Examples of Specific Solutions

In order to calculate $z(r)$ either analytically or numerically it is convenient to define a dimensionless variable $x$ such that

$$x \equiv \Upsilon_{0}\,r \;\;\;\Longrightarrow$$ (77)

$$\frac{d}{dr} = \Upsilon_{0}\frac{d}{dx}.$$ (78)

In terms of $x$, Equation (46), that determines $z(r)$, becomes

  $$z'(x) + \frac{\eta+2x^{3}}{2x(\eta+3x-x^3)}\, z(x) = \frac{3x^{3}}{2x(\eta+3x-x^3)},$$ (79)

where the primes indicate now derivatives with respect to $x$, and where we define

$$\eta \equiv x_{2}^3-3 x_{M},$$ (80)

$$x_{1} \equiv \Upsilon_{0}\,r_{1},$$ (81)

$$x_{2} \equiv \Upsilon_{0}\,r_{2},$$ (82)

$$x_{M} \equiv \Upsilon_{0}\,r_{M}.$$ (83)

Thus $x_{1}$, $x_{2}$ and $x_{M}$ correspond respectively to the internal radius $r_{1}$, the external radius $r_{2}$ and the Schwarzschild radius $r_{M}$, expressed in terms of the new variable $x$. The solution of Equation (79) is obtained by writing Equation (52) in terms of $x$,

  $$z(x) = \sqrt{\frac{\eta+3x-x^3}{x}} \left[ \sqrt{\frac{x_{2}... ...3}{2}\, \int_{x_{2}}^{x}dy\, \frac{y^{5/2}}{(\eta+3y-y^{3})^{3/2}} \right],$$ (84)

where, in order to remain within the matter region, we must have $x_{1}\le x\le x_{2}$. If we multiply both the numerator and the denominator of the integral in Equation (84) by $y^{3/2}$, define the polynomial $Q(y)=y\left(\eta+3y-y^{3}\right)$ and the rational function $S(y,Q)\equiv y^{4}/Q^{3}$, then the integral in Equation (84) can be rewritten as

  $$\int_{x_{2}}^{x}dy\, \frac{y^{5/2}}{(\eta+3y-y^{3})^{3/2}} = \int_{x_{2}}^{x} S\!\left[y,\sqrt{Q(y)}\,\right]dy.$$ (85)

The expression on the right-hand side of Equation (85) is by definition an elliptic integral [#!AbramowitzStegun!#] and cannot be expressed in terms of elementary functions except in two cases: 1) $S\!\left(y,Q^{1/2}\right)$ contains no odd powers of $y$; in our case this happens when $\eta=0$ and leads to the Schwarzschild interior solution; 2) the polynomial $Q(y)$ has two equal roots; this leads to the explicit solutions that we discuss next.