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Estimating the Mass of an Asteroid

For the Predicting Inter-Asteroid Collisions project, the mass needs to be known for each and every asteroid. Given that this information is only available for a few asteroids, there needs to be some method for estimating the masses of each of the others. Here is a proposed approximation (working in kg):

mn = 311 . m1 . exp( – √((yodn – yod1)/t0 ) ) . ( (qn – 1) / (q1 – 1) )2 . exp(H1 – Hn) + 2.8x1018

In effect, it states that asteroids that are only recently being discovered are either small, or distant, or not very reflective, or some combination of these. It allows the mass of the asteroid to be inferred using the observational data maintained in the MPC database.

The first term, m1, uses the mass of Ceres as the reference. The second term, the exponential, accounts for the year of discovery of asteroid-n, yodn, counting from the year of discovery of Ceres, yod1 (the time constant, t0, was included to make the argument to the exponential a dimensionless quantity, but appears, a little worringly, to have the value of unity). The third term takes the orbital distance into account. The fourth term uses the absolute magnitude to scale for the albedo of the asteroid surface.

Asteroid-n orbits at qn astronomical units (AU) from the sun at perihelion, and hence about (qn–1) AU from the earth at our closest approach over the past few centuries. Similarly, Ceres was (q1–1) AU from the earth at its closest approach. The apparent brightness of each scales with the square of these distances, and the last term uses the ratio of the absolute magnitudes to allow for the albedo (normally it would be expressed as an exponential to base 2.512, but has been expressed here to base e for convenience of combining in the other exponential term).

The justification for that other exponential term, though, 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 technology, for example). The square root in the argument, though, was determined by curve fitting.

Asteroid-Earth Diagram

This expression was found, empirically, to give a good fit (a coefficient of correlation of 0.93) with the other main-belt asteroids (2, 3, 4, 10, 11, 15, 16, 20, 45, 121, 140, 216, 243, 253, 433, 704, 951, 4979) whose masses have been determined elsewhere (Solarviews, SEDS, USNO Navy, Quasar, and http://www/ (link now broken)).

This preliminary result is already useful. However, it can be improved by taking the lines of regression on logarithmic axes. In this way, the result gives a more equally balanced weighting to small asteroids as to large ones.

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