Estimating the Mass of a Main-belt Asteroid
Malcolm Shute
Abstract
There are occasions, such as for simulation purposes, when the mass might be needed for each and every
asteroid known in the solar system. Given that this sort of information is only available for a few asteroids, it
would be useful to have a way of estimating the masses of the others, at least to an order of magnitude.
The Minor Planet Center (MPC), at the Smithsonian Astronomical Observatory of Harvard University,
maintains a publicly-readable, on-line database of the orbit parameters of all the known asteroids and comets,
but understandably does not include the mass amongst this data.
This paper presents a formula for estimating each mass, using only the data that is available in the MPC
database, and gives a brief analysis of the quality of the estimate obtained.
Introduction
There are at least two project proposals for which the mass would need to be known for each and every
asteroid in the main belt. Hilton (ref.1) proposes a project to compute the perturbation of the orbits of Mars and
Earth by the asteroids. Another project is proposed (ref.2) to predict the likelihood of inter-asteroid collisions,
with a view to preparing space missions for observing such events. The work of Scheeres (ref.3) implies other
possibilites for observing asteroids that have acquired satellites, and monitoring any chaotic behaviour for any
interesting consequences.
Given that the mass is only known for a few asteroids, it would be useful to have a formula that can give an
estimate for the mass of each of the others, at least to within plus or minus one order of magnitude.
The following expression is proposed as a first attempt at allowing the mass of the asteroid to be inferred
using the observational data that is maintained in the MPC database (ref.4):
mn = K1 . m1 . exp( – √((yodn – yod1) / t0) ) . ( (qn – 1) / (q1 – 1) )2 . exp(H1 – Hn) + K2
In effect, the formula states that asteroids that are only recently being discovered are either small, distant, or
not very reflective, or some combination of these.
The constants, K1 and K2, are derived from the closest fit straight lines (the lines of regression, in fact), as
explained later. The next term (measured in kilograms) uses the mass of Ceres, m 1, as the reference. The
next term, the exponential, accounts for the year of discovery of asteroid-n, yodn, counting from the year of
discovery of Ceres, yod1. The squared term takes the asteroid's minimum distance from the Earth into
account. Finally, the last exponential term uses the absolute magnitude of the asteroid to scale for its albedo.
Asteroid-n orbits at qn astronomical units from the Sun at perihelion, and hence about (q n–1) astronomical
units from the earth at our closest approach over the past centuries. Similarly, Ceres was (q 1–1) astronomical
units from the earth at its closest approach. The apparent brightness of each then scales with the square of
these distances.
The final exponential term uses the ratio of the absolute magnitudes to scale for the reflectivity of the surface.
Normally this would be expressed as an exponential to base 10, but has been expressed here to the natural
base for convenience, ready for combining it with the first exponential term.
The justification for the first exponentional term is not quite so clean. It is not surprising for it to be exponential,
since this is a good model for the rate of advancement in human knowledge (in telescope and survey-satellite
technology, for example). The square root in the argument, though, was determined by curve fitting.
The time constant, t0, was included just to get the dimensions right (and has the dimensions of years, to
ensure that the argument to the exponential function is dimensionless), but, for the present at least, it appears
to have a value of unity.
Spectral types and other parameters
The spectral type of each asteroid has an influence on the ease of observing it, and of initially discovering it,
Malcolm Shute
07 March 2016
1
from the earth. This parameter has not been included in the formula, though, for several reasons. One reason
is that the parameter is not yet listed in the MPC database, although there are other extensive sources of this
sort of information (ref.5). The aim was to be able to work exclusively with the data that is present in the MPC
database as the single resource. Another reason, though, is that the absolute brightness parameter implicitly
takes this data into account, already.
Similarly, the MPC database does not list the radius of each asteroid. Since larger asteroids might be
expected to be denser, due to gravitational compacting, it is to be expected that this would have an effect on
their optical properties, and hence on their observability. Again, for the purposes of this paper, this is
considered to have been accounted for by the absolute brightness, acting as a single lumped parameter.
Both of these are left as areas for future work (either directly, to be added to the formula, or used in the
reverse direction, for estimating of the radius or spectral type of an asteroid, based on the discrepency
between its predicted mass, and that obtained by observation).
