Estimating the fraction of ejecta travelling between velocity limits

[deltapants]

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!

 

[mfb]

Ejecta where?
Gaussian distribution for the velocity in 3D, or for the speed?

A Gaussian distribution has no minimal and maximal values.

 

[deltapants]

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.

 

[mfb]

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)?

 

[deltapants]

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.

 

[mfb]

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.

 

[deltapants]

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!

 

[mfb]

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.

 

[deltapants]

Perfect, thanks a lot for your help.