Fragment-Cloud Model

Overview

Implementation of the fragment-cloud model (FCM) for meteoroid atmospheric entry modeling. The theoretical model is described in detail in Wheeler et al. (2018) DOI:10.1016/j.icarus.2018.06.014

We implemented the model as a C++ extension for Python and added a number of capabilities:

• Computing the location of impact craters.
• Calculating the size of impact craters with scaling laws by Holsapple (1987) DOI:10.1016/0734-743X(87)90051-0.
• Calculating several characteristics of impact crater clusters, such as effective diameter, dispersion, aspect ratio (established in Daubar et al. (2019) DOI:10.1029/2018JE005857).
• Restricting the model to the separate fragments model (Passey and Melosh (1980) DOI:10.1016/0019-1035(80)90072-X or one of the "pancake-type" models.
• Choosing between all major "pancake-type" models: Chyba et al. (1993) DOI:10.1038/361040a0, Hills and Goda (1993) DOI:10.1086/116499, and Avramenko et al. (2014) DOI:10.1002/2013JD021028.
• Options for introducing additional randomness into the break up events (see Model section below)
• Using either the spherical or the flat planet approximation.
• Built-in utility to retrieve air density values from the Mars Climate Database Web Interface http://www-mars.lmd.jussieu.fr/mcd_python/

Installation

Prerequisites

You need to have the following packages installed:

• python3 v. 3.7 or later with numpy and the python development headers
• C++17 compiler, e.g. g++ or clang
• cmake v. 3.12 or later
• boost v. 1.70 or later, including the python3, numpy, and unit_test_framework components.

Instructions

1. Open command line and navigate to the fragment-cloud/ folder.
2. Inside the fragment-cloud/ folder, create a new folder called debug/ or release/ and cd into it.
3. Run the following command: cmake .. This will check all dependencies and create a file called setup.py inside the fragment-cloud/ folder.
4. Navigate to the fragment-cloud/ folder.

For the final step, there are a few different options:

• Install the fcm software like a pip package: python3 setup.py install. Then the fcm package will be available in your default python path.
• Only compile the C++ component: python3 setup.py build_ext --inplace. Then you have to add the fragment-cloud folder to sys.path in order to import the fcm module in python, or run scripts from within the fragment-cloud folder.
• Build the fcm package in a separate build folder: python3 setup.py build. This will create a build/ folder and place all package files in there.

Tests

Python test suite

Use pytest on the folder tests/python/.

On my machine, a signature is required for whatever reason: pytest tests/python/*_tests.py

C++ test suite

1. Navigate to an empty folder called debug/ or release/ inside the fragment-cloud/ folder.
2. Run cmake -DCMAKE_BUILD_TYPE=Debug -DBUILD_TESTS=1 .. This will instruct the compiler to use debug flags, which makes it possible to run them with a debugger.
3. cmake --build .
4. ctest

Example usage

Have a look at the scripts and the Jupyter notebook in the examples/ folder. Here is just a minimal example:

import fcm
import fcm.atmosphere as atm

# Load atmospheric density vs. elevation data
atmosphere = atm.US_standard_atmosphere()

# Default parameters
parameters = fcm.FCMparameters(g0=9.81, Rp=6371, atmospheric_density=atmosphere)

# Define a meteoroid
impactor = fcm.FragmentationMeteoroid(velocity=15, angle=45, density=3000, radius=1, strength=1e4)

# Simulate impact
results = fcm.simulate_impact(parameters, impactor, h_start=100)

# Visualise results
import matplotlib.pyplot as plt
from fcm import crater_tools

## 1. dE/dz
plt.plot(results.energy_deposition.to_numpy(), results.energy_deposition.index)
plt.xscale('log')
plt.xlabel('dE/dz (kt TNT/km)')
plt.ylabel('z (km)')
plt.show()

## 2. Crater cluster
crater_tools.plot_craters(results.craters)
plt.show()

The Model

A meteoroid is modeled as a homogenous sphere, described by the following parameters: mass , velocity , radius , strength . If the Wheeler et al. (2018) model is used, the meteoroid also has an inner structure as described in the paper, which only has an impact during break up. Before and after, it is still modeled as a homogeneous sphere.

When the meteoroid enters the planetary atmosphere at a given initial height, there are 3 distinct processes that are modeled: atmospheric descent, meteoroid break up, and impact crater formation.

