The Estimated Population of Small NEOs

Alan Harris

MoreData! Inc.

Target NEO 2

Washington, DC, July 9, 2013

Estimating completion from re-detection ratio

For a computer modeled survey with a known number of synthetic NEAs, we can tabulate the re-detection ratio vs. size (dm) as well as the real completion. For a real survey, we can measure re-detection ratio, but not completion, since we don’t know the total population.

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Survey sim. re-detectionsSurvey sim. completion

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Current survey completion

Estimating completion from re-detection ratio

The actual re-detection ratios for the combination of LINEAR, Catalina, and Siding Spring can be adjusted horizontally to match the model curve to within the uncertainties in the survey data. We thus adopt the model completion curve as adjusted to match the dm scale as representing current survey completion.

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Survey sim. re-detectionsSurvey sim. completionRe-detections, 2010-2012

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Extrapolation to very small sizeThe observed re-detection ratio becomes uncertain below about 0.1 (that is, H greater than about 22) due to the low number of re-detections. However, having “calibrated” the completion curve in the range of good re-detection statistics, we can extend to still smaller sizes by assuming that the computer completion curve accurately models actual completion. This works until the number of “detections” in the computer model falls below a statistically useful number, say about 100 “detections” out of the 100,000 model asteroids, or a completion of about 10-3. This corresponds to about dm = -4.0, or on the scaled curve to about H = 25.0.

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Computer model completionTheoretical log(C) -0.8dm

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Fortunately, below dm of ~3.0, detections are close to the Earth and can be modeled with rectilinear motion rather than accounting for orbital motion. An analytical completion function can be matched to the computer completion curve and extrapolated to arbitrarily small size.

With these extensions, we now have an estimate of completion over the entire size range of observed objects.

Differential Population

Plotted here are the numbers in each half-magnitude interval, in red the total number discovered as of August 2012, and in blue the estimated total population in that size range, based on the completion curves of the previous graphs.

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Population estimate

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Cumulative Population

The cumulative population is the running sum of the differential population, from the previous plot. The number N is the total number of NEAs larger than the specified size (H or Diameter).100

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Estimated, 2012Annual bolide eventConstant power lawDiscovered to August, 2012

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Population of very small mission-suitable NEAs

• Over half do not cross the Earth’s orbit or even approach very closely. Only about 1/5 are PHA-class (MOID < 0.05 AU).

• The median encounter velocity with the Earth (v) is ~20 km/sec; the fraction with suitably low v (< 2.6 km/sec) is at most a few tenths of a percent.

In the material presented so far, it can be seen that there are approximately 108 NEAs in the size range 5-10 m diameter, of which only about 100 have been found. This would imply that there are abundant numbers of suitable mission targets that could be discovered. But not all of the 108 NEAs are suitable mission targets, in fact only a very tiny fraction are:

Velocity (v) distribution of small Earth-crossing asteroids (ECAs)

We expect the velocity distribution of the entire population to be roughly homologous over size. The dramatic difference among discovered small objects is mostly or entirely due to the relative ease of discovering low v objects.

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> 1 km intrinsicdistribution (model)> 1 km discoveredfraction (actual)< 40 m discoveredfraction (model)< 40 m discoveredfraction (actual)Maxwellian distribution

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Velocity distribution of ECAs

Very low v NEAs

• “Thermalized” tail of the v distribution of regular NEAs from the main asteroid belt

• Lunar ejecta (Gladman et al.1996)• Space debris

Most ECAs arrive from the main asteroid belt through perturbations leading to multiple planet orbit crossings, hence most have relative encounter velocities with the Earth that are sufficient, if properly directed, to reach to the neighboring planets, Mars or Venus. That turns out to be, for both neighbors, about v 2.5 km/sec. Multiple encounters with only the Earth-moon system can evolve v to lower values, but require a “Maxwell’s Demon” (e.g., spacecraft navigation) to do so effectively. Thus, there is a “hole” in the velocity distribution below ~2.5 km/sec.

There are about 30 discovered objects with v 2.5 km/sec, so the phase space is not entirely empty. However, one must be mindful that there are three plausible sources of such objects:

Close-up of fitted Maxwellian distribution

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Close-up of low v range of fitted Maxwellian distribution

This suggests that the “thermalized” tail of the velocity distribution of real NEAs with v < 2.5 km/sec is only ~0.002 of the total population.

Earth collision time scale vs. v

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Minimum v to get to/from

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Earth collision time scale for low v ECAs

But the Earth collision time scale of v < 2.5 km/sec objects is shorter than the diffusion time scale to populate that phase space. So the Earth is gobbling up much of what chances to make it there, and the steady state population is likely even less than “Maxwellian”.

Evolution of orbits of Lunar ejecta

Gladman et al., Science 271, 1387-1392, 1996.

Orbits of low-v NEAs

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a,e distribution of low-v NEAs

On the left is the (a,e) distribution of discovered NEAs. On the right is a plot of Gladman et al. orbits of Lunar ejecta after 105 years, matched to the same scale. The lack of discovered NEAs interior to the Earth’s orbit is likely a selection effect against discovering such objects.

Evolution of v 2.5 km/sec orbits

Gladman et al. consider the transfer of Martian ejecta to the Earth, which yields some insight into the diffusion of orbits from the main belt into Earth-crossing orbits of low v. Unfortunately a coordinate is missing, inclination, so we can only infer minimum values of v, corresponding to zero inclination. Only one object in the above panels, the one at a = 0.95, e = 0.05 in the 106 year panel, would have a v < 3 km/sec even at zero inclination, and that one object would have v > 2.5 km/sec if i > 5. Thus, out of the 200 particles simulated, at most one, and probably none, evolved into an Earth-crossing orbit with v < 3 km/sec. The prospects for objects of low v coming from the main asteroid belt are perhaps even less likely.

Space Debris

We know at least one discovered object of low v has been confirmed to be space debris; several others are likely old rocket bodies.

Sources of ultra-low v NEAs

1. Lunar ejecta (most)

2. Space debris (some)

3. Main-belt asteroids (almost none)

4. Mars ejecta (almost none)

The take-home message is that in choosing a very low v target, you need to have very good physical characterization of the object if you want to be sure you aren’t bringing a piece of the moon back to its home, or even an old rocket body.

Binary and spin properties of small NEAs

• There are no binaries below ~300 m diameter.

• All objects smaller than ~20 m are super-fast rotators.

• Many small objects are tumblers.

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4596 single, PA rotators 33 Tumblers 146 Binaries

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4,775 reliable asteroid rotation periods (November, 2012)

Better know before you go.