contourlabels diagonally

Question

I have this contour plot.

I want to place contours diagonally in the center of the plot. or put custom labels where I want. how can I do it? I use this command for contourlabels. but it is not desirable.

ContourLabels -> (Text[#3, {#3, #2}, Background -> White] &)

The plot in the picture above is obtained using the function eq defined in this answer.

Answer 1

cp = ContourPlot[Evaluate@eq[Range[10]], {Ti, 1, 1000}, {Te, 1, 150}, 
  ContourStyle -> ColorData["IslandColors"] /@ Rescale[Range[10]], 
  PlotPoints -> 50]; 

contours = Cases[Normal@cp, l_Line, All];

intersections = RegionIntersection[#, Line[{{100, 0}, {1000, 150}}]] & /@ contours;

Show[cp, 
  Epilog -> MapIndexed[Text[Framed[#2[[1]], FrameStyle -> None, Background -> White, 
      ContentPadding -> False], #[[1, 1]]] &, intersections], 
  ImageSize -> 500]

Answer 2

Manual approach: make small variation of angle from main diagonal so that labels do not overlap; all numbers (300, 500, ...) are approximately taken from the figure.

SeedRandom[1234]; ang = 0.1 + RandomSample[Range[-0.04, 0.03, 0.01]];

ti = Ti /. 
FindRoot[(eq[#1] /. Te -> #2 Ti) == 
  0, {Ti, #3}] & @@@ (Transpose[{Range[1, 15, 2], 
 ang, {300, 500, 700, 800, 900, 1000, 1100, 1200}}])//Quiet

    ContourPlot[Evaluate@eq[Range[15]], {Ti, 1, 2000}, {Te, 1, 200}, 
 ContourStyle -> Black, 
 FrameStyle -> Directive[Black, Bold, 14, FontFamily -> "Times"], 
 FrameLabel -> {"\!\(\*SubscriptBox[\(T\), \(i\)]\)(keV)", 
   "\!\(\*SubscriptBox[\(T\), \(e\)]\)(keV)"}, PlotPoints -> 50, 
 LabelStyle -> Directive[Black, Bold, FontFamily -> "Times", 14], 
 Epilog -> (Text[
      Style[Framed[#1, Background -> White, FrameStyle -> None, 
        FrameMargins -> 0], 14, FontFamily -> "Times", Bold], #2] & @@@
     Transpose[{Range[1, 15, 2], Transpose[{ti, ang ti}]}])]