1. The Population of Near-Earth Asteroids Revisited
Alan Harris
MoreData! Inc.
DPS, Provo, UT, Oct , 2017

2. The re-visits:
Updated NEA population using survey data up to July, 2017, N(H<17.75) = 921.
New population estimate implies the number of remaining undiscovered NEAs of H < (nominally D > 1 km) is only about 36.
Transformation from magnitude to diameter, using NEOWISE determined albedo distribution, suggests N(D>1km) is ~830.

3. Correction of a “small” error
Estimating population versus size (absolute magnitude H) is done by binning the numbers of discovered asteroids, in my studies by half magnitudes. Unfortunately, the Minor Planet Center rounds most values of H to 0.1 magnitude. So, for example, a bin from H of 17.5 to 18.0 is really from to 18.05, or to 17.95, depending on which side of the bin you take “less than or equal to” rather than “less than”.
The problem came to light when another population study was submitted for publication (now published as Tricarico 2017, Icarus 284, ), which initially made the opposite “less than or equal to” assumption, and arrived at an estimate for N(H<17.75) about 10% less than our previous estimate (Harris and D’Abramo 2015, Icarus 257, ). We corrected the problem for the current analysis by choosing bin boundaries at .05 magnitudes, e.g to 17.75, so the 0.1 round-off thresholds naturally put objects in the right bin. When Tricarico and I each made these corrections, our population estimates fell into almost perfect agreement, as shown in the next slide.

4. Cumulative population estimate, 2017

5. Transforming from H to D, the NEOWISE albedo distribution
A plot of measured albedos versus diameter (left plot) does not reveal an obvious dependence of albedo on diameter. We make the initial rash assumption that the albedo distribution is constant over the full range of diameter, from ~0.1 km up to 10 km. The plot on the right is a histogram of that distribution, binned in albedo ranges that correspond to 0.5 magnitudes of H at a constant diameter.

6. Albedo distribution at constant H
Constant diameter Constant absolute magnitude
Even assuming the albedo distribution is uniform with respect to diameter, it is very different with respect to H magnitude, because at a given H magnitude, there are far more smaller objects of high albedo than larger objects of low albedo. The distortion of the distribution depends on slope of the size-frequency distribution, so the distribution is not uniform over H, even if it were over D.

7. n(H) to n(D) Sudoku
The numbers n(A,D) in each horizontal box must follow the fractions f listed. f
.0192
.1418
.2596
.1755
.1130
.0793
.1250
.0601
.0264
D / A
.5586
.3525
.2224
.1403
.0885
.0559
.0353
.0222
.0140
2.239
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
1.778
19.0
1.413
19.5
1.122
20.0
.8913
20.5
.7079
21.0
.5623
21.5
.4467
22.0
.3548
22.5
n(H) (known)
n(D) = ???
The numbers given in each box are the H magnitudes corresponding to each box diameter and albedo, and are to be replaced by estimates of n(A,D) for each diameter. The sums of the numbers along each diagonal of constant H must be equal the known n(H). Once these conditions are met as closely as possible, the resulting horizontal sums represent n(D).

8. Example solution n(18.5) = 688 (717) N(>1km) = 810f
.0192
.1418
.2596
.1755
.1130
.0793
.1250
.0601
.0264
.9999
N(>D/ 1.222)
D / A
.5586
.3525
.2224
.1403
.0885
.0559
.0353
.0222
.0140
n(D)
2.239
1.48
10.9
20.0
13.5
8.71
6.11
9.64
4.63
2.04
77.1
179.
1.778
2.43
18.0
16.0
16.5
17.0
17.5
18.5
19.0
127.
306.
1.413
3.88
28.7
52.5
35.5
22.8
25.3
12.1
5.33
202.
508.
1.122
5.80
42.9
78.5
53.0
34.2
24.0
37.8
18.2
7.98
302.
810.
.8913
8.38
61.9
113.
76.6
49.3
34.6
54.6
26.2
11.5
436.
1250.
.7079
12.4
91.8
168.
114.
73.2
51.3
80.9
38.9
17.1
647.
1890.
.5623
18.1
134.
245.
166.
107.
74.9
118.
56.8
24.9
944.
2840.
.4467
25.2
186.
341.
230.
148.
104.
164.
78.9
1310.
4150.
.3548
34.3
253.
464.
314.
142.
223.
47.2
1790.
5940.
N(>1km) = 810
The inversion can be back-checked analytically and exactly. The numbers horizontally should follow the f(A) fractions exactly, and the diagonal sums (along red line) should come to the input numbers n(H). This solution deviates by several percent from bin to bin of n(H), but follows pretty closely.

9. One transformation from H to D
Here is one attempt to transform n(H) to n(D), left, and N(<H) to N(>D), right, using the albedo distribution for all NEAs. In each plot, the H and D scales have been adjusted to get the best overlap of the respective curves. As can be seen, the D and H curves overlap well with an offset of D = 1 km equivalent very closely to H = This implies an effective mean albedo of 0.16.

10. Summary
Population estimates in terms of absolute magnitude H agree very closely, N(H<17.75) = 921 ± 20, current surveys are about 96% complete to this size, only ~36 left to find.
At the smallest size range, estimates agree within a factor of 2 or so with bolide frequency estimates.
Conversion to diameter size frequency is problematic, but roughly matched by equivalence of H = 17.6 to D = 1 km, thus an effective mean albedo of ~0.16, and implied N(D>1km) = 830.

11. Comparison with some recent publications