The Estimated Population of
Small NEOs
Alan Harris
MoreData! Inc.
Target NEO 2
Washington, DC, July 9, 2013
Estimating completion from re-detection ratio
C u rre n t s u rv e y c o m p le tio n
C o m p le tio n o r R e -d e te ctio n ra tio
1 .0
S u rv e y s im . re -d e te c tio n s
S u rv e y s im . c o m p le tio n
0 .8
0 .6
0 .4
0 .2
0
-8
-7
-6
-5
-4
-3
-2
-1
0
dm
1
2
3
4
5
6
7
8
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 redetection ratio,
but not
completion, since
we don’t know the
total population.
Estimating completion from re-detection ratio
C u rre n t s u rve y c o m p le tio n
C o m p le tio n o r R e -d e te ctio n ra tio
1 .0
S u rv e y s im . re -d e te c tio n s
S u rv e y s im . c o m p le tio n
R e -d e te c tio n s , 2 0 1 0 -2 0 1 2
0 .8
0 .6
0 .4
0 .2
dm
0
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
H m a g n itu d e
The actual redetection 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.
Extrapolation to very small size
The 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.
Fortunately, below dm of ~3.0,
detections are close to the Earth and
10
can be modeled with rectilinear motion
rather than accounting for orbital
10
motion. An analytical completion
function can be matched to the
computer completion curve and
10
C o m p u te r m o d e l c o m p le tio n
extrapolated to arbitrarily small size.
T h e o re tic a l lo g (C )  -0 .8 d m
-1
C o m p le tio n
-3
-5
10
With these extensions, we now have
an estimate of completion over the
entire size range of observed objects.
-7
-1 1 -1 0 -9
-8
-7
-6
-5
-4
-3
-2
dm = V
lim
-1
- H
0
1
2
3
4
5
6
7
Differential Population
D ia m e te r, k m
0 .0 1
10
0 .1
1
10
8
P
op
ul
io
6
at
n
10
es
at
n (H )
tim
e
10
10
10
4
2
T o ta l d isc o v e re d
0
31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9
H
Plotted here are
the numbers in
each halfmagnitude
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.
Cumulative Population
Im p a c t E n e rg y, M T
-1
10
8
2
10
5
10
8
E s tim a te d , 2 0 1 2
A n n u a l b o lid e e v e n t
C o n sta n t p o w e r la w
D is c o v e re d to A u g u s t, 2 0 1 2
10
10
10
K -T Im p a cto r
10
4
2
0
31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9
0 .0 1
0 .1
1
D ia m e te r, K m
10
10
10
A b s o lu te M a g n itu d e , H
10
2
6
N (< H )
10
0
10
4
6
8
Im p a ct In te rva l, ye a rs
10
10
T u n g u ska
10
10
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).
Population of very small missionsuitable NEAs
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:
• 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.
Velocity (v) distribution of small Earthcrossing asteroids (ECAs)
V e lo city d istrib u tio n o f E C A s
1 .0
F ra ctio n < v

0 .8
0 .6
> 1 km  in trin s ic
d istrib u tio n (m o d e l)
> 1 km d isco ve re d
fra c tio n (a ctu a l)
< 4 0 m d isco ve re d
fra c tio n (m o d e l)
< 4 0 m d isco ve re d
fra c tio n (a ctu a l)
M a xw e llia n d istrib u tio n
0 .4
0 .2
0
0
10
20
30
v

40
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.
Very low v NEAs
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:
• “Thermalized” tail of the v distribution of
regular NEAs from the main asteroid belt
• Lunar ejecta (Gladman et al.1996)
• Space debris
Close-up of fitted Maxwellian distribution
C lo s e -u p o f lo w v ra n g e o f fitte d M a xw e llia n d is trib u tio n

F ra ctio n < v

0 .0 1 0
M in im u m v to g e t to /fro m

V e n u s o r M a rs
0 .0 0 5
0
0
1
2
3
v
4

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
E a rth c o llis io n tim e s c a le fo r lo w v E C A s

E a rth c o llis io n tim e s c a le , y e a rs
10
10
10
10
8
7
6
5
M in im u m v to g e t to /fro m

10
10
V e n u s o r M a rs
4
3
0 .5
1
2
5
v

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
1 0 than “Maxwellian”.
Evolution of orbits of Lunar ejecta
Gladman et al., Science 271, 1387-1392, 1996.
Orbits of low-v NEAs
a ,e d is trib u tio n o f lo w - v N E A s

1 .3
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.
1 .2
a
1 .1
1 .0
0 .9
0 .8
0
0 .0 5 0 .1 0 0 .1 5
e
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 Earthcrossing 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.
2.
3.
4.
Lunar ejecta (most)
Space debris (some)
Main-belt asteroids (almost none)
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
4,775 reliable asteroid rotation perio ds (N ovem ber, 2 012)
0 .001
• There are no
binaries below
~300 m
diameter.
• All objects
smaller than
~20 m are
super-fast
rotators.
• Many small
objects are
tumblers.
R o ta tio n p e rio d , h o u rs
4596 single , P A rotators
33 T um b lers
1 46 B in aries
0.1
10
1000
0.001
0 .1
10
D iam eter, km
1000
Better know
before you go.