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Sky-Plane Velocity Distributions of Interstellar Objects and Implications for Their Detection

This paper presents an efficient analytic formula for calculating the sky-plane velocities of interstellar objects and demonstrates that their rapid motion, particularly for intrinsically dim bodies, likely hinders detection, suggesting that many more such objects may be traversing the Solar System undetected.

Original authors: Cassidy E. Walker, Darryl Z. Seligman

Published 2026-06-05

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CC BY 4.0

Original authors: Cassidy E. Walker, Darryl Z. Seligman

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the night sky as a giant, busy highway. Most of the cars (asteroids and comets) we see belong to our Solar System; they drive in predictable lanes and move at speeds we are used to tracking. But every now and then, a "visitor" from another star system zooms through. These are Interstellar Objects (ISOs). We have only spotted three so far: 1I/'Oumuamua, 2I/Borisov, and 3I/ATLAS.

This paper asks a simple but critical question: Are we missing a lot of these visitors because they are moving too fast for our cameras to catch them?

Here is the breakdown of their findings, explained through everyday analogies:

1. The "Blurry Photo" Problem

When you take a photo of a fast-moving car at night, it often comes out as a blurry streak rather than a clear car. In astronomy, this is called "trailing loss." If an object moves too quickly across the sky, it becomes too faint or too streaked for telescopes to recognize it as a distinct object.

The authors developed a new, super-fast math formula (like a shortcut calculator) to predict exactly how fast these interstellar visitors would appear to move across our sky, depending on their orbit and how far away they are. Instead of running a slow, complex computer simulation for every single object, this formula gives the answer instantly.

2. The "Flashlight" Analogy: Bright vs. Dim Objects

The paper simulated a massive crowd of 100,000 fake interstellar objects to see how fast they would be moving when our telescopes could finally see them. They split them into two groups:

The "Dim Flashlights" (Asteroid-like): These are like small, dark rocks that don't glow. To see them, they have to get very close to Earth, like a flashlight held right up to your face. Because they are so close, they zip across your vision incredibly fast.
- The Result: Most of these dim objects are moving so fast when they become visible that they might be too blurry for our surveys to catch. Only a tiny fraction of them get close enough to be seen, and they are often moving at record-breaking speeds.
The "Bright Flashlights" (Comet-like): These are like comets that have a glowing tail. They are bright enough to be seen from much further away, like a lighthouse visible from miles out.
- The Result: Because they can be seen from far away, they appear to move much slower across the sky. This makes them much easier to spot and track.

3. The "Speed Trap"

The paper found that the three objects we have found are actually a bit of a mixed bag:

1I/'Oumuamua was a "dim flashlight." It was discovered only because it got extremely close to Earth, and at that moment, it was moving at a blistering speed (about 6.6 degrees per day).
2I/Borisov and 3I/ATLAS were "bright flashlights" (active comets). They were spotted much further away, so they were moving at a more manageable, slower pace.

The authors warn that our current surveys (like the upcoming LSST telescope) have a "speed limit" for alerts. They generally won't flag objects moving faster than 10 degrees per day.

The Risk: 'Oumuamua was moving at 6.6 degrees when found, but just a few days earlier, it was moving at 12.2 degrees. If our telescopes had a strict 10-degree cutoff, we might have missed 'Oumuamua entirely.
The Conclusion: There are likely many more "dim" interstellar objects zooming through our solar system that we are missing because they are moving too fast to be linked together by our software.

4. The Silver Lining

The good news is that the "bright" comets (like Borisov and ATLAS) are easier to find because they are visible from far away, where they move slowly. The paper suggests that if we look back at old telescope data (a process called "precovery"), we might find these bright visitors in the archives, even if we missed them the first time they flew by.

In summary: The universe is likely full of interstellar travelers. However, the "dark" ones are like speedsters that blur past our cameras, while the "bright" ones are like slow-moving lighthouses that are easy to spot. We might be missing a huge population of the fast, dark ones simply because they move too quickly for our current detection methods to keep up.

Technical Summary: Sky-Plane Velocity Distributions of Interstellar Objects and Implications for Their Detection

Problem Statement
Since the discovery of 1I/'Oumuamua in 2017, only two additional macroscopic interstellar objects (2I/Borisov and 3I/ATLAS) have been identified. Theoretical models suggest a vast population of such objects traverses the Solar System, yet detection rates remain low. A primary hypothesis for this discrepancy is that the rapid sky-plane velocities of interstellar objects (ISOs) may preclude their discovery, particularly due to "trailing loss" and difficulties in linking moving sources in survey data. While previous studies have calculated sky-plane velocities for specific subsets of synthetic objects, there has been no comprehensive calculation of the typical sky motion rates for ISO populations as a function of their orbital parameters and limiting magnitudes. This paper investigates whether high sky-plane velocities constitute a fundamental barrier to detection, particularly for intrinsically dim objects.