Finding the lines of regression
Of the main-belt asteroids, the masses of thirty-eight of them (1, 2, 3, 4, 6, 7, 9, 10, 11, 15, 16, 17, 19, 20, 22,
24, 29, 31, 45, 48, 52, 54, 65, 88, 92, 121, 140, 216, 243, 253, 324, 444, 451, 511, 532, 704, 951, 4979) have
been obtained already from observational data, and the values published on the Internet (refs.1, 6, 7, 8, 9).
Taking all but the first of these values (since Ceres is to be used as the reference asteroid), and plotting the
value from the formula against these known masses, the lines of closest fit can be found.
This gives a value for the gradient, K1, between 251 and 385 (with a geometric mean value of 311) and for the
intercept, K2, between 7.7e+18 and -3.3e+18 kilograms (with a mean value of 2.8e+18 kg, for the line with
gradient 311).
The coefficient of correlation is good, at 0.81. Unfortunately, this is achieved because the statistics of the large
asteroids dominate those of the smaller ones by many orders of magnitude. It does, at least, though, give a
good starting point from which to launch a better analysis.
In particular, to remedy the problem of differing orders of magnitude of asteroid mass, it was considered
advantageous to find the lines of regression based on the logarithms of the masses, rather than directly on the
masses themselves. Thus, the following expression was used (where m is the unit vector of mass, and is
introduced just to make the quantities dimensionless before taking logarithms):
ln( mn/m ) = K3 . ( H1 – Hn – √((yodn – yod1) / t0) + ln( m1/m . ( (qn – 1) / (q1 – 1) )2 ) ) + K4
Plotting the logarithms of the masses of the thirty-seven known asteroids against this expression (with K3
temporarily set to 1, and K4 temporarily set to 0), and finding the lines of regression, the value for the gradient,
K3, was found to be between 0.7 and 0.9 (with a geometric mean value 0.8) and, that for the intercept, K 4, to
be between 18 and 10 (with a mean value of 14.4, for the line with gradient 0.8). The coefficient of correlation
is very good, at 0.87.
The formula for estimating mass therefore becomes:
mn = 1.8x106. m . (m1/m)0.8 . exp( 0.8 . ( H1 – Hn – √((yodn – yod1) / t0) ) ) . ( (qn – 1) / (q1 – 1) )1.6
Figure 1 shows the result of plotting the points using the derived values of K 3 and K4. (Common logarithms
have been used for the axes, for convience.)
Conclusion
The formula can be simplified to the following expression (where 7x1022 represents a reference mass,
measured in kilograms, that has no real physical significance, because, as well as the mass of Ceres, it also
incorporates a number of normalising constants, such as the one for converting absolute magnitudes from
base-10 to base-e):
mn = 7x1022 . exp( 0.8 . ( H1 – Hn – √(yodn – yod1) ) ) . ( (qn – 1) / (q1 – 1) )1.6
Tables 1 and 2 compare the value generated by the expression against the observed mass, for the thirtyseven asteroids where this is known from observations. Other columns in Table 2 list the figures for absolute
magnitude, orbital eccentricity, semi major axis, and year of first observation, as given in the MPC database,
for the first 20 asteroids, and then for a number of others that are of interest.
