April 3, 2011 | David F. Coppedge

Assuming Reality: Can Crater Dating Be Tested?

Two astronomers in Paris have come up with a new crater chronology for the moon and offered it as a way to date other objects in the inner solar system.  Their paper in Icarus,1 however, assumes so many unobservable things, the reader may wonder if it talks about the true history of the moon or some alternate reality in the imagination.  Here are some instances of assume in their paper (readers may wish to just scan the blue text to get a feel for the assumptions):

  1. The measured size-frequency distributions of lunar craters are reconciled with the observed population of near-Earth objects under the assumption that craters smaller than a few kilometers in diameter form in a porous megaregolith.
  2. The total predicted size-frequency distribution for any given time is obtained by multiplying the production function, assumed independent of age, by a time-variable constant.
  3. …the crater chronology method assumes that craters accumulate uniformly on the surface of the planetary body…
  4. Under the assumption of a steady state distribution of impactors, the distribution of craters on ~ 3 Ga old surfaces2 should be consistent with the present astronomically inferred cratering rates.
  5. Wiesel (1971) used a simplified asteroid population, and Bandermann and Singer (1973) used analytical formulations based on strongly simplifying assumptions in order to calculate impact locations on a planet.
  6. This formulation assumes that no correlations exist between the size of the object and its orbit, which is consistent 15 with the observations of Stuart and Binzel (2004) for diameters ranging from 16 ~10 m to ~10 km.
  7. This model assumes that the NEO population is in steady-state, continuously replenished by the influx coming from source regions associated with the main asteroid belt or the transneptunian disk.
  8. Various assumptions have led to all these estimates.  Among them, the assumed impact velocity and bolide density are only of moderate influence.
  9. Consequently, we simply fit a 10th-order polynomial to the entire dataset, assuming each data is error free, and that the average combination of all estimates gives a good picture of the impactor population.
  10. The size-frequency distribution of impactors is here assumed to be the same for all bodies in the inner solar system.
  11. ….the assumptions under which an encounter is considered to occur can be summarized as follows:
    1. An encounter between the target (Moon or planet) and impactor occurs at the geometrical point of crossing of the two orbits (the mutual node)….
    2. The relative encounter velocity does not account for the acceleration generated by the mass of the target….
    3. The impactor, as seen by the target, is treated as if it were approaching from an infinite distance, under only the gravitational influence of the target….
  12. For simplicity and without altering the results, it is assumed that the lunar orbit is circular about the Earth and possess a zero inclination with respect to the ecliptic.
  13. A major difference between our approach and previous investigations (Shoemaker and Wolfe, 1982; Zahnle et al., 1998, 2001) is that the argument of pericenter of the hyperbolic orbits is not assumed to precess uniformly within the Earth-Moon system, but is explicitly given by the encounter geometry.
  14. It is assumed that only the vertical component of the impact velocity, whose value is obtained from the impact angle, contributes to the crater size (Pierazzo et al., 1997), though other relations could be easily incorporated into this analysis.
  15. An increase of the transient crater diameter by wall slumping and rim formation is under the assumption of a constant impact flux over the last ~3 Ga.
  16. we assume in calculating dp that the density of the porous material is 2500 kg m-3
  17. We note that given the simplicity of our crater-scaling procedure in the transition zone, the correspondance [sic] between T and the actual megaregolith thickness should not be expected to be exact.
  18. By the use of a porous regime dictated by the properties of a megaregolith, our model production function reproduces the measured crater distributions in shape and in the absolute number of craters formed over the past 3 Ga, under the assumption of a constant impact flux.  We caution that our simple formulation of the porous / non-porous transition does not account for the temporal evolution of the megaregolith and that the inferred megaregolith thicknesses are only qualitative estimates.
  19. For illustrative purpose, Rc is shown for the inner planets in figure 3 by assuming that craters with diameters less than 10 km form in a porous soil on both the planet and Moon, while craters with greater sizes form in solid rocks (except for the Earth and Venus where only the non-porous regime is used).
  20. These calculations assume that the lunar obliquity stayed equal to its present value in the past.
  21. …we leave the implications for the contribution of secondary craters to further investigations.
  22. These authors used Öpik equations (Shoemaker and Wolfe, 1982) for hyperbolic orbits that were assumed to precess uniformly inside the planet-moon system.  We nevertheless point out that Zahnle et al. (2001) applied equation (20) to the moons of Jupiter, where this approximation might be valid.
  23. We further assume that the lunar obliquity was equal to its present value (nearly zero) for the entire time between 3.9 Ga and the present.
  24. The “vertical component” scaling appears to be the safest assumption for a single target body, though the impact angle dependence of the average crater efficiency may vary from planet to planet….
  25. Recently, Marchi et al. (2009) proposed a revised crater chronology.  The main differences with our approach (excluding the assumption of spatially uniform cratering rates in the latter) are the following:
    1. We use the orbital distribution of near-Earth objects of Bottke et al. (2002), modified for Mars, which is assumed to be in steady state and independent of bolide size….
    2. We assume that the size frequency distribution of objects impacting the planets is the same for all planets and that the probability of an object impacting a planet is independent of size….
    3. When converting transient crater diameters to final crater diameters, we use a multiplicative factor of 1.56 as suggested by Melosh (1989, 253 pp.) and Melosh (1998), whereas Marchi et al. (2009) assume that the transient crater diameter is equivalent to the final simple crater diameter for their preferred impact scaling law that is based on the equations in Holsapple and Housen (2007).
    4. Both studies treat the case of impact crater scaling in the porous megaregolith differently….
  26. It is difficult to quantify how each of these differences affect the final crater size-frequency distribution on a planetary object, and hence the derived ages of a surface.  Nonetheless, we note that the different bolide size-frequency distributions and the different crater scaling laws could be significant.
  27. Öpik probabilities assume that the argument of pericenter [omega] takes any value between 0 and 2 [pi] with an equal probability.
  28. Equations (A.27) and (A.29) come from the assumption that the projectile and Moon follow straight lines trajectories in the vicinity of the node, and are demonstrated in Öpik.
  29. Other assumed functional dependencies on the incidence angle can easily be used in place of equation (A.62).

