Sunday, April 24, 2022

Closing The Loop

In my last post I showed that there was a lunar seismic event on May 29th, 1972 that seems likely to have been caused by the impact of the Apollo 16 Lunar Module Orion. Knowing the time of this event is an extremely valuable clue to help locate the point of impact. I showed in an earlier post that the impact locations of a randomized set of simulations varied with time, shifting gradually westward on the Moon for later impact times. If we have a target impact time, we can run more trials, and look for impacts at the right time. Then we can check the area where these occur…and hopefully the area is small enough to make a visual search practical. 

There is a problem with this "shotgun" strategy though. If we generate the trials randomly the impact times with be spread out over days. Only a very small fraction of the runs will happen to hit the Moon around the desired time. Even though each simulation completes in just 8 minutes, it would take thousands of trials to build up a good database of timely impacts. We need a better way.

In looking at the data from the first trials, there is a pattern which might offer a way forward. Each simulation starts with six numbers that represent the initial state of Orion’s orbit. The numbers represent the position (latitude, longitude, and altitude) and velocity (speed and direction) of the spacecraft when it was jettisoned. To generate a set of simulations, each of these six parameters is perturbed randomly. The hope is that the random variations will make up for any errors of precision Orion’s initial state. Hopefully all the variation covers the true initial state. In looking over the six input variables, and comparing to the results, an interesting pattern emerges. 

Figure 1: Impact Date versus initial VMAG

Figure 1 is a plot of the speed parameter, VMAG, versus the resulting simulated impact date. Each dot in the figure represents one simulation run.  What is interesting is the trend in the data, as summarized by the dotted trend line. Simulations resulting in an earlier impact had, on average, smaller VMAG values, and those resulting in later impact had larger VMAG values. According to the trend line, on average a change in VMAG of just 0.000694 km/sec resulted in a one-day change in the impact. That’s 69 cm/sec, for a parameter that is in the range of 1.6 km/second. So a very slight nudge, less that 0.05%, leads to in a one-day shift in the impact date.

Therefore, the strategy is to first run a randomized set of simulations. Then calculate the number of days that the resulting impact “missed” the desired target impact time. Multiply this error by 69 cm/second, and add this “nudge” factor to the initial VMAG value, to generate a new VMAG value. Then re-run the simulation with the new VMAG value. By using the outcome to feed back into the initial conditions, we are "closing the loop". Let's give it a try. 

Figure 2: Impacts converging on the desired time after several rounds of VMAG nudging

Figure 2 shows the results using this technique. On the left side, after the initial run, the impact times range from 2 days early to 4 days late compared to the target time. For each case, I calculate the “nudge” value, update VMAG, and run the simulations again. After one round of this nudging, the results are much more concentrated around the target date. Everything is within +/-1 day of the target. Now I can repeat the process a second time, again adjusting the VMAG values by ~69 cm/sec per day of error. Since the errors are fractions of a day, the nudges are proportionally smaller. After two rounds, I have a large set of simulations which are impacting around the desired time.’s working well!

Isn’t this cheating? We started by generating random variations, but now we are selectively adjusting one of those values. VMAG is no longer random. That is true, but there is still value in this technique. Take a look at the progression of VMAG values in the trials, shown in Figure 3. (These are sorted from lowest to highest initial values.) The values for VMAG vary randomly with a total of 6 meters/sec variation around the nominal value. (This is a very generous variation, given that NASA said in 1972 that the doppler tracking enabled them to measure the spacecraft speed to within 0.5 feet/sec.) After two rounds of nudging, the variation in VMAG is less than 3 meters/sec. Focusing on the desired impact date has compressed the variation of VMAG, but we are still testing a generous set of its possible values. And we still have fully random variation of the other five parameters.

Figure 3: Initial VMAG values (blue) and final values (red) after nudging

OK, we are able to focus the impact times, but what about the impact locations? As hoped, as the impacts begin to cluster more tightly around the desired time on May 29th, they also begin to cluster more tightly on the surface of the Moon. As I showed in a previous post, initially the impacts are spread across a wide band of longitudes, from 62° E to 126° E. That band is over 1000 miles wide! Figure 4 shows the results after nudging towards the target impact time. The impacts are now clustered around a few prominent terrain features near 10° N and 105° E. Although this is still quite a large area, we are making progress. The total search area is greatly reduced, especially since the impacts are concentrated primarily along crater rims. I'm beginning to have hope that Orion's final impact crater might be found. 

Figure 4: Locations for simulated impacts that occur near the time of the May 29th seismic event

Perhaps there are other clues we can use to further constrain the impact time. Meanwhile, happy hunting!

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