Atmospheric Descent

Atmospheric descent is modeled with the standard meteoroid physics equations. They describe how the forces of drag, lift and gravity act on the meteoroid. In addition, the meteoroid ablates due to the heat of atmospheric entry, and there are some correction terms that account for the planet's curvature.

The following figures shows these forces and the coordinates that the model uses:

Lateral view (from Passey and Melosh, 1980)	Bird's eye view

is the downrange distance from the point of atmospheric entry, projected onto the planetary surface. is the height above standard 0, and expresses the deviation from the original straight downrange path after meteoroid break up. is the angle relative to the horizon at the current position and on the surface. is the trajectory angle projected on the -plane.

Using these coordinates, the meteoroid physics equations are:

• 
• 
• 
• 
• 
• 
• 

is the air density, the gravitational accelecation, are the coefficients of drag, ablation and lift respectively. is the planet's radius, and is the ground elevation above standard 0.

Break Up

When the ram pressure exceeds the meteoroid's strength, it breaks up into pieces. There are multiple ways to model the break up process. This library provides the three major ones, the separate fragments approach, the debris cloud approach, and the fragment-cloud approach, which is a combination of the two.

Separate fragments model

On break up, the meteoroid splits into a small number of fragments (some may be smaller, some larger), which then descend further into the atmosphere under separate bow shocks.

The following illustration from Passey and Melosh (1980) shows how exactly it works:

schematic	description

The transverse velocity is calculated like ( is the meteoroid density) and added perpendicularly to to both new fragments, while preserving momentum. The direction in the plane perpendicular to is chosen randomly. is a constant; (Passey and Melosh, 1980), (Artemieva and Shuvalov, 2001, DOI:10.1029/2000JE001264).

The mass ratio of the new fragments (and therefore and are set by the user as an input. Further, a random range can be specified, e.g. (50/50 to 80/20).

Finally, a new strength is calculated for the two fragments. Following the suggestion in Artemieva and Shuvalov (2001), we scale the strength like , where is a chosen from a random normal distribution around 0, with a user-defined standard deviation (default 0). is a constant, defined by the user. This relationship increases strength for smaller fragments, while also providing some randomness.

The simulation stops once the fragment would produce a crater that is too small to be detected.

Debris cloud model

The meteoroid splits into thousands of fragments which all continue descending under a common bow shock, and can therefore be simulated as a common entity, describing a cloud of small debris pieces that expands and ablates. The model provides three cloud models of this class:

The image is from McMullan and Collins (2019) DOI:10.1016/j.icarus.2019.02.013.

The simulation stops once the debris cloud has lost 99.99% of its original kinetic energy.

Fragment-cloud model

Both things happen: A few large pieces separate and form individual bow shocks, the rest is described with a debris cloud model. Furthermore, the meteoroid can be modeled as a rubble pile with an inner structure. On the first break up, these components of the inner structure, called structure groups, separate and descend under separate bow shocks.

The following figure from Wheeler et al. (2018) summarizes them:

Example of inner structure definition from Wheeler et al. (2018):

Impact crater formation

When a fragment or a debris cloud impacts the ground, scaling relations by Holsapple (1987) DOI:10.1016/0734-743X(87)90051-0 are used to calculate the crater diameter :

, where

The variables are defined as:

variable	description	variable	description
	impactor mass		terrain density
	gravitational accelecation		impactor radius
	impactor vertical velocity		impactor density
	terrain cohesive strength		rim-to-rim factor

The parameters are user-definable numbers in the Holsapple equations. The rim-to-rim factor expresses that from a top-down perspective, the crater size is measured including the rim around the crater formed by the ejecta. Defaults for two ground types are provided in the fcm module.

Crater merging

Because of the fact that all fragments impact the ground nearly simultaneously, two fragments impacting closeby typically form only one crater. Our model has a built-in thereshold below which two nearby craters are merged into one larger crater. The volume of the larger crater equals the sum of the two smaller crater volumes.

The threshold is the following: Let be the distance between the centers of two craters with radii and . The two craters are merged if

This is a weighted combination of two thresholds: means that two craters are only merged if one of their centers lays within the other crater. means that two craters are merged when they touch. The weighting of these two criteria reflects the majority of the data gathered by Daubar et al. (2019).