Malcolm Shute
07 March 2016
2
Table 1. Masses of selected asteroids, both observed and estimated
Ref Name
Estimated Observed
% Ref
Name
Estimated
Observed
mass
mass
mass
mass
(in kg)
(in kg)
(in kg)
(in kg)
%
2
Pallas
1.99E+20
3.15E+20
63
52
Europa
2.23E+19
2.52E+19
89
3
Juno
6.25E+19
1.80E+19
347
54
Alexandra
4.16E+18
5.15E+18
81
4
Vesta
3.50E+20
2.54E+20
138
65
Cybele
1.89E+19
1.15E+19
165
6
Hebe
7.84E+18
1.37E+19
57
88
Thisbe
5.24E+18
1.17E+19
45
7
Iris
2.14E+19
2.38E+19
90
92
Undina
9.87E+18
5.15E+18
192
9
Metis
1.16E+20
5.03E+18
2305
121
Hermione
5.99E+18
8.92E+18
67
10
Hygiea
7.27E+19
8.05E+19
90
140
Siwa
9.20E+17
1.50E+18
61
11
Parthenope
1.55E+19
5.69E+18
272
216
Kleopatra
1.75E+18
1.98E+18
88
15
Eunomia
3.86E+19
3.17E+19
122
243
Ida+Dactyl
3.41E+17
4.32E+16
788
16
Psyche
3.34E+19
1.35E+19
248
253
Mathilde
1.15E+17
1.03E+17
111
17
Thetis
4.62E+18
6.54E+17
707
324
Bamberga
9.94E+17
1.05E+19
9
19
Fortuna
7.19E+18
9.31E+18
77
444
Gyptis
6.35E+17
8.92E+18
7
20
Massalia
1.20E+19
4.76E+18
251
451
Patientia
2.70E+18
1.17E+19
23
22
Kalliope
2.35E+19
1.68E+19
140
511
Davida
2.68E+18
4.54E+19
6
24
Themis
1.51E+19
6.94E+18
218
532
Herculina
2.54E+18
2.28E+19
11
29
Amphitrite
1.77E+20
1.53E+19
1155
704
Interamnia
2.59E+18
2.52E+19
10
31
Euphrosyne
1.48E+19
1.68E+19
88
951
Gaspra
1.03E+16
1.00E+16
103
45
Eugenia
7.12E+18
5.94E+18
120
4979
Otawara
3.20E+14
2.00E+14
160
48
Doris
1.59E+19
1.21E+19
131
The percentage column in Table 1 shows that the original aim of this paper has not been met: many of the
estimated asteroid masses are within plus or minus one order of magnitude (that is within 10% and 1000%) of
the actual figure, but there are still exceptions (9, 29, 324, 444, 511). Metis (9) is the worst, being out by over
two orders of magnitude.
Even so, these results are considered encouraging. At the very least, such large discrepencies serve to
highlight those cases that might make for interesting further study, to find out why the estimated mass differs
so much from the observed mass (or rather, to investigate why the expected date of discovery, given the
mass, distance and albedo, differs from the actual date of discovery).
Table 2 goes on to show that the formula does at least generate fairly reasonable looking values even when it
is interpolating (5, 8, 12, 13, 14, 18, 87) and extrapolating (5535, 7755, 9969). As far as the formula is
concerned, the interpolation and extrapolation asteroids were effectively chosen at random – though, in actual
fact, they were chosen because they are described, in other contexts, on the Internet. This, at least, has the
advantage that they are well studied, and might lead to future data with which to test the findings of this paper.
On the other hand, it could be argued that this makes them a biased sample. Asteroid-7772 was chosen,
therefore, as a control (an asteroid that has attracted so little attention that it does not yet even have a
registered proper name).
Figure 1 shows the observed mass plotted against the estimated mass (with each represented on logarithmic
scales). It also shows the position of the two lines of regression, which intersect at the point (18.76,18,76),
which merely indicates that the geometric mean of the masses of the thirty-seven asteroids is 5.80e18 kg.
Figure 1. Observed mass plotted against estimated mass (logarithmic scales)
Malcolm Shute
07 March 2016
3
Measurement error and other uncertainties
This exercise has, therefore, centred on finding the lines of regression not for the expression to derive mass,
but for the one to derive log(mass). As indicated earlier, since the universe sets the mass, distance and
reflectivity of each asteroid, the formula is really making predictions about when human technology ought to
have been up to the task of discovering the asteroid.
Thus, the formula can be rearranged to reflect this (where t=(yod n – yod1) / t0):
t = ( H1 – Hn + ln( m1/ mn . ( (qn – 1) / (q1 – 1) )2 ) + K4 / K3 )2
The influence of measurement errors can be deduced by taking logs of this:
Δt/t = 2.( Δ(H1–Hn)/(H1–Hn) + ΔK4/K4 + ΔK3/K3 + ε
The addition of the ε term at the end allows for the contribution of taking logs of the summation term in the
original expression. However, it can be argued that the contribution of this term is so small that it can, in fact,
be neglected altogether, without spending more time on analysing it further.
