Independent scaling of arrowheads and vectors in vector plot

Question

I'd like to scale with independent parameters the vectors and the corresponding arrowheads in a vector plot. The basic code is as follows:

vecP = 9;
disk = 1.4;
vecScale = .1;
arrScale = .8;
thickn = 0.007;

toPlot[x, y]:=(4 E^(2 Im[ArcTan[x, y]] + 2 Re[-x^2 - y^2]) Abs[x^2 + y^2])/\[Pi];

vpm1 = VectorPlot[
   toPlot[x, y] {y, -x}, {x, y} \[Element] Disk[{0, 0}, disk], 
   VectorPoints -> vecP, 
   VectorScale -> {vecScale, arrScale}, 
   VectorStyle -> {Black, Thickness[thickn]}, 
   ]

Of course the above code doesn't work, as the "arrScale" parameter assigns a fixed dimension to each arrowhead in the plot (while vecScale is actually scaling the vectors).

What I'd like to do is to scale the arrowheads as well, but with a different scaling coefficient respect to the vector scaling.

I thought of doing something like:

VectorScale -> {vecScale, arrScale * #&}

as one would do in a ColorFunction approach:

ColorFunction -> (Blend[{White,Black}, #] &)

where the slot #& is referred to the plotted function's value point by point. However, this approach doesn't work in my specific case (if someone can explain me why I'd be glad :) ).

Answer 1

I think I might have found a solution: the second option of VectorScale has to be given with the Scaled function as follows:

VectorPlot[toPlot[x, y] {y, -x}, {x, y} \[Element] Disk[{0, 0}, disk],
 VectorPoints -> vecP, 
 VectorScale -> ({vecScale, Scaled[arrScale], Automatic}), 
 VectorStyle -> {Black, Thickness[thickn]}]

This "decouples" the vectors' length and the arrowheads' dimension. (This is not exactly true, as they are still somehow coupled, but one has enough parameters to customise the arrows as he wants!)