Calculations

Integrate[
      Pi ((x0 + Sqrt[r^2 - y^2])^2 - (x0 - 
                  Sqrt[r^2 - y^2])^2), {y, -r, r}]

Pi ((x0 + Sqrt[r^2 - y^2])^2 - (x0 - Sqrt[r^2 - y^2])^2) // 
      Simplify

Integrate[
      4 Pi x Sqrt[r^2 - (x - x0)^2], {x, x0 - r, x0 + r}]

f[x_] = Sqrt[r^2 - x^2]

4 Integrate[Sqrt[1 + f'[x]^2], {x, 0, r}]

Solve[x^2/a^2 + y^2/b^2 == 1, y] // FullSimplify

f[x_] = b/a Sqrt[a^2 - x^2]

1 + f'[x]^2 // Together

Sqrt[1 + f'[x]^2] // PowerExpand

4 Integrate[Sqrt[1 + f'[x]^2], {x, 0, a}]

\(x[t_] = a\ Cos[t]; 
y[t_] = b\ Sin[t]; \)

Integrate[Sqrt[x'[t]^2 + y'[t]^2], {t, 0, Pi/2}]

s1 = 
      8 Integrate[
          Sqrt[r^2 - (x - x0)^2] 
            Integrate[
              Sqrt[1 + (e^2 - 1) t^2/(t^2 - x^2)], {t, 0, x}], {x, 
            x0 - r, x0 + r}]

s1 = 
      8 Integrate[
          Sqrt[r^2 - (x - x0)^2] 
            Integrate[
              Sqrt[1 + (e^2 - 1) t^2/(t^2 - x^2)], {t, 0, x}], {x, 
            x0 - r, x0 + r}, 
          Assumptions 
            -> {0 < e < 1, Im[r] == 0, Im[x0] == 0, Re[x0] > Re[r], 
              r > 0, x0 ≥ 0}]

s1 = 
      8 Integrate[
          Sqrt[r^2 - (x - x0)^2] 
            Integrate[
              Sqrt[1 + (e^2 - 1) t^2/(t^2 - x^2)], {t, 0, x}], {x, 
            x0 - r, x0 + r}, Assumptions -> {0 < e < 1, r > 0, x0 > r}]

Series[EllipticE[z], {z, 0, 8}] // Factor

Series[EllipticE[e^2], {e, 0, 8}] // Factor

Integrate[Sqrt[1 + (e^2 - 1) x^2/(x^2 - r^2)], {x, 0, r}, 
      Assumptions -> {0 < e < 1, r > 0}]

1 + b^2 x^2/(a^2 (a^2 - x^2)) /. {a -> r, 
          b -> r Sqrt[1 - e^2]} // Simplify

Graphics

<< Graphics`Shapes`

locate[x0_, theta_, r_] := 
      TranslateShape[
        Graphics3D[Sphere[r, 10, 10]], {x0 Cos[theta], x0 Sin[theta], 0}]

locate[x0_, theta_, r_] := 
      TranslateShape[
        Graphics3D[Sphere[r, 8, 8]], {x0 Cos[theta], x0 Sin[theta], 0}]

p1 = 
      Show[{Graphics3D[Sphere[5, 15, 15]], 
          Table[locate[20, theta, 1.25], {theta, 0, 2  Pi, 2  Pi/59}]}, 
        Boxed -> False]

Do[Show[{Graphics3D[Sphere[5, 15, 15]], 
          Table[locate[20, theta, 1.25], {theta, 0, t 2  Pi/99, 2  Pi/99}]}, 
        Boxed -> False, 
        PlotRange -> {{-22, 22}, {-22, 22}, {-5, 5}}], {t, 1, 
        99}]

p2 = 
      Show[{Graphics3D[Sphere[5, 15, 15]], 
          Graphics3D[Torus[20, 1.25, 40, 15]]}, 
        Boxed -> False]

Show[GraphicsArray[{{p1, p2}}]]

<< Graphics`Shapes`

locateb[x0_, y0_, theta_, r_] := 
      TranslateShape[
        Graphics3D[Sphere[r, 10, 10]], {x0 Cos[theta], y0 Sin[theta], 
          0}]

p1 = 
      Show[{Graphics3D[Sphere[5, 20, 20]], 
          Table[locateb[30, 20, theta, 1.25], {theta, 0, 2  Pi, 2  Pi/59}]}, 
        Boxed -> False]

Do[Show[{Graphics3D[Sphere[5, 20, 20]], 
          Table[locateb[30, 20, theta, 1.25], {theta, 0, t 2  Pi/99, 
              2  Pi/99}]}, Boxed -> False, 
        PlotRange -> {{-38, 38}, {-23, 23}, {-5, 5}}], {t, 1, 
        99}]

Rotating Planets

A Graphics Array

anim1 = 
      Table[Show[{Graphics3D[Sphere[5, 15, 15]], locate[10, Pi - theta, 1.5], 
            locate[20, theta, 1.25]}, Boxed -> False, 
          PlotRange -> {{-25, 25}, {-25, 25}, {-25, 25}}, 
          DisplayFunction -> Identity], {theta, 0, 2  Pi, 
          2  Pi/15}]

Show[GraphicsArray[Partition[anim1, 4]]]

An Animation

Do[Show[{Graphics3D[Sphere[5, 15, 15]], locate[10, Pi + theta, 1.5], 
          locate[20, 2  theta, 1.25]}, Boxed -> False, 
        PlotRange -> {{-25, 25}, {-25, 25}, {-25, 
              25}}], {theta, 0, 6  Pi, 6  Pi/179}]

All Together

p1 = 
      Show[{Graphics3D[Sphere[5, 15, 15]], 
          Table[locate[20, theta, 1.25], {theta, 0, 2  Pi, 2  Pi/59}]}, 
        Boxed -> False]

p2 = 
      Show[{Graphics3D[Sphere[5, 15, 15]], 
          Graphics3D[Torus[20, 1.25, 40, 15]]}, 
        Boxed -> False]

Show[GraphicsArray[{{p1, p2}}]]

Elliptical Orbit

<< Graphics`Shapes`

locateb[x0_, y0_, theta_, r_] := 
      TranslateShape[
        Graphics3D[Sphere[r, 10, 10]], {x0 Cos[theta], y0 Sin[theta], 
          0}]

All Together

p1 = 
      Show[{Graphics3D[Sphere[5, 20, 20]], 
          Table[locateb[30, 20, theta, 1.25], {theta, 0, 2  Pi, 2  Pi/59}]}, 
        Boxed -> False]

A Graphics Array

anim2 = 
      Table[Show[{Graphics3D[Sphere[5, 20, 20]], 
            locateb[20, 10, Pi - theta, 2], locateb[25, 20, theta, 1.25]}, 
          Boxed -> False, 
          PlotRange -> {{-30, 30}, {-30, 30}, {-30, 30}}, 
          DisplayFunction -> Identity], {theta, 0, 2  Pi, 
          2  Pi/15}]

Show[GraphicsArray[Partition[anim2, 4]]]

An Animation

Do[Show[{Graphics3D[Sphere[5, 20, 20]], 
          locateb[20, 10, Pi + 2  theta, 2], locateb[25, 20, theta, 1.25]}, 
        Boxed -> False, 
        PlotRange -> {{-30, 30}, {-30, 30}, {-30, 
              30}}], {theta, 0, 6  Pi, 6  Pi/179}]