Continuing to constrain craters on Titan.

This will compose of two parts. First, an update of where I am at since the last post. Second, I’ll try to give a rough outline of where I hope to go over the span of the next few months.

Screenshot at Jun 22 17-04-06
Figure 1: An hypothetical crater on Titan with a diameter (D), and a distance of the profile D_p from the left rim to the right rim.

In the first figure, we see a hypothetical crater. For a perfect circle, if we know the distance from the center to the center of the profile (from rim to rim), we have a right triangle, mirrored on the other side. The radius is r=sqrt(d_c^2+(D_p/2)^2). I use this to plot where previous literature suggests that the crater rim is.

If we find the rim using the topography, the rim won’t always aline perfectly with the center, but with only 2 points we can can’t infer where the center should be. So in the event of a single profile, we have to average the radius using the distance from the center to the rim. Obviously, r_l and r_r probably won’t be equal. The easiest estimation is to average that distance. In Figure 2 I suggest another technique.

Screenshot at Jun 22 17-02-52
The two rims are hypothesized to be at the two solid green notch in the red line. Using the same logic as was used to find the rim based on past literature can be used by assuming the profile is aligned with the assigned center so that r_l and r_r would be the same, i.e. can be found using the pythagorean theorem.

In reality, the rims won’t match up to a perfect circle, but with only two values there isn’t much that can be done to approximate a good radius. I use the same idea that r=sqrt(d_c^2+(D_p/2)^2). I don’t actually shift the profile over, but by using the d_c and this equation, it assumes the two rims are centered around the given center. I suppose I could go a step further then find where the center would be for a circle of that radius using these two points. I didn’t do it in the Soi example (Figures 3 and 4), but I think I’ll be doing that because the center has to match up some how? But I feel like I don’t have much justification for it…so should I?

Figure 3: Rims (red), center (between rims, green), and the ‘+’ represent the range of the profile I tell my code to search for the rim (the max height) based on past literature and the radar images.

So in Figure 3 we can see what I was talking about applied to a single profile. The center of the profile is found by finding the closest point to the center (as defined by literature). Then I search for the rims using the max height over a defined profile range (based on past literature and the radar images). Soi is perplexing. Despite being perhaps the freshest crater its topography is absurdly not symmetrical. This upended what I was doing with Menrva and required I take a different approach. That was why I decided to have users look at this profile and input were to search for the rim.

Figure 4: Sartopo data in Figure 3 shown overlain a radar map of the surface.

I translate the data in Figure 4 to a radar view to see where the data is relative to the radar imagery. The red circle uses the center as defined by my the center point in past literature and the radius I solved for. Like I said before I could shift that to a center that matches the rim.

Screenshot at Jun 23 09-53-45
Figure 4.5: The results of the Soi approximation. The crater is found to be 83.15 (km) vs the 78 (km) in literature. The center is the same is literature.

Now, if we move onto another example, Menrva (Figures 5 and 6) we have multiple profiles to estimate the circle.

Figure 5: Menrva, see figure 3.

Here, I can ascertain a radius and center by creating a best fit circle to the data. Luckily, such a function already exists on the net. There are three beautiful profiles in the lower part of the crater (30123, 30124, 30134). These are very close though so the fit using these profiles produces a fairly large crater, more than 425km and outside what we see in the radar image. Now if we include the top profile, where only a few gaps exist, a much nicer fit occurs.

Figure 6: Menrva, see figure 4. The red circle is fit to the results in Figure 6.5.

The rims are in the same position as the others. It fits well with the radar image. With the bottom profile, not in Figure 5, the crater is shrank to below 380km and doesn’t fit the radar image well. I choose to stick to the top four profiles to get what we see in Figure 6. We get 395(km) using the top four.

Screenshot at Jun 23 09-58-07
Figure 6.5: Results from the Menrva approximation. The crater radius is smaller than previously though at 395.31(km) vs the 425 (km) in literature. The center is 20.1 Lat and 87.2 Long vs 19.6 Lat and 87 Long, very close to literature. The same range to search for the rim is used in each run.

The red circles I plot in figures 4 and 6 use the rectangle function with curvature. If we want to consider rectangular shapes there are probably approximations or fits I could do to rectangular shapes. I would need some feedback as to when and how to guide that.