Methodology
The authors develop and apply an analytic framework to quantify the apparent sky motion of ISOs, avoiding the computational overhead of direct numerical integration.

Analytic Formulation: The paper derives a closed-form analytic solution for the angular rate of sky motion ( $d\theta/dt$ ) for an object on an arbitrary orbit (specifically hyperbolic for ISOs) observed from an arbitrary location (Earth).
- The derivation begins with a generalized vector form subtracting the line-of-sight velocity component from the total velocity vector.
- This is expanded into a 2D function of orbital elements (semimajor axis $a$ , eccentricity $e$ , true anomaly $f$ ) assuming a circular Earth orbit.
- Finally, the model is extended to a full 3D function incorporating inclination ( $i$ ), argument of periastron ( $\omega$ ), and longitude of the ascending node ( $\Omega$ ) via rotation matrices transforming the perifocal frame to the ecliptic frame.
- The validity of the 3D analytic formula is confirmed by comparing its output against JPL Horizons ephemeris data for 1I and 2I, showing close agreement.
Synthetic Population Generation: Using the probabilistic method (Marčeta 2023) with M-star kinematics, the authors generate three synthetic populations of $\sim 10^5$ ISOs each, confined within heliocentric spheres of radii 1.2 AU, 3.0 AU, and 5.0 AU.
Magnitude and Motion Calculation:
- Asteroid-like Populations: Two intrinsic brightness populations are modeled using constant absolute magnitudes ( $H_V$ ): a "dim" population ( $H_V = 22.08$ , based on 1I) and a "bright" population ( $H_V = 17.1$ , based on 3I). Apparent magnitudes are calculated at 100 points along each trajectory.
- Comet-like Populations: To account for activity, two additional populations are modeled with radii of 100 m and 1 km, utilizing cometary brightening laws (Cook et al. 2016) where brightness scales with heliocentric distance ( $r$ ) and a photometric index ( $n=5.0$ ).
- For all populations, the sky motion is calculated at the specific points where objects reach a range of limiting apparent magnitudes ( $m_V$ from 17.0 to 27.0).

Key Results

Velocity vs. Brightness: Intrinsically brighter objects (and active comets) reach detection magnitude thresholds at significantly larger heliocentric distances than dim objects. Consequently, they are detected at lower sky-plane velocities.
Dim Objects: For the intrinsically dim population ( $H_V = 22.08$ ), the median sky motion at limiting magnitudes typical of current surveys (e.g., $m_V \approx 19.0$ ) is high. In the 1.2 AU sphere, the median motion is $\sim 1.56$ deg/day, and a significant fraction of these objects move faster than the discovery speed of 1I (6.64 deg/day).
Bright Objects: The intrinsically bright population ( $H_V = 17.1$ ) exhibits slower median sky motions at the same limiting magnitudes because they are detectable further away. For example, in the 3.0 AU sphere at $m_V = 19.0$ , the median motion is $\sim 0.92$ deg/day.
Cometary Activity: Active comets (modeled with $n=5.0$ ) show the slowest median sky motions. Their rapid brightening allows detection at large distances where their transverse velocity across the sky is minimal.
Distribution Tails: While median speeds for bright objects are lower, the tails of all distributions extend to speeds faster than the discovery motion of 1I. This suggests that even bright objects can occasionally be detected at high velocities.
Detection Limits: The paper notes that the LSST survey will not alert for objects moving faster than 10 deg/day. 1I was detected at 6.64 deg/day but was moving at 12.2 deg/day just days prior, illustrating the risk of missing fast-moving ISOs entirely.

Significance and Claims
The paper claims that the rapid sky-plane velocities of interstellar objects, particularly those that are intrinsically dim or inactive, likely contribute to their under-detection. The authors conclude that:

Detection Bias: The difficulty in linking rapidly moving objects, combined with trailing losses, implies that many ISOs traversing the Solar System remain undetected.
Survey Constraints: Current and future surveys may miss a significant fraction of the ISO population, specifically those that are dim and must pass very close to Earth to be bright enough to detect, at which point their sky motion is prohibitively fast.
Precovery Potential: The successful precovery of 2I and 3I in archival data (ZTF, LSST, TESS) suggests that objects with cometary activity (which are brighter and slower at detection) are more likely to be found in existing datasets.
Methodological Utility: The provided analytic formula offers a computationally efficient tool for estimating sky motion for any moving object from any observation point, bypassing the need for numerical integration in population studies.

The authors explicitly state they do not speculate on specific pipeline failures beyond sky motion but argue that the kinematic properties of ISOs themselves present a fundamental challenge to discovery, especially for the "dark comet" population analogous to 1I.

1. The "Blurry Photo" Problem

2. The "Flashlight" Analogy: Bright vs. Dim Objects

3. The "Speed Trap"

4. The Silver Lining

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