Davis to the Rescue

I was at the cosmology seminar today and I was complaining to Nic Ross about my correlation function woos and Marc Davis happened to be in earshot and told me that I was thinking about this all wrong. Basically what I should do is take my galaxies with coordinates of Ra and Dec and calculate their x,y,z coordinates (projecting them onto a unit sphere). Then I can calculate the 3D correlation function on these objects (which I already have code to do), where I can translate between their physical separation in x, y, z coordinates to their angular separation by the following:

Ra₁, Dec₁ → x₁, y₁, z₁
Ra₂, Dec₂ → x₂, y₂, z₂

Separation of objects (s) in Cartesian coordinates:

s = [(x₁ - x₂)² + (y₁ - y₂)² + (z₁ - z₂)²]^½

Geometry relates the separation angle (γ) to Cartesian separation (s):

s = 2sin(γ/2)

or using the dot product of the two vectors:

x₁x₂ + y₁y₂ + z₁z₂ = cos( γ)

Therefore, the only change I need to make is after I grid objects, and select which objects are 'close by' using above conversion from s to γ, is to convert the 3D separation of the objects to the angular separation. This shouldn't be very difficult to do using the 3D code I already have.

Thank you Marc Davis for making my life a lot better today!

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