This notebook contains additional material to the Mathematica GuideBooks  (Springer-Verlag, New York, 2004) http://www.MathematicaGuideBooks.org . It uses the stylesheet of the Mathematica GuideBooks, which is embedded.

Symbolics Volume, Chapter 1

Consider the Poynting vector |S|∼((|n×((n-v)×a)|)^2)(((1-n.v))^(-5)) from Section 2.2 of the Graphics volume. Calculate the volume of the emission cone bounded by |S|n. For a||n, find an implicit polynomial representation p(x,y,z} for the surface {x,y,z}=|S|n.

Poynting[φ_, ϑ_, v0_, α_] =
With[{(* direction *) n = {Cos[ϑ], Sin[φ] Sin[ϑ], Cos[φ] Sin[ϑ]},
      (* velocity *) v = {v0, 0, 0},
      (* acceleration *) a = {Cos[α], 0, Sin[α]}}, 
      (Cross[n, Cross[n - v, a]].Cross[n, Cross[n - v, a]])/((1 - n.v)^5)];

We obtain the volume 𝒱((Subscript[v, 0]),α) of the emission cone by carrying out the following integral:

𝒱((Subscript[v, 0]),α)∼(∫_0^π)(∫_0^(2π))(∫_0^(|S|))(r^2)sin(ϑ)dr dφ dϑ.

The r-integration is trivial and only the two angular integrations remain. Because ((|S|)^3)/3 is a rational function in cos(φ), sin(φ), cos(ϑ), and sin(ϑ), we will expand the integrand into products and integrate each term individually. For the φ-integration we have 438 terms to integrate. Because all terms are smooth functions of φ and ϑ, we calculate the indefinite integral and substitute limits. This is faster than using Integrate to calculate the definite integrals directly.

ℒ = Expand[Poynting[φ, ϑ, v0, α]^3/3];
Length[ℒ]

IndefIntφ = Integrate[#, φ]& /@ ℒ;
defIntφ = (IndefIntφ /. φ -> 2Pi) - (IndefIntφ /. φ -> 0);
Length[defIntφ]

We have 95 terms to integrate for the ϑ-integration. The resulting volume 𝒱((Subscript[v, 0]),α) is a relatively complicated function of (Subscript[v, 0]) and α.

IndefIntϑ = Integrate[# Sin[ϑ], ϑ]& /@ defIntφ;
defIntϑ = (IndefIntϑ /. ϑ -> Pi) - (IndefIntϑ /. ϑ -> 0);
emissionVolume[v0_, α_] = Factor[Together[defIntϑ]]

For (Subscript[v, 0])->1, 𝒱((Subscript[v, 0]),α) diverges as (((1-(Subscript[v, 0])))^11).

Series[emissionVolume[v0, α], {v0, 1, -9}] // Simplify

Here is a plot of the normalized 𝒱((Subscript[v, 0]),α)/𝒱((Subscript[v, 0]),0) as a function of (Subscript[v, 0]) and α.

Plot3D[emissionVolume[v0, α]/emissionVolume[v0, 0], 
       {v0, -0.8, 0.8}, {α, -Pi, Pi}, PlotRange -> All];

Now, let us derive the implicit form of |S|n for a||n. These are the three parametrized equations for x(ϑ), y(ϑ), and z(ϑ) (because of the rotational symmetry of the problem around the x-axis, we can use φ=0).

eqs = Numerator[Together[{x, y, z} - 
        Poynting[φ, ϑ, v0, 0] {Cos[ϑ], Sin[φ] Sin[ϑ], Cos[φ] Sin[ϑ]}]] /. φ -> 0

We convert the last equation into polynomial form by using the exponential representation for the trigonometric functions.

eqs1 = Numerator[Factor[Together[TrigToExp[eqs]] /. 
                 Exp[s_] :> (θ^(Coefficient[s, ϑ]/I))]]

Eliminating the parametrization variable θ yields an implicit polynomial of total degree 10 in x and z.

Resultant[eqs1[[1]], eqs1[[3]], θ, Method -> Modular] // Factor

Using the rotational symmetry, we have the following result for the surface of the emission cone.

PoyntingImplicit[{x_, y_, z_}, v0_] = Simplify[%[[-1]] /. z -> Sqrt[y^2 + z^2]]

Substituting the original expression for the Poynting vector in the last result shows its correctness.

Together[TrigToExp[PoyntingImplicit[Poynting[φ, ϑ, v0, 0]*
             {Cos[ϑ], Sin[φ] Sin[ϑ], Cos[φ] Sin[ϑ]}, v0]]]

We end with a set of contour plots of |S|n in the x,z-plane.

Show[{#, (* construct the lower half *)
      # /. Line[l_] :> Line[{1, -1}# & /@ l]}&[ 
     Table[Graphics @ 
            ContourPlot[Evaluate[PoyntingImplicit[{x, 0, z}, v0]],
              {x, -3/2, 3}, {z, 0, 3}, PlotPoints -> 160,
              Contours -> {0}, ContourShading -> False,
              ContourStyle -> {{Thickness[0.002], Hue[v0]}},
              DisplayFunction -> Identity, FrameTicks -> False],
      (* various velocity/c values *) {v0, 0, 0.5, 0.5/20}]], 
      DisplayFunction -> $DisplayFunction]