The accuracy of measuring the orbital distance of asteroids is extremely good, and even that for estimating the
masses has a mean uncertainty of 16% for the twenty-nine listed in (ref.9), reaching 62% in the worst
individual cases. Taking logarithms of a parameter with a range of possible values, gives a result with a greatly
reduced ranges of possible values.
Thus, the uncertainty is dominated by that in calculating the difference of the two absolute magnitudes, and
that in the two constants K3 and K4 (13% and 44%, respectively)
Δt/t = 2.( Δ(H1–Hn)/(H1–Hn) + ΔK4/K4 + ΔK3/K3 )
Malcolm Shute
07 March 2016
4
The two lines of regression in Figure 1 give an indication of this uncertainty when extrapolating back to the
smaller asteroids (reaching about plus or minus one order of magnitude at asteroids when extrapolating for a
mass of 108 kg, for example, and getting progressively worse for masses below that).
Acknowledgements and ideas for future work
My greatest thanks go to Dr. Sergio Ilovaisky, of the Observatoire de Haute Provence, for all of his help while
writing this paper: in reviewing drafts, advising on changes, and on its publication. I am also extremely grateful
to him for the enormous amount of computer support that he has given, that has enabled me to do this work.
I am also very grateful to Dr. Claude Chevalier, formerly of the Observatoire de Haute Provence, also for
reviewing drafts of this paper, and for her support during the period over which this work has been conducted.
I would also like to express my thanks to James Hilton who made very valuable comments and suggestions on
reviewing an earlier draft of this paper, including comments about the possible influence of spectral types on
the observed data, and of the radius on the compaction of the asteroid.
I acknowledge the Minor Planet Center (MPC), at the Smithsonian Astronomical Observatory of Harvard
University, for the major contibution made to this work by the public availability of their invaluable database.
References
1. Hilton, J.L. (1999). “Masses of the largest asteroids”, <http://aa.usno.navy.mil/hilton/asteroid_masses.htm>.
2. Shute, M.J. (2006). “Predicting inter-asteroid collisions”, <http://malcolm.shute.free.fr/rastro.htm>.
3. Scheeres, D.J. (2002). “Stability of binary asteroids”, Icarus, vol.159, pp.271-283.
4. MPC. (2003). “IAU: Minor Planet Center”, <http://cfa-www.harvard.edu/iau/mpc.html>.
5. Masi, G., S. Foglia and R.P. Binzel (2007). “Search for unusual spectral candidates among 40313 minor
planets from the 3rd release of the Sloan Digital Sky Survey Moving Object Catalog”. Submitted to Astronomy
& Astrophysics, June 2007.
6. Mytangledweb (c.2003). “Asteroids and meteorites”, <http://www/mytangledweb.co.uk/aster.htm> (link now
broken).
7. Hamilton, C.J. (2002). “Asteroid introduction”, <http://www.solarviews.com/eng/asteroid.htm>.
8. Arnett, W. (2003). “Asteroids”, <http://www.nineplanets.org/asteroids.html>.
9. Chernetenko, Yu, O. Kochetova, and V. Shor (2005). “Masses and densities of minor planets”,
<http://quasar.ipa.nw.ru/PAGE/DEPFUND/LSBSS/engmasses.htm>.
