Appendd

Final Report Appendix A Appendix B Appendix C

Appendix D - Detailed Models

D.1 Phase 1 Pre-collision.

D.1.1 Meteor Volume. In determining the volume of the meteor, a sphere was chosen as the shape of the meteor. This provides an adequate model which only depends on one parameter (radius). Typical values of radius are easy to find. The following is the model for a volume of a sphere:

where

V - volume of sphere (m³)

r₀ - initial radius of meteor (m)

This model for the volume of a sphere came from reference [1].

D.1.2 Meteor Mass. To find meteor mass, you need to the density of the meteor. There are three common densities of meteors. They are:

Rock - 3028.1 kg/m³

Iron-rock - 5440.0 kg/m³

Iron-nickel - 7580.0 kg/m³

The following is the model for the mass of a object:

where

m - mass of the object (kg)

V - volume of the object (m³)

d - density of the object (kg/m³)

This model came from reference [1].

D.1.3 Meteor Initial Kinetic Energy. Modeling the kinetic energy transferred from the meteor into a surface wave is the overall goal of Phase 1. By modeling the kinetic energy of the meteor before it enters the Earth's atmosphere and correcting for atmospheric losses, you can model the meteor's kinetic energy just before impact. This involves first finding the initial mass and velocity of the meteor. The following is the model for the kinetic energy of an object:

where

K₀ - initial kinetic energy of the object (joules)

m - mass of the object (kg)

v - velocity of the object (m/sec)

This model came from reference [1].

Common velocities of meteors are usually between 10000 and 30000 m/sec [5].

D.1.4 Atmospheric Energy Losses. The following is the model for kinetic energy losses due to the Earth's atmosphere for a meteor with a radius of over 200 m:

where

K_i - Kinetic energy at time of collision (joules)

K₀ - Kinetic energy of meteor before hitting the Earth's atmosphere (joules)

This model came from reference [5].

D.1.5 Inelastic Collision. An inelastic collision is a collision in which the colliding bodies do not rebound. An elastic collision is a collision in which the two bodies rebound.

An inelastic collision was chosen as the model for the meteor-Earth collision. This is the most likely collision result and produces a useful model. The meteor enters the ocean and the energy is totally transferred on impact.

Meteor kinetic energies are expected to be very large. Typically these energies are reported in the energy equivalent of Giga-tons of TNT.

The following is a model for the energy transfer from the meteor to the ocean in a totally inelastic collision:

where

K_c - Kinetic energy transferred from the meteor to the ocean (Giga-tons)

g - 4.25 x 10¹⁸ (joules/Giga-ton)

K_i - Kinetic energy of meteor right before collision (joules)

This model came from reference [1].

D.2 Phase 2 Collision.

D.2.1 Distance from the Impact Site to the Shore. The Haversine model takes two locations on a sphere and models the length of the arc between them. For Phase 2, this model was used to model the distance between the impact location and the shore. The following is the Haversine model:

dlon = lon2-lon1

dlat = lat2-lat1

a = (sin(dlat/2))^2 + cos(lat1) * cos(lat2) * (sin(dlon/2))^2

c = 2 * atan2(sqrt(a), sqrt(1-a))

where

lon2 - longitude of the shore (rad)

lon1 - longitude of the impact site (rad)

lat2 - latitude of the shore (rad)

lat1 - latitude of the impact site (rad)

a - intermediate number

c- distance between the two points (km)

This model came from reference [8].

D.2.2 Ocean Depth. Ocean depth is an important part of determining the height of a tsunami and determines which of the two models for tsunami propagation is used for a certain meteor. On the internet, reference [6] had a file called Etopo5 that had the depth of the ocean or elevation of the land at every five minutes of latitude and longitude on the Earth. The ocean depth is important in the model for wave propagation.

D.2.3 Wave Propagation. Wave propagation is important to this project. Wave propagation is the progression or regression of the height of a wave over the open ocean. There are two models for wave propagation. One is for deep water, or when the depth of the ocean at the impact site is more than twelve times the meteor diameter. This is the model for this impact:

where

h_w - peak-to-peak amplitude of surface wave (m)

K_i - meteor kinetic energy at impact (Giga-tons of TNT)

R - distance of shore from impact site (km)

If the ocean is not twelve times as deep as the diameter of the meteor, the collision is considered a shallow water impact. The model for that type of impact is as follows:

where

h_w - peak-to-peak amplitude of surface wave (m)

d - local ocean depth (m)

R - distance of shore from impact site (km)

K_i - kinetic energy of meteor at impact (Giga-tons of TNT)

This model came from reference [7].

D.2.4 Peak Amplitude of the Surface Wave. Peak amplitude of the surface wave is the height of the wave above sea level. The following is the model for the peak amplitude of the surface wave:

where

h - peak amplitude of surface wave (m)

h_w - peak-to-peak amplitude of surface wave (m)

This model came from reference [7].

D.3 Phase 3 Post-Collision.

D.3.1 Shoreline Depth. The shoreline depth of the tsunami is needed to determine the maximum distance inland the tsunami travels. The model needs the tsunami runup factor of the shore, which is how fast a wave rises in height after reaching the shore. The runup factor is usually 10. The following is the model for determining the shoreline depth of the tsunami:

where

h_s - shoreline depth (m)

t_r - tsunami runup factor

h - peak amplitude of the surface wave at shore (m)

This model came from reference [7].

D.3.2 Maximum Tsunami Surge Inland. The next model needed was the model to determine the maximum tsunami surge inland. The only input to this model is the shoreline height of the tsunami. The following is the model for the maximum tsunami surge inland:

where

X_m - maximum distance inland the water can surge (km)

h_s - shoreline depth of tsunami (m)

This model came from reference [7].

D.3.3 Distance between Shore and City. The Haversine model takes two locations on a sphere and models the length of the arc between them. For Phase 3 this model determines the distance between the shore and the city. The following is the Haversine model:

dlon = lon3-lon2

dlat = lat3-lat2

a = (sin(dlat/2))^2 + cos(lat3) * cos(lat2) * (sin(dlon/2))^2

c = 2 * atan2(sqrt(a), sqrt(1-a))

where

lon2 - longitude of the shore (rad)

lon3 - longitude of the city (rad)

lat2 - latitude of the shore (rad)

lat3 - latitude of the city (rad)

a - intermediate number

c- distance between the two points (km)

This model came from reference [8].

D.3.4 Flood Depth in City. The fourth model needed for this phase was the model for the height of the tsunami at the city.. The shoreline depth, the distance of the city from the shore, and the maximum tsunami surge inland are inputs for this model.. The model is defined as follows:

where

h_f - flood depth of tsunami at the city (m)

h_s - shoreline depth of the tsunami (m)

X - distance of the city inland (km)

X_m - maximum distance inland tsunami will surge (km)

This model came from reference [7].