In regards to crater shapes, I had hoped to read Catherines 2013 paper on crater topography by the end of this week but got lost in matlab. Moving forward I plan to read this next because I feel I need to understand how to identify where to look for rims. Because the basic shape of a rim and crater floor is not what we see here. I like what I have made above and hope to work from it moving forward, barring much pushback from Catherine or Mike.

I also need to estimate roughness. I have a few ideas on how best to do this. I could do basic statistics on the center of the crater, and I could do roughness parameters similar to what Kevin is doing. I’ll probably do some reading to figure out what past studies have done to characterize crater erosion/roughness and work with that. Depending on how time consuming that is, I’d like to finish up with working with Matlab in mid to lat July.

Finish work with Matlab and crater characterization (July 31st)

  • June 23rd-June 30th: Read about crater topography and crater erosion while continuing work with plot on rim and diameter estimations
  • July 1st-July 14th: Move on to reading about crater roughness as it relates to erosion literature review
    • Also 4th of July and begin prepare/review my Astronomy night presentation.
  • July 15th-July 21th: Apply what I learned about crater roughness to Matlab code.
  • July 22nd-July 28th: Finalize Matlab code and output figures.

Map Craters on Titan (August 31st)

  • July 29th-August 4th: Finish Titan radar map
    • I’m a little worried this may be more difficult than I expect. Will Mike be around? Does he know ISIS? I think I should be okay but still.
  • August 5th-August 31st: Map the craters on Titan
    • It’s difficult to asses how long this will take. I’ll probably start mapping while periodically updating you all getting feedback
    • I’ve reached out to Zibi again to try and make some progress with her. I’d like to meet with her a little in July, maybe early August (virtually). My goal with that is to have the code accepting the default by early August. (Consider reaching out to Mike or others?)

Finalize crater work and dig deeper into tekton again (October 31st)

  • 11th- September 22nd: Use the crater map to estimate crater parameters for all the craters (with data).

    • Begin with the easiest craters. In the last week, look at the hardest to try and estimate that data as well.
      • A large table will review all the results. As to what craters to highlight with more detailed figures will be decided later with feedback. These are the figures I would include in a paper/thesis.
    • Meet with Zibi again to ensure I understand how to work with input files in Tekton.
    • Prepare for Titan meeting and DPS.
    • Also begin having to work on the Impact Craters course (23rd to 30th)

Then meetings dominate October.

One Reply to “Continuing to constrain craters on Titan.”

  1. Hi Josh,

    You should not assume that r_left and r_right are the same, since many craters on Titan are not perfectly circular (e.g. Selk). So, you have two equations:

    r_left = sqrt(d_c^2 + (D_p/x)^2)
    r_right = sqrt(d_c^2 + (D_p/y)^2)

    The variables x and y are not exactly two unless the two radii are the same. So, we have two equations with five unknowns: r_l, r_r, d_c, x, and y. You could try a minimization process to get the ‘best guess’ for these values. Start with reasonable assumptions and see which combination produces the smallest chi-squared value when compared to the known value of D_p.

    D_p = x * sqrt(r_left^2 – d_c^2) = y * sqrt(r_right^2 – d_c^2)

    If, instead, you assume the two radii are the same, you have one equation:

    r = sqrt(d_c^2 + (D_p/2)^2)

    So, you have one equation and two unknowns: r and d_c. You could also do a minimization.

    I’m worried that your current method assumes too much. You have to guess at the crater centre (two variables) and assume that the crater is a perfect circle (when in fact it is not). I feel like there’s got to be a better way to do this.

    Also, don’t use Soi as your first example. It is barely visible in the SARTopo data. And besides, we have a stereo data set that would be much better for determining depth and diameter. It is not fresh – it’s one of the most degraded craters on Titan. Try Sinlap or Selk instead.

    Can you tell me more about the best fit circle function you’re using? I’m worried that it relies too much on the shape of the crater in Cassini RADAR data. It can be quite difficult to accurately identify the rim in that data set – you really need the topography information.

    I’m not sure what you mean by estimating roughness, but I wouldn’t put that at the top of your priority list. You should, however, think about ways to include uncertainty and error bars in your analysis. You should also develop your own method for calculating crater depths, based on the technique outlined in my 2013 paper.

    Best, Catherine


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