Table 2. Data for selected asteroids, with estimated and observed masses (where known)
Ref
Abs Mag Orbital Semi Major Year of
Name
Estimated mass Observed mass
(H)
Eccen
Axis (in AU)
Discovery
1
3.34
0.0798
2.7660
1802
Ceres
2
4.13
0.2300
2.7726
1839
Pallas
1.99E+20
3.15E+20
3
5.33
0.2586
2.6673
1839
Juno
6.25E+19
1.80E+19
4
3.20
0.0886
2.3614
1841
Vesta
3.50E+20
2.54E+20
5
6.85
0.1934
2.5736
1845
Astraea
1.42E+19
6
5.71
0.2012
2.4253
1869
Hebe
7.84E+18
1.37E+19
7
5.51
0.2300
2.3869
1850
Iris
2.14E+19
2.38E+19
8
6.49
0.1565
2.2015
1847
Flora
1.15E+19
9
6.28
0.1219
2.3865
1822
Metis
1.16E+20
5.03E+18
10
5.43
0.1193
3.1361
1849
Hygiea
7.27E+19
8.05E+19
Malcolm Shute
(in kg)
07 March 2016
(in kg)
8.70E+20
5
11
6.55
0.0996
2.4522
1850
Parthenope
1.55E+19
12
7.24
0.2203
2.3341
1850
Victoria
5.21E+18
13
6.74
0.0847
2.5761
1850
Egeria
1.60E+19
14
6.30
0.1683
2.5847
1851
Irene
1.70E+19
15
5.28
0.1855
2.6451
1851
Eunomia
3.86E+19
3.17E+19
16
5.90
0.1395
2.9199
1852
Psyche
3.34E+19
1.35E+19
17
7.76
0.1340
2.4699
1853
Thetis
4.62E+18
6.54E+17
18
6.51
0.2187
2.2953
1852
Melpomene
7.88E+18
19
7.13
0.1593
2.4413
1852
Fortuna
7.19E+18
9.31E+18
20
6.50
0.1433
2.4089
1852
Massalia
1.20E+19
4.76E+18
22
6.45
0.1031
2.9075
1852
Kalliope
2.35E+19
1.68E+19
24
7.08
0.1320
3.1309
1853
Themis
1.51E+19
6.94E+18
29
5.85
0.0727
2.5542
1825
Amphitrite
1.77E+20
1.53E+19
31
6.74
0.2258
3.1493
1854
Euphrosyne
1.48E+19
1.68E+19
45
7.46
0.0831
2.7203
1857
Eugenia
7.12E+18
5.94E+18
48
6.90
0.0748
3.1102
1857
Doris
1.59E+19
1.21E+19
52
6.31
0.1014
3.0986
1858
Europa
2.23E+19
2.52E+19
54
7.66
0.1965
2.7124
1858
Alexandra
4.16E+18
5.15E+18
65
6.62
0.1051
3.4333
1861
Cybele
1.89E+19
1.15E+19
87
6.94
0.0800
3.4888
1866
Sylvia
1.26E+19
88
7.04
0.1648
2.7674
1866
Thisbe
5.24E+18
1.17E+19
92
6.61
0.1006
3.1898
1871
Undina
9.87E+18
5.15E+18
121
7.31
0.1370
3.4550
1872
Hermione
5.99E+18
8.92E+18
140
8.34
0.2158
2.7330
1877
Siwa
9.20E+17
1.50E+18
216
7.30
0.2521
2.7935
1880
Kleopatra
1.75E+18
1.98E+18
243
9.94
0.0465
2.8604
1884
Ida+Dactyl
3.41E+17
4.32E+16
253
10.2
0.2651
2.6502
1885
Mathilde
1.15E+17
1.03E+17
324
6.82
0.3381
2.6826
1892
Bamberga
9.94E+17
1.05E+19
444
7.83
0.1729
2.7713
1899
Gyptis
6.35E+17
8.92E+18
451
6.65
0.0772
3.0612
1899
Patientia
2.70E+18
1.17E+19
511
6.22
0.1850
3.1696
1903
Davida
2.68E+18
4.54E+19
532
5.81
0.1786
2.7701
1904
Herculina
2.54E+18
2.28E+19
704
5.94
0.1478
3.0659
1910
Interamnia
2.59E+18
2.52E+19
951
11.5
0.1740
2.2094
1913
Gaspra
1.03E+16
1.00E+16
4979 14.3
0.1446
2.1680
1949
Otawara
3.20E+14
2.00E+14
5535 14.2
0.0633
2.2124
1942
Annefrank
6.08E+14
7755 12.1
0.1095
3.1509
1952
Haute-Provence
5.32E+15
7772 12.3
0.1800
2.8784
1955
1992 EQ15
2.71E+15
9969 15.8
0.4308
2.3452
1992
Braille
1.01E+13
Malcolm Shute
07 March 2016
5.69E+18
6