Le Feuvre and Wieczorek admitted in the abstract that “Our model may be inaccurate for periods prior to 3.5 Ga because of a different impactor population, or for craters smaller than a few kilometers on Mars and Mercury, due to the presence of subsurface ice and to the abundance of large secondaries, respectively.”  Nevertheless, they felt that their new revised chronology is better than earlier ones.  “Standard parameter values allow for the first time to naturally reproduce both the size distribution and absolute number of lunar craters up to 3.5 Ga ago, and give self-consistent estimates of the planetary cratering rates relative to the Moon.”
    While simplifying assumptions are commonly employed in lab work, they can usually be tested by experiment.  These assumptions involve an unobserved history of the solar system that cannot be observed, repeated, or tested.  The authors also did not state to what degree their parameters might have been chosen to reproduce a crater history that was also assumed.

1.  Mathieu Le Feuvre and Mark A. Wieczorek, “Nonuniform cratering of the Moon and a revised crater chronology of the inner solar system,” Icarus (article in press, accepted manuscript), March 31, 2011, DOI:10.1016/j.icarus.2011.03.010.
2.  Ga = giga-annum, billions of years.

We do not expect readers to wade through all the quotes above (feel free if that is your favorite form of self-flagellation).  It is the visual impact of the sheer number of assumptions that go into crater count dating that makes a powerful point: does their model have anything to do with reality?
    This is not to deprive Mssrs. Le Feuvre and Wieczorek of the convenience of some of their beloved assumptions.  Perhaps it really is only the vertical component of velocity that matters for an impact, and if it makes the math easier, fine.  But many of their assumptions seem naïve if not audacious.  How could they possibly know that the incoming impact rate has been in steady state for three billion years?  The impact rate could be episodic.  A few heavy episodes in short order could completely invalidate their model.  Further, they appeared to gloss over the big issue of secondary craters (03/22/2005, 10/20/2005, 06/08/2006, 09/25/2007, 01/17/2008, 03/25/2008), leaving that little difficulty to “further investigation.”  Well, guess what.  As the links above show (q.v.), that one difficulty alone could completely confound their imaginary chronology.
    Notice, too, that these authors invalidated other crater chronologies that were state-of-the-art for previous generations of scientists.  One could hardly get better than Gene Shoemaker in the 1990s, whose views they “revised” (overturned).  At least he got out there and did experiments firing rifles at rocks to see what happened.  They also showed how their assumptions differed from the assumptions of Marchi et al.  Well, whose assumptions are better, when nobody was there to watch?  Take your pick.
    The distinct possibility arises from these considerations that Le Feuvre and Wieczorek, bless their hearts, have done nothing but manipulate numbers to create an imaginary history that doesn’t match reality.  If so, why should anybody believe a word they said?  It reduces to an exercise in impressing their colleagues with mathematics and prose in a closed mutual admiration society that has nothing to say to people who want science to talk about reality that is really real.
    If they want to claim that their exercise was worthwhile because it is the best that can be done under the circumstances, they commit the best-in-field fallacy.  How do they know that ten years from now, some young upstarts from another university won’t refer to this paper as a misguided piece of balderdash?
    Popper explained that it is easier to falsify a hypothesis than to confirm it, but that was for observable, testable things, like the effect of Einstein’s relativity on starlight during a solar eclipse.  Observations will never be able to confirm this paper’s model about an unobservable history.  It may, however, be possible to falsify their model by arguing that their assumptions are unrealistic.  It is more likely, therefore, that this model will be falsified in the future than supported.
    You may or may not agree that scientific papers about unknowables, like this one, are worthwhile exercises.  After all, we can observe craters in the present, and they got there somehow at some time.  Let us all, however, take their caution seriously: “It is difficult to quantify how each of these differences affect the final crater size-frequency distribution on a planetary object, and hence the derived ages of a surface.”  Difficult, yes, in the sense of impossible.  There are some things that science can never know.  For some of those, other sources of information are required.

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