Estimating the fraction of ejecta travelling between velocity limits

In summary, the conversation discusses estimating the fraction of spalled impact ejecta that occupies a given velocity range. The individual asking the question has a reasonable estimate for the total number of particles ejected, and a minimum and maximum velocity. They assume a Gaussian distribution with a mean of 13.1 and a variance of 1. They are unsure how to integrate this to estimate the number of particles traveling between 11.7 and 12.7 km/s. The other individual suggests using a Gaussian distribution restricted to the range of 11.2 to 15 km/s and integrating between 11.7 and 12.7 to obtain the desired fraction. They also discuss the possibility of using a top hat distribution, but note that a
  • #1
deltapants
5
0
First of all,

THIS IS NOT HOMEWORK. It's related to my research.

And forgive me if this is rather elementary (sadly, I was something of an underachiever at school, which has left some gaps in my maths education that I've been working on since I returned to education) but I have a question about estimating the fraction of spalled impact ejecta that occupies a given velocity range. I have a reasonable estimate for the total number of particles ejected, and a minimum and maximum velocity (11.2 and 15).

I'm assuming that the ejecta follows a Gaussian distribution, and I'm assuming a variance of 1 and a mean of 13.1.

My question is - how would I actually integrate this, so I can estimate how many particles are traveling between 11.7 and 12.7 km/s?

I've been approaching the problem conceptually like I might with a QM problem, by considering the function as a probability distribution such that the integral between -∞ and +∞ = 1, except in this case it's between 11.2 and 15 as my limits. Does this make sense? How would I then go about integrating between the 11.7 and 12.7 limits? Do I set 11.2 = 0 and 15 = 1 or something?

Again, sorry if this is all very elementary, but some guidance would be appreciated!
 
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  • #2
Ejecta where?
Gaussian distribution for the velocity in 3D, or for the speed?

A Gaussian distribution has no minimal and maximal values.
 
  • #3
Hi, thanks for getting back to me.

Sorry, It's the speed distribution I'm interested in. And the ejecta is traveling through a vacuum.

I know the lower limit of the speed is 11.2 km/s, and the upper is 15. And I want to estimate what fraction of the total is traveling betweem around 11.7 and 12.7 km/s.

Is this easily doable? What do you mean the Gaussian doesn't have a min and max value? Does that mean I'm wrong to assume a Gaussian/normal distribution here?

Thanks in advance.
 
  • #4
I know the lower limit of the speed is 11.2 km/s, and the upper is 15. And I want to estimate what fraction of the total is traveling betweem around 11.7 and 12.7 km/s.
Well, you need some model for the velocity or speed distribution.
Is this easily doable?
Depending on the distribution, probably yes.

What do you mean the Gaussian doesn't have a min and max value?
What is unclear about "not having a min or max value"?

Does that mean I'm wrong to assume a Gaussian/normal distribution here?
Certainly. You could use a Gaussian distribution restricted to some range (like 11.2 to 15km/s). But then you still have to make clear what is distributed like that. The speed or the velocity (if yes, in which way)?
 
  • #5
It's just the speed I need to be concerned with.

So, would I restrict the Gaussian as you suggested, by setting 10 km/s to equal 0 while 15 equals 1. Then integrate between 11.7 and 12.7?

Would a top hat distribution be appropriate perhaps? It really is an estimation - a high degree of accuracy is not necessary.
 
  • #6
Then integrate between 11.7 and 12.7?
Yes, and integrate between 11.2 and 15 to get the normalization right.

Would a top hat distribution be appropriate perhaps?
I don't know your physical process, so I have no idea.
A symmetric speed distribution with a minimal and maximal value looks really unrealistic to me.
 
  • #7
Ahh, so integrate between 11.2 and 15 first, to get a value that is normalised to 1... THEN integrate between 11.7 and 12.7 to obtain the fraction of that value that I'm looking for?

Thanks so much for your help!
 
  • #8
deltapants said:
Ahh, so integrate between 11.2 and 15 first, to get a value that is normalised to 1... THEN integrate between 11.7 and 12.7 to obtain the fraction of that value that I'm looking for?
Right.
 
  • #9
Perfect, thanks a lot for your help.
 

What is the purpose of estimating the fraction of ejecta travelling between velocity limits?

The purpose of estimating the fraction of ejecta travelling between velocity limits is to understand the distribution of ejecta from explosive events, such as volcanic eruptions or meteorite impacts. This information can help in predicting the impact of these events and mitigating potential hazards.

How is the fraction of ejecta travelling between velocity limits calculated?

The fraction of ejecta travelling between velocity limits is calculated by dividing the volume of ejecta travelling at a specific velocity range by the total volume of ejecta produced during an explosive event. This can be done using various techniques, such as field measurements, remote sensing, or numerical simulations.

What factors can affect the accuracy of estimating the fraction of ejecta travelling between velocity limits?

There are several factors that can affect the accuracy of estimating the fraction of ejecta travelling between velocity limits. These include variations in the explosive event, such as the type and strength of the explosion, as well as environmental factors, such as wind and terrain. Additionally, the method used to estimate the fraction can also impact the accuracy.

Why is it important to estimate the fraction of ejecta travelling between velocity limits?

Estimating the fraction of ejecta travelling between velocity limits is important for understanding the potential impact of explosive events. It can help in predicting the spread of ejecta and assessing the potential hazards to human populations and infrastructure. This information can also be used to improve hazard mitigation and response strategies.

Can the fraction of ejecta travelling between velocity limits change over time?

Yes, the fraction of ejecta travelling between velocity limits can change over time. This can be due to various factors, such as changing environmental conditions, ongoing volcanic or meteorite activity, or the use of different estimation techniques. Therefore, it is important to regularly update and refine these estimates to ensure the most accurate information.

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