory requires large numbers of careful telescopic observations taken over a period of weeks, months, or even years. Though this process will certainly be initiated by astronomers around the globe as soon as Apophis reappears, it is very hard to predict how long a full computation of Apophis's path based solely on post-2027 observations will take. Instead, we ask a simpler question here: if Apophis reemerges into the nighttime sky on a trajectory perturbed enough to move it into an important keyhole or even to a 2029 Earth impact, how would its on-sky position differ from its current nominal ephemeris? What observations could quickly reveal whether it has been significantly perturbed?

For Earth-based observers, Apophis will emerge—barely—from the solar glare in late 2027 February, after having been largely unobservable for over 6 yr (Figure 5). The elongation of Apophis from the Sun reaches 60° as seen from Earth's geocenter on 2027 February 22 (JD 2461458.5). ⁶ At this point, the asteroid has an expected apparent magnitude of 21.0. But Apophis does not reach elongations above 62°, hovering near 60° until 2027 April before moving further into the daytime sky again. Apophis finally reemerges to elongations greater than 60° on 2027 December 6 (JD 2461745.5) at magnitude 19.7, remaining visible until 2028 June 30 (2461952.5). These time ranges will be used here to define "early" and "late" observing windows for Apophis.

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What deviation from the nominal ephemeris would be detectable? For a single night's observation from a single station, we will consider that a displacement of more than 1'' on the sky from the nominal R.A. or decl. will be detectable by ground-based telescopes, as this level is routinely achieved at professional observatories. On-sky positions can be measured to better than 1'' under good conditions, but given that we are considering observations of Apophis taken near the Sun, with higher-than-usual scattered light and potentially fewer visible stars for astrometric solutions, it is a reasonable if conservative assumption.

In Section 3.1, we saw that Farnocchia et al. (2013) determined that Apophis will intersect the target b-plane 200–1500 km from the most important keyholes. As a result, we will take those particles found to move on the b-plane by Δb > 200 km to represent those that move appreciably with respect to the keyholes. These cases do not necessarily represent particles that have been perturbed into a dangerous keyhole. In fact, the odds are very low that keyholes will be entered even under a large perturbation, but such a determination will require a full orbital analysis after Apophis reappears in 2027. Our goal here is simply to provide a framework in which cases of concern can be quickly identified for further study.

3.3.1. Keyholes

In Section 3.1, we saw that a Δv between 3 × 10⁻⁴ and 3 × 10⁻³ m s⁻¹ could move Apophis appreciably relative to keyholes. The effect of the larger of these Δv on the on-sky position of Apophis is shown in Figure 6.

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Figure 6 illustrates that, if a 1'' displacement of Apophis is the observational limit, then many scenarios involving an impulse large enough to move Apophis appreciably on the target plane might not be detectable from a single-night observation during either the early or late observing windows. But displacement Δθ in the on-sky position is not an unambiguous measure of Δb. To quantify this more carefully, we ask: if Apophis is observed to have some displacement Δθ from its nominal position, what is the probability that this case is associated with substantial movement on the target plane?

Figure 7 shows the probability of an observed on-sky offset Δθ being associated with Δb < 200 km, based on all the clones simulated in this work. Each clone is weighted by the probability of a small asteroid impact creating its associated Δv, using Equation (1) to determine the asteroid size d needed, then assuming a small asteroid flux onto Apophis proportional to that given by Brown et al. (2002) for Earth (Section 3.2). Though this procedure is only approximate, it provides a useful probability-weighted outlook at possible outcomes. For example, if a deviation of Apophis from its nominal ephemeris position of Δθ ≈ 01 was measured on 2027 April 12, Figure 7 reveals that the odds are about 50:50 that the case involves substantial motion of Apophis relative to the keyholes.

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From Figure 7, we conclude that single-night observations good to 1'' will be insufficient to determine if Apophis has moved relative to the important keyholes on the target plane. However, if Apophis is seen to be displaced by more than 02 from its ephemeris position at this time, then a substantial displacement of its intersection with the target plane has possibly occurred. This would not, however, mean that Apophis is on track to enter a dangerous keyhole; the odds are still in favor of a continued safe trajectory. But a displacement of Apophis of more than a few tenths of an arcsecond from nominal in 2027 would be cause for additional follow-up observations and analysis.

