Estimating the fraction of ejecta travelling between velocity limits

  1. 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 travelling 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!
  2. jcsd
  3. mfb

    Staff: Mentor

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

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

    Sorry, It's the speed distribution I'm interested in. And the ejecta is travelling 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 travelling 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.
  5. mfb

    Staff: Mentor

    Well, you need some model for the velocity or speed distribution.
    Depending on the distribution, probably yes.

    What is unclear about "not having a min or max value"?

    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)?
  6. 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.
  7. mfb

    Staff: Mentor

    Yes, and integrate between 11.2 and 15 to get the normalization right.

    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.
  8. 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!
  9. mfb

    Staff: Mentor

  10. Perfect, thanks a lot for your help.
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