How is it that the SPH cubic spline kernel in normalized?

[ElPimiento]

Hi,

(This is more of a math question but I thought Astronomy people would be more familiar with the equation and how it's used)
So in Monaghan 1992 (http://adsabs.harvard.edu/abs/1992ARA&A..30..543M, bottom of pg 554) a cubic spline in three dimensions is defined. I tried to integrate it (using a triple integral in spherical coordinates), expecting to get 1 but got $3\pi/4h^2$. Are there special considerations that need to be taken into account when your integrating this function? Here's my work:

(I tried to make the writing easier to see by fiddling with the contrast, it kind-of worked)
(Also, I'm an Undergrad doing research so I didn't know what level to make the thread ... I or A ... )

Thanks,
- Andrew

 

[AndyW]

Hi Andrew,

Thank you for your question! As an astronomer, I am familiar with the use of equations and integrals in our field of study. In response to your question, there are a few things to consider when integrating the cubic spline function in three dimensions.

Firstly, it is important to note that the cubic spline function is a piecewise polynomial function that is used to interpolate data. Therefore, when integrating it, you are essentially finding the area under the curve within a given range of values. This means that the result of the integration will depend on the specific range of values that you choose.

Secondly, the cubic spline function is defined in terms of a smoothing parameter, h, which determines the smoothness of the curve. This parameter can affect the result of the integration, as you have observed in your calculation.

Lastly, it is also important to double check your integration method and make sure it is appropriate for the function you are integrating. In this case, using a triple integral in spherical coordinates may not be the most suitable method, as the cubic spline function is defined in Cartesian coordinates.

I hope this helps clarify some of the considerations that need to be taken into account when integrating the cubic spline function in three dimensions. It's great to see you exploring this topic as an undergraduate researcher. Keep up the good work!