# Finding minimum fly-by radius between Mars and spacecraft from interpolating function

I've written an interplanetary trajectory solver/plotter that plots the path taken by a spacecraft on an Earth-Mars mission, but have run into a little trouble when the spacecraft actually reaches Mars. I'm hoping to be able to get the spacecraft into an orbit around Mars upon arrival, but before I can do that I need to find its closest approach in order to apply the correct delta-v at the right time (using the wonders of NDSolve's WhenEvent function). The trajectory plot relies on the output of an interpolating function from NDSolve, and it is this interpolating function that I am trying to work with. So far I know when the closest approach occurs to within a day, and it is between days 254 and 255 of the trip. I have put the Heliocentric x-y positions of Mars and the spacecraft (between days 254 and 255) into separate tables using the following:

MarsPosition = Table[{x[2][t], y[2][t]} /. Soln, {t, 254*86400, 255*86400, 86400/100}]
SpaceCraftPosition = Table[{x[3][t], y[3][t]} /. Soln, {t, 254*86400, 255*86400, 86400/100}]

And have then calculated their relative x-y positions using the following:

dxy = Sqrt[(MarsPosition - SpaceCraftPosition)^2]

But I am now trying to calculate the minimum approach radius (i.e. combining the x-y positions into a magnitude) from the above bits of code. My futile attempt of trying

dr = Min[Sqrt[dxy[1]^2 + dxy[2]^2]]

and other similar attempts have unfortunately ended in failure. Some sample output looks like this:

dxy = {{{2.00946*10^8, 9.69241*10^7}}, {{2.03432*10^8,
9.83663*10^7}}, {{2.05917*10^8, 9.98081*10^7}},...}

Where, for example, the first entry shows the x and y relative position as {2.00946*10^8, 9.69241*10^7} and I'm trying to combine those two components into a radius. Combining the x-y position pairs into a radius using Sqrt[x^2 + y^2] for the first 3 position pairs given above I would get

dr = {{2.231*10^8},{2.25965*10^8},{2.2883*10^8},...}

But this was done manually for each entry and is not practical for hundreds of entries. From this list of radius positions, I can then use Min[] to find the closes approach point for the spacecraft relative to Mars.

As can be seen below, the spacecraft gets very close to Mars (where the inner semi-circle is Earth's orbit, the outer semi-circle is Mars' orbit and the line joining the two is the spacecraft's trajectory):

And if propagated further we can see that it gets close enough for Mars' gravity to affect its trajectory:

So it looks pretty close, but exactly how close it what I'm hoping to find out. Any help would be greatly appreciated.

• You need to give us Soln first or all your flybys are belong to us :D PS: Look up Norm. – Yves Klett Mar 14 '14 at 12:33
• I was hoping someone would know a general way to work with tables where each entry gives an x-y position pair. I can give the code for Soln if needed, but it is a ton of code what will take some time to edit into a "stackexchange friendly" format. – InquisitiveInquirer Mar 14 '14 at 12:36
• yeah, but some data to work with would be great, otherwise (how) are we supposed to make that up? Did you already look up Norm or Nearest? – Yves Klett Mar 14 '14 at 12:38
• I would think you would be able to use Soln and FindMinimum. You could probably find the closest point with WhenEvent, too, if Soln is produced by NDSolve. Hard to tell. – Michael E2 Mar 14 '14 at 13:11
• Very true, my apologies, I've edited the original post and given some example output for the first 3 x-y position pairs from the dxy table as well as what I hope to get as a table for dr. Doing that now I realise that I should have just give a general example in the first place instead of writing out what I'm using it for. – InquisitiveInquirer Mar 14 '14 at 13:13

I guess I'll answer my own question in case someone else needs to use it for future reference. The following code was used to find the minimum approach radius:

MarsPosition = Table[{x[2][t], y[2][t]} /. Soln, {t, 253*86400, 256*86400, 60}] ;
SpaceCraftPosition = Table[{x[3][t], y[3][t]} /. Soln, {t, 253*86400, 256*86400, 60}] ;
(*xy positions for spacecraft relative to Mars*)
dxy = Sqrt[(MarsPosition - SpaceCraftPosition)^2];
(*Table used to hold approach radii*)
dr = Table[Norm[dxy[[i]]], {i, 1, Length[dxy]}]