3.3.2. Earth Impact

The probability of a small asteroid deflecting Apophis onto an Earth-intersecting orbit in 2029 is exceptionally low, less than one in 1 billion by the analysis of Section 3.2. But if this unlikely event occurs, how will it manifest itself observationally when Apophis first returns to visibility in 2027?

Most cases leading to an Earth impact show easily measurable on-sky position differences from the nominal ephemeris during the early observing window; these are shown in Figure 8. Earth impactors form a distinct subset; however, positions within the "red zone" of Figure 8 are not sufficient to identify Earth impactors unequivocally, as they overlap with other cases that have been heavily perturbed but will not reach our planet.

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Though most Earth impactors are significantly displaced from the nominal on-sky position of Apophis, not all are. Roughly 1% of Earth impactors appear less than 1'' from the nominal location on the sky when the early window opens, increasing to 3% at its closing. These mostly correspond to cases where the collision that created the velocity change happened within the 30 days prior to the date of observation: Apophis has not yet had time to move much from its nominal trajectory. Since the dust cloud and/or tail created by the DART impact was visible for at least 30 days after that event (Kareta et al. 2023; Opitom et al. 2023), we expect that the unlikely case of Apophis being both near its nominal on-sky position and on an Earth-intercepting trajectory will likely be distinguishable by the presence of a dust cloud or tail. We do find that a very small fraction of Earth impactors are located near the nominal on-sky position of Apophis despite having received an impulse more than 30 days before the simulated observation, though none are more than 180 days. This suggests that single-night observations of Apophis are likely to be very helpful in identifying dangerous scenarios but may not be sufficient on their own.

During the late observing window, the majority of the Earth-impacting clones are well away from the nominal ephemeris position. A few of these cases fall within 1'' of nominal on the sky (Figure 9); however, all such scenarios examined in this study occurred when the impact that deflected Apophis happened within the 20 days prior to the observation. Given the persistence of the resulting dust cloud, this case should be reliably identifiable from telescopic measurements.

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3.3.3. Other Observables

Since our goal is the rapid assessment of whether or not Apophis might have undergone a perturbation of concern, we consider briefly whether other simple observables could be used to assess Apophis's post-2027 orbit.

The apparent magnitude of an asteroid can be used as a proxy for its distance if its absolute magnitude is well known. But if a small asteroid does deflect Apophis, its surface properties and reflectivity could be changed by the impact. There could also be residual dust orbiting Apophis that makes using its apparent magnitude to make deductions about its trajectory difficult. But a distinct brightening of Apophis observed in 2027 could be the result of an impact and would indicate that further investigation was warranted.

Additional orbital information could be provided by parallax measurements of distance. Though not traditionally used in orbit determination, parallax-based techniques are becoming more prevalent (Heinze & Metchev 2015; Gural et al. 2018; Zhai et al. 2022). Two stations on opposite sides of the Earth observing Apophis when it emerges from the daytime sky in 2027 will see it at a distance of approximately 1 au and so would measure a parallax p,

$$\begin{eqnarray*}&&p=\displaystyle \frac{2{R}_{{\rm{E}}}}{1\,\mathrm{au}}\approx 18^{\prime\prime} ,\end{eqnarray*}$$

where R_E is Earth's radius. The parallax could be used to determine Apophis's distance if it could be measured precisely enough. A longer baseline would assist in parallax measurements. The Earth's orbital motion about the Sun provides roughly 2.5 million km of baseline per 24 hr period, but parallax measurements would be complicated by the motion of the asteroid during that period of time. A space telescope such as the JWST located at the Earth–Sun L2 point would provide a baseline of order 1 million km, much longer than that available to Earth observers alone. JWST is not designed to point within 85° of the Sun (Nella et al. 2004), but other current or future space telescopes may be able to contribute to a determination of Apophis's distance via parallax, in the unlikely event that circumstances warrant it.